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J. Appl. Phys. 129, 015305 (2021); https://doi.org/10.1063/5.0033772 129, 015305 © 2021 Author(s).Magnetic textures in hemispherical thin film caps with in-plane exchange bias Cite as: J. Appl. Phys. 129, 015305 (2021); https://doi.org/10.1063/5.0033772 Submitted: 19 October 2020 . Accepted: 07 December 2020 . Published Online: 06 January 2021 Andreea Tomita , Meike Reginka , Rico Huhnstock , Maximilian Merkel , Dennis Holzinger , and Arno Ehresmann COLLECTIONS This paper was selected as Featured This paper was selected as Scilight ARTICLES YOU MAY BE INTERESTED IN Doping of ultra-thin Si films: Combined first-principles calculations and experimental study Journal of Applied Physics 129, 015701 (2021); https://doi.org/10.1063/5.0035693 Spintronic terahertz emitter Journal of Applied Physics 129, 010901 (2021); https://doi.org/10.1063/5.0037937 Thickness-dependent thermoelectric properties of Si 1−xGex films formed by Al-induced layer exchange Journal of Applied Physics 129, 015303 (2021); https://doi.org/10.1063/5.0025099Magnetic textures in hemispherical thin film caps with in-plane exchange bias Cite as: J. Appl. Phys. 129, 015305 (2021); doi: 10.1063/5.0033772 View Online Export Citation CrossMar k Submitted: 19 October 2020 · Accepted: 7 December 2020 · Published Online: 6 January 2021 Andreea Tomita,a) Meike Reginka, Rico Huhnstock, Maximilian Merkel, Dennis Holzinger, and Arno Ehresmann AFFILIATIONS Institute of Physics and Center for Interdisciplinary Nanostructure Science and Technology (CINSaT), University of Kassel, Heinrich-Plett-Str. 40, 34132, Kassel, Germany a)Author to whom correspondence should be addressed: tomita@uni-kassel.de ABSTRACT Hemispherical caps of in-plane exchange biased IrMn/CoFe layer systems have been fabricated on top of regularly arranged spherical silica particles by magnetron sputtering, creating magnetic Janus particles. In this thin film layer system cap, the magnetic shape anisotropy of thetopographically non-flat hemispheres competes with the unidirectional anisotropy induced by the exchange bias. The magnetic properties of this non-trivial system have been investigated by longitudinal magneto-optical Kerr effect magnetometry, where a characterization method has been developed considering both the curved layer system and the signal contributions of flat parts of the sputtered thin filmsystem. Both remagnetization curves, from Kerr magnetometry and the magnetic force microscopy images, reveal an onion state in themagnetic caps of the ensemble. Additional micromagnetic simulations show a stabilization of the onion state due to the introducedunidirectional anisotropy also in individual hollow hemispheres as compared to the vortex state exhibited by purely ferromagnetic caps. Published under license by AIP Publishing. https://doi.org/10.1063/5.0033772 INTRODUCTION Particles designed with two sides of different physical or chemical characteristics, typically termed as Janus particles (JPs),have been the subject of interest for the last 30 years, starting with the synthesis of spherical glass particles with both a hydrophilic and a hydrophobic side. 1JPs possess a large variety of chemical, biological, and magnetic functionalization possibilities for one orthe other side and just as many fabrication strategies. 2–6One par- ticular class of JPs are magnetic Janus particles (MJPs) with one magnetic and one non-magnetic side. MJPs are typically used for controlled transport in liquids where an external magnetic fielddrives the actuation and/or directs their motion. 7–10This aspect renders them an immense application potential for lab-on-a-chipdevices, where they may be used as stirrers, 11transporters of bio- logical cargo,12,13or (bio)sensors,14among others.15,16When MJPs are fabricated by capping a non-magnetic spherical template particlewith a magnetic layer system, the set of magnetic characteristics ofthe layer system deposited on such a topographically non-flat surfaceposes by itself a research topic of fundamental interest. In such hemispherical caps, several competing effects and interactions are present rendering this system to be non-trivial. Magnetocrystallineanisotropy, shape anisotropy, exchange interactions between mag- netic moments in the individual layers or between moments of dif- ferent layers, interlayer exchange interactions, magnetostaticinteractions, and, for some materials, Dzyaloshinskii –Moriya interac- tions are present in addition to possible anisotropies induced by the non-trivial topology. 17Due to the deposition process, the relative strengths of interactions will, moreover, change as a function of posi- tion within the cap. This results from the existence of a thickness gradient from the cap ’s pole to its side. The morphology and texture of the films will change due to the varying tilt angles of the surfacewith respect to the incidence direction of the material during physi- cal vapor deposition. The local magnetic characteristics of the cap will, therefore, change as a function of position and will depend on the diameter of the spherical particle, even if only average quantities are concerned. In the past, magnetic characteristics of caps with neg- ligible or easy-plane, perpendicular, 18,19or uniaxial anisotropy,17as well as interlayer exchanged systems with perpendicular and negligi-ble anisotropies, 20have been studied. For soft magnetic permalloy (Py) caps (Ni 81Fe19), there are three possible ground state magnetic moment configurations: onion, vortex, and out-of-plane, depending on both the thickness of the cap and the diameter of the baseJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 015305 (2021); doi: 10.1063/5.0033772 129, 015305-1 Published under license by AIP Publishing.polymeric particle.21The onion state is characterized by an align- ment of the magnetic moments tangentially to the surface forming a macroscopic magnetization in remanence along a direction lying inthe tangential plane of the pole. A vortex state wraps the magneticmoments in concentric circles around the curved surface with amagnetic core of perpendicular magnetization at the pole of the cap. For the out-of-plane state, the averaged macroscopic magnetization points perpendicular to the tangential plane through the pole of thecap. The simulations, 21which considered particles in the sub- micrometer range with cap thicknesses of up to 60 nm formed ofsoft magnetic materials, resulted in a phase diagram where the out-of-plane and onion states dominate at very small scales but are completely dwarfed by the vortex state for larger particles. 22,23 Referring to the applications where these types of MJPs are of interest, the possibility to control their translatory and rotatorymotions by dynamically changing magnetic fields can be further enhanced by establishing specific magnetic properties. Therefore, utilizing a layer system as a hemispherical cap, which generates aremanent magnetization also for larger particles and for thickerlayers, will help access the particles ’degrees of freedom in a more precise manner. In order to achieve this, in the present work, we investigated the use of a hemispherical cap composed of an exchange bias (EB) layer system, 24where exchange biased thin films have been sputtered in the presence of an external magneticfield. This layer system introduces a strong unidirectional anisot- ropy, which will compete with shape and other magnetic anisotro- pies. Hemispherical particles capped with different EB layersystems have been previously investigated, where their macroscopicmagnetic characteristics have been determined (NiO/FeNibilayer) 25or where a vortex state has been created by zero-field cooling (Py).26For this work, the use of EB is to design particles where the magnetic cap presents an onion configuration in sizeranges where this would be energetically unfavorable for a pure fer-romagnetic cap. In order to verify the presence of this remanentonion state, experimental results from magnetic characterization techniques, i.e., Kerr magnetometry and magnetic force micros- copy, were complemented by micromagnetic simulations of themagnetic cap. FABRICATION OF MAGNETICALLY CAPPED JANUS PARTICLES Commercially available silica spheres of 1 μm in diameter, dis- persed in an aqueous solution, were assembled into a two-dimensional hexagonally ordered monolayer using entropy minimi-zation. 27Following the assembly, a magnetic thin film system is deposited on top of the particles via magnetron sputtering. The thin film layer system is an EB system composed of a 5 nm bufferlayer (Cu), a 30 nm antiferromagnetic layer (AF, Ir 17Mn 83), a 10 nm ferromagnetic layer (FM, Co 70Fe30), and a 10 nm capping layer (Si). The thicknesses are given for the pole of the hemisphere. EB was initialized by growing the layer system in an external in-plane magnetic field Hext≈28 kA/m. From an application point of view, a large unidirectional anisotropy, i.e., a large EB field, isexpected to extend the phase space for the onion state, and a large remanent magnetization is beneficial for large forces or torques acting on the MJPs by an external magnetic field. For the latter, ahigh saturation magnetization of the FM and a large FM layer thickness are advantageous. Although a large FM thickness is bene- ficial for a large magnetic moment, the EB strength scales areinversely proportional to the FM thickness. 24,28Therefore, these two counteracting influences have to be balanced. The design ofthe caps has to consider the layer thickness reduction from pole to side. Figure 1(a) shows the results of a test process, where spherical particles have been capped, cut by a focused ion beam (FIB), andthe thickness reduction of the deposited layers was characterized byscanning electron microscopy (SEM). Prior to milling, a protectivelayer derived from the decomposition of a carbon precursor gas was deposited on top of the JPs to achieve a high contrast between the metallic cap and this insulating protective layer. The thicknessat the particles ’sides was evaluated to be in the range of 30% –40% of that at the particles ’pole. In other words, if the layer thicknesses at the pole of the particles are as configured, Cu 5n m/IrMn30 nm/ CoFe10 nm/Si10 nm, the thicknesses at the equator of the particles result in Cu2n m/IrMn11 nm/CoFe4n m/Si4n m. As the exchange bias field is approximately constant or slightly rising above a certaincritical AF layer thickness, 29the remaining 11 nm of IrMn are still sufficient to induce an EB effect.30 CHARACTERIZATION OF THE MAGNETIC CAP The macroscopic magnetic behavior of the MJPs, self- assembled on a glass substrate, was investigated by the longitudinalmagneto-optical Kerr effect magnetometry (L-MOKE) with the Kerr sensitivity aligned parallel to the direction of the external field applied during deposition of the magnetic layers. As the monolayerof hexagonally ordered particles shows areas that are not fullycovered [ Fig. 1(b) ], the ratio between particle-covered areas and uncovered areas varies between the individual MOKE measure- ments, where the laser spot is set to different positions on the sample surface. In order to disentangle the contributions of themagnetic layer hemispheres and the contributions of the magneticlayers on the flat uncovered areas, MOKE hysteresis loops of the flat layer system with the same thickness as on the cap poles have been measured. A typical result is depicted in Fig. 2(a) as hollow circles, while the solid line represents an arctan fit function (see theAppendix ), which reflects the hysteresis loop and allows us to extract characteristic values, i.e., the exchange bias field and the coercivity. Figure 2(b) shows a set of three exemplary hysteresis curves measured at different locations on the sample. The shownloops have a characteristic shape that can be interpreted as thesuperposition of two signal components: one part with a rathernarrow magnetization reversal field range is indicative for the flat layer system and a second (wider) part we attribute to the magneti- zation reversal of the hemispherical magnetic particle caps. Thecoercivity and exchange bias shift of the signal from the flat layerpart agrees nicely with the magnetization reversal of the flat systemreference curves shown in Fig. 2(a) . The individual differences of the hysteresis loops are due to the relative signal contributions of areas covered and not covered by the MJPs within the differentlaser spot areas on the sample during the measurements. In orderto extract coercivity and EB shift values for both the flat layer and the hemispheres, and additionally the percentage of areal particle coverage from the measurements, an arctan function reflecting aJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 015305 (2021); doi: 10.1063/5.0033772 129, 015305-2 Published under license by AIP Publishing.double hysteresis was chosen and fitted to the measurements (see theAppendix ). The results are shown by the solid lines in Fig. 2(b) . Moreover, due to the curvature of the particles, only a reduced area at their poles fulfills the reflection condition in the MOKE setupand, therefore, contributes to the measurement signal associatedwith the particles. All these aspects have been considered for thedetermination of the signal ratio c JPbetween the particle-covered areas and the uncovered ones present in the fits. More details on the sensitivity of the method and the evaluation procedure aregiven in the Appendix . Figure 2(c) shows fitted magnetization reversal curves representing only the signal originating from the MJPs that have been extracted from Fig. 2(b) and scaled by the fitted signal ratio c JP. The results reveal that while the EB field strength of the layer system on the particles is very similar to that ofthe flat surface ( H EB,JP¼/C010:8+0:2 kA/m as compared to HEB,flat¼/C010:5+0:2 kA/m), the coercive field is significantly higher ( HC,JP¼18:9+0:2 kA/m as compared to HC, flat¼4:5+0:9 kA/m). The determined loops of the layer system on the MJPs show the expected EB shift and are thereforeindicative of an onion state rather than a vortex one. The increase of the coercivity can be attributed to the increased influence of the surface anisotropy. As shown, the areas of observed MOKE signalare limited to the top of the caps, where the surface can be consid- ered almost flat. Here, the emerging MOKE signal would reverse along the easy axis (in-plane EB direction) in a loop run. However, the FM magnetic moments in this observed area are linked to theones of the other, not directly observable areas by exchange cou-pling, i.e., the sides of the spheres, where the local “in-plane ”easy direction is increasingly tilted with respect to the horizontal easy direction of the topmost observable cap area. The components of the tilted anisotropies (at the sides of the particle cap), which areperpendicular to the applied in-plane field, act like magnetic harddirections for the latter and are linked to the probed area. As thecoercivity of the flat reference EB system ( H C, flat¼4:2+ 0:5 kA/m) and the coercivity extracted for the flat part from the fits of the superposed hysteresis ( HC, flat¼4:4+0:2 kA/m) from the particle-covered areas do not deviate significantly, this validates thefitting procedure. Furthermore, increasing particle coverage deter-mined from the fit procedure was seen to be in agreement with a reduction of the absolute reflected laser intensity at the photodi- ode/detector during the measurements. Finally, the three values 0.79, 0.56, and 0.37 for c JP[Fig. 2(c) ] only represent the relative signal contributions between the signal from the particles vs the signal from uncovered flat areas, while the areal coverage by the particles was determined to be 99.96%, FIG. 1. (a) Scanning electron micrograph of a focused ion beam cut section through two MJPs of 1 μm in diame- ter each, covered with the EB layer system. The dashed line marks the boundaries of the EB layer system as a guide to the eyes. T o enhance the contrast of the metalliclayer system, a carbon protective layer has been depos-ited on top of the spheres. (b) Light microscope image of assembled MJPs on a glass wafer showing regions of highly ordered particle monolayers as well as “defects. ”Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 015305 (2021); doi: 10.1063/5.0033772 129, 015305-3 Published under license by AIP Publishing.99.87%, and 99.72%, respectively. For these high coverage values, one would usually not expect to have any visible contribution of aresidual flat layer during a magnetization reversal measurement,but due to the magnetometer ’s low sensitivity for the particles, their signal contribution is strongly reduced too. Hence, these con- jectures underline the importance of a sound estimate for the tech- niques ’relative sensitivity for each contribution to the sum of the magnetic signal of the sample. Since the particles are measured in an ensemble rather than individually, the experimental results shown here have to be inter- preted respectively, e.g., the magnetic interaction between particles due to magnetostatic coupling 31should generally not be excluded. However, experimental measurements of the hard axis response[Fig. 2(d) ] indicate that while these interactions might aid the mac- roscopic assembly to populate onion states, they are not prominentenough to promote onion states along directions other than the EB direction. This can be further supported by the appearance of highirregularities in the lattice orientation as seen in Fig. 1(b) emphasiz- ing that the onion state is imprinted due to the introduction of theEB system. While hysteresis loops recorded by L-MOKE measurements display the Kerr rotation integrated over the whole material fromwhich the laser light is reflected into the detector as a function ofthe applied magnetic field, the local magnetic texture of individualMJP caps is not accessible. Therefore, MJP monolayers have been investigated via atomic and magnetic force microscopy (AFM/ MFM) to retrieve the magnetic texture of an individual cap in thearray by imaging the uncompensated magnetic net charge distribu-tion. AFM images [ Fig. 3(a) ] were taken in the non-contact mode, and the lowest point measured was set as zero. The scanned area FIG. 2. (a) Measured hysteresis of the flat layer system from a reference area uncovered by particles. The circles represent measured data points, while the s olid lines rep- resent the fit. (b) Hysteresis loops recorded at three different spots on the sample, noted as I, II, and III with different areal particle coverage rat ios, where the data were fitted by a double hysteresis loop fit with the signal ratio cJPas the ratio between the signal coming from the cap of the particle and the signal given by the flat areas around them. (c) Extracted loops from the material system deposited only on the JPs. (d) Measured hysteresis of MJPs covered area along the sample ’s hard axis (the sample rotated at an angle f¼90/C14). FIG. 3. (a) T opography of a MJP measured by AFM, 1μm diameter, assembled in a tightly packed monolayer; here, the color bar represents the height of the hemi- sphere where the lowest point has been set to 0. (b)Charge contrast image resulted from subtracting theimage measured with a south-magnetized MFM tip from the image measured with a north-magnetized MFM tip; the color bar shows the values between [ −1, 1] resulted from the subtraction.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 015305 (2021); doi: 10.1063/5.0033772 129, 015305-4 Published under license by AIP Publishing.size was chosen as 1.5 × 1.5 μm2in order to encompass a full MJP as well as parts of the neighboring particles. The MFM phase con- trast signal was obtained by following the surface topography in the contour mode with a magnetic tip, magnetized as the north poleand the south pole, respectively, i.e., the measurements were per-formed twice in order to ensure that there is no interactionbetween the tip and the surface. The relative scanning height wherethe MFM images were taken was 100 nm to ensure a measurementin charge contrast mode. Since the tip of the measurement system is sensitive to the out-of-plane fields, we observe strong opposite signals at adjacent poles of the JP, i.e., similar to a magnetic dipole,corresponding to the presence of stray fields generated by the mag-netic anisotropies in the cap. The phase contrast images recordedwith the inverted tip magnetization reveal a mirrored contrast,while the difference of both images shows the charge contrast[Fig. 3(b) ], in which susceptibility effects cancel out. From previous MFM measurements of curved magnetic surfaces, images of a vortex state 23,32,33displayed a radial contrast variation from the vortex core (the highest contrast) toward the edges of the particle.Images of an out-of-plane configuration 18showed uniform bright or dark contrast along the entire surface of each particle. In con-trast, the charge distribution observed in the present experiment iscomparable to that of a magnetic dipole and therefore indicates anonion configuration.MICROMAGNETIC SIMULATIONS Micromagnetic simulations were performed to correlate an underlying magnetic texture to the measured MFM image andstudy theoretically the influence of an additional biasing field inside a ferromagnetic cap. The simulation package Nmag 34was used with its finite element method numerical solver of theLandau –Lifshitz –Gilbert equation 35to compute the equilibrium magnetization state ~M.36Here, tetrahedral finite elements have been used for an accurate description of curved objects like the present MJPs. The caps were considered hollow hemispheres with constant thickness and an inner diameter of 1 μm, neglecting the thickness gradient observed in the FIB images. This simplificationof the geometry leads to a more robust convergence of the simula-tion while still allowing to theoretically compare the exchange biased and non-exchange biased caps. For an FM thickness of 10 nm, the unidirectional anisotropy exerted on the system due tothe EB is implemented in the simulation as a constant externalmagnetic field H EB, which is only present within the hollow hemi- sphere and points parallel to the anisotropy direction of the system (into the x-direction). The strength of this field was set to HEB¼12 kA/m, so that it compares to the experimentally deter- mined values. The maximum cell size was implemented as 5 nmsince the exchange length of CoFe is in the same order of magni- tude l ex¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2A/μ0M2 Sp /C255n m/C0/C1 . FIG. 4. Micromagnetic simulations of a 1 μm diameter hollow hemisphere with (a) and without (b) the added EB field; (a/b.1) projection of the spatial magnetization distri- bution ~M(~r) onto the (x,y)-plane (top view) and (a/b.2) onto the (x,z)-plane (side view); (a/b.3) surface charge distribution ρA(~r) seen from above and (a/b.4) from the side. ρA(~r) was normalized for (a) and (b) between −1 and 1. For contrast enhancement, the scale was cut off at [ −0.4, 0.4] (a) and [ −0.04, 0.04] (b).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 015305 (2021); doi: 10.1063/5.0033772 129, 015305-5 Published under license by AIP Publishing.For the simulations, the exchange coupling constant J¼3/C210/C011J/m for CoFe37and the damping constant α¼0:01838 were used. In addition, the values for the saturation magnetization MS¼1:23/C2106kA/m39and the uniaxial anisotropy constant KUAA¼4:5/C2104J/m340were taken from the literature. Upon convergence, the simulation provides data for the magnetic moment distribution and the magnetic scalar potential within the hollow hemisphere for the calculated equilibrium state. To compare the micromagnetic simulations qualitatively with our experimental results, we were particularly interested in the spatialdistribution of the magnetization ~M(~r)a n dt h es u r f a c ec h a r g e s ρ A(~r). Although the magnetic scalar potential, w(~r), can be obtained from the simulations, which are interconnected by /C0~∇w¼~M, the surface charges need to be determined via ρA¼~n/C1~Mwith the normal vector ~n(~r)¼(coswsinϑ,s i nwsinϑ,c o s ϑ)Tin spherical coordi- nates and the azimuthal and polar angles wandϑ, respectively.41 Utilizing arrows for visual representation, ~M(~r) is shown to wrap around the curvature of the hollow hemisphere similar tomeridian lines in the presence of exchange coupling, i.e., for EBlayer systems. This behavior is characteristic to the magnetic onionconfiguration, and it is shown in both Figs. 4(a) .1 and 4(a).2 as viewed from the top and from the side, respectively. Similarly, dis- played from the top [ Fig. 4(a) .3] and from the side [ Fig. 4(a) .4], ρ A(~r) indicates an accumulation of uncompensated magnetic net charges with their distribution diverging from one side of the hollow hemisphere (pictured here from the left) along the curvature and then converging at its opposite end (pictured here at the right)similar to meridian lines. As seen above, this representation is inqualitative agreement with the determined charge contrast obtainedvia MFM of assembled magnetic MJPs. In the absence of an EB field, the magnetic ground state of a hollow hemisphere of 1- μm diameter reverts to a vortex configura- tion as the most energetically favorable state. Figure 4(b) details the simulation of a system where the EB field is not present, in similarsteps as the simulation for the case where the EB is present. The magnetization, represented again by arrows, is shown in concentric circles along the parallels of the hollow hemisphere with one singleout-of-plane moment at the top [ Figs. 4(b) .1 and 4(b).2]. The charge distribution shows the biggest contrast on top of thehemisphere where the out-of-plane moment is present [ Figs. 4(b) .3 and4(b).4]. CONCLUSIONS The magnetic characteristics of a magnetic exchange biased thin film system covering spherical nonmagnetic particles and forming hemispherical magnetic caps have been investigated via MOKE magnetometry and MFM. A quantitative method to charac-terize the magnetization reversal of this material system byL-MOKE magnetometry and particularly to disentangle the signalcontributions from the layers on the spherical particles and from the layers on flat areas uncovered by particles has been developed. Only a small part of the thin film system covering the nonmagneticspherical particles is detected directly by the MOKE setup, namely,a small circular area around the pole of the cap that is defined by the geometry of the optical setup. Although only a small area is probed in these measurements, the magnetic characteristics of theother areas, which do not directly contribute to the measured signal, do also influence the magnetization reversal, as the magnetic moments within the magnetic layers of these areas are exchange-coupled to those of the directly probed area. Here, a strong influ-ence of magnetic hard directions, induced by the bent geometry ofthe hemispherical film system, has been observed in an enhanced coercivity and saturation fields as compared to the flat film. The values of the EB field obtained from these measurements were usedfor micromagnetic simulations of an individual exchange biasedcap in comparison to a purely ferromagnetic cap. The calculatedmagnetization and surface charge distribution on the exchange biased hollow hemisphere of 1 μm diameter resulted in the presence of an onion texture as the magnetic ground state, while contrast-ingly the absence of the additional unidirectional anisotropy led tothe formation of a vortex state for purely ferromagnetic caps of thissize. This finding can open an alternative route for fabricating par- ticles with defined remanent magnetic moments for a complete translator and rotatory motion control by dynamically changingexternal fields in liquids for lab-on-chip devices. ACKNOWLEDGMENTS We would like to thank Professor Dr. Thomas Kusserow for his generous contribution to the acquisition of the FIB images. Dr. Michael Vogel is gratefully thanked for the long and inspiring discussions. APPENDIX: MOKE DATA EVALUATION PROTOCOL For the evaluation of magneto-optically obtained magnetiza- tion reversal measurements, the knowledge of the technique ’s sensi- tivity for the investigated object is crucial. In the used L-MOKE setup, p-polarized light from a laser with a central wavelength of 632 nm illuminated the sample, while the reflected light is detectedafter passing an optical focusing unit and an analyzer to reveal theKerr angle, which is directly proportional to the sample ’s magneti- zation. In this study, the sample is composed of large areas of self- assembled Janus spheres and residual planar areas not covered by particles but by the planar thin film system only [see Fig. 1(b) ]. Due to the curvature of the particles, specularly reflected light onlyfrom a small area A caparound the top of the particle will reach the detector. This area is determined by the radius rlensof the lens focusing the rays onto the detector and the distance Lbetween it and the sphere (see Fig. 5 ). The maximum central angle describing Acapfrom which a Kerr signal can be expected and as depicted in Fig. 5 is calculated by β¼0:5/C1arctanrlens L/C0/C1 . Consequently, the reflected intensity of the MOKE measurements decreases with increasing coverage of the substrate with particles. As the flat partsof the layer system do contribute completely to the reflected inten-sity, it is important to estimate the intensity ratio between lightreflected from the MJPs and from the flat areas. Assuming a hexag- onally packed monolayer, the MOKE setup ’s sensitivity factor for JPs (as the ratio between the JP ’sa r e a A capfrom which light reflected to the area the particle covers Ahex/nhex) becomes fJP¼nhex/C1Acap Ahex¼πffiffiffi 3p(1/C0cos(β)), (A1)Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 015305 (2021); doi: 10.1063/5.0033772 129, 015305-6 Published under license by AIP Publishing.where Ahexis the hexagonal unit cell ’s area and nhex¼3, which is the number of particles within the unit cell. Consequently, theplanar thin film ’s sensitivity is by definition f flat¼1. When taking these sensitivity factors into account for the interpretation of theMOKE data, the areal particle coverage λof the sample can be deduced from the ratio between the Kerr signal originating from JPs and the one from the residual flat layer. For the analysis of therecorded data, a superposition of two hysteresis curves has beenfitted to each branch (ascending þ; descending /C0) of the loops, both normalized to the magnetization in saturation, M(H +)¼X ici/C12 π /C1arctan [ ai(H+/C0HEB,i+HC,i)]for i [{JP, flat}, (A2) Fit parameters are ai,HEB,i,HC,i, and ci, where cflat¼1/C0cJP. Here, the coercivity of each component iis extracted from the parameter HC,i, while the exchange bias shift is HEB,i. Furthermore, cJPand cflatcan be understood as relative signal contributions: cJP¼nJPAcap/Asignal and cflat¼1/C0cJP¼Aflat/Asignal, where Asignal is the area from which light is reflected to the lens and guided to the photodiode ( Asignal¼nJPAcapþAflat). Weighting these relative signal contributions ( cJPand cflat) by their respective sensitivity factors ( fJPandfflat) allows us to set up a linear system of equations, cJP fJP¼cJP nhexAcap Ahex¼nJPAhex nhexAsignal, (A3)cflat fflat¼Aflat Asignal, (A4) that can be solved for the areal coverage λi, where λJP¼nJP 3/C1Ahex AlaserandλJP¼Aflat Alaser. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1C. Casagrande and M. Veyssie, “‘Grains Janus ’: Réalisation et Premières Observations Des Propriétés Interfaciales, ”C. R. Acad. Sci. (Paris) II, 307 (1988). 2K. P. Yuet, D. K. Hwang, R. Haghgooie, and P. S. Doyle, “Multifunctional superparamagnetic Janus particles, ”Langmuir 26(6), 4281 –4287 (2010). 3J. Hu, S. Zhou, Y. Sun, X. Fang, and L. Wu, “Fabrication, properties and appli- cations of Janus particles, ”Chem. Soc. Rev. 41(11), 4356 (2012). 4J. Jeong, E. Um, J. K. Park, and M. W. 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Rivier, P. Leiderer, and A. Erbe, “Frustration-induced magic number clusters of colloidal magnetic particles, ” Phys. Rev. E 77(3), 031407 (2008). 11G. Steinbach, M. Schreiber, D. Nissen, M. Albrecht, S. Gemming, and A. Erbe, “Anisotropy of colloidal components propels field-activated stirrers and movers, ” Phys. Rev. Res. 2, 023092 (2020). 12L. Baraban, D. Makarov, R. Streubel, I. Mönch, D. Grimm, S. Sanchez, and O. G. Schmidt, “Catalytic Janus motors on microfluidic chip: Deterministic motion for targeted cargo delivery, ”ACS Nano 6(4), 3383 –3389 (2012). 13K. Lee, Y. Yi, and Y. Yu, “Remote control of T cell activation using magnetic Janus particles, ”Angew. Chem. Int. Ed. 55(26), 7384 –7387 (2016). 14V. B. Varma, R. G. Wu, Z. P. Wang, and R. V. Ramanujan, “Magnetic Janus particles synthesized using droplet micro-magnetofluidic techniques for protein detection, ”Lab Chip 17(20), 3514 –3525 (2017). 15Y. Song, J. Zhou, J. B. Fan, W. Zhai, J. Meng, and S. 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Arekapudi, and M. Albrecht, “Single vortex core recording in a magnetic vortex lattice, ”J. Appl. Phys. 115(6), 063906 (2014). 33M. V. Sapozhnikov, O. L. Ermolaeva, B. G. Gribkov, I. M. Nefedov, I. R. Karetnikova, S. A. Gusev, V. V. Rogov, B. B. Troitskii, and L. V. Khokhlova, “Frustrated magnetic vortices in hexagonal lattice of magnetic nanocaps, ”Phys. Rev. B 85(5), 054402 (2012). 34H. Fangohr. , M. Albert, and M. Franchin, “Nmag micromagnetic simulation tool—Software engineering lessons learned, ”in SE4Science ’16:Proceedings of the International Workshop on Software Engineering for Science (ACM Press, 2016), pp. 1 –7. 35M. Lakshmanan, “The fascinating world of the Landau-Lifshitz-Gilbert equa- tion: An overview, ”Philos. Trans. R. Soc. A 369(1939), 1280 –1300 (2011). 36W. Daeng-Am, P. Chureemart, A. Rittidech, L. J. Atkinson, R. W. Chantrell, and J. Chureemart, “Micromagnetic model of exchange bias: Effects of structure and AF easy axis dispersion for IrMn/CoFe bilayers, ”J. Phys. D Appl. Phys. 53(4) 045002 (2020). 37D. V. Berkov, C. T. Boone, and I. N. Krivorotov, “Micromagnetic simulations of magnetization dynamics in a nanowire induced by a spin-polarized current injected via a point contact, ”Phys. Rev. B 83(5), 054420 (2011). 38D. Y. Kim, B. Parvatheeswara Rao, C. O. Kim, M. Tsunoda, and M. Takahashi, “Annealing temperature dependence of microwave permeability in CoFe/MnIr bilayers, ”Phys. Status Solidi C 4(12), 4388 –4391 (2007). 39H. Huckfeldt, A. Gaul, N. David Müglich, D. Holzinger, D. Nissen, M. Albrecht, D. Emmrich, A. Beyer, A. Gölzhauser, and A. Ehresmann, “Modification of the saturation magnetization of exchange bias thin film systems upon light-ion bombardment, ”J. Phys. Condens. Matter 29(12), 125801 (2017). 40D. Holzinger, N. Zingsem, I. Koch, A. Gaul, M. Fohler, C. Schmidt, and A. Ehresmann, “Tailored domain wall charges by individually set in-plane magnetic domains for magnetic field landscape design, ”J. Appl. 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1.4862467.pdf

Spintronics of antiferromagnetic systems (Review Article) E. V. Gomonay and V. M. Loktev Citation: Low Temperature Physics 40, 17 (2014); doi: 10.1063/1.4862467 View online: http://dx.doi.org/10.1063/1.4862467 View Table of Contents: http://scitation.aip.org/content/aip/journal/ltp/40/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Low switching current in a modified exchange-biased spin valve via antiferromagnetic spin transfer torquea) J. Appl. Phys. 109, 07C915 (2011); 10.1063/1.3559481 Spin torque-driven switching of exchange bias in a spin valve J. Appl. Phys. 106, 073906 (2009); 10.1063/1.3236572 Effect of polarized current on the exchange bias in a current-in-plane spin valve J. Appl. Phys. 105, 07D106 (2009); 10.1063/1.3057796 Size-dependent alternation of magnetoresistive properties in atomic chains J. Chem. Phys. 125, 121102 (2006); 10.1063/1.2354080 Thermal decay of ferro/antiferromagnetic exchange coupling in Co/CrMnPt systems J. Appl. Phys. 86, 6305 (1999); 10.1063/1.371691 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.210.130.147 On: Tue, 08 Apr 2014 14:32:36Spintronics of antiferromagnetic systems (Review Article) E. V. Gomonaya)and V. M. Loktev National Technical University of Ukraine “KPI”, pr. Peremogi 37, Kiev 03056, Ukraine and N. Bogolyubov Institute of Theoretical Physics of NAS of Ukraine, Metrologicheskaya 14-b, Kiev 03143, Ukraine (Submitted August 22, 2013) Fiz. Nizk. Temp. 40, 22–47 (January 2014) Spintronics of antiferromagnets is a new and rapidly developing field of the physics of magnetism. Even without macroscopic magnetization, antiferromagnets, similar to ferromagnetic materials areaffected by spin-polarized current, and as in ferromagnets this phenomenon is based on a spin-dependent interaction between localized and free electrons. However, due to the nature of antiferromagnetic materials (complex magnetic structure, essential role of exchange interactions,absence of macroscopic magnetization) the study of possible spintronic effects requires new theoretical and experimental approaches. The purpose of this review is to systemize and describe recent developments in this area. After presenting the main features of structure and behavior ofantiferromagnets various microscopic and phenomenological models for description of the current-induced phenomena in heterostructures containing ferro- and antiferromagnetic layers are considered. The questions related to an effect of antiferromagnetic ordering on an electric current,as well as the questions of possible creation of fully antiferromagnetic spin valves are discussed. In addition, we briefly discuss available experimental results and try to interpret them. VC2014 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4862467 ] Introduction Spintronics (SPINelecTRONICS) is a new branch of physics and technology of electronic devices, in which the main role in the encoding, transfer and processing of infor- mation is played not by the charge but by the spin of an elec-tron. Components of spintronic devices are materials with different magnetic and electrical properties, which have as a rule sub-micron dimensions. Spintronics has appeared as a separate section of the physics of magnetism after discovering the effect of giant magnetoresistance (GMR) by Grunberg and coworkers 1,2and due to wide application of this effect in superdense informa- tion recording. The subsequent discovery of the effect of spin torque transfer (STT), predicted by Slonczewski3and Berger,4 has given a new impetus to the development of spintronics and led to the creation of fundamentally new high-speed spin- tronic devices which are controlled exclusively by current. To date, the main active components of spintronic ele- ments are ferromagnets (FM). Their behavior is being widely studied and is described in detail in the literature (see, forexample, reviews Refs. 5–7). However, an interest has recently been expressed in other magnetic materials, antifer- romagnets (AFM), as potential carriers of information. Forexample, a recent paper 8has demonstrated an ability to encode information in AFM-ordered nanostructures consist- ing of a small amount of iron atoms, as well as to read infor-mation using a spin-polarized tunneling current. It has been found that it is also possible to write and read information in spiral structures (practically, weak FM). 9 For application AFM have several advantages over FM. First, with a magnetic structure and a high magnetic suscep- tibility to external fields, AFM particles have zero or lowmagnetization. In other words, they do not create external magnetic fields and, consequently, interact weakly with each other. Second, the characteristic frequencies of AFMR andhence the characteristic frequencies of switching between different states of AFM states are several orders higher than those values for typical FM materials (for example, see Ref.10). This means a possibility to make high-speed devices operating not in gigahertz (as FM) but in terahertz range. Finally, an AFM order in semiconductors is observed moreoften at much softer conditions than an FM ordering (see Ref. 11) that makes it possible to combine in a single devise the advantages of both electronics (performance, easy han-dling) and spintronics (high sensitivity, low energy con- sumption). Note also that AFM can have properties of semimetals, 12i.e., exhibit properties of a conductor for one spin polarization and of an insulator for the other, which also makes them to be very attractive spintronic materials. For the first time the idea of using AFM materials as active components in typical spintronic devices—spin valves—was proposed in Refs. 13–15. The author have also performed ab initio calculations, clearly indicating the possi- bility of spintronic effects in a number of AFM metals (such as Cr, FeMn, NiMn). The dynamics of magnetic moments of AFM in the presence of spin-polarized current in the frame-work of a phenomenological approach has been investigated in Refs. 16–18, where it was shown for the first time that spintronic effects in AFM, as well as in FM, are related withthe transfer of spin angular momentum (magnetization), and their manifestation due to an exchange enhancement can be as pronounced as in FM. In parallel, there appeared a numberof experiments clearly indicating that there is an interaction between an auxiliary AFM layer, which was assumed to be passive, and a current of high density in a wide variety ofsystems: from spin valves with a metal (FeMn, IrMn) 19–27or non-conductive (NiCoO) AFM28to bilayer FM/AFM sys- tems.29,30The experiments,31,32showing the connection of electric and magnetic properties in chromium (the only known pure AFM metal), provided the basis for the further development of microscopic models of this system. 1063-777X/2014/40(1)/19/$32.00 VC2014 AIP Publishing LLC 17LOW TEMPERATURE PHYSICS VOLUME 40, NUMBER 1 JANUARY 2014 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.210.130.147 On: Tue, 08 Apr 2014 14:32:36Due to demands of an experiment, general thermody- namic approaches for the description of various aspects of spintronics of AFM, applicable to both discrete33–37and continuous38–40systems, have been developed. As it turned out, the phenomena of STT, spin pumping, GMR are inher- ent to both FM and AFM systems regardless of details of their magnetic structure. In general, it can be argued that the AFM spintronics is a “hot” topic, as evidenced by the appearance of a number of papers21,32,41,42with similar names, in each of which authors are trying to somehow formulate a concept of this new direc- tion. However, the main attention is paid to experimental aspects and barely addresses theoretical ones with a numberof specific features in AFM, which differs fundamentally AFM from FM. Therefore, we aim to present primarily the existing theoretical approaches. To do this, we will discussideas and methods related to both the description of the behavior of AFM materials in the presence of current and their usage as active elements in spintronic systems. Thus,we hope to supplement experimental results contained in the review literature. 1. Spintronics of ferromagnets Before proceeding to the main subject of this review, we recall the main effects that make the physical content of the so-called spintronics of FM. It is based on two interrelated effects, GMR and STT. In essence, they both are of the samenature: the interaction of band electrons, providing a current, with localized ones, responsible for the formation of mag- netic moments, or the so-called sd-exchange. This interac- tion leads, on the one hand, to the dependence of the scattering cross section (and, hence, resistance) of delocal- ized electrons on the direction of localized magneticmoments (i.e., GMR), and, on the other hand, to the exchange during the scattering process not only by energy but also by spin angular momentum between localized andfree electrons (i.e., STT). Let us consider both phenomena in more detail. When passing through a uniformly magnetized (with magnetizationM pol) FM layer, which serves as a polarizer, a flow of elec- trons is also acquiring the spin polarization and, conse- quently, the magnetization meljjMpolin the same way that the light becomes polarized when passing through an opti- cally anisotropic medium. If in the path of so polarized con- duction electrons there is another FM layer—ananalyzer—with the magnetization M an(see Fig. 1), the num- ber of electrons passed through the system (and, as a result, the resistance of the system) depends on the relative orienta-tion of the vectors M polandMan. This phenomenon is called GMR; qualitatively it is similar to decreasing an intensity of light when passing between two polarizing plates (the law ofMalus). In most materials the resistance is minimal when both FM vector are parallel to each other, M pol""Man, and maximal in the opposite case, Mpol"#Man, which is due to an increased scattering of electrons with the “wrong” spin ori- entation when entering the second FM (FM2) layer. The scattering of delocalized electrons at the FM2 layer can occur both with and without spin flip. In the first case, for the sufficiently high electron concentration (i.e., high cur- rent density) the above-mentioned phenomenon of STTschematically depicted in Fig. 2(a) takes place. In other words, the localized spins in FM2 experiencing recoil take on an excessive momentum (magnetization). In the lattercase, the free electrons transfer further through a sample the polarization obtained in FM1 layer and via the sd-exchange can generate a magnetic field H sdof an “exchange” nature, Fig. 2(b) (in addition to the Oersted field created by charge degrees of freedom). The magnetization dynamics in the presence of a current is related with the spin pumping phenomenon (a thermody- namically conjugate STT), namely the rotation of the mag- netization vector (created by localized spins) due to the samespin-dependent scattering can lead to the polarization of free electrons and thereby generate the spin-polarized current. This phenomenon is observed either in discrete systems (het-erostructures) consisting of FM and nonmagnetic (NM) layers, or in FM textures (for example, in nanowires with an inhomogeneous distribution of the magnetization along theaxis). Currently there are several (complementary) approaches to the description of spintronic effects in FM: from quantummechanical problem of scattering of conduction electrons byFIG. 1. FM spin valve device and the effect of GMR. Mpol""Man, electrons polarized in the fixed FM1 layer pass through the free FM2 layer without scattering; a high current (a); Mpol"#Man, most of the electrons scattered in the nonmagnetic (NM) layer; a low current (b). FIG. 2. STT effects in FM: under spin-dependent scattering of free electrons in the active FM2 layer localized moments experience a recoil in the formof torque T STT(a); the electrons passed without scattering contribute to the active layer an effective magnetic field Hsd(b). The dynamics of magnetiza- tionM/C17Manof the active (FM2) layer: TSTTturns Mand can compete with the internal friction ( G/M/C2_M),Heffcauses rotation of M(solid arrow) (c).18 Low Temp. Phys. 40(1), January 2014 E. V. Gomonay and V. M. Loktev This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.210.130.147 On: Tue, 08 Apr 2014 14:32:36a potential generated by localized moments43to semiclassi- cal models based on equations of balance charge and spin angular momentum.44In particular, to describe STT the most commonly used is the equation of magnetic dynamicsof Landau and Lifshitz with damping in the form of Gilbert supplemented by the Slonczewski term depending on the current I, _M¼cH FM/C2MþaG MsM/C2_MþTSTT; TSTT/C17rFMI MsM/C2ðM/C2epolÞ:(1.1) Here (in notations of Ref. 5),M/C17Manis the magnetization of the free (FM2) layer, the magnitude of which Ms¼jMjis assumed to be constant; cis the absolute value of the gyro- magnetic ratio. The first term on the right describes the rotation of the magnetization around the effective fieldH FM¼/C0@wFM/@M, defined by the bulk density of the mag- netic energy wFMof the free layer (see Fig. 2(c)). The second term in Eq. (1.1) describes the relaxation (internal friction), and the corresponding phenomenological constant aGis inversely proportional to the quality factor of FMR. Finally, STT (the term TSTT) is created by the current polarized along the direction epol¼Mpol/jMpolj. The quantity introduced in Eq.(1.1) rFM¼/C22hec 2jejMstFM(1.2) is determined by the electron charge e, the volume of the free layer tFM, as well as the efficiency of the spin polariza- tione(/C201);/C22his Planck’s constant. The friction of Gilbert and STT Eq. (1.1) represent the moments of dissipative forces which lead with the power dwFM dt¼/C0aG cMs_M2þaFMI cMsð_M/C1epol/C2MÞ; (1.3) to change of the energy density wFMof the system. Obviously, depending on sign and magnitude of the current I STT can play the role of either positive or negative friction. In the latter case, when the current exceeds some criticalvalue, I FM cr¼aG rFMxFMR; (1.4) which depends on the frequency xFMRof the ferromagnetic resonance, the initial state loses the stability, and the mag- netization either flips (changes to the opposite direction) or,depending on the configuration of the system goes into a steady precession state (so-called nano-oscillator). 2.sd-exchange and spintronic phenomena in ferromagnetic textures As mentioned, the phenomena of GMR and STT are based on the interaction between conduction electrons and localized magnetic moments. Let us consider a quasi-classical approach to the description of the exchange part of this interaction, the so-called sd-exchange defined by the Hamiltonian,^H sd¼/C0X nJsdðr/C0RnÞ^sðrÞ^SðRnÞ; (2.1) where Jsdis the exchange integral, ^sðrÞand ^SðRnÞare the spin operators of a free electron ( s) located at the point rand an electron ( d) localized on the site nwith the coordinate Rn. (Strictly speaking, for such an interaction a total angular momentum should conserve, and in systems with a strong spin-orbit coupling (for example, in semiconductors (Ref. 6) a part of the spin angular momentum of delocalized electrons can be transferred to the lattice. This leads to all sorts of magnetomechanical phenomena; the corresponding effectsapplied to AFM are described in Ref. 45.) Since the dynamics of localized spins in most magnetic metals can be considered to be slow in comparison with con-duction electrons, the operators ^SðR nÞcan be reduced to classical field vectors of magnetization M(r,t),46and instead of the operator ^sðrÞone can consider an average over all the states of free electrons, i.e., actually to use the classic vector of density of dimensionless magnetization of conduction electrons m(jmj¼1). Under such conditions, the Hamiltonian (2.1) corresponds to the semiclassical energy density of the sd-exchange, ^HsdðrÞ¼/C0Jsd Msm/C1M; (2.2) where Msis the saturation magnetization introduced above. On the one hand, the sd-exchange leads to a biasing of conduction electrons along M(r), and, on the other hand, to the rotation of the magnetization maround M(r). The first process in the presence of the current with the density je causes an adiabatic magnetization flux, ^P FM¼/C0lB ePje/C10M Ms/C17/C0lB ePje/C10epol; (2.3) where lBis the Bohr magneton, Pis the degree of spin polar- ization depending on material properties, eis the electron charge, and the symbol /C10denotes a direct tensor product. The second process is described by the standard equa- tion of precession. As a result, the quasi-classical equationfor the magnetization density of conduction electrons in a medium with a non-uniform distribution of the magnetiza- tionM(r) of localized electrons has the following form: _mþr ^P FM¼/C0Jsd Msm/C2M/C0C: (2.4) The last term on the right side describes the dissipative proc- esses and in the simplest case includes both diffusion (with the coefficient D) and collision contributions; moreover the second is parameterized by the spin-flip relaxation time ssf. As a result, the dissipation rate is described by the expression, C¼/C0DDmþ1 ssfðm/C0meqÞ; (2.5) in which the equilibrium spin density meq¼neqM/Ms depends on the concentration neqof carriers with a spin par- allel to M;Dis the Laplace operator. Usually the diffusion of the spin angular momentum is neglected, assuming that the characteristic size of a magnetic inhomogeneity is signif- icantly greater than the mean free path.Low Temp. Phys. 40(1), January 2014 E. V. Gomonay and V. M. Loktev 19 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.210.130.147 On: Tue, 08 Apr 2014 14:32:36Physically the phenomena of GMR and STT are provided by the non-equilibrium spin polarization, or by the non- equilibrium magnetization of free electrons dm¼m/C0meq, for which the expression in the applied approximations can beobtained from the Eq. (2.4) dm¼lBP eJsdMsð1þn2ÞM Ms/C2ðje/C1rÞMþnðje/C1rÞM/C20/C21 ;(2.6) where the parameter n¼/C22h/(Jsdssf)/C281. In order to obtain a closed equation for the vector M,w e take into account that the non-equilibrium correction (2.6) creates, due to the sd-exchange, the above-mentioned (see Fig. 2) exchange magnetic field Hsd¼Jsddm/Msand as a result the torque which depends on the current (cf. Eq. (1.1)), TSTT/C17cHsd/C2M¼lBP eMsð1þn2Þ /C2ðje/C1r ÞMþnM Ms/C2ðje/C1r ÞM/C20/C21 : (2.7) The first term in Eq. (2.7) , called an adiabatic STT, describes the kinematic effect: conduction electrons transfer the mag- netic energy of FM from one point of the texture to another, so that the total magnetic energy of localized spins is con-served. More interesting from the point of view of control- ling the state of FM medium is a second, much smaller (due to the smallness of the parameter n/C281), contribution which is called a non-adiabatic STT. It is associated with the exchange by energy between conduction electrons and local- ized moments. It is this contribution that provides a uniformmotion of domain boundaries in FM at a constant rate, v DW¼lBnP aGeMsð1þn2Þje; (2.8) depending on the current density.46 Finally, we mention the formalism for describing the phenomenon of spin pumping (rather detailed descriptioncan be found in Ref. 47). From the standpoint of non- equilibrium thermodynamics, dissipative processes in FM in the presence of the current can be considered in terms ofgeneralized fluxes—the current density j eand the magnetiza- tion flux (it should not be confused with the flux (2.3) of the magnetization of free electrons) _M, on the one hand, and their conjugate generalized forces, the electric field Eand the effective magnetic field HFM, on the other. If we use Ohm’s law, je¼rcondE(where rcond is the conductivity), then the Eq. (1.1) with STT in the form Eq. (2.7) is one of the linear Onsager relations between the flux _Mand the force E. Additionally, using the Onsager’s reciprocity rela- tions, one can write the inverse relation between the force E and the flux _M. As a result, we find that (for a given poten- tial difference, i.e., for a given field E) the electric current depends on the magnetization: jpump a¼/C22hrcond 2eP_M/C1ðM/C2r aMþnraMÞ: (2.9) The quantity jpumpis called the pump current, appearing when the time-varying magnetization leads to the movement of electric charges.3. Structure and features of antiferromagnets Here are some necessary information related to AFM. As is known, they are materials which have long-range order (observed, e.g., by neutron scattering methods), but have no(or almost no) macroscopic magnetization. Unlike FM the observed magnetic structures of AFM are much more numer- ous and varied. At the microscopic level the AFM ordering corresponds to the presence of non-zero spin moments on each magnetic atom. (In some AFM metals, particularly in chromium, theAFM ordering can be associated with the formation by con- duction electrons of a spin density wave which can have a period both commensurate and incommensurate with the lat-tice constant (Ref. 48)) However, from one atom to another the direction of the spin vector is changed so that within the magnetic unit cell the total spin and, accordingly, the totalmagnetization are zero (or in special cases of the so-called weak ferromagnetism are much smaller than the magnetiza- tion of each sublattice). The relative position and orientationof spins can be very diverse: from a simple alternation of oppositely directed vectors to complex non-collinear multi- sublattice structures and spirals; however, all such magneticcrystals form a class of AFM. At the macroscopic level AFM are often described as a system of interleaved N submagnetic sublattices. In practice, it means that the magnetic order is determined by several mag- netic vectors Mk(r)( w h e r e k¼1,2,Nsub) which are the field variables concentrated in the same point rof the medium (actually, in a physically small volume). The linear combina- tion of the vectors Mk(r), which are non-zero in the absence of external fields, are referred to as the AFM vectors or theN/C19eel vectors or the multicomponent AFM order parameter. For example, in the particular case of a two-sublattice collin- ear AFM the unique N /C19eel vector is determined by the differ- ence L¼M 1/C0M2. In addition, for a complete description of features of AFM the magnetization vector MAFM¼P kMk is also introduced that in the above-mentioned two- sublattice case means MAFM¼M1þM2. Note that unlike FM, in AFM there is no simple (semi- classical) transition from micro- to macro-description. So, ifin FM both the ground state and the dynamics of the macro- scopic magnetization vector in some cases are similar to the behavior of an average (in the quantum mechanical sense)vector of the spin at a site, then in AFM the macroscopic N/C19eel state has no quantum mechanical (nodal) analogue. In addition, upon the transition from a nodal microscopicdescription to a description in terms of a continuous medium the translation vector between the magnetic sublattices is “lost” (vanishes). As a consequence, the relationship ofresults of ab initio calculations and phenomenological equa- tions of motion of magnetic vectors in AFM is associated with certain difficulties which we will discuss below. 3.1. A description of the low-frequency dynamics of antiferromagnets A main energy parameter characterizing an AFM system is usually a strong exchange interaction between spins of sub-lattices, which is parameterized through a magnitude of their flop fields H ex; in this case an intrasublattice exchange (even if it is relatively large), the only in FM, is often not taken into20 Low Temp. Phys. 40(1), January 2014 E. V. Gomonay and V. M. Loktev This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.210.130.147 On: Tue, 08 Apr 2014 14:32:36account. In typical AFM, used in spintronics (e.g., FeMn, IrMn, NiO), the value of Hexestimated by the N /C19eel tempera- ture is /103kOe. The characteristic values of the anisotropy field Han, which defines the orientation of the AFM vector with respect to crystal axes, is substantially less, Han/0.1 kOe. Thus, in external fields, an effective value of which is substantially less than Hex, there may take place such a motion of vectors of magnetic sublattices Mkthat their rela- tive position remains almost unchanged. In other words, under such conditions, the low-energy dynamics of AFM can beregarded as “ solid-body rotation ” of the system of magnetic vectors. This fact, first noted and used in works of Andreev and Marchenko, 49as well as Bar’yakthar and Ivanov,50,51can greatly simplify the description of the low-lying excitations in AFM and their interaction with external fields. Recall that AFM have two types of magnetic oscillations: acoustic (when Nsub/C203), with a characteristic frequency /ffiffiffiffiffiffiffiffiffiffiffiffiffiffiHanHaxp,a n d exchange (forNsub>3) the frequency of which is determined solely by the exchange interaction. In this sense, acousticmagnetic oscillations can be considered as low-frequency (although by analogy with lattice ones they are sometimes divided into the most low-frequency—acoustic and severalmore high-frequency— optical ), whose dynamics is deter- mined by the “solid-body” motion. It should be borne in mind that the frequencies corresponding to them as a rule appear to be significantly (up to orders of magnitude) higher than the intrinsic frequencies of magnetic oscillations in FM materials. Now, without going into detail described in numerous original papers as well as in the monograph Ref. 52, we dis- cuss main features of the AFM dynamics that are essentialfor the description of spintronic phenomena. First of all, we note that, despite the stiffness provided by the intersublattice exchange interaction the motion of magnetic vectors in AFMis always accompanied by the appearance, though vanish- ingly small but nonzero magnetization M AFM, depending on the speed of rotation of the magnetic structure. (This fact isoften overlooked in the microscopic description of spintronic effects. Thus, in Refs. 14,15,53, and 54it was unreasonably believed that angles between magnetic vectors are strictlyfixed, thereby there was ignored a contribution to the mag- netization M AFMof conduction electrons. Because of this the substantial correction to the rotation of AFM momentsinduced by the current was missed (see blow)). Thus, in the simplest case of a collinear AFM in an external magnetic fieldH(see Refs. 50and51), M AFM¼Hex 2cMs½L/C2_LþcL/C2ðH/C2LÞ/C138; (3.1) where c, as above is the gyromagnetic ratio. In addition, it is assumed that at temperatures much lower than the Neel temperature, Ms¼jM1j¼jM2jand, as a consequence, MAFM?L. For AFM with an arbitrary number of sublattices the appearing dynamic magnetization is expressed in terms ofthe vector Xof the angular velocity of the “solid-body” magnetic structure (in the case of a collinear AFM _L¼X/C2L), M AFM¼^v cðXþcHÞ; (3.2)where the structure of the magnetic susceptibility tensor ^vis determined by the exchange symmetry of AFM.49Below, in considering multisublattice AFM, for the sake of simplicity we will use the approximation of isotropic susceptibility andsuppose that as in the two-sublattice case v/C17M s/Hex. The basic equation of the low-frequency dynamics of AFM follows from the equation of the magnetization (spinangular momentum) balance, dMAFM dt¼r/C1 ^PordMðaÞ AFM dt¼@Pab @xb; (3.3) substantially coinciding with it. The magnitude of MAFMmust be expressed through the AFM order parameter according toEqs. (3.1) or(3.2), and the tensor of rank ^Pdetermines the density of magnetization flux. Using Eq. (3.1),t h eE q . (3.3) can be expressed in terms of the N /C19eel vector Las follows: L/C2ð€Lþc_H/C2Lþ2cH/C2_L/C02c 2HexMsHLÞ ¼/C02cHexMsr/C1^P; (3.4) where HL/C17/C0 @UAFM/@Lis the effective field which depends on the magnetic energy density UAFM; it is assumed thatjLj/C252Ms. The dynamic state of multisublattice AFM is conven- iently determined by using the Gibbs vector u(r,t) ¼tg(h/2)e, which parametrizes the rotation of the magnetic lattice as a whole at an angle h(r,t) around the axis e(r,t). In this case the instantaneous value of the sublattice magnetiza- tion vector Mkis defined by a rotation with respect to an equilibrium (reference) direction Mð0Þ k(jMð0Þ kj¼jMkj¼Ms) in the following way: Mkðr;tÞ¼Mð0Þ kþ2 1þu2ðu/C2½u/C2Mð0Þ k/C138þu/C2Mð0Þ kÞ: (3.5) The above-introduced vector of the angular velocity also depends on the Gibbs vector X¼2_uþu/C2_u 1þu2: (3.6) In this notations the equation for description of the mag- netic dynamics of AFM takes the form d dtðXþcHÞþcH/C2Xþc2Hex 2Ms^k/C01@UAFM @u/C18/C19 ¼cHex Msr/C1^P; (3.7) where ^k/C01is the tensor which is reciprocal to the tensor kab¼dabþeacbuc 1þu2; (3.8) determining the metric of the Riemannian space of the group of three-dimensional rotations (see Ref. 49),eacbis the anti- symmetric Levi-Civita tensor. Finally, from Eqs. (3.4) and (3.7) it follows that the dynamic phenomena in AFM, similar to resonant ones, areLow Temp. Phys. 40(1), January 2014 E. V. Gomonay and V. M. Loktev 21 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.210.130.147 On: Tue, 08 Apr 2014 14:32:36exchange-enhanced. In particular, all the terms in the right- hand side (i.e., the generalized forces and the magnetization flux) are multiplied by a large constant Hexof the exchange origin. Hence, the relatively small magnetization flux canlead to spinning the AFM moment and cause, as will be shown below, a steady precession of the latter. It is also noteworthy that the equations of magnetic dynamics for FM and AFM differ in structure. The Landau- Lifshitz equation for the field M(r,t) of the FM magnetiza- tion is the precession equation and contains only the firsttime derivative. As a consequence this equation is invariant (in the absence of dissipative forces) under Galilean transfor- mations. The equation for the N /C19eel vector in a homogeneous AFM is equivalent to the Newton equation of a material point, as it contains the second time derivative. In the case of inhomogeneous distribution L(r,t) this equation, what is essential, is invariant under Lorentz transformations. The differences between magnetic dynamics of FM and AFM systems are evident enough in small oscillations nearthe equilibrium position. Let us compare for example the behavior of homogeneous easy-axis FM and two-sublattice AFM (an easy axis along the axis z) in the absence of exter- nal fields. The corresponding equations for small deviations m 6¼mx6imyof FM and l6¼lx6ilyof AFM vectors have the form _m6¼6ixFMRm6;€l6¼/C0x2 AFMR l6; (3.9) where xFMR,xAFMR are the frequencies of ferromagnetic and antiferromagnetic resonances, respectively. As can beseen from Eq. (3.9) , the intrinsic oscillations of the FM vec- tor are always circularly polarized (the vector Mhas a fixed length and thus moves along the surface of a sphere), whilethe eigenmodes in AFM can also be linearly polarized (the vector Loscillates in a plane). Such oscillations of the vector L(for example, in the plane xz) immediately cause oscilla- tions in the perpendicular plane of the ( y-th) component of the magnetization M AFM. Fig. 3schematically shows the motion of vectors of the magnetic sublattices for AFM at such oscillations and, for comparison, the synchronized motion of two independent FM vectors which at the initial moment were antiparallel (infact, they correspond to the sublattices between which there is no exchange interaction). We emphasize that, as noted above, in the presence of the intersublattice exchange theangle between magnetic sublattices in their motion is always different from 180. Therefore, the synchronous motion of the antiparallel (i.e., with M AFM¼0) magnetic vectors of sublattices within the accepted model of AFM is impossible(otherwise it would mean an infinitely large exchange, H ex!1 , which excludes any “noncollinear” dynamics). In conclusion we note that in the absence of dissipative forces, the dynamic equations (3.4) and(3.7) can be regarded as Euler-Lagrange equations for the Lagrangian function, LAFM¼1 c2MsHex½_L2/C0t2 magðrLÞ2/C138/C0ð_L;L;HÞ cMsHex þ1 4MsHex½L/C2H/C1382/C0UAFMðLÞ; (3.10) or LAFM¼Ms 2c2Hex/C20 ðXþcHÞ2/C0t2 magX aX2 a/C21 /C0UAFMðuÞ; (3.11) with the magnon velocity tmag, which determines the limit- ing speed of propagation of magnetic perturbation in a me- dium and depends (also) on the inhomogeneous exchangeconstant, and the vectors X acharacterize the change of the Gibbs vector in space and are obtained from Eq. (3.6) by the substitution _/!r a/;a¼x;y;z. 3.2. Features in the description of dissipative effects in antiferromagnets Since the spintronic effects are associated with the proc- esses of energy dissipation, a formal description of the mag- netic dynamics must necessarily include the internal losses.But if for FM the model of the Gilbert internal friction (the term with a Gin Eq. (1.1) ) works well enough, then the description of dissipative processes in AFM remains largelyan open problem. The main difficulty is due to the presence in AFM of so- called exchange relaxation, which is away from the N /C19eel point stabilizes the relative orientation of magnetic sublatti- ces. (We do not discuss issues related to the longitudinal relaxation of sublattice magnetizations, which is also deter-mined by the exchange interaction and plays an important role near the Neel point and spin-reorientation phase transi- tions (see Refs. 52,55, and 56), as well as in ultrafast dy- namics of magnets (Ref. 10)). Obviously, these processes (in particular, fluctuations of exchange origin) are related with appearance and “diffusion” of the magnetization M AFM, i.e., with additional degrees of freedom in AFM which are not reduced, in general, to motion of AFM vectors. In other words, within the Onsager formalism the vectors _MAFMand _Lof generalized fluxes should be considered as independent. Then generalized forces which generate them coincide with the effective fields HL(see Eq. (3.4) ) and HM¼/C0@UAFM/ @MAFM .57–59To construct the dissipative function it is neces- sary to find the Onsager coefficients between generalized fluxes and forces, which are restricted, apart from theOnsager relations and the symmetry requirements, by conditions arising from equivalence of absolute values or from normalization of the sublattice magnetizationsFIG. 3. Dynamics of vectors of magnetic sublattices M1,M2in AFM (a) and two independent initially antiparallel FM (b). In AFM the vectors M1 andM2synchronously move across the surface of a cone whose axis is par- allel to MAFM. The magnetizations of identical FM move across surfaces of different cones; the synchronicity of movement is possible for a specific choice of initial conditions.22 Low Temp. Phys. 40(1), January 2014 E. V. Gomonay and V. M. Loktev This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.210.130.147 On: Tue, 08 Apr 2014 14:32:36jM1j¼jM2j¼Ms. This leads to the fact that already in the exchange approximation the dissipation function of AFM depends on spatial and/or time derivatives of the generalized flux _L,60for example, RAFM¼kðexÞ 1 2½rðL/C2_LÞ/C1382þkðexÞ 2 2€L2þkðrelÞ 2_L2:(3.12) Here kðexÞ 1;2andkðrelÞare the Onsager coefficients describing exchange and relativistic relaxations. The last term in Eq.(3.12) essentially represents a relaxation of Gilbert and can be obtained directly from the Landau-Lifshitz-Gilbert equation (see Eq. (1.1) , provided that I¼0) for magnetic sublattices. The most convenient, from our point of view,is the expression for the dissipation function obtained in Ref. 38in the approximation jM AFMj/C28Ms(these are the author’s notations of the Onsager coefficients), RAFM¼G1 2_M2 AFMþG2 2_L2: (3.13) Note that the first—exchange—term (the constant G1)i n Eq.(3.13) coincides with the second term in Eq. (3.12) if we take into account the relationship between the magnetizationM AFMand the AFM vector (see Eq. (3.1) ). However, the question about the quantitative relation- ship between the Onsager coefficients of exchange and rela-tivistic origin, which requires the knowledge of microscopic relaxation mechanisms remains unclear. From general con- siderations (see, for example, the discussion of time scalesof magnetic processes in Ref. 61) one can only assume that the ratios of inverse relaxation times for different modes are of the same order as the ratios of the corresponding intrinsicfrequencies. Since the frequencies of acoustic magnetic oscillations (and we consider them here) are much smaller than the exchange ones, one may consider that the character-istic times of exchange relaxation are so small that their cor- responding processes can be excluded from the consideration (i.e., one can formally put that G 1¼0). (On the other hand, considering the lifetime of the exchange modes to be small enough, it would be necessary to take into account that G1!1 , which leads to high critical fields and currents for the transition in the FM state. If the interval of the used external fields and currents is significantly less than critical, the processes related to the flop of sublattices just donot have time to develop, and at this point _M AFM!0. At the same time, since the flux _MAFM is inextricably related with the rotation of the AFM vector, it is easier in Eq. (3.13) to put G1¼0.) Note however, that sometimes in the simula- tion for simplicity it is taken that G1¼G2(see, e.g., Refs. 62 and63), which automatically excludes from the considera- tion the hierarchy of time of various relaxation processes and can lead to artifacts (see Sec. 4.1). 3.3. A phenomenon of exchange bias A widespread application of AFM as supporting ele- ments in spintronic structures is associated with the discov- ery in 1956 by Meiklejohn and Bean of the phenomenon ofthe exchange bias, 64,65or the emergence in a bilayer system FM/AFM of an effective internal magnetic field Hbias(called the bias field), which leads to a shift of the magnetichysteresis loop. Since this effect is important for the inter- pretation of existing experiments on spintronics of AFM, we consider it in more detail. Let FM material is characterized by the macroscopic magnetization MFM, and AFM by the N /C19eel vector L. The bias effect appears due to the interaction of the exchange nature between spins of FM and AFM. The energy of this interactionW bias(per unit area) depends on the relative orientation of MFMandL, that can be represented in general form through the dependence of the scalar product MFM/C1L,i . e . , Wbias¼WbiasðMFM/C1LÞ: (3.14) Since near the interface the translational symmetry of both FM and AFM layer is broken, the presence of this term in the energy of the system does not contradict theprinciples of symmetry. The explicit form of the interaction energy is largely determined by the details of the structure of materials, their chemical reactivity, surface quality, etc.However, as experiments show, in the most cases the bias favors either parallel orientation of M FMandL(if, for example, the antiferromagnet surface adjacent to FM con-tains an uncompensated magnetic moment) or perpendicular (the case of so-called spin-flop when FM plays the role of an external field for the AFM layer (Ref. 66)). Accordingly, the bias energy (3.14) is modeled using the actual situation, so that W bias¼/C0JbiasMFM/C1L;MFMkL; ðMFM/C1LÞ2;MFM?L;( (3.15) where Jbias>0 is the phenomenological constant, providing such a relationship. Obviously that the energy Wbiasdoes not change when the directions of the magnetic vectors are inverted simultane- ously: MFM!–MFMandL!–L, but it is responsible for the creation in a bilayer system of the state with an effective“frozen” magnetic field H bias. The state with Hbias6¼0 is metastable and persists as long as the system temperature does not exceed the energybarrier (called the blocking temperature) between two equiv- alent states with opposite directions of the magnetic vectors. An initial state is formed in the manufacturing process of thefilm, when the system itself “chooses” one of the directions of ordering (either by accident or through an external field applied specially for this purpose). A stability of the finalstructure with respect to external fields is caused by the dif- ference between characteristic fields for the reorientation of FM and AFM layers. The magnitude of the spin-flop field(the reorientation field) in AFM is exchange enhanced, H s/C0f/ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi HAFM anHexp , and may exceed the anisotropy field HFM anof FM by several orders of magnitude. Thus, the exter- nal field H/HFM an,Hbias/C28Hs/C0fcan cause significant rever- sal of the vector MFM practically not affecting the orientation of L. This makes it possible to determine experi- mentally Hbias, and consequently Jbiasby measuring, for example, the magnetization curves MFM(H). If a thickness of the AFM layer is comparable with a typical thickness of the domain wall, the rotation of the FM vector caused by an external field may form a so-called exchange spring predicted by Mauri et al.67andLow Temp. Phys. 40(1), January 2014 E. V. Gomonay and V. M. Loktev 23 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.210.130.147 On: Tue, 08 Apr 2014 14:32:36experimentally observed in such structures (see, e.g., Refs. 68–70). The spring is formed by the AFM vector reversal in the interface FM/AFM and fixed by a strong effective anisot- ropy in the AFM layer. In an external field the spring returnsthe system to its initial state, even if the vector M FMwas turned by 180 (i.e., went to the equivalent in terms of FM layer equilibrium state). The magnitude and direction of the bias field depend on the state of the AFM layer, especially if the latter is multi- domain (see for example Ref. 71). Therefore, the reorienta- tion of AFM vectors caused, say, by spin-polarized current or temperature fluctuations can lead to a noticeable (and observed) change of the bias field. The bias effect has already found application in spin valves with two working FM layers to fix one of them (a polarizer). However, as shown by recentexperiments, 20,22–24,28–30,72the study of the dependence of the bias field on the drive current gives an indication of spin- tronic effects in the AFM layer too. 3.4. An effect of size and shape of antiferromagnetic particles on their properties Because objects used in spintronics typically have sub- micron dimensions, their properties may differ significantlyfrom those of bulk materials due to significant influence of the surface at small sizes (see Ref. 73). In particular, AFM nanoparticles always have a non-zero magnetization 74due to such surface phenomena as translational symmetry breaking, an additional (other than bulk) anisotropy, etc. This leads to the fact that in the macrospin approximation the magnetiza-tionM AFM of such an AFM must contain along with the dynamic part (see Eq. (3.1) ) also the static part. Furthermore, within this approximation the normalizationconditions for magnetic sublattices may be violated and the whole system is characterized by a larger number of degrees of freedom than in the case of bulk AFM. In other words, anAFM nanoparticle can behave like a weak FM. This, of course, complicates the interpretation of the behavior of the already rather complex spin-valve systems. However, asnoted above, the dynamics of AFM has specific, other than FM, features (very different ranges of characteristic frequen- cies and fields, different types of dynamic equations) whichgive an opportunity to distinguish a role in these effects of the AFM ordering only. Another feature of AFM nanoparticles is unusual mani- festation of shape effects, which have traditionally been neglected, associating them only with the existence of an uncompensated magnetic moment. At the same time,experiments 75–77have convincingly shown that the shape is a source of an additional and controllable by it magnetic ani- sotropy of AFM particles; the effect itself is not associatedwith a weak FM. A physically consistent and accurate inter- pretation of the shape effects in AFM is based on the assumption of a fundamental role in them of magnetoelasticinteractions, 78–80due to which the surface magnetic anisot- ropy appears. Formally, the shape effects within the macro- spin approximation can be taken into account by usingadditional terms in the energy of AFM. In particular, for a flat rectangular particle ( a x/C2ay) the corresponding contribu- tion has the formUshape AFM¼1 2Kshape 2ðax=ayÞðL2 x/C0L2 yÞ; (3.16) where the phenomenological constant Kshape 2ðax=ayÞdepends on the ratio of the axes ax/ay, the magnetoelastic interaction constant and the initial energy of the surface anisotropy. The shape should also be taken into account when describing multidomain states of AFM, realized in samples with bias. It is the shape determines the size and the ratiobetween a number of domains of different types. To describe the corresponding effects one can introduce the destressing energy which depends on quantities averaged over samplevolume hL 2 ji(j¼x,y,z). Thus, for the same flat plate Udest AFM¼1 2Kdest 2ðax=ayÞðhL2 xi/C0h L2 yiÞ2: (3.17) 4. Spin dynamics of antiferromagnets in the presence of current Let us consider now the behavior of AFM when an elec- tric current flows through it. Up to date there are twoapproaches proposed to describe spintronic phenomena in AFM: microscopic which takes into account details of the band structure and therefore focused on specific material,and phenomenological based on general principles of magneto-hydrodynamics and hence is “independent” of materials. A direct comparison of the results obtained on thebasis of these approaches is difficult because they use vari- ous simplifying assumptions. Below we give an integral pic- ture, based on semi-classical equations for free electrons andthe macroscopic dynamic equations of AFM, discussed in Sec. 3. Nevertheless, we must admit that, despite the diver- sity of methods and approximations, all the known experi-mental and theoretical studies in the field of spintronics of AFM indicate unequivocally the possibility to control a state of the AFM layer using current. In this section we consider two fundamentally different types of structures based on AFM 1. Discrete systems including layers of FM andAFM materi- als. In such systems, primal is the question about an effect of the current polarized by the ferromagnet state on the state of the AFM layer. 2. Discrete systems consisting of two different AFM layers (by analogy with spin valves FM/FM), as well as continu- ous AFM systems (textures, domain walls). Here, in par-ticular, the question arises whether the AFM structure can polarize the current and influence through it on a neigh- boring layer, and whether there is the spin pumping phe-nomenon in AFM. Of note are synthetic (or artificial) structures consisting of antiferromagnetically coupled FM layers. Such systems are characterized by a highly anisotropic exchange between “layers-sublattices,” as well as by a strong dependence of theAFM exchange on the thickness of a nonmagnetic layer between FM layers, which allows their use as intermediate (between FM and AFM) model systems. There is a reason tobelieve that spintronic effects in such systems have features atypical for the above-mentioned structures, 40,81but a detailed discussion of this issue is beyond the scope of thisreview.24 Low Temp. Phys. 40(1), January 2014 E. V. Gomonay and V. M. Loktev This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.210.130.147 On: Tue, 08 Apr 2014 14:32:364.1. A discrete heterostructure FM/AFM This system is considered to be simple for description and consists of flat (with the normal n) FM and AFM layers separated (to avoid direct interaction between FM and AFM) by a nonmagnetic material. Typically, the thickness of mag- netic layers is less than the characteristic length of magneticinhomogeneity that allows the description of the state of FM and AFM by magnetization M FMand N /C19eelLvectors, respectively, which are homogeneous over volume (i.e.,using the above-mentioned macrospin approximation). The current (with density j e), flowing through FM is spin-polarized in the direction epol¼MFM/jMFMj, parallel to the magnetization MFM, and thus creates the magnetic moment flux ^PFM(see Eq. (2.3)), which flows into the AFM layer. At the interface NM/AFM and inside of the AFM layer free elec-trons are scattered with the spin-flip by localized ones. By anal- ogy with FM an intensity of these processes can be characterized by the dimensionless tensor ^e, determining the degree of depolarization of the magnetic moment flux. Note that the contribution to the depolarization can be made by bothpassed and reflected particles, so the direction of the current may be either perpendicular (CPP) or parallel (CIP) to the het- erostructure plane. The tensor structure depends, in general, onthe symmetry of the AFM layer. In particular, for a collinear AFM in the exchange approximation nonzero are only the components transverse with respect to the AFM vector whichare equal to e, that allows us to represent it in the form ^e¼e^I/C0 L/C10L 4M2 s/C18/C19 ; (4.1) where ^Iis the unity matrix. As a result, the magnetization flux absorbed by the AFM layer takes the form ^PAFM¼/C0r 4M2 sje/C10½L/C2ðepol/C2LÞ/C138;r/C17lB eeP:(4.2) In addition, the current is a source of magnetic fields of two types: the Oersted field, which we neglect, and an effec- tive field created by the sd-exchange and described by the phenomenological constant Jsd, Heff¼bðje/C1nÞepol;b/C17Jsdrð1/C0eÞ: (4.3) Let us now use the balance equation for the magnetization of AFM Eq. (3.3), to the right side of which we substitute the expression (4.2) and integrate over the thickness dAFMof the AFM layer in the direction of the normal n,t h e nw eo b t a i n dMAFM dt¼/C0r 4M2 sdAFMðje/C1nÞ½L/C2ðepol/C2LÞ/C138: (4.4) Expressing in accordance with Eq. (3.1) the magnetiza- tionMAFMthrough the N /C19eel vector Land assuming that the external field for it is Eq. (4.3) , from Eq. (4.4) we find the dynamic equation for this vector in the presence of a current(cf. Eq. (3.4) ), L/C2ð€Lþ2c AFM_L/C02c2HexMsHL/C0FSTTÞ¼0; (4.5) where 2 cAFM/C17caGHex, as above, is the coefficient of inter- nal friction (the line width of AFMR), andFSTT¼cbðje/C1nÞð2epol/C2_Lþ_epol/C2LÞ þrc2Hex 2MsdAFMðje/C1nÞþcbd dtðje/C1nÞ/C18/C19 ðepol/C2LÞ; (4.6) depending on the current. Similarly to the case of FM the Eq. (4.5), due to the pres- ence in it of the “external” force Eq. (4.6), contains current- defined terms of two types: one proportional to the constant r is due to the effect of STT and similar to the “Slonczewskiterm” in FM; others proportional to the constant bare “field” terms. However, their form and influence on the dynamics of AFM are somewhat different from the case of FM. First of all, note that for AFM the Slonczewski term (from a mechanical point of view) is not a moment of force, but is a force that, in addition, should be taken into account inthe terminology. (Note that in the works of several groups 14,15,53,54,82the torque acting on each of the AFM sub- lattices separately was calculated, and thus the “FMterminology” was used. An attribution of the obtained expres- sions to a “force” requires, however, the transition from a description in terms of sublattices to a description in terms ofthe N /C19eel vector in accordance with the procedure described in Sec. 3.1.) This term is exchange-enhanced (contains a factor H ex). From this it follows immediately that the dynamics of AFM induced by the current is a large enough effect caused by intersublattice exchange, and does not depend on the mag- nitude of the macroscopic magnetization MAFM. In contrast to FM the field terms give a significant con- tribution only in the case of an AC current or a variable (time-dependent) magnetization MFMof a polarizer. This is due to the specific of the interaction between AFM and an external magnetic field. A constant field leads to a quadratic in its value effects (in the Eq. (4.5) , the corresponding terms are omitted as small), and linear in the field contribution arises only in the case of the alternating field. Note also that, recent experiments on optical excitation of oscillations ofthe AFM vector 83,84show that the influence of the alternat- ing field can indeed be substantial. There are two more features of field terms. First, the contri- butions of AC and DC currents in the Eq. (4.5) are identical in structure (the last term), although, as noted above, are caused by different (in some sense, mutually exclusive) mechanisms: thecontribution caused by rincreases with increasing e,o raf r a c - tion of one-electron spin-flop e vents, and the contribution caused by bdecreases. This opens up th e possibility in principle of experimental determination o f the value by a relative contri- bution of the different components of the current. 45Second, the field term, depending on time derivative _epol(i.e., in practice on _MFM), has no counterpart in the dynamics of FM. A contribu- tion of this term can be significant when the considered FM layer is active. In addition, it simulates the Langevin sourcerelated with the magnetiza tion fluctuations in FM. Of importance, however, is something else: the current terms in both AFM and FM are sources/collectors of energyof the system. In fact, as follows from the Eq. (4.5) , a rate of change of the self-energy, E AFM¼1 2c2Hex_L2þUAFMðLÞ; (4.7)Low Temp. Phys. 40(1), January 2014 E. V. Gomonay and V. M. Loktev 25 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.210.130.147 On: Tue, 08 Apr 2014 14:32:36of the AFM layer due to work of the current and internal friction is described by the expression dEAFM dt¼/C0cAFM c2Hex_L2 þr 2MsdAFMðje/C1nÞþb cHexd dtðje/C1nÞ/C18/C19 ðepol/C1L/C2_LÞ; (4.8) wherein for simplicity it is assumed that the magnetization of the polarizing layer does not change, i.e., _MFM¼0. In contrast to FM a contribution to the dissipation is not only given by the Slonczewski term, but, as can be seen inEq.(4.8) also by the field term. Let us compare the features of the behavior of AFM and FM caused by the spin-polarized current using uniaxial sys-tems as an example. As an example, let us consider a stabil- ity of small oscillations Eq. (3.9) in the presence of DC current assuming in both cases that the vector e polof the cur- rent spin polarization is parallel to the easy axis and jejjn. Then the dynamic equations linear with respect to small deviations m6bl6take the form (cf. Eq. (3.9) ): _m67ixFMRm6¼6iaG_m6þCFM curjem6; €l6þx2 AFMR l6¼/C02cAFMi6þiCFM curjel6;(4.9) where the constants CFM curandCAFM curintroduced for compact- ness describe the interaction with the current. An analysis of the Eq. (4.9) shows (see also Fig. 4(a)) that in the case of FM the current term may play a role of ei- ther positive or negative friction, which is determined by the direction of the current (for a fixed direction of polarization)or the polarization (for a fixed direction of the current). Thus, in one case ( j e<0) the equilibrium state is stable, and in the other ( je>0) an instability occurs when je>jFM cr ¼aGxFMR=CFM cur. The latter is due to the fact that the direc- tion of STT (or, equivalently, a sign of the product CFM curje) depends on the relative orientation of the vector epoland the equilibrium magnetization Meqjjepolof the active layer. When Meq"#epolthe energy is pumped into the system; for Meq""epolthe spin torque enhances its own friction.AFM have two modes of oscillations with opposite cir- cular polarization (Fig. 4(b)). One of them, as can be seen from Eq. (4.9) , is increased with the current, and the second one is damped down. If the direction of the current isreversed, the second mode is gained, and the first one is damped down. As a consequence, for any direction of the current the equilibrium state of AFM loses stability at jj ej >jAFM cr¼2cAFMxAFMR =CAFM cur(see Fig. 5(a)) As a result, we conclude that in FM the STT phenomenon depends on the current direction, and in AFM it does not. It is interesting that the critical current value resulting in a loss of stability of the equilibrium state in AFM can be less than the corresponding value in FM (provided of course thatmaterials are with the same quality factor). Indeed, calcula- tions show that the critical current in AFM, j AFM cr¼2cAFMxAFMRMsdAFM crHex/cAFM xAFMRHAFM an; (4.10) and in FM (in similar notations) is jFM cr/ðcFM=xFMRÞHFM an,5 i.e. in both cases is proportional to the magnetic anisotropy field Han, which in AFM typically has a lower value, HAFM an/C20HFM an. Above, we considered the case of uniaxial anisotropy, when modes of two oscillations in AFM are degenerate andtherefore can have any (including, circular) polarization. For biaxial AFM or AFM with the anisotropy of “easy-plane” type the frequencies x xandxyof intrinsic linearly polarized modes are different. Since there is no exchange of energy between the spin-polarized current and the linearly polarized modes (with Ljj_L), the loss of stability of the initial state occurs at larger values of the current,18 jAFM cr¼MsdAFM crHexffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ðx2 x/C0x2 yÞ2þ4cAFMðx2 x/C0x2 yÞq :(4.11) However, an external magnetic field Hat a certain ge- ometry reduces (or even eliminates) the difference between the frequencies xxandxy, and thus decreases the critical current value18(see Fig. 5(a)). There is another significant difference between the dy- namics of FM and AFM induced by the spin-polarized cur- rent, noted in Refs. 17,18,53,82, and 85. While in FM/FM structures the current tends to flip the magnetization so as to create a parallel ordering of magnetic moments in polarizing and active layers, in a FM/AFM system the current favorsperpendicular ordering of FM and AFM vectors. Moreover, as shown in Refs. 17and18, when the current exceeds a crit- ical value Eq. (4.11) the AFM vector goes into a state of steady precession (or self-oscillations, Fig. 5) in the plane perpendicular to the polarization vector of the current e poljjMFM. This result does not depend on the nature of mag- netic anisotropy of AFM layer (at least in the approximation of the isotropic sd-exchange). Conversely, in FM a stable precession state is only achieved at a certain geometry of thesystem which is difficult to realize in an experiment: the polarization vector should be tilted with respect to the easy axis of the active FM layer. 5,86,87 The steady precession frequency of AFM is of the order of or higher then the frequency xAFMR and at jjej>jAFM cr depends linearly on the current,17,18FIG. 4. A comparison of STT in FM and AFM: the torsion moment in FM TSTT, depending on the orientation of magnetization of an active layer M can both enhance the friction G(position 1) and weaken it (position 2), causing reorientation M(a). In AFM the force associated with STT FSTT always reduces the internal friction for one of circularly polarized modes (mode 1) and increases for the other (mode 2) (b).26 Low Temp. Phys. 40(1), January 2014 E. V. Gomonay and V. M. Loktev This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.210.130.147 On: Tue, 08 Apr 2014 14:32:36xprec¼xAFMRjjej/C0jAFM cr jAFM cr: (4.12) Taking into account that the characteristic values of xAFMR/C241 THz significantly exceed xFMR(10–100 GHz) and, accordingly, the frequencies of self-oscillations stimulated by the current, nano-oscillators based on AFM materials may be promising sources of microwaves in the terahertz range. The behavior of the AFM vector described above in the presence of the current is associated with the pumping of the angular momentum (magnetization) into the system. In view ofthe approximations made with respect to relatively small dynamic magnetization, in the Eq. (4.5) we took into account the energy transfer of the external flow of the spin ^Ponly in the degree of freedom associated with rotation of the AFM vector (for practically unchanged its modul e) with respect to crystallo- graphic axes. There is, however, a natural question about thepossibility to induce a spin-flop p rocess of the magnetic sublatti- ces by current (i.e., actually the transition to the FM phase). This problem is closely related with th e exchange relaxation in AFM, d i s c u s s e di nS e c . 3.2. For example, in Ref. 62there was investi- gated the dynamics of a two-sub lattice AFM in the presence of current without limitations on t he value of the magnetization vector M AFM. The coefficients of internal friction responsible for processes of both exchange and a nisotropic relaxations, were assumed to be identical, i.e., it was supposed that G1¼G2(see Eq.(3.13) ). A numerical integration of the equations of motion of the AFM vector, on the one hand, confirmed the presence ofa stable precession regime of the AFM vector, obtained in Refs. 17and18neglecting the exchange damping ( G1¼0). On the other hand, it turned out that in this model the spin flop of sublattices takes place, and the characteristic times ofstabilization of the both states (both the dynamic precession of the AFM vector and the achievement by the magnetiza- tionM AFM of the saturation) are similar in magnitude. The latter fact, apparently, is the consequence of the relation between the coefficients of internal friction. In general, the dynamics of AFM in the presence of spin-polarized current can be conveniently described within the Lagrangian formalism, expanding the Lagrangian func- tion(3.10) ,(3.11) by the current terms, LAFM¼1 c2MsHex½_L2/C0t2 magðrLÞ2/C138 /C0bðje/C1nÞ cMsHexð_L;L;epolÞ/C0UAFMðLÞ; (4.13) and introducing the Rayleigh dissipative function which takes into account the internal friction and the Slonczewski term, RAFM¼/C0cAFM 2c2Hex_L2þr 2MsdAFMðje/C1nÞepol/C1L/C2_L: (4.14) 4.2. Two antiferromagnetic layers: Is the polarization possible? Let us consider now a system that does not contain an FM layer at all. For example, two AFM separated by a FIG. 5. The precession of AFM vector induced by a current (based on Ref. 18). In the initial state MFM||L||Z. A phase diagram of states in the variables “external magnetic field H–current I” (a). AFM vector trajectory for I>Icr(b). An onset of a steady frequency Xof rotation of AFM vector (c). A time de- pendence of the projection LzAFM vector on the direction MFM(d) (Xandtare in dimensionless units).Low Temp. Phys. 40(1), January 2014 E. V. Gomonay and V. M. Loktev 27 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.210.130.147 On: Tue, 08 Apr 2014 14:32:36nonmagnetic layer, i.e., a virtually complete AFM analog of a spin valve. Are there spintronic phenomena when current passes through such a system? The possibility of creating spin valves based solely on AFM materials was first raised in Ref. 13and subsequently analyzed in the framework of micr oscopic approaches,14,15,54,85,88–90and on base of phenomenological models for continuous AFMtextures. 38–40(In this connection we should also mentioned the works91,92which studied the spintronic effects in spiral struc- tures without macroscopic magnetization.) A construction of spintronic elements consisting of two AFM instead of two FM layers which is attractive from a practical point of view faces two problems which are notcompletely solved today: (1) the problem of reading the in- formation using an effect similar to GMR effect; (2) the problem of operating the state of this element by the phe-nomenon similar to STT. Recent experiments, confirming the presence of anisotropic magnetoresistance in AFM mate- rials, we will discuss below, in Sec. 5, and here we dwell on the question of the mutual influence of localized and free electrons, which is related to STT. From symmetry considerations it is evident that a uni- formly ordered AFM with zero magnetization cannot create an excess spin polarization of the conduction electrons. The spin polarization of the current can occur if AFM is in an external magnetic field, resulting in turning the sublattice magnetic moments, 36,93or if the AFM vector moves, creat- ing a small but non-zero magnetization.40On the other hand, from the same considerations that such material characteris- tics of AFM as scattering coefficients should depend on ori-entation (but probably not on the direction) of the AFM vectors. Moreover, the experimentally determined dependen- cies of electrical resistivity 94and absorption coefficient of linearly polarized X-rays95,96on the orientation of the AFM vector makes it possible to observe not only AFM domains,97but details of the distribution of the AFM vector within the domain wall98and on the sample surface.99Thus, we can expect that the magnetic order in the system of local- ized spins somehow affect the distribution of the spin polar-ization of both equilibrium free electrons and non- equilibrium involved in spin transport. The main mechanism of interaction of free and localized electrons in AFM as well as in FM is the sd-exchange. Using a microscopic language, this means that the Hamiltonian of an AFM metal, among others, includes the term of the type(cf. Eq. (2.2) ). ^H sd¼/C0X k;n;r;r0JðsdÞ n^a† k;n;r^srr’^ak;n;r’Sn; (4.15) where ^a† k;n;r^ak;n;rare the creation and annihilation operators of an electron with a transverse (with respect to the currentdirection) wave vector kand a spin index r,nis the number of an atomic plane in the current direction, J ðsdÞ nis the con- stant of the exchange interaction with localized spins Sn which are considered as classical vectors, ^sis the Pauli mat- rices. In the expression (4.15) , it is assumed for simplicity that the AFM ordering is uniform in a plane perpendicular tothe current direction. Unlike FM (cf. Eqs. (2.1) and(2.2) ), the transition to the quasi-classical approximation in the Hamiltonian (4.15) is difficult for the reasons described above: althoughmacroscopic vectors of magnetic sublattices M n(r,t) are associated with site vectors Snthey are considered to be localized at one point of a continuous medium r, while clas- sical (but not macroscopic!) vectors Snrefer to different points in space. Accordingly, the question arises: how to describe within the semiclassical approximation the distribu- tion of the spin density of free electrons? One possible way is to formally divide all conduction electrons into groups, each of which interacts with “its” sub- lattice of localized spins (i.e., assuming that each sublatticeof AFM physically represents FM). 90Then, by analogy with FM one can introduce the normalized vectors of the spin density mn(r,t)(jmnj¼1,n¼1,2) as averages of the spin operator over states of conduction electrons. From symmetry considerations it is clear that each magnetic vector of system of localized spins corresponds to a similar combination ofvectors m n(r,t) with the same translational and commutation properties. Thus, in a two-sublattice collinear AFM the mag- netization vector MAFMis associated with the magnetization of electronic gas m¼m1þm2, and the AFM vector is asso- ciated with the vector l¼m1–m2. Of course, the possibility of separating a system of free electrons on “sublattices”requires justification and depends on the characteristics of transport in AFM. A sufficiently obvious application of such a model is in synthetic AFM, in which each magnetic sublat- tice is a layer of FM material. In particular, for chromium such an approach follows from results of microscopic calcu-lations 13,15(see Fig. 6). However, when the separation into sublattices is impractical, within this model the vector lcan be excluded from consideration. In order to supplement the equations of magnetic dy- namics (3.4) and(3.7) for localized moments with the equa- tions for spins of conduction electrons, we use thesemiclassical Eq. (2.4) for each of sublattices, _m nþr ^Pn¼/C0/C22hJsd Msmn/C2Mn: (4.16) Equation (4.16) must be supplemented by relaxation terms, which in accordance with the principles of non-equilibrium thermodynamics should be introduced according to the sym- metry conditions, i.e., after the transition to the linear combi-nations introduced above. As a result, for the vectors mandl we obtain the following equations: _m nþr ^Pm¼/C0/C22hJsd 2Msðm/C2MAFMþI/C2LÞ/C0Cm; iþr ^Pl¼/C0/C22hJsd 2Msðm/C2LþI/C2MAFMÞ/C0C1;(4.17) in which (cf. Eq. (4.2) ) we determine the magnetization flux and AFM vector as follows: ^Pm/C17/C0lB 2eMsPje/C10MAFM;^Pl/C17/C0lB 2eMsPje/C10L; (4.18) and the relaxation terms we represent in the form (cf. Eq.(2.5) ) Cm¼m sm/C0DmDm;C1¼l/C0leq s1/C0DlDl; (4.19)28 Low Temp. Phys. 40(1), January 2014 E. V. Gomonay and V. M. Loktev This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.210.130.147 On: Tue, 08 Apr 2014 14:32:36where sm,lare the relaxation times, Dm,lare the diffusion coefficients, and the vector leqcorresponds to its equilibrium value. Obviously, in a state of equilibrium in a uniform AFM with MAFM¼0 the Eq. (4.17) are identically satisfied ifleqjjL. Let us now consider a system consisting of two identical N/C19eel AFM vectors L1andL2separated by a thin layer of a non-magnetic metal. Let the magnetization of localized spinsin AFM layer is still zero. Then from the first equation in Eq.(4.17) it follows that in the ballistic regime (in the ab- sence of diffusion, D m,l¼0) in the system there can appear non-zero magnetization of free electrons mjjL1/C2L2. The emergence of spin polarization in a two-AFM sys- tem with a non-magnetic layer was predicted in Refs. 14,15, 54,88, and 89on the basis of microscopic calculations of electronic states with taking into account the sd-interaction in the form of Eq. (4.15) . Chrome in which the occurrence of magnetic order is associated with structural features of the Fermi surface was chosen as AFM.48The calculations dem- onstrated that the spin polarization vector mjjL1/C2L2is con- stant within AFM layers, and within a non-magnetic interlayer of the thickness of dNMits direction oscillates. The latter is associated with a phase coherency of wave functionsof free electrons in the whole system AFM/NM/AFM and disappears in the case of non-ideal interfaces, as well as with increasing temperature. However, if the coherence can bepreserved, the magnetization of free electrons creates an effective magnetic field H eff nand, as a consequence, a tor- sional moment acting on each of the localized spins Sn,15 TSTT j¼Sn/C2Heff n;Heff n¼lBI eð dkjjJðsdÞ nmn X rð dkjjIrðEFÞ;(4.20) where the integration is carried out over projections of wave vectors kjjin the current direction, and IrðEFÞis the spin de- pendent transmission coefficient at the Fermi energy EF.A similar result under the same approach was also obtained forthe system FeMn/Cu/FeMn. 85 In the phenomenological approach the emergence of the magnetization mthroughout the volume of the active AFMlayer (e.g., the layer 2, if in the layer 1 the orientation of L1 is fixed) should in accordance with the equation of magnetic dynamics (3.4) lead to a reorientation of L2to the state in which L2jjL1. The driving force for this behavior is both an effective magnetic field Heff nand, probably, a flux of the magnetization m(note that in Refs. 14,15,85, and 90, the second of these contributions was not taken into account). Thus, the system AFM/NM/AFM under the influence of the current may change the relative orientation of the AFM vectors, but in contrast to standard heterostructuresFM/NM/FM unstable is the state L 1jjL2(while in FM an unstable state is Mpol"#Man). Furthermore, as already men- tioned, the structure AFM/NM/AFM must be quite perfect(i.e., persist a quantum coherence). More resistant to processes breaking the ideality (for example, inelastic scattering) proved to be another spintroniceffect observed in a system Cr/Au, 15namely, the first atomic layer of AFM at the interface is served as a polarizer for con- duction electrons. Microscopic calculations show that theoccurrence of polarization is caused by the presence of reso- nant electronic states at the interface. Qualitatively, this can be understood if we assume such a structure of the systemwhere each atomic layer of Cr is ferromagnetically ordered (i.e., forms a spin sublattice), and in the current direction (and perpendicular to the film plane) sublattices alternate. The translational symmetry breaking at the interface leads to the appearance there of the uncompensated spin density ofcharge carriers m6¼0, and in the depths of AFM, where the translational symmetry exists, m!0 (see Fig. 6). As a con- sequence, the three-layer system Cr/Au/Cr behaves similarlyto a standard spin valve of the type of FM/NM/FM, and the role of FM layers is played by atomic AFM layers adjacent to both sides of the interface (Fig. 7). A quite general approach to the description of spintronic effects in AFM structures exclusively (without FM compo- nents) was proposed in Ref. 38using the principles of non- equilibrium thermodynamics. Following it, we introduce conjugate pairs of generalized thermodynamic fluxes and forces: “charge” j eandEand two “magnetic” discussed in Sec. 3.2are generalized fluxes of magnetization and AFM vector, _MAFMand _L, and their conjugate forces HMandHL. By analogy with the FM (see Sec. 2) we assume that in AFM there is also a phenomenon of spin pumping, i.e., fluxes of magnetic vectors _MAFM and _Lcan create an additional charge current with the density jpump. In this case, considering an exchange symmetry (in this case, the exchange symmetry group includes rotations in the space of magnetic vectors, translation permuting sublattices (here,M AFM!MAFM,Lfi–L) and the time-reversal operation which reverses signs of MAFM andL), the density of the pumping current and, correspondingly, an electric field canbe written as E a¼jpump a rcond¼gðL/C1_MAFM/C2r aLÞþfð_L/C1raLÞ;(4.21) where gandfare the phenomenological coefficients, similar to adiabatic and nonadiabatic spin transfer in FM (see Eq.(2.7) ). Coming in Eq. (4.21) with the use of the equa- tions of magnetic dynamics to generalized forces HMand HL, we obtain for pumping current. . . . . . FIG. 6. A layer-by-layer distribution of the normalized spin density mof free electrons in the system Cr (layers 1–12)/Au (layers 13–24), calculated on the basis of the density of states at the Fermi surface, obtained in Ref. 15. The arrows indicate the orientation of local magnetic moments in a corre- sponding layer. The vertical line indicates the position of the interface.Low Temp. Phys. 40(1), January 2014 E. V. Gomonay and V. M. Loktev 29 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.210.130.147 On: Tue, 08 Apr 2014 14:32:36jpump a¼crcond 1þG1G2/C2½ ðf/C0gG2ÞðL/C2r aLÞ/C1HM þðfG1þgÞðr aLÞ/C1HL/C138; (4.22) where the constants G1,G2determine the dissipative func- tion (3.13). Now using the Onsager’s reciprocity relations,one can find a force created by the charge current, acting on the AFM vector. As a result, we obtain an equation similar to Eq. (4.5) , in which the force created by the current is (cf. Eq.(4.6) ), F STT¼cðgþG1fÞ 1þG1G2dje dt/C1r/C18/C19 Lþ2c2fHexMs 1þG1G2ðje/C1r ÞL: (4.23) Equation (4.23) , obtained for a continuous medium, is easily transformed to the equation for a discrete system AFM/FM/AFM with a fixed AFM vector Lpolof one of the layers (of the polarizer) by using the substitution rL!Lpol/dAFM. For clarity we write the equation in a sim- plified form, i.e., for a DC current and neglecting anexchange relaxation ( G 1¼0), €Lþ2cAFM_L¼2c2HexMsHL/C0fje dAFMLpol/C18/C19 ; (4.24) then ask the question: “What are the effects which result from the assumption of the existence of the spin pumping in AFM?” In discrete systems, as seen from Eq. (4.24) , the cur- rent plays a role of an external force tending to order the active AFM layer so as to set LjjLpol. In other words, two AFM in this case will behave as two FM. The same conclu- sion can also be obtained from an analysis of Eq. (4.23) ,i n particular in such an AFM the current supports steadymotion of a domain wall with the speed 38(cf. Eq. (2.8) ) vDW¼/C0cf G2je: (4.25) To some extent, this response is also confirmed by ab initio calculations for an AFM system FeMn,85which predict (though when quantum coherence of wave functions ofconduction electrons is conserved) the existence of STT which can lead to domain wall motion. Note that the Eqs. (4.23) and (4.24) can be derived directly from the Eq. (4.17) and the expression, Hsd¼/C0Jsd 4MsðmMþILÞ; (4.26) for the sd-exchange in AFM. To do this, one should put sm¼sl¼ssf,Dm¼Dl¼0. Practically, this approach means that the ordering of free electrons created in a certain region of AFM ( l6¼0) persists and is transferred over the entire sample, thus leading to spintronic effects. This modeldescribes well artificial AFM structures, where the current is distributed along individual FM layers-sublattices, between which there is an exchange interaction of the AFM-type.However, one cannot exclude other relationships between the parameters s mandsl,Dm,andDl, describing the relaxa- tion properties of exchange and relativistic nature. As also inthe case of the relaxation constants for localized spins, dis- cussed in Sec. 3.2, this question, as far as we know, is still open. Based on the already cited works, 15,41,85one can assume that in natural AFM the diffusion of the non-equilibrium AFM spin density dlof carriers occurs so fast and at such small distances that it can be neglected and one can only con-sider the processes associated with the formation of the mag- netization dmof free electrons. In this case, an effect of the current on the system of localized spins is determined mainlyby the dynamic magnetization M AFM/L/C2_L(3.1), and the corresponding force depends on time and spatial derivatives of the AFM vector,40 FSTT¼lBP ceHexðje/C1r Þ ð L/C2_LÞ/C0L½L/C1ðje/C1r ÞL/C2_L/C138=4M2 sno þsmJsdP 4eHexM3 s2_L½_L/C1L/C2ðje/C1r ÞL/C138/C8 þ4M2 s€L/C2ðje/C1r ÞLþL½€L/C1L/C2ðje/C1r ÞL/C138g:(4.27) The dynamics of the AFM vector under the action of this force will obviously differ significantly from FM, and details of the behavior wait for further study. Thus, at this stage, one can conclude that the localized moments in AFM, in principle, can create a spin order of conduction electrons and thereby to initiate a spintronic effect. The appearance of such AFM effects is more diversethan in FM, and the nature of the phenomena depends strongly on the ideality of structures and the chemical purity of AFM materials. 4.3. A spin transfer from antiferromagnet to ferromagnet: A recoil effect Finally, we discuss the problem of transfer of spin order from AFM to FM, ignoring for a while the direct exchange interaction (RKKY) and the exchange bias (assuming, forexample, that between AFM and FM layers there is a thin NM layer). Microscopic calculations performed for systems Co/FeMn (Ref. 85) and Co/NiMn (Ref. 53) showed that the FM layer closest to the interface experiences STT (see Eq.(4.20) ), induced by spin transfer from AFM, but deeper in the ferromagnet this effect is weakened. Moreover, theFIG. 7. A layer-by-layer distribution of x,y,zcomponents of torque (4.23) in the system Cr (layers 1–12)/Au (layers 13–16)/Cr (layers 17–24), plotted according to Ref. 41. The arrows indicate the orientation of local magnetic moments of Cr in interface layers. The round arrows show the direction of torque rotation in the plane xz(dotted line) and in the direction y(solid line). The vertical line indicates the position of interfaces.30 Low Temp. Phys. 40(1), January 2014 E. V. Gomonay and V. M. Loktev This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.210.130.147 On: Tue, 08 Apr 2014 14:32:36STT transferred between AFM and FM ordered layers depends on the angle hbetween LandMFMas sin 2 h (Fig. 8(a), curves P1,P2), while for two FM layers a typical dependence is sin h(Fig. 8(a), curve P3). The dependence which includes even harmonics sin 2 nhwith different n¼1,2 was obtained phenomenologically in Ref. 82, where AFM is presented as a synthetic structure, i.e., as a set of FMlayers separated by a space each of which interacts with an active FM independently. From our point of view, the influence of AFM on FM induced by the conduction current is general in nature and is associated, as well as all the effects discussed in this work, with the transfer of spin angular momentum from conductionto localized electrons. Indeed, the current polarized by FM transfers the magnetization dmjjM FMinto AFM (Fig. 9). The component dmperpendicular to the vector L(in the case of a two-sublattice structure) is partially absorbed by an AFM layer, resulting in the appearance of the magnetization MAFMjj[L/C2(MFM/C2L)] and the rotation of localized moments (see Eqs. (4.2) and (4.4) ). A flux coming in the AFM layer is equal in magnitude and opposite in sign to that which goes out of the FM layer, which thereby experiences“recoil.” A torsional moment appearing in this case is deter- mined by the flux n/C1^P AFM, which is carried into the AFM and is taken with an opposite sign. Thus, for FM (see Eqs.(4.4) and(1.1) ) T STT¼rI 4M2 sMFM/C23FM½L/C2ðMFM/C2LÞ/C138; (4.28) where, as above, tFMis the volume of an FM layer. From Eq. (4.28) it is obvious that TSTT/sin 2h. Therefore, due to the recoil the magnetization of FM rotatesso as to be parallel Lwhen T STT¼0. If initially L?MFM (and TSTT¼0), then the recoil depending on the current direction leads to increase or decrease in the stability of theFM state with respect to the flipping of M FMby 180 (see Fig. 10). The recoil effect is retained even when accounting an exchange bias. Moreover, it is just the effect which canlead to the dependence of coercivity field and bias field on direction and magnitude of the current (see Sec. 5). 5. An experimental observation of spintronic effects in antiferromagnetic structures An observation of spintronic effects in AFM is a chal- lenging experimental task for several reasons. First, the val-ues of typical fields of reorientation and frequencies of AFMR are considerably higher than those in FM materials, which requires to use measuring methods fundamentally dif-ferent from those in the case of FM. Second, a magnetic structure of AFM in a multilayer sample depends essentially on dimensions (particularly, on a thickness) of the layer, aswell as on the chemical composition of adjacent layers and may differ from that of the bulk crystal. Finally, it is not yet clear if it is possible to observe the effect of the anisotropicmagnetoresistance in AFM (without FM layers) heterostruc- tures. (Such an effect similar to GMR would allow to detect reorientation of AFM vectors under the influence of the cur-rent. An attempt to find it was made in Refs. 22,100, and 101.) However, the information on an effect of the spin- polarized current on the state of AFM can be obtained bydefining (in the magnetization curve) the bias field in a bilayer structure FM/AFM at different current values. Another type of experiments based on measuring field andFIG. 8. A dependence of STT (per unit current) in the system Co/NiMn on the angle between magnetic vectors in AFM and FM layers (Fig. 9), based on data from Ref. 53. STT transferred through the planes P1,P2, and P3in the interface area: in the Mn layer magnetic moments of sublattices fully compensate each other, the Ni layer acquires a weak magnetic moment due to the proximity to the FM layer of Co (a). The effect of “recoil:” STT at theinterface, acting on AFM and FM layers are equal in magnitude but opposite in sign (b). The calculated data are points; solid lines are fits by the functions sinh, sin2 h.FIG. 9. The current induced dynamics of magnetic moments in the system AFM/FM. The broad arrows show the orientation of magnetic moments of sublattices M1,M2in AFM and the magnetization vector MFMin FM layers, (a). The current of electrons e, polarized in the FM layer, contributes to AFM the magnetic moment dmand gives rise to the magnetization MAFM and rotation of the vectors M1,M2in the plane perpendicular to MAFM.A sa result of the recoil due to the magnetization of carriers absorbed in the AFM layer— dm, the vector MFMrotates in the picture plane, (b). FIG. 10. An influence of the current on stability of the FM layer at L?MFM. An unstable state: in the FM region due to the recoil dm"#MFM(a); a stable state: in the FM region dm""MFM(b).Low Temp. Phys. 40(1), January 2014 E. V. Gomonay and V. M. Loktev 31 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.210.130.147 On: Tue, 08 Apr 2014 14:32:36current dependencies of GMR between two FM layers (free, FM2, and fixed by AFM, FM1) in a spin valve of the type of FM1/NM/FM2/AFM (see Sec. 1). An interpretation of the data obtained in this way is also difficult, because, amongother things, it requires to take into account the influence of bias and STT effects on a state of the FM system. Nevertheless, there is already a series of experiments clearly indicating the possibility of controlling the state of AFM layer by the spin-polarized (using FM) current. Their detailed description (except, perhaps, very recent workRef. 30) can be found in the review of MacDonald and Tsoi, 42so here we restrict ourselves to a brief description and interpretation of known experiments based on the theo-retical approaches discussed above. Calculations show that when a high-density current flows through FM/AFM the following phenomena may take place.First of all, the current polarized by an FM layer leads to a reorientation of the AFM vector and thereby to a change of the bias field. Such effects were observed by Tanget al. , 23–25,29,30and by Urazhdin and Anthony20in IrMn/NiFe and FeMn/NiFe systems. As an example, Fig. 11shows the dependence of the bias field on the magnitude of the currentflowing through the structure. As can be seen, the value of H biasstarts to decrease when the current exceeds a certain threshold value Icr, which allows us to correlate the latter with the magnitude of the critical current (4.10) . The dependence ofHbiason the current direction (whether it flows from FM to AFM or vice versa) makes it possible to exclude the influence of Joule heat,20,27and the measured temperature dependence of the critical currents20indicates the relation of the observed effects with the presence of an AFM layer. Further, the spin-polarized current can lead to the pre- cession of the AFM vector that is associated with the absorp-tion by the AFM layer of the energy carried by non- equilibrium carriers. Such an increase in attenuation and the associated suppression of the precession of the FM layer wasobserved in Ref. 20, and the corresponding threshold current is correlated with I crfor the field Hbias. Due to the processes of spin-dependent scattering at the interface FM/AFM there can appear a recoil effect (see Sec. 4), which, in our view, took place in experiments.102,103 In particular, in Ref. 102a spin valve, in which Co was used as both an active FM and a polarizer, was studied. It wasfound that when the biasing layer of an antiferromagnet IrMn was placed next to a polarizer, the critical current of switching between parallel and antiparallel ordering of FM layers signif- icantly decreased, despite the presence of the exchange inter-action (bias) between the AFM and FM layers (see Fig. 12). This observation can be interpreted as follows. IrMn used as AFM absorbs a spin angular momentum carried by free elec-trons from a fixed FM layer (FM1). In this case due to the recoil effect the magnetization of the FM1 layer turns, as shown schematically in Fig. 12.A c c o r d i n g l y ,t h ea n g l e between the magnetizations M polandManof fixed and active layers increases, and as a result TSTTincreases (see Eq. (1.1)). Thus, the magnetization of the active layer flips at currentsless than those in the system without an AFM layer. Finally, the current can cause the mutual motion of the magnetic vectors in the exchange-coupled (biased) AFM andFM layers. A detailed study of their dynamics with taking into account features of an AFM system remains an open problem, so we restrict ourselves by qualitative considera-tions. (An attempt to explain the behavior of a bilayer system with biasing was made in Refs. 104 and105. However, in the models used dynamics of the AFM layer is related withits small uncompensated moment, and the exchange enhancement effects considered above are not taken into account. Nevertheless, the results of calculations predict a change in the bias field and qualitatively correlate with experiments (Ref. 20).) Obviously, the behavior of FM/AFM system depends on the relations between values of the criti- cal currents and the damping coefficients of each component. Thus, if the FM layer is magnetically hard and, conse-quently, I FM cr/C29IAFM cr, then the current causes only a reorien- tation of the AFM vector, and as demonstrated above independently of the current direction (unlike the similar FIG. 11. A change in the magnitude Hbiasand the direction (given by the angle with respect to the initial one) of the bias field in a bilayer film IrMn/NiFe under the current I(according to Ref. 30). The current in the CIP configuration.FIG. 12. Recoil effect in the spin valve IrMn/Co/Cu/Co. The graph showsthe dependence of the resistance (normalized to the minimum value) on the current in the two cases: with the AFM layer and without it (according to Ref. 102). A deviation of the magnetization of the fixed layer FM1 due to the recoil (the solid arrow is with AFM, the dashed one is without AFM) is shown schematically at the top.32 Low Temp. Phys. 40(1), January 2014 E. V. Gomonay and V. M. Loktev This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.210.130.147 On: Tue, 08 Apr 2014 14:32:36situation in FM). However, because the frozen magnetization of the AFM layer providing the bias flips together with the AFM vector, then as a result the value of the bias field and the field of switching of the biased FM layer will grow forone direction of the current (when electrons move from the FM layer to the AFM), and decrease in the opposite (Figs. 13(a) and13(b) ). If hard is the AFM layer, i.e., I FM cr/C28IAFM cr, then the recoil effect plays an essential role. Thus, AFM works as a reflector absorbing a magnetic moment created by the FM layer. As a result, for one direction of the current(from AFM to FM layer) the field of switching does not depend on the current, while in the opposite case it decreases (Fig. 13(c) ). If the critical currents of AFM and FM layers are comparable, I FM cr/IAFM cr, the system AFM/FM can be considered as closed and its behavior in the presence of cur- rent depends on the state of the second (free) FM layer.However, in any case the presence of AFM increases the coercivity of the AFM/FM system, which leads to an increase in the switching field of the hard FM layer at thetransition from an antiparallel orientation of magnetizations of the FM layers to parallel one, and to a decrease for the reverse transition, independent of the current direction (Figs.13(d) and13(e) ). Almost all described scenarios of behavior of spin-valve systems FM1/NM/FM2/AFM were observed in experiments 19,21for different combinations of FM and AFM materials. In this case, if CoFe was used as FM with a large magnetic anisotropy, then the dependence of the type shownin Figs. 13(a) and13(b) were observed, and the slope of the line depended on the order of the layers. When using permal- loy (small magnetic anisotropy) the dependencies which areplotted schematically in Fig. 13(c) for AFM with a largecoefficient of spin-dependent scattering (IrMn) and in Figs. 13(d) and13(e) in the case of AFM with a smaller coefficient (FeMn) were observed. In any case, the dependencies of the switching field of a hard FM layer in a spin valve with bias-ing are qualitatively different from those of a standard sys- tem FM1/NM/FM2 (Fig. 13(f) ), which allows again to see an effect of the current on a state of the AFM layer. Conclusion We considered the main approaches to the description of spintronic phenomena in AFM systems—the microscopic and phenomenological. Both are based on the spin-dependent interaction between free and localized electrons. Microscopic models are mainly focused on the description of itinerant electrons, considering localized spins statically.(After the main part of the manuscript was finished we found the works, 90,106in which motion of localized moments is also taken into account within a microscopic model.) Inaddition, these models consider ballistic processes for real- ization of which the quantum interference of electronic func- tions must be kept. In turn, the phenomenologicalapproaches allow taking into account both non-dissipative and dissipative dynamics of localized spins on the basis of general thermodynamic considerations without detailing thesd-exchange. The phenomena predicted within such models, as for FM, are rather “rough” and do not require for observa- tion of ideal (in the sense of quality of interfaces, stoichiom-etry, etc.) samples. However, a full picture of the spin dynamics of AFM requires a combination of micro- and macro-approaches and is still far from complete. All these models are based on exchange nature of the sd- interaction, which automatical ly leads to the conservation of the total spin of the system of localized and itinerant electrons.Thus, the dynamics of AFM, as for FM, is determined by a magnetization flux created by non-equilibrium carriers. However, in AFM, unlike FM, the introduced magnetizationcan lead to (1) rotation of magnetic sublattices as a whole, (2) turn of magnetic sublattices, and (3) change in the magnitude of the sublattice magnetization vector. In Refs. 42and90it is noted that it is necessary to consider the processes (2) and (3) associated with the spin transfer between magnetic sublattices. We believe that such processes, because of their purelyexchange nature, are associated with a high-energy excitations and therefore significant at suf ficiently high densities of non- equilibrium free carriers (for example, for a non-resonant opti-cal pumping). In order to describe the behavior of AFM in this case it is necessary to reject a number of simplifying assump- tions (such as normalization conditions) and, among otherthings, to determine the spin relaxation mechanisms. In the case of spintronic phenomena caused by flowing electric cur- rent the main role is played by the low-energy processes (1),which justifies the use in such situations of the solid state approximation. For those interested in spintronics, we specify a number of open problems and possible directions of research in spin- tronics of AFM. First of all these are questions related to studying mechanisms of spin-dependent scattering of con-duction electrons and spin transport in AFM materials. Because of the strong spin-orbit interaction in AFM the approximation of the sd-exchange may not be sufficient and FIG. 13. A change of the switching field of hard (solid line) and free (dashed line) FM layer, depending on the current for a spin valve with a bias, FM1/NM/FM2/AFM (a)–(e) and without bias, FM1/NM/FM2 (f). The arrows indicate the direction of magnetization in FM layers, the doublearrow is AFM layer. (a) and (b) is the hard magnetic FM, I FM cr/C29IAFM cr, the structure FM1/NM/FM2/AFM (a) and the inverted one, AFM/FM2/NM/ FM1 (b), (c) is the soft magnetic FM, IFM cr/C28IAFM cr; (d) and (e) is the inter- mediate case, IFM cr/IAFM cr, different slopes correspond to measurements with decreasing (d) and increasing (e) a magnetic field.Low Temp. Phys. 40(1), January 2014 E. V. Gomonay and V. M. Loktev 33 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.210.130.147 On: Tue, 08 Apr 2014 14:32:36moreover a variety of magnetomechanical phenomena, both similar to those observed in FM107–109and different from them45cannot be excluded. This question needs attention of both theoreticians and experimentalists. The interesting and important for applications problems are related to accounting for magnetic noise and studying its influence on spintronic effects. Some steps in this direction,based on a general formalism (in particular, development of kinetic equations taking into account the specific features of AFM) were performed in Refs. 110and111. Unclear, how- ever, remains the question about mechanisms and channels of spin relaxation in AFM, which is especially important in the description of strongly non-equilibrium states. Scarcely explored is the joint dynamics of FM/AFM structures, which would take into account both features of behavior of an AFM layer and exchange bias and spin polar-ization effects. The results of such studies would allow not only to interpret already known data, but also to effectively operate by the properties of spin valves. An experimental observation of spintronic effects in AFM requires development and application of new methods in principle, and, in this respect, studies of ultrafast dynam-ics under action of femtosecond laser pulses seem to be promising. Possible difficulties in registration of the AFM layer state in a spin valve can be apparently overcome by using the effect of tunnel anisotropic magnetoresistance recently found in AFM. 72An effect of current on the behavior of inhomogeneous AFM systems (texture or hetero- structures), as predicted by both microscopic and phenome- nological theories was not possible to observe yet. Finally, in addition to metals FeMn, IrMn, Cr, and dielec- tric NiO (Mott) studied today, it would be worth to include to AFM investigated by spintronics AFM semiconductors112and semi-metals,12as well as, perhaps, antiperovskites with a non- collinear magnetic structure based on Mn. The last-named, as recent experiments113,114showed, reveal a significant magne- toresistance, and depending on stoichiometric composition and temperature can have FM, AFM or weak FM ordering. In addition, they can be grown as thin films and, more impor-tantly, can be both metals and semiconductors. We thank V. G. Bar’yakhtar, E. D. Belokolos, Yu. B. Gaydidey, B. A. Ivanov, V. P. Kravchuk, B. I. Lev, Yu. G.Naidyuk, S. M. Ryabchenko, A. Slavin, D. D. Sheka, V. Baltz, R. Duine, A. N. MacDonald, C. Schneider, and X. L. Tang, discussions with which helped to clarify the issues of descrip-tion and observation of spintroni c effects in antiferromagnets. 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1.3679760.pdf

Current-induced domain wall motion in a multilayered nanowire for achieving high density bit T. Komine, A. Ooba, and R. Sugita Citation: J. Appl. Phys. 111, 07D314 (2012); doi: 10.1063/1.3679760 View online: http://dx.doi.org/10.1063/1.3679760 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v111/i7 Published by the American Institute of Physics. Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 17 Jun 2013 to 18.7.29.240. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsCurrent-induced domain wall motion in a multilayered nanowire for achieving high density bit T. Komine,a)A. Ooba, and R. Sugita Department of Media and Telecommunications Engineering, Ibaraki University, 4-12-1 Nakanarusawa-cho, Hitachi, Ibaraki 316-8511, Japan (Presented 2 November 2011; received 23 September 2011; accepted 19 December 2011; published online 13 March 2012) We herein propose a multilayered nanowire in order to achieve a high density bit or short bit length. The multilayered nanowire consists of a continuous layer and a granular layer which are coupled by interlayer exchange and magnetostatic interactions. The continuous layer has a role of data transfer based on current-induced domain wall motion (CIDWM). The granular layer has a role of stabilizingdata, and the current flows only through the continuous layer. We demonstrate CIDWM in the multilayered nanowire by micromagnetic simulation. The domain wall width in the multilayered nanowire is narrower than that of the single-layer nanowire because of weak exchange coupling inthe granular layer. As a result, the smaller bit length can be obtained in the multilayered nanowire. Moreover, the critical current density in the multilayered nanowire almost equals to the critical current density of the single-layer nanowire with the same domain wall width as that of themultilayered nanowire. VC2012 American Institute of Physics . [doi: 10.1063/1.3679760 ] I. INTRODUCTION Current-induced domain wall motion (CIDWM)1in a nanowire has attracted much attention in recent years. Mag- netic race-track memory based on current-induced domain wall motion (CIDWM) was proposed.2Other applications of current-induced domain wall motion to memory3devices have been also presented. The reduction of the intrinsic criti- cal current density for current-induced domain wall motionis one of important issues for practical use. Moreover, in order to develop such devices, achievement of high density bit is inevitable as well as reduction of critical current den-sity to move domain wall. 4,5 The wall pinning site such as a notch has been expected as one of candidates to stabilize domain wall.6On the other hand, more accurate fabrication of such pinning sites by lithography may be strictly required as the bit becomes smaller. However, the possibility of high density bit in suchdevices has not been sufficiently discussed yet. The increase in the areal density of the hard disk has been achieved by adopting multilayered (ML) structure such ascoupled granular/continuous (CGC) structure 7and exchange- coupled composite.8,9The small grain in the granular layer enables us to obtain a small magnetic cluster or small bitlength in each ML structure. In this study, we propose a multi- layered nanowire (MLNW) which has coupled granular/con- tinuous structure, and discuss the possibility of high densitybit. Moreover, CIDWM in the multilayered nanowire is dem- onstrated by micromagnetic simulation. II. CALCULATION MODEL In order to simulate CIDWM, the Landau-Lifshitz - Gil- bert equation including the spin transfer torque effect wassolved.10Figure 1shows the schematic calculation model of a multilayered nanowire (ML-NW) proposed in this study.The nanowire consists of a continuous layer and a granularlayer which are coupled by interlayer exchange and magne-tostaic interactions. The continuous layer (transfer layer) hasa role of data transfer based on CIDWM. The granular layer(stabilization layer) has a role of stabilizing data. The currentflows only through the continuous layer, and the current doesnot flow through the granular layer because grains in thegranular layer are separated by insulating materials. Each grain in the granular layer is assumed to have a rectangular shape of 8 /C28/C25n m 3. The intragrain exchange constant Ain each layer was set to 1.3 /C210-6erg/cm. The interlayer exchange coupling constant is set to also1.3/C210 /C06erg/cm. The intergrain exchange coupling con- stant A0was varied from 0 to 1.3 /C210/C06erg/cm. The satu- ration magnetization in each layer is 400 emu/cm3. The nanowire has a perpendicular anisotropy and the anisotropy field is 7.5 kOe. The nanowire was divided into rectangle cells of 2 /C22/C25n m3for the micromagnetic calculation. The gyromagnetic ratio is 1.76 /C2107Hz/Oe and the Gilbert damping constant ais assumed to be 0.02. Although this value influences the wall velocity under the applied current, the intrinsic current density is independent on a. In order to investigate the intrinsic current density in ML-NWs, the non-adiabatic spin-torque parameter is set to 0. First, we investigated the possibility of high density bit in the nanowire by utilizing micromagnetic simulation. The minimum bit length was investigated for ML-NW comparingwith single-layer case by varying the intergrain exchangecoupling A 0. The single layer nanowire (SL-NW) has the same dimension as only transfer layer in ML-NW. Next, the relation between spin drift velocity uand wall velocity vof single wall in SL- and ML-NWs was investigated.a)Electronic mail: komine@mx.ibaraki.ac.jp 0021-8979/2012/111(7)/07D314/3/$30.00 VC2012 American Institute of Physics 111, 07D314-1JOURNAL OF APPLIED PHYSICS 111, 07D314 (2012) Downloaded 17 Jun 2013 to 18.7.29.240. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsIII. RESULTS AND DISCUSSION Figure 2shows magnetization distributions of SL- and ML-NWs. The initial magnetiz a t i o ni ne a c hN Wh a so n l yo n e bit or two domain walls, and the equilibrium state was obtained by varying bit length. The intergrain exchange constant A0in the granular layer is 1.3 /C210/C07erg/cm. The stable domain structure is Ne ´el wall even in the granular layer. The calculated minimum bit length in ML-NW was less than 80 nm while the bit length was about 106 nm in SL-NW. Reduction in the satu- ration magnetization can surpre ss magnetostatic interaction between walls, and enables us to o btain smaller bit length. The domain wall width of 20 nm was obtained in the continuous layer of ML-NW while the wall width was 24 nm in the SL- NW with the same magnetic parameters. Figure 3shows minimum bit length and domain wall width as a function of intergrain exchange coupling A0.T h e minimum bit length decreases as the intergrain exchange cou-pling decreases. If the grain size in the granular layer is reduced or the perpendicular anisotropy in each grain becomes higher, of course, smaller bit can be easily achieved. Figure 4shows wall displacement of a single wall in SL- and ML- NWs as a function of time, and also shows the in- plane component of averaged magnetization which correspondsto the magnetization direction at the wall. The initial domain wall structure is Ne ´el wall in each case, which is determined by dimensions of nanowire. In thi s case, the spin drift velocity uis about 43 m/s, which corresponds to the current density of 6.0/C210 7A/cm2when the spin polarization is 0.5. When the non-adiabatic spin torque parameter is 0, it is well-known that the magnetization at the domain wall rotates counterclockwise a ss h o w ni nF i g . 4(a). The averaged wall velocity vis about 27 m/s in SL-NW. The magnetization direction at the wall in ML- NW also rotates counterclockwise with the time evolution as FIG. 2. Magnetization distributions of SL- and ML-NWs. The minimum bit is positioned in the center of each NW. FIG. 3. Minimum bit length and domain wall width as a function of intergrainexchange coupling A 0. FIG. 4. Wall displacement in SL- and ML- NWs and the in-plane component of averaged magnetization as a function of time. The spin drift velocity is about 43 m/s. FIG. 1. Multilayered nanowire consisting of continuous layer and granular layer.07D314-2 Komine, Ooba, and Sugita J. Appl. Phys. 111, 07D314 (2012) Downloaded 17 Jun 2013 to 18.7.29.240. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionss h o w ni nF i g . 4(b). This wall motion mode in ML-NW is simi- lar to that of SL-NW even if the granular layer exists. The aver- aged wall velocity vis about 35 m/s in ML-NW, which is larger than the SL-NW case. This is because the domain wallwidth in the continuous layer of ML-NW is narrower than the SL- NW case due to weak intergrain exchange coupling A 0in the granular layer. The narrowe r domain wall width changes the hard-axis anisotropy in ML -NW. As a result, the narrower domain wall width changes the critical spin drift velocity uc. However, the critic al current density does not necessarily decrease because the hard-axis a nisotropy is related to nanowire dimensions.11 Figure 5shows relation between spin drift velocity u and wall velocity vin SL- and ML-NWs. The intergrain exchange constant A0is 1.3/C210/C07erg/cm. The correspond- ing domain wall width is about 20 nm. The critical currentdensity is also dependent on the domain wall width. In order to investigate influence domain wall width in ML-NW, CIDWM was also demonstrated in the other SL-NW withthe same domain wall width as that of the continuous layer in ML-NW. The other SL-NW has a different exchange con- stant Aof 9.2/C210 /C07erg/cm. Although the relation between uandvin ML-NW is different from that of the SL-NW with A,theu-vrelation in ML-NW is similar to that of SL-NW with different A. The critical spin drift velocity ucin ML-NW is about 14 m/s, which corresponds to 2.0 /C2106A/cm2when the spin polarization is 0.5. Also, ucin SL-NW with different Ais about 14 m/s, which is almost the same as the ML- NW case. This means that the critical current density in ML-NW canbe estimated by the critical current density in SL-NW with the same wall width as that of ML-NW. IV. CONCLUSION In this paper, we propose a multilayered nanowire in order to achieve a high density bit or short bit length. The ML-NW consists of a continuous layer and a granular layerwhich are coupled by interlayer exchange interaction. We demonstrate CIDWM in ML-NW by micromagnetic simula- tion. The domain wall width in the ML-NW is narrower thanthat of SL-NW because of weak exchange coupling in the granular layer. As a result, the smaller bit length can be obtained in ML-NW. Moreover, the critical current densityin ML-NW is slightly smaller than that of SL-NW with the same exchange stiffness constant as that of the continuous layer. The critical current density in ML-NW is about2.0/C210 7A/cm2if the spin polarization is 0.5. This value is almost equivalent to that of SL-NW with the same domain wall width as that of ML-NW. This means that the reductionof critical current density of ML-NW can be achieved by the same way of reducing the critical current density of SL- NW, and the smaller bit length can be simultaneouslyobtained in ML-NW. ACKNOWLEDGMENTS This research was supported in part by Grants-in-Aid from the Hoso-Bunka Foundation and from the Nippon SheetGlass Foundation for Materials Science and Engineering, and by a Grant-in-Aid for Scientific Research C (2256023) from the Japan Society for the Promotion of Science. 1L. Berger, Phys. Rev. B 54, 9353, (1996); J. Slonczewski, J. Magn. Magn. Mater. 159, L1, (1996). 2S. S. P. Parkin, U.S. Patent 6834005 (2004). 3N. Sakimura, T. Sugibayashi, T. Honda, H. Honjo, S. Saito, T. Suzuki, N. Ishiwata, and S. Tahara, IEEE J. Solid-State Circuits 42, 830 (2007). 4T. Komine, K. Takahashi, A. Ooba, and R. Sugita, J. Appl. Phys. 109, 07D503 (2011). 5H. Murakami, K. Takahashi, T. Komine, and R. Sugita, J. Phys. Conf. Ser. 200, 042018 (2010). 6M. Kla ¨ui, H. Ehrke, U. Ru ¨diger, T. Kasama, R. E. Dunin-Borkowski, D. Backes, L. J. Heyderman, C. A. F. Vaz, J. A. C. Bland, G. Faini, E. Cambril, and W. Wernsdorfer, Appl. Phys. Lett. 87, 102509 (2005). 7Y. Sonobe, D. Weller, Y. Ikeda, M. Schabes, K. Takano, G. Zeltzer, B. K. Yen, and M. E. Best, J. Magn. Magn. Mater. 235, 424 (2001). 8R. H. Victora and X. Shen, IEEE Trans. Magn. 41, 2828 (2005). 9Y. Inaba, T. Shimatsu, H. Aoi, H. Muraoka, and Y. Nakamura, O. Kitakami, J. Appl. Phys. 99, 08G913 (2006). 10Z. Li, J. He, and S. Zhang, J. Appl. Phys. 99, 08Q702 (2006). 11S.-W. Jung, W. Kim, T.-D. Lee, K.-J. Lee, and H.-W. Lee, Appl. Phys. Lett. 92, 202508 (2008). FIG. 5. Relation between spin drift velocity uand wall velocity vin SL- and ML-NWs.07D314-3 Komine, Ooba, and Sugita J. Appl. Phys. 111, 07D314 (2012) Downloaded 17 Jun 2013 to 18.7.29.240. This article is copyrighted as indicated in the abstract. 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1.3203936.pdf

Propagation of nonlinear acoustic plane waves in an elastic gas-filled tube Michal Bednarik and Milan Cervenkaa/H20850 Faculty of Electrical Engineering, Czech Technical University in Prague, Technicka 2, 166 27 Prague, Czech Republic /H20849Received 26 January 2009; revised 9 June 2009; accepted 17 July 2009 /H20850 This paper deals with modeling of nonlinear plane acoustic waves propagating through an elastic tube filled with thermoviscous gas. A description of the interactions between gas and an elastic tubewall is carried out by the continuity equation of a wall velocity. Simplification on the basis of thelocal reaction assumption enables to model an acoustic treatment on the tube wall by using a wallimpedance. Because there are considerable losses due to wall friction, the influences of the acousticboundary layer were also considered. Using certain assumptions a special form of the Burgersequation was derived which enables to describe the propagation of nonlinear waves in the elastictube. This model equation takes into account nonlinear, dissipative, and dispersion effects whichcompete each other. Characteristic lengths of the supposed effects and numerical results with respectto the source frequency were used for a qualitative analysis of the model equation. Applicability ofthis model equation was demonstrated by series of measurements. By application of the long-waveapproximation the Korteweg–de Vries–Burgers and Kuramoto–Sivashinsky equations were derivedfrom the modified Burgers equation.©2009 Acoustical Society of America. /H20851DOI: 10.1121/1.3203936 /H20852 PACS number /H20849s/H20850: 43.25.Cb, 43.20.Mv /H20851OAS /H20852 Pages: 1681–1689 I. INTRODUCTION The problems concerning with the interactions between acoustic oscillations in a fluid-filled tube and the vibrationsof its wall have received the attention of many investigators.Since time of Young who first found the pulse wave speed inhuman arteries, a number of scientists dealt with propagationof acoustic waves through elastic tubes filled with fluids.Their works differ each other by various assumptions, inparticular, as far as media, wave modes, wave amplitudes,and tube walls are concerned. Fay et al. 1carried out an analytical and experimental investigation of a water-filled acoustic impedance tube. Theirwork was motivated by the fact that water-filled tube wallscannot be assumed to be rigid for acoustic waves. The paperby Jacobi 2was also focused on the problems regarding the sound transmission through tubes filled with ideal liquid;however, in addition he considered the higher wave modes.In contrast to the above mentioned authors who consideredonly inviscid fluids, Morgan and Kiely 3took into account also viscosity of the liquid and internal damping in the tubewall. Sondhi 4for investigation of wave propagation in a lossy vocal tract used an approach which was based on alocal wall admittance model. Guelke and Bunn 5presented work which deals with application of the transmission linetheory to linear acoustic wave propagation through tube withyielding walls. They limited themselves to vibrations only inthe radial direction and considered only the breathing cir-cumferential motion. In a similar spirit, Fredberg et al. 6mod- eled mechanic oscillations of the respiratory system at highfrequencies. Elvira-Segura7dealt with the study of speed and attenuation of an acoustic wave propagating inside a cylin-drical elastic tube filled with a viscous liquid. This authorextended influence of the liquid viscosity by the boundarylayer effects. Because the flaws in the roundness of a tubeinduce coupling between the structural and acoustic modeswhich do not exist in the case where the cross-section isperfectly circular, Pico and Gautier 8presented a model which allows us to take into account the small imperfections in the tube circularity. Gautier et al.9besides a well-arranged bibliographical review presented a study on cylindrical mem-branes submitted to a static tension. The authors of theabove-cited papers supposed only the thin-walled tubes.Grosso 10considered the exact longitudinal and shear wave equations for the multimode axial acoustic propagation intubes of arbitrary wall thickness filled with inviscid liquid.Nonlinear effects were taken into account, e.g., by Yomosa 11 who described the propagation of weakly nonlinear waves inan infinitely long distensible thin-walled elastic tube filledwith an ideal fluid. Erbay and Dost 12investigated the propa- gation of weakly nonlinear waves in an infinitely long non-linear viscoelastic thin tube filled with incompressible, invis-cid fluid. By means of the long-wave approximation theyderived a number of nonlinear evolution equations represent-ing various regimes. Kamakura and Kumamoto 13presented a work which concerned with investigation of nonlinear planeacoustic waves through an elastic tube. The authors assumedthat an elastic tube wall reacts locally to the inner pressure.On the basis of their assumptions they derived a model equa-tion. Validation of the model equation was justified experi-mentally. Unfortunately, the comparison of theoretical andexperimental results was not presented in a way enabling toevaluate measure of the validation of their model equation a/H20850Author to whom correspondence should be addressed. Electronic mail: bednarik@fel.cvut.cz J. Acoust. Soc. Am. 126 /H208494/H20850, October 2009 © 2009 Acoustical Society of America 1681 0001-4966/2009/126 /H208494/H20850/1681/9/$25.00 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.21.35.191 On: Mon, 22 Dec 2014 00:41:27rigorously. Though the problems concerning propagation of acoustic waves through tubes were widely studied by manyauthors, investigation seems to be missing which would si-multaneously include most of effects that influence acousticwaves. A model equation, which takes into account moreeffects at the same time, can offer qualitatively new results.This fact determines the main object of this work. Hence this work is focused on the description of nonlin- ear plane acoustic waves propagating through an elastic tubefilled with thermoviscous gas. For this purpose, a specialform of the Burgers equation was derived on the basis of thelocal wall reaction hypothesis. This model equation extendsthe standard Burgers equation by terms which represent dis-persion and dissipative effects caused by the wall elasticityand acoustic /H20849Stokes /H20850boundary layer which plays an impor- tant role in the course of wave forming, in particular, forlower frequencies. Theoretical results were verified experi-mentally. The Korteweg–de Vries–Burgers /H20849KdVB /H20850and Kuramoto–Sivashinsky equations are derived from the modi-fied Burgers equation by means of the long-wave approxima-tion. These equations are supplemented by the term whichtakes into account boundary layer effects. Section II is dedicated to derivation of the model equa- tions. Analysis of numerical solutions of the modified Bur-gers equation is presented in Sec. III. Then in Sec. IV wecompare theoretical and experimental data to justify applica-bility of the model equations. II. MODEL EQUATIONS AND DISPERSION RELATIONS A. Derivation of the modified Burgers equation for a gas-filled elastic tube If we take into account the acoustic boundary layer ef- fects we can write the following one-dimensional continuityequation /H20849see Ref. 14/H20850: /H11509/H9267 /H11509t+/H11509/H20849/H9267v/H20850 /H11509x=2 r0/H20881/H9263 /H9266/H208731+/H9253−1 /H20881Pr/H20874/H9267/H20885 0/H11009/H11509v/H20849x,t−/H9273/H20850 /H11509xd/H9273 /H20881/H9273. /H208491/H20850 Here xis space coordinate in the direction of the tube axis, t is time, /H9267=/H9267/H11032+/H92670is density of fluid, where /H9267/H11032is the acoustic density and /H92670is density corresponding to the equilibrium fluid state, vis the acoustic velocity, /H9263is the kinematic vis- cosity, Pr= /H92670/H9263cp//H9282is the Prandtl number, /H9253=cp/cVis the ratio of specific heats /H20849cpandcVare the specific heats under constant pressure and volume, respectively /H20850,/H9282is the coeffi- cient of heat conductivity, and r0is an equilibrium tube inner radius. Using the continuity equation /H208491/H20850is conditioned by the following relations:15,16 /H9254/lessmuch/H9261,/H9254/lessmuchr0, where /H9254is a boundary layer thickness and /H9261is a wavelength. Further, when it is satisfied2 r0/H20881/H9263 /H9266/H208731+/H9253−1 /H20881Pr/H20874/H11011/H9262, where /H9262/H112701 is a small dimensionless parameter /H20849the peak Mach number of the source /H20850, then within the scope of the second order nonlinear theory we can replace the density /H9267 =/H9267/H11032+/H92670by an ambient fluid density /H92670in Eq. /H208491/H20850. After the replacement and using the relation between the acoustic ve-locity and the velocity potential v=/H11509/H9272//H11509xEq. /H208491/H20850can be written in the form /H11509/H9267 /H11509t+/H11509/H20849/H9267v/H20850 /H11509x=2/H92670 r0/H20881/H9263 /H9266/H208731+/H9253−1 /H20881Pr/H20874/H20885 0/H11009/H115092/H9272/H20849x,t−/H9273/H20850 /H11509x2d/H9273 /H20881/H9273 =2/H92670 r0/H20881/H9263 /H9266/H208731+/H9253−1 /H20881Pr/H20874/H115092 /H11509x2/H20885 0/H11009 /H9272/H20849x,t−/H9273/H20850d/H9273 /H20881/H9273. /H208492/H20850 With help of the linear one-dimensional wave equation /H115092/H9272 /H11509x2=1 c02/H115092/H9272 /H11509t2, it is possible to rewrite Eq. /H208492/H20850 /H11509/H9267 /H11509t+/H11509/H20849/H9267v/H20850 /H11509x=2/H92670 c02r0/H20881/H9263 /H9266/H208731+/H9253−1 /H20881Pr/H20874/H115092 /H11509t2/H20885 0/H11009 /H9272/H20849x,t−/H9273/H20850d/H9273 /H20881/H9273 =2/H92670 c02r0/H20881/H9263 /H9266/H208731+/H9253−1 /H20881Pr/H20874/H20885 0/H11009/H115092/H9272/H20849x,t−/H9273/H20850 /H11509t2d/H9273 /H20881/H9273. /H208493/H20850 The equality /H115092/H9272//H11509t2=/H115092/H9272//H11509/H92732enables us to express Eq. /H208493/H20850 as /H11509/H9267 /H11509t+/H11509/H20849/H9267v/H20850 /H11509x=2/H92670 c02r0/H20881/H9263 /H9266/H208731+/H9253−1 /H20881Pr/H20874/H20885 0/H11009/H115092/H9272/H20849x,t−/H9273/H20850 /H11509/H92732d/H9273 /H20881/H9273. /H208494/H20850 The integral on the right hand side /H20849rhs/H20850of Eq. /H208494/H20850can be rewritten as 1 /H20881/H9266/H20885 0/H11009/H115092/H9272/H20849x,t−/H9273/H20850 /H11509/H92732d/H9273 /H20881/H9273=1 /H20881/H9266/H20885 −/H11009t/H115092/H9272/H20849x,/H9273/H20850 /H11509/H92732d/H9273 /H20881t−/H9273. /H208495/H20850 The rhs of Eq. /H208495/H20850represents the fractional derivative /H20849A3/H20850 /H20849see, e.g., Refs. 17,15, and 16/H20850. We can express Eq. /H208493/H20850as follows: /H11509/H9267 /H11509t+/H11509/H20849/H9267v/H20850 /H11509x=2/H92670B c02/H115093/2/H9272 /H11509t3/2, /H208496/H20850 where B=/H20881/H9263 r0/H208731+/H9253−1 /H20881Pr/H20874 /H208497/H20850 is the boundary layer parameter. If we suppose that acoustic waves propagate through a tube with an elastic wall /H20849a variable cross-section /H20850and we do not take into account the boundary layer effects then we canexpress the continuity equation as 1682 J. Acoust. Soc. Am., Vol. 126, No. 4, October 2009 M. Bednarik and M. Cervenka: Nonlinear acoustic waves in elastic tubes Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.21.35.191 On: Mon, 22 Dec 2014 00:41:27/H11509/H20849/H9267S/H20850 /H11509t+/H11509/H20849/H9267Sv/H20850 /H11509x=0 , /H208498/H20850 where S=S/H20849x,t/H20850is an inner tube cross-section. Further, Eq. /H208498/H20850can be modified into the form /H11509/H9267 /H11509t+/H11509/H20849/H9267v/H20850 /H11509x+/H9267 SdS dt=0 , /H208499/H20850 where the operator d/dtis given as d dt=/H11509 /H11509t+v/H11509 /H11509x. /H2084910/H20850 If the elastic tube wall is assumed to be locally reacting /H20849see, e.g., Refs. 18and5/H20850then we can consider the inner cross- section Sonly as a function of time, i.e., S=S/H20849t/H20850=/H9266r2/H20849t/H20850, where r/H20849t/H20850=r0+r/H11032/H20849t/H20850is a total inner tube radius and r/H11032rep- resents a change in r. If we use the fact that r/H11032/H11270r0, then it is possible to use the following simplification: 1 SdS dt/H112291 S0dS dt=1 /H9266r02d/H20849/H9266r2/H20850 dt=2 r0dr dt=2 r0vw, /H2084911/H20850 where vwis a radial wall velocity. Because within the scope of the second order nonlinear theory we neglect terms which are of the third order orhigher, it is possible to simplify Eq. /H208499/H20850by using relation /H2084911/H20850 and adopting the supposition vw/H11011/H92622, /H11509/H9267 /H11509t+/H11509/H20849/H9267v/H20850 /H11509x=−2/H92670 r0vw. /H2084912/H20850 Using the linear relation /H11509p /H11509t=−/H92670/H115092/H9272 /H11509t2, /H2084913/H20850 we can write the following convolution integral for the radial wall velocity vw: vw=/H20885 −/H11009t kw/H20849t−t/H11032/H20850/H11509p/H20849x,t/H11032/H20850 /H11509t/H11032dt/H11032 =−/H92670/H20885 −/H11009t kw/H20849t−t/H11032/H20850/H115092/H9272/H20849x,t/H11032/H20850 /H11509t/H110322dt/H11032, /H2084914/H20850 where kw/H20849t/H20850is a kernel function representing behavior of the considered tube wall. If we take into account both the acoustic boundary layer /H20851Eq. /H208496/H20850/H20852and the tube elasticity /H20851Eq. /H2084912/H20850/H20852together with re- lation /H2084914/H20850we obtain the resulting continuity equation /H11509/H9267 /H11509t+/H11509/H20849/H9267v/H20850 /H11509x=2/H926702 r0/H20885 −/H11009t kw/H20849t−t/H11032/H20850/H115092/H9272 /H11509t/H110322dt/H11032+2/H92670B c02/H115093/2/H9272 /H11509t3/2. /H2084915/H20850 Using the basic equations of hydromechanics, namely, the Navier–Stokes equation of motion, the heat transfer equa-tion, the state equation /H20849see, e.g., Refs. 19–21/H20850, and the modi- fied continuity equation /H2084915/H20850we can derive in the second approximation the following one-dimensional modified Kuz-netsov’s equation 22for the velocity potential /H9272:/H115092/H9272 /H11509t2−c02/H115092/H9272 /H11509x2=b /H92670c02/H115093/H9272 /H11509t3−/H11509 /H11509t/H20875/H9252−1 c02/H20873/H11509/H9272 /H11509t/H208742 +/H20873/H11509/H9272 /H11509x/H208742/H20876 −2/H92670c02 r0/H20885 −/H11009t kw/H20849t−t/H11032/H20850/H115092/H9272 /H11509t/H110322dt/H11032−2B/H115093/2/H9272 /H11509t3/2, /H2084916/H20850 where c0is the small-signal sound speed, b=/H9256+4/H9257/3 +/H9282/H208491/cV−1 /cp/H20850is the coefficient of sound diffusivity, /H9256and /H9257are the coefficients of bulk and shear viscosity, and /H9252is the coefficient of nonlinearity. As the profile of simple waves varies slowly in the space we can search for the solution of Eq. /H2084916/H20850in the moving reference frame in the form /H9272=/H9272/H20873x1=/H9262x,/H9270=t−x c0/H20874. /H2084917/H20850 Then taking into consideration that b/H11011B/H11011/H9262we can derive the following modified Burgers equation from Eq. /H2084916/H20850 within frame of the second order nonlinear theory: /H11509v /H11509x−/H9252 c02v/H11509v /H11509/H9270+/H92670c0 r0/H20885 −/H11009/H9270 kw/H20849/H9270−/H9270/H11032/H20850/H11509v /H11509/H9270/H11032d/H9270/H11032+B c0/H115091/2v /H11509/H92701/2 −b 2/H92670c03/H115092v /H11509/H92702=0 . /H2084918/H20850 It is obvious that it is possible to use the model /H20851Eq. /H2084918/H20850/H20852 also for the nonlinear plane waves which propagate througha hard-walled tube. In this case we can consider the termrepresenting the elastic properties of the tube wall as equal tozero. 15 B. Derivation of dispersion relations Under the above mentioned restrictions, the tube can be regarded as consisting simply of a series of ring-shaped ele-ments whose radial motion is caused only by elastic circum-ferential stresses and radial inertia. As the tube wall yieldslocally to the inner pressure we can write the following ex-pression for the force acting on the ring: F/H20849t/H20850=p/H20849t/H20850A/H20849t/H20850=p/H20849t/H20850/H20851A 0+A/H11032/H20849t/H20850/H20852=p/H20849t/H208502/H9266r/H20849t/H20850/H9004l =p/H20849t/H20850/H208512/H9266r0/H9004l+2/H9266r/H11032/H20849t/H20850/H9004l/H20852, /H2084919/H20850 where pis an inner pressure and /H9004lis the width of the ring element. If we suppose that r/H11032is sufficiently small compared tor0then we can simplify relation /H2084919/H20850, F/H20849t/H20850/H11229p/H20849t/H20850A0. /H2084920/H20850 With using the complex representation we can rewrite Eq. /H2084920/H20850as Fˆ/H11229pˆA0. /H2084921/H20850 For the ring element we can use the equivalent electrome- chanic circuit which is sketched in Fig. 1. We suppose that air surrounds the ring and consequently the radiation load onthe outside of the ring can be neglected. 23 On the basis of the equivalent circuit we obtain J. Acoust. Soc. Am., Vol. 126, No. 4, October 2009 M. Bednarik and M. Cervenka: Nonlinear acoustic waves in elastic tubes 1683 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.21.35.191 On: Mon, 22 Dec 2014 00:41:27Fˆ=/H20873R+j/H9275M+1 j/H9275C/H20874vˆw=Zˆwvˆw, /H2084922/H20850 where Mis the inertance, Cis the compliance, Ris mechani- cal resistance, and Zˆwis the resulting wall mechanical im- pedance. Using relations /H2084921/H20850and /H2084922/H20850we can write vˆw/H11229A0 Zˆwpˆ=2/H9266r0/H9004l Zˆwpˆ. /H2084923/H20850 With the help of the convolution integral /H2084914/H20850we can also write that vˆw=j/H9275kˆwpˆ. /H2084924/H20850 By comparing Eqs. /H2084923/H20850and /H2084924/H20850we obtain kˆw=2/H9266r0 j/H9275Zˆm, /H2084925/H20850 where Zˆm=Zˆw /H9004l=Rm+j/H9275Mm+1 j/H9275Cm, /H2084926/H20850 where Rm=R//H9004lis the specific mechanical resistance /H20849kg m−1s−1/H20850,Mm=M//H9004lis the specific inertance /H20849kg m−1/H20850, andCm=C/H9004l/H20849kg−1ms2/H20850is the specific compliance. With help of expressions /H2084925/H20850and /H2084926/H20850it is possible to write /H20849see Ref. 13/H20850 2kˆw r0=4/H9266 j/H9275Zˆm=4/H9266Cm 1−/H20873/H9275 /H9275m/H208742 +j/H9275RmCm, /H2084927/H20850 where /H9275mrepresents the mechanical resonance angular fre- quency which is given by the following relation: /H9275m=1 /H20881CmMm. /H2084928/H20850 After linearizing modified Kuznetsov’s equation/H115092/H9272 /H11509t2−c02/H115092/H9272 /H11509x2−b /H92670c02/H115093/H9272 /H11509t3+2/H92670c02 r0/H20885 −/H11009t kw/H20849t−t/H11032/H20850/H115092/H9272 /H11509t/H110322dt/H11032 +2B/H115093/2/H9272 /H11509t3/2=0 , /H2084929/H20850 we can find the following dispersion relation: k=/H9275 c0/H208811−j/H9275b /H92670c02+2/H92670c02kˆw r0−2B/H20849j/H9275/H208503/2 /H92752, /H2084930/H20850 where kis the complex wave number. Expression /H2084930/H20850can be simplified on the basis of the binomial series k/H11229/H9275 c0/H208731−j/H9275b 2/H92670c02+/H92670c02kˆw r0+B /H20881j/H9275/H20874. /H2084931/H20850 Since kˆw/r0is complex, see relation /H2084927/H20850, we can write kˆw r0=R/H20873kˆw r0/H20874+jI/H20873kˆw r0/H20874=2/H9266Cm/H208751−/H20873/H9275 /H9275m/H208742/H20876 /H208751−/H20873/H9275 /H9275m/H208742/H208762 +Rm2Cm2/H92752 −j2/H9266RmCm2/H9275 /H208751−/H20873/H9275 /H9275m/H208742/H208762 +Rm2Cm2/H92752. /H2084932/H20850 It is possible to simplify formula /H2084932/H20850when the following relations are satisfied:13 /H9275/lessmuch/H9275m,/H9275RmCm/lessmuch1. /H2084933/H20850 We can write that kˆw r0/H112292/H9266Cm/H208751+/H20873/H9275 /H9275m/H208742/H20876−j2/H9266RmCm2/H9275/H208751+2/H20873/H9275 /H9275m/H208742/H20876. /H2084934/H20850 Expression /H2084934/H20850is valid only for lower frequencies; it means that the dispersion and dissipation effects dominate above thenonlinear ones. In the opposite case the higher harmoniccomponents, which arise in the course of the wave propaga-tion, do not satisfy conditions /H2084933/H20850. When we can take into account only small number of harmonics then it is furtherpossible to simplify relation /H2084934/H20850as follows: kˆ w r0/H112292/H9266Cm/H208751+/H20873/H9275 /H9275m/H208742/H20876−j2/H9266RmCm2/H9275. /H2084935/H20850 C. The KdVB and Kuramoto–Sivashinsky model equation for an elastic tube We can rewrite Eq. /H2084918/H20850into the form24 /H11509v /H11509x−/H9252 c02v/H11509v /H11509/H9270+/H20885 −/H11009/H11009 K/H20849/H9270−/H9270/H11032/H20850/H11509v/H20849x,/H9270/H11032/H20850 /H11509/H9270/H11032d/H9270/H11032=0 , /H2084936/H20850 where K/H20849/H9270/H20850represents a kernel function which describes as- sumed dispersion and dissipation properties. Suppose we know the dispersion relationRM C /hatwideF /hatwidevw FIG. 1. The equivalent electromechanic circuit, Mis the inertance, Cis the compliance, Ris mechanical resistance, and vˆwis radial wall velocity. 1684 J. Acoust. Soc. Am., Vol. 126, No. 4, October 2009 M. Bednarik and M. Cervenka: Nonlinear acoustic waves in elastic tubes Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.21.35.191 On: Mon, 22 Dec 2014 00:41:27D/H20849/H9275*,k/H11032/H20850=0 , /H2084937/H20850 which corresponds to the kernel function K/H20849/H9270/H20850,/H9275*=/H9275and k/H11032=k−/H9275/c0=/H9275 c0/H20873/H92670c02kˆw r0+B /H20881j/H9275−j/H9275b 2/H92670c02/H20874 /H2084938/H20850 is a complex wave number that is related to the independent variables /H9270,xin contrast to the complex wave number /H2084931/H20850 that is connected with the independent variables tand x. Provided that certain conditions, which are discussed later,are fulfilled then we can simplify the dispersion relation /H2084937/H20850 into the asymptotic form D a/H20849/H9275,k/H11032/H20850=0 . /H2084939/H20850 After linearizing Eq. /H2084936/H20850we can write25 /H11509v /H11509x+/H20885 −/H11009/H11009 Ka/H20849/H9270−/H9270/H11032/H20850/H11509v/H20849x,/H9270/H11032/H20850 /H11509/H9270/H11032d/H9270/H11032=0 , /H2084940/H20850 where the kernel function Ka/H20849/H9270/H20850corresponds to asymptotic dispersion relation /H2084939/H20850. We assume the solution of Eq. /H2084940/H20850 in the form v=vmexp /H20851j/H20849/H9275/H9270−k/H11032x/H20850/H20852 /H20849vmis the acoustic veloc- ity amplitude /H20850. We can substitute this solution into Eq. /H2084940/H20850 and we obtain −jk/H11032+/H20885 −/H11009/H11009 Ka/H20849/H9270−/H9270/H11032/H20850j/H9275exp /H20851−j/H9275/H20849/H9270−/H9270/H11032/H20850/H20852d/H9270/H11032=0 . /H2084941/H20850 Equation /H2084941/H20850can be rewritten into the form k/H11032 /H9275=F/H20851Ka/H20849/H9270/H20850/H20852=/H20885 −/H11009/H11009 Ka/H20849/H9270/H20850exp /H20849−j/H9275/H9270/H20850d/H9270, /H2084942/H20850 where F/H20851·/H20852represents the Fourier transform. The rhs of Eq. /H2084942/H20850is the Fourier transform of the given kernel Ka/H20849/H9270/H20850.O n the basis of the inverse Fourier transform we have Ka/H20849/H9270/H20850=F−1/H20875k/H11032 /H9275/H20876=1 2/H9266/H20885 −/H11009/H11009k/H11032 /H9275exp /H20849j/H9275/H9270/H20850d/H9275. /H2084943/H20850 Substituting expression /H2084934/H20850into relation /H2084938/H20850we obtain k/H11032/H11229/H9275/H208772/H9266/H92670c0Cm/H208751+/H20873/H9275 /H9275m/H208742/H20876 −j2/H9266/H92670c0RmCm2/H9275/H208751+2/H20873/H9275 /H9275m/H208742/H20876 +B c0/H20881j/H9275−j/H9275b 2/H92670c03/H20878. /H2084944/H20850 On the basis of relation /H2084943/H20850we can find the asymptotic kernel function Ka/H20849/H9270/H20850by using formula /H2084944/H20850. After substitu- tion of this asymptotic kernel function into Eq. /H2084936/H20850we get/H11509v /H11509x−/H9252 c02/H20873v−2/H9266/H92670c03Cm /H9252/H20874/H11509v /H11509/H9270+B c0/H115091/2v /H11509/H92701/2 −b+4/H9266/H926702Cm2Rmc04 2/H92670c03/H115092v /H11509/H92702−2/H9266/H92670c0RmCm /H9275m2/H115093v /H11509/H92703 +4/H9266/H92670c0RmCm2 /H9275m2/H115094v /H11509/H92704=0 . /H2084945/H20850 Equation /H2084945/H20850represents the Kuramoto–Sivashinsky /H20849Ben- ney /H20850equation26,27which is modified by the term representing the boundary layer effects. This equation can be simplifiedinto the KdVB equation /H20849see, e.g., Ref. 28/H20850by setting the last term equal to zero when propagation wave distances are rela-tively short or the source frequency is sufficiently low andthe dispersive and dissipative effects are insofar importantthat it is possible to take into account a reasonably smallnumber of harmonics. III. ANALYSIS OF NUMERICAL SOLUTIONS This section is devoted to qualitative analysis of numeri- cal results of the model equation /H2084918/H20850for a silicon rubber tube /H20849hose /H20850. This analysis enables to show different scenarios of possible wave evolution for various source frequenciesand hence demonstrates how the individual terms in themodel equation depend on the source frequency. The realphysical parameters are presented in Table Iand for these parameters the analysis of numerical solutions is carried out.In order to show frequency limits of the KdVB andKuramoto–Sivashinsky equation we can depict the courses of function kˆ w/r0in dependence on frequency according to relations /H2084932/H20850,/H2084934/H20850, and /H2084935/H20850, see Figs. 2and3. It is evident from Figs. 2and 3that in this case the Kuramoto–TABLE I. Real parameters for air and the used elastic tube /H2084923 °C /H20850. c0 /H20849ms−1/H20850/H92670 /H20849kg m−3/H20850b /H20849kg m−1s−1/H20850B /H20849s−1 /2/H20850Cm /H20849ms2kg−1/H20850Rm /H20849kg m−1s−1/H20850Mm /H20849kg m−1/H20850 345.22 1.193 4.578 /H1100310−50.7227 4.45 /H1100310−8154.0 0.118 /Ifractur/parenleftBig /hatwidekw/r0/parenrightBig (µms2kg−1)f(Hz) FIG. 2. Comparison of imaginary parts of relations /H2084932/H20850/H20849solid line /H20850,/H2084934/H20850 /H20849dashed line /H20850, and /H2084935/H20850/H20849dotted line /H20850. J. Acoust. Soc. Am., Vol. 126, No. 4, October 2009 M. Bednarik and M. Cervenka: Nonlinear acoustic waves in elastic tubes 1685 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.21.35.191 On: Mon, 22 Dec 2014 00:41:27Sivashinsky equation can be used in the frequency range approximately up to 1000 Hz, whereas the KdVB equationonly up to about 800 Hz. The model equation /H2084918/H20850takes into account three effects, i.e., the nonlinear, dissipative, and dispersion ones. There is acompetition between the nonlinear effect, on the one hand,and the dispersion and dissipative effect on the other hand. Inthis case we can estimate a wave behavior on the basis of thecharacteristic lengths /H20849see, e.g., Ref. 19/H20850. The nonlinear length is given as L N=c02 /H9252vm/H9275, /H2084946/H20850 the dissipative length can be expressed as LD=1 /H9251, /H2084947/H20850 and the dispersion or coherent length for the second har- monic component as LC=/H9266/H208792/H9275/H208751 cph/H208492/H9275/H20850−1 cph/H20849/H9275/H20850/H20876/H20879. /H2084948/H20850 It holds that the shorter characteristic length means the greater influence of the considered effect /H20849in our case the diffraction length tends to infinity because we suppose thatplane waves propagate in the tube /H20850. On the basis of the char- acteristic lengths we can outline several scenarios of possiblewave evolutions with respect to a chosen source frequency/H20849amplitude of the source is supposed to be constant, 4000 Pa /H20850. First, we suppose that the source frequency is chosen so thatL N/LC/H110221 and LN/LD/H110211. It means that the dispersion effects dominate the nonlinear ones and at the same time thenonlinear effects dominate the dissipative ones. For instance,the source frequency of 1000 Hz satisfies /H20849if we suppose thephysical parameters in Table I/H20850the above mentioned rela- tions. For this case the wave distortion is negligible /H20849the first harmonic component dominates /H20850which affirms that disper- sion effects dominate nonlinear ones and thus nonlinearacoustic interactions are ineffective. Thus, we could ignorethe nonlinear effects and use only the linearized form of Eq./H2084918/H20850. When the supposed relations are satisfied also for lower source frequencies, in our case for a smaller source ampli-tude, it is possible to use the linearized Kuramoto–Sivashinsky or KdVB equation. Further, we assume that the relations L N/LC/H110211 and LN/LD/H110211 are satisfied for at least the first ten harmonics. For this purpose we can choose the source frequency, e.g.,80 Hz. The satisfaction of these relations means that a saw-tooth wave forms during propagation. The boundary layerdispersion causes the waveform asymmetry which occurs be-hind the shock formation. This result is demonstrated in Fig.4. Furthermore, when the relations are satisfied then it is possible to replace Eq. /H2084918/H20850by the standard Burgers equation which is supplemented by the term representing the bound-ary layer effects. 15,16 As the next case, we assume that the relations LN/LC /H110211 and LN/LD/H110211 are approximately satisfied only for the first two or three harmonics; hence, the generation of higherharmonics is suppressed. Figure 5illustrates this situation. When the boundary layer effects are small and again the relations L N/LC/H110211 and LN/LD/H110211 are satisfied, we can ob- serve a gradual degeneration of the original waveform intoindividual solitons. 24,28,29The solutions for this case are de- picted in Figs. 6and7. In the figures we can see a train of the/Rfractur/parenleftBig /hatwidekw/r0/parenrightBig (µms2kg−1) f(Hz) FIG. 3. Comparison of real parts of relations /H2084932/H20850/H20849solid line /H20850and /H2084934/H20850 /H20849dashed line /H20850.0 T 2T−4000−3000−2000−100001000200030004000 x=0 m x=20 m x=40 mp/prime(Pa) t FIG. 4. Two periods of wave forms at different distances from the source; the source frequency is 80 Hz. 0 T 2T−4000−3000−2000−100001000200030004000 x=0 m x=5 m x=8 mp/prime(Pa) t FIG. 5. Two periods of wave forms at different distances from the source;the source frequency is 400 Hz. 1686 J. Acoust. Soc. Am., Vol. 126, No. 4, October 2009 M. Bednarik and M. Cervenka: Nonlinear acoustic waves in elastic tubes Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.21.35.191 On: Mon, 22 Dec 2014 00:41:27solitons which are gradually attenuated as the wave propa- gates through the considered tube. First, we can observeslight oscillations, see Fig. 6. This fact is related to the har- monics generation because dispersion effect starts to play amore important role for higher harmonics. Gradually theslight oscillations develop into a series of solitons whichbreak up the previous waveform, see Fig. 7. As the last case, we can consider the relations L N/LC /H112701 and LN/LD/H110211. It is obvious that kˆw/r0, see formula /H2084927/H20850, tends to zero for high source frequencies. It means that wecan ignore the radial wall vibrations for high source frequen-cies. The considered relations for the characteristic lengthsare satisfied for higher source frequencies. IV. EXPERIMENT To validate applicability of the model equations, in par- ticular, the local reaction hypothesis, we compared the theo-retical and experimental data. A. Experimental setup Traveling sound waves were driven in silicone-rubber hose /H20849with inner diameter of 16 mm, wall thickness of 2m m /H20850, length of the hose was 20 m, it was lengthened by 20 m of polyvinyl chloride hose of the same inner diameterin order to suppress standing waves due to reflections. Allmeasurements were made at room temperature of 23 °C. Harmonic driving signal was generated with direct- digital-synthesis function generator Motech FG 503 whichwas amplified by power amplifier Mackie M1400. High am-plitude acoustic waves were generated by two speakers B&C 10MD26 /H20849each of 350 W input /H20850that were screwed one against another with a plastic ring for fastening of the hose.At the place where the hose was fastened to the plastic ring,reference 1 /8 in. GRAS Type 40DP microphone was at- tached to measure the input acoustic pressure. In the hose,there were 13 small cuts distributed with 1 m distances formeasurement of acoustic pressure with probe-microphoneGRAS Type 40SA with stainless-steel 20 mm probe tube of1.25 mm outer diameter /H20849inner diameter of 1 mm /H20850. Measured signals were sampled by 16 bit National In- struments PCI-6251 plug-in computer data acquisition cardat the rate of 100 kS /s per channel. Software for data acqui- sition and its processing was written in LABVIEW system. On the basis of measurement of the silicone-rubber hose we found its material parameters which are shown in Table I. In the course of measuring it was necessary to take into account the fact that the driving speakers heated up exces-sively. For this reason it was not possible to realize measur-ing with the same driving pressure at all points; hence thedriving pressures were registered for individual measure-ments apart. The value of the driving pressure was around3600 Pa /H20849 vm=8.74 m s−1/H20850. We confined ourselves to the rela- tively low source frequency in order that the dispersion and nonlinear effects could dominate above the dissipative ones. B. Comparison of theoretical and experimental results The measured and theoretical /H20849computed numerically /H20850 data are depicted in Figs. 8–10. By comparison of measured and theoretical waveforms at different distances from theharmonic source it is clear that the model equation /H2084918/H20850en- ables to describe behavior of nonlinear traveling waves in theelastic tube and that the local reaction hypothesis is sufficientwhen the conditions mentioned in Sec. II are fulfilled. Fromthese figures it is obvious that the shock waveform did notarise. This result is given by the fact that the relationsL N/LC/H110211 and LN/LD/H110211 are fulfilled approximately only0 T 2T−5000−4000−3000−2000−100001000200030004000 x=0 m x=5 m x=8 mp/prime(Pa) t FIG. 6. Two periods of wave forms at different distances from the source; the source frequency is 200 Hz. 0 T 2T−6000−4000−2000020004000 x=20 m x=30 mp/prime(Pa) t FIG. 7. Two periods of two wave forms at different distances from thesource; the source frequency is 200 Hz.0 0.5 1 1.5 2 2.5 3−3000−2000−10000100020003000p/prime(Pa) ωt/2π FIG. 8. Comparison of theoretical /H20849solid line /H20850and measured /H20849dotted line /H20850 wave forms at the distance x=3 m, pm=3602 Pa; the driving frequency is 400 Hz. J. Acoust. Soc. Am., Vol. 126, No. 4, October 2009 M. Bednarik and M. Cervenka: Nonlinear acoustic waves in elastic tubes 1687 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.21.35.191 On: Mon, 22 Dec 2014 00:41:27for the first three harmonics; thus the generation of higher harmonics is suppressed. It means that the relation LN/LC /H110221 holds for higher harmonics. Figure 11shows development of the first four harmonics of acoustic pressure at the distance of 13 m when inputacoustic pressure /H20849f=333 Hz /H20850increases. It is apparent from the figure that even if dispersion is present, several higher harmonics appear and the amplitude of the first and secondharmonics is not proportional to the pressure amplitude at theinput. This effect is called acoustic saturation and it has connection with the nonlinear attenuation. It is obvious fromFig.11that the fourth harmonics and the third one depend on the pressure amplitude at the input almost linearly. It meansthat the cascade process of harmonics generation is inter-rupted or in other words, the characteristic coherent length isshorter than the nonlinear one /H20849L N/H11022LC/H20850for higher harmon- ics in contrast to the lower ones.V. CONCLUSION We derived the modified Burgers equation /H2084918/H20850which enables to describe nonlinear acoustic plane waves propagat-ing through the elastic tube filled with a thermoviscous gas.The model equation takes into account only the breathingcircumferential damped vibrations of the considered tubewall. Since the viscosity of gas is considered, it is necessaryto suppose that the thin acoustic boundary layer appears nearthe tube wall. The acoustic boundary layer affects the acous-tic field outside of this layer /H20849the mainstream /H20850in a consider- able way. For this reason the derived model equation alsocontains the term which represents the boundary layer ef-fects. The applicability of the modified Burgers equation wasverified experimentally. The realized measurements demon-strate the fact that the dispersion induced by the elastic tubewall causes nonlinear acoustic interactions which are almostineffective and further that it is not possible to ignore theinfluences of the acoustic boundary layer. It means that theshock formation does not arise and so we can observe only asmall number of harmonics. Because we consider nonlinear,dispersion, and dissipative wave effects, that compete eachother, it is possible to distinguish different regimes. Theseregimes are classified by means of the relations betweencharacteristic lengths of the supposed wave effects. The sup-posed regimes lead to different wave evolutions. In the casewhen the nonlinear effects dominate the dispersion and dis-sipative one then the original waveform degenerates gradu-ally into individual solitons. However, we can observe thesolitons only when the acoustic boundary layer effects areweak. Contrary if it is not possible to suppose that theseeffects are weak /H20849e.g., in the case of a small tube radius /H20850then the generation of solitons is suppressed. This fact shows howthe boundary layer plays important role. Using the long-wave approximation, the modified Bur- gers equation was reduced to the KdVB and Kuramoto–Sivashinsky equations, which were investigated by many au-thors; however, they have not studied influence of theboundary layer effects. Considering that acoustic boundary0 0.5 1 1.5 2 2.5 3−2000−1500−1000−500050010001500p/prime(Pa) ωt/2π FIG. 9. Comparison of theoretical /H20849solid line /H20850and measured /H20849dotted line /H20850 wave forms at the distance x=7 m, pm=3604 Pa; the driving frequency is 400 Hz. 0 0.5 1 1.5 2 2.5 3−1200−1000−800−600−400−2000200400600800 p/prime(Pa) ωt/ 2π FIG. 10. Comparison of theoretical /H20849solid line /H20850and measured /H20849dotted line /H20850 wave forms at the distance x=10 m, pm=3597 Pa; the driving frequency is 400 Hz.1000 2000 3000 4000 5000 60000100200300400500600700 PI(Pa)PO(Pa)1st harm. 2nd harm. 3rd harm. 4th harm. FIG. 11. Dependence of amplitude of the first four harmonics at distance of 13 m on input pressure amplitude. 1688 J. Acoust. Soc. Am., Vol. 126, No. 4, October 2009 M. Bednarik and M. Cervenka: Nonlinear acoustic waves in elastic tubes Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.21.35.191 On: Mon, 22 Dec 2014 00:41:27layer plays more important role for lower frequencies, we cannot neglect its influence in the case of long-wave approxi-mation. Considering that the KdVB equation can be used only for a relatively narrow frequency range we derived theKuramoto–Sivashinsky equation which can be used for awider frequency range and thus it can take into account morehigher harmonics correctly. We intend to extend further the presented results and methods for the case of nonlinear standing waves in elasticresonators. ACKNOWLEDGMENT This work was supported by GACR Grant No. 202/09/ 1509. APPENDIX: FRACTIONAL DERIVATIVE OPERATOR Let us introduce the following fractional integration op- erator of the order /H9251: I/H9251/H20851f/H20849t/H20850/H20852=/H20885 −/H11009t/H20849t−/H9273/H20850/H9251−1 /H9003/H20849/H9251/H20850f/H20849/H9273/H20850d/H9273, /H20849A1/H20850 where /H9003is the gamma function. The /H9251th fractional derivative is given as d/H9251f/H20849t/H20850 dt/H9251=In−/H9251/H20875dnf/H20849t/H20850 dtn/H20876, /H20849A2/H20850 where n=1+ /H20851/H9251/H20852, where /H20851/H9251/H20852represents the whole part of /H9251. On the basis of the mentioned definition of the fractional derivative we can write for the fractional derivative of theorder /H9251=3 /2 that d3/2f/H20849t/H20850 dt3/2=I2−3 /2/H20875d2f/H20849t/H20850 dt2/H20876=I1/2/H20875d2f/H20849t/H20850 dt2/H20876 =1 /H20881/H9266/H20885 −/H11009td2f/H20849/H9273/H20850 d/H92732d/H9273 /H20881t−/H9273, /H20849A3/H20850 where we used that /H9003/H208491/2/H20850=/H20881/H9266. 1R. D. Fay, R. L. Brown, and O. V. Fortier, “Measurement of acoustic impedances of surfaces in water,” J. Acoust. Soc. Am. 19, 850–856 /H208491947 /H20850. 2W. J. Jacobi, “Propagation of sound waves along liquid cylinders,” J. Acoust. Soc. Am. 21, 120–127 /H208491949 /H20850. 3G. W. Morgan and J. P. 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Vila, “Wave propagation in a fluid filled rubber tube: Theoretical and experimental results for Ko-rteweg’s wave,” Acta. Acust. Acust. 93, 333–344 /H208492007 /H20850. 10V. A. D. Grosso, “Analysis of multimode acoustic propagation in liquid cylinders with realistic boundary conditions—Application to sound speedand absorption measurements,” Acustica 24, 299–311 /H208491971 /H20850. 11S. Yomosa, “Solitary waves in large blood vessels,” J. Phys. Soc. Jpn. 56, 506–520 /H208491987 /H20850. 12S. E. H. A. Erbay and S. Dost, “Wave propagation in fluid filled nonlinear viscoelastic tubes,” Acta Mech. 95, 87–102 /H208491992 /H20850. 13T. Kamakura and Y. Kumamoto, “Waveform distortions of finite amplitude acoustic wave in an elastic tube,” in Frontiers of Nonlinear Acoustics:Proceedings of the 12th ISNA, edited by M. F. Hamilton and D. T. Black-stock /H208491991 /H20850, pp. 333–338. 14W. Chester, “Resonant oscillations in closed tubes,” J. Fluid Mech. 18, 44–64 /H208491964 /H20850. 15S. N. Makarov and E. V. Vatrushina, “Effect of the acoustic boundary layer on a nonlinear quasiplane wave in a rigid-walled tube,” J. Acoust.Soc. Am. 94, 1076–1083 /H208491993 /H20850. 16M. Bednarik and P. Konicek, “Propagation of quasiplane nonlinear waves in tubes and the approximate solutions of the generalized Burgers equa-tion,” J. Acoust. Soc. Am. 112, 91–98 /H208492002 /H20850. 17I. M. Sokolov, J. Klafter, and A. Blumen, “Fractional kinetics,” Phys. Today 55, 48–54 /H208492002 /H20850. 18M. C. Junger and D. Feit, Sound, Structures, and Their Interaction /H20849MIT Press, Cambridge, 1986 /H20850. 19O. V. Rudenko and S. I. Soluyan, Theoretical Foundations of Nonlinear Acoustics /H20849Consultants Bureau, New York, 1977 /H20850. 20K. Naugolnykh and L. Ostrovsky, Nonlinear Wave Processes in Acoustics /H20849Cambridge University Press, Cambridge, 1998 /H20850. 21M. F. Hamilton and D. T. Blackstock, Nonlinear Acoustics /H20849Academic, New York, 1997 /H20850. 22S. Makarov and M. 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Kruskal, “Interactions of solitons in a collision- less plasma and the recurrence of initial states,” Phys. Rev. Lett. 15, 240– 243 /H208491965 /H20850. J. Acoust. Soc. Am., Vol. 126, No. 4, October 2009 M. Bednarik and M. Cervenka: Nonlinear acoustic waves in elastic tubes 1689 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.21.35.191 On: Mon, 22 Dec 2014 00:41:27

1.4976581.pdf

Energy consumption analysis of graphene based all spin logic device with voltage controlled magnetic anisotropy Zhizhong Zhang , Yue Zhang , Zhenyi Zheng , Guanda Wang , Li Su , Youguang Zhang , and Weisheng Zhao Citation: AIP Advances 7, 055925 (2017); doi: 10.1063/1.4976581 View online: http://dx.doi.org/10.1063/1.4976581 View Table of Contents: http://aip.scitation.org/toc/adv/7/5 Published by the American Institute of PhysicsAIP ADV ANCES 7, 055925 (2017) Energy consumption analysis of graphene based all spin logic device with voltage controlled magnetic anisotropy Zhizhong Zhang,1,2,aYue Zhang,1,2,3,a,bZhenyi Zheng,1,2Guanda Wang,1,2 Li Su,1,2Youguang Zhang,1,2and Weisheng Zhao1,2,3,b 1Fert Beijing Research Institute, Univ. Beihang, Beijing 100191, China 2School of Electrical and Information Engineering, Univ. Beihang, Beijing 100191, China 3Beijing Advanced Innovation Center for Big Data and Brain Computing (BDBC), Univ. Beihang, Beijing 100191, China (Presented 3 November 2016; received 23 September 2016; accepted 13 November 2016; published online 14 February 2017) All spin logic device (ASLD) is a promising option to realize the ultra-low power computing systems. However, the low spin transport efficiency and the non-local switching of the detector have become two key challenges of the ASLD. In this paper, we analyze the energy consumption of a graphene based ASLD with the ferromagnetic layer switching assistance by voltage control magnetic anisotropy (VCMA) effect. This structure has significant potential towards ultra-low power consumption: the applied voltage can not only shorten switching time of the ferromagnetic layer, but also decreases the critical injection current; the graphene channel enhances greatly the spin transport efficiency. By applying the approximate circuit model, the impact of mate- rial configurations, interfaces and geometry can be synthetically studied. An accurate physic model was also developed, based on which, we carry out the micro-magnetic simulations to analyze the magnetization dynamics. Combining these electrical and magnetic investigations, the energy consumption of the proposed ASLD can be esti- mated. With the optimizing parameters, the energy consumption can be reduced to 2.5 pJ for a logic operation. © 2017 Author(s). All article content, except where oth- erwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). [http://dx.doi.org/10.1063/1.4976581] I. INTRODUCTION As the scale of electronic device is more and more miniature, the energy consumption has become a bottleneck for improving the devices’ performance. A lot of new technologies are proposed in order to retain the Moore’s law or transcend it.1,2Among them, spintronics is considered a promising solution to realize the ultra-lower power memory and computing systems. Magnetic Random Access Memory (MRAM) shows a successful memory application of spintronics.3,4There have also been various concepts and structures proposed in aspect of spin logic. Nevertheless, most of them need the frequent spin-charge conversions.5–7In 2010, B. Behin-Aein et al. proposed the concept of all spin logic device (ASLD).8In ASLD, all the operations are well done only with the spin degree of freedom, which will save greatly the energy consumption. However, due to the relatively low spin transport efficiency, the ASLD requires a certain high injection current which is harmful to the ultra- low power performance.9In this paper, we employ the voltage control magnetic anisotropy (VCMA) effect to assist magnetization switching,10–15aiming to reduce the injection current and shorten the switching time. A voltage assisted graphene based ASLD (VG-ASLD) is focused to demonstrate its aThese authors contributed equally to this work. bCorresponding Authors’ emails: yz@buaa.edu.cn, weisheng.zhao@buaa.edu.cn 2158-3226/2017/7(5)/055925/5 7, 055925-1 ©Author(s) 2017 055925-2 Zhang et al. AIP Advances 7, 055925 (2017) advantage.16,17By applying lumped spin circuit model and physical model, we carry out electric and micro-magnetic simulations to analyze the energy consumption. II. MICROMAGNETIC AND ELECTRIC MODELING As shown in Fig. 1, the VG-ASLD structure consists of four parts, i.e. the free ferromagnetic layer (e.g. CoFeB), the tunnel barrier (e.g. MgO), the graphene channel and the isolator. Interfacial perpendicular magnetic anisotropy (PMA) can be induced in CoFeB/MgO layers structure.18,19The vertical isolator is used to prevent from the permeation of spin current. There are also two crucial interfaces: the one between the free layer and the tunnel barrier, the other between the tunnel barrier and channel. According to the lumped spin circuit model, the electric property of material can be represent by a matrix equation as follows20 (Ic1 Is1) =1 L2666641 p p p2+cosech L sf377775(
VC
Vs) +1 L2666640 0 0tanh L sf377775(0 Vs1) (1) where is the conductivity, pis the spin polarization, sfis the spin diffusion length, Lis the channel length. Ic1is the charge-component of injected current, Is1is the spin-component of injected current.
Vc=Vc2Vc1is the difference of charge voltage, where Vc2is charge-component of detected voltage and Vc1is charge-component of injected voltage.
Vs=Vs2Vs1is the difference of spin voltage, where Vs2is spin-component of detected voltage and Vs1is spin-component of injected voltage. This equation is suited for all the components of the ASLD. Once the parameters are assigned according to the property of the structure, the equation can be used to describe the quantitative relation of the injected voltage/current and the detected voltage/current. To calculate the energy consumption of the VG-ASLD, we consider that all of the energy is dissipated by spin injection step. Although there are three main steps for a logic operation, i.e. spin injection, spin transport and the magnetization switching of detection ferromagnetic layer (FM2 in Fig. 1a), the spin transport and voltage-assisted magnetization switching consume barely energy. In this case, the energy consumption can be calculated by the formula E=Vc1Ic1Tswitching , where Tswitching is the switching time of FM2. Particularly, Ic1is calculated by using the aforementioned lumped spin circuit model, Tswitching can be obtained through micromagnetic simulation including spin transfer torque (STT) effect. Assuming that spin-torque term dominates the magnetization switching process, Tswitching is thus inversely proportional to the absorbed spin current.21Here the spin transport efficiency can be introduced as FIG. 1. (a) The schematic of VG-ASLD. The oxide layer added is for applying the VCMA. The ferromagnetic layer consists of 3 parts. Only part A will absorb spin current, besides part B cannot absorb spin current from channel directly because of the existence of isolator and part C can be used as the injector for the next VG-ASLD. The tunnel barrier is inserted between the FM and the channel, aiming to improve the spin injection. (b) The energy consumption analysis diagram of the VG-ASLD.055925-3 Zhang et al. AIP Advances 7, 055925 (2017) =Is2 Ic1(2) where Is2is the absorbed spin current of the detection ferromagnetic layer. Moreover, magnetic anisotropy energy is one of the crucial material parameters that influence the magnetization switching time. With the VCMA effect, we have22 KS,V=KS,0V d2(3) where KSis the magnetic anisotropy energy, is the coefficient of VCMA effect, d is the thickness of the ferromagnetic layer, V is the voltage applied. Here we focus on partial spin absorption induced magnetization switching. In this case, the dimension of the ferromagnetic layer is 600 nm 100 nm 1 nm, we assume that only a part of it can absorb the pure spin current from the channel, and we set this part to 400 nm 100 nm 1 nm (part A in Fig. 1). As these dimensions exceed the conditions for single domain assumption, we neglect the impact of the channel width on spin transport. For the FM2, when the magnetization of the part A begins to switch under the STT effect, its magnetization does not align with those of the part B and part C. The exchange torque will then take effect on these parts. In this process, it is the exchange energy that induces the switching of the part B and part C. However, magnetic anisotropy competes with exchange torque. That is why we apply VCMA to assist the switching by decreasing the anisotropy energy. III. RESULTS AND DISCUSSIONS By using the above-mentioned micromagnetic and electric models, we perform co-simulations to analyze the energy consumption of the proposed VG-ASLD with the parameters listed in Table I. The relationship between the tunnel barrier resistance and the energy consumption is studied firstly. The spin diffusion length of channel is set to 8 m, once the applied voltage is fixed, the increase of the tunnel barrier resistance will affect the energy consumption in two ways. In one way, the increase of the tunnel barrier resistance improves the spin transport efficiency by compensating the interface conductance mismatch.23In the other way, the increase of the tunnel barrier resistance augments the total ohmic resistance of the system and consumes more energy by Joule heating. As shown in Fig. 2a, when the tunnel barrier resistance is smaller than 0.25 , the spin transport efficiency continues to increase. The energy consumption will decrease with the increase of the tunnel barrier resistance. However, the situation changes when the tunnel barrier resistance is larger than 0.25 . Owing to the saturation of the spin transport efficiency, the energy consumption increases rapidly due to the Joule heating. When the voltage is increased to 9.8 V , the energy consumption can be reduced to 2.5 pJ. Obviously, the injection current has a significant influence on the energy consumption. Larger injection current will make the FM of detector absorbing more spin current and shorten the switching time. However, it will also consume the energy. The dependence of the injection current versus energy is calculated when the applied voltage is 1.6 V and 2 V respectively. The results are shown in TABLE I. Parameters applied in the simulations.18 Parameter Description Default Value  Gilbert damping constant 0.03 Ms Saturation magnetization 6.6 x 105A/m Gyromagnetic ratio 1.76 x 1011Hz/T Ks,0 Uniaxial anisotropy energy 6.5 x 105J/m3 t Ferromagnetic layer thickness 1 nm L Channel length 500 nm N Channel Resistivity 1 x 10-8
m F Ferromagnetic Resistivity 1 x 10-7
m A Exchange Constant 3 x 10-11J/m055925-4 Zhang et al. AIP Advances 7, 055925 (2017) FIG. 2. (a) The dependence of energy consumption on the resistance of tunnel barrier. (b) The dependence of energy consumption on injection current for different applied voltages. FIG. 3. The dependence of energy consumption on applied voltage for different spin diffusion length of the channel. Inset is the relationship between the spin transport efficiency and spin diffusion length. Fig. 2b, the energy consumption increases with the increase of the injection current. It implies that the injection current plays a critical role for the energy consumption. There are two reasons to explain the results. First, the energy consumption is proportional to the square of the injection current. Second, it is difficult to decrease the switching time significantly only by increasing the injection current. The influence of the spin diffusion length of the channel has also been studied. Fixing the injection current to 320 A, we change the spin diffusion length of the channel and calculate the corresponding energy consumption of the system. It is clear that longer spin diffusion length of the channel possess higher spin transport efficiency and can decrease the injection current. Unfortunately, the benefit from lengthening the spin diffusion length has a ceiling. As shown in Fig. 3, when the structural and material parameters are fixed, the spin transport efficiency will be saturated with regards to relatively long spin diffusion lengths, e.g. 6.5 m and 8 m. IV. CONCLUSION In summary, we analyze the energy consumption of VG-ASLD. The voltage control mecha- nism can not only assist magnetization switching of the detector, i.e. reducing the injection current and shortening the switching time, but also not cost additional energy consumption. By implement- ing micro-magnetic and electric co-simulation, we validate the functionality of the VG-ASLD. The dependences of energy consumption on applied voltage, spin diffusion length of the channel, resis- tance of the tunnel barrier are investigated respectively. With optimizing material configurations and applied voltage, the energy consumption can be reduced to 2.5 pJ/operation. This VG-ASLD can be potential candidate for building ultra-low power computing system.055925-5 Zhang et al. AIP Advances 7, 055925 (2017) ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (No.61504006 and 61571023), the International Collaboration Project (No. 2015DFE12880) from the Ministry of Science and Technology of China. 1R. K. Cavin, P. Lugli, and V . V Zhirnov, Proc. IEEE 100, 1720 (2012). 2S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, and S. V on Molna, Science 294, 1488 (2001). 3Y . Huai, AAPPS Bull. 18, 33 (2008). 4C. J. Lin, S. H. Kang, Y . J. Wang, K. Lee, X. Zhu, W. C. Chen, X. Li, W. N. Hsu, Y . C. Kao, M. T. Liu, W. C. Chen, Y . Lin, M. 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Amiri, P. Upadhyaya, S. S. Cherepov, J. Zhu, M. Lewis, R. Dorrance, J. A. Katine, J. Langer, K. Galatsis, D. Markovic, I. Krivorotov, and K. L. Wang, Proc. Electron Devices Meeting (IEDM), IEEE International (2012) , pp. 29–5. 13U. Bauer, L. Yao, A. J. Tan, P. Agrawal, S. Emori, H. L. Tuller, S. van Dijken, and G. S. D. Beach, Nat. Mater. 14, 174 (2014). 14X. Zhang, C. Wang, Y . Liu, Z. Zhang, Q. Y . Jin, and C.-G. Duan, Sci. Rep 6, 18719 (2016). 15Z. Zhang, Y . Zhang, L. Yue, L. Su, Y . Shi, Y . Zhang, and W. Zhao, Proc. Nanoarch, IEEE International (2016) , pp. 141–142. 16W. Han, R. K. Kawakami, M. Gmitra, and J. Fabian, Nat Nano 9, 794 (2014). 17L. Su, Y . Zhang, J.-O. Klein, Y . Zhang, A. Bournel, A. Fert, and W. Zhao, Sci. Rep. 5, 14905 (2015). 18S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan, M. Endo, S. Kanai, J. Hayakawa, F. Matsukura, and H. Ohno, Nat. Mater. 9, 721 (2010). 19S. Peng, M. Wang, H. Yang, L. Zeng, J. Nan, J. Zhou, Y . Zhang, A. Hallal, M. Chshiev, K. Wang, Q. Zhang, and W. Zhao, Sci. Rep. 5, 18173 (2015). 20S. Srinivasan, V . Diep, B. Behin-Aein, A. Sarkar, and S. Datta, eprint arXiv: 1304.0742 (2013). 21Y . Zhang, W. Zhao, Y . Lakys, J. Klein, J. Kim, D. Ravelosona, and C. Chappert, IEEE Trans. Electron Devices 59, 819 (2012). 22Duan, J. P. Velev, R. F. Sabirianov, Z. Zhu, J. Chu, S. S. Jaswal, and E. Y . Tsymbal, Phys. Rev. Lett. 101, 137201 (2008). 23A. Fert and H. Jaffr `es, Phys. Rev. B 64, 184420 (2001).

1.5018637.pdf

Magnonic waveguide based on exchange-spring magnetic structure Lixiang Wang , Leisen Gao , Lichuan Jin , Yulong Liao , Tianlong Wen , Xiaoli Tang , Huaiwu Zhang , and Zhiyong Zhong Citation: AIP Advances 8, 055103 (2018); doi: 10.1063/1.5018637 View online: https://doi.org/10.1063/1.5018637 View Table of Contents: http://aip.scitation.org/toc/adv/8/5 Published by the American Institute of Physics Articles you may be interested in On the radiated EMI current extraction of dc transmission line based on corona current statistical measurements AIP Advances 8, 055001 (2018); 10.1063/1.5018328 Circulation of spoof surface plasmon polaritons: Implementation and verification AIP Advances 8, 055002 (2018); 10.1063/1.5020670 Optimization design of wireless charging system for autonomous robots based on magnetic resonance coupling AIP Advances 8, 055004 (2018); 10.1063/1.5030445 Effects of the magnetic field variation on the spin wave interference in a magnetic cross junction AIP Advances 8, 056619 (2018); 10.1063/1.5007164 Comparison of swirling strengths derived from two- and three-dimensional velocity fields in channel flow AIP Advances 8, 055302 (2018); 10.1063/1.5023533 Thermal analysis of epidermal electronic devices integrated with human skin considering the effects of interfacial thermal resistance AIP Advances 8, 055102 (2018); 10.1063/1.5029505AIP ADV ANCES 8, 055103 (2018) Magnonic waveguide based on exchange-spring magnetic structure Lixiang Wang, Leisen Gao, Lichuan Jin, Yulong Liao, Tianlong Wen, Xiaoli Tang, Huaiwu Zhang, and Zhiyong Zhonga State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science and Technology of China, 610054 Chengdu, China (Received 8 December 2017; accepted 24 April 2018; published online 2 May 2018) A soft/hard exchange-spring coupled bilayer magnetic structure is proposed to obtain a narrow channel for spin-wave propagation. Micromagnetic simulations show that broad-band Damon-Eshbach geometry spin waves are strongly constrained within the channel and propagate effectively with a high group velocity. The beam width of the bound spin waves is almost independent from the frequency and is smaller than 24nm. Two side spin beams appearing at the low-frequency excitation are demonstrated to be coupled with the channel spins by dipole-dipole interaction. In contrast to a domain wall, the channel formed by exchange-spring coupling is easier to be realized in exper- imental scenarios and holds stronger immunity to surroundings. This work is expected to open new possibilities for energy-efficient spin-wave guiding as well as to help shape the field of beam magnonics. © 2018 Author(s). All article content, except where oth- erwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/1.5018637 INTRODUCTION Magnonics is an emerging technology for low-power signal transmission and data process- ing based on spin waves (magnons) propagating in magnetic materials.1–4Due to its nanometer wavelengths and Joule-heat-free transfer of spin information over macroscopic distances,6–8such magnon-based computing concept is actively studied and undergoes benchmarking in the framework beyond-CMOS strategies.5For magnonics,9–14the energy-efficient propagation of spin waves is crit- ical for any form of circuit design, and is crucial for wave processing schemes that rely on spin wave interferences.13–17In the last few years, magnonic crystals have been widely studied,10,18–24which is periodically patterned at the nanoscale to control the transmission of spin waves through magnetic materials. However, it is very challenge to precisely fabricate magnonic crystals at the submicron or nanometer scale. Recently, individual domain walls were functionalized as nanoscale magnonic conduits that allowed for a rewritable nanocircuitry.25–28Here a magnetic domain wall acts as a potential well, and spin waves can be strongly confined at its center when propagating along the wall without extra fields. Unfortunately, these specific domain walls are usually difficult to realize in experimental scenarios. Further, the domain wall structure is fragile and can be easily destroyed by stray fields from surroundings. Here, we provide an alternative method to overcome these drawbacks. We present a paradigm for spin-wave propagation that relies on a domain-wall-like magnetic channel for magnonic waveguides. The channel is naturally induced on a soft/hard bilayer magnetic structure via exchange-spring coupling interaction, thereby can be easily realized in laboratory and be robust from outside disturbance. We show through micromagnetic simulations that a Damon-Eshbach (DE) propagation geometry sent to the channel can be strongly confined in a narrow area with a broad frequency band. These magnetic structures are very promising to open new perspectives for efficient spin-wave propagation towards magnonic nanocircuitry. aAuthor to whom correspondence should be addressed. Electronic mail: zzy@uestc.edu.cn 2158-3226/2018/8(5)/055103/8 8, 055103-1 ©Author(s) 2018 055103-2 Wang et al. AIP Advances 8, 055103 (2018) MICROMAGNETIC MODELING Fig. 1(a) sketches the geometry sizes of the model we used. The model includes a soft magnetic layer (SL) that is exchange coupled to a hard magnetic layer (HL). The thicknesses of two layers are tSL= 12nm and tHL= 8nm, respectively. The HL magnetization was initially oriented in the plane- normal direction with stripe domains of a width 100nm along the xdirection.30Periodic boundary condition in the ydirection was used to model an infinite array of stripe domains. The SL magnetization was randomized with an in-plane random vector field, and was then allowed to equilibrate at zero applied fields. The whole simulation process contains two steps: (1) stabilizing the system and (2) propagating spin waves. In (2) an out-of-plane field pulse was used to locally excite spin dynamics at the channel’s position at a distance of 32nm from the left edge (green area in Fig. 1b). The OOMMF micromagnetic code31was used to carry out simulations by solving the Landau-Lifshitz-Gilbert equation. Parameters of the SL layer were taken from NiFe thin film for the numerical simulations,32the magnetic saturation, namely, MS S=8.0105A=m, the exchange stiffness AS= 131012J/m and the anisotropy, KS= 0 J/m3. The standard values of magnetic saturation of [Co/Pd] 5,MH S=3.6105A=m, the perpendicular anisotropy, KH= 6.3105J/m3, and the exchange stiffness, AH= 6.01012J/m3,32 were taken for the HL. The exchange between the SL and HL was modeled with an intermediate value ASH= 9.5 1012J/m3. The size of the unit cell was 4nm 2nm4nm. The model size in thexdirection was set to 3 m to minimize boundary effects and provide sustainable spin-wave propagation. The damping parameter was set at = 0.5 in step (1) to lead to rapid convergence to get grounded magnetization state but = 0.01 (were taken from NiFe) in step (2) to support spin waves for a long-distance propagation. FIG. 1. (a) Schematic illustration of the magnetic structure used in simulations. Red-blue areas in the HL denote the magnetic moments directing up and down, respectively. (b) Stabilized magnetization configuration on the top surface of the SL, with the discussed channel (indicated by the blue dashed box) obtained from the OOMMF simulation. The color of vectors codes myvalues as indicated by the color bar. Spin waves were locally excited at the position indicated by the green dot. (c) The magnetization and internal-field distribution over the waveguide width.055103-3 Wang et al. AIP Advances 8, 055103 (2018) RESULTS AND DISCUSSIONS Here, we focus on the possibility of bringing about an expected channel for spin-wave propagation based on such soft/hard magnetic structure. Fig. 1b shows the remnant magnetization configuration of the structure after stabilization of the system. The orientation of arrows represents the magnetization vectors, with red-green color codes the normalized magnetization component myparallel to the yaxis of the model. In Fig. 1b, one observes a noticeable strip area (indicated by the dashed box) aty= 200nm along the xaxis, which separates two almost opposite domain patterns. The area is characterized by inside magnetic moments that are strictly oriented to the ydirection. It is particularly similar to a 180Ne´vel wall, but the magnetization within it is in a better order. For exchange-spring magnets, the hard-magnetic grains provide the high anisotropy and coercive fields while the soft- magnetic grains the high saturation magnetization. Such magnets are characterized by enhanced remanent magnetization since the soft grains can be firmly pinned to the hard grains by the exchange interaction at zero applied field.32–34Therefore, the channel presented on the studied structure should exhibit stronger immune to stray fields compared with a domain wall. The variation of myand the Hy component of the effective field on the SL over the model width are plotted in Fig. 1c. Focus on the Hyprofile, one obtains the especially inhomogeneous field distribution, by which the potential well are created at the strip area. In the following, we utilize the well as a transmission channel to carry out the spin-wave propagating studies. Since the channel was formed naturally and exhibits strong sustainability, no external bias fields were needed in our simulations to either induce the channel or propagate spin waves. To verify the channeling effect, monochromatic spin waves propagating along the channel were investigated first. An oscillating magnetic field of hz=H0sin(!t) with H0= 100mT, to render higher phase resolution between neighboring simulation cells, was applied to the marked position in Fig. 1b to generate spin dynamics. The amplitude profiles of the mzmagnetization component, showing spin-wave propagating nature, are plotted at 2.0ns in Fig. 2a for the labeled frequencies. For the lower excitation frequencies of 5 and 12GHz, spin waves exist only inside the channel as expected for a bound mode, justifying that the channel formed by the studied structure can be indeed served as a waveguide for spin-wave propagation. Similar to a domain wall, the channeling effect can be understood as following.29The perfectly ordered magnetization within the channel separates two opposite domains, giving rise to opposite magnetic volume charges on the two sides of the channel. These charges generate a strong magnetostatic field antiparallel to the magnetization within the channel, resulting in a locally decreased effective field which forms a magnetic potential well for bond modes. As presented in Fig. 2a, the well in the internal field is so deep that spin waves can be completely confined within the channel with much pronounced strength. It is noteworthy that the magnetization of the channel is oriented along the ydirection and thus perpendicular to the propagation direction of the spin waves, which forms the DE propagation geometry. It is well known that DE spin waves can be easily generated and controlled, have positive dispersion and, more importantly, high group velocity.35The slight decrease in amplitude for long-distance propagation is related to the introduction of Gilbert damping in the simulations (here = 0.01 was used). In contrast, once the excitation frequency reaches 25GHz, the spin waves break free and reach out to the entire layer, then the channeling effect disappears. The observation can be analogized to the spin waves propagating in a domain wall:26–28spin dynamics with a high frequency lie in the allowed band of the extend spin-wave modes and, thus are mixed with the extend modes, resulting in the loss of the channeling effect. Here, we focus on the spin-wave mode confined within the channel. The beamwidth of the confined spin waves for various frequencies, which is defined as the full width at half maximum (FWHM) of the well in Mz, was studied below. As shown in Fig. 2b, all the calculated data of the beamwidth is smaller than 24nm and is almost irrelevant to the excitation frequency, which is in good agreement with what is expected from a waveguide width at nanoscale. Thanks to the perfect confining effect of the channel, spin waves are strongly squeezed to a narrow alley, result- ing in the weakness of magnetization precession at the lateral edges.38In other words, spin waves propagating along the channel are insensitive to the edge, as the light traveling in a fiber. Such optic- fiber-like waveguide should have great potential for magnonic transmission,25,26since the negligible055103-4 Wang et al. AIP Advances 8, 055103 (2018) FIG. 2. (a) Spin-wave propagation patterns at t = 2.0ns for the labeled frequencies. The color codes the out-of-plane component of the magnetization mzas indicated by the color bar. (b) The beamwidth as a function of frequency. The beamwidth is defined as the FWHM of the SW amplitude distribution over the waveguide width. (c) Dispersion relation of the well-confined mode calculated by 2D FFT (the bottom panel) and the illustration of spin-wave nonreciprocity (the top panel). boundary scattering experienced by the channeled spin waves will lead to an increased propagating length. In addition to spatial and spectrum characteristics, we also shed light on the spin-wave dispersion to get a deeper insight into the well-confined mode. A symmetric sincfield pulse of hz(t)=H0sin(2fct) 2fct, ranging from 0 to 25GHz, was used to the excite spin waves within a frequency range. The simula- tion is run for 8ns and the data are recorded every 12.5ps. Two-dimensional fast Fourier transform (2D FFT) of Mzat the center of the channel is performed along the propagating direction (from x=40 to 2960nm) to plot the dispersion curve.37Fig. 2c shows the resulting positive dispersion that enables the transport of information via spin waves propagating within the channel. The red-white color bar codes the normalized logarithmic Fourier amplitude, which indicates the spin-wave intensity at particular values of the wave number and frequency. Parabolic dispersion relation with large wave vectors (corresponding to small wavelength) suggests that the bond mode is mainly dominated by exchange energy. However, one notes that the dispersion curve is not exactly symmetric with respect to the yaxis, suggesting the nonreciprocal nature of spin-wave propagation owing to the antisymmet- ric Dzyaloshinskii–Moriya interaction (DMI) arising from spin-orbit scatting of itinerant electrons,29 as is shown in the top panel in Fig. 2c. One observes two spin beams with the same frequency but different strength propagating oppositely along the channel, as indicated by the different color of left and right dispersion branches. In fact, the difference on wave vectors also exists between two beams but is not noticeable, so it is not easy to be observed in the figure. The wide frequency range of the dispersion curve illustrates that the critical frequencies, within which spin waves are permitted to be055103-5 Wang et al. AIP Advances 8, 055103 (2018) confined in the channel, can be close to 0 and 22GHz, respectively. The observation of well-defined wave vectors along the propagation path is a crucial precondition for numerous applications that rely on the interference of spin waves and highlights the potential of the channel in magnonic circuits for data processing. We further investigate the group velocity of the bound mode. An illustration of wave packet propagating along the channel at the labeled times is shown in Fig. 3a. The wave packets were generated with a field pulse that comprises a sine wave oscillation at the frequency f= 5GHz. The temporal evolution of the normalized mzcomponent of the wave packet is shown for three instants after the application of the pulse field. By relating the evolution time
t to the propa- gated distance
x, we can extract the wave packet velocity, which vg=
x=
tis approximately 1440m/s for the given frequency. In fact, the upward curve plotted in Fig. 3b reflects a positive correlation between the group velocity and frequency, which can be derived from the spin-wave dispersion relation by vg=@!(k)/@k. The group velocity can exceed 1km/s at 1GHz, while over 8GHz the velocity exceeds 1.5km/s. Large value of the propagation velocity brings competitive- ness to the well-defined mode for computing technology, since the velocity determines the speed of calculation.7 When the excitation frequency applied in the channel was reduced to a very low value, for example 2GHz, an attractive observation emerges: two other spin-wave beams parallel to the channel are presented at y= 100nm and 300nm respectively (see Fig. 4a for details). The amplitude of this two beams is almost equal but obviously smaller than that of the above discussed one (about 1/4 of the later). Clearly, the appearance of them suggests the existence of other propagating channels at the corresponding positions. To get a deeper insight of these channels, we extracted the xcom- ponent of the internal field Hx effacross the ydirection and Fig. 4b (the bottom panel) shows the result. Two equal-height but antisymmetric peaks on the curve prove that channels for spin-wave propagation indeed exist and the potential wells are equal-deep but their magnetization is oppo- site (see the top panel in Fig. 4b). Below we label this two channels as side channels (SCs) while the previous one as middle channel (MC) to distinguish from each other. Note that the magnetic moments in the SCs are oriented to the x( x) axis and thus parallel to the propagation direction of spin waves, which forms the backward-volume-wave propagation.35It is quite different from the MC mode. Again, we should be noted that the SC modes appear only when the excitation frequency is lower, they will vanish at a high excitation frequency (Fig. 2a), as indicated by their dispersion curve FIG. 3. (a) Illustration of wave packet propagating along the channel at various time. (b) The group velocity as a function of frequency. Green circles are simulation results obtained from OOMMF, while red dashed line is corresponding analytical results calculated by vg=@!(k)/@k.055103-6 Wang et al. AIP Advances 8, 055103 (2018) FIG. 4. (a) Spin-wave propagation pattern at t= 2.0ns for f= 2GHz. (b) The xcomponent distribution of the internal field over the width (the bottom panel) and the 3D view of the spin configuration (the top panel). (c) The dispersion curve of the SC modes for d= 100nm. (d) Energy spectrum of spin waves across the ydirection for d= 100 and 80nm. in Fig. 4c. We propose to give the explanation as following. It is well known that spin dynam- ics are dominated by dipole or exchange energy depending on the wavelength. For the studied mode with a low frequency, which corresponds to a large wavelength, spin waves in the MCs are dominated by dipole-dipole interaction. The dynamic dipole energy characters a wide inter- acting range that can cross the barriers between the channels, powering the spin precession in the SCs, then the SC beams emerge. However, when the frequency is at a high value (correspond- ing to a small wavelength), exchange interaction charactering narrow interacting range will take place of the dominating role of dipole-dipole interaction, resulting in the loss of driving force on the SCs, hence the SC modes disappear. To verify the words above, we decreased the distance between the channels from d= 100nm to 80nm and the cut-off frequency of the SC modes increased as expected (shown in Fig. 4d). It can be understood that shorter distance between the adjacent channels allows narrower-range interaction, which corresponds to higher frequency, to drive the SC modes, accordingly resulting in the higher cut-off frequency. The appearance of such SC modes is a noteworthy observation, it might provide an unprecedented possibility for spin-wave filter design.055103-7 Wang et al. AIP Advances 8, 055103 (2018) CONCLUSION In summary, we have demonstrated that a soft/hard exchange-spring coupling bilayer magnets with certain magnetization can cause a deep potential well for spin-wave channeling. Because the channel is primarily governed by intrinsic interplay between two layers, it is less sensitive to exper- imental scenarios. Spin waves with a DE propagation geometry exhibit a well-defined wave vector along the channel, enabling data transport and processing using wave properties. The beamwidth of the bound mode is smaller than 24nm and is almost independent from frequency, which can avoid the boundary scattering36caused by edge irregularity and the extra dispersion.38In addition, a rela- tive high group velocity exceeding 1km/s promises the channeled mode a candidate for computing technology. Finally, we have addressed and qualitatively verified the appearance of the SC modes, which provides a new train of thought for spin-wave filter design. These observations pave the way for the realization of nano-sized, energy-efficient, reconfigurable magnonic circuits. ACKNOWLEDGMENTS This paper is supported by the National Key Research and Development Plan (No. 2016YFA0300801); the National Natural Science Foundation of China under grant Nos. 61734002, 61571079 and 51702042, and the Sichuan Science and Technology Support Project (Nos. 2016GZ0250 and 2017JY0002). 1A. V . Chumak, V . I. Vasyuchka, A. A. 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Wang et al. , “Micromagnetic study of spin wave propagation in bicomponent magnonic crystal waveguides [J],” Applied Physics Letters 98(15), 153107 (2011). 25J. Lan, W. Yu, R. Wu et al. , “Spin-wave diode [J],” Physical Review X 5(4), 041049 (2015). 26X. Xing and Y . Zhou, “Fiber optics for spin waves [J],” NPG Asia Materials 8(3), e246 (2016). 27K. Wagner, A. K ´akay, K. Schultheiss et al. , “Magnetic domain walls as reconfigurable spin-wave nanochannels [J],” Nature Nanotechnology 11(5), 432–436 (2016). 28F. Garcia-Sanchez, P. Borys, R. Soucaille et al. , “Narrow magnonic waveguides based on domain walls [J],” Physical Review Letters 114(24), 247206 (2015). 29C. S. Davies and V . V . Kruglyak, “Graded-index magnonics [J],” Low Temperature Physics 40(10), 91–983 (2015). 30L. Tryputen, F. Guo, F. 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1.4826447.pdf

Weibel instability with nonextensive distribution Hui-Bin Qiu and Shi-Bing Liu Citation: Physics of Plasmas (1994-present) 20, 102119 (2013); doi: 10.1063/1.4826447 View online: http://dx.doi.org/10.1063/1.4826447 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/20/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Particle in cell simulations of Buneman instability of a current-driven plasma with q-nonextensive electron velocity distribution Phys. Plasmas 21, 092307 (2014); 10.1063/1.4896243 Is the Weibel instability enhanced by the suprathermal populations or not? Phys. Plasmas 17, 062112 (2010); 10.1063/1.3446827 Weibel instability with semirelativistic Maxwellian distribution function Phys. Plasmas 14, 072106 (2007); 10.1063/1.2749254 Weibel instability with non-Maxwellian distribution functions Phys. Plasmas 14, 022108 (2007); 10.1063/1.2536159 Covariant kinetic dispersion theory of linear waves in anisotropic plasmas. II. Comparison of covariant and noncovariant growth rates of the nonrelativistic Weibel instability Phys. Plasmas 12, 022104 (2005); 10.1063/1.1844511 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.205.114.91 On: Tue, 20 Jan 2015 12:38:27Weibel instability with nonextensive distribution Hui-Bin Qiu and Shi-Bing Liua) Strong-field and Ultrafast Photonics Lab, Institute of Laser Engineering, Beijing University of Technology, Beijing 100124, China (Received 28 June 2013; accepted 7 October 2013; published online 24 October 2013) Weibel instability in plasma, where the ion distribution is isotropic and the electron component of the plasma possesses the anisotropic temperature distribution, is investigated based on the kinetic theory in context of nonextensive statistics mechanics. The instability growth rate is shown to be dependent on the nonextensive parameters of both electron and ion, and in the extensive limit, theresult in Maxwellian distribution plasma is recovered. The instability growth rate is found to be enhanced as the nonextensive parameter of electron increases. VC2013 AIP Publishing LLC . [http://dx.doi.org/10.1063/1.4826447 ] I. INTRODUCTION In space plasmas, it has been observed that particle velocity distributions typically possess a non-Maxwellian high-energy tail which can be effectively modeled by kappa distribution introduced by Vasyliunas1in 1968, and since then, kappa distributions have been utilized in numerous stud- ies of the solar wind2–5and planetary magnetospheres,6–9 observations of ions in the outer heliosphere10,11and theoreti- cal analyses of the ion and energetic neutral atoms.12,13 Although kappa distribution is widely used, its statistical basis is not clear until the nonextensive entropy is proposed,14 which is a milestone in statistical mechanics. Decades later, Silva et al.15gave the nonextensive distribution function, and Leubner16gave a nonextensive entropy approach to kappa dis- tribution, which unified the one dimensional kappa and nonex- tensive distribution function, and Liu et al.17proposed the three dimensional (3D) nonextensive distribution function,which is different from the one proposed by Silva et al. , 15but is equivalent to the 3D kappa distribution. For nonextensive parameter q<1, nonextensive distribution has suprathermal power-law tails at high energies, which have been proved to be equivalent to kappa distribution; for q>1, nonextensive distribution is suitable for the description of systems contain-ing only low-speed particles; and in the extensive limit ðq¼1Þ, Maxwellian distribution is recovered. Much attention has been paid to the physics of nonextensive distributedplasma. For instance, the nonextensive statistical mechanics has been used to analyze plasma oscillations, 18stellar plasma and solar neutrinos,19and non-equilibrium plasma systems with Coulombian long-range interactions.20Some theoretical studies on ion-acoustic mode,21electro-acoustic mode,22and arbitrary amplitude kinetic Alfven solitons23in nonextensive plasma have been reported, as well as experimental laboratory observations.24 Weibel instability is first investigated in 1959,25in which the mechanism depends on the free energy stored in an anisotropic particle distribution in velocity space.26Such distributions in velocity space with an enhanced temperature along some direction are quite common in both space andlaboratory plasmas,27and Weibel instability is considered to explain the generation of magnetic field in the vicinity of gamma ray burst sources,28supernovas, and galactic cosmic rays.29Weibel’s work inspired series of further investiga- tions in both relativistic30–32and nonrelativistic33regimes in the unmagnetized plasma. Recently, Califano et al.34have investigated the Weibel instability where the role of tempera-ture anisotropy is taken by two counter streaming electron populations, and the multispecies Weibel instability for the intense charged particle beam propagating through back-ground plasma and the kinetic theory of electromagnetic ion waves in the relativistic plasma region were reported. 35Most recently, Mahdavi et al.36considered the Coulomb collision effect of electron-ion on the Weibel instability. Plasma is a complex system, and its long-range Coulomb interaction is relevant to nonextensive effects37 which will appeared in numerous aspects, e.g., instability of plasma, which is this study aimed to consider. In detail, an investigation of the effects of nonextensive parameters onWeibel instability in plasma where the ion distribution is iso- tropic and the electron component of the plasma possesses the anisotropic temperature distribution are carried out basedon the kinetic theory, which will be contributed to an under- standing of the interactions in relativistic laser plasma. The paper is organized as follows. In Sec. II, we obtain the Weibel instability growth rates with Maxwellian, nonex- tensive, and kappa distributions, respectively. Sec. IIIshows numerical results and discussions, and summary and conclu-sion are given in Sec. IV. II. WEIBEL INSTABILITY GROWTH RATE In order to analyze the Weibel instability, we can begin with the dispersion relation equation38 k2/C0x2 c2e11¼0; (1) where k,x, and care wave number, frequency, and velocity of light in vacuum, respectively; e11is a component of dielectric tensor in an anisotropic unmagnetized plasma, of which the general expression reads38 a)sbliu@bjut.edu.cn 1070-664X/2013/20(10)/102119/4/$30.00 VC2013 AIP Publishing LLC 20, 102119-1PHYSICS OF PLASMAS 20, 102119 (2013) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.205.114.91 On: Tue, 20 Jan 2015 12:38:27eijðx;kÞ¼dijþX a4pe2 a xðdp x/C0k/C1vvi@f0a @pl1/C0k/C1v x/C18/C19 dljþklvj x/C20/C21 ; (2) where k,ea, and f0aare wave vector, the charge, and distribu- tion function of a(¼e;idenotes electron and ion, respec- tively) element in the plasma, respectively; momentum pis related to the particle mass mand velocity vbyp¼mv.I n the following, we will consider the Weibel instability withMaxwellian and nonextensive distribution, respectively. A. Maxwellian distribution Now, we consider a transverse wave propagating in an unmagnetized, collisionless plasma, in which components of the plasma possess the Maxwellian anisotropic temperature distribution f¼n0 p3=2h2 ?hkexp/C0v2 k h2 kþv2 ? h2 ? !"# ; (3) where n0is the particle number density, and the thermal speed his related to the particle temperature Tby h2 k;?¼Tk;?=m. Substituting Eq. (3)into Eq. (2), and from Eq.(1), we obtain c2k2¼x21/C0X ax2 pa x21þT?a Tka½ZðnaÞ/C01/C138/C26/C27 ! ; (4) where xpa¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4pe2 ana0=map is the plasma frequency of com- ponent a,na¼x=khkais the phase velocity normalized to the thermal velocity hka, and ZðnaÞis the dispersion function ZðnaÞ¼1ffiffiffippðþ1 /C01na na/C0xaþi/expð/C0x2 aÞdxa; (5) where xa¼pk=mhkaand i/comes from Landau rules (d!þ 0Þ.39When na/C291,40 ZðnaÞ/C251þ1 2n2 aþ3 4n4 a/C0iffiffiffippnaexpð/C0n2 aÞ; (6) and when na/C281,40 ZðnaÞ/C252n2 a/C04 3n4 a/C0iffiffiffippnaexpð/C0n2 aÞ: (7) For simplicity, we assume that the ion distribution is iso- tropic and that the electron component of the plasma pos-sesses the anisotropic temperature distribution, and kh Ti/C28x/C28khke. In this case, T?i¼Tki, and from Eqs. (6) and(7), we obtain ZðneÞ/C252n2 e/C0iffiffiffippne; (8) ZðniÞ/C251: (9) Substituting Eqs. (8)and(9)into Eq. (4), and under the con- dition that x2/C28c2k2, we obtain38Imx¼/C0ffiffiffip 2rkvTke x2 peTke T?e/C18/C19 c2k2þx2 piþx2 pe1/C0T?e Tke/C18/C19 /C20/C21 ; (10) where vTke¼hke=ffiffiffi 2p . In the plasma where the wave vector k is small enough, the quantity Imxcan become positive for T?e/C29Tke, which corresponds to an instability. B. Nonextensive distribution Let us begin with recalling some basic facts about Tsallis statistics. In Tsallis statistics, the form of the en-tropy 14is Sq¼jB1/C0P ipq i q/C01; (11) where jBis the Boltzmann constant, piis the probability of the i-th microstate, and qis a parameter describing the degree of nonextensivity. It recovers the B-G entropy in the limit q!1. The basic property of Tsallis entropy is the non- extensivity which refers that the nonextensive entropy isnonadditive for q6¼1, namely, for one system composed of two parts A and B, the total nonextensive entropy of the composite system A þBi s S qðAþBÞ¼SqðAÞþSqðBÞþð 1/C0qÞSqðAÞSqðBÞ:(12) The 3D nonextensive distribution function reads17 fq¼n0Aq p3=2h2 ?hk1/C0ðq/C01Þv2 k h2 kþv2 ? h2 ? !"# 2/C0q q/C01 ; (13) where Aqis the constant of normalization; the thermal speed his related to the particle temperature Tbyh2 k;?¼ ð3q/C01Þv2 Tk;?with v2 Tk;?¼Tk;?=m. As one may check, for q/C20/C01, the distribution is unnormalizable; it has power-law tails for /C01<q<1 and exhibits a thermal cutoff on the maximum value allowed for the velocity of the particles for q>1, given by vmax¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v2 T=ðq/C01Þp ; in addition, in the limit of q!1, Maxwellian distribution is recovered. This means that the range of integration over vx,vy, and vzcon- taining the term fqshould be from /C01 toþ1 when /C01<q/C201, and from /C0vmaxtoþvmaxwhen q/C211. The con- stant of normalization Aq¼ffiffiffiffiffiffiffiffiffiffiffi1/C0qp C1 1/C0qðÞ C1 1/C0q/C01 2ðÞwhen /C01<q /C201;Aq¼1þq 2ffiffiffiffiffiffiffiffiffiffiffiq/C01p C1 q/C01þ1 2ðÞ C1 q/C01ðÞ. Here, Cis the gamma function. Substituting Eq. (13) into Eq. (2), and from Eq. (1),w e obtain c2k2¼x21/C0X ax2 pa x21þqa 2þT?a TkaZqðnaÞ/C01þqa 2/C20/C21 () ! ; (14) where ZqðnaÞis the generalized plasma dispersion function in the context of nonextensive statistical mechanics102119-2 H.-B. Qiu and S.-B. Liu Phys. Plasmas 20, 102119 (2013) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.205.114.91 On: Tue, 20 Jan 2015 12:38:27ZqðnaÞ¼Aqaffiffiffippðxamax /C0xamaxna na/C0xaþi/½1/C0ðqa/C01Þx2 a/C1382/C0qa qa/C01dxa; (15) where xamax¼þ 1 , when /C01<q/C201;xamax¼1=ffiffiffiffiffiffiffiffiffiffiffiq/C01p, when q>1. When na/C291,17 ZqaðnaÞ/C25ð1þqaÞ 2þ1 2n2 aþ3 2ð3qa/C01Þn4 a /C0iffiffiffippnaAqa½1/C0ðqa/C01Þn2 a/C1382/C0qa qa/C01; (16) and when na/C281,17 ZqaðnaÞ/C25ð1þqaÞð3/C0qaÞ 2n2 aþð1þqaÞð3/C0qaÞð3qa/C05Þ 6n4 a /C0iffiffiffippnaAqa½1/C0ðqa/C01Þn2 a/C1382/C0qa qa/C01: (17) When consider the same situation as Maxwellian case, namely, assume that the ion distribution is isotropic and thatthe electron component of the plasma possesses the aniso- tropic temperature distribution, and kh Ti/C28x/C28khke, then T?i¼Tki, and from Eqs. (16) and(17), we obtain ZqeðneÞ/C25ð1þqeÞð3/C0qeÞ 2n2 e/C0iffiffiffippneAqe½1/C0ðqe/C01Þn2 e/C1382/C0qe qe/C01; (18) ZqiðniÞ/C25ð1þqiÞ 2: (19) Substituting Eqs. (18) and(19) into Eq. (4), and under the condition that x2/C28c2k2, we obtain Imx¼/C0ffiffiffip 2rkvTke Aqex2 peTke T?e/C18/C19 c2k2þx2 pi1þqi 2/C18/C19 þ x2 pe1þqe 2/C18/C19 1/C0T?e Tke/C18/C192 66643 7775: (20) As expected, in the limit q!1, Eq. (20) reduces to Eq. (10) being the standard result in B-G statistics, and we also notethat the instability growth rate depends not only on the tem- perature and plasma frequency of components but also on the nonextensive parameter q a. When qa¼1/C01 ja, where ja is the spectral index of kappa distribution1 fj¼n0Aj p3=2h2 ?hk1þ1 jv2 k h2 kþv2 ? h2 ? !"#/C0ðjþ1Þ ; (21) in which Aj¼CðjÞ j1=2Cðj/C01=2Þ; (22) Imxmust reduce to the growth rate in kappa distributed plasmaImxj¼/C0ffiffiffip 2rkvTke Ajex2 peTke T?e/C18/C19 c2k2þx2 pi1/C01 2ji/C18/C19 þ x2 pe1/C01 2je/C18/C19 1/C0T?e Tke/C18/C192 66643 7775: (23) Equation. (23) is contained in Eq. (20) due to Eq. (21) being only the q<1 part of Eq. (13). III. NUMERICAL RESULTS AND DISCUSSIONS In this section, we will numerically investigate the effects of nonextensive parameters on instability growth ratein plasma with nonextensive distributed electrons. According to Eq. (20), the instability growth rate is related to temperature, nonextensive parameter, plasma frequency ofelectron and ion. We choose nominal values that v Tke=c ¼0:0556, Tke=T?e¼1=15, and x2 pi=x2 pe¼1=1836, under which the effect of qiwill be limited, so qi¼1 is assumed for simplicity. With these values, we plot the curves of the normalized instability growth rate Imx=xpeas a function of the normalized wave number ck=xpewith selected values of the nonextensive parameters qein Figure 1. Figure 1shows that the system will excite the Weibel instability. In small wave number region, the instabilitygrowth rate increases as the wave number increases and reaches the maximum growth rate in middle wave number region, and then decreases until the instability growth rateconverts to damping rate due to energy of wave with shorter wavelength is easier to translate to particles for wave- particle collisions; the instability growth rate enhances as thenonextensive parameter of electron q eincreases, but damp- ing rate which has the opposite properties decreases when qe is increasing. The mechanism of Weibel instability is that the curvature of the electrons in the fluctuating magnetic field causes a momentum flux, which in turn effects hvi(and hence hJi) in such a way as to increase the fluctuation field,26 andhviis relevant to parameter qe, and we come to the FIG. 1. Normalized Weibel instability growth rate as a function of the nor- malized wave number for different values of qe.102119-3 H.-B. Qiu and S.-B. Liu Phys. Plasmas 20, 102119 (2013) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.205.114.91 On: Tue, 20 Jan 2015 12:38:27conclusion that the nonextensive nature of electron distribu- tion has a promoting effect on the Weibel instability. IV. SUMMARYAND CONCLUSION We have obtained the Weibel instability growth rate in a collisionless, unbounded, unmagnetized plasma modeled bynonextensive anisotropic temperature distribution with dif- ferent q-parameters for electrons and ions based on the ki- netic theory. As expected, in the extensive limit ( q¼1), the results reduce to the case of Maxwellian distribution, and with the substitution q a¼1/C01 ja, conveniently, the Weibel instability growth rate with kappa distribution which is con-tained in nonextensive distribution is obtained. We found that the instability growth rate enhances as the nonextensive parameter of electron q eincreases, namely, the nonextensive nature of electron distribution has a promoting effect on the Weibel instability. The nonextensive nature in plasma has been experimen- tally verified,41and Weibel instability has played an impor- tant role in laser plasma interactions, e.g., magnetic field generation at the expense of the thermal energy of an aniso-tropic electron population, 42which provides the justification of this investigation who also can be employed to compre- hend the nonextensive effect on Weibel instability in theplasma regions of atmospheres of pulsars 43,44and Earth magnetotail45,46and so on. ACKNOWLEDGMENTS The author thanks Professor S. Q. Liu and X. Q. Li for helpful suggestions. This work was supported by theNational Natural Science Foundation of China (Grant No. 10974010). 1V. M. Vasyliunas, J. Geophys. Res. 73, 2839, doi:10.1029/JA073i009p02839 (1968). 2G. 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Lavagno and P. Quarati, Nucl. Phys. B (Proc. Suppl.) 87, 209 (2000). 20J. Du, Phy. Lett. A 329, 262 (2004). 21L. Liu and J. Du, Physica A 387, 4821 (2008). 22A. S. Bains, M. Tribeche, and T. S. Gill, Phys. Lett. A 375, 2059 (2011). 23Y. Liu, S. Q. Liu, and B. Dai, Phys. Plasmas 18, 092309 (2011). 24Z. Tang, Y. Xu, L. Ruan, G. Buren, F. Wang, and Z. Xu, Phys. Rev. C 79, 051901(R) (2009). 25E. S. Weibel, Phys. Rev. Lett. 2, 83 (1959). 26B. D. Fried, Phys. Fluids 2, 337 (1959). 27E. A. Startsev and R. C. Davidson, Phys. Plasmas 10, 4829 (2003). 28M. Medvedev and A. Loeb, Astrophys. J. 526, 697 (1999). 29L. O. Silva, R. A. Fonseca, J. W. Tonge, J. M. Dawson, W. B. Mori, and M. V. Medvedev, Astrophys. J. 596, L121 (2003). 30P. H. Yoon and R. C. Davidson, Phys. Rev. A 35, 2718 (1987). 31P. H. Yoon, Phys. Fluids B 1, 1336 (1989). 32U. Schaefer-Rollfs, I. Lerche, and R. Schlickeiser, Phys. Plasmas 13, 012107 (2006). 33R. Schlickeiser and P. K. Shukla, Astrophys. J. 599, L57 (2003). 34F. Califano, T. Cecchi, and C. Chiuderi, Phys. Plasmas 9, 451 (2002). 35M. Marklund and P. K. Shukla, Phys. Plasmas 13, 094503 (2006). 36M. Mahdavi and H. Khanzadeh, Phys. Plasmas 20, 052114 (2013). 37E. G. D. Cohen, Physica A 305, 19 (2002). 38A. F. Alexandrov, L. S. Bogdankevich, and A. A. Rukhadze, Principles of Plasma Electrodynamics (Springer, New York, 1984), pp. 162–165. 39E. M. Lifshitz and L. P. Pitaevskii, Physical Kinetics , translated by J. B. Sykes and R. N. Franklin (Pergamon Press, Oxford, 1981). 40X. Q. Li, Collapsing Dynamics of Plasmons (Chinese Science and Technology Press, Beijing, 2004). 41J. M. Liu, J. S. De Groot, J. P. Matte, T. W. Johnston, and R. P. Drake, Phys. Rev. Lett. 72, 2717 (1994). 42F. Califano, F. Pegoraro, and S. V. Bulanov, Phys. Rev. Lett. 84, 3602 (2000). 43M. A. Ruderman and P. G. Sutherland, Astrophys. J. 196, 51 (1975). 44Y. Kazimura, F. Califano, J. Sakai, T. Neubert, F. Pegoraro, and S. Bulanov, J. Phys. Soc. Jpn. 67, 1079 (1998). 45C. S. Wu, P. Yoon, L. F. Ziebell, C. Chang, and H. K. Wong, J. Geophys. Res. 97-A1 , 141, doi:10.1029/91JA02373 (1992). 46A. Lui, P. Yoon, and C. Chang, J. Geophys. Res. 98-A1 , 153, doi:10.1029/92JA02034 (1993).102119-4 H.-B. Qiu and S.-B. Liu Phys. Plasmas 20, 102119 (2013) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.205.114.91 On: Tue, 20 Jan 2015 12:38:27

1.2435812.pdf

Coupling of spin-transfer torque to microwave magnetic field: A micromagnetic modal analysis L. Torres, L. Lopez-Diaz, E. Martinez, G. Finocchio, M. Carpentieri, and B. Azzerboni Citation: Journal of Applied Physics 101, 053914 (2007); doi: 10.1063/1.2435812 View online: http://dx.doi.org/10.1063/1.2435812 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/101/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Proposal for a standard problem for micromagnetic simulations including spin-transfer torque J. Appl. Phys. 105, 113914 (2009); 10.1063/1.3126702 Spin-current pulse induced switching of vortex chirality in permalloy Cu Co nanopillars Appl. Phys. Lett. 91, 022501 (2007); 10.1063/1.2756109 Effect of the classical ampere field in micromagnetic computations of spin polarized current-driven magnetization processes J. Appl. Phys. 97, 10C713 (2005); 10.1063/1.1853291 Spin-polarized current-driven switching in permalloy nanostructures J. Appl. Phys. 97, 10E302 (2005); 10.1063/1.1847292 Spin-transfer effects in nanoscale magnetic tunnel junctions Appl. Phys. Lett. 85, 1205 (2004); 10.1063/1.1781769 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.12.30.104 On: Wed, 26 Mar 2014 15:46:59Coupling of spin-transfer torque to microwave magnetic field: A micromagnetic modal analysis L. T orres,a/H20850L. Lopez-Diaz, and E. Martinez Departamento de Fisica Aplicada, Universidad de Salamanca, Plaza de la Merced s/n 37008 Salamanca, Spain G. Finocchio, M. Carpentieri, and B. Azzerboni Dipartimento di Fisica della Materia e Tecnologie Fisiche Avanzate, University of Messina,Salita Sperone 31, 98166 Messina, Italy /H20849Received 3 May 2006; accepted 12 December 2006; published online 13 March 2007 /H20850 Micromagnetic computational spectral mapping technique is applied to analyze the magnetic oscillation modes excited by either a microwave circularly polarized magnetic field or a spinpolarized current flowing through Permalloy /H20849Py/H20850spin valves. A complete study has been carried out on multilayers Py /H2084910 nm /H20850/Cu /H208495n m /H20850/Py /H208492.5 nm /H20850with rectangular cross section /H2084960/H1100320 nm 2/H20850. The magnetic normal modes obtained agree with recent analytical spin wave models in patterned nanostructures. When both excitations, microwave field and spin polarized current, are applied at thesame time a complex coupling process is observed. The detailed micromagnetic analysis of thecoupling shows three different stages: /H20849i/H20850The initial stage in which the magnetic normal modes are dominant, /H20849ii/H20850an intermediate stage showing an incoherent behavior, and /H20849iii/H20850the final stage where a persistent domain wall oscillation is present. Micromagnetic spectral mapping technique is shownto be an adequate tool for describing the temporal evolution of the magnetization spatial patterns innanostructures. © 2007 American Institute of Physics ./H20851DOI: 10.1063/1.2435812 /H20852 I. INTRODUCTION Spin polarized current /H20849SPC /H20850driven magnetization dy- namics in pillar nanostructures has been investigated inten-sively in the last years. 1–15The main interest of the research- ers lies on the possibility of using the features of the SPC-driven magnetization dynamics for improving the technicalcharacteristics of the magnetic random access memories,such as, switching time or energy losses. 3–7On the other hand, SPC-driven magnetization dynamics also presents os-cillatory states depending on the current and field applied.This fact is being used for producing tunable microwavenano-oscillators which, as it has been recently demonstratedby the experimenters, can be synchronized. 8,9 In this context, understanding in detail the way in which these nanostructures behave at microwave frequencies andunder the action of SPC torques is a crucial point. Someprimary questions arise at once: Are the nanostructures al-ways behaving as uniformly magnetized particles? If not, arethere any normal modes of the system which are alwayspresent? Or perhaps incoherent behavior is the most impor-tant one leading the dynamics? Since the nanosecond time-domain experimental tech- nique with nanometer spatial resolution is not available at themoment, micromagnetic computations are an adequate tech-nique in order to show the internal aspects of the nanostruc-ture dynamics in the nanosecond regime. 10–15 As pointed out by some recent studies,16,17a complete micromagnetic analysis of the modes excited in nanostruc-tures is a formidable computational task which requires anoptimized micromagnetic and mathematical processing code.In this work we will use our own three-dimensional /H208493D/H20850 micromagnetic code which has been developed over the lastten years. 18–20This code allows us to compute the magneto- static field, along the entire nanostructure. This code alsoallows us to compute the Ampere classical field produced bythe current. This can be done for any pillar shape, by meansof a tensor specifically developed to treat eddy current prob-lems in micromagnetics. 21 Our micromagnetic technique can be applied to any shape and material. In previous works it has been used in avariety of nanopillars of Permalloy /H20849Py/H20850and Cobalt with different sizes and circular, elliptical, or rectangular crosssections. 15,20,22Complete information about the dynamics in- duced by the coupling of low amplitude microwave magneticfields and SPC can be predicted. In order to show in detail the inner magnetization dynamics in a single case, we willnot present a systematic study in this paper, instead, we willfocus on the Py nanostructure with a rectangular shape andthe typical size used in the experiments. Simulations per-formed in different kinds of nanopillars provided similarqualitative behavior, but the rectangular shape was finallychosen in order to test our technique against recent analyticalspin wave models in rectangular magnetic nano-elements. 23–25 Regarding the organization of the paper, in Sec. II the main features of the model and the spectral processing pro-cedures are described. Section III presents the results ob-tained when /H20849i/H20850just microwave field is applied and /H20849ii/H20850when the SPC torque is also present. Finally Sec. IV includes asummary and our conclusions about the usefulness of thismicromagnetic technique to analyze complex magnetizationdynamics. a/H20850Electronic email: luis@usal.esJOURNAL OF APPLIED PHYSICS 101, 053914 /H208492007 /H20850 0021-8979/2007/101 /H208495/H20850/053914/8/$23.00 © 2007 American Institute of Physics 101 , 053914-1 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.12.30.104 On: Wed, 26 Mar 2014 15:46:59II. MODEL AND PROCESSING TECHNIQUES A. Micromagnetic model Our study will be focused on a pillar nanostructure simi- lar to the one sketched in Fig. 1. There are three rectangular layers /H2084960 nm /H1100320 nm /H20850of Py, Copper, and Py with a thick- ness of Py 10 nm/Cu 5 nm/Py 2.5 nm. The thick Py layer will be denoted as the “fixed layer” and its magnetization asP.While the thin Py layer will be called the “free layer” and its magnetization as M. Previous computations, 20with just an external applied field, at room temperature showed thatboth are parallel /H20849Mparallel to P/H20850and antiparallel states were metastable equilibrium states. The fixed layer washarder to switch, as was expected, due to its higher volume.No anisotropy is included as it corresponds to the Py nano-structures, the anisotropy present is simply induced by theshape due to the magnetostatic energy. A complete description of our micromagnetic model can be found in previous works. 15,18–22We present, just as a brief summary to help the reader, a 3D time domain finite differ-ences technique to be used. The sample is divided in pris-matic cells and the magnetization is assumed uniform in eachcell. The Landau-Lifshitz-Gilbert /H20849LLG /H20850equation is com- puted by adequate numerical techniques. 15,18–22In order to take into account the effect of SPC, an additional first-principles term as deduced by Slonczewski, 1is added to the torque in the Gilbert equation. This term gives rise to twonew terms in the equivalent LLG equation d M dt=−/H9253/H11032M/H11003Heff−/H9251/H9253/H11032 MsM/H11003/H20849M/H11003Heff/H20850 −2/H9262BJ /H208491+/H92512/H20850dMs3eg/H20849M,P/H20850M/H11003/H20849M/H11003P/H20850 +2/H9262B/H9251J /H208491+/H92512/H20850dMs2eg/H20849M,P/H20850M/H11003P, /H208491/H20850 where Heffis the effective field, /H9253/H11032=/H9253//H208491+/H92512/H20850,/H9253is the elec- tron gyromagnetic ratio, Msis the saturation magnetization, and/H9251is the dimensionless damping parameter. Regarding the SPC term, /H9262Bis the Bohr’s magnetron, Jis the current density, dis the thickness of the free layer, and eis the electron charge /H20849positive scalar /H20850. The scalar function g/H20849M,P/H20850 was deduced by Slonczewski1g/H20849M,P/H20850=/H20851−4+ /H208491+/H9257/H208503/H208493+MP/MS2/H20850/4/H92573/2/H20852−1, /H208492/H20850 with/H9257the polarizing factor, which for Py is /H9257=0.3.14In the computations presented in this work, the effective field in-cludes the following contributions: Heff=Hexch+Hext+HM+HAmp+HMC, /H208493/H20850 such that Hexch,Hext, and HMare the standard micromag- netic contributions from the exchange, external, and demag-netizing fields. H AmpandHMCare the Ampere field and the magnetostatic coupling /H20849MC /H20850field due to the fixed layer on the free layer. In this paper our aim is to study the dynamics of the free layer so that the MC has been calculated in 3D assuming thefixed layer was always saturated. Once the MC is calculatedin 3D, our computations of the free layer were performed intwo-dimensional /H208492D/H20850with prismatic cells of 1.25 /H110031.25 /H110032.5 nm size. This size is small enough to ensure enough precision in the solution according to the standard criterionin micromagnetics which consist on using cell sizes smallerthan the exchange length /H208495.7 nm for Py 14/H20850. We have per- formed computational tests with cubic cells of 2.5 nm andthe qualitative results were the same although some of thespatial modes would not have been detected, as will beshown in the results. Smaller cell sizes would not have aphysical sense from the micromagnetic point of view be-cause the number of ions in the cell would not be enough todefine the magnetization as a continuum function, and thenan atomic model would be required. Regarding the time interval used for solving the LLG equation, /H9004t=16.5 fs was used throughout the paper and damping constant was /H9251=0.02. Smaller time steps were tested producing exactly the same solutions. All the results ofthe paper have been obtained without taking into account theeffect of the temperature /H20849i.e., T=0 K /H20850. A detailed analysis of the modes at room temperature would require the solution of the stochastic Langevin-LLG equation and averaging overa sufficient number of realizations. Although this is possiblewith our code, 19the computational effort would increase by the number of realizations /H20849typically around one hundred /H20850. We have performed some tests at T=300 K and qualitatively the results were similar to the ones shown here. No newnormal modes were excited by the temperature, probably be-cause the strength of the random room temperature thermalfield is small compared to the other fields involved in ourexperiment. Another possible experiment would have been toset our sample in an equilibrium state and then try to excitethe magnetic normal modes just by the temperature, with noother external fields present, but this would be the matter ofa different work and then we could not study the effect ofSPC on these modes. B. Micromagnetic spectral mapping technique In order to describe in detail the spatial dependence of the magnetic normal modes and compare them with analyti-cal models, several authors have used quite similartechniques 16,17which could be called “micromagnetic spec- tral mapping techniques.” The temporal evolution of themagnetization for each cell M/H20849r k,t/H20850is stored for each time FIG. 1. /H20849Color online /H20850Sketch of the analyzed pillar nanostructure.053914-2 T orres et al. J. Appl. Phys. 101 , 053914 /H208492007 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.12.30.104 On: Wed, 26 Mar 2014 15:46:59step /H20849/H9004t=16.5 fs /H20850. In this way we have a record of the tem- poral evolution of the magnetization at each point and then it is possible to construct the local power spectrum for eachcell. For the xcomponent, by using a discrete Fourier trans- form, it would be S x/H20849rk,f/H20850=/H20879/H20858 jMx/H20849rk,tj/H20850ei2/H9266ftj/H208792. /H208494/H20850 In order to obtain a general overview of the magnetization dynamics in the frequency domain, the averaged power spec-trum is obtained by summing over every cell. /H20849Note that is different from applying Fourier transform to the averagedmagnetization. /H20850 Sx/H20849f/H20850=/H20858 kSx/H20849rk,f/H20850. /H208495/H20850 This averaged power spectrum will present peaks corre- sponding to magnetic modes which arise, due to the fact thatsome cells are oscillating at the same frequency with mea-surable amplitude. The spatial configuration of the mode atthe frequency of one peak /H20849f peak /H20850is achieved by plotting /H20849density plot, 3D /H20850the power S/H20849rk,fpeak /H20850of each cell at the frequency of the peak. In this way it is possible to check if it is a normal linear mode /H20849spin waves /H20850of the system which can even be described by analytical models or they are justmagnetic nonlinear modes compatible with a maximum inthe averaged power spectrum. 23 III. RESULTS AND DISCUSSION A. Excitation by a microwave circularly polarized field We will begin by analyzing the modes excited by a cir- cularly polarized microwave external field hy=h0sin/H208492/H9266ft/H20850, hz=h0cos /H208492/H9266ft/H20850with h0=1mTandfand f=15.6 GHz to- gether with an external static field HDC=220 mT /H11003uxap- plied both to an initial parallel state, no SPC current ispresent at the moment. These values have been chosen be-cause of the coupling with small angle SPC oscillation as itwill become apparent further in the discussion. The temporalevolution of the average magnetization ycomponent is shown in Fig. 2. Following the procedure previously de- scribed, the averaged power spectrum S/H9252/H20849f/H20850has been calcu- lated for the three components /H20849/H9252=x,y, and z/H20850as can be observed in Fig. 3. Then, following again the spectral map- ping technique we have plotted the intensity of the spectralpower in each cell at the frequencies labeled in Fig. 3. In thisway we can see the spatial distribution of the mode at each peak frequency. The results are displayed in Fig. 4. The variation of Myis around 3 %/H20849Fig.2/H20850, therefore, in this case we are clearly dealing with a perturbation on theuniformly magnetized parallel state along x. This is the ad- equate scenario for defining magnetic normal modes, i.e.,spin waves 23 M/H20849r,t/H20850=Msux+/H9255, /H208496/H20850 where /H9255is a low amplitude spin wave propagating along x direction which solutions could be expressed as /H9255y=/H92550sin/H9275tcoskxxcoskyycoskzz; /H9255z=/H92550cos/H9275tcoskxxcoskyycoskzz, /H208497/H20850 being kx,y,zthe wave vector components and /H92550the spin wave amplitude. If we consider an infinite medium where just ex-change is taken into account then the well-known quadraticdispersion relation /H9275/H20849k/H20850=/H9253/H208492A//H92620Ms/H20850k2is obtained.23On the other hand, and concerning the spin valve nanostructures studied here, it has been observed experimentally that thespin wave spectrum is quantized 24–27and some recent works have proposed analytical expressions for the dispersion rela-tion based on different approximations. 24,25Although our mi- cromagnetic model directly provides the spatial distributionof the magnetization at the frequency of the main modes, it isa good test for both analytical and micromagnetic models tocompare their results. Since the free layer is thin enough /H208492.5 nm /H20850and we have carried out 2D computations, in our case we are concerned only about the quantization of k xandky; k2=kx2+ky2/H20849kz=0/H20850. Considering the dispersion relation for an infinite thin film17but including also the effect of external, magnetostatic and MC fields, we would have /H20873/H9275/H20849k/H20850 /H9253/H208742 =/H20873Hext+HM+HMC+Ms/H208491−Nk/H20850+2A /H92620Msk2/H20874 /H11003/H20873Hext+HM+HMC+MsNkky2 k2+2A /H92620Msk2/H20874, /H208498/H20850 where HMand HMCare calculated averaging spatially the micromagnetic fields of Eq. /H208493/H20850,Ais the exchange constant, and Nk=/H208491−e−kd/H20850kdis a demagnetization factor which de- pends on kand film thickness d17. In order to compare with FIG. 2. Temporal evolution of Myin the free layer. HDC=200 mT. hy =h0sin/H208492/H9266ft/H20850;hz=h0cos /H208492/H9266ft/H20850.h0=1 mT. f=15.6 GHz. FIG. 3. Averaged power spectrum for x/H20849dotted /H20850,y/H20849dashed /H20850,a n d zcompo- nents /H20849solid line /H20850under the same conditions of Fig. 2.053914-3 T orres et al. J. Appl. Phys. 101 , 053914 /H208492007 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.12.30.104 On: Wed, 26 Mar 2014 15:46:59the modes obtained in Fig. 4, a first step would be to choose quantized wave vectors knx=nx/H9266/Lxand kny=ny/H9266/Lywith nxy=0,1, 2... being Lx,Ly, and Lzthe dimensions of the free layer . The labels of the modes of Fig. 3correspond to the indexes /H20849ny,nx/H20850, note that the power spectrum is squared /H20851Eq./H208494/H20850/H20852so that the indexes are the number of half-wavelengths which fit in the nanostructure. The comparison of the micro-magnetic modes to Eq. /H208498/H20850is presented in Fig. 5, good agree- ment is found taking into account that there is no free pa-rameter to fit in Eq. /H208498/H20850. Although the modes observed in Fig. 4seem to be basi- cally unpinned ones /H20849nonzero amplitude at the ends /H20850, a more sophisticated model, following Guslienko 24,25which includes dipolar pinning at the ends has been also tested. For rectan-gular magnetic elements, the quantization on xdepends on the fulfillment of an integral quantization condition 25and it would not be considered here, but the quantization of ky leads to an effective sample width Ly,effwhich results to be slightly larger than the real Lyproviding some space for the magnetization to reduce the magnetic pole density.24,25Ac- cording to these calculations kny,eff=ny/H9266/Ly,eff;Ly,eff=Ly/H20849t//H20849t−2 /H20850/H20850; t=2/H9266//H20851/H20849Lz/Ly/H20850/H208491−2l n /H20849Lz/Ly/H20850/H20850/H20852. /H208499/H20850 As it can be noted in Fig. 5this model leads to very similar frequencies in the dispersion relation for our samples, in fact,the reduced size of our sample implies that the exchangeinteraction prevails over the magnetostatic one and then thequantization of Eq. /H208499/H20850provides wavevectors very close to the naive election of k n,yx=nx,y/H9266/Lx. It is also to be noted that the strong magnetostatic effect due to the reduced thick-ness of the free layer is taken into account in Eq. /H208498/H20850by means of N kand also the magnetostatic field along xis com- puted in the contribution of HMand HMC, this is why an unpinned naive quantized election of the wavevectors agreeswith the micromagnetic computation. B. Coupling of spin-transfer torque to microwave circularly polarized field After checking that our technique allows us to obtain the magnetic normal modes of the free layer in our spin valve,the coupling of the previously analyzed field to SPC smallangle oscillations is addressed in this section. The problem isinteresting also from the technological point of view sincethis coupling leads to an asymmetry of the magnetizationoscillation and it can be used as a strategy to decrease theswitching time in fast switching processes. 28 Figure 6displays the Myvariation due to the coupling of both microwave and SPC small angle oscillation. The spin FIG. 4. /H20849Color online /H208503D plot of the spatial modes obtained at the frequen- cies labeled in Fig. 3. FIG. 5. Dispersion relation obtained by the micromagnetic model /H20849stars /H20850 compared to the analytical expressions given by Eq. /H208498/H20850/H20849squares /H20850and Eq. /H208499/H20850/H20849triangles /H20850.053914-4 T orres et al. J. Appl. Phys. 101 , 053914 /H208492007 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.12.30.104 On: Wed, 26 Mar 2014 15:46:59valve is under the action of an external field HDCx=220 mT and a spin current J=−0.4 /H110031012A/m2, the microwave field is the same reported in previous section. If just the SPCcurrent is applied the variation of the magnetization is thatshown in the top inset of Fig. 6, the amplitude of the oscil- lation is extremely small /H2084910 −7/H20850and the main frequency of its spectrum is f=15.6 GHz which is the frequency chosen for the microwave coupling signal. The normal modes excitedare the same than for the microwave case /H20849see Figs. 3–5/H20850. The main mode is also /H208490,1 /H20850, which is excited at the same frequency than in the microwave case. The behavior of the small oscillation has been checked to 130 ns, neither differ-ent modes nor chaotic behavior is present. Detailed informa-tion of the dynamic stability diagram, when just SPC is ap-plied to this kind of nanostructure, can be found in Ref. 29. In the bottom inset we have included the temporal evolutionof the magnetization when just microwave signal is applied,in this case /H20849see also Fig. 2/H20850the amplitude is around 10 −2. The temporal evolution of the coupling can be divided in three stages: /H20849i/H20850Initial stage /H208490–30 ns /H20850,/H20849ii/H20850intermediate stage /H2084930–90 ns /H20850, and /H20849iii/H20850final stage /H2084990–130 ns /H20850. The specific temporal intervals chosen are the ones in which micromag-netic spectral mapping technique has been applied. In factthe system continuously changes from small oscillation to amore incoherent behavior which becomes dominant from 50ns and there is no a quantitative clear criterion to separate thestages. Computational trials have been performed consider-ing the initial interval extended to 40 ns and the same resultshave been found for either the spatial distribution or the fre-quency of the modes. The following three intervals are ana-lyzed in detail: /H20849i/H20850Initial stage /H208490–30 ns /H20850: In the initial nanoseconds of the computation the magnetic normal modes are still present.In fact we think that the coupling takes place through themode /H208490,1 /H20850, which is by far the most intense of all the modes found by the micromagnetic spectral technique when either the microwave signal or the SPC small angle oscillation arepresent separately. Following the same procedure of previoussection of the paper, the spectrum is obtained and the sameexcited modes are presented /H20849Fig. 4compared to Fig. 7/H20850.I t can be observed how the spatial distribution of the modes isslightly deformed but still the normal structure is clearly de-tected. The /H208490,1 /H20850mode consists of small oscillations of the spins on the right and left boundaries of the nanostructure which turn together in phase as shown in the arrow plot ofFig.8/H20849top /H20850. This mode is excited at 15.6 GHz when SPC is injected alone but when a microwave signal of that samefrequency is also applied, then the angle of the small oscil-lation is augmented and the dynamics deviate from the smallangle oscillation framework going into a different dynamicstate /H20849intermediate stage /H20850. FIG. 6. Temporal evolution of Myin the free layer external DC and micro- wave fields like in Fig. 2andJ=−0.4 /H110031012A/m2. Top inset presents the temporal evolution without microwave field in the first 5 ns. Bottom inset isthe temporal evolution without current like in Fig. 2. FIG. 7. /H20849Color online /H208503D plot of the spatial modes obtained at the frequen- cies labeled in Fig. 3under the same conditions of Fig. 6.053914-5 T orres et al. J. Appl. Phys. 101 , 053914 /H208492007 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.12.30.104 On: Wed, 26 Mar 2014 15:46:59/H20849ii/H20850Intermediate stage /H2084930–90 ns /H20850: As shown in Fig. 6 the dynamics of the average magnetization seems quite inco-herent. This can be checked looking at the arrow plots of themagnetization which presents very nonuniform patterns asshown in Fig. 8/H20849second from the top /H20850. If the temporal evo- lution of the spatial distribution of the magnetization is ob-served in detail, it results in being a complex dynamics be-tween two metastable states which consists on the formationof a 360° oscillating domain wall either on the right or theleft part of the nanostructure. Figure 9shows the frequency spectrum in this stage, where the incoherence becomes ap-parent. Although there are some peaks clearly defined in thespectrum, no well defined spatial modes are obtained at thosefrequencies by means of the micromagnetic mapping tech-nique. It is to be noted also that the frequency of the mainpeak of Fig. 9is around 7.4 GHz which is not the frequency of the main normal mode /H208490,1 /H20850. This is indicating that the nanostructure is migrating to a different oscillation dynamic mode that cannot be defined as a perturbation of the uniformmode, i.e., it is no longer a spin wave or a magnetic normalmode as defined by Eq. /H208496/H20850. 23Figure 10shows the variation of the average magnetization Mz/H20849My/H20850. The magnetization is oscillating between the two metastable states referred previ- ously, until the magnetization falls in one of them /H20849final stage /H20850. /H20849iii/H20850Final stage /H2084990–130 ns /H20850: In this stage, the frequency spectrum is again coherent presenting well defined peaks asshown in Fig. 11. After applying the micromagnetic mapping at the labeled frequencies, the spatial modes of Fig. 12are obtained. The oscillation modes detected do not present thetypical standing wavelike structure of the normal modes.This was to be expected since the magnetization pattern isnonuniform and cannot be treated as an spin wave. Analyz-ing the temporal evolution of the magnetization pattern inthis case, it is observed that the magnetization dynamics in-volves the oscillation of a 180° domain wall in the right partof the nanostructure as shown in Fig. 8/H20849two bottom figures /H20850. The 3D plot of the average magnetization temporal evolutionis presented in Fig. 13.M xis always positive indicating that the wall oscillation takes place in the right part while Myand Mztake positive and negative values revealing that the wall oscillates out of plane pointing up and down as shown in thearrow plots of Fig. 8/H20849two bottom figures /H20850. The main har- monic of the wall oscillation lies at 7.3 GHz and it corre-spond to the mode labeled “1” in Fig. 12which shows maxi- mum amplitude in the wall spatial location. The modes“2–4” are the higher order harmonics at 14.6, 21.9, and 29.2GHz, their spatial distribution is complicated and it could berelated to the dipolar spatial spreading of the wall oscillation.The mode “5” is found at 58 GHz like the normal mode/H208491,3 /H20850of Figs. 4and7. It has been included to show that some reminiscences of high frequency normal modes can still be found even in this final stage. The position of the wall FIG. 8. /H20849Color online /H20850Arrow plots of the spatial configuration of the mag- netization in the free layer. From top to bottom: snapshots of initial stage,intermediate stage and two in the final stage. FIG. 9. Averaged power spectrum for x/H20849dotted /H20850,y/H20849dashed /H20850, and zcompo- nents /H20849solid line /H20850in the intermediate stage of the coupling. FIG. 10. Mz/H20849My/H20850during the intermediate stage. FIG. 11. Averaged power spectrum for x/H20849dotted /H20850,y/H20849dashed /H20850,a n d zcom- ponents /H20849solid line /H20850in the final stage of the coupling.053914-6 T orres et al. J. Appl. Phys. 101 , 053914 /H208492007 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.12.30.104 On: Wed, 26 Mar 2014 15:46:59oscillation at the right part is related to the characteristics of the external actions exerted on the magnetization, for ex-ample, we have tested to change the sense of the microwavecircular polarization and in that case the wall oscillation iscarried out in the left part of the nanostructure. It is to benoted also that, from the standard micromagnetic point ofview, 18the size of the sample would be too small to allocatea 360° domain wall. However, it has to be taken into account that with the spin torque term in the LLG equation, the stan-dard micromagnetic way of thinking is not correct and someconfigurations arise which would not be expected 13,14with- out the SPC term. IV. SUMMARY AND CONCLUSIONS The coupling of SPC oscillations to microwave circu- larly polarized magnetic field has been analyzed in detailboth in the time and frequency domain. /H20849i/H20850The spatial distribution of the normal modes excited by the microwave field and the mixed modes excited by thecombined action with the spin torque is reported. The mixedmodes are excited at the same frequencies than the normalmodes and their spatial distribution is just slightly perturbedin the first stage of the coupling. /H20849ii/H20850The frequency of the normal modes obtained by the micromagnetic spectral mapping technique is compared torecent analytical models. This is the first checking of thistechnique against these analytical models and represents animportant step in the validation of both methods. /H20849iii/H20850In the last stage of the coupling, final persistent wall oscillation is found. This oscillation would give rise to anincrease of the magnetoresistance signal in the multilayerwhich could be used to build microwave resonators. In thisstage, it is shown how the usefulness of our technique goesbeyond the normal modes and it can be a way of understand-ing persistent oscillation modes which present a coherentfrequency spectrum. 1J. C. Slonczewski, J. Magn. Magn. Mater. 159,L 1 /H208491996 /H20850. 2L. Berger, Phys. Rev. B 54, 9353 /H208491996 /H20850. 3J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers, and D. C. Ralph, Phys. Rev. Lett. 84, 3149 /H208492000 /H20850. 4S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J. Schoe- lkopf, R. A. Buhrman, and D. C. Ralph, Nature 425, 380 /H208492003 /H20850. 5S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, M. Rinkoski, C. Perez, R. A. Buhrman, and D. C. Ralph, Phys. Rev. Lett. 93, 036601 /H208492004 /H20850. 6I. N. Krivorotov, N. C. Emley, A. G. F. Garcia, J. C. Sankey, S. I. Kiselev, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett. 93, 166603 /H208492004 /H20850. 7I. N. Krivorotov, N. C. Emley, J. C. Sankey, S. I. Kiselev, D. C. Ralph, and R. A. Buhrman, Science 307, 228 /H208492005 /H20850. 8S. Kaka, M. R. Pufall, W. H. Rippard, T. J. Silva, S. E. Russek, and J. A. Katine, Nature 437, 389 /H208492005 /H20850. 9F. B. Mancoff, N. D. Rizzo, B. N. Engel, and S. Therani, Nature 437,3 9 3 /H208492005 /H20850. 10J. Miltat, G. Albuquerque, A. Thiaville, and C. Vouille, J. Appl. Phys. 89, 6982 /H208492001 /H20850. 11Z. Li and S. Zhang, Phys. Rev. B 68, 024404 /H208492003 /H20850. 12H. Xi and Z. Lin, Phys. Rev. B 70, 092403 /H208492004 /H20850. 13K.-J. Lee, A. Deac, O. Rendon, J.-P. Nozieres, and B. Dieny, Nat. Mater. FIG. 12. /H20849Color online /H208503D plot of the spatial modes obtained at the fre- quencies labeled in Fig. 11. FIG. 13. 3D evolution of the magnetization during the final stage.053914-7 T orres et al. J. Appl. Phys. 101 , 053914 /H208492007 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.12.30.104 On: Wed, 26 Mar 2014 15:46:593,8 7 7 /H208492004 /H20850. 14D. Berkov and N. Gorn, Phys. Rev. B 71, 052403 /H208492005 /H20850. 15L. Torres, L. Lopez-Diaz, E. Martinez, M. Carpentieri, and G. Finocchio, J. Magn. Magn. Mater. 286, 381 /H208492005 /H20850. 16M. Grimsditch, G. K. Leaf, H. G. Kaper, D. Karpeev, and R. E. Camley, Phys. Rev. B 69, 174428 /H208492004 /H20850. 17R. D. McMichael and M. D. Stiles, J. Appl. Phys. 97, 10J901 /H208491999 /H20850 18L. Torres, L. Lopez-Diaz, and J. Iñiguez, Appl. Phys. Lett. 73, 3766 /H208491998 /H20850. 19L. López-Díaz, L. Torres, and E. Moro, Phys. Rev. B 65, 224406 /H208492002 /H20850. 20E. Martinez, L. Torres, L. Lopez-Diaz, M. Carpentieri, and G. Finocchio, J. Appl. Phys. 97, 10E302 /H208491999 /H20850 21E. Martínez, L. Torres, and L. López-Díaz, IEEE Trans. Magn. 40, 3240 /H208492004 /H20850. 22M. Carpentieri, G. Finocchio, B. Azzerboni, L. Torres, L. Lopez-Diaz, E.Martinez, J. Appl. Phys. 97, 10C713 /H208491999 /H20850 23C. Kittel, Introduction to Solid State Physics /H20849Wiley, New York, 1986 /H20850. 24K. Y. Guslienko, S. O. Demokritov, B. Hillebrands, and A. N. Slavin, Phys. Rev. B 66, 132402 /H208492002 /H20850. 25K. Y. Guslienko, R. W. Chantrell, and A. N. Slavin, Phys. Rev. B 68, 024422 /H208492003 /H20850. 26G. Gubbiotti, G. Carlotti, T. T. Okuno, T. Shinjo, F. Nizzoli, and R. Zivieri, Phys. Rev. B 68, 184409 /H208492003 /H20850. 27L. Giovannini, F. Montoncello, F. Nizzoli, G. Gubbiotti, G. Carlotti, T. Okuno, T. Shinjo, and M. Grimsditch, Phys. Rev. B 70, 172404 /H208492004 /H20850. 28G. Finocchio, I. N. Krivorotov, M. Carpentieri, G. Consolo, B. Azzerboni, L. Torres, L. Lopez-Diaz, and E. Martinez, J. Appl. Phys. 99, 08G507 /H208492006 /H20850. 29M. Carpentieri, L. Torres, B. Azzerboni, G. Finocchio, G. Consolo, and L. Lopez-Diaz, J. Magn. Magn. Mater. /H20849in press /H20850.053914-8 T orres et al. J. Appl. Phys. 101 , 053914 /H208492007 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.12.30.104 On: Wed, 26 Mar 2014 15:46:59

1.3316794.pdf

Self-assembly of amphiphilic peanut-shaped nanoparticles Stephen Whitelam1,a/H20850and Stefan A. F . Bon2 1Molecular Foundry, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, California 94720, USA 2Department of Chemistry, University of Warwick, Coventry CV4 7AL, United Kingdom /H20849Received 4 January 2010; accepted 22 January 2010; published online 16 February 2010 /H20850 We use computer simulation to investigate the self-assembly of Janus-like amphiphilic peanut-shaped nanoparticles, finding phases of clusters, bilayers, and micelles in accord with ideasof packing familiar from the study of molecular surfactants. However, packing arguments do notexplain the hierarchical self-assembly dynamics that we observe, nor the coexistence of bilayers andfaceted polyhedra. This coexistence suggests that experimental realizations of our model canachieve multipotent assembly of either of two competing ordered structures. © 2010 American Institute of Physics ./H20851doi:10.1063/1.3316794 /H20852 Components are said to “self-assemble” when they orga- nize to form stable patterns or aggregates without externaldirection. Self-assembly is driven by interactions as differentas weak covalent bonds and capillary forces, involves com-ponents ranging in size from angstroms to centimeters, andoccurs both in inorganic settings and in living organisms. 1–9 Mimicry of the self-assembly seen in the natural world promises the development of new, functional materials pat-terned on the nanometer scale. 10,11In pursuit of this goal, we take inspiration from the self-assembly of molecular surfac-tants to investigate using computer simulation the behaviorof a simple model of their colloidal counterparts. Molecularsurfactants, comprising chemically linked hydrophobic andhydrophilic groups, are of central importance in biology andindustry, able to form a plethora of phases in water or mix-tures of water and oily liquids. These phases includemicelles, 12bilayers and vesicles, as well as numerous bicon- tinous phases13that serve as internal cellular packaging14and are the bane of many a plumber. Here we ask: What might self-assemble in aqueous so- lution from amphiphilic, peanut-shaped colloidal nanopar-ticles? Such particles can now be prepared in large quanti-ties, starting from spherical, crosslinked polystyrene “seed”particles of radius /H1101150 nm. Mixing these seeds with styrene initiates monomer-polymer phase separation and creates par-ticles with a peanutlike shape characterized by two fusedspherical lobes of controllable relative size. Additional treat-ment of the seed particle renders peanuts amphiphilic, withone lobe hydrophobic and the other hydrophilic; 15–17the re- sulting body can be regarded as a generalized Janusparticle. 18/H20849We note that colloidal silica dumbbells can be synthesized by other routes,19and that micrometer-scale pea- nuts show potential as Pickering stabilizers of oil-in-wateremulsions. 20/H20850In an attempt to answer our posed question we have constructed a model of interacting peanuts whose mini-mal character is motivated by the insight into self-assemblyafforded by similarly simple model systems. 21–39Our model can be evolved with computational efficiency sufficient toallow observation of collective, thermally-driven dynamics on timescales of seconds. Its construction rests upon twoassumptions: first, that peanuts in aqueous solution experi-ence a thermodynamic driving force that causes hydrophobiclobes to attract each other; and second, that functionalizationof peanuts’ hydrophilic lobes renders them chemically pas-sive. We assume solution conditions to be such that electro-static interactions between peanuts mediate only short-ranged repulsions. Model geometry is shown in the Appendix /H20849Fig.5/H20850. Each peanut consists of a spherical hydrophobic lobe of radius R 0 fused with a spherical hydrophilic lobe of radius R1. The centers of the two lobes are separated by a distance /H9254, which we quantify in dimensionless form via the parameter /H9280 /H11013/H9254//H20849R0+R1/H20850. We fixed R0throughout /H20849we consider R0to be approximately 50 nm /H20850, and investigated the behavior of the model as we varied /H9280andR1/R0. We required /H20841R1−R0/H20841//H20849R1 +R0/H20850/H11021/H9280/H113491; the lower inequality stipulates that one lobe may not be completely buried in the other, while the upper inequality requires lobes to be in contact. We define the ori-entation vector S iof peanut ito be the unit vector whose origin is the center of the hydrophobic lobe and which pointsdiametrically away from the center of the hydrophilic lobe.Neighboring peanuts interact via hard-core excluded volumeinteractions—nothing may approach closer than R 0/H20849respec- tively R1/H20850to the center of each hydrophobic /H20849resp. hydro- philic /H20850lobe—and via a short-ranged, pairwise interaction modeling the attraction between solvated nanoscalehydrophobes 40/H20849we do not consider solvent explicitly in our model /H20850. We assume the hydrophobic interaction to be attenu- ated on a scale of approximately 5 nm, and impose an attrac-tive interaction between the hydrophobic lobes of peanuts i andjof the form U ij=/H9280b/H9008/H20849rc−rij/H20850A/H20849/H9258ij/H20850A/H20849/H9258ji/H20850/H20849L/H9251/H20849rˆij/H20850−L/H9251/H20849rˆc/H20850/H20850. /H208491/H20850 Here/H9280bis a binding energy; xˆ/H11013x/R0−1 is a shifted and scaled distance; rijis the distance between the centers of the hydrophobic lobes of iand j, respectively C iand C j;rc /H110132.5R0is a cutoff length; and L/H9251/H20849x/H20850/H110134/H20849x−2/H9251−x−/H9251/H20850is a gen- eralized Lennard–Jones function. We take /H9251=15 to ensure ana/H20850Electronic mail: swhitelam@lbl.gov.THE JOURNAL OF CHEMICAL PHYSICS 132, 074901 /H208492010 /H20850 0021-9606/2010/132 /H208497/H20850/074901/8/$30.00 © 2010 American Institute of Physics 132 , 074901-1attraction of sufficiently short range. In simulations we var- ied the attractive binding energy /H9280bbetween limits of 4 kBT and 8 kBT: corresponding forms of the radial component of the hydrophobic interaction potential are plotted in Fig. 5/H20849a/H20850. In experiment, variation of the strength of the hydrophobicdriving force may be achieved by variation of temperature orsolvent composition. The factors Ain Eq. /H208491/H20850parameterize an angular modu- lation of the attractive interaction, with /H9258ijbeing the angle between Siand the vector pointing from C ito C j. This modu- lation enforces the tendency of two hydrophobes of girthexceeding 1 nm to maximize their surface-to-surfacecontact. 40Its form is given and derived in the Appendix. We performed simulations of collections of peanuts of given geometry, defined by their values of R1/R0and/H9280, for a range of values of the hydrophobic attraction strength /H9280b.W e used 1000 peanuts in a periodically-replicated cubic simula-tion box of side 64 R 0, corresponding to fixed mole fraction of colloid /H20849this choice reflects the low concentrations that we intend to use in experiments; in exploratory simulations weverified that the phases seen here also form at several otherchoices of /H20849low/H20850concentration /H20850. The bulkiest /H20849R 1/R0=2,/H9280 =1/H20850and most compact /H20849R1/R0=0.2, /H9280=0.8 /H20850peanuts we considered occupied volume fractions of about 14% and 1.5%, respectively /H20849see Appendix /H20850. Starting from configura- tions in which peanuts were randomly mixed and oriented,we evolved each system according to the version of thevirtual-move Monte Carlo algorithm 41described in the Ap - pendix of Ref. 42. This algorithm is designed to mimic an overdamped dynamics by making collective moves of par-ticles according to the potential energy gradients, or forces,they experience; such collective moves are neglected bystandard single-particle Monte Carlo algorithms. In brief,one particle is selected and subjected to a translation or arotation. If changes in pairwise potential energies betweenthat particle and its neighbors are favorable then the chosenparticle moves independently; if not, neighbors are recruitediteratively and experience a collective displacement or trans-lation. The acceptance rate for each move is chosen to pre- serve detailed balance and to reflect, in an approximate fash-ion, the /H20849short-ranged /H20850hydrodynamic drag suffered by the collective body. We scaled collective translation acceptancerates by the reciprocal of an approximate hydrodynamic ra-dius of the moving body, appropriate for its instantaneousdirection of motion 41/H20849identical for forward and reverse moves, as required to satisfy detailed balance /H20850, and scaled acceptance rates for rotations by the cube of a similar factor,accounting for an aggregate’s instantaneous axis of rotation/H20849identical for forward and reverse rotations /H20850. While these damping factors are approximate, control of aggregate diffu-sion constant scalings– and particularly enforcement of thenotion that an aspherical body does not translate equally rap-idly in all directions—provides one advantage of this methodover conventional integration of overdamped equations ofmotion. We drew translation magnitudes from a uniform dis-tribution with maximum 0.3 R 0,55and drew rotation angles from a uniform distribution with maximum 13.6° /H20849rotations were performed about a randomly chosen axis through themidpoint of the line joining the centers of hydrophobic andhydrophilic lobes /H20850. This ability to make large trial transla- tions and rotations of individual particles in the face of at-tractions and repulsions that vary rapidly with distance andangle is not shared by straightforward numerical integrationschemes. Based on the diffusivity in water of a body of ra-dius 50 nm, we estimate that our basic simulation timestep/H20849an average of one attempted virtual move per particle /H20850cor- responds to /H1101110 −6s. Our longest simulations exceeded 107 timesteps, implying that we probe ‘real’ timescales in excess of a second. Our results are summarized by the “kinetic phase dia- gram” of Fig. 1/H20849a/H20850. This diagram identifies those self- assembled products, whether equilibrated or kineticallytrapped, accessible to dynamical simulation starting fromwell-mixed initial conditions. We expect by extension thatsuch products will be accessible to experiments starting fromsimilar conditions. The horizontal and vertical axes of thisdiagram label the quantities R 1/R0and/H9280, respectively; pea- nuts with small /H20849resp. large /H20850hydrophilic lobes are found to the left /H20849resp. right /H20850of the diagram. We identify regions of compact clusters /H20851squares; see Fig. 1/H20849b/H20850/H20852, nonspherical mi- celles /H20849triangles /H20850, spherical micelles /H20851circles; see Fig. 1/H20849e/H20850/H20852 and bilayers /H20851crosses; see Fig. 1/H20849c/H20850/H20852. Our classification scheme and a summary of the computational resources wedeployed are discussed in the Appendix. We find these phases to be localized in peanut shape space largely in ac-cord with a simple estimate of peanuts’ molecular packingparameters 43,44/H20849see Appendix /H20850: we have labeled the diagram with lines of packing parameter equal to 1 /H20849the regime in00.250.50.751 /epsilon1 00.20.40.60.81 1 .52 R1/R0NS micelles clusters S micelles bilayers (a) (c)11 21 3(e) (d) (b) FIG. 1. Kinetic phase diagram identifying products of self-assembly. /H20849a/H20850 Kinetic phase diagram in the space of /H9280/H11013/H9254//H20849R0+R1/H20850andR1/R0identifying the products found following dynamic simulations of peanuts with specifiedgeometries /H20849see text for classification of bilayers, micelles and clusters /H20850. Examples of such products: /H20849b/H20850crystalline cluster at thermodynamic state /H20849R 1/R0,/H9280,/H9280b/H20850=/H208490.5,0.5,4 kBT/H20850/H20849peanuts shown reduced in size /H20850;/H20849c/H20850bilayers at /H20849R1/R0,/H9280,/H9280b/H20850=/H208490.9,0.2,5 kBT/H20850;/H20849d/H20850faceted polyhedron at state /H20849R1/R0,/H9280,/H9280b/H20850=/H208490.96,0.04,4.75 kBT/H20850/H20849peanuts shown reduced in size /H20850; and /H20849e/H20850spherical micelles at /H20849R1/R0,/H9280,/H9280b/H20850=/H208492,0.8,8 kBT/H20850/H20849bottom right shows hydrophilic lobes reduced in size /H20850. Contours of constant packing parameter /H20849values indicated /H20850are dotted lines; the dashed lines indicate the two branches satisfying /H9280=/H20841R1−R0/H20841//H20849R1+R0/H20850. Below these lines, one lobe of the “peanut” is completely buried within the other.074901-2 S. Whitelam and S. A. F . Bon J. Chem. Phys. 132 , 074901 /H208492010 /H20850which one expects bilayers /H20850; 1/2 /H20849the cylindrical micelle re- gime /H20850; and 1/3 /H20849the spherical micelle regime /H20850. The phase most interesting from the materials scientist’s perspective is perhaps the bilayer, because extended two-dimensional structures that do not require templating by asubstrate are attractive candidates for device fabrication. Ourobservation that amphiphilic nanoscale peanuts can form bi-layers, and that bilayer formation is localized to a specificregion of peanut shape space, will facilitate the experimentalsearch for this phase. We also observed bilayers to coexistwith faceted polyhedra composed of peanuts arranged with ahigh degree of local order /H20851see Figs. 1/H20849d/H20850and6/H20849d/H20850/H20852. Like micelles, and unlike extended clusters, polyhedra present hy-drophilic surfaces to solution and are consequently size-limited. We do not know why bilayers and polyhedra coexist:they may represent comparable minima of free energy, orone may embody a particularly stable kinetic trap. Thiscoexistence—which resembles simultaneous micellizationand phase separation 39—will be the focus of future work, but its observation nonetheless suggests that experimental real- izations of this system might achieve multipotent self-assembly, wherein components form coexisting, orderedphases of strikingly different symmetry. Some protein com- plexes appear to achieve such multipotent assembly by ex-ecuting conformational changes. 45 Of the other phases observed, micelles are found abun- dantly for many peanut shapes, especially when the thermo-dynamic driving force for association is large and the forma-tion of ordered structures like bilayers is hindered by kinetictraps. Micelle polydispersity may be controlled by varyingR 1/R0/H20849Fig. 2/H20850, suggesting a route to the synthesis of size- controlled nanometer scale assemblies. In addition, clusterformation is possible when the hydrophilic lobe is verysmall. Peanuts whose shapes lie on the left branch of the line /H9280=/H20841R1−R0/H20841//H20849R1+R0/H20850in Fig. 1/H20849a/H20850are isotropically attractive spheres, and form close-packed, crystalline clusters. Peanuts with small hydrophilic lobes also form aggregates whose in-nermost particles make 12 pairwise contacts, and extendedcrystalline order is possible even when the hydrophilic lobeis moderately sized /H20851see Fig. 1/H20849b/H20850/H20852.While considerations of packing predict the phases we have found, with the exception of faceted polyhedra, theygive little insight into the complex dynamics of assembly wehave observed. We focus on the hierarchical dynamics asso-ciated with peanut state /H20849R 1/R0,/H9280,/H9280b/H20850=/H208490.9,0.2,5 kBT/H20850. First, fluctional micelles appear. These can rearrange in a collec- tive fashion to form a flattish, metastable protobilayer com-posed of about seven close-packed peanuts per face, plusattendant edge particles /H20851see Fig. 3/H20849a/H20850, left, circled /H20852. Such nuclei do not grow immediately, but do so only after anadditional collective rearrangement that sees an abrupt in-crease in the number of bilayerlike particles comprising theircore /H20851see Figs. 3/H20849b/H20850and3/H20849c/H20850/H20852. As the nucleus exceeds a criti- cal size, bilayer growth proceeds readily; the largest bilayerswe observed in systems of 1000 peanuts exceeded 800 par-ticles in size /H20849in exploratory simulations of 6000 peanuts we observed bilayers comprising in excess of 2000 peanuts /H20850. The time of the appearance of the first bilayerlike nucleusexceeding 50 particles in size is long and drawn from a broaddistribution /H20849from 20 independent simulations of /H110112.5 /H1100310 7timesteps we observed the appearance of such struc- tures in 17 cases; of this set, mean nucleation time was 1.2/H1100310 7timesteps, with a standard deviation of 9 /H11003106 timesteps /H20850. We have not explored in detail the dynamics of bilayer formation at other points in peanut shape space, butwe present one example in Fig. 4. Here a bilayer and a mi- celle merge before the former grows at the expense of thelatter. Finally, we note that cluster formation by “peanuts” whose hydrophilic lobes are of vanishing size proceeds via“two-step” crystallization: 46–49first, dense amorphous blobs nucleate from vapor; second, crystalline order nucleates within these blobs /H20849not shown here /H20850. We observed a similar dynamics /H20849not shown /H20850for crystal-forming peanuts /H20851see Fig. 1/H20849b/H20850/H20852possessing hydrophilic lobes of moderate size. It would make for an interesting theoretical study to determine thedynamics of assembly as one proceeds into the regime ofincreasing hydrophilic lobe size: what is the nature of clusterformation as the crystal structure becomes suppressed orthermodynamically disfavored with respect to the dense amorphous phase? Clusters in our simulations generally coa-lesce upon contact, suggesting that in experiment such as-semblies would not be soluble, and would not be of greatpractical interest. We have demonstrated that dynamical simulation of a simple model of nanoscale amphiphilic peanut-shaped col-loids generates phases of clusters, micelles and bilayers inaccord with expectations based on simple ideas of packing.We have also found certain ordered structures and observedcomplex dynamical mechanisms that cannot be so explained.We expect this model and its behavior to be prototypical of aclass of amphiphilic structures now being synthesized inlarge quantities, and therefore to help guide the experimentalsearch for new, functional nanostructured materials. Futuretheoretical work involving this model will focus on furtherquantifying the dynamical pathways observed in this study;determining the effect upon assembly of liquid-liquidinterfaces; 50and studying the design potential of multilobed peanut generalizations.050100150200n(s) 02468 1 0sR1/R0=1.4 R1/R0=1.6 R1/R0=1.8 R1/R0=2 1.4 2 FIG. 2. Micelle polydispersity. Number of aggregates of size s,n/H20849s/H20850, for micelle-forming peanuts satisfying /H9280=1/2 and /H9280b=8kBT. Varying R1/R0, which is achievable synthetically, controls micelle polydispersity. Each linewas obtained using 10 independent simulations evolved for /H110115/H1100310 6 timesteps.074901-3 Peanut-shaped nanoparticle self-assembly J. Chem. Phys. 132 , 074901 /H208492010 /H20850Note: In a recent study, Miller and Cacciuto explored the self-assembly of spherical amphiphilic particles using mo-lecular dynamics simulation. 37Interestingly, despite the dif - ferences in interaction range and particle shape between theirmodel and ours, those authors also observed the coexistenceof bilayers and faceted polyhedra. This implies that multipo-tent assembly of this nature is not dependent upon fine de-tails of particle-particle interactions, but can be understoodon more general grounds. Furthermore, it may be that suchmultipotent assembly has already been seen in experiment:one of the referees of this paper pointed out that surfactantbilayers can form icosahedral structures in salt-free cationic solution. 51One possible origin of these structures is the elec - trostatic force;52it is also conceivable, based on our results and the results of Ref. 37, that their existence might be ex- plained in purely geometrical terms.ACKNOWLEDGMENTS This work was performed at the Molecular Foundry, Lawrence Berkeley National Laboratory, and was supportedby the Director, Office of Science, Office of Basic EnergySciences, of the U.S. Department of Energy under ContractNo. DE-AC02—05CH11231. We thank Sander Pronk for acritical reading of the manuscript, and Andrea Pasqua, LutzMaibaum and Phillip Geissler for useful discussions. APPENDIX: DETAILS OF PEANUT INTERACTION POTENTIAL AND ASSEMBLY CLASSIFICATIONSCHEME 1. Angular modulation of the hydrophobic interaction The factors Ain Eq. /H208491/H20850have the following origin /H20849refer to Fig. 5for geometry /H20850. We assume the strength of the hy- (a) (b)( c) 0255075100nb 0 50 100 150nnbn 0200400600n, n b 3.5×1061×1071.5×107 t0255050 025 FIG. 3. Bilayer self-assembly from one simulation at thermodynamic state /H20849R1/R0,/H9280,/H9280b/H20850=/H208490.9,0.2,5 kBT/H20850./H20849a/H20850Time-ordered configuration snapshots. Left /H20849t/H110152.1/H11003106/H20850: a micelle establishes a bilayerlike configuration, but fails to grow and eventually evaporates. Bilayerlike particles /H20849see Appendix /H20850are shown in pink. Center /H20849t/H110156.8/H11003106/H20850: a similar structure has undergone a collective rearrangement and attained a critical size, and as a result grows readily. Right /H20849t/H110151.6/H11003107/H20850: it is eventually joined by a second bilayer; these grow until they possess about 800 of the 1000 peanuts in the simulation box. /H20849b/H20850As a function of number of simulation timesteps twe plot number of particles nand number of bilayerlike particles nbcomprising the structure seen in the center panel /H20849which occasionally merges with the second bilayer seen in the right panel /H20850. Inset: early-time behavior. These quantities are plotted in /H20849c/H20850with tas a parameter. (a)(b) 0200400600800n, n b 05 ×1061×107 tnbn FIG. 4. Bilayer self-assembly from one simulation at thermodynamic state /H20849R1/R0,/H9280,/H9280b/H20850=/H208490.8,0.25,4.75 kBT/H20850./H20849a/H20850Configurations at t=/H208494,4.5,9 /H20850/H11003106/H20849left to right /H20850: a bilayer and a micelle merge, with the former eventually consuming the latter. Bilayer-like particles are shown in pink. Only aggregates possessin g more than 5 constituents are shown. /H20849b/H20850Size nand number of bilayer-like particles nbcomprising the most bilayerlike aggregate in the simulation box, as a function of t.074901-4 S. Whitelam and S. A. F . Bon J. Chem. Phys. 132 , 074901 /H208492010 /H20850drophobic attraction on the lengthscales we consider to be proportional to the contacting surface area between two hy-drophobic bodies. 40We define two surfaces to be “in con - tact” if they lie no further than 5 nm apart, reflecting ourassessment of the likely range of the hydrophobic force. Iftwo spheres of radius R 0/H1101550 nm are placed in contact, the plane half-angle subtended by the limit of the “contacting” surface area at the center of either sphere is /H9274 /H11013arccos /H208491–5//H208492/H1100350/H20850/H20850/H1101518.1° /H20849peanuts are azimuthally symmetric about their orientation vector, and so we consider only plane angles, as sketched in Fig. 5/H20850. Hence we argue that the hydrophobic lobe of peanut ipresents its maximum possible surface area to a similar body B if the angle betweenS iand the vector pointing from C ito the center of B is not within an angle /H9274of the line joining C ito the nearest surface obstruction. We next define the angle /H9258maxto be the angle between Si and the line joining C ito the largest obstruction provided by each hydrophobic lobe’s hydrophilic partner. This angle isgiven by geometrical considerations. The angle between S i and the line joining C ito the intersection of hydrophobic and hydrophilic lobes is /H9258intersec =/H9266−arccos /H20849/H5129/R0/H20850, where /H5129 /H11013/H9254/2−/H20849R12−R02/H20850//H208492/H9254/H20850/H20849note that /H5129may be negative /H20850. The angle between Siand the line joining C ito the greatest di- ameter presented by its hydrophilic partner is /H9258girth=/H9266 −arctan /H20849R1//H9254/H20850. We take /H9258max=min /H20849/H9258intersec ,/H9258girth/H20850,unless the largest diameter of the hydrophilic lobe is buried in its hy- drophobic counterpart /H20849corresponding to R1/H11021R0and/H5129/H11022/H9254/H20850; in this case we take /H9258max=/H9258intersec . To construct the function A/H20849/H9258/H20850we argue as follows. Let /H9258be the angle between Siand the line joining C ito the center of a similar body B. For /H9258=0 the peanut ipresents its maxi- mum possible hydrophobic surface area to B. Conversely, thegreatest obstruction of the hydrophobic lobe caused by thehydrophilic lobe occurs when /H9258=/H9266, i.e., when the body B approaches the peanut from its hydrophilic side. This ob-struction can be total, if the hydrophilic lobe is sufficiently large, or less than total, if the hydrophilic lobe is smalland/or well-buried within its hydrophobic partner. We takethe smallest possible “signal” presented by the hydrophobicsurface of peanut ito the body B to be A min=max /H208490,1− /H20849/H9266 −/H9258max/H20850//H9274/H20850. Finally, we assume for simplicity that the signal interpolates linearly with /H9258between its largest and smallest values, i.e. A/H20849/H9258/H20850/H11008−/H9258if B lies in the penumbra cast by the hydrophilic lobe /H20849when/H9258exceeds /H9258max−/H9274/H20850. These arguments imply the following piecewise linear form for the angularmodulation function in Eq. /H208491/H20850, A/H20849 /H9258/H20850=/H209021 /H20849/H9258/H11021/H9278/H20850 max/H20873Amin,1 −1 2/H9274/H20849/H9258−/H9278/H20850/H20874/H20849/H9258/H11350/H9278/H20850,/H20903/H20849A1/H20850 where /H9278/H11013/H9258max−/H9274. In Fig. 5/H20849b/H20850we plot A/H20849/H9258/H20850for two peanut geometries. We note that for the nanoscale particles we have studied here, identification of the length and angular scalesover which the hydrophobic effect operates reveals thatmultibody forces between particles are neither required norwarranted. We note that our estimate of the range of the hydropho- bic attraction is rough /H20849see, e.g., Ref. 53/H20850. If our assessment of the range of the hydrophobic force is inaccurate, then ourresults will describe nanoparticles whose size is not the 50nm assumed here. We have not performed systematic simu-lations using different ranges of attraction, but exploratorysimulations of peanuts whose hydrophobic lobes’ radial at-tractions have a square well form of range equal to the loberadius can form at least one phase not accessible to particlespossessing the shorter range of attraction studied in the text.This phase consists of micellelike blobs /H20849of many particle diameters in girth /H20850possessing dense, liquidlike interiors.00.20.40.60.81A(θ) R1/R 0=0.8 R1/R 0=0.1 θ0 π/2 πθmaxθ/prime max 1) 2)−8−404U(r) 22.12.25 2 .52 .75 r/R 0/epsilon1B=4kBT /epsilon1B=8kBT(a) δ R1R0 /primeθmax θ2ψBθmax θmax +ψ umbraB θpenumbra(b)r FIG. 5. Peanut-peanut interaction potential. /H20849a/H20850Peanut geometry with hydrophobic /H20849resp. hydrophilic /H20850lobe colored blue /H20849resp. yellow /H20850, together with the radial component of the blue-blue interaction potential. /H20849b/H20850Angular modulation function, A/H20849/H9258/H20850, designed to account for the degree of contact made by hydrophobic lobes of adjacent peanuts /H20849see Eq. /H20849A1/H20850/H20850./H9258is the plane angle between the peanut orientation vector and the vector pointing from the center of the hydrophobic lobe /H20849blue /H20850to the center of a similar body, B /H20849shown reduced in size /H20850. We plot A/H20849/H9258/H20850for peanuts of geometry /H208491/H20850/H20849R1/R0,/H9280/H20850=/H208490.8,0.85 /H20850and /H208492/H20850/H20849R1/R0,/H9280/H20850 =/H208490.1,0.85 /H20850. “Penumbra” and “umbra” identify the regions for which, respectively, partial and total occlusion of the blue lobe is caused by the yellow lobe. See text for details.074901-5 Peanut-shaped nanoparticle self-assembly J. Chem. Phys. 132 , 074901 /H208492010 /H20850These blobs can grow through their mutual coalescence and subsequent restructuring. We expect therefore that smallnanoparticles /H20849of girth, say /H110115n m /H20850can form phases that larger nanoparticles /H20849of size, say /H1101150 nm /H20850cannot. 2. Peanut volume The volume presented to solvent by two fused spheres of radii R0and R1whose centers are /H9254apart /H20849/H20841R1−R0/H20841/H11021/H9254 /H11349/H20849R1+R0/H20850/H20850is given from geometrical considerations by Vpeanut =4 3/H9266/H20849R03+R13/H20850−/H9266/H20885 /H5129R0 dx/H20849R02−x2/H20850 −/H9266/H20885 /H9254−/H5129R1 dx/H20849R12−x2/H20850 =/H208491+/H9280/H208502 12/H9280/H9266/H9018/H208493/H90042+/H9280/H208492−/H9280/H20850/H90182/H20850. /H20849A2/H20850 Here /H9280/H11013/H9254//H20849R0+R1/H20850,/H5129/H11013/H9254/2−/H20849R12−R02/H20850//H208492/H9254/H20850,/H9018/H11013R0+R1 and/H9004/H11013R0−R1. This formula may be used to calculate the fraction of the simulation box occupied by peanuts of arbi-trary geometry. 3. Computational details To obtain Fig. 1/H20849a/H20850we performed simulations for times sufficient to observe what we believe to be steady-state be-havior at each location in peanut shape space. In the regimeR 1/R0/H113501.2 /H20849the micelle-forming regime /H20850this is straightfor- ward: for each point in the /H20849R1/R0,/H9280/H20850plane we performed one simulation at each of /H9280b=/H208495,6,7,8 /H20850kBTfor /H110115/H11003106 timesteps /H2084960 CPU hours per simulation /H20850. In the cluster- forming regime of R1/R0/H113490.4 we performed 5 independent simulations at each labeled point for each value of /H9280b =/H208494,4.5,5,5.5 /H20850kBT, also for /H110115/H11003106timesteps each /H2084960 CPU hours per simulation /H20850. In the intermediate regime 0.4 /H11021R1/R0/H110211.2 we found some bilayers to form only after very long times. We therefore performed 5 independentsimulations of /H110115/H1100310 6timesteps each /H2084960 CPU hours per simulation /H20850at each point for each value of /H9280b =/H208494,4.5,4.75 ,5,5.5 /H20850kBT, and in addition performed three further independent simulations of /H110112.5/H11003107timesteps /H20849300 CPU hours per simulation /H20850at each point for each of the five values of /H9280b. Along the line R1/R0=1 we performed three additional simulations of /H110112.5/H11003107timesteps for each value of /H9280b=/H208496,6.5 /H20850kBT. We can of course not rule out the likelihood of nucleation and growth of structures on times- cales longer than those we probed, nor rule out the appear-ance of other interesting ordered structures at values of /H9280b that we did not consider. To obtain Fig. 2we performed 10 additional simulations of /H110115/H11003106timesteps at each of the four thermodynamic states shown /H2084960 CPU hours per simu- lation /H20850. To obtain nucleation statistics for bilayers similar to those seen in Fig. 3/H20850we performed 20 additional independent simulations of /H110112.5/H11003107timesteps /H20849300 CPU hours per simulation /H20850at thermodynamic state /H20849R1/R0,/H9280,/H9280b/H20850 =/H208490.9,0.2,5 kBT/H20850.4. Classification of self-assembled structures We classified the products of peanut self-assembly by analyzing the final configuration of each simulation as fol-lows. We define contacting peanuts to be those possessing apairwise energy of interaction of − k BTor less. We define an ‘aggregate’ to be a contiguous set of contacting particles. Weidentified locations in peanut shape space to contain “clus-ters” /H20849compact structures with some degree of close packing /H20850 if any particle possessed 12 contacts. We identified micellesby calculating for each aggregate the number M /H11013N a−1/H20858i=1NaSi·/H20849rCM−ri/H20850. Here Nais the number of peanuts comprising the aggregate, rCMis the center of mass of the aggregate, and riis the position of aggregate constituent i/H20849all coordinates are corrected for periodic boundaries /H20850. If any lo- cation in shape space contained at least three aggregates ofsize three or greater that scored M/H113500.9 then we considered that state to contain spherical micelles, and marked that statewith a circle. Any location with three or more aggregates ofsize three scoring 0.5 /H11349M/H110210.9 is considered to contain nonspherical micelles, and was marked with a triangle. Bi-layer order was identified on the basis of the order parameterB/H110132/H20849N a/H20849Na−1/H20850/H20850−1/H20858/H20849ij/H20850/H20849Si·Sj/H208502, where the sum runs over all particle pairs in the aggregate /H20849B=1 3for collections of randomly-oriented peanuts /H20850. Configurations possessing an aggregate of /H2084925,50,200 /H20850constituents or more having a score ofB/H113500.4 were marked with an /H20849orange, red, purple /H20850cross. In Figs. 3and4, we show for certain aggregates the number of their constituents possessing local bilayerlike order, nb.T o compute this quantity we formed the dot product of the ori-entation vector of a given particle and each of its neighbors’orientation vectors. We define a “bilayerlike” particle as onepossessing three or more neighbors with which it hasorientation-orientation dot product /H110220.8, and 1 or more neighbors with which it has dot product /H11021−0.8. This order parameter is not useful for examining clusters, whose inter-nal particles associate in an orientationally disordered wayand can test “false positive” for local bilayer ordering. How-ever, it does successfully distinguish between regions of lo-cal order and disorder in the bilayer-forming regime of pea-nut shape space. For most conditions considered, it ispossible to find amorphous structures answering to none ofthe above descriptions. This is most often true when the hy-drophobic driving force is very strong, and kinetically frus-trated aggregates form. We have ignored these structures,focusing on the ordered products we have described. Whilethe order parameters and the threshold numbers we have ap-plied are arbitrary, and the extent of each regime varies as thestructural classification criteria are varied, we consider thetrends we have identified to be representative of the nature ofthe self assembly we observed. Some additional structuresare shown in Fig. 6. We used VMD 54to render simulation configurations in this paper. 5. Packing parameter Considerations of packing have been used with success to predict the phases formed by molecular surfactants.43,44 We can define a packing parameter for an aggregate as Pagg/H11013Vagg//H20849aaggLagg/H20850, where Vaggis aggregate volume, aagg074901-6 S. Whitelam and S. A. F . Bon J. Chem. Phys. 132 , 074901 /H208492010 /H20850its surface area, and Lagga characteristic length. For a cylin- der of radius Rand length /H5129, for example, Vcylinder =/H9266R2/H5129, Lcylinder =R, and acylinder =2/H9266R/H5129, giving Pcylinder =1/2. Analo- gously, Psphere=1/3 and Pbilayer =1. If we now assume such shapes to be composed of amphiphilic components, then,loosely, a aggcorresponds to the total area presented to sol- vent by hydrophilic groups, while VaggandLaggare the vol- umes and characteristic lengths of the hydrophobic units. Ifone assumes that P aggcan be estimated from the geometry of asingle amphiphile /H20849see, however, Ref. 44/H20850, then values of Psingle /H11013Pnear 1, 1/2 or 1/3 suggest self-assembly of bilay- ers, cylindrical micelles or spherical micelles, respectively.We can make a rough estimate of Pfor an individual peanut. If we assume that hydrophobic and hydrophilic lobes touch but do not interpenetrate, then V= 4 3/H9266R03,L=2R0and a =/H9266R12. We thus have P=2 3/H20849R0/R1/H208502. To this approximation, therefore, contours of constant Pare vertical lines on Fig. 1/H20849b/H20850. Furthermore, P=1⇒R1/R0=/H208812/3/H110150.82 and P =1/3⇒R1/R0=/H208812/H110151.41. From Fig. 1/H20849b/H20850, we indeed find bilayers and spherical micelles in the corresponding regimes.A more refined estimate based on peanut geometry alonemay be made by accounting for the volume of the hydropho-bic lobe buried within the hydrophilic lobe. We have V=4 3/H9266R03−/H9266/H20885 /H5129R0 dx/H20849R02−x2/H20850, /H20849A3/H20850 L=R0+/H5129, and a=/H9266R12/H20849valid when the hydrophilic lobe’s greatest diameter is not buried within the hydrophobic lobe;if this is nottrue, then a= /H9266/H20849R02−/H51292/H20850/H20850. Contours of P=1, P=1/2 and P=1/3 to this approximation are plotted in Fig. 1/H20849a/H20850. 1J. 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Glotzer, Nano Lett. 3, 1341 /H208492003 /H20850. 24A. Wilber, J. Doye, A. Louis, E. Noya, M. Miller, and P. Wong, J. Chem. Phys. 127, 085106 /H208492007 /H20850. 25Z. Zhang, A. Keys, T. Chen, and S. Glotzer, Langmuir 21, 11547 /H208492005 /H20850. 26M. Hagan and D. Chandler, Biophys. J. 91,4 2 /H208492006 /H20850. 27G. Villar, A. Wilber, A. Williamson, P. Thiara, J. Doye, A. Louis, M. Jochum, A. Lewis, and E. Levy, Phys. Rev. Lett. 102, 118106 /H208492009 /H20850. 28T. Ouldridge, I. Johnston, A. Louis, and J. Doye, J. Chem. Phys. 130, 065101 /H208492009 /H20850. 29F. Sciortino, E. Bianchi, J. Douglas, and P. Tartaglia, J. Chem. Phys. 126, 194903 /H208492007 /H20850. 30E. Bianchi, P. Tartaglia, E. La Nave, and F. Sciortino, J. Phys. Chem. B 111, 11765 /H208492007 /H20850. 31E. Bianchi, J. Largo, P. Tartaglia, E. Zaccarelli, and F. Sciortino, Phys. Rev. Lett. 97, 168301 /H208492006 /H20850. 32C. De Michele, S. Gabrielli, P. Tartaglia, and F. Sciortino, J. Phys. Chem. B110, 8064 /H208492006 /H20850. 33T. Chen, Z. Zhang, and S. Glotzer, Proc. Natl. Acad. Sci. U.S.A. 104, 717 /H208492007 /H20850. 34L. Hong, A. Cacciuto, E. Luijten, and S. Granick, Langmuir 24,6 2 1 /H208492008 /H20850. 35O. Elrad and M. Hagan, Nano Lett. 8, 3850 /H208492008 /H20850. 36C. Iacovella and S. Glotzer, Nano Lett. 9, 1206 /H208492009 /H20850. 37W. Miller and A. Cacciuto, Phys. Rev. E 80, 021404 /H208492009 /H20850. 38B. Mladek, G. Kahl, and C. Likos, Phys. Rev. Lett. 100, 028301 /H208492008 /H20850. 39F. Sciortino, A. Giacometti, and G. Pastore, Phys. Rev. Lett. 103, 237801 /H208492009 /H20850. 40D. Chandler, Nature /H20849London /H20850437, 640 /H208492005 /H20850. 41S. Whitelam and P. Geissler, J. Chem. Phys. 127, 154101 /H208492007 /H20850. 42S. Whitelam, E. Feng, M. Hagan, and P. Geissler, Soft Matter 5, 1251 /H208492009 /H20850. 43J. Israelachvili, D. Mitchell, and B. Ninham, J. Chem. Soc., Faraday Trans. II 72, 1525 /H208491976 /H20850. 44R. Nagarajan, Langmuir 18,3 1 /H208492002 /H20850. 45S. Whitelam, C. Rogers, A. Pasqua, C. Paavola, J. Trent, and P. Geissler, Nano Lett. 9, 292 /H208492008 /H20850. 46P. Wolde and D. Frenkel, Science 277, 1975 /H208491997 /H20850. 47B. Chen, H. Kim, S. Keasler, and R. Nellas, J. Phys. Chem. B 112,4 0 6 7 /H208492008 /H20850. 48J. van Meel, A. Page, R. Sear, and D. Frenkel, J. Chem. Phys. 129, 204505 /H208492008 /H20850. 49A. Fortini, E. Sanz, and M. Dijkstra, Phys. Rev. E 78, 041402 /H208492008 /H20850. 50D. Cheung and S. Bon, Phys. Rev. Lett. 102, 066103 /H208492009 /H20850. (a) (b)( c) (d) FIG. 6. Additional configurations. /H20849a/H20850Micelles of various morphologies found at thermodynamic state /H20849R1/R0,/H9280,/H9280b/H20850=/H208491,0.8,5.5 kBT/H20850;/H20849b/H20850coexisting bilayers and micelles at state /H20849R1/R0,/H9280,/H9280b/H20850=/H208490.8,0.5,5 kBT/H20850;/H20849c/H20850disordered wormlike micelle at state /H20849R1/R0,/H9280,/H9280b/H20850=/H208490.6,0.8,5 kBT/H20850; and /H20849d/H20850coexisting polygon and bilayer at state /H20849R1/R0,/H9280,/H9280b/H20850=/H208490.96,0.04,4.75 kBT/H20850.074901-7 Peanut-shaped nanoparticle self-assembly J. Chem. Phys. 132 , 074901 /H208492010 /H2085051M. Dubois, B. Demé, T. Gulik-Krzywicki, J. Dedieu, C. Vautrin, S. Désert, E. Perez, and T. Zemb, Nature /H20849London /H20850411, 672 /H208492001 /H20850. 52G. Vernizzi and O. de La Cruz, Proc. Natl. Acad. Sci. U.S.A. 104, 18382 /H208492007 /H20850. 53Q. Lin, E. Meyer, M. Tadmor, J. Israelachvili, and T. Kuhl, Langmuir 21, 251 /H208492005 /H20850.54W. Humphrey, A. Dalke, and K. Schulten, J. Mol. Graph. 14,3 3 /H208491996 /H20850. 55Our simulation results were qualitatively unchanged upon varying the maximum displacement from 0.05 R0to 0.6 R0. Additionally, we also ob- served the phases reported here to form when we used only single-particle moves /H20849although some dynamical pathways involving the colli- sions of large aggregates were suppressed /H20850.074901-8 S. Whitelam and S. A. F . Bon J. Chem. Phys. 132 , 074901 /H208492010 /H20850The Journal of Chemical Physics is copyrighted by the American Institute of Physics (AIP). Redistribution of journal material is subject to the AIP online journal license and/or AIP copyright. For more information, see http://ojps.aip.org/jcpo/jcpcr/jsp

1.355565.pdf

Relaxation in magnetic continua V. L. Sobolev, I. Klik, C. R. Chang, and H. L. Huang Citation: Journal of Applied Physics 75, 5794 (1994); doi: 10.1063/1.355565 View online: http://dx.doi.org/10.1063/1.355565 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/75/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Magnetic diffusion and current relaxation AIP Conf. Proc. 1558, 2183 (2013); 10.1063/1.4825971 Temperature relaxation in a magnetized plasma Phys. Plasmas 20, 102518 (2013); 10.1063/1.4827206 Relaxation of the magnetization in magnetic molecules J. Appl. Phys. 99, 08D101 (2006); 10.1063/1.2162329 Relaxation in magnetic nanostructures J. Appl. Phys. 97, 10A702 (2005); 10.1063/1.1847854 Nuclear Magnetic Relaxation in Magnetite J. Appl. Phys. 36, 1020 (1965); 10.1063/1.1714081 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 137.30.242.61 On: Sun, 13 Jul 2014 17:16:11Relaxation in magnetic continua V. L. Soboiev, I. Klik, C. R. Chang, and H. L. Huang Department of Physics, National Taiwan lJniversi& Taipei, Taiwan, Republic of China Within a recently developed axiomatic formalism for construction of stochastic field e.quations of motion a new form of the equation for damped dynamics of magnetization in ferromagnets is proposed. It takes into account spatially inhomogeneous dissipation due to exchange interactions and replaces the customary dissipation constant (due to relativistic effects) by a dissipation operator, This operator is constructed here by comparison of the resultant phenomenological spin wave spectra with results of microscopic calculations. Dynamic description of relaxation of the modulus of magnetization is discussed. A consistent description of the damped dynamics of the magne.tization vector M(r,tj in magnetic continua is a prob- lem of considerable theoretical and practical interest, yet, to date., possibly due to the difficulty of solving the resultant equations, relatively little effort has been devoted to a gen- eralization of the original Landau-Lifschitz-Gilbert’*2 (LLGj theory which is distinguished by the presence of a single dissipation constant E. In a recent significant development,’ however, Bar’yakhtar (Ref. 3 provides a sum- mary of his theory) argued that the observed discrepancy between the value of E deduced experimentally from domain-wall drag and from ferromagnetic resonance (C is here much smaller) is due to spatially inhomogeneous dissi- pative. processes. He derived an e.quation of motion for mag- netization in which dissipative strength depends not only on the vector M proper, but also on its spatial derivatives and then recovered the exchange approximation expression for spin wave damping which cannot be obtained from the LLG theory. Dissipative forces, however, are invariably ass0ciate.d with randomly fluctuating thermal fields whose combined action drives the system in question towards the stationary state of thermal equilibrium. The existence of this stationary state, neglected by Bar’yakhtar, can only be included in the phenomenological theory within the framework of stochastic differential equations (SDE, see, e.g. Ref. 5). To every SDE one may then construct a (Stratonovich) Fokker-Planck op- erator and one demands that its stationary state is thermal equilibrium. We have recently derived” such a phenomenological SDE for damped magnetization dynamics in continua using a real time modification of the celebrated Caldeird-Leggett formalism.7 In this stochastic field theory the dissipation con- stant is replaced by a dissipation operator constructed in such a way as to satisfy a generalized fluctuation-dissipation (FD) theorem which guarantees that thermal equilibrium is the stationary state of the corresponding Fokker-Planck equa- tion. It is easy to write down a phenomenological equation of motion which has a given spatial symmetry, but the dissipd- tive operator is not a priori known and must be reconstructed a posteriori by comparison with microscopic theory and ex- perimental data on spin wave dispersion. Here we carry out this program for the special case of an easy axis ferromagnet and then also discuss the more complicated easy plane model of high degeneracy. In Ref. 6 we proposed that the damped dynamics of magnetization be described by the equation M,= yoMx[-Nnz-A?;,,M,], (1) where, for simplicity, we omitted the stochastic (noise) terms irrelevant to the present analysis of damped dynamics. In this equation M,=dM/dt, HM= S%/sM (.% is the Hamiltonian), ‘ya is the gyromagnetic ratio, and A:,, is the above mentioned dissipation operator which replaces here the familiar Gil- bert’s dissipation constant eg. In general A& may be an operator in both the magne.tization components Mi and the spatial variable r. In our previous study6 we used the simple form Aelll=irg rC- YJ$ where A is the Laplacian and ?,rK is a newly introduced dissipation constant associated with propa- gation of inhomogeneities through the medium. By construc- tion the magnitude of magnetization remains conserved. This assumption is not altogether justified and we shall return to it in our concluding remarks. Equation (lj, however, is but a special case of a more general class of equations, to wit M,= yoMx [ -H,-&fJ, (2) where b,, is a symmetric tensor (a 3X3 matrix) and the components of the vector f are polynomials in M”’ of de- sired spatial symmetry. It can be shown@ that the stochastic version of Eq. (2j describes evolution towards thermal equi- librium, however, in the underdamped limit only. The gen- eral form of this equation, valid for any dissipation strength remains unknown. We shall assume here that f=M and tind a scalar opera- tor A,, such that the damped system described by Eq. (1) not only goes to thermal equilibrium (which must be the station- ary value of the Fokker-Planck operator associated with the stochastic equations of motion) but also yields the correct dispersion relation for damped spin waves in a uniaxial fer- romagnet. Let us first consider the simplest microscopic model of a ferromagnet with uniaxial anisotropy. The Hamiltonian may be written in the following form: ST= -$ C .J(R,,,)(f$.S,)-k KC (@))2 - I,m 1 -gfiu,Ho*G s,, 5794 J. Appl, Phys. 75 (lo), 15 May 1994 0021-8979/94/75(10)/5794/3/$6.00 Q 1994 American Institute of Physics [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 137.30.242.61 On: Sun, 13 Jul 2014 17:16:11whe.re JQI?,,) is an exchange integral, R is the anisotropy constant, g is the g-factor, prl is the Bohr magneton, and the sum is taken over all lattice sites. The quantization procedure for this Hamiltonian is described, for example, in Ref. 9. Within this model, in the case of the so-called easy axis type of magnetic anisotropy (i.e., if in the absence of external magnetic field magnetization is parallel to the easy axis) the expression for the spin wave spectrum has the simple form E,,.(k) =&+Dk”, (4) where k=lkj is the wave vector modulus. Further, Erj=EA +tEll with E,=SR= yflA (~e=g,u~ and fi=l) and EM= r&r, being the anisotropy and Zeeman energy terms, respectively. The exchange stiffness constant is D=S.TOa’ where S is the spin of an ion, a is lattice constant, and J(O)-J(k)=Jo(ak)’ in the long wavelength approximation uk91. It is common practice to consider also the limiting exchange appro‘ximation in which one sets E,--+O and E,-+O so that the spin wave spectrum (4) takes on the form e&k)-Dk’. Spin wave dampinglO may then be represented Xi (5) where Bc =S.To for brevity. Thus, in the long wavelength limit the k dependence of the spin wave spectrum and damp- ing should be y,,(kj~k” and E,,J’k)iJk2, respectively. These results are of primary importance to test the correct choice of the relaxation term in the phenomenological equations for damped dynamics of magnetization. Let us again recall that we also demand that the stochastic form of these equations” leads to the state of thermal equilibrium. In the case of an easy axis ferromagnet, for example, our Eq. (3j, the spin wave damping is determined by a process of four magnon interaction. It may be described by the Hamiltonian (see, e.g., Refs. 9 and ‘11) Hint= C ~(S,2;3,4,r:clc,c,A(k,+k2-k3-k;t) 1,2,3,4 “I- KC. (6) in which i=k, ) the summation is carried out over the wave vectors of individual magnons, and 1 9(1,2;3,4)= -- i f D(k,~k,+k,.~)+E,z, 2SN *L i7) The resultant expression for spin wave damping within this model (assuming k--+1)) has the simple form JQ” -- Yzw’ ,#” a: * (8) Dipole-dipole, magnetoelastic, and other interactions are ne- glected. The problem now is to find such a tensorial operator A’&] and vector f in Eq. (2j such that the resultant phenomeno- logical equations of motion are compatible with Eqs. (4), (5), and (8) derived microscopically. For simplicity we consider here Eq. (J,) only, i.e., we assume that f=M and that the dissipation operator is a scalar quantity. The phenomenologi- cal counterpart of Eqs. (4) and (5) is then obtained if one sets A&=M-‘+K&= K1y,&c,-/c~a2A), (9) where M = IM(r,t)I = const. The dimensionless operator K,,,, combines the relativistic and exchange contributions to dis- sipation and yields #‘h’(k)=(Eo+Dk2)(/c SW r + K a2k2) ex .Y iw where E, = E, + E, , for the dissipative part of the spin wave spectrum. Comparing the phenomenological expression (lo), in which, in the exchange approximation, we must set K, = 0, with the microscopic result (5) we identify K~“-(2T)-s- 3T 7 (6)’ ln’[ &lkjf] Beyond the exchange approximation, with EA#O, a com- parison of Eqs. (10) and Eq. (3) yields The foregoing results show that for an easy axis ferro- magnet it is possible to reconcile the demands of micro- scopic and phenomenological, thermodynamic theories. So far, however, we have not succeeded in carrying out this program for the highly degenerate case of an easy plane ferromagnet.4 In this case, namely, the dissipative operator and the vector f cannot be chosen to be isotropic, as was conveniently done here, but their choice must reflect the easy plane symmetry. This is a somewhat intractable problem since the theory then contains two nontrivial quantities to be determine.d ad hoc, the vsctor f and the dissipation operator A&,. We have little doubt, however, that a consistent theory of a uniaxial (both easy plane and easy axis) ferromagnet is feasible within the proposed formalism of Eq. (2) which should, in principle, also allow to develop phenomenological models of even more complicated dissipative couplings.’ In deriving Eqs. (1) and (2) we assumed that IM(r,t)l=llf =const. This assumtion allowed us to introduce a conjugate pair of action angle variables’ and to write down canonical equations of motion. Problems arise if the above condition is violated and the rapid relaxation of M towards its equilib- rium value is incorporated into a theory which then contains three dynamical variables (the components Mij, rendering the canonical formalism inapplicable. By the study of Callen” the relaxation of the magnetization modulus is re- lated to the number of magnons with nonzero k vector and we expect that the missing dynamical variable is related to a collective effect in this system. This research was sponsored by the National Science Council of Republic of China under Grant No. NSC-83- 0405EO2-001. ‘L. D. Landau, Collected Papers, edited by D. ter Haar (Pergamon, Oxford, 1965). ‘T. L. Gilbert, Phys. Rev. 100, 1243 (195s). ‘IJ. G. Bar’yakhtaq Sov. Phys. ETP 60, 863 (1984). ‘V. G. Bar’yakhtar, Physica B 159, 20 (1989). J. Appl. Phys., Vol. 75, No. 10, 15 May 1994 Sobolev et al. 5795 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 137.30.242.61 On: Sun, 13 Jul 2014 17:16:11‘C, W. Gardinw, Handbook of Stochastic Methods, 2nd ed. (Springer, Ber- ‘A. I. Akhiezer, V G. Bar’yakhtar, and S. V. Peletminskii, Spin \%ws lin, 19X.5!, Chaps. 4 and 5. (North-Holland, Amsterdq 1968), Chap. 7. ‘I. Klik and C. R. Chang, Phys. Rev. E 48, 607 (1993). “‘F. Dyson, Phys. Rev. 102, 1217 (19%). ‘A. 0. Caldeira and A. J. Isggett, Ann. Phys. 149. 374 i1983j. “V Kashcheev and M. Krivoglaz, Fiz. Tverd. Tela 3, 1541 (1961). ‘I. I(lik, J. Stat. Phys. 66, 635 t.1992). “H. B. Callen, J. Phys. Chem. Solids 4, 255 (,1%4). 5796 J, Appl. Phys., Vol. 75, No. 10, i5 May 1994 Sobolev ef al. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 137.30.242.61 On: Sun, 13 Jul 2014 17:16:11

1.4932092.pdf

Electric-field induced nonlinear ferromagnetic resonance in a CoFeB/MgO magnetic tunnel junction E. Hirayama , S. Kanai , J. Ohe , H. Sato , F. Matsukura, , and H. Ohno Citation: Appl. Phys. Lett. 107, 132404 (2015); doi: 10.1063/1.4932092 View online: http://dx.doi.org/10.1063/1.4932092 View Table of Contents: http://aip.scitation.org/toc/apl/107/13 Published by the American Institute of Physics Electric-field induced nonlinear ferromagnetic resonance in a CoFeB/MgO magnetic tunnel junction E.Hirayama,1S.Kanai,1,2J.Ohe,3H.Sato,2,4F.Matsukura,1,2,5, a)and H. Ohno1,2,4,5 1Laboratory for Nanoelectronics and Spintronics, Research Institute of Electrical Communication, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan 2Center for Spintronics Integrated Systems, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan 3Department of Physics, Toho University, 2-2-1 Miyama, Funabashi, Chiba 274-8510, Japan 4Center for Innovative Integrated Electronic Systems, Tohoku University, 468-1 Aramaki Aza Aoba, Aoba-ku, Sendai 980-0845, Japan 5WPI-Advanced Institute for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan (Received 10 July 2015; accepted 19 September 2015; published online 29 September 2015) We investigate the rf power dependence of homodyne-detected ferromagnetic resonance (FMR) spectra of a nanoscale CoFeB/MgO magnetic tunnel junction, in which the FMR is induced by the electric-field modulation of the magnetic anisotropy. The increase of the rf power changes the spec-tral lineshape and decreases characteristic frequency, at which drastic change in spectrum is observed. The behavior is consistent with nonlinear magnetization precession with a large preces- sional angle at high powers. From the rf power dependence of FMR spectra, we determine electric-field modulation ratio of magnetic anisotropy energy density to be 78 fJ/Vm, which is in agreement with the reported values. VC2015 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4932092 ] Since the discovery of electric-field effects on magnetic semiconductors,1–3extensive studies on the effects have been done on a variety of material systems.4Among them, the works on magnetic tunnel junctions (MTJs) realized elec- tric-field-induced magnetization switching,5–8which is a promising switching scheme for developing low-power high- performance spintronics devices with reduced dimensions compatible with the integrated circuits.9,10Homodyne- detected ferromagnetic resonance (FMR) is very often utilized to study the physics and to evaluate the magnetic properties of MTJs.11–21It was recently demonstrated that FMR can be excited not only by spin torque but also by the electric-field modulation of magnetic anisotropy.14,15,18–21 The reports on the electric-field induced FMR, so far, have been limited to a linear regime of magnetization precession. Many spintronics devices, such as magnetization switching devices and oscillators, however, operate through the nonlin-ear dynamics. 5–8,22Hence, the understanding of the nonlin- ear dynamics induced by electric fields is important to incorporate the electric-field effect into practical devices. Inthis work, we investigate the electric-field induced FMR in a nanoscale CoFeB/MgO MTJ, and reveal the characteristics of nonlinear dynamics excited by applying a high-power rfsignal to the MTJ. A circular MTJ with a diameter of 100 nm is fabricated on a coplanar waveguide by electron-beam lithography andAr ion milling from a stack (from the substrate side), Ta(5)/ Ru(10)/Ta(5)/Co 18.75Fe56.25B25(1)/MgO(1.3)/Co 18.75Fe56.25 B25(1.8)/Ta(5)/Ru(5), deposited on a sapphire substrate by dc/rf magnetron sputtering. The numbers in parentheses arenominal thickness in nanometers. The MTJ is annealed in vacuum at 300/C14C for 1 h under a perpendicular magnetic field of 0.4 T. The two CoFeB layers have perpendicularmagnetic easy axis 23and the top layer is the free layer. The magnetic anisotropy in CoFeB is determined by the sum of the interfacial perpendicular anisotropy at the CoFeB/MgOinterface, bulk anisotropy, and shape anisotropy. 23Because the contribution from the interfacial anisotropy to the mag- netic anisotropy field is inversely proportional to the CoFeB thickness, the present two thin CoFeB layers show large dif-ference in their magnetic anisotropy fields, which is consist-ent with the previous report. 23A tunnel magnetoresistance (TMR) ratio rTMR, defined as rTMR¼(RAP–RP)/RPwith junc- tion resistance at parallel and antiparallel magnetization con-figuration ( R PandRAP), is/C24120%, and the resistance-area product RPAis 16Xlm2. Before FMR measurements, the magnetization configu- ration of the free and reference layers is set to parallel. An rfsignal is applied to the MTJ through the rf port of a bias teeconnected to the waveguide, and its power P rfis varied from 1 to 100 lW. The dc voltage Vdcthrough the dc port of a bias tee is detected by lock-in amplifier (inset in Fig. 1(a)), where the amplitude of the input rf signal is modulated betweenzero and P rfat 1.1 kHz with a duty ratio of 0.5. We measure FMR spectra as functions of an in-plane magnetic field Hin (l0Hin/C20220 mT, l0is the permeability of vacuum) and Prf by sweeping the rf frequency ffrom 2 to 10 GHz. FMR is induced in the free layer in the frequency range used here. The resonance frequency of the reference layer is between 30 and 40 GHz, and thus its perpendicular magnetic anisot-ropy field is /C241T . 16To exclude the background not related to the FMR, the spectrum at Hin¼0 is subtracted from spec- tra measured at finite Hin. The spectrum at Hin¼0 does not include the FMR-related characteristics, because thea)Author to whom correspondence should be addressed. Electronic mail: f-matsu@wpi-aimr.tohoku.ac.jp. Tel.: þ81-22-217-5555. Fax: þ81-22- 217-5555. 0003-6951/2015/107(13)/132404/4/$30.00 VC2015 AIP Publishing LLC 107, 132404-1APPLIED PHYSICS LETTERS 107, 132404 (2015) magnetization precession in the free layer does not result in time-dependent junction resistance.14,16,19 Figure 1(a) shows FMR spectra under various Hinat Prf¼1lW, which is the lowest Prfin this study. We fit the spectrum by a sum of symmetric and anti-symmetric Lorentzfunctions, V dc¼[Df2VSþ4Df(fr/C0f)VA]/[4(fr/C0f)2þDf2], to determine the resonance frequency fr, the linewidth (the full width at half maximum) Df, and the amplitudes of the sym- metric and anti-symmetric Lorentzians, VSand VA. For homodyne-detected FMR, the symmetric component is known to be brought about by the spin-transfer torque(STT), and the anti-symmetric one by the field-like torque (FLT) and/or electric-field modulation of the magnetic ani- sotropy. 11,14Figure 1(b) shows the in-plane magnetic field Hindependence of fr. The resonance condition in a linear re- gime is derived from the Landau-Lifshitz-Gilbert (LLG) equation as fr¼cl0ðH1H2Þ1=2=2p (1) with H1/C17HinsinhFþHstrcoshFþHeff K1cos2hF/C0HK2cos4hF; H2/C17HinsinhFþHstrcoshFþðHeff K1/C0HK2=2Þcos 2 hF /C0ðHK2=2Þcos 4 hF: Here, cis the gyromagnetic ratio, hFis the magnetization angle in the free layer measured from the film normal, and Hstris the perpendicular component of a stray field from the reference layer. The effective first order perpendicular mag-netic anisotropy field H K1effand the second order magnetic an- isotropy field HK2are expressed as HK1eff¼/C0 MS(3Nz/C01)/ (2l0)þ2(K1þ2K2)/MSand HK2¼4K2/MS, respectively, where K1andK2are the first- and second-order perpendicular magnetic anisotropy energy constants, Nzthe perpendicular demagnetizing factor, and MSthe spontaneousmagnetization.24The Land /C19egf a c t o r ginc¼glB//C22his adopted to be 2.1 ( lB: Bohr magneton, /C22h: Dirac constant) determined for CoFeB layers with similar thicknesses.24We fit Eq. (1) to the in-plane magnetic field Hindependence of frat l0Hin/C20200 mT by treating HK1eff,HK2,a n d Hstras adjustable parameters, and the value of hFis determined from the equi- librium condition of d H1/dhF¼0 during the fitting procedure. Here, we limit ourselves to the region of l0Hin/C20200 mT, where we can approximate that the direction of magnetization in the reference layer is along the film normal and the perpen- dicular component of the external field due to the small mis-alignment of the sample position is negligible. 21As o l i dl i n e in Fig. 1(b) is a fit with l0HK1eff¼165 mT, l0HK2¼28 mT, andl0Hstr¼30 mT, which reproduces the experimental result well at l0Hin/C20200 mT, and the values of anisotropy and stray fields determined from the fitting are consistent with thosedetermined from the analysis of the static measurement on the in-plane magnetic field dependence of the junction resist- ance. 21Figure 1(c)shows the in-plane magnetic field Hinde- pendence of VSandVA, which indicates that the present FMR spectra are dominated by the anti-symmetric component. Thenonmonotonic behavior in V Ain Fig. 1(c) indicates that the FMR is excited mainly by the electric-field effect, which results in a maximum in VAathF/C2455/C14(the maximum in sin2hFcoshF).14The influence of the FLT is expected to be small, because for the CoFeB/MgO MTJs, it was reported thatthe magnitude of the FLT is almost the same as or even less than that of the STT. 12 Figure 2(a)shows FMR spectra as a function of Prfrang- ing from 1 to 100 lWa t l0Hin¼100 mT ( hF/C2433/C14), where each spectrum is normalized by corresponding Prf. With increasing Prf, the frequency fC, at which Vdcintersects with the horizontal axis, shifts to lower frequency, and the line- shape is distorted from the anti-symmetric Lorentzian. Thebehavior is known as a foldover effect, and indicates that the magnetization precession enters a nonlinear regime as P rfFIG. 1. (a) Homodyne-detected ferromagnetic resonance spectra measured at rf power Prf¼1lW as a function of in-plane external magnetic fields l0Hinranging from 40 to 220 mT with 20 mT step. Inset is measurement configuration. In-plane magnetic field Hindependence of (b) resonance fre- quency frand (c) symmetric and anti-symmetric components, VSandVA,o f the spectra. Curve in (b) is a fit by Eq. (1)and that in (c) is a fit by VA0sin2hFcoshF, where VA0is a proportionality constant and hFis the mag- netization angle in the free layer with respect to film normal. FIG. 2. (a) Homodyne-detected ferromagnetic resonance spectra normalizedby rf power P rfwith Prfranging from 1 to 100 lW under in-plane magnetic field l0Hin¼100 mT. The rf voltage Vrfdependence of (b) the magnitude hof electric-field modulation of magnetic anisotropy field obtained from the analy-ses and (c) the frequency f C,a tw h i c h Vdcintersects with a horizontal axis obtained from the experiments and the analyses. Line in (b) is a linear fit.132404-2 Hirayama et al. Appl. Phys. Lett. 107, 132404 (2015) increases.25–31We do not observe the hysteresis between the spectra with upward and downward sweeps of fwithout lock-in detection. The reason for the absence of the hystere-sis that is expected for the foldover effect is not clear yet, 25–27,29,30but it may be related to the thermal effect on the magnetization dynamics.26The tails of Vdc/Prfspectra coalesce into a single curve, showing that Vdc/Prfin the tail region. Because the detected Vdcis expressed by the product of the rf-modulated junction resistance and the input rf current Irf, the magnitude of Vdcfor the electric-field induced FMR is expressed as14 Vdc/C24ffiffi ffi 2p pR2 0 4RAPrTMR1/C0C2 ðÞ IrfVrf@Heff K1 @Vv0 MScoshFsin2hF; (2) where C¼(R0/C0Z0)/(R0þZ0) is the reflection coefficient of Vrffrom the MTJ with the junction resistance R0athFand characteristic impedance Z0¼50Xof the waveguide, Vrf (¼(PrfZ0)1/2) is the input rf voltage, and v’ is the real part of dynamic magnetic susceptibility along the film normal. Theprefactor is dependent on the detection method, and is 2 1/2/p for the signal detected by lock-in amplifier. Because Prf¼IrfVrfand Vdc/Prf, the result indicates that the voltage-modulation ratio of the effective magnetic anisot-ropy field ( @H K1eff/@V) is independent of Vrf, which is con- sistent with the previous result.14,19,24 The increase of Prfenlarges the cone angle hfor the magnetization precession. The dynamic component mof the magnetization normalized by MSis expressed as m¼sinh¼v0 MSh¼Vrf1þCðÞ@Heff K1 @Vv0 MSsinhFcoshF;(3) where we approximate the magnetization precession in the free layer has a circular trajectory because the magnitude of mvaries only less than 10% during the precession. By com- paring the results in Fig. 2(a)with Eqs. (2)and(3), we evalu- ate the values of hC(Prf),hatfCandPrf,a shC(32lW)/C246.3/C14 andhC(100 lW)/C246.5/C14. We calculate v’ from the LLG equa- tion with a damping constant aof 0.0185,14,32which is deter- mined to reproduce the FMR linewidth at Prf¼1lW. Then, we evaluate the rf field h exciting FMR, h/C24Vrf(1þC)(@HK1eff/@V)sinhFcoshF. Figure 2(b) shows the rf voltage Vrfdependence of h. The dependence is almost lin- ear, and from the slope we determine the voltage-modulation ratio of the effective anisotropy field l0(@HK1eff/@V)t o be/C2457 mT/V, which corresponds to the anisotropy energy density modulation per electric field of /C2478 fJ/Vm for MS¼1.46 T. The value is consistent with those of the previ- ous reports.19,24 The precessional axis of the magnetization is along the direction of the effective magnetic field, which consists of the magnetic anisotropy and external fields, and thus theincrease of hincreases magnetic potential energy nearly pro- portional to sin 2h. The precessional frequency of magnetiza- tion is proportional to the second derivative of the potential energy, and thus the resonance frequency does not depend onhin the linear regime at low Prf, where one can approxi- mate sin h/C24h. In the nonlinear regime at high Prf, the effectof the high-order terms in sin hshifts the resonance fre- quency. By solving the LLG equation with high-order terms, one obtains the difference between frandfCas f2 r/C0f2 C/C25/C03 16p2bm2; (4) where bm2is an m2term in /C0c2l02H10H20, where H10andH20 areH1andH2with an additional h-related term and trigono- metric functions with hFþhinstead of hF. By using handh obtained above, we calculate numerically the values of fC. Figure 2(c) compares the experimentally obtained fCwith calculated one, and shows reasonably good agreementbetween them. We compute the microwave-power dependence of V dc from micromagnetics simulation. For the simulation, we use the unit cell size of 2 /C22/C21.8 nm3and a magnetic stiffness constant of 19 pJ/m.33The other structural and magnetic pa- rameters are the same as those designed or obtained from theexperiments. We apply the rf-modulated magnetic anisot- ropy energy to describe the electric-field modulation of the magnetic anisotropy, and determine handh Fin each unit cell. By using the obtained angles, we calculate the rf- modulated tunnel magnetoresistance in each unit cell from the scalar product of the magnetizations in the free and refer-ence layers. The rf-modulated junction resistance is then obtained as the parallel combined resistance of all the unit cells. The V dcis obtained as the time-averaged value of the product of the rf-modulated resistance and the input rf cur- rent, as adopted in the derivation of Eq. (2). The simulated result in Fig. 3describes the power dependence of the spec- tral shape observed in the experiment, indicating that the va- lidity of the analysis method adopted here for the nonlinear FMR spectra. The simulation is performed without consider-ing thermal agitation at fixed frequencies, and does not include hysteresis effect. In summary, we investigate the rf power dependence of homodyne-detected ferromagnetic resonance spectra in a 100 nm-diameter CoFeB/MgO magnetic tunnel junction to investigate the magnetization dynamics induced by theelectric-field effect. The frequency shift and lineshape defor- mation of the spectra at high powers reflect the magnetiza- tion dynamics in a nonlinear regime. The behavior can bereproduced by micromagnetics simulations. The analyses of FIG. 3. Computed ferromagnetic resonance spectra normalized by rf power Prfas a function of Prfranging from /C241t o/C24100lW under in-plane mag- netic field l0Hin¼100 mT, where magnetization dynamics is simulated by micromagnetics simulation.132404-3 Hirayama et al. Appl. Phys. Lett. 107, 132404 (2015) the spectrum intensity and the frequency shift give the electric-field modulation ratio of the magnetic anisotropy energy density of /C2478 fJ/Vm, which is consistent with the values obtained from the other methods. The authors thank S. Ikeda, M. Yamanouchi, T. Hirata, H. Iwanuma, Y. Kawato, K. Goto, C. Igarashi, Y. Iwami, and I. Morita for their technical supports and discussion.This work was supported in part by JSPS through FIRST program, R&D Project for ICT Key Technology of MEXT, Grants-in-Aid for Scientific Research from JSPS (No.26889007) as well as MEXT (No. 26103002), and ASPIMATT program from JST. 1H. Ohno, D. Chiba, F. Matsukura, T. Omiya, E. Abe, T. Dietl, Y. Ohno, and K. Ohtani, Nature (London) 408, 944 (2000). 2D. Chiba, M. Yamanouchi, F. Matsukura, and H. Ohno, Science 301, 943 (2003). 3D. Chiba, M. Sawicki, Y. Nishitani, Y. Nakatani, F. Matsukura, and H. Ohno, Nature (London) 455, 515 (2008). 4F. Matsukura, Y. Tokura, and H. Ohno, Nat. Nanotechnol. 10, 209 (2015). 5Y. Shiota, T. Nozaki, F. Bonell, S. Murakami, T. Shinjo, and Y. Suzuki, Nat. Mater. 11, 39 (2012). 6W . - G .W a n g ,M .L i ,S .H a g e m a n ,a n dC .L .C h i e n , Nat. Mater. 11, 64 (2012). 7S. Kanai, M. Yamanouchi, S. Ikeda, Y. 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1.4868782.pdf

Micromagnetic simulation dynamics of susceptibility spectrum in diamond-shaped ferromagnetics Ismail, D. Djuhana, and D.-H. Kim Citation: AIP Conference Proceedings 1589, 200 (2014); doi: 10.1063/1.4868782 View online: http://dx.doi.org/10.1063/1.4868782 View Table of Contents: http://scitation.aip.org/content/aip/proceeding/aipcp/1589?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Micromagnetic calculation of dynamic susceptibility in ferromagnetic nanorings J. Appl. Phys. 105, 083908 (2009); 10.1063/1.3108537 Micromagnetic simulation of the magnetic spectrum of ferromagnetic nanowire J. Appl. Phys. 103, 013910 (2008); 10.1063/1.2829817 Conduction in a two‐phase plane with diamond‐shaped tiling J. Math. Phys. 32, 1958 (1991); 10.1063/1.529213 Fraunhofer diffraction by diamond‐shaped apertures: A theoretical and experimental study Am. J. Phys. 56, 551 (1988); 10.1119/1.15550 Formation of Diamond‐Shaped Prismatic Loops in Quenched fcc Metals J. Appl. Phys. 34, 3363 (1963); 10.1063/1.1729193 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.237.29.138 On: Tue, 11 Aug 2015 01:40:31Micromagnetic Simulation Dynamics of Susceptibility Spectrum in Diamond-Shaped Ferromagnetics Ismail1, D. Djuhana1*, and D. -H. Kim2 1Department of Physic s, Faculty Mathematic s and Natural Sciences , University of Indonesia, Depok 16424, Indonesia . 2Department of Physics, Chungbuk National University, Cheongju * Email: dede.djuhana@sci.ui .ac.id Abstract. We have investigated dynamics susceptibility spectrum in diamond -shaped ferromagnetics Cobalt and Permalloy by means of micromagnetic simulation based on Landau -Lifshitz -Gilbert equation. Interesting, it is found that the susceptibility spe ctrum exhibits similar peak frequency as the diagonal of diamond -shaped increases and tends to deceases as the thickness increases. We have also examined the susceptibility spectrum of diamond - shaped ferromagnetics using the Kittel ’s formula for frequency resonance. Our simulation results show agree well to the formula . Keywords: micromagnetic, ferromagnetics, susceptibility, diamond -shaped, Kittel’s equation PACS: 75.78. Cd, 75.78. Fg, 76.50.+g INTRODUCTION In decades , magnetic nanostructures have been interesting since its potential for spintronics application and magnetic memory devices[1]. Understanding and controlling the magnetization dynamics is important to realize the high -speed magnetic device technologies that operating in gigahertz[2 -4]. Several studies have been reported the magnezation dynamics of magnetic nanostructures in term of magnetitc susceptibility spectrum both experiment[5 -7] and simulation[8 -10]. However, few studies have been devoted the magnetic susceptibility spectrum in diamond -shaped ferromagnetic In this work, we have investigated the dynamics susceptibility of diamond -shaped ferromagnetic by means of micromagnetic simulation. Interestingly, the susceptibility spectrum exhibits similar peak frequency as the diagonal diamonds -shaped increases. The peak frequency tends to decrease as the thickness increases. Our calculation is found same result with Kittel’s formula. METHODS We have examined the dynamics suscpetibility spectrum of diamond -shaped ferromagnetic by means of public m icromagnetic simulation software, OOMMF [11] based on the Landau -Lifzhitz- Gilbert equation [12]. Figure 1 shows the dimension and geometry of diamond -shaped ferromagnetic. The length of diagonal ab=from 100 nm to 500 nm with increamen t 100 nm. The cellsize is 5 5 nm t×× with respect to the thickness t = 5 nm, 10 nm, and 15 nm. The damping factor is 0.1α= . The materials FIGURE 1. Dimension and geometry of the diamond- shaped elements with diagonal length a = b varies from 100 nm to 500nm . The initial spin configuration + y direction and the exponential magnetic pulse is systematically applied along + x direction. are used in this simulation, Permalloy (P y) and Cobalt (Co). The material parameter of Py and Co, such as saturation magnetization Ms, constant exchange A, and the anisotropy constant K, as shown in Table 1[13] . Then, the exponential magnetic pulse ()91000exp( 10 ) Ht t =− is applied to the dia mond -shape d ferromagnetic in +x direction with the initial spin configuration in +y direction [14]. TABLE 1. Material parameters are used in this simulation Permalloy (Py) and Cobalt (Co) . Material MS (A/m) A (J/m) K (J/m3) Permaloy Cobalt 8.6 × 105 14 × 105 13 × 10-12 30 × 10-12 0 530 × 103 4th International Conference on Mathematics and Natural Sciences (ICMNS 2012) AIP Conf. Proc. 1589, 200-202 (2014); doi: 10.1063/1.4868782 © 2014 AIP Publishing LLC 978-0-7354-1221-7/$30.00 200 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.237.29.138 On: Tue, 11 Aug 2015 01:40:31RESULT AND DISCUSSIONS The magnetic susceptibility spectrum is determined using the following equation ()() ()() ()'' ' M Hωχω χ ω χ ωω== − (1) where ()χωʹ′ is the real part and ()χωʹ′ʹ′ is the imaginary part of susceptibility spectrum as function of frequency ω. Accordingly from the Eq (1), ()Mωand ()Hωare magnetization and applied field in the frequency domain, respectively. They are calculated by FFT analyzed. In this simulation we used the magnetization in +y direction yM. The result of magnetic susceptibility spectrum of Py and Co diamond -shaped with respect to thickness is shown in Figure 2. FIGURE 2. Imaginary part of susceptibility as a function frequency with respect to thickness t = 5~1 5 nm. (a) Py and (b) Co. As the figure, ()χωʹ′ʹ′ occurs in GHz regime. The frequency of susceptibility spectrum are various around 20 GHz to 40 GHz as the diagonal length of diamond -shaped increases. The calculation of frequency peaks o f susceptibility spectrum is depicted in Table 2. We observed that all frequency peak shows in a single peak. It can be understand that the frequency peak is originated from the uniformly magnetization tranverse to the microwave regime[1 5]. It is also foun d that diamond- shaped pattern shows reversal magnetization starts in the interior part of elemens[1 6]. It is clearly that the frequency peak originated from the main part of domain structure. Very i nterestingly, the susceptibility spectrum exhibit decreasi ng as the thickness increases in same diagonal length , as shown in Figure 3. The energy system is determined by energy total minus the Zeeman energy[1 7]. We observe the frequency peak shows to shift slightly to low frequency as the thickness increases. The frequency peak is originated from the demagnetization energy rather than the exchange energy . The demagnetization energy comes f rom the dipolar in teraction and spin -wave mode [18] . . FIGURE 3. The curve of m agnetization energy and frequ ency peak from imaginary susceptibility spectrum for (a) Py and (b) Co with respect to thickness t = 5~15 nm. We also have calculated using Kittel’s formula [19] for Py and Co diamond -shaped as following the equation : TABLE 2. The frequency resonance peaks of the diamond -shaped ferromagnetics material with a diagonal length and thickness variation . Diagonal Length (nm) Frequency Peaks Py (GHz) Frequency Peaks Co (GHz) Frequency Kittel (GHz) d = 5 nm d = 10 nm d = 15 nm d = 5 nm d = 10 nm d = 15 nm Py Co 100 36,68 35,28 34,23 26,32 25,72 23,39 39.9 29,8 200 25,55 24,21 22,85 29,81 26,41 25,21 300 34,92 34,09 33,79 28,73 27,39 26,63 400 39,66 35,01 32,82 28,72 28,00 27,71 500 34,32 33,83 33,49 29,95 29,29 28,77 201 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.237.29.138 On: Tue, 11 Aug 2015 01:40:31()[]()002Kx z s Ky z sHH NN MHH NN Mγωπ=+ + − + + −⎡⎤⎣⎦ (2) Where H0 is the external magnetic field. 102KsHK Mµ= is the effective magnetocrystalline anisotropy field. Nx, Ny, and Nz are the demagnetizing factors[20]. The calculation shows the frequency peak similar behavior from micromagnetic simulation. CONCLUSIONS In the conclusion, we have investigated the dynamics susceptibility spectrum of diamond-shaped ferromagnetics by means of micromagnetic simulation. The susceptibility spectrum shows in GHz regime. The frequency peak of susceptibility spectrum decreases as the thickness increases in same diagonal length. The dipolar and spin-wave mode interaction contribute to the frequency peak of susceptibility spectrum ACKNOWLEDGMENTS This work was supported by Jambi Province Government for Master Program 2011 and Kegiatan Insentif Riset Sistem Inovasi Nasional (Sinas) grant (No.1.66/Sek/IRS/PPK/1/2012). REFERENCES 1. S. A. Wolf, et al. “Spintronics: A Spin Based Electronics Vision for the Future.” Science 294, 14888 (2001) 2. S. E. Russek, et al. “High Speed Switching and Rotational Dynamics in Small Magnetic Thin Film Devices” in Hillebrands (Eds.): Topics Appl. Phys. 87, 93-156, Springer, Berlin, Heidelberg (2003). 3. N. Dao, et al. “Dynamic Susceptibility of Nanopillars”. Nanotechnology, 15, S634-S638 (2004) 4. R. Liu, J. Wang, Q. Liu, H. Wang, and C. Jiang. “Micromagnetic Simulation of the Magnetic Spectrum of Ferromagnetic Nanowire”. J. Appl. Phys. 103, 013910 (2008). 5. C. J. Oates, F. Y. Ogrin, S. L. Lee, P. C. Riedi, and G. M. Smith. “High Field Ferromagnetic Resonance Measurements of the Anisotropy Field of Longitudinal Recording Thin-Film Media”. J. Appl. Phys. 91, 3 (2002). 6. S. A. Manuilov, A. M. Grishin, and M. Munakata. “Ferromagnetic Resonance, Magnetic Susceptibility, and Transformation of Domain Structure in CoFeB Film with Growth Induced Anisotropy”. J. Appl. Phys. 109, 083926 (2011). 7. A. W. Harter, G. Mohler, R. L. Moore, and J. W. Schultz. “Calculating Magnetic Susceptibility Over Multiple Length Scales” Proceeding IEEE-Nano, 499-502 (2002). 8. O. Gérardein, H. Le Gall, M. J. Donahue, and N. Vukadinovic. “Micromagnetic Calculation of the High Frequency Dynamics of Nano-Size Rectangular Ferromagnetic Stripes”. J. Appl. Phys. 89, 11 (2001). 9. V. Castel, et al. “Perpendicular Ferromagnetic Resonance in Soft Cylindrical Elements : Vortex and Saturated States”. Phys. Rev. B. 85, 184419 (2012). 10. S. K. Kim. “Micromagnetic Computer Simulation of Spin Wave in Nanometre-Scale Patterned Magnetic Elements”. J. Appl. Phys. D. 43, 264004 (2010). 11. M. J. Donahue and D. G. Porter, OOMMF User’s Guide, http://math.nist.gov/oommf(2002). 12. T. L. Gilbert. “Phenomenological Theory of Damping in Ferromagnetic Materials”. IEEE Trans. Magn. 40, 3443 (2004). 13. F. López-Urías, J. J. Torres-Heredia, and E. Munõs-Sandoval. “Magnetization Pattern Simulation of Fe, Ni, Co, and Permalloy Individual Nanomagnets”. J. Magn. Magn. Mater. 294 (2005). 14. C. W. Bing, H. M. Gui, Z. Hao, O. Yu, and D. L. Jiang. “Micromagnetic Simulation on the Dynamic Susceptibility Spectra of Cobalt Nanowire Arrays: the Effect of Magnetostatic Interaction”. Chin. Phys. B. 19, 087502 (2010). 15. A. Kaya and J. A. Bain. “High Frequency Susceptibility of Closure Domain Structure Calculated Using Micromagnetic Modeling”. J. Appl. Phys. 99, 08B708 (2006). 16. C. König, et al. “Shape-Dependent Magnetization Reversal Processes and Flux_Closure Configurations of Microstructured Epitaxial Fe (110) Elements”. Appl. Phys. Lett. 79, 3648-3650 (2001). 17. M. Hayashi, L. Thomas, C. Rettner, R. Moriya, X. Jiang, and S. P. Parkin. “Dependence of Current and Field Driven Depinning Domain Walls on Their Structure and Chirality in Permalloy Nanowires”. Phys. Rev. Lett. 97, 207205 (2006). 18. S. Jung, J. B. Ketterson, and V. Chandrasekhar. “Micromagnetic Calculations of Ferromagnetic Resonance in Submicron Ferromagnetic Particles”. Phys. Rev. B. 66, 132405 (2002). 19. C. Kittel. “On the Theory of Ferromagnetic Resonance Absorption”. Phys. Rev. Vol. 73, 2 (1948). 20.O. Kohmoto. “Effective Demagnetizing Factors in Ferromagnetic Resonance Equations”. J. Magn. Magn. Mater. 262, 280-288 (2003). 202 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.237.29.138 On: Tue, 11 Aug 2015 01:40:31

1.1452188.pdf

Large-scale computer investigations of finite-temperature nucleation and growth phenomena in magnetization reversal and hysteresis (invited) M. A. Novotny, G. Brown, and P. A. Rikvold Citation: Journal of Applied Physics 91, 6908 (2002); doi: 10.1063/1.1452188 View online: http://dx.doi.org/10.1063/1.1452188 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/91/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Direct observation of individual Barkhausen avalanches in nucleation-mediated magnetization reversal processes Appl. Phys. Lett. 95, 182504 (2009); 10.1063/1.3256188 Magnetic reversal phenomena in pseudo-spin-valve films with perpendicular anisotropy J. Appl. Phys. 101, 09D116 (2007); 10.1063/1.2712942 Systematic study of the magnetization reversal in patterned Co and NiFe Nanolines Appl. Phys. Lett. 84, 759 (2004); 10.1063/1.1645332 Magnetic structure and hysteresis in hard magnetic nanocrystalline film: Computer simulation J. Appl. Phys. 92, 6172 (2002); 10.1063/1.1510955 Coherent magnetization reversal of nanoparticles with crystal and shape anisotropy J. Appl. Phys. 89, 507 (2001); 10.1063/1.1323519 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Sat, 20 Dec 2014 03:04:16Micromagnetic Modeling I Thomas Schrefl, Chairman Large-scale computer investigations of finite-temperature nucleation and growth phenomena in magnetization reversal and hysteresis invited M. A. Novotnya) Department of Physics and Astronomy, Mississippi State University, Mississippi State, Mississippi 39762 G. Brown School of Computational Science and Information Technology, Florida State University, Tallahassee, Florida 32306, and Center for Computational Science, Oak Ridge National Laboratory, Oak Ridge,Tennessee 37831 P. A. Rikvold School of Computational Science and Information Technology, Florida State University, Tallahassee,Florida 32306, and Center for Materials Research and Technology and Department of Physics,Florida State University, Tallahassee, Forida 32306 An overview of some of the results obtained from long-time dynamic simulations of models of magnetic nanoparticles and thin magnetic films is presented. The simulation methods includedynamic Monte Carlo simulations and micromagnetic simulations. The effects of nucleation andgrowth due to the finite temperature display similar behaviors for various models of magneticmaterialsandcomputationalapproaches.Thesebehaviorscanonlybeseencomputationallywiththeuse of advanced algorithms and massively parallel computers. Several different modes of reversalare seen at finite temperature, and crossovers from one mode of reversal to another for the samesystem can be seen as the temperature or applied field is changed. Statistical interpretations of bothfield-reversal experiments and hysteresis experiments are shown to be necessary to understand thefinite-temperature behavior of magnetic nanoparticles and thin magnetic films. © 2002 American Institute of Physics. @DOI: 10.1063/1.1452188 # I. INTRODUCTION The field of nanoscale magnetism is currently extremely active. There are a number of reasons for this activity. Fromthe applications side, the quest for higher storage densitiesfor magnetic recording requires smaller and better character-ized nanoparticles, and for small nanoparticles the thermalenergy can be comparable to other energies, such as ex-change or dipole–dipole energies. From the experimentalside, new processing techniques allow researchers to makewell-characterized nanoscale magnets. Simultaneously, newand refined measurement techniques enable magnetic mea- surements on samples with smaller volumes and in a fashionthat resolves the time-dependent spatial structure during achange in the magnetization. From the computational pointof view, better algorithms and massively parallel computersallow simulations to be performed with simulation times andsystem sizes becoming comparable to those that are techno-logically relevant and experimentally accessible. This article outlines some results from large-scale com- puter simulations of models of thermal effects in nanoscaleferromagnets. Both magnetic particles and thin magneticfilms are discussed. Two main ideas will be emphasized.First, when thermal effects are important in nanomagnets,understanding finite-temperature nucleation and growth re- lated to escape from metastable states is of paramount im-portance in order to be able to obtain either qualitative orquantitative understanding of the finite-time behavior. Sec-ond, many aspects of nucleation and growth can be under-stood from consideration of very idealized models for mag-netic nanoparticles and thin films, and this understanding canbe used to provide a foundation for understanding the morecomplicated behaviors of the time dependence of more real-istic models of nanomagnets and of experiments. II. BRIEF OVERVIEW OF NUCLEATION AND GROWTH Here we summarize the classical theory of homogeneous nucleation and growth, as applied to uniaxial magnets. Forfurther details, see Refs. 1–12, where it is shown thatdroplet-theory predictions match simulation results ex-tremely well. Our understanding of such simple systems willbe applied to the analysis of more complicated systems andsimulations of more realistic models of nanomagnets. Twodifferences between this finite-temperature description andthe zero-temperature descriptions that are also called ‘‘nucle-ation’’ should be kept in mind. First, for nonzero tempera-tures there exists a fluctuation, which is called a droplet, thatcauses escape over a saddle point. These droplets continuallyfluctuate and evolve in time, and they should not be confused a!Electronic mail: man40@ra.msstate.eduJOURNAL OF APPLIED PHYSICS VOLUME 91, NUMBER 10 15 MAY 2002 6908 0021-8979/2002/91(10)/6908/6/$19.00 © 2002 American Institute of Physics [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Sat, 20 Dec 2014 03:04:16with static magnetic domains. Second, to escape from a metastable state at zero temperature, the free energy must bechanged in such a way that the saddle point is no longerpresent.Forfinite-temperatureescapefromametastablestatethe free energy remains approximately unchanged, and ran-dom thermal fluctuations lead to escape over the saddlepoint. In a metastable phase, very small fluctuations are ener- getically costly and tend to decay quickly. For systems withshort-range interactions ~like exchange !, the fluctuations are compact droplets of radius R. The droplet free energy has two competing terms: a positive surface term } s(T)Rd21 and a negative bulk term }uHuRd, wheredis the spatial dimension, s(T) is the surface tension of the droplet wall, andHis the applied magnetic field. The competition yields a critical droplet radius, Rc(H,T)}s(T)/uHu. Droplets with R,Rcmost likely decay, while droplets with R.Rcmost likely grow to complete the switching process. The free-energy cost of the critical droplet ( R5R c)i s DF(H,T) }s(T)d/uHud21. Nucleation of critical droplets is a stochas- tic process with its nucleation rate per unit volume given byan Arrhenius relation I ~H,T!}exp@2bDF~H,T!#[exp@2bJ~T!/uHud21#, ~1! where b51/kBT~kBis Boltzmann’s constant !, and J(T)i s theH-independent part of DF.~For brevity, we here neglect an important prefactor in the nucleation rate, which has beenincluded in our calculations. 2,3,9! A second important length, R0, arises because large droplets grow at a finite velocity,13v’nuHu. This length is the average distance between independent droplets and isrelated to vandIby expressing a corresponding time scale in two different ways as R0dI5v/R0, yielding R0(H,T) 5(v/I)1/(d11). Comparing the system size LwithRcandR0, one finds different dynamic regimes; here we discus two. ForRc,L,R0, the first droplet to nucleate most likely completes the switching process before other droplets cannucleate. This is the single-droplet ~SD!regime, where the switching is a stochastic Poisson process. The metastablelifetime tis then inversely proportional to the system vol- ume, tSD(H,T,L)’L2dI21, and the probability of not switching by time tisPnot(t)5exp(2t/tSD). ForL.R0, many droplets contribute to the switching process. This is the multi-droplet ~MD!regime, in which the dynamics are almost deterministic and well described by theKolmogorov–Johnson–Mehl–Avrami theory of metastabledecay in large systems. 14–16Here tis independent of Land depends on vandIastMD(H,T,L)}(Ivd)21/(d11), while Pnot(t)’erfc@(t2t)/D(H,T,L)#, where D(H,T,L) is the standard deviation of the switching time.4,8 The switching field ~or coercive field !,Hsw(t,T,L), is found by solving the equation for t(H,T,L) in the relevant regime ~SD or MD !forH. III. MODELS AND METHODS The finite-temperature dynamics of magnetic materials are very difficult to model effectively using computations.The first complication is that magnetic systems obtain their behavior from quantum effects, so a first-principles calcula-tion would require simulating the dynamics of a finite-time,finite-temperature quantum system. This is not currently fea-sible. Consequently, approximations must be performed toobtain simplified models that contain the essential physicsbut are computationally reasonable.Two such models will bedescribed in this article. The ferromagnetic Ising model is computationally the simplest model of a magnetic material. For highly aniso-tropic uniaxial magnetic materials each region of the ferro-magnet may be considered to have two states correspondingto two discrete directions. This gives an Ising model as anapproximate Hamiltonian for ferromagnetic nanoparticles. 4 Starting from a quantum Hamiltonian of spin 1/2 particlesinteracting with a heat bath, it is possible to derive in certainlimits a physical dynamic for the model. 17,18This dynamic has two parts: first a spin is chosen uniformly among all thespins, then that spin is flipped with a probability p flipthat depends on the temperature Tand the applied magnetic field H. One algorithmic step is called a Monte Carlo step ~mcs! and corresponds to one spin-flip attempt. For a system withNIsing spins, the simulation time is often given in Monte Carlo steps per spin ~MCSS !, which corresponds to Nmcs. With weak coupling to a fermionic heat bath, 17the probabil- itypflipis the Glauber transition probability.19In this case pflip5e2bEnew/@e2bEnew1e2bEold#, whereEnewis the energy of the configuration with the Ising spin flipped, and Eoldis the energy of the current spin configuration. ~A different probability, more applicable to magnetic materials, whichalso satisfies detailed balance, has recently been derived forcoupling to a d-dimensional bath of phonons. 18Then the time in MCSS is on the order of an inverse phonon fre-quency. !The Glauber transition probability is used in the present article, which presents results of simulations of thenearest-neighbor ~nn!Ising model with Hamiltonian H52J ( ^i,j&sisj2H( isi. ~2! HereJ.0 is the ferromagnetic coupling constant, the first sum runs over all nn pairs, and the second sum runs over allNspins. We consider only the case of periodic boundary conditions.Amodified Ising Hamiltonian has also been usedto study dynamics of magnetic sesquilayer systems. 9,20 The other simulation method used in this article is finite- temperature micromagnetic simulations. Here the ‘‘spins’’are ‘‘coarse grained’’ to obtain magnetization vectors M(r i) that represent the total magnetization in a cell of a certainvolumeVcentered at r i. This model is reasonable at tem- peratures low compared to the critical temperature Tc, but high enough that quantum effects are unimportant.21The vectors have fixed magnitude given by the bulk saturationmagnetization density M s. The time evolution of each spin is given by the damped precessional motion of the Landau–Lifshitz–Gilbert ~LLG!equation 22,23 dM~ri! dt5g0 11a2M~ri!3FH~ri!2a MsM~ri!3H~ri!G, ~3!6909 J. Appl. Phys., Vol. 91, No. 10, 15 May 2002 Novotny, Brown, and Rikvold [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Sat, 20 Dec 2014 03:04:16where the electron gyromagnetic ratio is g051.76 3107Hz/Oe,23andais a phenomenological damping pa- rameter. The sign of the undamped precession term followsthe convention of Brown. 22The field Hi, seen by the mag- netization vector of each cell, includes contributions fromboth an external field and from the field generated by all theother cells. These fields are due to effects that include ex-change interactions, dipole–dipole interactions, and crystal-line anisotropy. To include thermal effects, a stochastic fieldH n(ri), which is assumed to fluctuate independently for each spin,24is included.The fluctuations are assumed to be Gauss- ian, each with ^Hn(ri,t)&50 and second moment given by ^Hnm~ri,t!Hnm8~ri8,t8!&52akBT g0MsVd~t2t8!dm,m8di,i8 ~4! withHnmindicating one of the Cartesian components of Hn. The LLG equation is integrated using discrete time steps Dt, and advanced methods such as the fast-multipole method areused to handle the dipole–dipole interactions. 25In the un- physical limit a@1 the LLG integrations correspond to a Monte Carlo simulation of a classical Heisenberg model.26 IV. FIELD REVERSAL AT FINITE TEMPERATURE The simulations for field reversal match as closely as possible those which can be performed experimentally. Thesystem is started in a state with all spins in the zdirection and the external magnetic field in the 2zdirection. An ex- ample showing magnetization configurations at three differ-ent times for a model of a magnetic nanoparticle undergoinga field reversal is shown in Fig. 1. The picture, shown inthree-quarter cutaway view, has the long axis of the magneticnanopillar along the zdirection and shows the zcomponentof the magnetization. The light shades correspond to the metastable orientation and the dark shades to the equilibriumorientation. This figure shows an individual pillar on a 737349 lattice, with parameters corresponding to an iron pillar 5.2 nm on each short side and 88.4 nm along the zaxis. The average lifetime tis defined to be the time from when the field is reversed to the time when the zcomponent of the magnetization equals zero. Note that in Fig. 1 the nucleationstarts near one end of the pillar and propagates with a finitegrowth velocity to reverse the magnetization of the pillar.This decay mode would not be seen in a zero-temperaturesimulation, since the two ends are symmetric and would bothreverse at the same field value where both metastable stateswould no longer exist. For the square-lattice nn Ising model dynamic Monte Carlo results for t~multiplied by the volume, so the units are in mcs !are shown versus the inverse temperature in Fig. 2. Where the two different system sizes fall on the same curve,the system is in the SD regime.Without multiplication by thevolume, the lifetimes would overlap in the MD regime. Thecrossover for each system size between the SD regime ~at lowT!and the MD regime has been called the dynamic spinodal ~DSp!. 1–3The low-temperature results27–31in Fig. 2 correspond to L2t5exp$b@8Jl22uHu~l22l11!#% ~5! with l5b2J/uHuc11 and the symbol bcdenoting the integer part. In Fig. 2 the applied field is uHu53J/4, so l53. The correction due to a prefactor to the exponential28,31,32is also shown in Fig. 2. Only through the use of advanced compu-tational algorithms 10,29,30can the long-time simulations be obtained in realistic amounts of computer time. The DSpcurves for the system sizes shown in Fig. 2 are included inFig. 3. The location of the DSp is where the growth time fora single nucleating droplet becomes comparable to t. For the FIG. 1. Pictures of an iron pillar switching for H51800 Oe and T520 K from LLG simulations. FIG. 2. The average lifetime t, multiplied by the volume L2, is shown for a model of a uniaxial thin film at uHu53J/4.6910 J. Appl. Phys., Vol. 91, No. 10, 15 May 2002 Novotny, Brown, and Rikvold [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Sat, 20 Dec 2014 03:04:16simulations shown here, the equivalent criterion on the stan- dard deviation of the lifetime, st, was used: st5t/2. Using Eq.~5!for the nucleating droplet, neglecting the prefactors, and using a reasonable interface velocity for the dynamicMonte Carlo simulation, enables calculation of the DSp 33for l51. The further assumption that tis continuous when l changes, allows calculation of the DSp for all l. These are shown as light curves in Fig. 3 for L520 and 200. With a lattice spacing and Jappropriate for bulk iron, the DSp can be calculated within this model for any lattice size. Since Eq.~5!is valid only for low temperatures, this expression for the DSp is expected to be experimentally relevant for low tem-peratures only. Figure 3 shows that for experimentally rea-sonable systems a micron on a side, as well as for muchlarger magnetic thin films, both the SD and MD regimesshould be observable. Figure 3 also shows as a heavy dashedline the crossover, called the mean-field spinodal, 3between the MD regime and the strong field regime where the ideasof nucleation and growth no longer are valid. 1–3Similar behaviors for tandstare seen in Fig. 4. The data shown in this figure are from micromagnetic simula-tions of a model for magnetic iron pillars with an aspect ratioof 17. This simplified model has the cross section of a pillarequal to the cross section of a cell. 34Despite the fact that nucleation for these pillars in the SD regime occurs almostexclusively at the two ends, the crossover between the SDand MD regimes is still seen. A similar phase diagram fromdynamic Monte Carlo of the classical Heisenberg model andlarge- aLLG simulations has recently been found.35 V. HYSTERESIS AT FINITE TEMPERATURE The phenomenon of hysteresis is common to both mag- netic systems and other nonlinear systems subject to an os-cillating field H5H 0sin(vt). The effect occurs when the dy- namics are too sluggish to keep pace with the oscillatingfield. The term was coined by Ewing in 1881 in the contextof magnetoelasticity. 36For finite-temperature magnetic nano- particles the hysteresis loops can look very different fromthose seen when thermal effects are unimportant. Such a loopis shown in Fig. 5. Note that the frequency shown is notrealistic for bulk magnetic materials. However, it is reason-able for magnetic nanoparticles which require only localizedexternal fields, and also it corresponds to desired switchingtimes in future magnetic storage devices. Although the mag-netization is often near the saturation magnetization, thereare excursions between these states that follow complicated,random trajectories. These transitions occur because the os-cillating field makes the two phases of saturated magnetiza-tion alternately stable and metastable, and the random ther-mal fluctuations are needed to switch from one orientation toanother. To analyze such hysteresis loops statistical interpre-tations are required. Among the earliest topics in hysteresisto receive lively interest was the area of the hysteresis loopover one period of the applied field. This is given by A FIG. 3. The ‘metastable phase diagram’ for a model of a uniaxial thin-film magnet is shown. The vertical line marks the critical temperature. FIG. 4. The mean switching time and standard deviation of the switching time vs inverse applied field at T520 K and T5100 K. This figure shows data for a simplified micromagnetic model for nanopillars with an aspectratio of 17. FIG. 5. Hysteresis loops for two periods taken from a micromagnetic simu- lation of an iron pillar at T520 K with H052000 Oe and n5100 MHz. The trace starts at the upper right, proceeds counterclockwise through oneloop, during which switching is not completed before the field is againpositive. It then proceeds through another loop, during which the switchingis completed. The pillar remained in the negatively oriented state for manyperiods after this switch.6911 J. Appl. Phys., Vol. 91, No. 10, 15 May 2002 Novotny, Brown, and Rikvold [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Sat, 20 Dec 2014 03:04:1652rm(H)dHand corresponds to the energy dissipation per period of the applied field. Questions about the dependenceofAon vandH0were among the first ones raised about hysteretic systems. Even today, experiments on ultrathin Feand Co films with Ising-like anisotropy have obtained differ-ent results for the frequency dependence of A. 37–40Figure 6 shows the behavior of the average hysteresis loop area ^A& for the aspect ratio 17 magnetic iron pillars as a function ofthe dimensionless frequency 1/ R. Note that there exists a maximum in ^A&. A similar maximum has been observed in dynamic Monte Carlo simulations of the nearest-neighborsquare-lattice Ising model 5,41,44in both the SD and MD re- gimes. Furthermore, an analytical analysis of the dependenceof ^A&onvis possible for the nearest-neighbor Ising ferromagnet.42–44This shows that the low-frequency behav- ior of ^A&is ultimately logarithmic in vdue to the exponen- tial dependence of ton 1/ uHud21. However, such a logarith- mic behavior is not seen for reasonable frequencies; rather, acrossover between the maximum in ^A&and the logarithmic behavior is observed. Such a crossover can easily be misin-terpreted as a power law over a limited range ~say three or four decades !in v.42Consequently, the low-frequency be- havior in Fig. 6 is a crossover effect rather than a power law.The same should be true for experiments on magnetic sys-tems, and the power laws seen experimentally indeed varydrastically from material to material, from sample to sample,and from one frequency range to another. VI. CONCLUSIONS The ideas of nucleation and growth as they relate to finite-temperature escape from a metastable state for nano-magnets have been reviewed. The explicit models and meth-ods described here include both dynamic Monte Carlo simu-lations of Ising models and micromagnetic simulations ofnanoscale magnetic pillars. In both instances the ideas ofnucleation and growth are required in order to understand thefinite-temperature behavior seen in field-reversal simulationsand in hysteresis. One consequence of the finite-temperature effects is that there are different possible decay regimes,even when the size of the system remains unchanged. Whichdecay regime the system utilizes to escape from the meta-stable state depends on the temperature, the applied field, andthe system size. In hysteresis the effect of finite temperaturedemands that the hysteresis loop area ultimately be logarith-mically dependent on frequency in the low-frequency limit.However, in realistic simulations and in experiments inwhich the range of frequencies is limited to a few decades,usually only a portion of a crossover curve for the hysteresisloop area can be seen. Other novel effects in hysteresis atnonzero temperatures can include stochastic resonance 43and a dynamic phase transition.44–47 ACKNOWLEDGMENTS The authors gratefully acknowledge the collaborators in our magnetic studies: M. Kolesik, G. Korniss, H. L. Rich-ards, S. W. Sides, D. M. Townsley, and C. J. White. Usefulconversations with S. Wirth and S. von Molna ´r are also ac- knowledged. Supported in part by U.S. National ScienceFoundation Grant Nos. DMR-9871455 and DMR-0120310.Supercomputer time supplied in part by NERSC. 1H. Tomita and S. Miyashita, Phys. Rev. B 46, 8886 ~1992!. 2P.A. Rikvold, H. Tomita, S. Miyashita, and S. W. 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Brown, Micromagnetics ~Wiley, New York, 1963 !. FIG. 6. The mean of the hysteresis loop area, ^A&, vs the dimensionless frequency 1/ R5vt(H0,T)/2p, shown on a log–log scale. These data are from micromagnetic calculations for a model of iron pillars.6912 J. Appl. Phys., Vol. 91, No. 10, 15 May 2002 Novotny, Brown, and Rikvold [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Sat, 20 Dec 2014 03:04:1623A. Aharoni, Introduction to the Theory of Ferromagnetism ~Clarendon, Oxford, 1996 !. 24W. F. Brown, Phys. Rev. 130,1 6 7 7 ~1963!. 25G. Brown, M. A. Novotny, and P. A. Rikvold, Phys. Rev. B 64, 134422 ~2001!. 26U. Nowak, R. W. Chantrell, and E. C. Kennedy, Phys. Rev. Lett. 84, 163 ~2000!. 27E. Jorda˜o Neves and R. H. Schonmann, Commun. Math. Phys. 137, 209 ~1991!. 28M. A. 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5.0020843.pdf

J. Chem. Phys. 153, 141104 (2020); https://doi.org/10.1063/5.0020843 153, 141104 © 2020 Author(s).Cherry-picking resolvents: A general strategy for convergent coupled-cluster damped response calculations of core-level spectra Cite as: J. Chem. Phys. 153, 141104 (2020); https://doi.org/10.1063/5.0020843 Submitted: 04 July 2020 . Accepted: 10 September 2020 . Published Online: 09 October 2020 Kaushik D. Nanda , and Anna I. Krylov ARTICLES YOU MAY BE INTERESTED IN From orbitals to observables and back The Journal of Chemical Physics 153, 080901 (2020); https://doi.org/10.1063/5.0018597 JCP Emerging Investigator Special Collection 2019 The Journal of Chemical Physics 153, 110402 (2020); https://doi.org/10.1063/5.0021946 Molecular second-quantized Hamiltonian: Electron correlation and non-adiabatic coupling treated on an equal footing The Journal of Chemical Physics 153, 124102 (2020); https://doi.org/10.1063/5.0018930The Journal of Chemical PhysicsCOMMUNICATION scitation.org/journal/jcp Cherry-picking resolvents: A general strategy for convergent coupled-cluster damped response calculations of core-level spectra Cite as: J. Chem. Phys. 153, 141104 (2020); doi: 10.1063/5.0020843 Submitted: 4 July 2020 •Accepted: 10 September 2020 • Published Online: 9 October 2020 Kaushik D. Nandaa) and Anna I. Krylova) AFFILIATIONS Department of Chemistry, University of Southern California, Los Angeles, California 90089-0482, USA a)Authors to whom correspondence should be addressed: kaushikdnanda@gmail.com and krylov@usc.edu ABSTRACT Damped linear response calculations within the equation-of-motion coupled-cluster singles and doubles (EOM-CCSD) framework usually diverge in the x-ray regime. This divergent behavior stems from the valence ionization continuum in which the x-ray response states are embedded. Here, we introduce a general strategy for removing the continuum from the response manifold while preserving important spectral properties of the model Hamiltonian. The strategy is based on decoupling the core and valence Fock spaces using the core–valence separation (CVS) scheme combined with separate (approximate) treatment of the core and valence resolvents. We illustrate this approach with the calculations of resonant inelastic x-ray scattering (RIXS) spectra of benzene and para -nitroaniline using EOM-CCSD wave functions and several choices of resolvents, which differ in their treatment of the valence manifold. The method shows robust convergence and extends the previously introduced CVS-EOM-CCSD RIXS scheme to systems for which valence contributions to the total cross section are important, such as the push–pull chromophores with charge-transfer states. Published under license by AIP Publishing. https://doi.org/10.1063/5.0020843 .,s Equation-of-motion coupled-cluster (EOM-CC) theory1–6pro- vides a robust single-reference framework for computing multiple electronic states. EOM-CC affords a balanced description of states of different characters and a systematic improvement of results by incremental improvements in treating the electron correlation. The EOM-CC hierarchy of approximations is based on a standard hier- archy of CC models for the ground state, such as CC singles (CCS), approximate CC singles and doubles7(CC2), CC with singles and doubles2,8–10(CCSD), CC with singles, doubles, and triples (CC311 and CCSDT12–14), and so on. The EOM-CC formalism naturally extends to state and transition properties. Together, these features make EOM-CC an ideal framework for modeling spectroscopy. EOM-CC can be used to compute solvatochromic shifts,15transi- tion dipole moments,2spin–orbit16–19and non-adiabatic20–22cou- plings, photoionization cross sections,23,24and higher-order proper- ties25such as two-photon absorption cross sections26–29and static and dynamic polarizabilities.30–33The EOM-CC framework is being vigorously extended to the x-ray regime for modeling x-ray absorption (XAS), photoionization, and emission spectra,34–39as well as multi-photon phenomena such as resonant inelastic x-ray scattering40–44(RIXS) (Fig. 1). The suc- cessful extensions of EOM-CC to the x-ray domain exploit the core– valence separation (CVS) scheme,45which effectively addresses the key challenge in the theoretical treatment of core-level states: their resonance nature due to the coupling with the valence ionization continuum. Being embedded in the continuum, core-level states, strictly speaking, cannot be treated by methods developed for iso- lated bound states with L2-integrable wave functions.46–48Practi- cally, the coupling with the continuum leads to erratic and often divergent behavior of the solvers, a lack of systematic conver- gence of results with the basis-set increase, and often unphysical solutions.48,49 The CVS scheme allows one to separate the continuum of valence states from the core-level states through a deliberate pruning J. Chem. Phys. 153, 141104 (2020); doi: 10.1063/5.0020843 153, 141104-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsCOMMUNICATION scitation.org/journal/jcp FIG. 1 . In the coherent RIXS process, an x-ray photon of energy ω1(resonant with a core-excited state) is absorbed and another x-ray photon of energy ω2 is emitted. The difference between the incoming and outgoing photon energies equals the excitation energy of the final valence state frelative to the initial state g. The process is often described in terms of a transition via a virtual state (black dashed line), which represents collective contributions from all electronic states of the system, including valence bound states (solid black), valence resonances (green), core-excited states (blue), and valence continuum states (ultrafine gray dashes). of the Fock space, i.e., by removing configurations that can couple core-excited (or core-ionized) configurations with the valence con- tinuum (see the supplementary material for the analysis of different decay channels). CVS can be described as a diabatization proce- dure that separates the bound part of the resonance from the con- tinuum, akin to the Feshbach–Fano treatment of resonances.50–52 In contrast to other methods46–48for describing resonances, this approach, in which the continuum is simply projected out, does not involve state-specific parameterization (such as, for example, tuning the strength of complex absorbing potential for each state) and offers an additional benefit of eliminating the need to compute lower-lying electronic states. Although separation of the full Hilbert space into bound and continuum parts is not well defined in general, it is possible in the case of core-level states because they are Feshbach resonances, which can only decay by a two-electron process in which one elec- tron fills the core hole, liberating sufficient energy to eject another electron.38,49This special feature allows one to separate the contin- uum decay channels from the core-level states in the Fock space, by removing the configurations that couple the resonances with the continuum. Such pruning of the Fock space is equivalent to omitting the couplings between the core and valence excited (or ionized) determinants from the model EOM-CC Hamiltonian. The eigenstates of this reduced EOM-CC Hamiltonian are either purely valence or purely core excited (or ionized). The pure core-excited (or ionized) states become formally bound because of the decou- pling from the valence determinants forming the continuum. Con- sequently, diagonalization of the core block of this CVS-EOM-CCHamiltonian yields core-excited (or core-ionized) states without any convergence issues. Initially used to describe transition energies and strengths of core-level states, the CVS scheme was recently extended into the response domain41–43to enable calculations of RIXS transi- tion moments within damped linear response theory53–56and the EOM-CCSD method for excitation energies (EOM-EE-CCSD). In this approach, the response equations are solved in the truncated Fock space spanning the singly and doubly excited determinants in which at least one orbital belongs to the core,41,42in the same manner as done in CVS-EOM-CCSD calculations of core-excited or core-ionized states. Thus, all valence excited states are excluded from the response manifold, which can be justified by the resonant nature of the RIXS process. This truncation of the response mani- fold works well in many situations, but is expected to break down when off-resonance contributions to the RIXS cross section from the valence states become important. At least one class of systems where this happens is push–pull chromophores featuring low-lying charge-transfer states.43 Here, we address this limitation of the previous formula- tion of RIXS theory within the CVS-EOM-CCSD framework and present a general strategy for including valence contributions to the RIXS signal while preserving smooth convergence of the RIXS response states. This strategy can also be applied to the modeling of other multi-photon x-ray processes, such as x-ray two-photon absorption. By using this approach, we provide a quantitative illus- tration of the significance of the valence contributions to RIXS spectra. The derivation of the equations for RIXS transition moments between the initial ( g) and final ( f) states starts from the Kramers– Heisenberg–Dirac formula,57,58which translates to the following sum-over-states (SOS) expressions40–44within the EOM-EE-CCSD damped response theory: Mf←g xy(ω1x+iε,−ω2y−iε) =−∑ n(⟨Φ0Lf∣¯μy∣RnΦ0⟩⟨Φ0Ln∣¯μx∣RgΦ0⟩ En−Eg−ω1x−iε +⟨Φ0Lf∣¯μx∣RnΦ0⟩⟨Φ0Ln∣¯μy∣RgΦ0⟩ En−Eg+ω2y+iε) (1) and Mg←f xy(−ω1x+iε,ω2y−iε) =−∑ n(⟨Φ0Lg∣¯μx∣RnΦ0⟩⟨Φ0Ln∣¯μy∣RfΦ0⟩ En−Eg−ω1x+iε +⟨Φ0Lg∣¯μy∣RnΦ0⟩⟨Φ0Ln∣¯μx∣RfΦ0⟩ En−Eg+ω2y−iε). (2) Here,Tis an excitation operator containing the CCSD ampli- tudes, and ¯μ=e−TμeTis the similarity-transformed dipole moment. LnandRnare the EOM-CCSD left and right excitation opera- tors, respectively,59for state nwith energy En. Their amplitudes and eigenenergies are found by diagonalizing the EOM-CCSD similarity-transformed Hamiltonian H=e−THeTin the appropriate sector of the Fock space (i.e., singly and doubly excited determinants J. Chem. Phys. 153, 141104 (2020); doi: 10.1063/5.0020843 153, 141104-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsCOMMUNICATION scitation.org/journal/jcp in EOM-EE-CCSD) as follows: HRn=EnRnandLnH=LnEn. (3) iεis the phenomenological damping (or inverse lifetime) term from damped response theory. ω1xandω2yare the x-polarized absorbed andy-polarized emitted energies, respectively, satisfying the RIXS resonance condition, ω1−ω2=Ef−Eg. (4) We drop the Cartesian indices of the photon energies for brevity. The Fock space of EOM-EE-CCSD comprises the Slater deter- minants Φρ, whereρ∈{reference (0), CV,OV,COVV ,CCVV , OOVV } and C,O, and Vdenote the core occupied, valence occupied, and unoccupied orbitals, respectively. Upon inserting the identity operator ( 1=∑ρ|Φρ⟩⟨Φρ|) in Eqs. (1) and (2), we obtain Mf←g xy(ω1+iε,−ω2−iε) =−∑ ρ(⟨˜Df y∣Φρ⟩⟨Φρ∣Xg x,ω1+iε⟩+⟨˜Df x∣Φρ⟩⟨Φρ∣Xg y,−ω2−iε⟩) (5) and Mg←f xy(−ω1+iε,ω2−iε) =−∑ ρ(⟨˜Dg x∣Φρ⟩⟨Φρ∣Xf y,ω1−iε⟩+⟨˜Dg y∣Φρ⟩⟨Φρ∣Xf x,−ω2+iε⟩), (6) where the amplitudes of the intermediate ˜Dk yand the first-order response wave function Xk x,ω1+iεfor state kare given by26,41 ⟨˜Dk y∣Φρ⟩=⟨Φ0Lk∣¯μy∣Φρ⟩ (7) and ⟨Φρ∣Xk x,ω+iε⟩=∑ n⟨Φρ∣RnΦ0⟩⟨Φ0Ln∣¯μx∣RkΦ0⟩ En−Eg−(ω+iε). (8) The response wave function in Eq. (8) is complex-valued and given by a linear combination of all EOM-CCSD states. The amplitudes of this response wave function, expressed in the basis of Slater determinants, are solutions of the following response equation: ∑ ρ⟨Φν∣H−Eg−(ω+iε)∣Φρ⟩⟨Φρ∣Xk x,ω+iε⟩=⟨Φν∣Dk x⟩, (9) where the amplitudes of intermediate Dk xare given by ⟨Φν∣Dk x⟩=⟨Φν∣¯μx∣RkΦ0⟩. (10) Equation (9) is the direct result of using the identity operator 1=∑n∣RnΦ0⟩⟨Φ0Ln∣and the following resolvent: ∑ n⟨Φρ∣RnΦ0⟩⟨Φ0Ln∣Φν⟩ En−Eg−ω−iε=⟨Φρ∣(H−Eg−ω−iε)−1∣Φν⟩. (11) In practice, response equations such as Eq. (9) are solved iteratively using standard procedures, e.g., the Direct Inversion in the Iterative Subspace60(DIIS), which rely on diagonal pre- conditioners. In the x-ray regime, such response equations oftendiverge for the following reasons.41,42The RIXS response wave func- tion in Eq. (8) includes large contributions from high-lying core- excited states that are nearly resonant with the absorbed photon’s energy. These high-lying states are embedded in the valence ion- ization continuum and are Feshbach resonances that are metastable with respect to electron ejection. Mathematically, these core-excited states (and consequently, the response state) are strongly coupled to the continuum via a subset of doubly excited determinants, { OOVV } (formally, singly excited { OV} determinants can also couple core- excited states to the continuum, but one can show that the respective matrix elements are expected to be very small; this is discussed in the supplementary material). This coupling to the continuum leads to the oscillatory behavior of the leading response amplitudes with large magnitudes in the course of the iterative procedure. The prob- lem is exacerbated by the coupling of these doubly excited determi- nants to the valence resonances, which leads to the erratic behavior of pure valence doubly excited response amplitudes. Moreover, for x-ray energies, the diagonal preconditioner for the valence doubly excited amplitudes is no longer a good approximation to the valence doubles–doubles block of H−E0−(ω+iε), which further cripples the convergence. We note that lower-level theories such as CIS (configuration interaction singles)61and TD-DFT (time-dependent density func- tional theory)62–65do not have valence double excitations in the exci- tation manifold, so that the issue with the convergence of response equations in the x-ray regime does not arise, simply because there is no coupling between Feshbach resonances and the valence con- tinuum. Similarly, this issue does not arise for theories such as ADC(2)66,67and CC2 in which the doubly excited configurations are not coupled with each other41,42(the doubles–doubles block is diagonal), so the diagonal preconditioner for the doubles–doubles block is exact and the effect of double excitations is evaluated perturbatively rather than iteratively. The first practical solution to these convergence problems, which stem from the physics of core-level states and plague higher- level many-body approaches, was introduced in Refs. 41 and 42 and implemented within the CVS-EOM-EE-CCSD framework. The tar- get EOM states obtained by diagonalizing the similarity-transformed CVS-EOM-EE-CCSD Hamiltonian ( H) are either purely valence excited or purely core excited. To distinguish between the full EOM- CCSD Hamiltonian, eigen- and response states from those of the CVS-EOM-CCSD method, we use different notations: H,R,L,E, andXfor full EOM-CCSD and H,R,L,E, and Xfor CVS-EOM- CCSD. Equation (8) in terms of these CVS-EOM-EE-CCSD states is given by the following direct sum: ⟨Φρ∣Xk x,ω+iε⟩≈⟨Φξ∣Xk,val x,ω+iε⟩⊕⟨Φλ∣Xk,core x,ω+iε⟩; (12) ⟨Φξ∣Xk,val x,ω+iε⟩=val ∑ n⟨Φξ∣RnΦ0⟩⟨Φ0Ln∣¯μx∣RkΦ0⟩ En−Eg−(ω+iε); (13) ⟨Φλ∣Xk,core x,ω+iε⟩=core ∑ n⟨Φλ∣RnΦ0⟩⟨Φ0Ln∣¯μx∣RkΦ0⟩ En−Eg−(ω+iε), (14) whereξspans the reference and the valence excitation manifold (OV andOOVV configurations), λspans the core-excitation man- ifold ( CV,COVV , and CCVV configurations), and ρdenotes the J. Chem. Phys. 153, 141104 (2020); doi: 10.1063/5.0020843 153, 141104-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsCOMMUNICATION scitation.org/journal/jcp entire manifold of the reference and singly and doubly excited deter- minants. The LandRin the sum over core-excited states are the left and right CVS-EOM-CCSD operators for the core-excited state nwith energy Enthat diagonalize the core CVS-EOM-EE-CCSD similarity-transformed Hamiltonian ( Hcore=e−THcoreeT).Tis the CCSD operator expressed in the space of singly and doubly excited determinants (either full space or valence only if the frozen-core variant of the theory is used). Similarly, the sum over valence states (including the reference CCSD state with energy Eg) involves the EOM operators that diagonalize the valence Hamiltonian ( Hval) with the core states projected out. In terms of resolvents, the amplitudes of the response state in Eq. (12) are given by ⟨Φξ∣Xk,val x,ω+iε⟩=∑ σ⟨Φξ∣(Hval−Eg−ω−iε)−1 ∣Φσ⟩⟨Φσ∣Dk x⟩, (15) ⟨Φλ∣Xk,core x,ω+iε⟩=∑ κ⟨Φλ∣(Hcore−Eg−ω−iε)−1∣Φκ⟩⟨Φκ∣Dk x⟩, (16) whereσspans the reference and the valence excitation manifold and κspans the core-excitation manifold. The core resolvent is given according to ⟨Φλ∣(Hcore−Eg−ω−iε)−1∣Φκ⟩=core ∑ n>0⟨Φλ∣RnΦ0⟩⟨Φ0Ln∣Φκ⟩ En−Eg−ω−iε =⟨Φχ∣(H−Eg−ω−iε)−1∣Φτ⟩⊖⟨Φ0∣Φ0⟩⟨Φ0(1 +Λκ)∣Φτ⟩ E0−Eg−ω−iε, (17) whereχandτspan the reference and the core-excitation space.41 In what follows, we discuss several approximations to the resolvent or, equivalently, to the response wave function: (i) com- plete exclusion of all pure valence excited determinants from the response manifold (CVS-0 approximation); (ii) exclusion of the pure valence doubly excited determinants from the response manifold (CVS-uS approximation); (iii) approximating the valence resolvent by CCS (CVS-CCS approximation); and (iv) approximating the valence resolvent by CC2 (CVS-CC2 approximation). Configura- tional analysis of the response states and various contributions to the RIXS cross sections, given in the supplementary material, shows that singly excited valence determinants are expected to play a dom- inant role while the effect of doubles is indirect, i.e., they enter the response states and affect the amplitudes of single excitations, but their contribution to the RIXS moments is small. These new approaches—CVS-uS, CVS-CCS, and CVS-CC2—are implemented in a development version of the Q-Chem electronic structure pack- age.68,69Below, we provide a detailed description of and describe the rationale behind each approximation and compare their abil- ity to recover contributions from the valence states to the RIXS moments. In the approach presented in Ref. 41 (we call it the CVS-0 approach here), the sum in Eq. (12) spans over core-excited states only. This truncation is justified because the contributions of nearly resonant core-excited states are dominant for most RIXS transitions, owing to the resonant nature of RIXS. The numerical convergence ofthe response state, now approximated as ⟨Φρ∣Xk x,ω+iε⟩≈⟨Φλ∣Xk,core x,ω+iε⟩, (18) is smooth because the continuum has been projected out by CVS. Numeric benchmarks have shown that for systems for which the response equations could be converged with the standard EOM-EE- CCSD resolvent, this scheme results in the RIXS spectra that agree well with the full, untruncated calculation.41 Whereas this approach is justified for cases in which the dom- inant contributions to RIXS moments arise from nearly resonant core-excited states (the most common scenario), Ref. 43 has identi- fied cases in which valence states significantly contribute to the RIXS moments; these counterexamples are push–pull chromophores such aspara -nitroaniline and 4-amino-4′-nitrostilbene. The CVS-EOM- CCSD calculation with the CVS-0 resolvent omits these contribu- tions and is insufficient for modeling the RIXS spectra of such systems. The analysis of the various contributions to the response states, given in the supplementary material, identifies the terms that can become large for charge-transfer transitions and shows that these terms originate in singly excited valence configurations. Here, we provide a general strategy for including the contributions from valence states to the damped linear response in the x-ray regime, while preventing the direct coupling of response states with the con- tinuum and thus preserving the robust convergence of response equations. The response equation for Xk,valis given by ∑ ξ⟨Φσ∣Hval−Eg−(ω+iε)∣Φξ⟩⟨Φξ∣Xk,val x,ω+iε⟩=⟨Φσ∣Dk x⟩. (19) As explained above, the convergence of this response equation is compromised due to the coupling of valence resonances with the continuum and the use of a diagonal preconditioner for the OOVV – OOVV block of Hval. The continuum can be projected out by removing the singles–doubles and doubles–singles blocks from the CVS-EOM-EE-CCSD Hval. By further ignoring the doubles–doubles block of Hval, the corresponding preconditioner is no longer needed in the iterative procedure. Indices ξandσnow span just the refer- ence and the OVexcited configurations, reducing the resolvent such thatXk,valis given as a linear combination of states obtained by the action of EOM operators R=r0+R1andL=L1on the CCSD ref- erence. The response equation for this CVS plus valence uncoupled singles approach (we call it the CVS-uS approach), which effectively ignores the doubles amplitudes of Xk,val, is given by 0,OV ∑ ξ⟨Φσ∣Hval−Eg−(ω+iε)∣Φξ⟩⟨Φξ∣Xk,val x,ω+iε⟩=⟨Φσ∣Dk x⟩. (20) Note that it is not sufficient to simply remove the OOVV block fromDk xin Eq. (19)—this does not project out the valence reso- nances from the continuum nor does it exclude the use of the prob- lematic doubles–doubles preconditioner. Therefore, a viable strat- egy for the inclusion of valence contribution to the RIXS moments must involve the modification of the response wave function and the model Hamiltonian (or the resolvent). J. Chem. Phys. 153, 141104 (2020); doi: 10.1063/5.0020843 153, 141104-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsCOMMUNICATION scitation.org/journal/jcp Another way to include the contributions from valence states approximated by single excitations is to express the response state in Eq. (13) in terms of the valence CVS-EOM-EE-CCS intermedi- ate states with the CCS reference. This amounts to replacing the valence CVS-EOM-EE-CCSD resolvent in Eq. (13) by the valence CVS-EOM-EE-CCS resolvent (note the change from H,E, and Xto H,E, and X), ⟨Φξ∣Xk,val x,ω+iε⟩≈⟨Φξ∣Xk,val x,ω+iε⟩, (21) 0,OV ∑ ξ⟨Φσ∣Hval−E0−(Eg−E0)−(ω+iε)∣Φξ⟩⟨Φξ∣Xk,val x,ω+iε⟩=⟨Φσ∣Dk x⟩, (22) where Hvalis the CVS-EOM-EE-CCS similarity-transformed Hamil- tonian, E0is the energy of the CCS reference, and Xk,valis the corre- sponding approximate Xk,val. Because the doubly excited configura- tions are not present in this low-level CVS-EOM-EE-CCS Hamil- tonian, its eigenstates do not couple with the continuum, which, as discussed above, precludes the problematic convergence of RIXS response states. We call this the CVS-CCS approach. Yet another alternative entails replacing the valence CVS- EOM-EE-CCSD resolvent by the valence resolvent from the CVS- EOM-EE-CC2 model; we call this the CVS-CC2 approach. The valence CVS-EOM-EE-CC2 Hamiltonian has a diagonal doubles– doubles block; therefore, the doubly excited configurations do not couple with each other and their contribution to the response is evaluated perturbatively. Indeed, as noted in Refs. 41 and 42, the CC2 linear response RIXS solutions do not diverge. We exploit this feature of CC2 to compute the valence response amplitudes in Eq. (22). The strategy of replacing the valence CVS-EOM-EE-CCSD resolvent by a resolvent of a lower-level method that avoids the erratic convergence of RIXS response solutions can be extended to CIS, ADC(2), and DFT resolvents, for example. Importantly, the core and valence resolvents can be cherry-picked separately after enforcing CVS. This “cherry-picking-of-resolvents” strategy exploits the better convergence of x-ray response equations within the frame- work of the lower-level method, while using the better energies and wave functions of the initial and final states computed at the higher EOM-CCSD level of theory. We note that, because currently there is no method that can provide a higher-level treatment of RIXS within CC theory, there is no simple way to benchmark the differ- ent approximate treatments of the valence contributions to the RIXS moments. However, the variations between different models pro- vide a rough estimate of the uncertainties in theoretical treatments and of the magnitude of valence contributions to the RIXS spec- tra. Moreover, the configurational analysis of response states given in the supplementary material provides theoretical justification for hierarchical expansion of the response Fock space in the CVS-0 → CVS-uS/CCS →CVS-CC2 series. We begin by comparing the RIXS spectra computed using different resolvents for a well-behaved case: benzene. The RIXS spectrum of benzene with the CVS-0 approach is discussed in detail in Ref. 41. Figure 2 shows the spectra computed for resonant exci- tation of the two brightest XAS peaks, peak A corresponding to a core→π∗transition and peak B corresponding to a core →Ry transition. As one can see, the CVS-0 approach captures all the FIG. 2 . RIXS emission spectra for benzene computed using different resol- vents within the CVS-EOM-EE-CCSD framework with (top) pumping peak A (285.97 eV) and (bottom) pumping peak B (287.80 eV). Scattering angle θ= 0○, 6-311(2+,+)G∗∗(uC) basis.41,70The spectra are convoluted using normalized Gaussians (FWHM = 0.25 eV). main features in the emission following peak-A excitation, and the inclusion of the valence excitations has negligible effect. For peak- B excitation, small differences appear in minor features relative to the CVS-0 results; the off-resonance valence contributions become more important due to the Rydberg character of the particle orbital and the smaller oscillator strength for the peak-B core excitation. The CVS-CCS and CVS-uS approaches yield similar peak intensi- ties and differ slightly from the CVS-CC2 results. Another exam- ple (water) is given in the supplementary material. In this case, the response equations converge without approximations,40–42and one can compare the RIXS spectrum computed with the full untrun- cated EOM-CCSD resolvent against CVS-0, CVS-uS, CVS-CCS, and CVS-CC2. In agreement with the previous observation41that CVS-0 yields a RIXS spectrum that is very similar to the one obtained in the full treatment, we observe close agreement between CVS-0, CVS- uS, CVS-CCS, and CVS-CC2; thus, in water, the effect of valence contributions is minor. J. Chem. Phys. 153, 141104 (2020); doi: 10.1063/5.0020843 153, 141104-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsCOMMUNICATION scitation.org/journal/jcp FIG. 3 . RIXS emission spectra for para-nitroaniline for the incident photon res- onant with the lowest core excitation (XA 1→B2) computed within the CVS- EOM-EE-CCSD framework using different resolvents. Scattering angle θ= 0○, 6-311++G∗∗(uC) basis. The spectra are convoluted using normalized Gaussian functions (FWHM = 0.25 eV). Figure 3 compares the RIXS emission spectra for para - nitroaniline41computed using the CVS-EOM-EE-CCSD framework with different resolvents. Here, the incident photon’s energy is res- onant with the lowest XA 1→B2core excitation at 285.88 eV, which corresponds to the dominant x-ray absorption peak (see the supplementary material). The RIXS spectrum computed with the CVS-0 approach shows a few small inelastic features. These features correspond to the XA 1 →1B2, XA 1→2A2, XA 1→3B1, and XA 1→4A2transitions at 4.68 eV, 5.91 eV, 6.42 eV, and 6.95 eV energy loss, respectively; XA 1→1B2being the dominant transition (see the supplementary material for raw data). The spectra computed with the CVS-uS and CVS-CCS approaches are similar. However, these two approaches give differ- ent dominant features compared to the CVS-0 approach. Although the XA 1→1B2transition—dominant in the RIXS spectra with the CVS-0 approach—is still important, it is no longer the dominant feature with these approaches. Rather, the two dominant features arise from the XA 1→3B1and XA 1→2B1transitions at 6.46 eV and 5.96 eV, respectively. Other transitions, such as the XA 1→1B1 at 4.64 eV, XA 1→4B2at 6.79 eV, and XA 1→5B2at 7.21 eV, also give non-negligible contributions to the RIXS spectra. The difference between CVS-0 and CVS-uS/CCS arise from the contributions from valence singly excited determinants to the response states, which result in large contributions to the RIXS moments in the case of charge-transfer final states (this is explained in the supplementary material). Next, we compare the spectra obtained with the CVS-0 and CVS-CC2 approaches. Whereas the former spectrum is dominated by the XA 1→1B2transition, the latter shows additional transitions such as XA 1→1B1, XA 1→2B1, XA 1→4B2, XA 1→3A2(at 6.85 eV), and XA 1→5B2. We note that the relative cross sections of these RIXS transitions also differ, with the cross section of the XA 1→2B1 transition being the largest.The comparison of the RIXS spectra of para -nitroaniline in Fig. 3 highlights the significance of off-resonance contributions from the valence states. The RIXS spectra computed with the CVS- CC2 approach differs significantly from the CVS-CCS and CVS-uS approaches. This is not surprising, because, as it is well known from previous benchmarks,71the valence two-photon absorption cross sections with CCS and CC2 response theory show significant dif- ferences. The choice of the valence resolvent within this framework, in addition to comparison with experiments, is subject to the ability of the model valence Hamiltonian to provide a balanced description of the full spectrum of states of the system. In conclusion, we have presented a novel general strategy for including valence contributions into RIXS moments while pre- serving the smooth convergence of the response states in the x- ray regime. Whereas the iterative procedure for computing EOM- EE-CCSD response states typically diverges in the x-ray regime due to the coupling of response states with the continuum, our strategy mitigates this issue by exploiting the CVS scheme that decouples the valence excitation block from the core-excitation block of the EOM-EE-CCSD Hamiltonian, which facilitates com- puting their contributions to the RIXS response separately. Refs. 41 and 42 have previously presented an EOM-EE-CCSD damped response theory approach that employs a damped CVS-EOM-EE- CCSD resolvent for computing the contribution from the core- excited states. Here, we introduce a more general strategy to also include the contributions from valence excited states. This strat- egy involves the replacement of the valence CVS-EOM-EE-CCSD resolvent in the expression of the response state with a resol- vent from a lower-level theory (such as CVS-EOM-CCS or CVS- EOM-CC2) for which the response equations do not diverge or the restriction of the valence CVS-EOM-EE-CCSD resolvent to the singly excited determinant space. We demonstrated the sig- nificance of including this off-resonance valence contribution to the RIXS cross section by comparing the RIXS emission spectra of para -nitroaniline computed with and without the different valence resolvents. The supplementary material contains the configurational anal- ysis of RIXS response states and their contributions to the RIXS moments; raw data for RIXS spectra of water, benzene, and para - nitroaniline; RIXS spectra for para -nitroaniline computed with dif- ferent basis sets; XAS data for para -nitroaniline; and the geometries and basis sets used. 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1.5007182.pdf

Merging magnetic droplets by a magnetic field pulse Chengjie Wang , Dun Xiao , and Yaowen Liu Citation: AIP Advances 8, 056021 (2018); View online: https://doi.org/10.1063/1.5007182 View Table of Contents: http://aip.scitation.org/toc/adv/8/5 Published by the American Institute of Physics Articles you may be interested in The design and verification of MuMax3 AIP Advances 4, 107133 (2014); 10.1063/1.4899186 Effect of external magnetic field on locking range of spintronic feedback nano oscillator AIP Advances 8, 056010 (2018); 10.1063/1.5007324 Size and temperature dependence of M-H loop for Pt/CoFe/IrMn heterojunction AIP Advances 8, 056012 (2018); 10.1063/1.5006322 Spin-orbit torques and Dzyaloshinskii-Moriya interaction in PtMn/[Co/Ni] heterostructures Applied Physics Letters 111, 182412 (2017); 10.1063/1.5005593 Eigenmodes of Néel skyrmions in ultrathin magnetic films AIP Advances 7, 055212 (2017); 10.1063/1.4983806 Making the Dzyaloshinskii-Moriya interaction visible Applied Physics Letters 110, 242402 (2017); 10.1063/1.4985649AIP ADV ANCES 8, 056021 (2018) Merging magnetic droplets by a magnetic field pulse Chengjie Wang, Dun Xiao, and Yaowen Liua Shanghai Key Laboratory of Special Artificial Microstructure Materials and Technology, School of Physics Science and Engineering, Tongji University, Shanghai 200092, China (Presented 9 November 2017; received 1 October 2017; accepted 14 November 2017; published online 2 January 2018) Reliable manipulation of magnetic droplets is of immense importance for their appli- cations in spin torque oscillators. Using micromagnetic simulations, we find that the antiphase precession state, which originates in the dynamic dipolar interaction effect, is a favorable stable state for two magnetic droplets nucleated at two identical nano- contacts. A magnetic field pulse can be used to destroy their stability and merge them into a big droplet. The merging process strongly depends on the pulse width as well as the pulse strength. © 2017 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/1.5007182 I. INTRODUCTION Spin torque oscillators (STOs) with nanoscale electrical contacts (NCs) have attracted great attentions due to their application in microwave generation.1–5Under sufficiently large currents, the intrinsic damping torque can be compensated by the spin-transfer torque (STT)6,7generated in such contacts in the free layer of spin valves or magnetic tunnel junctions, giving rise to coherent oscillations. For a STO magnetized in the film plane, a self-localized “bullet mode” can be excited depending on the applied magnetic field angle.8,9Besides, another type of magnetic vortex-based oscillation mode can also be excited.10In contrast, when the free layer is magnetized perpendicularly to the film plane, Slonczewski predicted that radially propagating spin waves can be excited.11More recently, a strongly self-localized oscillation mode the dissipative droplet soliton has been theoreti- cally predicted12,13and experimentally observed in a NC-STO that has a free layer with perpendicular magnetic anisotropy (PMA).14,15In this case, the STT can fully or partially switch the magnetization beneath the NC, resulting in a coherent magnetization precession in a ring-shape region bounding the reversed magnetization.16–22It can enhance the output power by a factor of 40 compared to the uniform excitation mode,14suggesting the droplet-based STOs being promising candidate for STO applications. Physically, the magnetic droplet is not a topologically stable state and its stability is mainly dominated by a balance between the damping dissipation and the energy from the current- induced STT effect.17We found that two magnetic droplets nucleated at two identical NCs could lose stability and merge into a larger single droplet when the energy balance is destroyed.23In this paper, we systematically investigate the stability of a droplet pair, and find that the antiphase precession state is a favorable stable configuration. We further study the merging process triggered by a magnetic field pulse. II. MODEL As shown in Fig. 1(a), we consider a NC-STO geometry based on a pseudo spin-valve struc- ture that consists of a spin polarizer layer (PL) and a free layer (FL). Both the PL and FL have a strong PMA. The lateral dimension is 700 512 nm2. The FL thickness is 2 nm, with two 70-nm circular NCs (namely, NC1 and NC2) on the top. The separation dvaries from 200 to aCorresponding author. Email: yaowen@tongji.edu.cn 2158-3226/2018/8(5)/056021/5 8, 056021-1 ©Author(s) 2017 056021-2 Wang, Xiao, and Liu AIP Advances 8, 056021 (2018) FIG. 1. (a) Schematic diagram of a STO with two NCs. (b) The applied bias magnetic field H 0, pulsed field H P, and current density J. (c) A droplet pair generated at the two NCs with d=190 nm. is the polar angle of the in-plane magnetization of droplets. (d) Schematic diagram of magnetic charge distribution .is the spatial azimuthal coordinate to designate the location of magnetic charges . (e) Time dependent phase difference
for droplet pairs with different separations. (f) Dipolar interaction energy distribution as a function of ( 1,2). 260 nm. The positive current is defined as the current flowing from PL to FL, which supports the antiparallel magnetization configuration. The magnetic field H0is applied in +z direction. A magnetic field pulse Hpis used to trigger the merging process of droplets. The micromagnetic simulations are performed using Mumax324and a unit cell of 2 22 nm3is used. The magneti- zation dynamics of the free layer is described by the Landau-Lifshitz-Gilbert-Slonczewski (LLGS) equation:6,25 dm dt= mHeff+mdm dt+aJm(mp) (1) where magnetization m=M/Msand M sis the saturation magnetization, is the gyromagnetic ratio,  is the Gilbert damping constant, Heffis the total effective field that includes the exchange, anisotropy, demagnetizing, and external magnetic fields. The last term of Eq. (1) describes the STT effect with the torque factor aJ. The following parameters are used for the free layer:14,16Ms= 7.16 105A/m, A=310-11J/m (exchange stiffness), Ku=4.47 105J/m3(uniaxial anisotropy constant), P=0.5 (spin polarization) and =0.03 (Gilbert constant). The current applied into each NC is 8 mA (i.e current density J= 2.08 1012J/m2) and0H0=0.9 T, unless something else is stated. We ignored thermal effects and the current-induced Oersted field. III. SIMULATION RESULTS A. Antiphase precession state First, we find that the antiphase precession state is a favorable state for a droplet pair nucle- ated at two identical NCs. This feature is independent of the NC size, separation distance, as well as the initial phase angle of magnetization. Here, the phase difference,
=|1-2|, is used to identify the stability of the droplet pair, where 1(2) is the polar angle of the in-plane magnetization component at NC1 (NC2) perimeter region, as illustrated in Fig. 1(c). The final steady state is achieved after running enough time ( >100 ns). Fig. 1(e) shows the evolution of phase difference
for droplet pairs with different separation dranging from 200 to 260 nm. The simulations start from an in-phase initial precession state (
= 0) and an antiphase preces- sion state (
=) will be achieved if the simulation time is long enough. The required transfer time from in-phase to antiphase precession state becomes longer when the separation distance increases.056021-3 Wang, Xiao, and Liu AIP Advances 8, 056021 (2018) The favorable antiphase state of a droplet pair can be roughly explained on the basis of a “rigid” droplet model. Theoretically, as illustrated in Fig. 1(d), we assume that the droplet structure can be divided into three parts. The center part within the NC is marked as the part-I, in which the mag- netization has been reversed to -z direction by the STT effect. The part-II region is located at the perimeter region of the NC at which the magnetization has an in-plane component and precesses around the z-axis. The rest outside of the NC is defined as the part-III, in which magnetization remains along the initial +z direction. We consider that the magnetic dipolar (magnetostatic) interac- tions affecting the magnetic dynamics of the droplet pair mainly originates from the spins at the part-II region.26The dipolar interactions can therefore be calculated by integrating the transient magneto- static energy due to the magnetic charges generated by the in-plane magnetization.27It can be further simplified as:28 ES=kX i,j1,i2,j=ri,j,(k>0) (2) where the magnetic charges caused by the magnetization at perimeters of NC1 and NC2 can be written as 1,i=0MScos(1 i) and 2,j=0MScos(2 j).k>0 is a prefactor associated with the total magnetization. Fig. 1(f) shows the calculated interaction energy distribution as a function of the magnetic droplet phase ( 1,2). Apparently, the magnetostatic energy has minima at
=(i.e. antiphase state), supporting the conclusion that without other external driving sources the magnetic droplet pair prefers the antiphase precession state. B. Merging the droplet pair However, the antiphase precession of a droplet pair will in principle reduce the output sig- nal of STOs, which is bad for applications. One approach to solving this problem can be done by using phase-locking technique through a microwave oscillating magnetic field to synchronize the multiple droplets.26However, another study reported that droplets cannot be locked to an external microwave ac current signal.29On the other hand, merging multiple droplets into a large droplet may also enhance the output signal because the precession region (part-II) is enlarged.23In the following, we will show that a magnetic field pulse Hpapplied in –z direction can merge the droplet. Fig. 2(a) and (b) show the merging process of a droplet pair into a large droplet. In this study, we performed a series of simulations by using field pulses with a fixed strength ( 0Hp= -0.9 T) and FIG. 2. (a) and (b) The pulse duration dependent merge process for a droplet pair with antiphase precession state. (c) The velocity of size dilation for an individual droplet driven by different biased H0and pulsed Hp.056021-4 Wang, Xiao, and Liu AIP Advances 8, 056021 (2018) FIG. 3. The dependence of the critical pulse duration con the applied pulse strength H pand the bias magnetic field H 0for two droplets with initial (a) antiphase precession state and (b) in-phase precession state. For a given H 0, a field pulse taken from the parameter value larger than each ccurve can trigger the merging process. various pulse durations . The merging process is characterized by the averaged z-component <mz> of the whole free layer magnetization. All simulations start from an initial antiphase droplet state. We found that there is a critical pulse duration c(=11 ns) for the merging process. For <c, the field pulse is not strong enough to destroy the initial equilibrium of the droplet pair. Therefore, only a slight decrease in <mz>is observed during the field pulse, e.g. see the curve at =10.9 ns. The size of the two droplets has dilated [snapshots at t=100 and 110.9 ns in Fig. 2(b)]. After the pulse, the droplet pair quickly recovers to the initial antiphase precession state. Further increasing the pulse duration (e.g. =11.3 ns) will drive the droplet pair merging into a big-sized droplet. This merged big droplet is a metastable state, which can remain after the field pulse. Also, an obvious drop in <mz> is observed after the end of the pulse. The size dilation of droplets is a key factor for this merging process. Fig.2(c) shows the size dilation velocity of an individual droplet excited by different H0andHp. Here the dilation velocity vis defined as the averaged radius change of the droplet during a period of the pulse field. The net magnetic field 0Hnet(=0H0+0Hp) is used to characterize this parameter. Clearly, the dilation velocity increases with the bias field and decreases with increasing net magnetic field. Compared with the merging process for the droplet pair with the initial in-phase precession (
= 0),23the antiphase droplet pair (
=) requires a relative wide pulse duration. Fig. 3(a) shows the dependence of the critical duration con the constant field H0and pulsed field Hp. For comparison, we also show the duration for a droplet pair with an initial in-phase precession state, see Fig. 3(b). The typical features have been observed as follows: i) For a given H 0,cdecreases with increased pulse strength H p. ii) For a given combination of H0andHp, a droplet pair with antiphase state usually requires longer pulse duration. For example, at 0H0= 1.1 T and 0Hp=1.0 T, cis 20 ns for the initial antiphase state, while c= 2 ns for the in-phase case. This mainly originates from the fact that the antiphase droplet pair requires a longer incubation period to break their energy equilibrium, because the phase transformation from an antiphase droplet pair (
=) to a merged big droplet (
= 0) is more difficult. Especially, the antiphase droplet pair needs to break a Bloch wall formed between the in-side of two droplets before merging.21iii) Interestingly, the critical duration of the anti-phase pair can also be reduced down to 5 ns if the net field 0Hnet<0. In addition, there is a singular point at 0Hnet= 0.01 T for all the curves shown in Fig. 3(a), at which the cis at least 43 ns. This singular point is still not understood. IV. CONCLUSION We show that antiphase precession state is a favorable configuration for a droplet pair nucleated at two identical NCs. The droplet pair can be emerged into a larger droplet by a magnetic field pulse when the pulse duration and strength are sufficiently larger than critical values.056021-5 Wang, Xiao, and Liu AIP Advances 8, 056021 (2018) ACKNOWLEDGMENTS The authors would like to thank Yan Zhou for valuable discussions. This work is supported by the National Basic Research Program of China (2015CB921501) and the National Natural Science Foundation of China (Grant No. 51471118 and No.11774260). 1W. H. Rippard, M. R. Pufall, S. Kaka, S. E. Russek, and T. J. Silva, Phys. Rev. Lett. 92, 027201 (2004). 2S. Kaka, M. R. Pufall, W. H. Rippard, T. J. Silva, S. E. Russek, and J. A. Katine, Nature 437, 389 (2005). 3F. B. Mancoff, N. D. Rizzo, B. N. Engel, and S. Tehrani, Nature 437, 393 (2005). 4T. J. Silva and W. H. Rippard, J. Magn. Magn. Mater. 320, 1260 (2008). 5T. Chen, R. K. Dumas, A. Eklund, P. K. Muduli, A. Houshang, A. A. Awad, P. D ¨urrenfeld, B. G. Malm, A. Rusu, and J. Åkerman, Proc. IEEE 104, 1919 (2016). 6J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). 7L. Berger, Phys. Rev. B 54, 9353 (1996). 8A. Slavin and V . Tiberkevich, Phys. Rev. Lett. 95, 237201 (2005). 9E. Iacocca, P. D ¨urrenfeld, O. Heinonen, J. Åkerman, and R. K. Dumas, Phys. Rev. B 91, 104405 (2015). 10V . S. Pribiag, I. N. Krivorotov, G. D. Fuchs, P. M. Braganca, O. Ozatay, J. C. Sankey, D. C. Ralph, and R. A. Buhrman, Nat. Phys. 3, 498 (2007). 11J. C. Slonczewski, J. Magn. Magn. Mater. 195, L261 (1999). 12M. A. Hoefer, T. J. Silva, and M. W. Keller, Phys. Rev. B 82, 054432 (2010). 13M. A. Hoefer, M. Sommacal, and T. J. Silva, Phys. Rev. B 85, 214433 (2012). 14S. M. Mohseni, S. R. Sani, J. Persson, T. N. A. Nguyen, S. Chung, Y . Pogoryelov, P. K. Muduli, E. Iacocca, A. Eklund, R. K. Dumas, S. Bonetti, A. Deac, M. A. Hoefer, and J. Åkerman, Science 339, 1295 (2013). 15F. Maci `a, D. Backes, and A. D. Kent, Nat. Nano. 9, 992 (2014). 16E. Iacocca, R. K. Dumas, L. Bookman, M. Mohseni, S. Chung, M. A. Hoefer, and J. Åkerman, Phys. Rev. Lett. 112, 047201 (2014). 17S. Chung, S. M. Mohseni, S. R. Sani, E. Iacocca, R. K. Dumas, T. N. Anh Nguyen, Y . Pogoryelov, P. K. Muduli, A. Eklund, M. Hoefer, and J. Åkerman, J. Appl. Phys. 115, 172612 (2014). 18S. M. Mohseni, S. R. Sani, R. K. Dumas, J. Persson, T. N. Anh Nguyen, S. Chung, Y . Pogoryelov, P. K. Muduli, E. Iacocca, A. Eklund, and J. Åkerman, Physica B 435, 84 (2014). 19S. Chung, S. Majid Mohseni, A. Eklund, P. D ¨urrenfeld, M. Ranjbar, S. R. Sani, T. N. Anh Nguyen, R. K. Dumas, and J. Åkerman, Low Temp. Phys. 41, 833 (2015). 20S. Chung, A. Eklund, E. Iacocca, S. M. Mohseni, S. R. Sani, L. Bookman, M. A. Hoefer, R. K. Dumas, and J. Åkerman, Nat. Commun. 7, 11209 (2016). 21M. Carpentieri, R. Tomasello, R. Zivieri, and G. Finocchio, Sci. Rep. 5, 16184 (2015). 22D. Xiao, V . Tiberkevich, Y . H. Liu, Y . W. Liu, S. M. Mohseni, S. Chung, M. Ahlberg, A. N. Slavin, J. Åkerman, and Y . Zhou, Phys. Rev. B 95, 024106 (2017). 23D. Xiao, Y . Liu, Y . Zhou, S. M. Mohseni, S. Chung, and J. Åkerman, Phys. Rev. B 93, 094431 (2016). 24A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. G. Sanchez, and B. V . Waeyenberge, AIP Advances 4, 107133 (2014). 25X. Li, Z. Zhang, Q. Y . Jin, and Y . Liu, New J. Phys. 11, 023027 (2009). 26C. Wang, D. Xiao, Y . Zhou, J. Åkerman, and Y . Liu, AIP Advances 7, 056019 (2017). 27H. H. Chen, C. M. Lee, Z. Z. Zhang, Y . W. Liu, J. C. Wu, L. Horng, and C. R. Chang, Phys. Rev. B 93, 224410 (2016). 28H. H. Chen, C. M. Lee, J. C. Wu, L. Horng, C. R. Chang, and J. H. Chang, J. Appl. Phys. 115, 13 (2014). 29Y . Zhou, E. Iacocca, A. A. Awad, R. K. Dumas, F. C. Zhang, H. B. Braun, and J. Åkerman, Nat. Commun. 6, 8193 (2015).

1.165958.pdf

Transient chaos in room acoustics Fabrice Mortessagne, Olivier Legrand, and Didier Sornette Citation: Chaos 3, 529 (1993); doi: 10.1063/1.165958 View online: http://dx.doi.org/10.1063/1.165958 View Table of Contents: http://chaos.aip.org/resource/1/CHAOEH/v3/i4 Published by the American Institute of Physics. Additional information on Chaos Journal Homepage: http://chaos.aip.org/ Journal Information: http://chaos.aip.org/about/about_the_journal Top downloads: http://chaos.aip.org/features/most_downloaded Information for Authors: http://chaos.aip.org/authors Downloaded 17 Sep 2012 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://chaos.aip.org/about/rights_and_permissionsTransient chaos in room acoustics Fabrice Mortessagne, Olivier Legrand, and Didier Sornette Laboratoire de Physique de 10 Matiere Condensee, CNRS URA 190 Faeulte des Sciences, Pare Valrose, 06108 Nice Cedex 02, France (Received 18 June 1993; accepted for publication 13 September 1993) The decay of sound in an auditorium due to absorption is central to the theory and practice of room acoustics. Within geometrical acoustics, this problem involves the partial trapping of chaotic ray trajectories in billiards, hence transient chaos. We first present a theoretical and numerical analysis of the decay of rays in 2-D chaotic billiards (2-D room acoustic mOdels) and show that the existence of fluctuations (in the mean-free path and in the rate of phase space exploration) leads to modifications from the standard statistical theory. An ergodic wave theory of room acoustics based on a wave formulation is then discussed and tested by direct numerical calculations of the eigenmodes in 2-D billiards. Finally, we present a semiclassical calculation of the acoustic Green's functions in the time domain, based on a summation over rays, viewed as generalized wave impulses, and successfully compare its predictions with a direct numerical integration of the wave equation. This formalism provides a framework to link the geometrical ray description to the ergodic wave theory. I. INTRODUCTION Room acoustics is the study of the transient behavior of sound waves in an enclosure, be it a living room, a conference room, a theater, or a concert hall. In contrast to optics, where all wavelengths are very short with respect to life-size objects, in acoustics, one must face both numerous length scales which coexist in an auditorium and a wide band of excitation frequencies, from 20 Hz to 20 kHz (fre quency interval of the human auditory sensitivity) corre sponding to wavelengths from 17 m to 17 mm, with an air sound speed of 340 ms-l. Due to various causes of dissi pation, a sound impulse decays after being created within a characteristic time (called the reverberation time) of less than 1 s for office rooms to more than 10 s for Gothic churches (well suited for Gregorian chant), corresponding to ray length trajectories between a few hundred meters to kilometersl We thus indeed deal with transient phenomena but which can have, in some cases, quite long durations. The sound energy may be viewed (keeping only the first order of the eikonal approximation of the wave equation) as concentrated around geometrical trajectories, or as re sulting from a complex superposition of characteristic modes. Thus, room acoustics may be investigated through many different physical approaches. However, it is noteworthy to remark that there is up to now no fundamental theory of room acoustics based on first principles. The best that exist can be found in two extremes: (I) in rectangular or other simple room shapes, modal analysis analogous to comparable separable quan tum mechanical problems has been nicely worked out by Morse and Bolt I for instance, extending the formalism of quantum mechanics to account for dissipation; (2) in the other extreme case of very irregular shapes at very high frequencies or with very incoherent wave fields, a statistical analysis can be developed successfully. However, "real life" is in neither of these two extremes, and one would like at least a general formulation which would enable one to go from one regime to another, even if the practical com putations cannot be carried out directly from it. This glo-bal perspective would provide a better understanding of the various approximations involved in a given regime and help in delineating its corresponding domain of validity. It should also be able to suggest new useful methods of com putation. The nature of the difficulty in a direct wave analysis comes from various aspects: (1) the interaction between the sound wave and the walls is quite complex in general (role of wall roughness, dependence upon angle of inci dence, frequency, ... ); (2) the shape of the boundary is not simple (seats, rough ceiling, presence of performers and audience members) and, often, one has to take into ac count the effect of diffraction; (3) even in relatively simple shapes, it is known that the acoustic equation (wave or Helmholtz) is not separable and leads to the phenomenol ogy of quantum chaotic problems in the high-frequency regime. In view of these complexities, it is important to divide the problem into several simpler ones and consider simpli fied regimes. It was thus to be expected that geometrical acoustics should have been the first theory used by early workers. The acoustical engineer still uses, with a great deal of "common sense," geometrical acoustics (often even in cases where it should become suspect theoretically) and is quite successful in designing satisfactory auditoria and office rooms. As Knudsen2 said 60 years ago: "The approx imate theories; when used with caution and understanding) have served satisfactorily for the practical purposes of acoustical designing, and they will continue to do so until they are superseded by more exact theories." It should also be noted that there is practically no general wave approach to the problem. We feel that now is the time for more exact theories inspired and generalized from the field of statisti cal physics and chaotic dynamics. It is thus the purpose of the present work to present some introductory remarks toward a theoretical formulation of room acoustics which could provide the sought unifying perspective. We will first present a brief historical perspective which allows one to grasp better the complexities and spec ificities of room acoustics (Sec. II A). This story is mainly CHAOS 3 (4), 1993 1 054·1500/93/3(4)/529/13/$6.00 @ 1993 American Institute of Physics 529 Downloaded 17 Sep 2012 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://chaos.aip.org/about/rights_and_permissions530 Mortessagne. Legrand, and Sornette: Transient chaos in room acoustics focused on the ray geometrical formulation of acoustic propagation. We then give results (Sec. II B) of a theoret ical and numerical analysis of two-dimensional (2-D) bil liards within the geometrical ray limit, which provide well defined paradigms for real room enclosures. This study allows us to test the various hypotheses and results usually accepted in the acoustical community. We find that the existence of fluctuations [in the mean-free path (Sec. II C) and in the rate of phase space exploration (Sec. II D) 1 leads to modifications from the standard theory. We then present a first attempt (Sec. III A) to develop a theory of room acoustics based on a wave formulation, in the limit where one can assume that the eigenmodes have the prop erty of ergodicity. This hypothesis is tested by direct nu merical calculations of the eigenmodes in 2-D rooms. We find that the existences of "scars" or local resonances may drastically alter the validity of the theory. Finally, we present a semiclassical theory for the calculation of acous tic Green's functions in the time domain (Sec. III B). It is based on a summation over rays, viewed as wave packet trajectories, taking into account the phase accumulated by the rays upon propagation. This formalism provides a framework to link the pure geometrical ray description to the pure ergodic wave theory. II. REVERBERATION WITHIN GEOMETRICAL ACOUSTICS A. A brief historical survey The main practical question in this field is to determine what are the objective properties a given room should pos sess in order to be suitable for its use, be it an office room or an auditorium.3 It is clear that the acoustic quality of a room obviously depends on aesthetic and psycho physiological criteria, but the acoustical engineer will worry about more objective factors which may serve to establish the concept of a "good" room on firm grounds. The hope is to possess a single variable, or at least a finite number of them,3 which allow one to characterize the acoustic quality of a room, and which provide unambigu ous criteria for deciding what makes a "good" auditorium. In the beginning of the century, the American physi cist Wallace Clement Sabine partly answered this query: 4 he experimentally showed that an important criterion is the more and less persistence of a sound after the source is stopped, which he called reverberation. This transient re sponse of a room corresponds to a regular damping of sound, and is not of the same nature as an echo. In order to measure this important effect, Sabine defined the rever beration time: the duration for the mean square pressure of a suitably chosen distribution of sound waves to diminish to one-millionth of its original intensity. One can see in Fig. 1 the important role played by the reverberation time for the quality of hearing. Nowadays, acousticians are also interested in other quantities which could improve the as sessment of the quality of a room. For instance, the ratio of the acoustic energy transmitted directly from the source to the audience to that transmitted after reflections (the so-FIG. 1. Sound level recording of reverberation processes during a concert (Beethoven's Coriolanus Overture, Op. 62, measures 9-13). The ordinate scale for the sound pressure level is in dB (i.e., logarithmic) and the abscissa axis is linear in time. The exponential decay after a chord burst is shown by a straight line. lethe decay time (proportional to the so·called reverberation time) was twice as large, for instance, the following rest would disappear resulting in a drastically altered auditory perception of the passage. called diffuse acoustic field) is thought to control the per ception of acoustic "depth.,,3 With the help of a stopwatch and of his ears, Sabine "showed" that the level, below which the audibility of the sound produced by a pistol shot is lost, is reached after about the same time at all/ocations in the room. Thus, this time is a good candidate for characterizing the room as a whole. To quantify the time of reverberation, Sabine used an organ and studied the increase in duration of reverber ation as an increasing number of pipes is used, always blown with the same air pressure. He noticed that a change from one to two pipes will lead to the same absolute in crease in duration as a change from two to four pipes. This behavior reveals an exponential decay law of the energy: dW dt -W=-; (I) W=Woe-th• To this experimental result, Sabine added certain sta tistical assumptions, namely: (I) uniform, diffuse distribu tion of energy throughout the room, (2) equal probability for all directions of propagation of sound, and (3) contin uous absorption of sound at the boundaries. With these assumptions, using a power and energy balance argument, Sabine wrote the now most famous formula of acoustics, called after his name, which yields the reverberation time T for a room of volume v.. 4V T= (61n IO)T= (61n 10) "" . <! ' c .. aJ¢Jk (2) where c is the velocity of the sound in the air, and aiP k is the equivalent absorption area for an absorber of true area Sk and absorption coefficient ak. The equivalent absorp tion area would be the actual area of the absorbing bound ary surface in the case of total absorption, or, as Sabine expressed it, the equivalent absorption area of an "open window." The factor 6 In 10 stems from the decibel units CHAOS. Vol. 3, No.4. 1993 Downloaded 17 Sep 2012 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://chaos.aip.org/about/rights_and_permissionsMortessagne, Legrand, and Sornette: Transient chaos in room acoustics 531 used in acoustics and is such that T is the time needed for an impulse to decrease in amplitude by 60 dB. This formula, also derived by Franklin,5 is quite re markable because the reverberation time is only deter mined by some global characteristics of the room (total volume and equivalent absorption area) and is independent of how the absorption is distributed on the enclosure and what the specific shape of the room is. It is important to stress the "universality" embedded in Sabine's law, i.e., the independence of T with respect to room shape or location of absorption. This relation is a tool of great interest for acoustic designers and acoustical engineers, but unsatisfac tory for the physicist. The main fault of Sabine's equation is that it gives no insight into how some possible departure from the above-mentioned assumptions would influence the reverberation and break down the universality. Fur thermore, this formula (2) is considered by practitioners as only a first-order approximation which does not account in detail for observations in many cases of practical inter est. The assumptions used by Sabine are clearly in the spirit of the geometrical limit of room acoustics. In this limit, one can consider the problem of sound energy decay as a deterministic problem of energy particles, which travel along geometrical sound rays and lose part of their energy each time they hit an absorbing surface. Thus one is led to introduce the concept of mean-free-path length between rebounds on the enclosure. In acoustic literature, Jaeger6 was the first to give the correct value of the mean-free-path iength (/): (I) =4 VIS, (3) where S is the total area of the boundary surface. Twenty years before Jaeger, Clausius,7 in the context of the newborn kinetic gas theory, had given the derivation ofEq. (3) which may be understood as an average over all particles in the room during the same time interval (en semble average); some five years before Clausius' deriva tion, Czuber' had given a purely geometrical proof of Eq. (3) (see also Ref. 9). Sabine tried to evaluate this mean-free-path length from the reverberation time that he measured in rooms for which he could calculate the equivalent absorption area on the basis of other experiments. For this purpose, he defined a mean sound absorption coefficient 0 as the arithmetic mean of the different absorbing surfaces: The reverberation time becomes (I) I T=61n 10--. c 0 (4) (5) A comparison of Eqs. (2) and (5) with (4) immediately yields relation (3) but neither Sabine nor Franklin came to this conclusion. We cite a brief anecdote taken from Cremer and Miiller:lO "( ... ) Sabine evaluated the mean-free-path length for two rectangular rooms of different size but similar shape: (I) =0.623,fV. (6) He expressed the length dimension by Vl/3, since, consid ering the very different shapes of auditoriums, theaters, churches, and so on, there was no reason to choose the dimension in any particular direction. But he did not rule out the possibility that thefactor 0.62 in Eq. (6), which he determined for the ratios 6:3:2 of length: width: height, might depend on the particular shape." If you do the com putation with this particular shape, Eq. (3) leads to (I) = 2 and Eq. (6) to (I) =2.04! Eyring",l2 proposed the first important modification of Sabine's law: (I) I T=61n 10 C [ -In(l-0) l . (7) The importance of Eyring's formula is well beyond the simple fact to replace 0 by [-In (1-0) l: this expression requires dropping the assumption of continuous absorption and replacing it by a discontinuous process of reduction of energy, occurring essentially at each rebound. Each sound ray possesses an initial energy, at each reflection the energy of the ray is multiplied by the constant factor (1-0). A very straightforward method for deriving Eq. (7) was first proposed by Norris:13 the average number of reflections in a time t being n=ctl(I), the decay law for the total sound energy (taken as the sum of the individual energies) reads [cln(l-0) 1 w(t) = Wo(l_0)ctl(/) = Wo exp IJ\ t (8) L \~, J which yields the expression (7) according to (I) and the first equality of relation (2). B. The chaotic billiards as paradigm of 2-D ergodic auditorla Let us avoid a misleading interpretation of the termi nology "sound rays" used above: in the true geometrical acoustic limit, we consider rays as trajectories of pointlike particles which carry energy. With a specular law of re flection (angle of incidence equals angle of reflection), the assumption of uniform and isotropic distribution of sound energy in the room is fulfilled when the (deterministic) dynamic of the rays trajectories is chaotic. In this regime where deterministic ray trajectories present such a strongly chaotic behavior mimicking so well a truly random Mar kovian dynamic, it is natural to expect that Sabine's law should hold. Indeed, about 15 years ago, Joyce" showed that "for almost any initial sound distribution, a sufficient condition of validity of Sabine's reverberation-time expres sion is that the enclosure be mixing and the absorption be weak. " The researchers, who study the decay of sound within geometrical acoustics,15-20 use various reflection laws rang ing from perfectly diffusing (Lambert'S law), assuming that the roughness of the wall is at a scale much smaller than the acoustic wavelength, to mirror reflecting. Here we will not use the assumption of diffuse reflection but treat the problem of sound reverberation through the investiga tion of the deterministic ray dynamics in chaotic 2-D CHAOS, Vol. 3, No.4, 1993 Downloaded 17 Sep 2012 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://chaos.aip.org/about/rights_and_permissions532 Mortessagne, Legrand, and Sarnette: Transient chaos in room acoustics 2 L "'5SIJ Reflections from: . ' FIG. 2. Ceiling profile for the Gennan Opera in Berlin. Top: proposal based on ray construction. Bottom: ceiling as built (architect, Borne mann; acoustical consultants, Cremer and Gabler). model rooms.2I,22 By comparison to a fully three dimensional (3-D) study, this restriction does not lead to any serious lack of generality. First, most of the billiards we consider preserve their chaotic properties under rota tion around one of their natural axes;23 second, the com~ mon acoustical engineering practice subdivides the room in various typical sections (Fig. 2), each of them being rep resented by a 2-D shape. Since the early work of Sinai,24 "chaotic billiard" was synonymous to "dispersing enclo sure," but such concave outward 2MD domains are not pre cisely reminiscent of sections of actual concert halls. As Bunimovich proved,23 most 2-D billiards with boundaries containing only focusing and neutral regular components (real auditoria often possess similar sections) display a truly deterministic ray dynamic which is strongly chaotic. Among the various shapes studied in the literature, the Bunimovich stadium23,2s is the paradigm of a nondefocus ing billiard which is chaotic in the strongest sense, namely ergodic, mixing, K-complex, and B-complex. At this stage, a brief explanation of this terminology is pertinent, since the corresponding ergodic properties underlie the estab lishment of Eq. (7) in the true geometrical acoustic limit. In dynamical systems, an "ergodic hierarchy" allows one to encompass their compiexities in universality classes:26 from the most regular to the most chaotic, this hierarchy reads: ergodic <= mixing <=K-system <= B-system. The billiard we consider here has the highest level of com plexity. The first level, namely ergodicity, is the most fa-miliar notion to physicists: equality between ensemble and time average. In an ergodic system, any phase space ele mentary volume tends asymptotically to wander, in a ho mogeneous way, through the whole phase space. In a mix ing system, any phase space elementary volume tends asymptotically to spread homogeneously over the whole space. This property expresses the rapid deformation of any initial volume element and may be compared to the process of dilution. A so-called chaotic system must at least be mixing. Indeed, ergodicity does not exclude inte grability. K-systems are mixing and in addition unstable in that orbits diverge with an average exponential rate in time. This property, often referred to as extreme sensitivity upon initial condition, destroys, from a practical point of view, the determinism of the system. A B-system stands as the ideal tool to mimic the ~~rouleiie" game. The break down of predictability is so important that, in such a sys tem, one is able to define a phase space partition such that points representative of an orbit fall in cells of the partition with no memory of the previous impact. Such a dynamical complexity as viewed in the abstract phase space is also present in configuration space, as is shown in Fig. 3. In our study, we use the "standard" stadium billiard, shown in Fig. 4, and made of two semicircular caps of radius R, joined by two rectilinear parts of!ength 2R. In an our simulations, we launch 106 particles isotropically from a single source point. We have checked that our results do not qualitatively depend on the position of the source. We switch on the absorption after a time long enough to ensure sufficient ergodization of the particles positions and veloc ities. For large absorption, this caution allows one to avoid very short transients which depend on the position of the source point. The total energy W(t) of all particles at a time t after turning on the absorption is then measured. In the 2-D case, expression (3) must be replaced by7,8,21 (I) =rrS/P, (9) where Sand P are, respectively, the surface and the perim eter of the enclosure. c. Generalization of exponential reverberation law Let us start with simple cases where the absorption is uniformly distributed over the enclosure. Figure 5 pre sents, for various values of the absorption a, the numeri cally obtained In[W(t)/Wol as function of tIT, where l' is the decay time as defined in (1) and relates to the rever beration time by the first equality in Eq. (2). In this rep resentation, the classical reverberation law (8) corre sponds to the straight "antidiagonal" line joining the left upper corner to the right lower corner of the picture, since by definition ct «rrS/P){II[ -In(l-a)]})' (10) The five curves are close to or even indistinguishable from the "antidiagonal," thus demonstrating the validity of the geometrical assumptions. The corresponding reverberation CHAOS, Vol. 3, No.4, 1993 Downloaded 17 Sep 2012 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://chaos.aip.org/about/rights_and_permissionsMortessagne. Legrand, and Barnette: Transient chaos in room acoustics 533 (c) FIG. 3. An illustration of the ergodic hierarchy. (a) A characteristic trajectory inside the circular billiard is plotted, exhibiting a caustic which delimits a region of the configuration space, which will never be visited by the trajectory; (b) a typical trajectory inside a stadium billiard with straight sides oflength E= 10-2 R (R is the radius of caps), this trajectory exemplifies the first level of ergodicity; (e) and Cd) are an illustration of the mixing property of the same billiard as (b): from a unique source point, 100 particles are launched within an angular sector of 10-12 fad; the initial state is shown in (e), corresponding to a very small area in phase space representation (equal positions and narrow sector angle); positions and propagating directions of the particles after an average number of rebounds of 4000 are shown in (d): the memory of initial conditions is lost (note that this effect is observable for any nonzero value of f). times predicted by Eyring's formula are (with R = 10 m and c=34O ms-'): T=88.2 s (a=O.OI); T=29.1 s (a =0.03); T=17.3 s (a=0.05); T=12.2 s (a=0.07); T=8.4 s (a=O.I). With lower values of a (corresponding to unrealistic values of 1'), the transient response of the room is large enough to ensure uniform and isotropic distribution of E=2R FIG. 4. The "standard" Bunimovich stadium (parallel sides with length € equal to twice the radius R of the circular caps). o 2 4 6 8 -2 -4 -6 -8L-----------------------~ FIG. 5. Numerically obtained In[W(t)/WoJ. where W(t)/Wo is the frac tion of the total energy of all rays which remains at time t, as a function of tiT where T(a) is the decay time as given by Eq. (10). Each curve is associated to a different value of the unifonn absorption coefficient. namely. a=O.OI. 0.03, 0.05, 0.07, 0.10. Note that this representation allows us to collapse all the curves. The five curves are close to or even indistinguishable from the "antidiagonal," thus demonstrating the valid ity of the geometrical assumptions leading to a pure exponential decay. sound. To reach a given energy level, the sound rays must encounter the walls more often when a is lower, thus all the average quantities used within the Sabine-Eyring ap proach are well defined. When a increases, since the dura tion of the transient response decreases, asymptotic prop erties are less relevant and deviations from the Sabine Eyring prediction due to fluctuations can be expected, as we now discuss. In Fig. 6, the result of our simulation is plotted for a=30% (T=2.5 s). We observe a pure exponential decay law, as expected, but with a decay rate lower than the one predicted by Eyring's law. This corresponding increase of the reverberation time may be understood following a typ- o -2 -4 - 6 2 4 , , , , , 6 , , 8 , , , -8.L--------- __________ ~ FIG. 6. Same as Fig. 5 with a=30%. We observe a pure exponential decay law. as expected, but with a decay time longer than the one pre dicted by the classical law (broken line) as given by Eq. (10). The correct decay time is now given by Eq. (13). CHAOS, Vol. 3, No.4, 1993 Downloaded 17 Sep 2012 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://chaos.aip.org/about/rights_and_permissions534 Mortessagne, Legrand, and Sornette: Transient chaos in room acoustics FIG. 7. Illustration of the fluctuations of free-path lengths between re bounds away from the average mean-free-path length, for a typical tra jectory in the standard stadium. ical trajectory, like the one shown in Fig. 7. It is obvious that the lengths from reflection to reflection exhibit large fluctuations. For a given trajectory length, some rays may have encountered the boundary less than others. At a given time t, the total amount of sound energy W(t) is the sum of the individual energies of all the particles. Thus the influence of the ray trajectories which have encountered the enclosure less often than the average dominates the decay rate of the total sound energy. In the derivation of expression (7), the "mean field" relation L=n(I) is used, which does not take into account 0=1 '" • " :E :c ~ '" 0 ~ '" 0 (a) trajectory length in units of DR n = 10 o 4 (b) trajectory length in units of DR the fluctuations of the lengths between successive re bounds. We are thus interested in characterizing the dis tribution Pn(L) of the length L with n rebounds. Given this distribution, the reverberation law becomes: -dW(t=~)=Wo~Pn(L)(I-a)ndL; (11) Pn(L) is plotted in Fig. 8 for increasing values of n. As n increases, the distribution tends to a Gaussian distribution. This can be accounted by the following argument.22 In the stadium, the decay of correlations is fast enough27 to allow the central limit theorem to apply for the variable L(n),28 leading to the normal distribution: Pn(L) I [(L-n(l) )2] ~21Tna(n)exp 2nQ2(n) ' (12) where if(n) is the normalized variance. For large values of n, if(n) converges to a constant ~ (for a more detailed account see Refs. 15 and 29). We solve Eq. (II) with the asymptotic expression (12) and obtain the new decay time, using a saddle point method: (e) (I) -,----....!r __ - c ~1-2?ln(!-a)-1 n = 100 .,ili o trajectory length in units of DR n = 1000 1111111111111 ",,'111111 trajectory length in units of DR (l3a) 4 70S 8 -+P rn FIG. 8. The distribution Pn(L) is shown for increasing values of n. Using a large enough sample of trajectories, the histogram of number of trajectories with lengths between Land L+I::J.L at the nth rebound is numerically evaluated (80 bins are used in the shown range of normalized lengths); (a) n= 1; (b) n= 10; (c) n= 100; (d) n= 1000. As n increases, the distribution tends to a normal Gaussian law. CHAOS, Vol. 3, No.4, 1993 Downloaded 17 Sep 2012 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://chaos.aip.org/about/rights_and_permissionsMortessagne, Legrand, and Sarnette: Transient chaos in room acoustics 535 1000.-----------------------, , 100 't""ill. Y c ";1+2/1n(1 -Ct) -I 10 1~~--~--~--~----~ 0,00 0,10 0,20 0,30 0,40 0,50 0,60 absol"ption FIG. 9. Effect of fluctuations in the number of encounters of cay trajec tories with the absorbing walls in the stadium. The square symbols give the dependence of the numerically obtained decay time as a function of the absorption coefficient a. In aU investigated cases, the energy decay is very well fitted by an exponential law. The continuous curve is the theo retical prediction for the decay time 7", given by Eqs. (13), with r=CTj(I). (13b) Our numerical estimation of the decay time, for different values of the absorption coefficient, is plotted on Fig. 9 and shows an excellent agreement with expression (13). Let us give another vision of reverberation. The rever beration time may be viewed as the lifetime of an initial metastable state which has a probability a to overpass a barrier. In this spirit, the mean-free-path length (divided by c) becomes the average time interval between successive encounters with the barrier. Within Sabine (Eyring) as sumptions the reverberation time depends, in an exponen tial way, on the absorption; equivalently, within standard mean field theory, the lifetime depends on the height of the barrier. Indeed, such physical situations appear in various contexts30,31 such as the various forms of nuclear radioactivity,32-36 atomic disintegrations,32,33 molecular dissociations, as a result of ergodicity theory in statistical mechanics,8,9,24 in nucleation phenomena,37 in transient chaos,38,39 and in coarse-grained properties of classical orbits.30,39,4O We have shown above how the reverberation time should be corrected due to the effect of fluctuations in the number of encounters of ray trajectories with the ab sorbing walls. In Ref. 29, we point out that fluctuations in , the number of encounters with the barrier renormalize the lifetime and call for a correction to the mean field predic tion, in these various physical situations. D_ Departure from the pure exponential reverberation law We address the role of the spatial distribution of ab sorption and test the hypothesis of invariance under spatial distribution of the absorption, which underlies the use of 0 in (7). Figure 10 presents, for a mean sound absorption value 0=2%, the time dependence of the total energy stored in the 2-D stadium. The curve (a) corresponds to an absorption uniformly spread on the perimeter of the stadium with au=2%; curve (b) corresponds to absorp tion localized on a short arc of length r in the right cir cular portion of the stadium, with an absorption coefficient ar=50%, where r is determined by the condition o to 2,0 30 -5 -10L-----~~------------~ FIG. 10. In[W(t)/W o] as a function of tiT, for two types of absorption distributions with the same value for the mean sound absorption coeffi cient 0=0.02. Curve "au.": absorption unjfonnly spread on the perimeter of the stadium with au.=0.02. Curve "ar": absorption localized on a short arc of length lr in the right circular portion of the stadium perimeter whh an absorption coefficient ar=O.5. (14) ensuring an equivalent decay time, according to Eq. (7). Note that in the presence of a highly nonuniform spatial distribution of absorption, as is the case here, acoustical engineers with their "common sense" do not use Eyring's formula but the Millington-Sette expression,41,42 which re lies on another absorption averaging. Both forms rely on Sabine geometric conditions,17 give the same qualitative results, and can be used alternatively for our purpose in this section. Curve (b) exhibits a power-law decay of the energy: W(t) ~rl. One must realize what would be the conse quence in acoustics of such a slow decay: a speech in such a room will be completely unintelligible, due to overlap between sentences separated in time (see the first figure). We notice here the particular effect played by special ray trajectories (called "bouncing balls"), which represent a family of neutrally stable nonisolated orbits.43 An example of such an orbit is shown in Fig. 11. A particle, which bounces back and forth between the two parallel sides of the stadium, remains trapped in a reduced fraction of phase space for a very long time. This feature does not contradict the ergodic properties of the stadium since the most extreme dynamical complexity would not forbid tran sient regUlarity. As mentioned above, the ergodic proper ties of chaotic billiards are asymptotic ones: therefore, dur ing a finite time, a trajectory may be trapped inside a finite part of phase space. The asymptotic ergodicity occurs since, after leaving the trapping region in phase space, the particle spends equal times in equal areas of phase space. Therefore, all the trajectories densely cover both the phase space and the configuration space, hence all the particles will be, at a moment of their history, between the parallel sides of the stadium with a direction close to their normal. Thus at each time, there will be, on the average, a small CHAOS, Vol. 3, No.4, 1993 Downloaded 17 Sep 2012 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://chaos.aip.org/about/rights_and_permissions536 Mortessagne, Legrand, and Sornette: Transient chaos in room acoustics FIG. 11. Illustration of the transient trapping of a trajectory in the vi. cinity of the neutrally stable "bouncing-ball" orbits. A particle, which bounces back and forth between the two parallel sides of the stadium. remains trapped in a reduced fraction of phase space for a very long time. According to the ergodicity of the dynamics in the stadium, after leaving the trapping region in phase space, the particle spends equal times in equal areas of phase space. fraction NofJf)121r of particles, or equivalently a small frac tion of energy, close to the trapping direction. Those par ticles will remain trapped between the parallel sides of the stadium for a very long time of the order of t-RltJf). The reverberation is then driven by the escape rate d(tJf)ldt, which yields for large t dW(t) d(tJf) 2 ----"'--"'t-dt dt (15) which gives upon integration the power-law tail shown in Fig. 10, wet) ",rl. (16) Irrespective of the location of the localized absorption, the same qualitative results are observed.22 The two effects, found in Sees. II C and II D, illustrate the limits of the mean-field Sabine-Eyring theory of sound reverberation. When the absorption is weak and the tran sient duration is large, the assumptions underlying the clas sical expressions, based on asymptotic properties, are ful filled. However, for stronger absorptions and thus shorter transients, fluctuations of the mean-free path renormalize the reverberation time and fluctuations of the rate of phase space exploration may produce a critical slowing down of the sound energy decay. Iff. TOWARD A WAVE APPROACH OF REVERBERATION A. Asymptotic modal analysis Room acoustics is first of all a wave problem and a rigorous treatment of sound reverberation should be devel oped using a genuine wave-theoretical approach. Among the earliest pioneers, one can cite the comprehensive work by Morse and Bolt on "Sound Waves in Rooms," I The conceptual consistency of geometrical acoustics and modal analysis has been investigated by many authors since then,44,45 inasmuch as the basic assumptions of geometrical acoustics, namely energy equipartition and isotropy, are validated. As a simple introductory example, consider the exactly solvable case of a parallelepipedic room with ab sorbers evenly distributed over walls, ceiling and floor. In the weak absorption limit, the decay constant tJ for a char acteristic mode of complex frequency (j) -ill is given by C[ ,cos{}x ,cos{}y ,COS{}z] tJ=2: ax---z:-+ay---z:-+az-:r:- ' x y z (17) where Lx> Ly' and Lz are the side lengths of the parallel epiped and {}x (resp. {}y and {}z) is the angle of incidence of the mode on both walls perpendicular to the x axis (resp. y and z axes). In expression (17), the absorption exponent a' is used, defined as a'=-ln{1-a) where a is the ab sorption coefficient, and the subscript x (resp. y and z) specifies the normal direction to the wall since a depends on the angle of incidence. Of course, a possible dependence of a on the frequency is implicitly assumed but should not bother us if it is weak in the frequency range of excitation. In general, the, reverberation process, even for a narrow~ band excitation, involves a variety of modes with different angles of incidence. One should thus not expect a simple exponential decay except if the absorption exponent is in versely proportional to the cosine of the angle of incidence. In fact, this condition is approximately fulfilled (except at grazing incidence) for a small homogeneous (complex) specific wall admittance {3 defined as (normal air velocity at Wall) (3-pcx - air pressure at wall ' (18) where p is the density of air. The absorption coefficient, as a function of the angle of incidence {}, then reads as44 a({}) (19) where {3R denotes the real part of {3. For small {3Rlcos {} (hard walls and no grazing incidence), the absorption ex ponent has the following expansion: 4{3R [ (4 {3~ )( {3R)2 1 a'({})=cos{} 1+ 3"Tt3r- 1 cos{} + .... (20) If one considers modes with nongrazing incidence, one may then keep the leading-order term in the last expression CHAOS, Vol. 3, No.4, 1993 Downloaded 17 Sep 2012 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://chaos.aip.org/about/rights_and_permissionsMortessagne, Legrand, and Sornette: Transient chaos in room acoustics 537 for a' and replace it in expression (17) to obtain the fol lowing Sabine-like form of the decay constant for the am plitude: (21) where V is the volume and the sum runs over the six walls with areas Sj. Note that formula (21) would yield a pure exponential decay if it were not restricted to a particular set of characteristic modes excluding the so-called axial and tangential modes (i.e., modes with wave vectors par allel to four and two walls, respectively) which yield widely varying decay rates thus invalidating a strict expo nential decay of the Sabine type. Most actual auditoria have complex geometries and a great variety of absorbing materials on their boundaries so that the general wave problem for them is intricate because of nonseparability in real space coordinates and of nontriv ial interactions of sound waves with objects like panels, curtains, or seats. Since, from a pure geometrical point of view, generic auditoria shapes produce chaotic ray dynam ics, one is led to consider the fate of geometrical chaos in a wave formulation. This motivates us to investigate, through a perturbation calculation, the problem of the de termination of the decay constant of characteristic modes within a room where the wave problem is nonseparable and with a boundary condition close to the condition of perfectly rigid walls (weak absorption limit). We consider a 2-D room which presents chaotic rays dynamics in the pure geometric limit and assume a small homogeneous specific wall admittance {3. Using the veloc ity potential "', the pressure p and the air velocity u are obtained by the equations a", p= P7ii' U= -grad ",. (22) Thus, in terms of the specific admittance at the boundary for a wave depending on time as e-iwt, the boundary con ditions reads Un {3=pc-, p (23) where Un is the air velocity normal to the wall. Hence, the condition for a perfectly rigid wall simply is un=O corre sponding to the Neumann boundary condition an",= (grad "') • n=O, where n is the unit vector directed inward normal to the wall. First consider the unperturbed Neumann problem for which the characteristic functions </>j satisfy ri' 2 (f} j V </>j+--cz</>j=O, (24) with an</>j=O on the boundary. Then, to first order in the small parameter {3, the per turbed characteristic functions CPj satisfying 2 2 'l/j 0 V CPj+(7Pj= , (25) with anrpj=i(w/c){3CPj on the boundary, have a character istic complex frequency given by • 0 c ¢i{3</>7 dl 'l/j'=Wj-,{Jj""Wj 2 f </>7 dS . (26) Here, we assume that mode coupling is negligible (i.e., {J j'~ Aw with /!"w denoting the mean spacing between two consecutive eigenfrequencies around w) and this can be justified to a certain extent for wave problems in chaotic enclosures such as the stadium or even in nonseparable enclosures where it has been shown 46-48 that level repulsion is the rule at high frequencies. At this stage, we will use a qualitative argument pro posed by Berry49 to characterize irregular modes in a clas sically chaotic system. If an irregular mode is supported by chaotic rays, it can be viewed as a random superposition of plane waves of different phase and direction but with the same wave vector magnitude k. Berry showed that such a mode has Gaussian random amplitudes and an isotropic correlation function given by a Bessel function: (27) This argnment was proposed before the existence of scars was discovered by Heller50 through a phase space descrip tion, and further described in a coordinate space represen tation by Bogomolny.51 A scar can be viewed as a weak localization of mode amplitude in the vicinity of an unsta ble periodic orbit. This effect being related to the phase space structure of the chaotic system is best identified by using a phase space representation (for instance by pro jecting the mode onto the linearized evolution of a Gauss ian wave packet)." Whether the scarring phenomenon should be of chief importance for an asymptotic modal approach of reverberation cannot be decided since little is known about the fraction of heavily scarred modes in a local frequency domain. Another effect of strong localiza tion of amplitude is due to the nonisolated bouncing-ball periodic orbits in the stadium, described in the previous section, leading to "bouncing-bail" modes which roughly look like modes of a rectangle [see Fig. 12(b)] but the fraction of such modes vanishes as the frequency tends to infinity. Now if we consider the case of a single ideal irregular mode, averaging the absorption coefficient on the walls is equivalent to averaging it over all directions of incidences (since a typical chaotic trajectory explores phase space uni formly). For small {3, expression (19) averages to (a) = 27r{3R . (28) The decay constant can then be rewritten c ¢</>2dl 5"" 47r (a) H2 dS (29) which precisely yields Sabine's formula (since r= 1/25) provided that (I/P)¢</>2 dl 5 ( I/S)f </>2 dS 2 CHAOS, Vol. 3, No.4, 1993 Downloaded 17 Sep 2012 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://chaos.aip.org/about/rights_and_permissions538 Mortessagne, Legrand, and Sornette: Transient chaos in room acoustics (a) (0) FIG. 12. Three odd-odd modes for the Neumann problem in the stadium with circular endeaps of radius R and a rectilinear portion of length E=R. The squared amplitude is shown on a quarter of the stadium. For the three modes. the wave number, defined as k=(I)/c. is given in units of l/R and the ratio 5 is numerically evaluated as: (a) k=80,099. ratio=4.45; (b) k~80.259, ratio~1.27; and (c) k~80.278, ratio~2.17. and (30) (a)=0, where P is the perimeter of the enclosnre, S the total area of the 2-D room, and 0 the mean sound absorption coef ficient defined in Sec. II A. This is what is expected for an irregular mode. 53 Furthermore, even if absorption is local ized on a fraction of the boundary of length L, Eq. (30), with P replaced by L, is still expected to hold for an irreg ular mode provided that L be much larger than the wave length. We checked how Eq. (30) is fulfilled for genuine Neumann modes in the stadium. Our numerical evalua tions show that the ratio 5 is in fact highly fluctuating from mode to mode. This is illustrated in Fig. 12 where are shown three different types of characteristic odd--{)dd modes in the quarter of the stadium with rectilinear por tion of length equal to the radius of the caps. These par ticular modes have their wave number around k=80 (in the units where the radius of the circular caps is I) and were chosen to illustrate the three classes of modes men tioned above, namely (a) scarred mode exhibiting a "whis pering gallery," (b) localized "bouncing-ball" mode, and (c) irregular mode. Corresponding values of the ratio 5 for these three modes are (a) 4.45, (b) 1.27, and (c) 2.17. A mode of wave number k has typically k degrees of freedom (for instance, the number of nodes on the perimeter is of order k). We thus expect the variance of the fluctuations of 5 around the value 2 to decrease as k-1/2 for large k. This very slow convergence rationalizes our observations. Obviously, the first-order perturbation calculation per formed above is valid only up to a cutoff frequency beyond which one enters a different regime where modes overlap and therefore can only be excited incoherently: this is the so-called regime of Ericson54 or of Schroeder. 55 Here, we will not pursue a thorough discussion of this regime but for the sake of completeness we simply recall the main features relevant to statistical wave acoustics. In this regime, the spectrum of the frequency response is statistically de scribed by assuming modal overlap (which means that the mean amplitude decay rate IJ is larger than the average spacing between adjacent resonance frequencies) and a re verberation time slowly varying with frequency. Then, from very general considerations based on the theory of random matrices, one may determine the statistical prop erties of the transmission function of a reverberant room. The transmission function T is defined as the squared mod ulus of the ratio of the reverberant sound field pressure to the volume velocity of the sound source at a given fre quency "'. To obtain the relative variance of this function, one is led to perform an average over the frequency of excitation and possibly also on the positions of the source and receiver. Denoting the average over '" by angular brackets, Schroeder's prediction for the "frequency auto correlation function" in the limiting case of large modal overlap reads: (T("'H) T(",)} -(T(",) >' (T(",)}2 (31) where IJ is the decay rate (assumed constant) of modal amplitudes. Corrections to this equation may be evaluated, in the case of finite modal overlap, by assuming a specific level repulsion given by the random matrix theory of the Gaussian orthogonal ensemble as shown by Weaver. 56 The result [Eq. (31)] is far from intuitive since it es tablishes that the fluctuations of T ("') have the same order of magnitude as its mean value. This result stems from an effect of spatial coherence. Indeed, if, for instance, one averages T over receiver and source positions, these fluc tuations completely vanish due to the incoherent character of the attenuation process. However, one is naturally led to question the validity of this theory since certain regUlarities in the spectra, for instance due to periodic orbits, are not accounted for in random matrix theory and yet must have their signature in the frequency autocorrelation function. This must be the object of further investigations. B. A geometrical building of interferences It is obvious that certain aspects of the problem of room acoustics can only be understood by taking into ac count the wave nature of sound. Specific wave phenomena, namely interference and diffraction, are completely ne glected in the true geometrical limit presented in Sec. II. Nevertheless, it would be quite convenient to keep the in tuitive and pictorial power of geometrical ray propagation and its easy numerical implementation. Here, we show that the knowledge of typical wave quantities, such as the im pulse response of 2-D room or a membrane, may be de duced from an improved pure geometrical description. The relationship between waves and geometrical rays lies within the theoretical framework of the semiclassical limit developed in the context of quantum mechanics. Indeed, using de Broglie's wavelength A,B' Schr6dinger's stationary wave equation may be written CHAOS, Vol. 3, No.4, 1993 Downloaded 17 Sep 2012 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://chaos.aip.org/about/rights_and_permissionsMortessagne, Legrand, and Semette: Transient chaos in room acoustics 539 ( 2 4"') V +-;q; 1/1=0, (32) which is formally equivalent to the stationary Helmholtz wave equation, when interpreting AB as the acoustic wave length A. This provides the equivalence between the two asymptotic limits, namely AB~O, which defines the semi classical approximation of quantum mechanics, and A~O, representing the high-frequency limit of wave problem. In the study of the semiclassical approximation to the SchrMinger equation, much effort has been directed to the spectral problem with the rigorous methods produced by Maslov,57 the trace formula of Gutzwiller,58 and the mul tiple reflections formalism developed by Balian and Bloch.59 Reverberation being essentially a dynamical tran sient process, one must face the problem of adopting a time-dependent approach. In the time domain, the most important results concerning semiclassical approximations were obtained by Heller and co-workers who developed, for the parabolic SchrMinger equation, an extraordinarily accurate method based on an imprOVed version of linear ized wave packet dynamics using a summation over repre sentative orbits (in phase space) called "heteroclinic orbits.,,52 Since the time-dependent wave equation is sec ond order in time, these previous semiclassical propagation methods are not directly relevant. Neither would spectral split-operator methods or more simply decomposition on a finite set of modes allow us to reach the desired long time accuracy or even to capture the physical process of absorp tion at the wall. Our approach to the wave problem follows the same philosophy as Heller and is also based on a summation of wavelets associated to particular subgroups of trajectories in coordinate space, hereafter called "plateaus," for a rea son that will become clear below. These entities allow us to describe the finite time impulse propagation from a source point to a small receiver region (in practice a disk of di ameter d in 2-D billiards), where the response is evaluated on a coarse-grained scale which is chosen much smaller than the smanest wavelength associated to the highest fre quency of the source spectrum. Due to extreme sensitivity on initial conditions in chaotic billiards (leading to an av erage exponential rate of divergence between initially neighboring trajectories), these plateans develop in a com plex way with time, related to what is known as a chaotic repel/er in the field of chaotic scattering.39 To illustrate this complexity, we show in Fig. 13 the length of the trajecto ries which first return to within a sman distance O.IR of the source point as a function of the launching angle, inside the billiard made of one-fourth of the Sinai billiard shown in Fig. 14. A similar result is obtained when plotting the lengths of the trajectories which first reach the receiver position as a function of the launching angle. The figure allows one to make precise the notion of a "plateau": two launching angles in the same plateau correspond to two recurrent orbits of approximately equal lengths. Plateaus are thus bundles of ray trajectories,each bundle being characterized by a compact launching angle interval and a given orbit length. ~ u ~ " g ~ = ~ >. -S u ~ 'ii' !: 0 2n (a) angle 103 ~ u ~ " g ~ = ~ >. -~ ~ 'i? !: 10 4 (b) angle FIG. 13. Length of the trajectory as a function of the launching angle for a disk or radius r= 10-1& (a) in the interval [O,21T]; (b) in the interval l4; 4+ 10-7], This Jigure shows the existence of plateaus: two launching angles in the same plateau correspond to two recurrent orbits of approx imately equaiiengths. The hierarchy of the plateaus widths, which occurs at all scales of angle, is reminiscent of a Cantor set. We now present our attempt to develop an asymptotic method for the calculation of the time-dependent Green's function of the hyperbolic wave equation (e.g., impulse response of a membrane), which is based On a summation over rays or more correctly over wave packet trajectories, FIG. 14. 2-D model ofa strongly chaotic room: one-fourth portion of the Sinai billiard, where the Sinai billiard is made of a square of side length 4R in the center of which a disk of radius R is removed. CHAOS, Vol. 3, No.4, 1993 Downloaded 17 Sep 2012 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://chaos.aip.org/about/rights_and_permissions540 Mortessagne, Legrand, and Sornette: Transient chaos in room acoustics taking into account the phase accumulated by the rays upon propagation, due to reflections, convergence to focal points, caustics when they exist, etc. The time-dependent Green's function is defined by the following equation: [ -~:;+ V~ jg(q,q"t-t') =8(q-q')8(t-t'), (33) subject to certain homogeneous boundary conditions on the surface S of the domain n of q, q'. When the domain n extends over the whole real space, corresponding to the free propagation case, the impulse response, which is a function of T=t-t', is obtained by performing the inverse Laplace transform of the time independent Green's functions (i.e., the Green's function in the frequency domain), r/[,J, 3-D: Gr,,,,(q,q';,,,z) = --4-;ImCP) > 0, (34a) TTr i 2-D: Gr,,,,(q,q';,,,z) = -41101)( pr);Im( p) >0, (34b) where, r= I q-q' I and 1101) is the Hankel function of ze roth order of the first kind.6O With respect to analytic prop erties of the time-independent Green's function, the inverse Laplace transform is achieved by using a suitable integra tion contour in the", complex plane."! We will use the retarded Green's function g" (T) which satisfies the rela tion: gII(T) =0 for T<O. For the free case, gII(T) reads C 3-D: gf",,(q,q';T) = -41Tr8(cT-r), (35a) C I 2-D: gf",,(q,q';T) = -Y(cT-r)2TT ~f:2;t-;:z' (35b) where Y is the Heaviside's step function. The 2-D expression of gf, .. can be obtained by integrat ing the 3-D gf,,,, over the component perpendicular to the plane of propagation. Indeed, as mentioned in Ref. 62, a point source of a 2-D space is equivalent to a uniformly distributed source along a line perpendicular to the plane of propagation. As far as free propagation is involved, the establish ment of expressions (34) and (35) is straightforward. Here, we are interested in building a time-dependent Green's function within finite 2-D domains, namely cha otic billiards. Details about the method that we have de veloped will be published elsewhere."' The expression of the 2-D time-dependent Green's function, that is finally found, reads (36) where the summation runs over all the plateaus, with a length 1 <CT, connecting the source point A and the small disk of diameter d around B. Here N/ is the number of particles in a given plateau of length 1, giving a measure of the energy in this ray bundle, and /(/ denotes a factor asso-'A [ . ~ , o 10 time in units of RIc FlO. 15. The time response of a membrane with the shape of one-fourth of a Sinai billiard is numerically calculated at a location away from a short duration impulse source. Comparison is shown between results ob tained by a finite·difference scheme (dots) and our geometrical method based on a summation over plateaus (continuous line), ciated to a given plateau and depends on its particular history through specific boundary conditions and the pos sible occurrence of focal points. Note that, relying on the fact that the wavelets used to construct the Green's func tion can be considered locally as plane waves, one can account for a possible absorption at the walls via the ab sorption coefficient concept used in the geometric ap proach. To validate this method, we compare the time response it predicts with the impulse response we get from a direct numerical solution of the time-dependent wave equation using a finite-difference scheme, with Dirichlet boundary conditions on the boundaries of a billiard having the shape of one-fourth of the Sinai billiard shown in Fig. 14. The results presented in Fig. 15 give the wave amplitude as a function of time at the receiver, created by a pointlike source launching a narrow Gaussian envelope impulse. The result obtained from Eq. (36) is represented by the continuous line and can be compared with the "exact" result (dotted line) obtained from the finite-difference scheme of direct integration of the wave equation. The agreement is excellent and extends to times much larger than those presented here, the only limitation being the numerical cost in sampling the exponentially increasing number of plateaus which become relevant at large times. We note that the wave amplitude at times of the order of a few Ric (and this is all the more true at longer times) already results from the interference of many ray trajecto ries involving quite different numbers of reflections at the wall: we are thus dealing with a complex N-body problem in the sense that it is hopeless to attempt to describe the sound amplitude from the knowledge of one or a few dom inating orbits. If this is true at very early times, the situa tion, very rapidly, gets much more involved with increas ing time and results from the complex interference between many ray trajectories. We note that it seems quite possible to develop a statistical theory of the distribution of wave amplitudes and space-space as well as time-time ampli- CHAOS; Vol. 3, No.4; 1993 Downloaded 17 Sep 2012 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://chaos.aip.org/about/rights_and_permissionsMortessagne, Legrand, and Sornette: Transient chaos in room acoustics 541 tude correlation functions, based on the mathematical for mulation of the present theory and relying on nonasymp totic properties of chaotic orbits. Our approach is open to improvements in order to take into account the nonasymptotic corrections to the phase shifts associated with focal point crossings, which seem important in our preliminary tests. The presence of absorp tion is also easily implemented by affecting an exponential weight (l_a)n, where a is the absorption coefficient of the boundary and n is the number of reflections, to each term in the sum of (36) to account for the decay of the energy at each rebound. The corresponding results will be pre sented elsewhere. Qualitatively, our preliminary calcula tions indicate that the time dependence shown in Fig. 15 will be modified by the appearance of a systematic decay of the observed oscillations. To conclude, we note that the wave amplitude shown here corresponds closely to the quantity measured in the experimental situation used by Sabine in his seminal work on the establishment of the reverberation formula: a local ized source (a pistol shot) and receiver (his ears) and the analysis of the time decay of an impulse sound bouncing off the wall of a room while being attenuated at each rebound. Note that modernized versions of this experiment consti tute the standard present-day method to measure the decay of sound fields, using microphones linked to a computer. Our asymptotic analysis in the time domain thus appears to be central to the relevant questions of room acoustics and at the same time opens the avenue for a deeper under standing of room acoustics, since it provides a framework to link the pure geometrical ray description to the pure wave theory. ACKNOWLEDGMENTS We would like to acknowledge very stimulating discus sions with M. V. Berry, O. Bohigas, D. Delande, and C. Schmit at various stages of our work. Ip. M. Morse and R. H. Bolt, Rev. Mod. Phys. 16, 69 (1944). 'V. O. Knudsen, Rev. Mod. Phys. 6, 1 (1934). 3L. L. Beranek, J. Acoust. Soc. Am. 92, 1 (1992). 4W. C. Sabine, Collected Papers on Acoustics (Harvard University Press, Cambridge, MA, 1992) (reprinted by Dover, New York, 1964). sw. S. Franklin, Phys. Rev. 16, 372 (1903). 60. Jaeger, Sitzungsber. Akad. Wiss. Wien. Math. Naturwiss. Kl. 2 A 120,613 (1911). 7R, Clausius, Die Kinetische Theorie der Gase (Yieweg, Brauchschweig, 1889), Chap. II. sE. Czuber, Sitzungsber. Akad. Wiss. 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Phys. 65, 295 (1979). 24y. Sinai, Usp. Mat. Nauk. 25, 141 (1970); Russ. Math. Surveys 2.5, 137 (1970). .. 2sL. A. Bunimovich, Funct. Anal. Appl. 8, 73 (1974); Physica D 33,58 (1988). 26y. I. Arnold and A. Avez, Ergodic Problems of Classical Mechanics (Addison-Wesley, Reading, MA, 1989). "L. A. Sunimovich, Sov. Phys. JETP 62,842 (1985). 2SJ. P. Bouchaud and A. Georges, Phys. Rep. 195, 127 (1990). 29p. Mortessagne, O. Legrand, and D. Sornette, Europhys. Lett. 20, 287 (1992). 3°0. Legrand and D. Sornette, Phys. Rev. Leu. 66, 2172 (1991). 31W. Bauer and G. F. Bertsch, Phys. Rev. Lett. 66, 2173 (1991). 32L. Valentin, Physique Subatomique (Hermann, Paris, 1982). 3JC. Cohen-Tannoudji, B. Diu, and F. Laloe, Mecanique Quantique, Col lection Enseignement des Sciences 16 (Hermann, Paris, 1977). 34W. Heitler, The Quantum Theory of Radiation (Oxford, University Press, London, 1966). "J. H. Van Vleck and D. L. Huber, Rev. Mod. Phys. 49, 939 (1977). 36W. Bauer and G. F. Bertsch, Phys. Rev. Lett. 65, 2213 (1990). 37J. D. Gunton and M. Droz, Lect. Notes Phys. 183, 1 (1983). 3ST. Tel, in Experimental Study and Characterization of Chaos, edited by H. Bai-Lin (World Scientific, Singapore, 1990). 390. Legrand and D. Sornette, Europhys. Leu. 11, 583 (1990). "0. Legrand and D. Somette, Physica D 44, 229 (1990). 410. Millington, J. Acoust. Soc. Am. 4, 69 (1932). 42ut T ""_u_ T A __ .. _~ C' __ A_ A 1n'l/1n'l'l\ TY. J. I3I¥LLC, J • .Mo\AlUM. ~Ul,,; • .Moill. '+,17':> \17':>':>,. "M. V. Berry, Eur. J. Phys. 2. 91 (1981). 44p. M. Morse and K. U. Ingard, Theoretical Acoustics (McGraw-Hili, New York, 1968), pp. 576-599. "Y. Kubota and E. H. Dowell, J. Acoust. Soc. Am. 92,1106 (1992). 460. Bohigas, M. J. Giannoni, and C. Schmit, in Quantum Chaos and Statistical Nuclear Physics, edited by T. H. Seligman and H. Nishioka (Springer-Yerlag, Berlin, 1986). 47R, L. Weaver, J. Acoust. Soc. Am. 85,1005 (1989). 480. Bohigas, O. Legrand, C. Schmit, and D. Sornette, J. Aeonst. Soc. Am. 89, 1456 (1991). 49M. V. Berry, J. Phys. A. 10, 2083 (1977). '"E. J. Heller, Phys. Rev. Lett. 53, 1515 (1984). SIE. B. Bogomolny, Physica D (Amsterdam) 31, 169 (1988). S2S. Tomsovic and E. J. Heller, Phys. Rev. E 47,282 (1993); Phys. Rev. Lett. 70,1405 (1993). "0. Legrand and D. Sorne!!e. Lect. Notes Phys. 392, 267 (1991). s4T. Ericson, Phys. Rev. Lett. 5, 430 (1960); Ann. Phys. 23, 390 (1963). SSM. R. Schroeder and K. H. Kutruff, J. Acoust. Soc. Am. 34, 76 (1962); M. R. Schroeder, J. Acoust. Soc. Am. 34, 1819 (1962). lOR. L. Weaver, J. Sound Vib. 130,450 (1989). S7y. P. Maslov and M. Y. Fedoriuk, Semiclassical Approximations in Quantum Mechanics (Reidel, Dordrecht, 1981). SSM. C. Gutzwiller, J. Math. Phys. 12, 343 (1971) and references therein. s9R. Balian and C. Bloch. Ann. Phys. (NY) 69,76 (1972) and references therein. 6OM. Abramowitz and I. A. Stegun, Handbook 0/ Mathematical Functions (Dover, London, 1965). 61 E. N. Economou, Green's Function in Quantum Physics (Springer Verlag, Berlin, 1979). 62p. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. I. pp. 842-848. 63p. Mortessagne, O. Legrand, and D. Sornette (to be published). CHAOS. Vol. 3, No.4. 1993 Downloaded 17 Sep 2012 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://chaos.aip.org/about/rights_and_permissions

1.4945958.pdf

Magnetization reversal in 3D nano-structures of different shapes T. Blachowicz, and A. Ehrmann Citation: AIP Conference Proceedings 1727 , 020003 (2016); doi: 10.1063/1.4945958 View online: http://dx.doi.org/10.1063/1.4945958 View Table of Contents: http://aip.scitation.org/toc/apc/1727/1 Published by the American Institute of PhysicsMagnetization Reversal in 3D Nano-Structures of Different Shapes T. Blachowicz1, a) and A. Ehrmann2 1Silesian University of Technology, Institute of Physics - Center for Science and Education, ul. Konarskiego 22B, 44-100 Gliwice, Poland 2Bielefeld University of Applied Sciences, Institute of Engineering Sciences and Mathematics, Interaktion 1, 33619 Bielefeld, Germany a) Corresponding author: tomasz.blachowicz@polsl.pl Abstract. Magnetic nano-particles have been intensively studied during t he last decade due to their potential utilization in various applications. An importa nt topic is the dependence of magnetic properties on the exact samples shape. After demonstra ting the influence of shape distortions in magnetic nano half-spheres on magnetization reve rsal processes and hysteresis shapes, a series of different 3D nano-objects from permalloy wi th shape modifications has been examined with respect to their magnetic properties. Modificatio ns are performed by cutting parts of diverse samples between the extrema of a cuboid and a half-s phere. Simulations of these samples have been performed by Magpar, using external magnetic fields along two different a x e s , s w e p t w i t h t w o d i f f e r e n t s p e e d s . D e p e n d i n g o n t h e o r i g i n a l particle shape and its modifications, several phenomena can be found: Cutting a hole i n a cuboid can switch the hard axis from out-of-plane to the in-plane direction. In some nano- particles, strong oscillations occur which can be suppressed by appropriate shape modifications. In some of the nano-objects, the magnetization reversal mechanism is completely altered by a cha nge in the field sweeping speed. The article gives an overview of the different possibilities to tailor magnetic properties of nano- systems. Keywords: 3D nano-magnet; Magnetic anisotropy; Shape modifications; Simulati on. PACS: 75.30.Gw, 75.50.Bb, 75.60.Jk, 75.75.-c INTRODUCTION Magnetic nano-systems show a broad variety of reversal mechanis ms and dynamics. Since this large spectrum offers possibilities of tai loring magnetic properties of nano-particles with respect to the desired applic ations, such nano- structured elements were intensively studied over the past deca des. Several different shapes were investigated, such as nano-dots and nano-rings in w hich, e.g., vortexes, flux-closed states, and onion states can be found [1-7]. The ex amination of a flux- closed vortex state is of special interest for storage applicat ions, since in this magnetization state the stray fields are minimized [4, 5]. In a ddition to the aforementioned examinations of two-dimensional nano-magnets, th ree-dimensional half-spheres were also probed experimentally, giving rise to ne w magnetization reversal mechanisms [8, 9]. This study thus examines the influence of shapes and shape dist ortions on the magneti c properti es of nano-parti cles. Simulating a seri es of t hree-dimensional nano- International Advances in Applied Physics and Materials Science Congress & Exhibition (APMAS ’15) AIP Conf. Proc. 1727, 020003-1–020003-11; doi: 10.1063/1.4945958 Published by AIP Publishing. 978-0-7354-1374-0/$30.00020003-1objects with geometry modifications, the influences of the basi c shapes as well as the shape modifications are investigated, also taking into account different field sweeping speeds to accommodate the need of short switching times in data storage applications etc. [10, 11]. SIMULATION Ferromagnetic permalloy (Py) nano-particles with dimensions bet ween 12.5 nm and 50 nm were modeled. An overview of the different sample geometr ies is given in T a b l e 1 . E a c h b a s i c f o rm A 1 -F 1 i s m o d i f i e d b y a h ol e i n z - d i r e c tion (A2-F2), by a cylindrical top cut (A3-F3), and by both of the aforementioned cuts (A4-F4). The basic forms themselves range from a cube (A) to a half-ball (F), with different asymmetries and edge forms in between. The sample dimensions in x-, y-, and z-direction are depicted in Table 2. For the simulation, the “Parallel Finite Element Micromagnetics Package (MAGPAR)” was used [12]. This program is based on solving the Landau-Lifshitz- Gilbert (LLG) equation of motion for a mesh built from finite e lements enabling precise design of shape details. These tetrahedral finite eleme nts are not larger than 3.7 nm, which is smaller than the Py exchange length of 5.7 nm [13] . The other values used in the simulation – ܣ = 1.05 ڄ 10−11 J/m, magnetic polarization at saturation ݏܬ = 1T, and the Gilbert damping constant ߙ = 0.01 – are typically used for Py samples and also suggested by the MAGPAR simulator. 020003-2TABLE 1. Sample geometries. 1 2 3 4 A B C D E F TABLE 2 . Sample dimensions. For the half-ball F, the sample height z i s reduced by the shape modifications. x (nm) y (nm) z (nm) A 28.5 28.5 12.5 B 50.0 28.5 12.5 C 50.0 50.0 12.5 D 50.0 50.0 12.5 E 50.0 28.0 12.5 F1 50.0 50.0 25 The external magnetic field was swept between -600 kA/m and + 6 00 kA/m for a field orientation along the z-axis and between -450 kA/m and + 450 kA/m for a field orientation along the x-axis, to reach a saturated state and to avoid minor loops. To test the influence of the field sweeping speed – which is an important parameter for the x z y 020003-3possible use of such nano-particles in magneto-electronic appli cations – t h e t w o different field sweeping speeds of 1 kA/(m ns) and 10 kA/(m ns) were used. RESULTS Figure 1 shows hysteresis loops, simulated for the external ma gnetic field oriented along the x-axis (upper panels) and the z-axis (lower panels), using field sweeping speeds of 1 kA/(m ns) (left panels) and 10 kA/(m ns) (right pan els), respectively. Figure 1. Hysteresis loops for the four modifications of sample A for an external magnetic field along x (top panels) and z (bottom panels) and for field sweeping speed s of 1 kA/(m ns) (left panels) and 10 kA/(m ns) (right panels). From [15], modified. While the hard axis of samples A1 and A3 is the z-direction, th is direction becomes an easy axis by cutting the cylindrical hole for samples A2 and A4. The hysteresis loops of samples A2 and A4, simulated for the fi eld sweeping speed of 10 kA/(m ns), and of sample A2, simulated for the field swee ping speed of 1 kA/(m ns), with the magnetic field oriented along the x-axis, show a step around Hext = 0 which corresponds to a flux-closed vortex state as seen in simi lar samples before (e.g. [14]). Former examination of the influence of the field sweeping proce ss in a half-ball showed only quantitative differences between hysteresis loops s imulated with different field sweeping speeds [15]. The cube modeled here, however, dep icts also qualitative differences for both sweeping speeds. For A1 with the field alo ng z, e.g., the reversible 020003-4part of the hysteresis loop detected with 1 kA/(m ns) vanishes completely for a higher switching speed of 10 kA/(m ns). Similarly, the shape of the hy steresis loop of sample A4 for an external field along the x-axis also changes qualitat ively. Figure 2. Hysteresis loops for the four modifications of sample B for an external magnetic field along x (top panels) and z (bottom panels) and for field sweeping speed s of 1 kA/(m ns) (left panels) and 10 kA/(m ns) (right panels). From [15], modified. Hysteresis loops simulated for sample B are depicted in Figure 2. Here the z- direction is always a hard direction, while hysteresis loops si mulated in the x-y-plane mostly show the typical shape of easy axis loops. For B2 and B4 , the hysteresis loops simulated for a field sweep in z-direction are nearly identical . The curves for B1 and B3 along z also differ only slightly. All loops along the z-dir ection are only slightly altered by the different field sweeping speeds. For a field sweep along the x-direction, sample B2 shows a loop with the broad step w h i c h i s t y p i c a l f o r a v o r t e x s t a t e . T h i s v o r t e x s t a t e i s i n t r o duced with strong oscillations, similar to the respective simulations performed o n samples A2 and A4. It should be mentioned that sample B does not have fourfold symmet ry, which is why a vortex state could not be expected to occur in this set of samp les. Opposite to A4, there is no such vortex state visible in B4. Ap parently, here the cylindrical top cut has a stronger influence than in the larger sample A. Additionally, the absence of a vortex state in this particle can be attribute d to the twofold symmetry of sample B, opposite to the pure fourfold symmetry of sample A . 020003-5 Figure 3. Hysteresis loops for the four modifications of sample C for an external magnetic field along x (top panels) and z (bottom panels) and for field sweeping speed s of 1 kA/(m ns) (left panels) and 10 kA/(m ns) (right panels). From [15], modified. Unlike sample sets A and B, sample C1 has a rotational symmetry . As depicted in Figure 3, all hysteresis loops simulated for the external field oriented along the x- direction show the typical easy-axis shape, independent of the existence of shape modifications. For the higher field sweeping speed of 10 kA/(m ns), however, samples C1 and C2 show strong oscillations. This behavior is significan tly reduced by the horizontal cut in samples C3 and C4. The coercive fields and lo op shapes nevertheless are not significantly influenced by the field sweeping speed. Simulations for the external field oriented along the z-directi on, the hysteresis loops mostly have the typical hard-axis form. Small irreversible part s of the loop, as visible for sample C1 simulated with 1 kA/(m ns), are similar to the ir reversible magnetization reversal processes also shown by sample A1. The i rreversible part of the hysteresis loop is again strongly increased by an increase of the field sweeping speed to 10 kA/(m ns). The strong oscillations visible here at the transitions between reversible and irreversible parts, however, do not occur in the respective loop taken for sample A1. All other sample modifications show very similar behavior for b oth field sweeping speeds. Sample D only differs from sample C by the diminution of the la tter in z-direction. This small difference is taken into account due to typical feat ures of lithographically produced nano-structures which can have conical walls as a resu lt of the etching 020003-6process. Figure 4 shows the results of the respective simulatio ns. If the external field is oriented along x, both sets of samples result in nearly the sam e coercive fields and loops shapes, for field sweeping speeds of 1 kA/(m ns) as well as 10 kA/(m ns). For the external magnetic field being swept along the z-directi on, the hysteresis loops are also quite similar to those simulated for samples C1- C4, with only small numerical deviations of the external fields for which saturatio n and reversible parts of the hysteresis loops are reached, respectively. Apparently, a possible conical wall shape of real nano-structur ed samples can be ignored in case of round nano-structures with or without shape modifications. Figure 4. Hysteresis loops for the four modifications of sample D for an external magnetic field along x (top panels) and z (bottom panels) and for field sweeping speed s of 1 kA/(m ns) (left panels) and 10 kA/(m ns) (right panels). From [15], modified. In a similar way as sam pl e D i s correl ated wi th sam pl e C, sam pl e E differs from sam pl e B only by th e dimi nuti on of th e l atter i n z-di recti on . T hus, similar results as those simulated for sample B can be expected here. Figure 5 depicts the results of the respective simulations. Ind e e d , t h e h y s t e r e s i s loops for a field sweep along the z-direction look very similar to those found for sample B. The values for which positive and negative saturation are reached are nearly identical. For a field sweep in x-direction, most coercive fields and hyst eresis loop shapes are again quite similar to sample B. Nevertheless, for sample E2 sw ept with 10 kA/(m ns) a significant difference is visible. While the magnetization re versal process in B2 simulated for the same field sweeping speed occurred by an inte rmediate vortex state, 020003-7depicted by broad steps in the hysteresis loop, this step is no t visible here, and the magnetization snapshots of different stages of the reversal pro cess (not depicted here) do not show a vortex state, either. Instead, after crossing the c o e r c i v e f i e l d , a reversible state is reached. This example shows clearly that the stability of magnetization reversal processes against small deviations of nano- particles’ forms should not be taken for granted, when simulation results are to be translated into experiments. Apparently, even small changes in the shape can sometimes significantly influence the magnetic properties of magnetic nano-structure. This can result in undesired changes o f the magnetization reversal processes, interacting with the shape modifications in troduced to tailor the magnetic properties in a favored way. Figure 5. Hysteresis loops for the four modifications of sample E for an external magnetic field along x (top panels) and z (bottom panels) and for field sweeping speed s of 1 kA/(m ns) (left panels) and 10 kA/(m ns) (right panels). From [15], modified. Figure 6, finally, depicts hysteresis loops simulated for the h alf-ball sample F, as the other extreme opposite to sample A. Here, the well-known st rong oscillations for field sweeps in x-direction can be recognized, as depicted for half-balls of other dimensions and with different shape modifications before (e.g. [14]). Opposite to most other nano-particle under examination in this study, severe dif ferences of the hysteresis loop shapes and coercive fields occur, depending on the field sweeping speed in x-direction. For the slower field sweeping with 1 kA/( m ns), none of the four samples F1-F4 reaches a vortex state, opposite to the results o f the higher field sweeping speed of 10 kA/(m ns). This finding is exactly opposit e to the behavior of 020003-8sample E2, for which the higher field sweeping speed led to a s uppression of the vortex state which was visible for the lower field sweeping spe ed. If the external magnetic field is swept in z-direction, not onl y the fields necessary to reach saturation are influenced by the field sweeping speed. Fo r sample F1 – t h e unaltered half-ball – the magnetization reversal process is completely changed: Whil e the hysteresis loop of F1 simulated for 10 kA/(m ns) is quite s imilar to that of F3 or that of C1 and D1 at the same field sweeping speed, a two-step magnetization reversal process can be recognized for the smaller field sweeping speed o f 1 k A / ( m n s ) . A similar hysteresis loop does not occur for any other sample in this investigation nor for the larger half-balls simulated in Refs. 7 and 14. To examine this special case in detail, Figure 7 shows snapshot s of the magnetization reversal process. Starting at positive saturation (1st panel), all magnetic moments are oriented in positive z-direction, which is shown by red arrows. When the magnetization reversal begins, a vortex in the x-y-plane is for med (green arrows in the 2nd panel), while the vortex core is still oriented in positive z- direction. When the external magnetic fields become smaller the vortex core diamete r is reduced, and the vortex core starts to precess. Figure 6. Hysteresis loops for the four modifications of sample F for an external magnetic field along x (top panels) and z (bottom panels) and for field sweepi ng speeds of 1 kA/(m ns) (left panels) and 10 kA/(m ns) (right panels). In the first panel, the x-axis is different from the values used before. From [15], modified. 020003-9 Figure 7. Magnetization reversal of sample F1 for the external magnetic f ield swept along the z- direction with a speed of 1 kA/(m ns). Colors correspond to M z = 1 (red) / 0 (green) / - 1 (blue). While in larger half-balls this precession leads to a flip of t he core [14], here another mechanism occurs: The vortex core moves to the half-bal l border, resulting in the whole magnetization laying approximately in the x-y-plane. The precession of the vortex core now is continued by a wave-like precession of the w hole sam ple’s magnetization. After some precession periods, the magnetization i s m o r e a n d m o r e oriented along negative z-direction, which is visible by a slow transfer from the dominant red regions to the blue ones. During this process, the precession of the whole sample’s magnetization continues. With larger negative external magnetic fields dominating the form anisotropy, the final switch into negative saturation occurs. The main difference between the samples examined here and in [1 4] is the particle size, differing about a factor of 2. Thus, the form anisotropy has a larger influence here, which leads to a stronger difference between the easy x-y -plane and the hard z- direction. This domination of the form anisotropy results in a stable intermediate state with the magnetization oriented in the x-y-plane during magneti zation reversal along the z-direction. Such a phenomenon, if it can also be found for higher field swe eping speeds, can also support the idea of more than two stable states at remanen ce which has also been examined in nano-wire systems [16] – in this way, three stable states could be realized. However, since this effect has only been found for the lower fi eld sweeping speed, is has not been further analyzed. 020003-10It should be noted, however, that apparently only for a small r ange of parameters t h i s e f f e c t c a n b e g e n e r a t e d . I f i t i s t o b e u s e d i n f u t u r e m a g neto-electronic applications, sample shapes as well as field sweeping speeds ha ve to be defined carefully to avoid undesired deviations of the magnetic propert ies of such nano- structures. CONCLUSION To conclude, we have shown the significant influence of the sha pe of magnetic nano-particles on the magnetization reversal processes, resulti ng in possibilities to tailor the magnetic properties of nano-systems with respect to the desired applications. ACKNOWLEDGMENTS This work was partially supported by the local BK-251/RIF/2014 research program. REFERENCES 1. W. K. Hiebert, A. Stankiewicz, and M. R. Freeman, Phys. Rev. Lett. 79, 1134 (1997). 2. R. P. Cowburn, D. K. Koltsov, A. O. Adeyeye, M. E. Welland, and D. M. Tricker, Phys. Rev. Lett. 83, 1042 (1999). 3. J. G. Zhu, Y. F. Zheng, and G. A. Prinz, J. Appl. Phys. 87, 6668 (2000). 4. F. Q. Zhu, D. L. Fan, X. C. Zhu, J. G. Zhu, R. C. Cammarata, an d C. L. Chien, Adv. Mater. 16, 2155 (2004). 5. W. Zhang and S. Haas, Phys. Rev. B 81, 064433 (2010). 6. K. He, D. J. Smith, and M. R. McCartney, J. Appl. Phys. 107, 09D307 (2010). 7. T. Blachowicz, A. E hrmann née Tillmanns, P. Steblinski, and L. Pawela , J. Appl. Phys. 108, 123906 (2010). 8. M. M. Soares, E. de Biasi, L. N. Coelho, M. C. dos Santos, F. S. de Menezes, M. Knobel, L. C. Sampaio, and F. Garcia, Phys. Rev. B 77, 224405 (2008). 9. E. Amaladass, B. Lu descher, G. Schütz, T. Tyliszczak, M. -S. Lee, and T. Eimüller, J. Appl. Phys. 107, 053911 (2010). 10. B. D. Terris and T. Thomson, J. Phys. D: Appl. Phys. 38, R199 (2005). 11. J. Akerman, Science 308, 508 (2005). 12. W. Scholz, J. Fidler, T. Schrefl, D. Suess, R. Dittrich, H. For ster, V. Tsiantos, Comp. Mat. Sci. 28, 366 (2003). 13. N. Smith, D. Markham, and D. la Tourette, J. Appl. Phys. 65, 4362 (1989). 14. T . Blachowicz and A. Ehrmann, Journal of Physics: Conference Series 574, 012054 (2015). 15. A. Ehrmann, “Examination and simulation of new magnetic materials for the possible application in memory cells ” Logos Verlag, Berlin / Germany, 2014. 16. T. Blachowicz and A. Ehrmann, J. Appl. Phys. 110, 073911 (2011). 020003-11

1.1288129.pdf

Semiclassical analysis of level widths for one-dimensional potentials Gert-Ludwig Ingold, Rodolfo A. Jalabert, and Klaus Richter Citation: American Journal of Physics 69, 201 (2001); doi: 10.1119/1.1288129 View online: http://dx.doi.org/10.1119/1.1288129 View Table of Contents: http://scitation.aip.org/content/aapt/journal/ajp/69/2?ver=pdfcov Published by the American Association of Physics Teachers Articles you may be interested in Addendum to: “The one-dimensional harmonic oscillator in the presence of a dipole-like interaction” [Am. J. Phys. 71 (3), 247–249 (2003)] Am. J. Phys. 71, 956 (2003); 10.1119/1.1592514 One-dimensional laser cooling of an atomic beam in a sealed vapor cell Am. J. Phys. 70, 71 (2002); 10.1119/1.1419098 Wave packet revivals and quasirevivals in one-dimensional power law potentials J. Math. Phys. 41, 1801 (2000); 10.1063/1.533213 Strong asymptotics of Laguerre polynomials and information entropies of two-dimensional harmonic oscillator and one-dimensional Coulomb potentials J. Math. Phys. 39, 3050 (1998); 10.1063/1.532238 Laser cooling of atoms to the Doppler limit Am. J. Phys. 65, 1120 (1997); 10.1119/1.18740 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 202.28.191.34 On: Fri, 20 Feb 2015 12:49:38Semiclassical analysis of level widths for one-dimensional potentials Gert-Ludwig Ingold Institut fu¨r Physik, Universita ¨t Augsburg, Universita ¨tsstraße 1, D-86135 Augsburg, Germany Rodolfo A. Jalabert Universite ´Louis Pasteur, IPCMS-GEMME, 23 rue du Loess, F-67037 Strasbourg Cedex, France Klaus Richter Max-Planck-Institut fu ¨r Physik komplexer Systeme, No ¨thnitzer Straße 38, D-01187 Dresden, Germany ~Received 20 September 1999; accepted 16 May 2000 ! We present a semiclassical study of level widths for a class of one-dimensional potentials in the presenceofanohmicenvironment.Usingasemiclassicalapproachforthedipolematrixelementweobtain the level widths within the golden rule approximation. For potentials with an asymptoticpower-lawbehavior,whichmayinadditionbelimitedbyaninfinitewall,wefindauniversalresult:The level widths are proportional to the corresponding quantum number. © 2001 American Association of Physics Teachers. @DOI: 10.1119/1.1288129 # I. INTRODUCTION The coupling of a system to environmental degrees of freedom plays an important role in many areas of physics.Already on a classical level it leads to fluctuations, as inBrownian motion, and to damping. 1In addition, in quantum mechanics the environmental coupling induces decoherence,which, for example, is of interest in the discussion of Schro ¨- dinger cats 2and quantum computation.3 An isolated quantum system has stable eigenstates and its density of states is given by a series of delta functions. Cou-pling to external degrees of freedom renders the originalstates unstable, since in general we obtain a new set of eigen-states for the entire system. When the number of degrees offreedom introduced is very large and the coupling suffi-ciently weak, the new spectrum, consisting of a series ofclosely spaced delta functions, will be signed by the originalone. In particular, the reduced density of states, describingthe system coupled to a large number of degrees of freedom,results from a smearing of that of the uncoupled case. Thesmearing of the original eigenenergies due to the coupling toan environment can be expressed as a level width which isrelated to the lifetime of the state. The example of an atom coupled to the electromagnetic modes of the radiation field may help to illustrate the conceptof level width. If the atom were isolated from the field, therewould be no transitions from excited states to states lower inenergy. This changes if we take the coupling to the radiationfield into account. Then transitions between states may oc-cur, and the finite lifetime of the excited states broadens thespectral lines associated with the transition. As a conse-quence of this mechanism a modification of the mode spec-trum may influence the lifetime of atomic states. 4 In the following discussion of level widths we will fix the environmental spectrum to be ohmic ~for a definition see Sec. II below !. On the other hand, the system considered, a particle moving in a one-dimensional power-law potential,will be quite general. While the level width of a systemeigenstate will increase with increasing coupling to the envi-ronment, the properties of the eigenstate will also be of rel-evance. This leads to an interesting question: How do thelevel widths depend on the quantum number of the eigen-state? As we will see, such a rather general question has asurprisingly simple answer if one restricts one’s self to the limit of large quantum numbers where semiclassical methodsare applicable. Semiclassical approaches were essential at the advent of quantum mechanics and have ever since remained a privi-leged tool for learning this subject, for developing our physi-cal intuition on new problems, and for performing analyticalcalculations. 5,6For a particle confined in a one-dimensional ~1D!potential, the semiclassical @Wentzel–Kramers– Brillouin ~WKB !#approximation yields closed expressions for the eigenenergies and eigenfunctions.7The applicability of the WKB approximation is restricted to large quantumnumbers, where the confining potential varies smoothly onthe scale of the de Broglie wavelength of the particle. In thislimit, the quantum properties of the system can be obtainedby means of classical trajectories. A particularly simple case is that of a power-law potential where the eigenenergies follow a simple scaling with thequantum numbers ~or the classical actions !. 8,9Restricting ourselves to power-law potentials, and in the presence of anohmic environment, we are able to extend the scaling ofRefs. 8 and 9 to level widths and demonstrate that they aresimply proportional to quantum numbers. It is interesting tonote that this scaling of level widths with the quantum num-ber is even simpler than that of the eigenenergies, despite thefact that the latter are more basic quantities than the former. The paper is organized as follows. We first present the general formalism for describing the dissipative environmentand its effect on the level widths ~Sec. II !, and we recall the well-known case of a particle in a harmonic potential. In Sec.III, we establish the central result of this work, proving thelinear dependence of level widths on quantum number underthe assumptions specified above. In the concluding sectionwe analyze the experimental implications of our findings andtheir possible extensions to higher dimensions. The exampleof a confining potential with the shape of a half-harmonicoscillator is discussed in detail in the Appendix. II. LEVEL WIDTHS IN A DISSIPATIVE ENVIRONMENT As our model we consider a particle of mass Mmoving in a one-dimensional potential V(q). The spectrum of the cor- responding Hamiltonian 201 201 Am. J. Phys. 69~2!, February 2001 http://ojps.aip.org/ajp/ © 2001 American Association of Physics Teachers This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 202.28.191.34 On: Fri, 20 Feb 2015 12:49:38HS5p2 2M1V~q! ~1! is assumed to consist of a discrete part at low energies which may be followed by a continuous part at higher energies. It ison the discrete part of the spectrum ~consisting of eigenen- ergiesE n,n50,1,2,...) that we focus our analysis. Since we consider the limit of large quantum numbers, we require that the number of discrete eigenstates is infinite or atleast can be made sufficiently large. This includes, for ex-ample, the radial part of the Coulomb problem but excludesthe Morse potential. The eigenstates acquire a finite width if we weakly couple the particle to environmental degrees of freedom. We assumethat the environment consists of a set of harmonic oscillatorscoupled bilinearly to the particle. This leads to the fullHamiltonian H5H S1( j50‘Fpj2 2mj1mjvj2 2Sxj2cj mjvj2qD2G, ~2! implying a coupling between system and environment through the Hamiltonian HI52( j50‘ cjxjq. ~3! By eliminating the environmental degrees of freedom, we obtain an effective operator equation of motion,1,10 q¨1E 0t dsg~t2s!q˙~s!11 MdV dq51 Mj~t!, ~4! with damping kernel g~t!52 ME 0‘dv pJ~v! vcos~vt!, ~5! a spectral density of bath oscillators J~v!5p( j50‘cj2 2mjvjd~v2vj!, ~6! and a fluctuating force j(t) which we do not need to specify further. The special case of J(v)5Mgvis of great importance since the damping kernel becomes local in time: g(t) 52gd(t). Noting that in Eq. ~4!only half of the delta func- tion contributes ~the integral ends at s5t), the second term becomes gq˙(t), describing the well-known classical damp- ing proportional to the particle velocity. This type of damp- ing is often referred to as ohmic because such a term alsoappears in equations for electrical circuits containing anohmic resistor. The previous approach provides a microscopic model for dissipation in quantum systems in the sense that dissipationis due to coupling to additional degrees of freedom. How-ever, we should not conclude that in a real resistor we canidentify environmental oscillators microscopically. TheHamiltonian ~2!allows us to treat analytically the effect of the environment and also provides a good description ofmany realistic systems. It has been studied over the years 10 and more recently became known as the Caldeira–Leggettmodel 11in the context of macroscopic quantum phenomena.Assuming a weak coupling between the particle and its environment, we calculate the zero temperature width of then-th level by means of the Fermi golden rule G n52p \( m,j50‘ u^m,1juHIun,0j&u2d~En2Em2\vj!.~7! This expression describes the decay of the state nto an en- ergetically lower state mby one excitation of the j-th envi- ronmental mode that changes its occupation number from 0to 1. Inserting the dipole matrix element ^1juxju0j&5S\ 2mjvjD1/2 ~8! of thej-th environmental oscillator we get Gn5p( m,j50‘cj2 mjvjudnmu2d~En2Em2\vj!, ~9! wherednm5^muqun&is the dipole matrix element of the sys- tem. The properties of the environmental modes appearing in Eq.~9!can be expressed in terms of their spectral density ~6!, and we may write for the level width Gn52 \( m50n21 udnmu2JSEn2Em \D. ~10! The sum over the system eigenstates is restricted since an environment at zero temperature cannot excite the systeminto states of higher energy. The result ~10!is valid for arbi- trary bath density. If we used a cubic frequency dependence forJ( v), appropriate for the electromagnetic field, we would obtain the natural decay width of an excited atomic stateduetospontaneousemission ~apartfromprefactorsaris- ing from a proper treatment of the polarization of the emittedphotons !. In the sequel we will concentrate ourselves on the impor- tant case of ohmic damping where the level widths can bewritten as G n52Mg \2( m50n21 udnmu2~En2Em!. ~11! This expression constitutes the starting point for a calcula- tion of level widths that will be performed in the followingsection. It represents, up to the factor g, a sum over oscillator strengths fnm52M \2udnmu2~En2Em!. ~12! The finiteness of the upper limit prevents application of the standard Thomas–Reiche–Kuhn sum rule12for oscillator strengths: (m50‘fnm51. For sufficiently simple confining potentials V(q) the evaluation of the level widths can easily be done. For in- stance, a harmonic potential with frequency v0has dipole matrix elements that only couple nearest neighbor states, dnm5A\ 2Mv0~An11dm,n111Andm,n21!. ~13! This leads to the well-known result for the level widths of a damped harmonic oscillator13 Gn5ng. ~14! 202 202 Am. J. Phys., Vol. 69, No. 2, February 2001 Ingold, Jalabert, and Richter This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 202.28.191.34 On: Fri, 20 Feb 2015 12:49:38In this case one can even go beyond the level widths and calculate the reduced density of states r(E) in closed form.14 As an illustration we show in Fig. 1 the result for ohmic damping with g/2v050.1. A delta function corresponding to the stable ground state has been omitted. The levels broaden with increasing energies, according to Eq. ~14!. For the rela- tively large coupling chosen, the regime where the eigen-states of the system are completely smeared out is reachedfor rather small quantum numbers. The strength of coupling of a system to its environment can vary widely according to the physical problem underconsideration. The theoretical analysis that we pursue in thefollowing addresses the situation where the coupling is muchweaker than that in Fig. 1. Thus we expect a large series ofbroadened eigenstates before the level widths become of theorder of the mean level spacing. III. LEVEL WIDTHS FOR POWER-LAW POTENTIALS In this section we establish the main result of our work, the proportionality of level widths to quantum numbers forthe model described in Sec. II and power-law confining po- tentials of the form V(q)5A uqua. The amplitude Aand the exponent ashould have the same sign to allow for bound states. The accessible classical region might be limited by aninfinite potential wall, and we assume that such a wall is present at q50 whenever a,0. The case a50 will be ex- cluded because it requires two walls and thus reduces to the exactly solvable case of a particle in a box ~see also a50 in Table I !. Furthermore, we will restrict the exponent to a.22. At a522 the action becomes independent of en- ergy as will become clear from Eq. ~21!below. Therefore we must exclude this pathological case. We emphasize that an attractive 1/ q2potential for small qcan never appear in a radial equation of motion in d.1 after elimination of the angular degrees of freedom. Since our semiclassical approach requires sufficiently large energies, the discussion applies also to confinementswhich effectively behave like a power-law potential at higherenergies, regardless of the shape at the bottom of the poten-tial. For example, the quartic double-well potential is in- cluded in our discussion since it has the form Aq 4for large energies. According to Eq. ~11!then-dependence of the level widths is determined by the dipole matrix element dnmand the energy difference En2Em. We start the analysis of these quantities by recalling some relations concerning energy quantization in the semiclassical limit. The central quantity isthe action S ~E!5MRdqq˙, ~15! taken over one cycle of the classical periodic motion. The periodTitself may be obtained by taking the derivative of S with respect to energy, T5dS/dE. Within semiclassical quantization, the eigenenergies are determined by S~En!52p\~n1m!, ~16! where mis a constant depending on the details of the poten- tial. We are interested in large quantum numbers nandm wheren,m@l5n2m. Then we obtain for the energy dif- ference En2Em’]E ]nl52p\ Tl. ~17! The dependence of the energy Eon quantum number n can be obtained by using the scaling properties of the clas-sical energy conservation condition E5M 2q˙21Aqa. ~18! To this end we introduce dimensionless coordinate and time q85SuAu ED1/a q,t85uAu1/aE~a22!/2a M1/2t, ~19! which simplifies Eq. ~19!to the dimensionless form 151 2q˙821sign~A!q8a. ~20! Here,q˙85dq8/dt8. The quantization condition ~16!in scaled variables reads S~E!5M1/2E~21a!/2a uAu1/aRdq8q˙852p\n, ~21! where again the integral runs over one period. On the right- hand side we have omitted the constant mintroduced in ~16! Fig. 1. Density of states for a harmonic oscillator of frequency v0coupled to an ohmic environment with g/2v050.1. The delta function correspond- ing to the stable ground state at energy E05\v0/2 is not shown.Table I. Eigenenergies, dipole matrix elements, and level widths in the semiclassical limit for 1D box ( a50 with two walls !, half oscillator ( a 52), and radial part of the Coulomb problem ( a521). aEn dn,n2l Gn/gn 0 ;n2;l22 7 p2z~3!50.85. . . 2 ;n ;n1/2l228 p250.81. . . 21 ;n22;n2l25/361/3 p2@G~2/3!#2z~7/3!50.47. . . 203 203 Am. J. Phys., Vol. 69, No. 2, February 2001 Ingold, Jalabert, and Richter This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 202.28.191.34 On: Fri, 20 Feb 2015 12:49:38which becomes irrelevant for large n. WithS85rdq8q˙8we then find for the energy eigenvalues E5FS2p\ S8D2auAu2 MaG1/~21a! n2a/~21a!, ~22! in agreement with previous results for power-law potentials8,9whereS8has been evaluated explicitly. Equa- tion~22!is still correct for sufficiently large nif the potential behaves like a power-law only asymptotically. Then, in gen- eral,S8can no longer be evaluated analytically. In the semiclassical limit the dipole matrix elements can be related to the Fourier components of the classical motionof the particle: 15 dn,n2l521 plE q1q2dqsinS2plt~q! TD 51 T~E!E 0T~E! dtq~t,E!cosS2plt T~E!D. ~23! Here the second line has been obtained by means of the substitution dq5q˙dtand partial integration. Equation ~23!, first found by Debye in 1927, extends Bohr’s correspondence principle, which states that in the limit \!0 the frequencies of an atomic transition should agree with electrodynamics, to intensities which are related to the square of dipole matrix elements, Eq. ~12!. Equation ~23!is the leading-order approximation in \. Higher order corrections have been derived in Ref. 16 but we will notmake use of them here. In the presence of hard walls the WKB approximation is still applicable and an extra phase in the semiclassical wavefunction takes into account the infinite potential. 5Therefore, the semiclassical approximation to the dipole matrix elementis the same as for smooth potentials. After scaling, the dipole matrix element ~23!reads d n,n2l5SE uAuD1/a dl8, ~24! with dl8521 plE q18q28dq8sinS2plt8~q8! T8D, ~25! whereT8denotes the period Tscaled according to Eq. ~19!. The entire dependence of the dipole matrix element on the quantum number nis now contained in the energy factor E1/a. In view of Eqs. ~22!and~24!, the level widths can be expressed as Gn5g8p2 S8221a 2aF( l51n ldl82Gn. ~26! Apart from the factor nthis result still depends on the state numbernvia the upper limit of the sum. As a last step, we therefore have to consider the convergence properties of thissum. For a.0 and in the absence of an infinite potential wall, the velocity of the particle as a function of time is continuous and consequently the scaled dipole moment ~25!will decay at least as l22. This still holds for a wall with finite potentialon one side since in that case the reflection will lead to a triangular cusp singularity in the trajectory and the dipole moment will decay as l22. The case of negative exponent awith an infinite potential wall atq50 where the potential diverges is more interesting. Close toq50 we may neglect the constant on the left-hand side of ~20!. Assuming that the reflection happens at t850 we find for the trajectory close to the reflection point q8 ;ut8u2/(22a). For sufficiently large l, this singular part yields the asymptotic behavior dl8;l2(42a)/(22a)for the scaled di- pole matrix element. For a.22 it follows that dl8decays always faster than l3/2. As a consequence, the argument of the sum in Eq. ~26! decays faster than 1/ l2for all potentials under consideration. Neglecting terms of order 1/ n, as is consistent with our pre- vious approximations, we may extend the upper summation limit to infinity and arrive at our final result Gn5g8p2 S8221a 2aF( l51‘ ldl82Gn. ~27! For sufficiently large energies, the level widths are therefore proportional to the state number n. We point out that the proportionality constant depends on aandgbut not on M andA. Apart from the harmonic oscillator discussed at the end of Sec. II there exist a few more systems for which the levelwidths can be evaluated exactly. Table I summarizes the re-sults for the box, the radial part of the Coulomb problem~with the dipole matrix element given in Ref. 17 !, and the half-harmonic oscillator for which we sketch the calculationin the Appendix. All quantities are given for large quantumnumbern. In addition, for the dipole matrix element the limit of largelwithn@lis taken, as was the case in the general derivation given above. For these three examples, the table shows that the linear dependence of the level widths onquantum number results in a nontrivial way from then-dependences of the eigenenergies and the dipole matrix elements. While these properties are special to the case of ohmic damping, an extension to other bath densities along the linespresented here is straightforward. Often one assumes apower-law behavior for the spectral density of bath oscilla- torsJ( v);vb. The behavior at large frequencies may lead to divergencies and one is often forced to introduce a cutoff which might be sharp or exponential in nature. If the cutofffrequency is much larger than all other energies of interest,one finds along the lines indicated above for the level widths G n;n12~b21!~22a!/~21a!. ~28! This clearly shows that the universality found above is spe- cial to the case of ohmic damping ( b51). For larger expo- nent bthe level widths may even decrease with increasing quantum number nas is well known from the stability of Rydberg atoms. IV. CONCLUSIONS Making use of a semiclassical expression for the dipole matrix element in terms of the Fourier transform of the clas-sical paths, we have shown that the level widths of a particlein a power-law potential coupled to an ohmic environmentare proportional to the number nof the state. This result is 204 204 Am. J. Phys., Vol. 69, No. 2, February 2001 Ingold, Jalabert, and Richter This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 202.28.191.34 On: Fri, 20 Feb 2015 12:49:38valid for sufficiently large nand can therefore be extended to potentials possessing only an asymptotic power-law behav-ior. The cases with one or two hard walls are also shown toobey such a scaling. The proportionality of level widths withthe state number known for the harmonic oscillator is thusgeneralized to a large class of one-dimensional potentials.The prefactor of such a linear law depends on the specificpotential. The applicability of our results to physically realizable situations is limited by the following restrictions: ~i!the mo- tion of the particle whose state may decay has to be one-dimensional; ~ii!large quantum numbers are involved; ~iii! the coupling giving rise to the decay has to be dipolar andthe environment of ohmic density of bath modes; ~iv!the coupling has to be sufficiently weak in order to have welldefined states at large n. These conditions could be met on the one hand in experiments on mesoscopic electronic de-vices, which enable one to build quantum confined systemsof reduced spatial dimensionality. Other candidates aretrapped atoms: They can, e.g., be highly excited into elon-gated ~quasi 1D !Stark-type states and their coupling to the environment can be measured with high precision. Hence,experimental scenarios to test the simple general scaling ofthe widths with quantum number appear possible. One condition that would be interesting to be relaxed among the above requirements is the one concerning the di-mensionality of the system. In three dimensions there existsa semiclassical treatment of radiative lifetimes in hydrogen-like atoms 18which is relevant for spectroscopy of Rydberg atoms. Apart from this special case, we are not aware of anysystematic study of level widths for a whole class of poten-tials in the limit of large quantum numbers. Multidimensional systems allow us to take into account the rich behavior that the underlying classical mechanicsyields. Semiclassics is appropriate to treat such cases ~in the limit of large energies !and allows us to address the differ- ences arising from the chaotic or integrable nature of theclassical motion. Therefore, an extension of the approachpresented here would contribute to the understanding of theinterplay between quantum chaos and dissipation. ACKNOWLEDGMENTS We have benefited from discussions with J.-Y. Bigot, H. Grabert, S. Kohler, and S. Otto. Part of this work has beencarried out while one of us ~G.L.I. !was at the Centre d’Etudes de Saclay with financial support from theVolkswagen-Stiftung. APPENDIX: HALF-OSCILLATOR We discuss in this Appendix the case of a half-harmonic oscillator, where the confining potential has the form of a harmonic oscillator for q.0 and an infinite wall at q50~the particle is then confined to q>0). The interest of treating this nontrivial example separately stems from the fact that both approaches, the direct calculation and the semiclassicalapproximation, are feasible and can be compared. In contrastto the case of the harmonic oscillator, the dipole matrix ele-ments of the half-oscillator couple not only nearest neighborstates. The eigenstates of the half-oscillator are given by theodd eigenstates of the harmonic oscillator with the prefactoradjusted to account for the restricted interval of normaliza- tion. The dipole matrix element d nmmay then be evaluatedby expressing the Hermite polynomial with the higher quan- tum number n.mby means of the Rodrigues formula, dnm52S\ MvpD1/21 2n1m@~2n11!!~2m11!!#1/2 3E 0‘ djjH2m11~j!d2n11 dj2n11exp~2j2!. ~A1! After repeated partial integration and use of dk djk~jH2m11~j!!U j505~21!m112k/22k21k~2m11!! ~m112k/2!!, ~A2! which holds for keven and yields zero for kodd, one arrives at dnm52S\ MpvD1/2S~2m11!! 22~m1n11!~2n11!!D1/2 3~21!n2m11( l51m11 22ll@2~n2l!#! ~m2l11!!~n2l!!.~A3! By induction one can show that dnm5S\ MpvD1/2~21!n2m11 2n1m21 3@~2n11!!~2m11!!#1/2 n!m!1 4~n2m!221.~A4! The semiclassical evaluation of the dipole matrix element using Eq. ~23!is considerably easier and leads to dn,n2l524 pF\ MvSn21 4DG1/21 4l221. ~A5! Forn@lthis agrees with the exact result up to an irrelevant sign. For the level widths we then obtain Gn58 p2gn.0.81gn ~A6! and thus the proportionality to the quantum number. 1See, e.g., Ulrich Weiss, Quantum Dissipative Systems ~World Scientific, Singapore, 1993 !; Thomas Dittrich, Peter Ha ¨nggi, Gert-Ludwig Ingold, Bernhard Kramer, Gerd Scho ¨n, and Wilhelm Zwerger, Quantum Trans- port and Dissipation ~Wiley-VCH, Weinheim, 1998 !,C h a p .4 . 2M. Brune, E. Hagley, J. Dreyer, X. Maı ˆtre, A. Maali, C. Wunderlich, J. M. Raimond, and S. Haroche, ‘‘Observing the progressive decoherence of the‘Meter’ in a quantum measurement,’’ Phys. Rev. Lett. 77, 4887–4890 ~1996!. 3David P. DiVincenzo, ‘‘Quantum computation,’’ Science 270, 255–261 ~1995!. 4See, e.g., Serge Haroche and Daniel Kleppner, ‘‘Cavity quantum electro- dynamics,’’ Phys. Today 42, 24–30 ~January 1989 !. 5Martin C. Gutzwiller, Chaos in Classical and Quantum Mechanics ~Springer, Berlin, 1990 !. 6Matthias Brack and Rajat K. Bhaduri, Semiclassical Physics , Frontiers in Physics ~Addison-Wesley, Reading, 1997 !, Vol. 96. 7For recent work on accurate WKB wave functions for 1D potentials see: H. Friedrich and J. Trost, Phys. Rev. Lett. 76, 4869–4873 ~1996!; Phys. Rev. A54, 1136–1145 ~1996!. 8Uday P. Sukhatme, ‘‘WKB energy levels for a class of one-dimensional potentials,’’ Am. J. Phys. 41, 1015–1016 ~1973!. 9J. F. Carin˜ena, C. Farina, and Ca ´ssio Sigaud, ‘‘Scale invariance and the Bohr-Wilson-Sommerfeld ~BWS !quantization for power law one- dimensional potential wells,’’ Am. J. Phys. 61, 712–717 ~1993!. 10As an early reference we mention V. B. Magalinskiı ˇ, ‘‘Dynamical model 205 205 Am. J. Phys., Vol. 69, No. 2, February 2001 Ingold, Jalabert, and Richter This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 202.28.191.34 On: Fri, 20 Feb 2015 12:49:38in the theory of the Brownian motion,’’ Zh. Eksp. Teor. Fiz. 36, 1942– 1944 ~1959!@Sov. Phys. JETP 9, 1381–1382 ~1959!#. Further references can be found for example in Ref. 1. 11A. O. Caldeira and A. J. Leggett, ‘‘Quantum tunneling in a dissipativesystem,’’ Ann. Phys. ~N.Y.!149, 374–456 ~1983!. 12C. Cohen-Tannoudji, B. Diu, and F. Laloe ¨,Quantum Mechanics ~Wiley, New York, 1978 !, Vol. 2, p. 1318. 13See, e.g., Claude Cohen-Tannoudji, Jacques Dupont-Roc, and Gilbert Grynberg, Atom-Photon Interactions ~Wiley, New York, 1992 !. 14A. Hanke and W. Zwerger, ‘‘Density of states of a damped quantum oscillator,’’ Phys. Rev. E 52, 6875–6878 ~1995!.15See, e.g., L. D. Landau and E. M. Lifshitz, Quantum Mechanics , Sec. 48 ~Pergamon, New York, 1959 !. 16Robert Karrlein and Hermann Grabert, ‘‘Semiclassical theory of vibra- tional energy relaxation,’’ J. Chem. Phys. 108, 4972–4983 ~1998!. 17S. M. Susskind and R. V. Jensen, ‘‘Numerical calculations of the ioniza- tion of one-dimensional hydrogen atoms using hydrogenic and Sturmian basis functions,’’ Phys. Rev. A 38, 711–728 ~1988!. 18Hermann Marxer and Larry Spruch, ‘‘Semiclassical estimation of the ra- diative mean lifetimes of hydrogen like states,’’ Phys. Rev. A 43,1 2 6 8 – 1274 ~1991!. UNIVERSAL GRAVITATION Stanley and Emin and their followers trekked for several months to the east African coast, reach- ing the sea at a small German post in today’s Tanzania. A German battery fired an artillery salute in their honor, and officials gave the two of them a banquet at the local officers’ mess. A naval band played; Stanley, Emin, and a German major gavespeeches. ‘‘The wines were choice and well-selected and iced,’’ writes Stanley. Then the near-sighted Emin, who had been moving up and down the banquet table, chatting with the guests anddrinking champagne, stepped through a second-floor window that he apparently thought openedon a veranda. It didn’t. He fell to the street and was knocked unconscious. He had to remain in alocal German hospital for two months, and Stanley was unable to bring him back to Europe intriumph. Most embarrassing of all for Stanley was that Emin Pasha, once he recovered, went towork neither for his British rescuers nor for Leopold, but for the Germans. Adam Hochschild, King Leopold’s Ghost—A Story of Greed, Terror, and Heroism in Colonial Africa ~Houghton Mifflin, Boston, 1998 !,p .1 0 0 . 206 206 Am. J. Phys., Vol. 69, No. 2, February 2001 Ingold, Jalabert, and Richter This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 202.28.191.34 On: Fri, 20 Feb 2015 12:49:38

1.3567005.pdf

A new method for wideband characterization of resonator-based sensing platforms Farasat Munir, Adam Wathen, and William D. Hunt Citation: Rev. Sci. Instrum. 82, 035119 (2011); doi: 10.1063/1.3567005 View online: http://dx.doi.org/10.1063/1.3567005 View Table of Contents: http://rsi.aip.org/resource/1/RSINAK/v82/i3 Published by the AIP Publishing LLC. Additional information on Rev. Sci. Instrum. Journal Homepage: http://rsi.aip.org Journal Information: http://rsi.aip.org/about/about_the_journal Top downloads: http://rsi.aip.org/features/most_downloaded Information for Authors: http://rsi.aip.org/authors Downloaded 17 Jul 2013 to 132.174.255.3. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://rsi.aip.org/about/rights_and_permissionsREVIEW OF SCIENTIFIC INSTRUMENTS 82, 035119 (2011) A new method for wideband characterization of resonator-based sensing platforms Farasat Munir, Adam Wathen, and William D. Hunt School of Electrical and Computer Engineering, Georgia Institute of Technology, 791 Atlantic Dr ., Atlanta, Georgia 30332, USA (Received 15 January 2011; accepted 12 February 2011; published online 29 March 2011) A new approach to the electronic instrumentation for extracting data from resonator-based sensing devices (e.g., microelectromechanical, piezoelectric, electrochemical, and acoustic) is suggested and demonstrated here. Traditionally, oscillator-based circuitry is employed to monitor shift in the resonance frequency of the resonator. These circuits give a single point measurement at the frequency where the oscillation criterion is met. However, the resonator response itself is broadband and contains much more information than a single point measurement. Here, we present a methodfor the broadband characterization of a resonator using white noise as an excitation signal. The resonator is used in a two-port filter configuration, and the resonator output is subjected to frequency spectrum analysis. The result is a wideband spectral map analogous to the magnitude of the S21parameters of a conventional filter. Compared to other sources for broadband excitation (e.g., frequency chirp, multisine, or narrow time domain pulse), the white noise source requires no design of the input signal and is readily available for very wide bandwidths (1 MHz–3 GHz). Moreover, itoffers simplicity in circuit design as it does not require precise impedance matching; whereas such requirements are very strict for oscillator-based circuit systems, and can be difficult to fulfill. This results in a measurement system that does not require calibration, which is a significant advantageover oscillator circuits. Simulation results are first presented for verification of the proposed system, followed by measurement results with a prototype implementation. A 434 MHz surface acoustic wave (SAW) resonator and a 5 MHz quartz crystal microbalance (QCM) are measured using theproposed method, and the results are compared to measurements taken by a conventional bench-top network analyzer. Maximum relative differences in the measured resonance frequencies of the SAW and QCM resonators are 0.0004% and 0.002%, respectively. The ability to track a changing sensor response is demonstrated by inducing temperature variations and measuring resonance frequency simultaneously using the proposed technique in parallel with a network analyzer. The relativedifference between the two measurements is about 5.53 ppm, highlighting the impressive accuracy of the proposed system. Using commercially available digital signal processors (DSPs), we believe that this technique can be implemented as a system-on-a-chip solution resulting in a very low cost, easyto use, portable, and customizable sensing system. In addition, given the simplicity of the signal and circuit design, and its immunity to other common interface concerns (injection locking, oscillator interference, and drift, etc.), this method is better suited to accommodating array-based systems.© 2011 American Institute of Physics . [doi: 10.1063/1.3567005 ] I. INTRODUCTION AND BACKGROUND Resonator-based sensing platforms have been widely studied for use in chemical and biological applications. A notable historic example is the development of the quartzcrystal microbalance (QCM) for mass sensing first reported by Sauerbrey in 1959. 1Since then, a variety of other types of resonators have been developed, taking advantage ofmicroelectronic fabrication techniques. Examples of more modern resonators include membrane resonators, 2cantilever resonators,3–5and high frequency surface and bulk acoustic wave (SAW and BAW) devices.6–8 The sensing functionality is achieved by coating the res- onator with a chemically sensitive or selective layer. A change in physical properties of this layer due to the sensing event affects the resonator response. An electronic system is re-quired to interface with the resonator and track the changing resonator response. The response parameters (e.g., resonancefrequency) are further converted to physical parameters (e.g., mass density and viscosity) of the thin film coating by usingequivalent circuit models. 9,10 Traditionally, oscillator circuits are used for monitoring the shift of the resonance frequency, fo, of the resonator. The resonator is employed in the feedback loop of the oscillator as a frequency-determining element. These circuits provideonly a single point measurement at the frequency where the oscillation criterion is met. However, the resonator response itself is broadband and contains much more information thana single point measurement. As an example, resonators op- erating in liquid phase experience a strongly damped Qas compared to gaseous phase. Extreme phase and circuit stabil-ity are required to obtain stable single frequency oscillation. 11 Hence, various modifications to simple oscillator circuits arereported which use automatic gain control 10,12or parallel ca- pacitance compensation techniques.13It is very difficult to 0034-6748/2011/82(3)/035119/7/$30.00 © 2011 American Institute of Physics 82, 035119-1 Downloaded 17 Jul 2013 to 132.174.255.3. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://rsi.aip.org/about/rights_and_permissions035119-2 Munir, Wathen, and Hunt Rev. Sci. Instrum. 82, 035119 (2011) maintain the accuracy of the measurements under reduced Q conditions and, therefore, a careful calibration of the oscil- lator circuit is required. This calibration in oscillator basedcircuits may require expensive instruments such as network analyzers. 10Phase locked loop (PLL) based techniques have been proposed14–16to overcome these issues but are relatively more complex and also require calibration. This calibration in PLL based circuits is achieved by employing additional refer- ence phase locked loops, thus resulting in manifold increase in circuit complexity. Another type of interface circuit relies on impulse excitation of the resonator. A popular example isthe ring-down technique. 17In addition to the resonance fre- quency, this method does provide an additional measurement point (energy dissipation, D, or quality factor, Q=1/D), but still relies on oscillator circuitry and suffers from the same problems previously mentioned. Measurement of the shift in the resonance frequency and quality factor is not always enough for complete determi- nation of the physical parameters of the tested sample. In particular, for biosensors operating in liquid loading condi-tions, there are more than two unknown physical parame- ters (such as mass density, shear modulus, film thickness) of the sample. 18Therefore, a complete characterization of the impedance spectrum is useful, which can be only performed using network analyzers that are sophisticated, bulky, and ex- pensive. Hence, they are not suitable for in situ techniques. Systems operating on the principle of network analysis, but with smaller and less expensive electronics, have been devel-oped for quartz crystal resonators. 19,20Unlike network ana- lyzers, they operate in a narrow range around the resonance frequency. As an alternative to impedance spectra, systemsmeasuring a “voltage transfer function,” dependent on sensor- impedance, have been reported; however, the transfer function method requires further fitting to application-specific modelsto extract load data. 21 Simultaneous detection of multiple biomarkers requires multielement sensor array systems. Such a system is veryuseful for screening of diseases like cancer or sepsis where single biomarker detection is not conclusive. 22,23Arrays of resonators have some unique electrical interface circuit re-quirements. In particular, for oscillator-based circuit designs, each amplifier–resonator pair must be designed separately to measure the shift in resonance frequency. However, each individual oscillator loop in the array is prone to interference and frequency pulling effects from neighboring resonators inthe array. Therefore, it is difficult to extract and compare the orthogonal responses of individual resonators. Excitation of a resonator with a sufficiently wideband signal gives an output that will contain only the filtered frequencies representative of the transfer function of the resonator. Previously, different types of wideband excitationsignals have been used with resonators (i.e., multifrequency chirp and narrow time domain impulse). 24,25Each of these signals requires careful design of the signal itself as well asthe circuit layout for the specific type, resonance frequency, and bandwidth of the resonator. These signals have their individual limitations for use with resonators: the chirpedsignal requires larger measurement time and the time domain pulse excitation signal is not well suited to very high Q FIG. 1. (Color online) Spectral analysis of noise excited resonator. resonators25such as a QCM ( Q∼30 ,000). Moreover, the response of reduced Qresonators (as experienced under liquid loading conditions) spans a larger bandwidth. The timedomain input pulse, then, must be very narrow in time and, correspondingly, its power is dispersed over a larger band- width. This will reduce the power at the resonance frequency down to a level which may fail to excite the resonance. 25 Another wideband signal is a multisine signal which is commonly used for frequency domain system identifi- cation and characterization. Its application to resonant sys- tems has not been reported. However, the circuit design formultisine generation is very complicated, especially at high frequencies. 26A multisine signal provides wideband excita- tion but its spectrum is discrete and will therefore be resolu-tion limited and ill-suited for high Qresonators. In this paper, we attempt to circumvent these issues and present an argument for and simulation and experimental re-sults of a new approach to the electronic instrumentation for extracting data from resonator-based sensing devices based on white noise excitation. An electromechanical resonator isused in a two-port filter configuration and its output is then subjected to frequency domain spectral analysis. This pro- posed method is shown schematically in Fig. 1. We further show that it offers a very simplistic interface design with several advantages over conventional methods of parameterextraction and tracking. The white noise source requires no design of the input signal and is readily available commer- cially for very wide bandwidths (1 MHz–3 GHz). Moreover,it offers simplicity in circuit design, as it does not require precise impedance matching; whereas such requirements are very strict for oscillator circuit systems, and are hard to fulfill.This results in a measurement system that does not require calibration, which is a significant advantage over oscillator circuits. The measurement output of this method is experi-mentally compared to that of a bench-top network analyzer and is shown to agree within 0.002%. Given the simplicity in the signal and circuit design, and its immunity to other com-mon interface concerns (injection locking, oscillator interfer- ence, drift, etc.), this method is better suited to accommodat- ing array-based systems. II. THEORETICAL UNDERPINNINGS AND SIMULATION RESULTS In deterministic signal scenarios, the input and output of a linear system are directly related through the transferfunction. In the case of the excitation of linear systems with stochastic inputs, such direct characterization does not exist. The white noise input signal proposed here can be assumed wide-sense stationary. The second-order moments of such a signal (such as autocorrelation) can be used to characterize Downloaded 17 Jul 2013 to 132.174.255.3. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://rsi.aip.org/about/rights_and_permissions035119-3 Munir, Wathen, and Hunt Rev. Sci. Instrum. 82, 035119 (2011) the system transfer function. The Fourier transform of the au- tocorrelation gives the power spectral density (PSD), which is used for frequency domain characterization. For a randominput signal, the input and output power density spectrums of the resonator are related a /Gamma1 yy=/Gamma1xx|H(F)|2(1) where, /Gamma1yyis the output PSD, /Gamma1xxis the input PSD, |H(F)| is the resonator transfer function in the frequency domain, |H(F)|2is thus the energy density spectrum of the resonator response. If the input is a white noise signal with power σ2 x, then we have /Gamma1yy=σ2 x|H(F)|2. (2) Thus, for white noise, the output PSD is proportional to the energy density spectrum of the resonator frequency re- sponse scaled by the power of input noise. The output, then,gives us a measure of the magnitude of the frequency response of the resonator. We cannot measure the phase of the resonator response with this method, which is very essential for oscilla- tor based interface methods. The phase measurement, though holds a much lesser significance here, because the frequencyresponse measured by this method can give a direct measure of the resonance frequency, f r, and the half-band-half-width (HBHW), bandwidth of the resonator. Conventionally, thesetwo parameters are used for characterizing the resonator, and are measured using network analyzers. It has been shown by various groups for acoustic as well as MEMS resonators thatthe viscoelastic properties of the surface perturbation sam- ple can be deduced by analyzing the resonance frequency and bandwidth. 21,27–31Resonance frequency and dissipation (measured by oscillator based interface circuits coupled with impulse excitation) have also been extensively used to inter- pret the viscoelastic properties of the surface coating layer andthe tested sample. 32,33Both of these methods are equivalent as Johannssmann has given a relation that describes dissipation, D, in terms of the HBHW bandwidth.34Furthermore, conver- gence relationships of dissipation analysis and the analysis based on /Delta1fand/Delta1HBHW, have also been reported.35 The proposed method described here gives not only in- formation about frand/Gamma1, but also the wide-bandwidth trans- fer function of the resonator (with a much simpler setup thana network analyzer). Potentially, this could offer more useful information than just relying on the f rand/Gamma1. The setup of Fig. 1was simulated using SIMULINKR/circlecopyrt software by MathWorksR/circlecopyrt. The simulation setup is shown in Fig. 2. The “white noise” block generates a signal with a uniform frequency distribution as an input to the resonator. To model the resonator response accurately in the simulation, FIG. 2. Setup for simulating noise excited response of a resonator. FIG. 3. (Color online) (a) Measured S-parameters of a SAW resonator. (b) Simulated FFT scope output showing both the input noise spectrum and the output of the resonator. a measured S-parameters file of an actual SAW resonator is used. The FFT scope is used for power spectrum density analysis. The results are shown in Fig. 3. Notice that in this simulation setup, we used a two-port resonator and the output frequency response is similar to the S21 response. However, one-port devices can also be used in this techniqueby employing them as a through element in a two-port measurement system. The results in Fig. 3are for an FFT of 1024 points. We did the simulation for an FFT size of256, 512, 1024, and 2048. The relative error between the S21 and FFT output decreased with increasing FFT size. However, there was minimal improvement between 1024 and2048 point FFT. We expect that the FFT size required for a minimal relative error is dependent on the resonator quality factor and the particular sensing application. The purpose ofpresenting this simulation is to give a proof-of-concept for the proposed method, which is clearly demonstrated by the results shown in Fig. 3. Next, we demonstrate the scalability of the proposed method to resonator arrays. Figure 4(a) shows the setup for exciting a two-element resonator array simultaneously with a single noise source. The resonators are again modeled using their S-parameters files with their individual resonance peaks (in the S21 Downloaded 17 Jul 2013 to 132.174.255.3. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://rsi.aip.org/about/rights_and_permissions035119-4 Munir, Wathen, and Hunt Rev. Sci. Instrum. 82, 035119 (2011) FIG. 4. (Color online) (a) Simulation setup for two- element resonator array. (b)The noise excited spectral output of a two-element resonator array. response) at 248.8 and 252 MHz. The simulated FFT of the combined output shows that the individual responses of each resonator element are reproduced fairly accurately in the spectral output. Based on these simulation results, we suggestthat the proposed method can be easily extended to resonator arrays. This method provides a significant advantage over all other methods for array systems by eliminating the need formultiple input sources or a switching method between array elements. However, it requires that the resonance frequency of each element in the resonator array is offset from the otherelements in the array. Implementing such an array system has been reported by Yang et al. 24 III. MEASUREMENT RESULTS To further investigate the proposed system, we built a prototype using a surface mount noise source, SMN-7114- C2A by MicroneticsR/circlecopyrtInc. This noise source excites a res- onator and the output spectrum is measured using an E4404BAgilent Spectrum Analyzer (SPA). The noise source and its frequency spectrum are shown in Fig. 5(a). The prototype setup is shown in Fig. 5(b). Using this setup, we measured two different types of resonators—a 434 MHz SAW resonator and a 5 MHz QCM. A. SAW resonator We measured a 434 MHz SAW resonator with the pro- totype setup shown in Fig. 5(b). A comparison of the SAW resonator output as measured by the SPA and as measured by the Vector Network Analyzer (VNA) is shown in Fig. 6.I ti s FIG. 5. (Color online) (a) SMN-7114-C2A noise source and its spectrum. (b) Prototype setup snapshot. evident that both responses are comparable in frequency con- tent. The characteristic resonance peaks are replicated at thesame frequencies, and the relative strengths of frequency con- tent are preserved and have been explicitly marked. It should be noted that the measurements with the proposed method are done without any calibration. The remarkable accuracy of the proposed system is further highlighted in Fig. 7, where both responses are plotted in the same graph. There is no scaling employed on frequency or magnitude axis, but both responses are plotted after subtracting the respective meanpower levels. The frequency resolution for VNA measure- ments and for SPA measurements is 3 and 10 kHz, respec- tively. The resonance frequencies (taken as the frequency atthe minimum point in the S21 amplitude) measured by VNA and SPA are 433.996875 and 434.000000 MHz, respectively. This gives a relative difference of about 0.0004%. The meanamplitude difference over the entire bandwidth of measure- ment, between the two methods is 0.3 dB with a variance of 0.09 dB. Further accuracy can also be obtained by improv-ing the frequency resolution of the SPA, but at the cost of increased measurement time. Downloaded 17 Jul 2013 to 132.174.255.3. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://rsi.aip.org/about/rights_and_permissions035119-5 Munir, Wathen, and Hunt Rev. Sci. Instrum. 82, 035119 (2011) FIG. 6. (Color online) Noise excited resonator output. (a) Measured by a spectrum analyzer. (b) Measured by a vector network analyzer. FIG. 7. (Color online) Measured responses of SAW resonator by VNA andthe proposed method after mean level subtraction. FIG. 8. Measured responses of QCM by the VNA and the proposed methodafter mean level subtraction. B. Quartz crystal microbalance The same circuit setup shown in Fig. 5(b) without any modification can be used for measuring low frequency orhigh frequency resonators, as well as one-port resonators. However, one-port resonators will be needed to plug into the two-port circuit setup as through elements. To demon-strate these properties of the proposed system, we measured a 5 MHz QCM (a very low frequency, one-port resonator) with the same circuit that was used for the SAW resonator men- tioned above. We measured the two-port S-parameters of the QCM with the VNA as well as with our proposed method.Figure 8shows the two responses in a single plot with the mean power level for each response subtracted from its respective response. Here, we present a comparison of QCM measurement with the two methods. There are two distinct points in the QCM response which corresponds to the maximum ampli-tude point (Max Point) and minimum amplitude point (Min Point). Each of these could be considered a measure of resonance frequency. At the maximum, the resonance fre-quencies measured by the VNA and the SPA are 4.9991 and 4.9990 MHz, respectively. This gives a relative difference in resonance frequencies of about 0.002%. The mean amplitudedifference between the two methods (over the entire band- width of measurement) is 0.58 dB with a variance of 1.4 dB. However, these numbers are misleading because the SPA re- sponse is not following the VNA-measured results in the neg- ative dip at minimum. This error is due to the fact that theQCM response has a very large dynamic range ( ∼80–90 dB), which takes the negative dip well below the noise floor of the SPA. Therefore, the QCM response is truncated at thenegative dip as it has approached the noise floor of the SPA (∼–124 dBm). This problem can be taken care of by properly amplifying the input noise signal level to raise the output ofthe resonator well above the SPA noise floor. C. Frequency tracking ability We measured the temperature curve of the SAW device with both a VNA and the proposed method. The temperature Downloaded 17 Jul 2013 to 132.174.255.3. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://rsi.aip.org/about/rights_and_permissions035119-6 Munir, Wathen, and Hunt Rev. Sci. Instrum. 82, 035119 (2011) FIG. 9. (Color online) Temperature curve measurement of SAW device with VNA and with noise-excited spectral measurement. was varied from 0 to 100◦C and at each temperature point the devices were allowed to settle for 2 min. The changes in device response with changing temperature were tracked bymeasuring the maximum amplitude point in the frequency response obtained by the spectral analyzer and in the |S21| response measured by the VNA. An important detail to notehere is that VNA measurements were taken after a tedious two-port calibration of the VNA, whereas the measure- ments with our proposed method were taken without anycalibration. Results are shown in Fig. 9. The temperature curves measured by both techniques are in agreement with a maximum difference of 5.53 ppm. This experiment demon-strates the ability of the proposed method to accurately track changes in the frequency response of the resonator. IV. DISCUSSION The proposed system described above gives a wideband response of the resonator with a very simple setup. The wide-band frequency response has more information than just the measurement of resonance frequency and bandwidths. This extra information could be potentially useful for extraction ofthe physical parameters of the surface load. It is extremely difficult to find and track the resonance frequency shifts of the resonator when heavily loaded by liquid. The oscillationsmay even fail to occur due to high damping. Therefore, un- der heavy liquid loading conditions the proposed method has an advantage over other methods as it is not just a single point measurement. Moreover, because of noise being used as the input signal, significant advantages are obtained overconventional single frequency excitation systems such as os- cillators. Circuit design for the proposed method is relatively independent of resonance frequency variations. Hence, thesame circuit designed for a particular center frequency may be used for a very large bandwidth around the center fre- quency. We demonstrated the use of same circuit for 5 and434 MHz devices without using any calibration for either de- vice. The proposed method is relatively immune to parasitic effects compared to oscillator-based systems. It is also im-mune to background noise fluctuations which appear as a jitter in the oscillator-based systems and result in limited frequency resolution. Wideband white noise has a continuous frequencyspectrum and hence, the resolution is, theoretically, not lim- ited. In practice, though the resolution will be limited by the sampling rate of the analog to digital converter (ADC), and the FFT resolution. There are some additional benefits for noise excitation of resonator arrays. It does not require individual circuits tuned for individual sensors. A single input source drives all the sensors simultaneously with no switches required (switchingcircuits introduce an additional complexity at high frequen- cies). In our proposed method, each resonator outputs its own frequency response and is not in an oscillator configurationwhere loop dynamics can lock onto external signal (from a neighboring resonator) injected into the loop. 36Therefore, it will be immune to interference from neighboring resonators(Table I). TABLE I. Summarized comparison of oscillator based systems and the proposed method. Oscillator-based interface circuitry for resonators Noise excited spectral analysis of resonators Measured information content •Gives a single point measurement •Gives a wideband frequency response, holding much more of the resonance frequency only information than just the resonance frequency Circuit design •Strict requirements of phase stability •No phase compensation requirements •Requires precise impedance matching •Tolerant to mismatch over a broadband •Circuit design requires modification •Same circuit can operate over a wide with the change in resonance frequency range of resonance frequencies•Phase noise appears as jitter and •Immune to phase noise and limited limits the frequency resolution in frequency resolution only by FFT size Reduced Qconditions •Difficult to excite oscillations •No Oscillatory behavior needed •Requires complex circuit modifications •Same setup can be used for both reduced for accurate measurements Qand high Qconditions Array systems •Requires individual circuit tuned for •Array is excited as a whole and hence does individual elements of the array not require individual circuit for each element•Each element is prone to interference (due to injection •No oscillator loops are involved and hence interference locking) from neighboring resonators in the array from neighboring elements is not a concern Downloaded 17 Jul 2013 to 132.174.255.3. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://rsi.aip.org/about/rights_and_permissions035119-7 Munir, Wathen, and Hunt Rev. Sci. Instrum. 82, 035119 (2011) V. CONCLUSION We have presented a novel method to measure and track a resonator’s response and extract its characterization parameters. This method measures the wideband frequency response of the resonator with a much simpler setup comparedto conventional methods. We have suggested and demon- strated the use of a white noise signal as a viable signal for broadband excitation of resonator-based sensing platforms.We have also established, through simulation and prototype measurements, the feasibility of the proposed method. The accuracy and speed of the system can be further improvedby FFT-based digital implementation of the spectral analysis system. This will allow for a low-cost and compact solution in the form of a system on a chip. ACKNOWLEDGMENTS This work was supported in part by the V Foundation and by the National Cancer Institute (NCI) (Grant No. 2106AZX). The authors also wish to thank Mohammad Omer for several helpful discussions regarding the MATLAB R/circlecopyrtsimulations. 1G. Z. Sauerbrey, Physik 155, 206 (1959). 2R. Abdolvand, H. Zhili, and F. Ayazi, in Proceedings of the 5th IEEE Con- ference on Sensors , 1297–1300, 2006. 3R. Raiteri, M. Grattarola, H. J. Butt, and P. Skladal, Sens. Actuators B 79(2–3), 115 (2001). 4T. Liu, J. Tang, M. Han, and L. Jiang, Biochem. Biophys. Res. Commun. 304(1), 98 (2003). 5J. Verd, A. Uranga, G. Abadal, J. L. Teva, F. Torres, J. L. Lopez, E. Perez- Murano, J. Esteve, and N. Barniol, IEEE Electron Device Lett. IEEE 29(2), 146 (2008). 6S. Rey-Mermet, R. Lanz, and P. Muralt, Sen. Actuators B 114(2), 681 (2006). 7F. Bender, P. Roach, A. Tsortos, G. Papadakis, M. I. Newton, G. McHale,and E. Gizeli, Meas. Sci. Technol. 20(12), 124011 (2009). 8D. D. Stubbs, L. 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Schroder, R. Borngraber, R. Lucklum, and P. Hauptmann, Rev. Sci. In- strum. 72(6), 2750 (2001). 20R. Schnitzer, C. Reiter, K. C. Harms, E. Benes, and M. Groschl, IEEE Sens. J.6(5), 1314 (2006). 21F. Eggers and T. Funck, J. Phys. E: J. Sci. Instrum. 20(5), 523 (1987). 22K. R. Kozak, F. Su, J. P. Whitelegge, K. Faull, S. Reddy, and R. Farias- Eisner, Proteomics 5(17), 4589 (2005). 23D. D. Taylor, S. Atay, D. S. Metzinger, and C. Gercel-Taylor, Gynecol. Oncol. 116(2), 213 (2010). 24Y . T. Yang, C. Callegari, X. L. Feng, K. L. Ekinci, and M. L. Roukes, Nano Lett. 6(4), 583 (2006). 25R. P. Wali, P. R. Wilkinson, S. P. Eaimkhong, J. Hernando-Garcia, J. L. Sanchez-Rojas, A. Ababneh, and J. K. Gimzewski, Sens. Actuators B 147(2), 508 (2010). 26N. B. Carvalho, J. C. Pedro, and J. P. Martins, IEEE Trans. Microwave Theory Tech. 54(6), 2659 (2006). 27B. Du and D. Johannsmann, Langmuir 20(7), 2809 (2004). 28H. L. Bandey, S. J. Martin, R. W. Cernosek, and A. R. Hillman, Anal. 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1.3126779.pdf

A combining rule calculation of the ground state van der Waals potentials of the mercury rare-gas complexes X. W. Sheng, P. Li, and K. T. Tang Citation: The Journal of Chemical Physics 130, 174310 (2009); doi: 10.1063/1.3126779 View online: http://dx.doi.org/10.1063/1.3126779 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/130/17?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Mobility of singly-charged lanthanide cations in rare gases: Theoretical assessment of the state specificity J. Chem. Phys. 140, 114309 (2014); 10.1063/1.4868102 Interactions between anionic and neutral bromine and rare gas atoms J. Chem. Phys. 128, 064317 (2008); 10.1063/1.2830031 Orbital-free embedding applied to the calculation of induced dipole moments in CO 2 X ( X = He , Ne, Ar, Kr, Xe, Hg) van der Waals complexes J. Chem. Phys. 123, 174104 (2005); 10.1063/1.2107567 Ab initio intermolecular potential energy surface, bound states, and microwave spectra for the van der Waals complex Ne–HCCCN J. Chem. Phys. 122, 174312 (2005); 10.1063/1.1888567 On the validity of different intercombination rules for the parameters of the ground state ( X 1 O + ) potential AIP Conf. Proc. 386, 227 (1997); 10.1063/1.51850 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 194.47.65.106 On: Sun, 12 Oct 2014 15:58:39A combining rule calculation of the ground state van der Waals potentials of the mercury rare-gas complexes X. W. Sheng,1P . Li,1and K. T . T ang1,2, a/H20850 1Institute of Atomic and Molecular Physics, Sichuan University, Chengdu, Sichuan 610065, People’ s Republic of China 2Department of Physics, Pacific Lutheran University, Tacoma, Washington 98447, USA /H20849Received 26 February 2009; accepted 8 April 2009; published online 7 May 2009 /H20850 The ground state van der Waals potentials of the Hg–RG /H20849RG=He,Ne,Ar,Kr,Xe /H20850systems are generated by the Tang–Toennies potential model. The parameters of the model are calculated from the potentials of the homonuclear mercury and rare-gas dimers with combining rules. The predictedspectroscopic parameters for these mercury rare-gas complexes are in good agreement withavailable experimental values, except for Hg–He. In the repulsive and potential well regions, thepredicted potential energy curves agree with the available experimental hybrid potentials, but theydiffer in the long range part of the potential. On the other hand, the present potentials are inagreement with the ab initio CCSD /H20849T/H20850calculations in the long range part of the potential, but there are some differences in the short repulsive regions. According to the present theory, the reducedpotential curves of these five systems, including Hg–He, are almost identical to each other. Thisreduced potential curve can also describe, within a few percent, the five reduced potentials obtainedfrom the ab initio CCSD /H20849T/H20850calculations. These reduced potentials have a potential bowl that is wider than that of the rare-gas dimers, but narrower than the mercury dimer. © 2009 American Institute of Physics ./H20851DOI: 10.1063/1.3126779 /H20852 I. INTRODUCTION The interaction potentials between rare-gas atoms are known for some time.1,2Recently the van der Waals potential of the mercury dimer was successfully described by theTang–Toennies potential 3with a new set of dispersion coefficients.4In this paper we investigate the possibility of determining the interaction between a mercury atom and arare-gas atom from the potentials of the homonuclear mer-cury and rare-gas dimers with combining rules. In the early 1980s, the efficient cooling by supersonic expansion made it possible to form weakly bound van derWaals molecules. Since then, laser spectroscopy has beenused to study mercury rare-gas complexes 5in many labora- tories around the world.6Vibrotational analysis of laser ex- citation spectra allow the determination of the parameters ofthe assumed potential functions for the ground and appropri-ate excited states. Theoretical calculations for such weakly bound van der Waals molecules are still a challenge, although withmodern computers, quasirelativistic pseudopotential valenceab initio calculations are possible. The calculations of Hg–RG /H20849RG=He,Ne,Ar,Kr,Xe /H20850potentials by Czuchaj et al. 7–9with coupled cluster single, double, and perturbative triple CCSD /H20849T/H20850excitations were able to generate potential curves that are comparable to experiments. In spite of these progresses in both experiment and theory, the potentials governing the elementary collisions be-tween mercury and rare gases are still not known with anaccuracy comparable to that achieved for the rare-gas andrare-gas interactions. Part of the reason is that for the van der Waals Hg–RG potentials, even the van der Waals dispersioncoefficients are not known with high precision. For the interactions of two rare-gas atoms, the dispersion coefficients are known to a high degree of accuracy. 10–12In addition, a variety of experimental data made it possible forthe well depth /H20849D e/H20850and equilibrium distance /H20849Re/H20850of the rare- gas potentials to be unequivocally determined.13As a conse- quence, we are now in command of a remarkably accurate knowledge of the potential curves of these systems, and allthese potential curves can be described by the Tang–Toennies potential model. 1,2 Recently, a new set of dispersion coefficients for the mercury dimer, considerably larger than previously acceptedvalues, was derived 4from the index of refraction data of mercury vapor measured in the modern laser experiments.14 This set of coefficients, together with the experimental De andReuniquely specify the parameters of the Tang–Toennies potential for the mercury dimer. This potential was shownto be in excellent agreement with the CCSD /H20849T/H20850calculations, and the predicted frequency /H9275eand anharmonicity /H9273e/H9275e are also in very good agreement with spectroscopicmeasurements. 4 The quest for reliable combining rules for predicting the parameters of the interaction potential of the mixed system ij from the parameters of the like systems iiandjjhas a long history.15Such rules have the obvious advantages that they can extend the potentials of a few like systems to a largernumber of unlike systems. Under certain circumstances,some of these rules are very useful. For example, very accu-rate combining rules were developed for calculating the well a/H20850Electronic mail: tangka@plu.edu.THE JOURNAL OF CHEMICAL PHYSICS 130, 174310 /H208492009 /H20850 0021-9606/2009/130 /H2084917/H20850/174310/9/$25.00 © 2009 American Institute of Physics 130 , 174310-1 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 194.47.65.106 On: Sun, 12 Oct 2014 15:58:39depth Deand equilibrium distance Reof the van der Waals potentials of the mixed rare-gas systems from the corre-sponding values of the homogeneous dimers. 1However, these rules are premised on the fact that the potentials of all rare-gas dimers are conformal. /H20849That is, when the reduced potentials U/H20849x/H20850=V/H20849xRe/H20850/Deare plotted against x=R/Re, the curves are identical to each other /H20850. Unfortunately, the poten- tial of rare-gas dimers has a different shape from that ofmercury dimer /H20849See Fig. 4/H20850. These rules are not valid in the present case. On the other hand, there exist since quite sometime a reliable set of combining rules for the dispersion co-efficients describing the long range attractive part of the po-tential. These rules have a firm theoretical foundation 16and are shown to be accurate to better than one percent and valid for all systems.16–20 To construct the entire potential energy curve, a knowl- edge of both the long range attractive and the short rangerepulsive potentials as well as the effect of charge overlapare required. All three contributions are accounted for in theTang–Toennies potential model, 3which is successful in de - scribing the interaction potentials of both mercury dimer andrare-gas dimers. 2,4The effect of charge overlap is expressed in terms of the damping on the dispersion series. The damp- ing functions depend only on the range parameter of theshort range repulsive potential. Therefore to obtain the po-tential energy curve of the mercury rare-gas system, we needto find a reliable combining rule for the short range repulsivepotential. In the Tang–Toennies model, the short range repulsive potential is expressed in terms of the Born–Mayer formAexp /H20849−bR /H20850. There are several combining rules for Aand b. 21–24Among them, the Böhm–Ahlrichs rules24seem to have a stronger theoretical underpinning. These rules are used in this paper for the determination of the Born–Mayerparameters for the interaction of mercury and rare gases fromthe corresponding parameters of homogeneous mercury andrare-gas dimers. These parameters and the dispersion coeffi-cients make it possible to predict the potential energy of allfive mercury and rare-gas interactions. In Sec. II, we start with a brief review of the potential model and the combining rules. Then the five parameters/H20849A,b,C 6,C8,C10/H20850for the Hg–RG complexes calculated from the homonuclear mercury and rare-gas dimer potentials are presented. The spectroscopic constants De,Re,/H9275e, and/H9273e/H9275e obtained from the predicted potentials are compared with the previous experimental and theoretical determinations. Thepredicted potential curves are also compared with the ab ini- tioCCSD /H20849T/H20850calculations and with the experimental hybrid potentials. This paper closes with a discussion of the results. Atomic units will be used in all calculations. For com- parison with experiments, energy unit cm −1and length unit Å will also be used. II. METHOD A. The potential model The potential model proposed by Tang and Toennies /H20849TT /H20850in 1984 /H20849Ref. 3/H20850will be used to generate the ground state interaction potential between a mercury atom and arare-gas atom. In this model, the short range repulsive Born– Mayer potential is added to the long range attractive poten-tial which is given by the damped asymptotic dispersionseries, V/H20849R/H20850=Ae −bR−/H20858 n=3nmax f2n/H20849bR /H20850C2n R2n, /H208491/H20850 where Ris the internuclear distance and C2nare the disper- sion coefficients. Based on the form of the exchange correc-tion to the dispersion series, Tang and Toennies were led tothe conclusion that the damping function is an incompletegamma function f 2n/H20849bR /H20850=1− e−bR/H20858 k=02n/H20849bR /H20850k k!, /H208492/H20850 where bis the same as the range parameter of the Born– Mayer repulsion on the ground that both the repulsion anddispersion damping are consequences of the wave functionoverlap. For simple systems, such as H 2,H e 2, and HHe, damped dispersion can be numerically calculated,25–27 these accurately calculated damping functions are all in very good agreement with Eq. /H208492/H20850. Since its introduction, the model potential of Eq. /H208491/H20850. was tested successfully for several chemically different types of van der Waalsinteractions, 3,28,29including homonuclear mercury dimer4 and rare-gas dimers.2Moreover, this model has been shown to have a firm foundation within the generalized Heitler– London theory.30The model potential is determined by five parameters A,b,C6,C8, and C10. In principle, the dispersion series should go on to infinity and including the third orderterms which starts with C 11and with an opposite sign. Thus, if the series is terminated at 2 n=10, the errors of the ne- glected higher second order terms are compensated some-what by the neglected third order terms. These effects arerelatively small in any case. If C 12,C14, and C16are neces- sary, they can be estimated with approximate recursion rela-tions. However, if there are uncertainties in the first threecoefficients, relatively large errors may appear in thesehigher coefficients. In some applications, with D eand Re fixed, these errors can be compensated by changes in the short range parameters Aandb. In the present case, Aandb are calculated from the combining rules which are indepen-dent from the dispersion series. Therefore as an approxima-tion, the upper limit of the dispersion series in Eq. /H208491/H20850is set atnmax=5. B. Combining rules for Aandb There are several reasonably accurate combining rules for short range Born–Mayer repulsive potential parameters A andb.21–24They all give similar results. To be specific, the Böhm–Ahlrichs24rules will be used in this paper. They are derived from an energy dependent hard core model Rij/H20849V/H20850=1 2/H20851Ri/H20849V/H20850+Rj/H20849V/H20850/H20852, where R/H20849V/H20850denotes the internuclear distance at a given po- tential V, and a single index indicates the potential parameter174310-2 Sheng, Li, and T ang J. Chem. Phys. 130 , 174310 /H208492009 /H20850 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 194.47.65.106 On: Sun, 12 Oct 2014 15:58:39for the like system. This assumption leads to the following combining rules: bij=2bibj bi+bj, /H208493/H20850 Aij=/H20851Ai1/biAj1/bj/H20852bij/2. /H208494/H20850 Böhm and Ahlrichs found these rules useful by testing them against the repulsive interactions obtained from the SCF cal-culation which is without correlation. In the present applica-tion, these rules are used for the entire repulsive potential.This may be justified by the observation that the small addi-tional repulsion due to correlation is more or less propor-tional to the SCF repulsion. 3 The input data Aandbof the like systems are listed in Table I. Those of the mercury dimer are taken directly from Ref. 4, and those of the rare-gas dimers are taken directly from Ref. 1. The Aandbfor the interaction of a mercury atom and a rare-gas atom calculated from Eqs. /H208493/H20850and /H208494/H20850are shown in Table II. C. Combining rules for C6,C8, and C10 The combining rules for C6,C8, and C10follow from the Casimir–Polder theory of dispersion interaction.31,32In this theory, the individual dispersion coefficients are made up of different terms arising in the multipole expansion of theperturbation operator C 6=Cij/H208491,1 /H20850, /H208495/H20850 C8=Cij/H208491,2 /H20850+Cij/H208492,1 /H20850, /H208496/H20850 C10=Cij/H208491,3 /H20850+Cij/H208492,2 /H20850+Cij/H208493,1 /H20850, /H208497/H20850 where Cij/H208491,1 /H20850is the dipole-dipole, Cij/H208491,2 /H20850the dipole- quadrupole, Cij/H208492,2 /H20850the quadrupole-quadrupole, and Cij/H208491,3 /H20850the dipole-octupole interaction. They are given by the exact formula31–33 Cij/H20849l1,l2/H20850=/H208492l1+2l2/H20850! 4/H208492l1/H20850!/H208492l2/H20850!/H208732 /H9266/H20874/H20885 0/H11009 /H9251l1i/H20849i/H9275/H20850/H9251l2j/H20849i/H9275/H20850d/H9275,/H208498/H20850 where /H9251l1i/H20849i/H9275/H20850and/H9251l2j/H20849i/H9275/H20850are respective dynamic multipole polarizabilities at the frequency /H9275of atom iand j, respec- tively. Note that Eq. /H208498/H20850is an integral over imaginary fre- quency of the product of the dynamic polarizabilities of theinteracting atoms. This formulation reduces the original twocentered problem to a one centered problem of evaluating thefrequency dependent polarizabilities. To approximate the dy- namic dipole polarizability /H92511/H20849i/H9275/H20850, a one term approximant /H20851/H92511/H20849i/H9275/H20850/H20852is introduced16 /H20851/H92511/H20849i/H9275/H20850/H20852=/H92511 1+ /H20849/H9275//H90241/H208502, /H208499/H20850 where /H92511is the static polarizability /H92511/H208490/H20850and/H90241is an effec- tive energy. With this approximation, C6of a homogeneous dimer is given by C6i=3 /H9266/H20885 0/H11009 /H20851/H92511i/H20849i/H9275/H20850/H208522d/H9275=3 /H9266/H20885 0/H11009/H20875/H92511i/H90241i2 /H90241i2+/H92752/H208762 d/H9275. /H2084910/H20850 With the well known mathematical identity34 2 /H9266/H20885 0/H11009ab /H20849a2+/H92752/H20850/H20849b2+/H92752/H20850d/H9275=1 a+b. Equation /H2084910/H20850can be solved for /H90241i, /H90241i=4 3C6i /H20849/H92511i/H208502. /H2084911/H20850 Clearly, /H92511i/H20849i/H9275/H20850and /H20851/H92511i/H20849i/H9275/H20850/H20852intersect at /H9275=0. By Tang’s theorem, /H20851/H92511i/H20849i/H9275/H20850/H20852with/H90241igiven by Eq. /H2084911/H20850must intersect /H92511i/H20849i/H9275/H20850once more.16As far as the integral is concerned, the amount that /H92511i/H20849i/H9275/H20850is overestimated by /H20851/H92511i/H20849i/H9275/H20850/H20852before the point of intersection is compensated by the amount that /H92511i/H20849i/H9275/H20850is underestimated by /H20851/H92511i/H20849i/H9275/H20850/H20852after the point of inter- section. This led Tang to conclude16that the combining rule obtained by C6ij=3 /H9266/H20885 0/H11009 /H20851/H92511i/H20849i/H9275/H20850/H20852/H20851/H92511j/H20849i/H9275/H20850/H20852d/H9275 =3 2/H20875/H92511i/H92511j/H90241i/H90241j /H90241i+/H90241j/H20876=2/H92511i/H92511jC6iC6j /H20849/H92511i/H208502C6j+/H20849/H92511j/H208502C6i/H2084912/H20850 should be very accurate and he used a few representative TABLE II. Potential parameters for the mercury rare-gas systems derived from the combining rules, all in a.u. System AbC6 C8 C10 Hg–He 17.715 1.6316 22.148 456.87 13 550.0 Hg–Ne 33.330 1.6300 45.621 983.35 30 069.0Hg–Ar 56.523 1.5037 158.09 4323.7 154 440.0Hg–Kr 81.265 1.4935 231.48 6445.3 228 970.0Hg–Xe 73.573 1.3971 361.48 11429. 448 000.0TABLE I. Input data for the combining rule calculation. Parameters of the mercury dimer are taken directly from Ref. 4, and parameters of the rare-gas dimers are taken directly from Ref. 1, all in a.u. System Ab /H92511 /H92512 /H92513 /H90241 /H90242 /H90243 Hg 16.63 1.2426 33.75 243.2 3010 0.4589 0.3867 0.4897 He 19.99 2.3751 1.383 2.443 10.6 1.0185 1.2285 1.3561Ne 125.42 2.3685 2.66 6.42 30.4 1.2946 1.0692 1.1083Ar 368.30 1.9036 11.1 52.4 490 0.7272 0.6360 0.6723Kr 886.29 1.8714 16.7 92.7 793 0.6789 0.4591 0.6344Xe 496.55 1.5955 27.3 170 2016 0.6083 0.5872 0.5381174310-3 Mercury rare-gas potential J. Chem. Phys. 130 , 174310 /H208492009 /H20850 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 194.47.65.106 On: Sun, 12 Oct 2014 15:58:39interactions to show that is indeed the case.16This was con - firmed with further testing by Kramer and Herschbach,17by Zeiss and Meath,18and by Kutzelnigg and Maeder.19 Thakkar20carried out the most comprehensive testing with 210 interactions and found a rms error of only 0.52%. Based on the same principle, Tang and Toennies1derived similar combining rules for C8and C10, which should be similarly accurate. Using approximants /H20851/H9251l/H20849i/H9275/H20850/H20852=/H9251l 1+ /H20849/H9275//H9024l/H208502, with l=1, 2, and 3, they found Cij/H208491,2 /H20850=15 4/H20875/H92511i/H92512j/H90241i/H90242j /H90241i+/H90242j/H20876, /H2084913/H20850 Cij/H208491,3 /H20850=7/H20875/H92511i/H92513j/H90241i/H90243j /H90241i+/H90243j/H20876, /H2084914/H20850 Cij/H208492,2 /H20850=35 2/H20875/H92512i/H92512j/H90242i/H90242j /H90242i+/H90242j/H20876. /H2084915/H20850 Thus, the C8ijand C10ijcoefficients can then be calculated from Eqs. /H208496/H20850and /H208497/H20850. In this paper, these expressions are used to calculate the dispersion coefficients of the unlike systems. The input data, /H92511,/H92512,/H92513,/H90241,/H90242, and/H90243of the homogeneous rare-gas dimers are taken directly from Ref. 1and they are listed in Table I. Those of the mercury dimer are taken directly from Ref. 4and they are also listed in Table I. The dispersion coefficients of the interactions between a mercury atom and arare-gas atom calculated from these combining rules areshown in Table II. D. Spectroscopic parameters Experimentally, the shape of the potential is best charac- terized by the spectroscopic parameters of vibration and ro-tation. The parameters are usually given by the coefficients /H20849known as Dunham coefficients 35/H20850of a power series expan - sion of the energy respect to /H20849v+1 /2/H20850kand /H20851J/H20849J+1 /H20850/H20852l, where vandJare the quantum numbers of vibration and rotation, respectively. That is, E/H20849v,J/H20850=/H9275e/H20849v+1 2/H20850+BeJ/H20849J+1 /H20850−/H9275e/H9273e/H20849v+1 2/H208502 −/H9251e/H20849v+1 2/H20850J/H20849J+1 /H20850+¯. /H2084916/H20850 By expanding the potential function near the equilibrium into a power series with respect to the relative distancex/H20849x=R/R e/H20850, all Dunham coefficients can be expressed in terms of the vibrational frequency /H9275eand the rotational con- stant Be. LetU/H20849x/H20850be the reduced potential, U/H20849x/H20850=1 DeV/H20849xRe/H20850. /H2084917/H20850 The coefficients anof the power series expansionU/H20849x/H20850=−1+ a0/H20849x−1 /H208502/H208491+a1/H20849x−1 /H20850+a2/H20849x−1 /H208502+¯/H20850 /H2084918/H20850 is given by a0=1 2U/H208492/H20850/H208491/H20850,a1=1 3U/H208493/H20850/H208491/H20850 U/H208492/H20850/H208491/H20850,a2=1 12U/H208494/H20850/H208491/H20850 U/H208492/H20850/H208491/H20850, ... , /H2084919/H20850 where U/H20849n/H20850/H208491/H20850=/H20879dnU/H20849x/H20850 dxn/H20879 x=1. /H2084920/H20850 Since the TT potential is an analytic expression, all de- rivatives can be calculated directly. Explicit formulas forthese derivatives are given in the Appendix of Ref. 4. These derivatives can be used to find the following three dimen-sionless combinations of spectroscopic parameters: 4,36 DeBe /H9275e2=1 4a0, /H2084921/H20850 /H9251e/H9275e Be2=−6 /H208491+a1/H20850, /H2084922/H20850 /H9273e/H9275e Be=−3 2/H20873a2−5 4a12/H20874, /H2084923/H20850 where /H9275eis the vibrational frequency expressed in its energy equivalent h/H9271, and Beis the rotational constant. In atomic units, it is given by Be/H20849a.u. /H20850=1 2/H9262Re2, /H2084924/H20850 where both the equilibrium distance Reand the reduced mass /H9262are in a.u. /H20849If/H9262is in amu, then it needs to be multiplied by a factor of 1822.8, since 1 amu=1822.8 mewhere meis the mass of the electron which is equal to one in atomic units /H20850.I f Beis to be expressed in cm−1, as customarily done in spec- troscopy, then another conversion factor has to be multiplied, Be/H20849cm−1/H20850=Be/H20849a.u. /H20850/H110032.194 74 /H11003105. /H2084925/H20850 With De,Re,a0,a1, and a2, the spectroscopic parameters /H9275e,/H9273e/H9275e, and /H9251ecan be easily determined from Eqs. /H2084921/H20850–/H2084923/H20850. III. RESULTS AND COMPARISON WITH PREVIOUS DETERMINATIONS The van der Waals potentials of mercury and rare-gas systems are calculated from the TT model of Eq. /H208491/H20850with the parameters listed in Table II. The full potential curves for all these five systems are shown in Fig. 1. It is interesting to compare this figure with Figs. 1 and 2 in Ref. 7where the corresponding potentials obtained from the CCSD /H20849T/H20850calcu- lations are shown. These two sets of curves have very similarshape. They also follow the same trend in that the depth ofthe potential well D efor the Hg–RG molecules increases regularly with the polarizabilities of the rare-gas atoms goingfrom He to Xe. The corresponding equilibrium position R e174310-4 Sheng, Li, and T ang J. Chem. Phys. 130 , 174310 /H208492009 /H20850 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 194.47.65.106 On: Sun, 12 Oct 2014 15:58:39increases only slightly with the size of the rare-gas atom, except for He. Here there is a difference. The Reof Hg–He in theab initio calculation of Ref. 7is greater than that of Hg–Ne, Hg–Ar, and Hg–Kr, whereas in the present combin-ing rules results, the R eof Hg–He is only greater than that of Hg–Ne and is smaller than that of the rest of the Hg–RGsystems. The predicted depth of the potential well D e, the equi- librium distance Reand the rotational constants Beare listed in Table III. They are all in atomic units. The three dimen- sionless combinations of spectroscopic parameters calculatedfrom Eqs. /H2084921/H20850–/H2084923/H20850are also shown in Table III. The vibra- tional frequency /H9275e, the anharmonicity /H9273e, and the rotation- vibration interaction constant /H9251ecan be easily obtained from these three dimensionless ratios. Since all experimental results are reported in the units of cm−1and Å, we compare in Table IVthe present De,/H9275eand /H9273e/H9275ein cm−1andRein Å with the available experimental and theoretical results. The theoretical results are from theCCSD /H20849T/H20850calculations of Czuchaj and Krosnicki. 7The theo - retical /H9275eis generated by solving the radial Schrödinger equation, therefore it also includes anharmonic effects. Thereare a large number of laser experiments, many of them usethe same ground state potentials. A few representative onesare listed in Table IVto facilitate discussion. Some experi- mental results have error bars, others do not. We quote themas they were originally reported. In the case of Hg–He, there are large differences be- tween the present and the previous results. It is possible thatthe combining rules we used for the repulsive potential arenot accurate enough for this system because of the smallnumber of electrons in the helium atom. However, as notedbefore, 7the bound of the Hg–He molecule is so weak that both theory and experiment are often encumbered with rela- tively large errors. This is clear from the large differencesbetween various experimental and theoretical determinationsof the well depths of the Hg–He potential shown in Table IV. Also seen in the table, the well depths of the CCSD /H20849T/H20850po-tentials for the Hg–RG pairs are all shallower than the ex- perimental values, except for Hg–He. On the other hand, thepresent well depths are all deeper than the CCSD /H20849T/H20850poten- tials, including Hg–He. Although the present results ofHg–He follow the general trend, the differences betweenvarious determinations are too large for us to draw any un-equivocal conclusion. Among all Hg–RG systems, experimentally Hg–Ne and Hg–Ar are the most often investigated. For Hg–Ne, theagreement between the present results and the previous ex-periments is very good. The first three experimental resultslisted in Table IVwere obtained by means of the free-jet expansion technique. The average D eof these three experi- ments is 46.1 /H110064.7 cm−1which is in excellent agreement with the present result of 45.4 cm−1. The equilibrium dis- tance Refrom these experiments is even closer to each other ranging from 3.87 to 3.90 Å. The present Reof 3.86 Å is very close to this narrow range. The more precisedetermination of the dissociation energy is probably fromthe photofragment excitation spectroscopy. 43The dis- sociation energy D0of the ground state Hg–Ne was deter- mined to be 35.6 cm−1. The well depth can be estimated as De=D0+/H208491/2/H20850/H9275e−/H208491/4/H20850/H9273e/H9275e=44.6 cm−1which is again in very good agreement with the present result. The present results are also in good agreement with De=42.3 cm−1 and Re=3.99 Å obtained from the ab initio CCSD /H20849T/H20850 calculations. For Hg–Ar, the present results are also in very good agreement with experiments. The present Defrom the TT model falls in the experimental range from 133.7 to142.0 cm −1. The determination of Quayle et al.45is espe - cially interesting. They started out with the potential of Ref.41which is a Morse potential with D e=142.0 cm−1. Since the Morse potential converges to zero too quickly, they re-placed the long range part of the Morse potential with a morerealistic van der Waals tail of − C 6/R6−C8/R8, similar to the TT model. After this modification, they found the value ofthe well depth should be changed to 135.2 cm −1, which is very close to the present value of 134.3 cm−1. The dissocia- tion energy D0determined from photofragment excitation spectroscopy is 123.7 cm−1.43The well depth De, based on this value of D0, is estimated to be 134.8 cm−1, which is in excellent agreement with the present result. The present welldepth is deeper than the ab initio well depth of 118 cm −1. The present Reof 3.96 Å is about the same as the experi- mental value of 4.0 Å and is smaller than the theoreticalvalue of 4.10 Å by about 3%. To explain the experimental data, Koperski 44proposedTABLE III. Potential characteristics for the mercury rare-gas systems pre- dicted by the Tang–Toennies model with Born–Mayer parameters and dis-persion coefficients derived from the combining rules. System D e/H2084910−4au/H20850Re/H20849au/H20850Be/H2084910−7au/H20850DeBe//H9275e2/H9251e/H9275e/Be2/H9273e/H9275e/Be Hg–He 0.892 7.44 12.554 0.007 497 30.732 37.031 Hg–Ne 2.075 7.30 2.8068 0.007 776 30.328 36.345Hg–Ar 6.118 7.49 1.4677 0.008 346 28.884 33.640Hg–Kr 8.363 7.57 0.8098 0.008 128 28.807 33.379Hg–Xe 10.28 7.85 0.5609 0.008 532 28.025 31.975 FIG. 1. The potential energy curves for the five mercury rare-gas complexes according to the present theory.174310-5 Mercury rare-gas potential J. Chem. Phys. 130 , 174310 /H208492009 /H20850 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 194.47.65.106 On: Sun, 12 Oct 2014 15:58:39the following hybrid potential of Hg–Ar for the entire range. After the obvious misprints are corrected, it is given by V/H20849R/H20850=133.7 n/H11569−6/H208756/H208733.99 R/H20874n/H11569 −n/H11569/H208733.99 R/H208746/H20876, /H2084926/H20850 with n/H11569= 11.3 + 10.8 /H20849R/3.99 − 1 /H20850 for the interval of 2.8 Å /H11349R/H113493.5 Å. In addition V/H20849R/H20850= 133.7 /H20851/H208491−e−1.5011 /H20849R−3.99 /H20850/H208502−1 /H20852/H20849 27/H20850 forR/H110223.5 Å. Here Vis in cm−1andRis in Å. We plot this potential together with the present potential and theCCSD /H20849T/H20850potential in Fig. 2/H20849a/H20850. Although this hybrid poten- tial is discontinuous at R=3.5 Å, /H20851V/H208493.5 /H20850=9.246 accordingto Eq. /H2084926/H20850and V/H208493.5 /H20850=24.16 according to Eq. /H2084927/H20850/H20852, this discontinuity is not discernible in the scale of Fig. 2/H20849a/H20850.I ti s seen that the present potential curve is essentially in com-plete agreement with this experimental potential in the repul-sive and well regions, especially if the experimental error baris taken into consideration. /H20849In Koperski’s report, 6the error bar of Deis put at /H110062.0 cm−1, and that of Reis/H110060.01 Å /H20850. However, in the long range region, there are some differ-ences. Both potential curves are below the CCSD /H20849T/H20850curve in the short range repulsive region but above it in the long range attractive region. To compare the shape of these potentials, we plot the respective reduced potentials U/H20849x/H20850against xin Fig. 2/H20849b/H20850, /H20849x=R/R e,U/H20849x/H20850=V/H20849Rex/H20850/De/H20850. It is both surprising and grati- fying to see that the present reduced TT potential curve is almost identical with the reduced CCSD /H20849T/H20850potential curve.TABLE IV . Comparison of the well depth, the equilibrium distance, the vibrational frequency, and the anhar- monicity of the ground state potentials of the mercury and rare-gas complexes. SystemDe /H20849cm−1/H20850Re Å/H9275e /H20849cm−1/H20850/H9273e/H9275e /H20849cm−1/H20850 Hg–He Present 19.58 3.94 26.9 10.3 Expt.a11.4 Expt.b8.0/H110061.0 4.6 Theoryc6.2 4.66 Theoryd13.7 4.19 Hg–Ne Present 45.4 3.86 19.1 2.26 Expt.e,f41.4/H110061.1 3.89 /H110060.01 20.6 /H110060.5 2.56 /H110060.03 Expt.g51.0 3.87 19.0 1.5 Expt.h46.0 3.90 18.5 1.6 Expt.i44.6 Theoryd42.3 3.99 13.5 Hg–Ar Present 134.3 3.96 22.8 1.09 Expt.j,f133.7/H110062.0 3.99 /H110060.01 24.7 /H110060.04 1.14 /H110060.02 Expt.h142.0 3.99 23.5 1.06 Expt.k135.2 4.01 23.5 1.06 Expt.i134.8 Theoryd118 4.10 19.4 Hg–Kr Present 183.5 4.01 20.1 0.60 Expt.l,f178.6/H110060.06 4.03 /H110060.02 20.7 /H110060.2 0.60 /H110060.10 Expt.g178 4.07 20.0 0.54 Expt.m200/H1100620 3.95 /H110060.20 Expt.n175/H1100610 Theoryd166 4.16 17.4 Hg–Xe Present 225.6 4.15 18.1 0.39 Expt.o254.0 4.25 18.3 /H110060.3 0.33 /H110060.05 Expt.p240/H1100610 4.10 /H110060.03 Expt.n220/H1100620 4.20 Theoryd213 4.32 16.6 aReference 37. bReference 38. cReference 39. dReference 7. eReference 40. fReference 6. gReference 41. hReference 42.iReference 43. jReference 44. kReference 45. lReference 46. mReference 47. nReference 48. oReference 49. pReference 50.174310-6 Sheng, Li, and T ang J. Chem. Phys. 130 , 174310 /H208492009 /H20850 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 194.47.65.106 On: Sun, 12 Oct 2014 15:58:39This reduced potential curve differs from the reduced experi- mental hybrid potential in the long range attractive region. For Hg–Kr, the accepted value of the well depth Dein laser spectroscopy is 178 cm−1.46,41The present result of 183.5 cm−1is only 3% larger than this value. Furthermore, the present well depth fits in the overlap of the two error barsof 200 /H1100620 cm −1/H20849Ref. 47/H20850and 175 /H1100610 cm−1/H20849Ref. 48/H20850ob- tained earlier from the temperature dependence of the ab-sorption spectra in the vicinity of the mercury resonancelines. From their experimental data, Koperski et al. 46proposed a hybrid potential for the ground state of Hg–Kr, /H20849Vin cm−1, Rin Å /H20850 V/H20849R/H20850=178.6 n/H11569−6/H208756/H208734.03 R/H20874n/H11569 −n/H11569/H208734.03 R/H208746/H20876, /H2084928/H20850 with n/H11569= 11.39 + 10.5 /H20849R/4.03 − 1 /H20850 for the interval of 2.85 Å /H11349R/H113493.55 Å,V/H20849R/H20850= 178.6 /H20851/H208491−e−1.4505 /H20849R−4.03 /H20850/H208502−1 /H20852/H20849 29/H20850 for 3.55 /H11349R/H110217.14 Å, and V/H20849R/H20850=−0.749/H11003106 R6/H2084930/H20850 for R/H113507.14 Å. This potential has two discontinuities /H20851V/H208493.55 /H20850=0.2657 according to Eq. /H2084928/H20850butV/H208493.55 /H20850=2.2198 according to Eq. /H2084929/H20850V/H208497.14 /H20850=−3.9029 from Eq. /H2084929/H20850but V/H208497.14 /H20850=−5.6532 from Eq. /H2084930/H20850/H20852. This potential includes a long range van der Waals tail. In Fig. 3/H20849a/H20850, this hybrid poten- tial, the present predicted potential and the theoreticalCCSD /H20849T/H20850potential are plotted together. It is seen that, within the differences discussed above, the present potential and thehybrid potential are very similar in the repulsive and wellregions. In the long range, in spite of the van der Waals tailof the hybrid potential, there are still considerable differ-ences. This is because in the hybrid potential, the pointwhere the van der Waals tail is taking over is too large andtheC 6used in Eq. /H2084930/H20850is substantially smaller than the present C6/H20849see Table II/H20850. FIG. 2. /H20849a/H20850The potential energy curves for Hg–Ar complex. The solid line is the present combining rule result. The dashed line is the experimentalhybrid potential /H20849Ref. 44/H20850. The dotted line is the ab initio potential /H20849Ref. 7/H20850. The black dots are the actual ab initio points from the CCSD /H20849T/H20850calculation. /H20849b/H20850The reduced potential energy curves for Hg–Ar complex. The present reduced potential /H20849solid line /H20850is almost identical to the reduced potential of theab initio CCSD /H20849T/H20850calculation /H20849Ref. 7/H20850. The black dots are actual ab initio data points. The present reduced potential agrees with the reduced hybrid potential /H20849dashed line /H20850of Ref. 44in the repulsive and potential well regions, but they differ in the long range part of the potential. FIG. 3. /H20849a/H20850The potential energy curves for Hg–Kr complex. The solid line is the present combining rule result. The dashed line is the experimentalhybrid potential /H20849Ref. 46/H20850. The dotted line is the ab initio potential /H20849Ref. 7/H20850. The black dots are the actual ab initio points from the CCSD /H20849T/H20850calculation. /H20849b/H20850The reduced potential energy curves for Hg–Kr complex. The present reduced potential /H20849solid line /H20850is almost identical to the reduced potential of theab initio CCSD /H20849T/H20850calculation /H20849Ref. 7/H20850. The black dots are actual ab initio data points. The present reduced potential agrees with the reduced hybrid potential /H20849dashed line /H20850of Ref. 46in the repulsive and potential well regions, but they differ in the long range part of the potential.174310-7 Mercury rare-gas potential J. Chem. Phys. 130 , 174310 /H208492009 /H20850 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 194.47.65.106 On: Sun, 12 Oct 2014 15:58:39It is interesting to note that, except for the energy scale, Figs. 3/H20849a/H20850and 2/H20849a/H20850are almost identical. Both the present potential and the experimental hybrid potential are below theab initio CCSD /H20849T/H20850potential in the short range and above it in the long range. In Fig. 3/H20849b/H20850, the three reduced potentials for Hg–Kr are plotted. Again the situation is the same as in Hg–Ar. Thepresent reduced potential and the reduced ab initio potential almost coincide with each other. The reduced experimentalhybrid potential and the present reduced potential are in ex-cellent agreement in the repulsive and the potential well re-gions but they differ in the long range region. In Hg–Xe, there is a discrepancy of 13% between the present D e/H20849225.6 cm−1/H20850and the one /H20849254.0 cm−1/H20850 from laser spectroscopy.49Although earlier results of /H20849240/H1100610 cm−1/H20850/H20849Ref. 50/H20850and /H20849220/H1100620 cm−1/H20850/H20849Ref. 48/H20850 from the temperature dependence of line broadening experi- ments are in better agreement with the present result, gener-ally the data from laser spectroscopy is considered more ac-curate. Unfortunately in the case Hg–Xe, we can find onlyone set of laser spectroscopic data for the ground state po-tential. The well depth D eof this potential was calculated from the observed /H9275eand/H9273e/H9275ewith an assumed Morse po- tential, in which the Reis estimated from Kong’s combining rule. It is important to keep in mind that the /H9275eand/H9273e/H9275eare the quantities more directly measured in the experiment. Asseen in Table IV, the experimental /H9275e=18.3 /H208493/H20850cm−1and /H9273e/H9275e=0.33 /H208495/H20850cm−1are actually in very good agreement with the present predictions. It is not clear whether a better analysis with a more realistic potential, similar to whatQuayle et al. 45did in the case of Hg–Ar, will resolve the differences in DeandRe. The theoretical well depth De=213 cm−1from CCSD /H20849T/H20850 calculations is less than 6% smaller than the present predic-tion and is farther away from 254 cm −1obtained in the laser spectroscopy. The relationship between the present TT poten-tial and the CCSD /H20849T/H20850potential for Hg–Xe is the same as for the other four Hg–RG systems. Again the reduced TT poten-tial for Hg–Xe is almost identical to the reduced CCSD /H20849T/H20850 potential. This reduced potential is somewhat different fromthe reduced Morse potential, especially in the long rangeregion. IV. CONCLUSION AND DISCUSSION The short range and long range parameters of the Tang– Toennies potentials for the mercury rare-gas complexes arecalculated from the combining rules with input data from thecorresponding parameters of the homonuclear mercury andrare-gas dimer potentials. With the exception of Hg–Hewhere there is no reliable experimental data, the predictedpotentials are in good agreement with the corresponding ex-perimental potentials in the short repulsive region and in themiddle part of the potential well region. Since the longrange part of the present potential is determined by thevan der Waals dispersion coefficients which are calculated bythe well established accurate combining rules, it is likely thatthe present potentials are accurate over all internucleardistances.The other advantage of the present potential over the experimental hybrid potentials is that there is no discontinu-ity anywhere in the present potential. Mathematically thepresent potential is an entire function, derivatives exist ev-erywhere and to all orders. It can be easily used in theoriesrequiring an analytic extension into the complex plane, suchas theories of predissociation of van der Waals complexes. 51 It is interesting to note that the predicted reduced poten- tials of these five systems /H20849including Hg–He /H20850have the same shape. The reduced potentials of the Hg–RG systems areshown as the continuous solid line in Fig. 4. For each sys- tem, the present results for U/H20849x/H20850/H20849=V/H20849xR e/H20850/De/H20850are plotted against x/H20849=R/Re/H20850. The five reduced potentials are essentially identical within a few percent. With the scale of Fig. 4, the small differences are not noticeable. This invariance withrespect to the shape of the potential is not to be construed asresulting from the use of the same TT model in all cases. Ithas been shown that this model is capable of predicting thevastly different potential shapes for different systems. 3In Fig. 4, the reduced TT potentials for mercury dimer and for rare-gas dimers, which also conform with each other, are alsoshown. It is seen that the reduced potentials for the mercuryrare-gas complexes have a potential bowl that is wider thanthat of the rare-gas dimers, but narrower than the mercurydimer. It is gratifying to find that the five Hg–RG potentials obtained from the ab initio CCSD /H20849T/H20850calculations also have the same reduced form. Furthermore, this common reducedpotential curve is almost identical with the reduced potentialcurve found in the present combining rules calculations. Inother words, the reduced potential curves of Hg–RG systemsin Fig. 4can also describe, within a few percent, the reduced potentials from the CCSD /H20849T/H20850calculations. FIG. 4. Comparison of the reduced potentials of mercury dimer /H20849dashed line /H20850, rare-gas dimer /H20849dotted line /H20850and mercury rare-gas complexes /H20849solid line /H20850. All these potentials are represented by the same Tang–Toennies poten- tial function, yet they grouped into three different shapes. The results of thepresent theory show that the reduced potentials for Hg–He, Hg–Ne, Hg–Ar,Hg–Kr, and Hg–Xe are almost identical. Within a few percent, the solid lineof the present theory can also describe the five reduced potentials of Hg–RGobtained from the ab initio CCSD /H20849T/H20850calculations /H20849Ref. 7/H20850.174310-8 Sheng, Li, and T ang J. Chem. Phys. 130 , 174310 /H208492009 /H20850 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 194.47.65.106 On: Sun, 12 Oct 2014 15:58:39It is remarkable that the ten sets of completely indepen- dent calculations /H20851five from the present combining rules and five from ab initio CCSD /H20849T/H20850/H20852should give almost the same result. This leads us to conclude that the true shape of theground state potential of Hg–RG complexes must be close towhat is predicted in this paper. With the shape of the potential fixed, the present poten- tial is essentially a two-parameter model. This is particularlyconvenient for the interplay between theory and experiment.Should the predictive requirements exceed the present levelof accuracy, adjustments of D eandRecan be made either by proportionally scaling the potential, or by slightly changingthe short range parameters Aandbin such a way that the long range structure of the potential is maintained. 52 ACKNOWLEDGMENTS We thank Professor J. Koperski for sending us his latest paper and informing us that their experiments cannot see thelong range part of the potential. 1K. T. Tang and J. P. Toennies, Z. Phys. D: At., Mol. Clusters 1,9 1 /H208491986 /H20850. 2K. T. Tang and J. P. Toennies, J. Chem. Phys. 118, 4976 /H208492003 /H20850. 3K. T. Tang and J. P. Toennies, J. Chem. Phys. 80,3 7 2 6 /H208491984 /H20850. 4K. T. Tang and J. P. Toennies, Mol. Phys. 106, 1645 /H208492008 /H20850. 5K. Fuke, T. Saito, and K. Kaya, J. Chem. Phys. 79, 2487 /H208491983 /H20850. 6J. Koperski, Phys. Rep. 369, 177 /H208492002 /H20850. 7E. Czuchaj and M. Kro śnicki, J. Phys. B 33, 5425 /H208492000 /H20850. 8E. Czuchaj, M. Kro śnicki, and H. Stoll, Chem. Phys. 263,7 /H208492001 /H20850. 9E. Czuchaj, M. Kro śnicki, and J. Czub, Mol. Phys. 99, 255 /H208492001 /H20850. 10K. T. Tang, J. M. Norbeck, and P. R. Certain, J. Chem. Phys. 64, 3063 /H208491976 /H20850. 11A. Kumar and W. J. Meath, Can. J. Chem. 63, 1616 /H208491985 /H20850. 12A. J. Thakkar, H. Hettema, and P. E. S. Wormer, J. Chem. Phys. 97, 3252 /H208491992 /H20850. 13R. A. Aziz, in Inert Gases , edited by M. Klein /H20849Springer, Berlin, 1984 /H20850, p. 5. 14D. Goebel and U. Hohm, J. Phys. Chem. 100, 7710 /H208491996 /H20850. 15G. C. Maitland, H. Rigby, E. B. Smith, and W. A. Wakeham, Intermo - lecular Forces ,/H20849Clarendon, Oxford, 1981 /H20850. 16K. T. Tang, Phys. Rev. 177,1 0 8 /H208491969 /H20850. 17H. L. Kramer and D. R. Herschbach, J. Chem. Phys. 53, 2782 /H208491970 /H20850. 18G. D. Zeiss and W. J. Meath, Mol. Phys. 33,1 1 5 5 /H208491977 /H20850.19W. Kutzelnigg and F. Maeder, Chem. Phys. 35,3 9 7 /H208491978 /H20850. 20A. J. Thakkar, J. Chem. Phys. 81, 1919 /H208491984 /H20850. 21T. L. Gilbert, J. Chem. Phys. 49, 2640 /H208491968 /H20850. 22T. L. Gilbert, O. C. Simpson, and M. A. Williamson, J. Chem. Phys. 63, 4061 /H208491975 /H20850. 23F. T. Smith, Phys. Rev. A 5, 1708 /H208491972 /H20850. 24H. J. Böhm and R. Ahlrichs, J. Chem. Phys. 77, 2028 /H208491982 /H20850. 25A. Koide, W. J. Meath, and A. R. Allnatt, Chem. Phys. 58,1 0 5 /H208491981 /H20850. 26M. Gutowski, J. Verbeek, J. H. van Lenthe, and G. Chalasinski, Chem. Phys. 111, 271 /H208491987 /H20850. 27W. Meyer and L. Frommhold, Theor. Chim. Acta 88, 201 /H208491994 /H20850. 28K. T. Tang and J. P. Toennies, Chem. Phys. 156, 413 /H208491991 /H20850. 29H. Partridge, J. R. Stallcop, and E. Levin, J. Chem. Phys. 115, 6471 /H208492001 /H20850. 30K. T. Tang, J. P. Toennies, and C. L. Yiu, Int. Rev. Phys. Chem. 17, 363 /H208491998 /H20850. 31C. Mavroyannis and M. J. Stephen, Mol. Phys. 5, 629 /H208491962 /H20850. 32A. D. McLachlan, Proc. R. Soc. London, Ser. A 271, 387 /H208491963 /H20850. 33A. Dalgarno, Advances in Chemical Physics /H20849Wiley, New York, 1967 /H20850, V ol. 12, p. 143. 34See, for example, K. T. Tang, Mathematical Methods for Engineers and Scientists /H20849Springer, Berlin, 2007 /H20850, V ol. I. 35J. L. Dunham, Phys. Rev. 41, 721 /H208491932 /H20850. 36D. R. Herschbach and V . W. Laurie, J. Chem. Phys. 35,4 5 8 /H208491961 /H20850. 37R. D. van Zee, S. C. Blankespoor, and T. S. Zwier, Chem. Phys. Lett. 158, 306 /H208491989 /H20850. 38M. C. Duval, C. Jouvet, and B. Soep, Chem. Phys. Lett. 119,3 1 7 /H208491985 /H20850. 39E. Czuchaj, H. Stoll, and H. Preuss, J. Phys. B 20, 1487 /H208491987 /H20850. 40J. Koperski, J. B. Atkinson, and L. Krause, Chem. Phys. 186,4 0 1 /H208491994 /H20850. 41K. Fuke, T. Saito, and K. Kaya, J. Chem. Phys. 81,2 5 9 1 /H208491984 /H20850. 42K. Yamanouchi, S. Isogai, M. Okunishi, and S. Tsuchiya, J. Chem. Phys. 88, 205 /H208491988 /H20850. 43T. Tasaka, K. Onda, A. Hishikawa, and K. Yamanouchi, Bull. Chem. Soc. Jpn. 70, 1039 /H208491997 /H20850. 44J. Koperski, Chem. Phys. 211, 191 /H208491996 /H20850. 45C. J. K. Quayle, I. M. Bell, E. Takacs, X. Chen, K. Burnett, and D. M. Segal, J. Chem. Phys. 99, 9608 /H208491993 /H20850. 46J. Koperski, J. B. Atkinson, and L. Krause, J. Mol. Spectrosc. 207,1 7 2 /H208492001 /H20850. 47T. Grycuk and E. Czerwosz, Physica 106C , 431 /H208491981 /H20850. 48C. Bousquet, N. Bras, and Y . Majdi, J. Phys. B 17, 1831 /H208491984 /H20850. 49M. Okunishi, H. Nakazawa, K. Yamanouchi, and S. Tsuchiya, J. Chem. Phys. 93, 7526 /H208491990 /H20850. 50T. Grycuk and M. Findeisen, J. Phys. B 16,9 7 5 /H208491983 /H20850. 51S. I. Chu, J. Chem. Phys. 72, 4772 /H208491980 /H20850. 52There is a simple program in the appendix of Ref. 1that converts Deand ReintoAandb, without changing the long range dispersion coefficients.174310-9 Mercury rare-gas potential J. Chem. Phys. 130 , 174310 /H208492009 /H20850 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 194.47.65.106 On: Sun, 12 Oct 2014 15:58:39

1.1747486.pdf

The InfraRed Spectrum of Nitrosyl Chloride John H. Wise and Jack T. Elmer Citation: The Journal of Chemical Physics 18, 1411 (1950); doi: 10.1063/1.1747486 View online: http://dx.doi.org/10.1063/1.1747486 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/18/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in The Vibrational Spectra and Structure of Inorganic Molecules. III. The InfraRed Spectra of Nitrosyl Chloride and Nitrosyl Bromide from 2.0 to 25μ J. Chem. Phys. 18, 1669 (1950); 10.1063/1.1747559 The InfraRed Spectrum of Nitrosyl Fluoride J. Chem. Phys. 18, 1516 (1950); 10.1063/1.1747531 The InfraRed Spectrum of Acetylene J. Chem. Phys. 18, 1382 (1950); 10.1063/1.1747483 The InfraRed Spectrum of Ketene J. Chem. Phys. 15, 552 (1947); 10.1063/1.1746591 The InfraRed Spectrum of Furan J. Chem. Phys. 10, 660 (1942); 10.1063/1.1723639 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.174.21.5 On: Fri, 19 Dec 2014 20:20:20THE JOURNAL OF CHEMICAL PHYSICS Letters to the Editor THIS section will accept reports of new work, provided these are terse and contain few figures, and especially few half-tone cuts. The Editorial Board will not hold itself responsible for opinions expressed by the correspondents. Contributions to this section should not exceed 600 words in length and must reach the office of the Managing Editor not later than the 15th of the month preceding that of the issue in which the letter is to appear. No proof will be sent to the authors. The usual publication charge ($8.00 per page) will not be made and no reprints will be furnished free. The Infra-Red Spectrum of Nitrosyl Chloride JOHN H. WISE AND JACK T. ELMER Department of Chemistry. Stanford University. Stanford. California July 31. 1950 THE effect of nitrosyl chloride as a possible contaminant due to reaction between various nitrogen compounds and the rock salt windows on infra-red gas cells has led to a re-examination of the infra-red spectrum of nitrosyl chloride. Bailey and Cassie' have reported the spectrum, but serious doubt of their assignment of fundamental frequencies was raised by the results of Beeson and Yost.2 The latter authors proposed a different assignment based on the electron diffraction data of Ketelaar and Palmer3 and some calculations involving Lechner's formulas.' In the present research, nitrosyl chloride was prepared in two ways: by passing N20, over a column of damp KCI, and by a direct combination of CI2 with NO. The first procedure produced an impure, condensable, mixture consisting of N20., CINO, and CIN02. This observation confirms the results of Ogg and Wilson6 on the oxidation of CINO by O2 since oxygen was not rigidly ex-' cluded during the preparation. In the second procedure, tank chlorine was purified by the method described by Beeson and Yost,! and the nitric oxide was generated by a similar procedure to that of Johnston and Giauque.6 The gases were collected in graduated traps, and then evaporated into the evacuated system. The mixture of gases, containing an excess of NO, was frozen out with liquid air and allowed to stand for about 36 hours in a bath originally containing dry ice and acetone. After refreezing in liquid air, the excess NO was removed by pumping from a dry ice bath. Unfortunately, a trace of N20. was detected in the product. The spectra were taken on a Perkin-Elmer Model 12-C Spectrometer using a 10-cm glass cell with NaCI or KBr windows. The observed spectrum of nitrosyl chloride differs somewhat from that reported by Bailey and Cassie,' particularly their band at 633 cm-I• A comparison of the various assignments is given in Table I. The assignment offered as a result of this work is similar to Beeson and YOSt'S2 proposal of the most reasonable choice based on Bailey and Cassie'sl results. Definite indication of absorption in the 650 cm-l region is found, but the position was not accur- TABLE 1. Vibrational frequencies of CINO. Bailey and Cassie Beeson and Y 06t propoaals This research • (em -.) Inten. Assign. • (em') Assign. • (em -.) Assign •• (em -.) Inten. Assign. 317 ., 290 ., 462 " 633 6 ., 633 .. 633 2" 594 VB .. 670 W7 2., 923 10 '1 923 1-2+V, 923 2" 921 B Jl2+"1 1200 3 2., 1185 W 2 .. or Vi-va 1832 7 PI 1832 PI 1832 •• 1800 VB PI 2155 4 2.,+ .. 2130 B "1+V2 2390 M V1+V, 3562 8 2 .. >4000 M 7 VOLUME 18. NUMBER 10 OCTOBER. 1950 ately determined. The calculated value of the entropy of CINO using Beeson and YOSt'S2 Eq. (16) and substituting 327, 594, and 1800 cm-l for the fundamentals gives: SOCINo=4R1nT+Sm+S m+S18oo+15.460 E.U. =62.466 E.U. at 298°K. This compares favorably with the value 63.0±0.3 cal./deg. found from thermal data. . • Bailey and Cassie. Proc. Roy. Soc. (London) A145. 336 (1934). , Beeson and Yost. J. Chern. Phys. 7. 44 (1939). 3 Ketelaar and Palmer. J. Am. Chem. Soc. 59. 2629 (1937). 'Lechner. Monatshefte fiir Chern. 61. 385 (1932). 'Ogg and Wilson. J. Chern. Phys. 18. 900 (1950). • Johnston and Giauque. J. Am. Chern. Soc. 51. 3194 (1933). "Glass Electrode Behavior in Acid Solutions" MALCOLM DOLE Department of Chemistry. Northwestern University. Evanston. nUnois July 31. 1950 UNDER the above title, Sinclair and Martelll have replied to my criticism2 of their earlier paper3 in which these authors advocated a negative ion theory for the acid solution errors of the glass electrode. In this most recent note Sinclair and Martell object to my description of the attack of hydrogen fluoride on glass as having no bearing on the interpretation of the results of their work in which they used sulfuric and hydrochloric acids. In my letter, I quoted our former work with hydrogen fluoride as a spectacular example of the way in which chemical attack on the glass surface can destroy the reversible glass electrode e.m.f.'s. I did not, of course, state or imply that the action of hydrofluoric acid is chemically similar to that of sulfuric or hydrochloric acid. I shall now explain in detail what is believed to be the action of strong acids such as sulfuric and hydrochloric on glass surfaces. In this connection I would like to refer to the excellent new book of Kratz,' in which his extensive work6 on the chemical properties of glass surfaces is summarized. When Corning 015 glass elec trodes are immersed in water, alkali is leached out of the surface leaving a layer or "skin" of hydrated silicic acid gel as residue. In contact with strong acids in which the vapor pressure of water has been significantly lowered, the surface film of silicic acid is dehydrated, the asymmetry potential is increased, the rate at which the glass gives up alkali to pure water is increased, the electrical resistance of the glass membrane is increased, and the hydrogen electrode function of the glass surface impaired. This is what I meant in my note by chemical attack, and its effects on the reversibility of glass electrode e.m.f.'s. I hope that this ex planation of the mechanism of chemical attack clarifies my criti cism of the negative ion theory of Sinclair and Martell. • E. E. Sinclair and A. E. Martell. J. Chern. Phys. 18.992 (1950). 'M. Dole. J. Chern. Phys. 18,573 (1950). • E. E. Sinclair and A. E. Martell. J. Chern. Phys. 18.224 (1950). 'L. Kratz. "Die Glaselektrode und ihre Anwendungen" Wissenschaftfiche Forschungsberichte. Vol. 59 Frankfurt. 1950. • L. Kratz. Glaslechn. Ber. 20. 15.305 (1942). Normal Modes of Ethane T. S. G. KRISHNAMURTY AND S. SATYANARAYANARAO Andhra University. Waltair. India August 9. 1950 THE theory of groups has been successfully used to obtain the normal modes and frequencies of molecules, and the selec- tion rules for spectral lines in the Raman effect and infra-red ab- sorption. In these applications, only symmetry operations of the type rotations, reflections and rotation-reflections are usually 1411 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.174.21.5 On: Fri, 19 Dec 2014 20:20:201412 LETTERS TO THE EDITOR TABLE J. Infra- E 2C, 3C, 3Uh 4T 9uf} 2TC, 6S, 6TC2 ni ni'Raman red Al I I I I I I I I I 3 3 P P A, I I I -I I -I I -I I I I f f A, I I -I 1 1 -1 1 1 -1 1 0 f f A. 1 I -1 -I I I I -I -I 3 2 f P El 2 2 2 0 -I 0 -I 0 -I 0 0 f j E, 2 2 -2 0 -I 0 -I 0 I 0 0 f E, 2 -I 0 -2 -I 0 2 1 0 0 0 f f E. 2 -I 0 2 -I 0 2 -I 0 0 0 f f G 4 -2 0 0 1 0 -2 0 0 4 3 P P 24 0 0 0 48 36 0 0 0 18 0 6 0 36 36 0 0 12 12 0 12 12 24 36 0 24 24 6 0 0 0 12 18 0 0 0 taken into account. However, in some molecules, it may be pos sible to consider other elements of symmetry, as for example, a twist through 1200 of one of the CHa groups against the other in the molecule C2H6• Using the point group Dah' Howardt obtained the normal modes, etc., of this molecule. In this paper, assuming that the potential energy has the symmetry of the molecule in cluding the twist also, the symmetry characters of the modes and the selection rules for their appearance in the Raman effect and the infra-red absorption are given. The results are tabulated in Table I. Details regarding the normal frequencies will be published separately. t J. B. Howard. J. Chern. Phys. 5, 442 (1937). The Vibrational Energy of H2S* HARRY C. ALLEN, JR. AND PAUL C. CROSS Department of Chemistry and Chemical Engineering, University of Washington, Seattle, Washington AND GILBERT W. KING Arthur D. Little, Inc., Cambridge, Massachusetts August I, 1950 THE approximate location of four new band centers in the infra-red spectrum of hydrogen sulfide has enabled the evaluation of the complete quadratic expression for the unper turbed vibrational energies as (E-Eo) cm-I = 2651nu+2635n.+1189n8-26n u2 _24n".2-6n82-90nun".-19nun8-20n".n8. (1) Equation (1) yields the unperturbed levels shown in column two of Table r. From the observed perpendicular type bandsI.2 at 9911 cm-I and 10194 em-I, the Darling-Dennison a parameter l' may be evaluated as 47 em-I. The calculated energy values for the two pairs of interacting levels are listed in column 3 of the table. The constants in (1) and the Darling-Dennison interaction parameter were evaluated from the unstarred levels in Table r. nu n" 0 0 0 0 0 I I 0 I 0 I I 0 2 2 0 I 1 2 2 1 3 3 1 0 4 4 0 n8 I 2 0 0 I 0 0 0 1 0 0 0 0 0 TABLE J. Unperturbed 1183' 2354' 2611' 2625' 3789' 5146' {5174} 5198 6290' *10012' i10044} 10060 10156 10188' D D resonance *5138' *5234' 99111 10194' The absorption in the region9 of 5146 cm-l shows an overlapping spectrum centered at approximately 5143 cm-l as compared to the calculated value 5138 cm-l for the perturbed (020) band. The appearance of this parallel type band is presumably due to the borrowing of intensity through rotational interaction from the neighboring (110) perpendicular type band just as has been previ ously indicated in the appearance of the (400) band2 at 10188 em-I. The analysis of the 8,u-region reveals that recent investigators have not detected the P branch.lo-l2 The early investigation of Mischke,la however, shows a weak P branch extending from 1180 cm-I to nearly 1000 em-I. As would reasonably be expected the results described by (1) are analogous to a similar treatment of the water vapor system.' The most conspicuous feature of this analysis is the low frequency of the bending fundamental, from 50 to 100 cm-1 less than it has recently been considered to be.14 * The work herein reported was supported in part by the ONR under Contract N80nr 52010. I P. C. Cross, Phys. Rev. 47, 7 (1935). , Grady, Cross, and King, Phys. Rev. 75, 1450 (1949). 3 B. T. Darling and D. M. Dennison, Phys. Rev. 57, 128 (1940). • G. M. Murphy and J. E. Vance, J. Chern. Phys. 6, 426 (1938). 'R. M. Hainer and G. W. King, J. Chern. Phys. 15, 89 (1947). 6 Allen, Cross, and Wilson, J. Chern. Phys. 18,691 (1950). , To be reported shortly. • Not yet observed. • Unpublished data of M. K. Wilson. 10 A. D. Sprague and H. H. Nielsen, J. Chern. Phys. 5, 85 (1937). 11 R. H. Noble, Thesis, Ohio State University (1946), 12 E. A. Wilson, Thesis, Brown University (1947). .. W. Mischke, Zeits. f. Physik 67, 106 (1931). 14 G. Herzberg, Infrared and Raman Spectra of Polyatomic Molecul .. (D. Van Nostrand Company, Inc" New York, 1945), p. 283. The Quantum-Mechanical Equation of State ROBERT W. ZWANZIG Gates and Crellin Laboratories of Chemistry,* California Institute of Technology, Pasadena, California July 18, 1950 IN the last few years there has been some disagreement about the equivalence of the thermodynamic and kinetic pressure of a quantum liquid, with special emphasis on liquid helium II. The kinetic pressure is obtained from the quantum-mechanical virial theorem. Equations of state which do not appear to be identi cal have been derived from statistical thermodynamics by H. S. Greenl and J. de Boer.2 In this note, a simple derivation will be given, which may expose more clearly the assumptions required. By differentiating the partition function with respect to volume, we get _p= vA =2": J3CA-E;)vE; VV i VV (1) where all symbols have their usual significance. The dependence of the energy levels of the system on its volume is needed. To get this, we introduce a scale factor, depending on the volume, into the Schrodinger equation for the system, and determine the de pendence of the energy levels on this scale factor. This device is related to the one used by H. S. Green to get the classical equation of state, and by de Boer for the quantum-mechanical case. The Hamiltonian operator for the system is h2 N H=-- 2": V'rk2+V(r,,·· ·rN). (2) 2m k~1 The potential is assumed to become infinite at the walls. This wall potential is taken into account by requiring eigenfunctions to vanish at the walls. We define a new coordinate system, (3) where v is the volume of the system. A variation in v corresponds to an expansion or contraction of the system, without any change in its relative shape or configuration. The equation of state that will result will be valid only for such changes in volume. However, when the dimensions of the system are large compared with the This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.174.21.5 On: Fri, 19 Dec 2014 20:20:20

1.462179.pdf

Reversible electron transfer dynamics in nonDebye solvents Jianjun Zhu and Jayendran C. Rasaiah Citation: The Journal of Chemical Physics 96, 1435 (1992); doi: 10.1063/1.462179 View online: http://dx.doi.org/10.1063/1.462179 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/96/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Electron transfer reaction dynamics in non-Debye solvents J. Chem. Phys. 109, 2325 (1998); 10.1063/1.476800 Electron transfer reactions in a nonDebye medium with frequencydependent friction J. Chem. Phys. 104, 9408 (1996); 10.1063/1.471706 Outersphere electron transfer reactions in nonDebye solvents. Theory and numerical results J. Chem. Phys. 91, 2869 (1989); 10.1063/1.456957 Solvent dynamical effects in electron transfer: Predicted consequences of nonDebye relaxation processes and some comparisons with experimental kinetics J. Chem. Phys. 90, 912 (1989); 10.1063/1.456117 Dielectric friction and the transition from adiabatic to nonadiabatic electron transfer in condensed phases. II. Application to nonDebye solvents J. Chem. Phys. 88, 4300 (1988); 10.1063/1.453789 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.114.34.22 On: Sat, 22 Nov 2014 00:17:06Reversible electron transfer dynamics in non-Debye solvents Jianjun Zhua) and Jayendran C. Rasaiah Department o/Chemistry, University 0/ Maine, Orono, Maine 04469 (Received 28 June 1991; accepted 23 September 1991) The general solutions obtained earlier [J. Chem. Phys. 95,3325 (1991)] for the coupled diffusion-reaction equations describing reversible electron transfer reactions in Debye solvents, governed by Sumi-Marcus free energy surfaces, are extended to non-Debye solvents. These solutions, which depend on the time correlation function of the reaction coordinate ~(t), are exact in the narrow and wide window limits for Debye and non-Debye solvents and also in the slow reaction and non-diffusion limits for Debye solvents. The general solution also predicts the ~eha~ior between these limits and can be obtained as the solution to an integral equation. An Iterative method of solving this equation using an effective relaxation time is discussed. The relationship between ~(t) and the time correlation function S(t) of Born solvation energy of the reacting intermediates is elucidated. I. INTRODUCTION In a previous study,l an approximate general solution was obtained for two coupled diffusion-reaction equations governing reversible electron transfer (ET) reactions in a Debye solvent which are characterized by a single dielectric relaxation time. The solutions for reversible and nonreversi ble electron transfer reactions in Debye solvents have four limits; 1.2 the narrow and wide reaction window limits, as well as the slow reaction and nondiffusion limits. Solvent dynam ics play an important role in these reactions except in the slow reaction and wide window limits, when it can be ne glected. In this paper we extend our theoretical analysis of reversible ET reactions to non-Debye solvents which are characterized by mUltiple dielectric relaxation times. The free energy surface used in our earlier work was suggested by Sumi and Marcus2 and includes contributions from solvent reorganization and ligand vibrations of the reacting species. The model is similar to the one introduced earlier by Kestner, Logan, and Jortner3 who treated the problem quantum mechanically without reference to the sol vent dynamics. Sumi and Marcus' original discussion2 ofET reactions in Debye solvents ignored the reverse reaction which simplified the mathematical analysis. However, the presence of a finite barrier for the reverse reaction I and the existence of mUltiple relaxation times for the solvent in which the reactions often take place4 can have a significant effect on the rates of these reactions. In an earlier paper we addressed the problem of including reversibility in the analy sis of ET reactions in Debye solvents; here we consider the analysis the same reactions in non-Debye solvents which ex hibit multiple relaxation times. The task of linking solvent relaxation in non-Debye solvents with the kinetics of reversi ble electron transfer reactions in these solvents is greatly simplified by the existence of a close relationship between the time correlation function S(t) for the free energy of sol vat ion of the reacting intermediates and the time correlation function ~ (t) of the reaction coordinate for the ET reaction. This is also discussed at length in this paper. .) Present address: Department of Chemistry, Michigan State University, East Lansing. Michigan 48824. The dynamics of electron transfer reactions have been studied by many workersS-!2 who considered primarily the contribution of solvent reorganization to the free energy of activation. It is well known that electron transfer dynamics in Debye solvents are governed by the longitudinal relaxa tion time 'TL .4(d) Quite typically in non-Debye solvents how ever, the time correlation function ~(t) of the reaction coor dinate appears instead of e -t IrL (where t is the elapsed time) in the expressions for the survival probabilities of the react ing species in ET reactions. This was shown by Hynes8(a) who studied single outer-sphere electron transfer reactions and by Fonseca8(b) who investigated the corresponding re versible reactions. Our analysis however deals with reversi ble ET reactions described by the Sumi-Marcus free energy surface. This considers ligand vibrational contributions as well as contributions from fluctuations in the solvent polar ization to the activation energy of the reacting species. We find that the time correlation function along the reaction coordinate ~ (t) continues to playa key role in the dynamics of these model reactions and that an approximate solution for the survival probabilites of reactants and products is ob tained by the substitution of ~(t) for e -tlrL in the expres sions we have derived previously for the survival probabili ties in Debye solvents.! We also show that in this model this substitution is exact for non-Debye solvents in the narrow and wide reaction window limits. By applying linear response theory to the dynamics of ET reactions in continuum solvents, we find that ~(t) is identical to the time correlation function S(t) of the free energy of solvation of the reacting intermediates. This is so even when the solvent displays multiple relaxation times which are typical of non-Debye solvents4 and are readily observed and measured in time-dependent fluorescence Stokes (TDFS) shift experiments.13-15 This identity pro vides a useful link between TDFS experiments and the mea sured rates of electron transfer reactions in the same solvent. Our paper is organized as follows. The Sumi-Marcus free energy surface, the general reaction diffusion equations and a summary of pertinent results for reversible electron transfer reactions in Debye solvents are given in Sec. II. In Sec. III we discuss the dynamics of the polarization coordi nate and in Sec. IV we present our results for reversible ET J. Chern. Phys. 96 (2). 15 January 1992 0021-9606/92/021435-09$06.00 © 1992 American Institute of Physics 1435 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.114.34.22 On: Sat, 22 Nov 2014 00:17:061436 J. Zhu and J. C. Rasaiah: Reversible electron transfer dynamics reactions in non-Debye solvents as the solution to an integral equation for the survival probabilities. Methods of solving these integral equations are treated in Sec. V followed by a short discussion in Sec. VI. An Appendix e~amines the time correlation function S(t) of the Born solvation free energy for the reacting intermediates and its relationship to a (t). II. FREE ENERGY SURFACES AND SOLUTIONS OF THE DIFFUSION-REACTION EQUATIONS FOR ELECTRON TRANSFER IN DEBYE SOLVENTS A. Potential surfaces and diffusion-reaction equations The Sumi-Marcus2(a) free energy surfaces for reactants and products are VI (q,x) = aq2/2 + x2/2, V2 (q,x) = a(q -qo )2/2 + (x -Xo )2/2 + aGo, (2.1a) (2.1b) where q and x are the vibrational and polarization coordi nates, respectively, a = /-Uj)2 is assumed to be the same for reactants and products (p is the reduced mass and liJ is the vibrational frequency of the ligand), aGo is the reaction free energy and the coordinate x is related to the outer solvent polarization pex(r) byl x2 = (41T/C) J Ipex(r) -p?ex(rW dr, (2.2a) where peX(r) = per) _ poe (r). (2.2b) per) and poe (r) are the total polarization and electronic polarization respectively of the solvent, while p?,ex(r) is the equilibrium value of this polarization at r due to the charge distribution of the reactants. Both pex(r) and p?,ex(r) can have contributions from the translation and rotation of the solvent molecules. In Eq. (2.2a), C = lIE"" -l/Eo (2.3) and aq~/2 and x~/2 in Eq. (2.1b) are contributions from intramolecular ligand vibration and outer solvent polariza tion, respectively, to the reorganization energy. The total reorganization energy is the sum of these ..1,=..1,0 + Aq, (2.4) with ..1,0 = x~/2 = (21T/C) J IP~,ex(r) -p?,ex(r) 12 dr = (C/81T) J ID~ (r) -D? (rW dr, Aq = a%/2, (2.5a) (2.5b) (2.6) where p~,ex(r) is the equilibrium polarization at r due to the charge distribution of the products and the relation po,ex(r) = (C/41T)Do(r) (2.7) between the equilibrium polarization pO,ex(r) and the elec tric displacement DO(r) has been used. The ligand vibrational motion is much faster than the relaxation of the solvent polarization and electron transfer can take place at each value of x, leading to coordinate de pendent rate coefficients 1-3 ki(x) =Vq exp[ -,8aG,,(x)] (i= 1,2), (2.8) in which aGf(x) = (l/2)(Ao/Aq)(X-Xlc)2, aG!(x) = (l/2)(Ao/Aq)(x-x 2c)2, are the free energies of activation, and Xlc = (A + aGo)/(U o) 112, X2c = (A + aGo -Uq )/(2..1,0) 1/2. The normalization constant in Eq. (2.8) Vq = ko [21TAq/(,8Ao)] -112, (2.9a) (2.9b) (2.9c) (2.9d) (2.10) where ko is determined by whether the reaction is adiabatic or nonadiabatic.1 In the narrow reaction window limit when Aq -+0 the vibrational contribution to the reorganization en ergy is neglected and the rate coefficients are approximated by delta functions I k(x) = kl (x) = k2 (x) = k08(x -xc), where Xc = (..1,0 + aGo)/(u o )1/2 is identical to xlc and X2c in this limit. (2.11a) (2.11b) The time dependence of reversible ET reactions is de scribed by the following coupled diffusion reaction equa tions: aPI/at = [LI (t) -kl (x) ]PI + k2 (x)P2, ap2/at = [L2 (t) -k2 (x) ]P2 + kl (x)PI, (2.12a) (2.12b) where PI = PI (x,t) and P2 = P2 (x,t) are the probabilities of reactants and products, respectively, LI (t) and L2 (t) are generalized Fokker-Planck operators defined by Li =D(t) ~+,8D(t) ~ [dVi(X)] (i= 1,2), ax2 ax dx (2.13 ) in which D(t) is a time dependent diffusion coefficient, ,8 = (kB n -I where kB is BoItzman constant, Tis tempera ture and Vi (x) is given by the second term ofEq. (2.1), i.e., VI (x) = x2/2, V2 (x) = (x -Xo )2/2 + aGO. (2.14a) (2.14b) The structure of the diffusion reaction Eqs. (2.12) is strik ing: the diffusion terms describe the diffusion of the polariza tion coordinate x, while the reaction terms k; (x) are rate coefficients averaged over the vibrational coordinate q at a particular x. The reactants are initially considered to be at thermal equilibrium, so that PI (x,O) = exp[ -,8VI (x)]/ J exp[ -,8VI (x) ]dx, (2.15a) P2 (x,O) = O. (2.15b) The survival probabilities Q; (t) for the reactants and prod ucts are obtained from the solutions ofEq. (2.12) by Q;(t) = f: oe P;(x,t)dx (i = 1,2). (2.16) J. Chern. Phys., Vol. 96, No.2, 15 January 1992 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.114.34.22 On: Sat, 22 Nov 2014 00:17:06J. Zhu and J. C. Rasaiah: Reversible electron transfer dynamics 1437 B. The solutions In a Debye solvent In a Debye solvent, the diffusion constant is time inde pendent and D(t) = D is related to the longitudinal dielec tric relaxation time 1" L by 1-3 (2.17) where 1" D is Debye relaxation time, and E 00 and Eo are the high frequency and static dielectric constants, respectively. In our previous paper we showed that an approximate but general solution of Eq. (2.12) for a Debye solvent leads to the Laplace transforms of the survival probabilities given byl QI (s) = 1/s -Q2 (s), Q2 (s) = kIJ{~[1 + asl (s) + as2 (s) n. Here the Laplace transform aSi (s) is defined by (2.l8a) (2.18b) as; (s) = k;; I (gdk; (s + H;) -Ik, Ig;) (i = 1,2) (2.19a) with k;e = (g; (x) Ik, (x) Ig, (x». (2.19b) The operator (i = 1,2) (2.20) is similar to the Hamiltonian operator for a harmonic oscil lator with potentials given in Eq. (2.14). The eigenvalues are En = nr L I (n = 0,1,2, ... ) (2.21) and I un.;) are the eigenkets of Hi with eigenfunctions En which implies that Hilun.i) = Enlun•i)· There is no zero point energy and the lowest order eigenfunction Iuo';> = g, (x) = exp( -[3V,(x)/2)1 J exp( -[3V;(x)/2)dx (2.22) so that H,g, (x) = O. Inserting Eq. (2.22) into Eq. (2.19) and making use ofEqs. (2.8)-(2.10) one finds kle = vexp[ -[3(A + aGo)2/4A], (2.23a) (2.23b) . where l' = l'q [Aql A ] 112. For a Debye solvent we have shown that! as; (s) = k ;; I I Cn•i (s + En) -I (i = 1,2), (2.24a) n=O where cn,; = (un,; Ik, Ig, )2. The inverse Laplace transform of Eq. (2.24a) is a, (t) = k'e + k i; I I Cn., exp( -Ent) (i = 1,2), n=l (2.24b) which can be written in closed form by making use of the density matrix of the Harmonic oscillator,l,2 a! (t) = kle(1 -A 2e-2tirL) -1/2 Xexp [3xL e , [ A2 -tlrL ] 1 + Ae -tlrL (2.25a) Xexp [3(x2c -XO)2 e . (2.25b) [A 2 -tlrL ] 1 + Ae -t/rL Here (2.26) reflects the size of the reaction window. For example, in the narrow window limit, Aq ~A.o and A;::::: 1 while in the wide window limit, Aq >,1.0' A ;:::::0. The generalization to non-De bye solvents is discussed in Sec. IV but before that we will review certain limiting cases of importance to our analysis. Equation (2.18), with ai (t) displayed in Eq. (2.25) or (2.24), become exact in different limits. In the slow reaction limit (k, (x) ~1"L)' thermal equilibrium of the polarization coordinate x is always maintained and the time scale in which the reaction takes place is much larger than 1"L' It follows from Eq. (2.2) that ai(t) = kie which is equivalent to asi(s) = kiels. Substituting in Eq. (2.18) and taking the inverse Laplace transform we have QI(t)=1-Q2(t), (2.27a) Q2 (t) = [klel(k le + k2e)] {1 -exp [ -(kle + k2e)t n· (2.27b) In the wide reaction window limit (Aq >Ao), A = 0 and asi (s) = k;lswhere ki has the same form as Eq. (2.23) with v = Vq and A = Aq• The survival probabilities are the same as Eqs. (2.27) with kj replacing kje. In the narrow reaction window limit (Aq ~Ao), A = 1, and Eqs. (2.12) reduce tol JPI/Jt=LIP I -koo(x-xc)(P I -P2), (2.28a) JP2IJt=L2P2 +koo(x-xc)(P I -P2), (2.28b) wherexc = (Ao + 6.Go)/(U o) 1/2. The Laplace transforms of the survival probalities arel QdS)=S-l_Q2(S), (2.29a) Q2 (s) = koPI (xc,D)/{~ [1 + koGsl (xc Ixc's) + kOGs2 (xc -Xo Ixc -xo,s)]}, (2.29b) where koGsl(xclxc's) and kOGs2(xc -xolxc -xo,s) are precisely the Laplace transforms of al (t) and a2 (t), respec tively, with A = 1. Equations (2.29) are exact but have been solved I only in certain approximations or limits. Examples are the long and short time approximations and barrierless reactions. 1 In the nondiffusion limit k, (x) > 1" L I and the survival probabilities show a multiexponential time depend ence which is discussed in detail in Ref. 1. Away from these limits an interpolation formula be tween the long and short time limits of ai (t) leads to an expression which reproduces different types of single expo nential time dependences found for the survival probabili ties. This expression has the same form as Eq. (2.27) except that kje is replaced by k; = k;e1a, where a= 1 +al1"L/lxlcl +a21"Lllx2c -xol (2.30) and al and a2 are constants.! It is seen that al = a2 = 0 in the slow reaction limit and a I = a2 = 1 in the narrow reac tion window limit. I Multiexponential time dependence of the survival probabilities is found when terms beyond the leading term in the expansion of asi (s) are taken into ac- J. Chern. Phys., Vol. 96, No.2, 15 January 1992 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.114.34.22 On: Sat, 22 Nov 2014 00:17:061438 J. Zhu and J. C. Rasaiah: Reversible electron transfer dynamics count. Explicit expressions for the time dependence of the survival probabilites are given in Ref. 1. Numerical solutions of the reaction-diffusion equations were found to agree satis factorily with the analytic results. I For a non-Debye solvent, the diffusion operators are generally time dependent, and the solution of Eq. (2.12) becomes more involved. We show in Sec. IV that Eq. (2.29) is exact in the narrow reaction window limit (A = 1) with koGi(xclxc,t) =ai(t) ofEq. (2.25) except that e-th·l. is replaced by Do(t), the time correlation function of the reac tion coordinate. More generally, when A =1= 1, we find that Eqs. (2.18) is a useful solution to Eq. (2.12) for non-Debye solvents provided ai (t) is similarly redefined with Do (t) re placing e -t/TL in the expressions for ai (t) given in Eq. (2.25). However, the interpolation formula and other ap proximations discussed in the previous paragraph can be carried over to non-Debye solvents only if we use an effective relaxation time 71ff to characterize Do (t). This is discussed in the next section where we investigate the dynamics of the polarization coordinate x. III. DYNAMICS OF THE POLARIZATION COORDINATE x Before we consider the solution of Eq. (2.12) for ET in non-Debye solvents, we will first describe the dynamics of the polarization coordinate x when the the vibrational coor dinate q is neglected. This should help to clarify our argu ment without the added complication ofligand or inner sol vation shell vibration. During electron transfer of the reacting intermediates, the polarization coordinate x at time t takes on a value x2(t) = (41Tlc) f lPex(r,t) -p?eX(r) 12 dr, (3.1) which is an obvious extension ofEq. (2.2). The time depen dent polarization pex(r,t) is linearly related to the "effec tive" charges e~ff (t) on the ions with which it would be in equilibrium and e~ff(t) = er + z(t)(ei -en (i = 1,2), (3.2) where er and ei are the charges on the reactant and products, respectively, and z(t) changes from 0 to 1 as reactants are completely transformed into products. It follows that at any time t, pex(r,t) _p?ex(r) =z(t)[p~·ex(r) _P?·ex(r)]. (3.3) Equation (3.3) simply mirrors (Eq. 3.2). Inserting this in Eq. (3.1) and using Eqs. (2.3), (2.5), (2.7), and (2.l4a), we see that the potential energy for the reactants is given by VI = x2(t)/2 = Aor(t), (3.4a) where Ao is defined in Eq. (2.5). Likewise for the product potential energy, given in Eq' (2.l4b), we have V2 = (1/2) [x(t) -xo] 2 + DoGo (3.4b) The second of these relations in each of these equations has also been derived by Hynes.8(a) Linear response theory predicts that the nonequilibrium dielectric polarization pex (r,t) is related to the displacement field D(r,t) byl6 peX(r,t) = (41T) -I f~ "" c(t -7)D(r,t)d7 = (41T) -I f~ "" c( 7)D(r,t -7)d7. (3.5) The response function c(t) has the Laplace transform c(s) = 1/E"" -l/E(s), (3.6a) where E"" is the high frequency dielectric constant and E(S) is the frequency dependent dielectric function. If D(r,t -7) = DO(r) for t>O and is zero for t<O, pex(r,t) -+pex(r) as t-+ 00, and Eq. (3.5) reduces to Eq. (2.7). It follows from this that C=1/E"" -1/E= Sa"" c(7)d7. (3.6b) In a Debye solvent E(S) = E"" + (Eo -E"" )/(1 + S7D). (3.7) Inserting this in Eq. (10.1.6a), taking the inverse Laplace transform and recalling the definition of the longitudinal re laxation time 7 L = E"" 7 DI Eo one finds c(t) = (C/7L) exp( -t IrL), (3.8) which is the response function for a Debye solvent. 16 We now consider a thought experiment, similar to one suggested by Hynes,8(a) in which we start with the reactant charge distribution and let it be transformed instantaneously at t = 0 to the product charge distribution. The medium then rearranges by translation and rotation to a new charge distribution. The change in the solvent polarization with time is pex(r,t) -p?ex(r) = (41T) -I l' c( 7)d7 . X [D~(r) -D?(r)], which on combining with Eq. (3.1) gives x(t) =xoc-I [1' C(7)d7] ' (3.9) (3.10) where Xo = x( 00) = (Uo) 112. This equation is equivalent to Hynes' Eq. (2.12) for z(t).8(a) The Laplace transform of Eq. (3.10) gives xeS) = Xo (cs) -IC(S) = Xo (cs) -I [1/E "" -l/E(s)]. (3.11 ) Defining the deviation c5x(t) = x(t) -xo, it follows that the normalized time correlation function Do(t) = (c5x(t)c5x(O» I (c5x2(0» -(c5x(t»lx o -c-I [1' C(7)d7] + 1, (3.l2a) (3.12b) (3.l2c) where we have made use of the fact that c5x(O) = -Xo and c5x2(0) = x~. J. Chern. Phys .• Vol. 96, No.2, 15 January 1992 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.114.34.22 On: Sat, 22 Nov 2014 00:17:06J. Zhu and J. C. Rasaiah: Reversible electron transfer dynamics 1439 On taking the Laplace transforms one finds that ox(s) = (xo/s)[c(s)/c -1] (3.13) and 1l(s) = (1/s) [1 -c(s)/c] (3.14a) =E"" [(Eo -E(S»/(E o -E"" )]/(sE(S» (3.14b) = E"" [1 -E(s) ]I{s[ E"" + E(S)(Eo -E",,) p, (3.14c) where E(s) = [E(S) -E"" ]/(Eo -E"" ). (3.15 ) Equation (3.14c) for the forward reaction which has been derived earlier by Hynes.8(a) If we define the reaction coor dinate x' (I) = Xo -xU) for the backward reaction, a simi lar equation can be written for ox' (t). For a Debye solvent, it follows from Eqs. (3.7), (3.14), and (3.15) that E(s) = 1/0 + S1"D)' ~(s)=[s+rLI]-1 or the inverse Laplace transform ~(t) = exp( -t IrL). (3.16) (3.17) (3.18 ) Non-Debye solvents are characterized by a frequency de pendent longitudinal dielectric relaxation time, when 1" L I in Eq. (3.17) is replaced by a "frequency dependent" 1"ds) -I. In the Appendix we show that in a continuum solvent, ~(t) is identical to the time correlation function S(t) of the Born solvation energy for the reacting species. This provides another source of information on ~ (t) since there are experi mental probes which determine S(t); for example, time-de pendent fluorescence Stokes shift measurements (TDFS).13-15 The dynamics of the reaction coordinate can be studied using either a Langevin-type equation or a probability diffu sion equation. In Sec. II the Fokker-Planck diffusion equa tion was used to describe the dynamics of the polarization coordinate x in Debye solvents. This is, as discussed below, consistent with the use of the Langevin equation in the over damped limit. For non-Debye solvents one uses either a gen eralized Fokker-Planck equation or a generalized Langevin equation. The two approaches are equivalent, but the former provides a natural extension of our previous discussion I of reversible electron transfer reactions to non-Debye solvents. Since the polarization coordinate x(t) is related to Hynes' reaction coordinate z(t) through Eq. (3.4), the discussion which follows in this section is similar to his. S(a) A. Generalized langevin equation approach For non-Debye solvents the generalized Langevin equa tion for ox(t) is d 2[OX(t) lldt 2 = -wiox(t) -L {;(t -r)Ox( 1') dr, (3.19) where ox(t) = d[ox(t) ]ldt, (;(t) is the frequency depen dent friction and WL is given by wi = 1/mL, which follows from the harmonic potential given in Eq. (3.4), and m L is the reduced mass. On taking the Laplace transform of Eq. (3.19) and using Eq. (3.12a) we have 1l(s) = s + (;(s) (3.20) ? + wi + s{;(s) In the overdamped limit d2[ox(t) ]ldt2 = 0 we have, in stead of Eqs. (3.20), the relation ~(s) = [s + 1" L I(S)] -1, (3.21) where the frequency dependent longitudinal dielectric relax ation time 1"L (s) is defined bys 1"L (s) = {;(s)/wi. By com paringEq. (3.21) withEq. (3.14) onefindsthatS 1"L (s) = E"" [1 -E(s)]I [sEoE(s)] , (3.22) which provides a relation, in the overdamped limit, between the 1"L (s) and the measured dielectric response function E(s). The relationship between 1l(s) and E(s) follows from Eq. (3.23). For Debye solvents E(s) is given by Eq. (3.16) and we see that 1"L (s) = E"" 1"D/Eo = 1"L' which is indepen dent of the frequency. For non-Debye solvents with multiple dielectric relaxation times E(s) = 2:1: (1 + sr;) -t, (3.23 ) ; where 1: is a constant, the inverse of Eq. (3.23) becomes more complicated than that of Eq. (3.16). For instance, a double exponential form has been used by HynesS(a) for ~(I) in n-propyl alcohol to reproduce the dielectric relaxa tion data. B. The Fokker-Planck equation approach For a non-Debye solvent, the generalized diffusion equations with no reaction are (3.24) where the generalized Fokker-Planck operator L; (t) is giv en in Eq. (2.9). The solutions ofEq. (3.24) for parabolic potential wells are well knowns and are PI (x,t) = [21TkB T(1 _ ~2)] -1/2 [ {3(x -X(0)~)2] Xexp - , 2(1 _ ~2) (3.25a) P2 (x,t) = [21TkB T(1 _ ~2)] -1/2 { {3[(x-xo) -(x(O) _XO)~]2} Xexp - , 2(1 _ ~2) (3.25b) wherex(O) is the initial value of x, and ~ = ~(t) is related to the diffusion constant by D(t) = -{3 -Id [In ~(t) ]ldt. (3.26) This solution is easily verified by direct substitution.8(a) For a Debye solvent ~(t) is given, in the overdamped limit, by Eq. (3.18) and we have D(t) = D = ({3rL) -I, (3.27) which is independent of time and is just the result given in Eq. (2.17). This implies that the solutions given in reference 1 for ET reactions in Debye solvents using Fokker-Planck operators with a constant diffusion coefficient Dare consis- J. Chern. Phys., Vol. 96, No.2, 15 January 1992 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.114.34.22 On: Sat, 22 Nov 2014 00:17:061440 J. Zhu and J. C. Rasaiah: Reversible electron transfer dynamics tent with the simple Langevin equation in the overdamped limit. IV. SOLUTIONS OF THE GENERALIZED DIFFUSION REACTION EQUATIONS FOR REVERSIBLE ET REACTIONS IN NON-DEBYE SOLVENTS For non-Debye solvents, we need to solve Eq. (2.12) together with the initial conditions Eqs. (2.15). Since the diffusion coefficient D and the longitudinal relaxation time 7 L are now frequency dependent, the slow diffusion and slow reaction limits are not clearly defined unless we can identify an effective D and 7L' but the narrow and wide reaction window limits still hold. We will first discuss the exact solu tions in these two limits before considering the solutions for the general case. A. Solutions in the wide and narrow window limits In the wide window limit, one can still use the same argument given in Ref. 1 for Debye solvents and the solu tions, for the survival probabilities, are single exponentials QI (t) = 1 -Q2 (t), (4.1a) Q2(t) = [kl/(kl +k2)]{1-exp[ -(kl +k2)t]), ( 4.1b) with kl = Vq exp [ -f3(Aq + AGo)2/4Aq], (4.2a) k2 = kl exp [f3AG 0]. (4.2b) In the narrow reaction window limit, Eqs. (2.12) reduce to I JPIIJt=LI(t)P I -koo(x-xc)(PI -P2), (4.3a) JP2/Jt =L2 (t)P2 +koo(x-xc)(PI -P2), (4.3b) where Xc is given by Eq. (2.11 b). The Laplace transforms of the survival probalities are given by Eq. (2.29) where the Green's functions GI (xc Ixc,s) and G2 (xc -Xo Ixc -xo,s) are now the Laplace transforms of the solutions to the gener alized diffusion equations (4.4 ) JG2 (x -Xo Ixc -xo,t) = LI (t)G2 (x -Xo Ixc -xo,t), (4.5) with initial conditions GI (xlxc,O) = o(x -xc) and G2 (x -Xo Ixc -Xo ,0) = 0 [(x -xo) -(xc -xo) ]. The operator LI (t), which has a time dependent diffusion coeffi cient D(t), is defined in Eq. (2.13) and the solutions ofEqs. (4.4) and (4.5) are the same as Eq. (3.25) exceptthatx(O) has to be replaced by Xc> GI (xclxc,t) = [21TkBT(1 -A2)] -1/2 [ f3x~(1 -A)2] Xexp - , 2( 1 -A2) (4.6a) G2 (xc -Xo Ixc -Xo ,t) = [21TkBT(1- A2)] -1/2 Xex [_f3(Xc -Xo)2(1-A)2]. p 2(1 -A2) (4.6b) Note that A = AU). These results together with Eqs. (2.29) are exact for non-Debye solvents. They have been given ear lier by Fonseca8(b) but the argument used in its derivation here is exact since we do not replace L; (t) by an effective time independent operator L 7ff to prove this result, see Ref. 8(b). It is seen that the generalized Green's functions de pend on the time correlation function A (t) of the reaction coordinate which was discussed extensively in the preceding section. For Debye solvents the Green's functions reduce to a;(t), defined in Eq. (2.25), divided by ko as discussed in Sec. II. B. Approximate solutions to the generalized diffusion reaction equations We now come to our discussion of approximate general solutions of the reaction diffusion equations (2.12) in non Debye solvents. Because the generalized Fokker-Planck op erators are time dependent, the Laplace transform technique used in Ref. 1 becomes too complicated to use. This diffi culty can be formally avoided by replacing the generalized Fokker-Planck operators by effective operators L ~ff in which the effective diffusion constant D eff is time indepen dent. This argument was used by Fonseca8(b) in deriving Eqs. (4.6) which we have shown to be exact. The use of operators L ~ff instead of L; implies that H; defined in Eq. (2.20) can then be replaced by H7ff with eigenvalues c,.ff = nl7 ~ff, where the effective relaxation time may be de fined by 7"! = So"" A(t)dt (4.7) 7 ~ff = 7L for Debye solvents. In this way, the final solutions have the sameform as Eqs. (2.18) except that the generalized a; (t), which include the vibrational contribution to activa tion as well, are given by al (t) = kle (1 -A2A2) -1/2 eXP[f3xic A2A] (4.8a) 1 +AA ' a2 (t) = k2e (1 -A2A2) -1/2 eXP[f3(x2C _ XO)2 A2A ], 1 +AA (4.8b) where A = A(t) and A = AoIA. This should be compared with the corresponding relations given in Eq. (2.25) for De bye solvents to which it is similar. In deriving Eq. (4.8), we began from the effective opera torsL 7ff and used the approximation A(t) ;::::exp( -t!'T ~ff), which is consistent with L 7ff• But we also expect Eq. (4.8) to be an excellent approximation for any A(t} not limited to exp( -t 17 ~ff). Indeed it gives the exact results in the limit ing cases! For example, in the narrow reaction window limit kl(x) =k2(x) =koo(x-x c), A=AoIA= 1, Eq. (2.19b) leads to kle = ko (21TkB n -1/2 exp( -f3x~/2), k2e = ko (21TkB n -1/2 exp( -f3(xc -Xo )2/2), (4.9a) (4.9b) and asl (t) and as2 (t) reduce to koGI (xlxc,t) and kOG2 (x -Xo Ixc -xo,t), respectively, as seen from the Green functions given in Eqs. (4.6). This result is exact. In the wide reaction window limit, A;::::O, Aq;::::A, as;(s) ;::::k/s, J. Chern. Phys., Vol. 96, No.2, 15 January 1992 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.114.34.22 On: Sat, 22 Nov 2014 00:17:06J. Zhu and J. C. Rasaiah: Reversible electron transfer dynamics 1441 the inverse of Eq. (2.1S) is identical to Eq. (4.1). Also for Debye solvents, as shown in Eq. (3.1S), A(t) =exp( -t!rL)' and Eqs. (4.S) is again exact and consistent with Eqs. (2.25). The Laplace transforms of Eq. (4.S) are quite compli cated functions even when aCt) for non-Debye solvents has the simple form of the sum of exponentials. 8 This has lead us to try to solve Eq. (2.1S) in real space. Multiplying Eq. (2.1Sb) by {.r[l + asJ (s) + as2 (s)]} and taking the in verse Laplace transform, we see that the survival probabili ties can also be written as QJ (t) = 1 -Q2 (t), (4. lOa) Q2 (t) = kJet -f aCt -r)Q2 (r)dr, (4.lOb) where aCt) = al (I) + a2 (I) in which al (I) and a2 (t) are given in Eqs. (4. S). Equation (4.1 Ob) is an integral equation for Q2 (t). Any solution, numerical or analytical, for a non Debye solvent would necessarily require detailed informa tion about a (t). This is discussed in Sec. III [see Eqs. (3.14), (3.20), and (3.21)] and the Appendix. Once an explicit analytic form of A(t). [or equivalently D(t)], is known, Eq. (4.10) can be solved analytically (see Sec. V) or numerically using Eq. (4.S). The accuracy of these solutions can be checked by comparison with direct numerical solutions of the reaction-diffusion equations giv en in Eq. (2.12) from which Eq (4.10) is derived. Ref. 1 discusses the numerical solution of Eq. (2.12) for Debye solvents and compares it with the limiting solutions and oth er results (e.g. the interpolation formula and the double ex ponential approximation) to Eq. (4.S). V. SOLUTION OF THE INTEGRAL EQUATION FOR THE SURVIVAL PROBABILITIES Here we present a method of solving the integral equa tion (4.lOb) for non-Debye solvents in the context of an I -(co + Eo) -CI -C2 -Co -(CI + €I) -C2 C= -Co -C1 -(C2 + €2) -Co -CI -C2 In Eq. (5.7), Ko is a constant vector to be determined from the initial conditions F(O) = O. It is apparent that the solu tion ofEq. (4.10), which up to this point is exact for a Debye solvent, can become quite complicated even though we know, in principle, how to solve it. An approximate solution could be obtained by an itera tive procedure. Starting from a first order approximation by taking only the first term n = 0 in the sum of Eq. (5.6), we have dFo (t)ldt = klet -coFo (t), which has the solution (5.9) effective relaxation time r el defined in Eq. (4.7). For Debye solvents the method of solution presented here is, in princi ple, exact since there is only one relaxation time rL• The kernal in Eq. (4.Sb) for Debye and non-Debye solvents can then be expressed as aCt) = al (t) + a2 (t) = L cn exp( -tE""lI') (i = 1,2), n=O (5.1) where E';.II' = nlr 111' and cn = Cn,l k 1-;; 1+ Cn,2k 2-;; I. (5.2) Substituting Eq. (5.1) in Eq. (4.lOb) we find Q2 (t) = klet -L cnFn (t), (5.3 ) n=O where Fn (t) = f exp{ -E",,1I'(t -r)}Q2 (r)dr. (5.4) Differentiating Eq. (5.4) and making use ofEq. (5.3), we have (5.5) which on combining with (5.3) leads to a set of linear ordi nary differential equations for n = 0,1,2, ... , dFn (t)ldt = klet -€nFn (t) -L cnFn (t), (5.6) n=O with constant coefficients. The initial condition Fn (0) = O. The general solution ofEq. (5.6) isl7 F(t) = etC (f' e-TCG(t)dt + Ko), (5.7) where F(t) and G(t) are vectors whose components are Fn (t) and klet, respectively, and etC is the fundamental ma trixl7 in which the matrix C, with m--00, is given by -Cm -Cm -Cm (5.S) -(Cm + Em) Fo (t) = klecO-2[ -1 + cot + exp( -cot)]. (5.10) Substitution in Eq. (5.3) yields Q2(t)-;::::,klek -1[I-exp( -kt)], (5.11) where Co = k = k Ie + k2e. Equation (5.11) is just the slow reaction limit discussed in Sec. II. 1,2 Using this approximate result for Q2 (t) in Eq. (5.5) and solving the differential equation, we obtain Fn (t) = klek -I{[ 1-exp( -€,',lI't)]/E~t -[exp( -kt) -exp( -E",,1I't) ]/(€,',II' -k)} (n = 0,1,2, ... ). (5.12) J. Chern. Phys., Vol. 96; No.2, 15 January 1992 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.114.34.22 On: Sat, 22 Nov 2014 00:17:061442 J. Zhu and J. C. Rasaiah: Reversible electron transfer dynamics This is in accord withFo (t) given in Eq. (5.10). Substitution in Eq. (5.3) leads to Q2(t)-zklek -I[l-exp( -kt)] -2:. cnFnCt), (5.13) n=1 where Fn (t) is given in Eq. (5.12). The terms beyond the first provide systematic corrections to the slow reaction limit outside this region. One can iterate again by substituting Eq. (5.13) into Eq. (5.5) to find an improved solution for Fn (t), and so on. VI. DISCUSSION Electron transfer reactions are usually characterized in the literature by their rate constants. This assumes that the reaction dynamics is sufficiently well known to identify a unique rate constant which is the case when the reactants show a simple exponential time decay. One can then distin guish between adiabatic and nonadiabatic reactions as dis cussed in the text and in Ref. 1. When this decay is multiex ponential, however, the rate constant becomes ambiguous except when the dynamics can be described in terms of an effective relaxation time 7 ~ff or when attention is focused on the decay at very long times when a residual single exponen tial time dependence remains. In this case the solvent dy namics affects electron transfer in complicated ways which have been elucidated by us in several limiting cases for De bye solvents. I For instance in the narrow reaction window limit the rate constant k = 0.8337 L I for barrierless reac tions while if the barrier /3t::.G! for the reverse reaction is large and the forward reaction is barrierless k = [0.6 + (17'1/3t::.G!) 1/2] -17 L I. Many electron transfer experiments in Debye solvents confirm the proportionality between k and the inverse longitudinal relaxation time4 which has also been discussed theoretically.5,7,9 A distribu tion of relaxation times for the solvent however would not generally produce such simple behavior or permit a full de scription of the kinetics by a simple rate constant.4 It re quires instead a more complete analysis of the survival prob abilities of the reacting species which we have attempted. Equations (2.18) or (4.10), with aiCt) given by Eqs. (4.8), are our main results for ET reactions in non-Debye solvents when the free energy surfaces are described by the Sumi-Marcus model, see Eq. (2.1). The equations also ap ply to Debye solvents in which case t::.(t) = exp( -t hI.). The solutions are exact in the narrow and wide reaction win dow limits for Debye and non-Debye solvents and also in the slow reaction and nondiffusion limits for Debye solvents. The behavior between these limits, is predicted by the general solutions. An iterative method of solving these equations is discussed, which requires the identification of an effective relaxation time 7 et. The interpolation formula and other approximations derived for barrierless reactions in Debye solvents can be carried over to non-Debye solvents with the use 7et. This paper provides an explicit method of calculation of the survival probabilities in ET reactions when the solvent reorganization energy /3Ao, the ratio A = Ao I A of this to the total reorganization energy, the constant ko which depends on the reaction adiabaticity, the reaction free energy /3t::.Go and the time correlation function along the reaction coordi nate t::.(t), are known. In certain limiting cases however one or more of these quantities is no longer an independent vari able. For example, in the narrow and wide window limits A = 0 and 1, respectively. Linear response theory shows that t::.(t) is identical to the time correlation function Set) of the solvation free energy of the reacting intermediates. While oUr derivation applies strictly for a continuum solvent we expect the result to hold accurately even in a discrete molec ular solvent. This provides a useful link between time de layed fluorescence measurements of Set) for a solvent and the rates of electron transfer reactions in the same solvent. 18 ACKNOWLEDGMENTS We thank Professor Robert Dunlap for his interest in our work and for a critical review of the manuscript. Jianjun Zhu acknowledges a University Research Fellowship. APPENDIX: THE TIME CORRELATION FUNCTION OF THE BORN SOLVATION ENERGY In recent years solvation dynamics, which plays an im portant role in the kinetics ofET reactions, has been studied extensively both experimentallyl3-15 and theoretically. 19-22 The dynamics of solvation is measured by time dependent fluorescence Stokes shifts (TDFS) of chromophores form ing suitable charge transfer complexes and is related to the time correlation function Set) of the Born solvation ener gy.18 In this appendix we will discuss the relationship be tween S ( t) and the time correlation function t::. (t) of the ET reaction along the reaction coordinate. The Born solvation energy is defined by22(a) EB(t) = -(1/2) f D(r)'P(r,t)dr, (AI) where D(r) is the bare field of the reacting ions and P(r,t) is the total polarization of the medium, which is related, by linear response theory, to the field D(r) by P(r,t) = P"'(r) + (417') -1 f c(r,7)D(r,t -7)d7. (A2) Here poe (r) is the electronic polarizability, which follows the field instantaneously and is given by P"'(r) = (417') -1(1-liE", )D(r) (A3) c(r,7) is a response function, which is determined by the microscopic structure of the medium surrounding the ions. Substituting Eqs. (A3) and (A2) into Eq. (AI), we have t::.EB (t) = EB (t) -E~ where -(817') -1 f dr f D(r) 'c(r,7)D(r,t -7)dr, (A4) E~ = -(817') -1(1 -liE",,) f D(r)' D(r)dr, (AS) The Laplace transform of Eq. (A4) is t::.EB (s) = -(817') -1 f dr D(r) ·c(r,s)D(r,s). (A6) J. Chem. Phys., Vol. 96, No.2, 15 January 1992 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.114.34.22 On: Sat, 22 Nov 2014 00:17:06J. Zhu and J. C. Rasaiah: Reversible electron transfer dynamics 1443 To pursue this further we need information about the re sponse function e(r,s) from a detailed molecular theory such as the dynamic mean spherical approximation (MSA).19-23 For a continuum solvent, however, the re sponse function e(r,s) is e(s) = IlEac -1/e(s), (A7) which is independent ofr and was given earlier in Eq. (3.6a). If the charges on the ions are suddenly switched on, D(r,t -r) = D(r) for t> 0 and is zero otherwise. Equation (A4) then reduces to AEB(t) = -(81T)-1 J drD(r)'D(r) Le(r)dr = E~C-I L c(r)dr, (A8) where e is defined in Eg. (3. 6b) and the Born solvation ener gy at equilibrium24 E~ = AEB( (0) = -(81T) -Ie J dr D(r) ·D(r). (A9) The time correlation function of the Born solvation en ergy E B ( t) is defined as S(t) = [EB(t) -EB(oo)]/[EB(O) -EB(oo)] (AlOa) = [AE B (t) -AE B ( 00 ) ] / [ AE B (0) -AE B ( 00 ) } (AlOb) (AlOc) where we have used AEB (0) = O. Substituting (A9) and (A8) in (AlOe) we have Set) = _c-J f c(r)dr+ 1. On comparing with Eg. (3.12c), we see that AU) = S(t). (All) (AI2) I J. Zhu and J. C. Rasaiah, J. Chern. Phys. 95, 3325 (1991). [Errata: In the denominator ofEq. (4.35b) replace -a,1 by + a,1 ).J 2 (a) H. Surni and R. A. Marcus, J. Chern. Phys. 84,4894 (1986). (b) W. Nadler and R. A. Marcus, ibid. 86, 3906 ( 1987). J N. R. Kestner, J. Logan, and J. Jortner, J. Phys. Chern. 78, 2148 (1974). 4(a) M. McGuire and G. McLendon,J. Phys. Chern. 90, 2549 (1986); (b) H. Heitele, M. E. Michel-Beyerle, and P. Finckh, Chern. Phys. Lett. 138, 237 (1987); (c) H. Hetele, F. Pollinger, S. Weeren, and M. E. Michel Beyerle, Chern. Phys. 143, 325 (1990); (d) E. M. Kosower and D. Hup pert, Chern. Phys. Lett. 96, 433 (1983). , (a) L. D. Zusrnan, Chern. Phys. 49, 295 (1980); 80, 29 (1983); 119, 51 (1988); (b) 1. V. Alexandrov, ibid. 51,449 (1980). 6 (a) H. L. Friedman and M. D. Newton, Faraday, Discuss. Chern. Soc. 74, 73 (1982); (b). B. Ternbe, H. L. Friedman, and M. D. Newton, J. Chern. Phys.76, 1490 (1982); (c) H. L. Friedman and M. D. Newton, J. Elec troanal. Chern. 204, 21 (1986). 7 (a) D. F. Calef and P. G. Wolynes, J. Phys. Chern. 87, 3387 (1983); (b) J. Chern. Phys. 78, 470 (1983). 8 (a) J. T. Hynes, J. Phys. Chern. 90, 3701 (1986); (b) T. Fonseca, J. Chern. Phys. 91, 2869 (1989). 9 (a) 1. Rips and J. Jortner, J. Chern. Phys. 87, 2090 (1987); (b) 87,6513 (1987); (c) 88, 818 (1988); (d) Chern. Phys. Lett. 133,411 (1987). lO(a) R. I. Cukier,J.Chem. Phys. 88, 5594 (1988); (b) D. Y. YingandR. 1. Cukier, ibid. 91, 281 (1989). II M. Sparpaglione and S. Mukamel, J. Chern. Phys. 88, 3263 (1988).88, 4300 (1988). 12M. Weaver, Acc. Chern. Res. 23, 294, (1990). IJ J. D. Simon, Acc. Chern. Res. 21, 128, (1988). 14 (a) M. Maroncelli, J. MacInnes, and G. R. Fleming, Science 243, 1674 (1989); (b) G. R. Fleming and P. G. Wolynes, Physics Today 43,36 (1990). I' (a) P. F. Barbara and W. Jarzeba, Acc. Chern. Res. 21,195 (1988); (b) Adv. Photochem. 15, 1 (1990); (c) W. Jarzeba, Gilbert Walker, A. E. Johnson, and P. F. Barbara, Chern. Phys. 152, 57 (1991). 16R. Zwanzig, J. Chern. Phys. 38,1603 (1963); 38,1605 (1963). 17 Tyn Myint-U, Ordinary Differential Equations (North-Holland, Amster dam, 1978). l8The Set) relevant to ET experiments is the time correlation function for the free energy of solvation of the reacting intermediates. Time resolved fluoresence measurements with electronically excited polar fluorescent probes yields the time correlation function for the free energy of solvation of these probes. These measurements show only a slight dependence on the nature of the probes, see Ref. 15. This indicates thatS(t) for the react ing intermediates is essentially the same as S( t) obtained from these ex periments. The relationship between time dependent fluorescence shift and dielectric friction has been discussed by van der Zwan and Hynes [1. Phys. Chern. 89, 4181 (1985) J. 19p. G. Wolynes, J. Chern. Phys. 86, 5133 (1987). 20 A. L. Nichols and D. F. Calef, J. Chern. Phys. 89, 3783 (1988). 21 (a) I. Rips, J. Klafter,andJ. Jortner,J. Chern. Phys. 88, 3246 (1988); (b) 89,4288 (1989). 22 (a) B. Bagchi, Annu. Rev. Phys. Chern. 40, 115 (1989) and references therein; (b) A. Chandra and B. Bagchi, J. Phys. Chern. 94, 3152 (1990). 23 (a) F. O. Raineri, Y. Zhou, H. L. Friedman, and G. Stell, Chern. Phys. 152,201 (1991); (b) Y. Zhou, H. L. Friedman, and G. Stell, Chern. Phys. 152,18 (1991). "The equilibrium Born solvation energy of an ion of radius a in a continu um solvent is E~ = -(1 -1/E~ )e2/2a. However, in electron transfer reactions E~ is the equilibrium solvation energy of the reacting interme diates -see Ref. 23. The details of this do not matter since E~ does not appear in the final expression for S(t), see Eq. (A 11). J. Chern. Phys., Vol. 96, No.2, 15 January 1992 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.114.34.22 On: Sat, 22 Nov 2014 00:17:06

1.4975998.pdf

Electrical detection of single magnetic skyrmion at room temperature Riccardo Tomasello , Marco Ricci , Pietro Burrascano , Vito Puliafito , Mario Carpentieri , and Giovanni Finocchio Citation: AIP Advances 7, 056022 (2017); doi: 10.1063/1.4975998 View online: http://dx.doi.org/10.1063/1.4975998 View Table of Contents: http://aip.scitation.org/toc/adv/7/5 Published by the American Institute of Physics Articles you may be interested in Current-driven skyrmion motion along disordered magnetic tracks AIP Advances 7, 056017 (2017); 10.1063/1.4975658 The influence of the edge effect on the skyrmion generation in a magnetic nanotrack AIP Advances 7, 025105 (2017); 10.1063/1.4976726 Mobile Néel skyrmions at room temperature: status and future AIP Advances 6, 055602 (2016); 10.1063/1.4943757 Skyrmion-based high-frequency signal generator Applied Physics Letters 110, 112402 (2017); 10.1063/1.4978510 Current-driven skyrmion dynamics in disordered films Applied Physics Letters 110, 132404 (2017); 10.1063/1.4979316 Room temperature skyrmion ground state stabilized through interlayer exchange coupling Applied Physics Letters 106, 242404 (2015); 10.1063/1.4922726AIP ADV ANCES 7, 056022 (2017) Electrical detection of single magnetic skyrmion at room temperature Riccardo Tomasello,1Marco Ricci,1,aPietro Burrascano,1Vito Puliafito,2 Mario Carpentieri,3and Giovanni Finocchio4,a 1Department of Engineering, Polo Scientifico Didattico di Terni, University of Perugia, Terni I-50100, Italy 2Department of Engineering, University of Messina, I-98166 Messina, Italy 3Department of Electrical and Information Engineering, Politecnico di Bari, via E. Orabona 4, I-70125 Bari, Italy 4Department of Mathematical and Computer Sciences, Physical Sciences and Earth Sciences, University of Messina, Messina I-98166, Italy (Presented 3 November 2016; received 23 September 2016; accepted 11 November 2016; published online 8 February 2017) This paper proposes a protocol for the electrical detection of a magnetic skyrmion via the change of the tunneling magnetoresistive (TMR) signal in a three-terminal device. This approach combines alternating spin-transfer torque from both spin- filtering (due to a perpendicular polarizer) and spin-Hall effect with the TMR sig- nal. Micromagnetic simulations, used to test and verify such working principle, show that there exists a frequency region particularly suitable for this achieve- ment. This result can be at the basis of the design of a TMR based read-out for skyrmion detection, overcoming the difficulties introduced by the thermal drift of the skyrmion once nucleated. © 2017 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). [http://dx.doi.org/10.1063/1.4975998] INTRODUCTION Magnetic solitons1–5are fascinating particle-like magnetization textures with promising proper- ties for the use in spintronic devices. Among them, skyrmions are attracting a large interest for their fundamental properties, such as topological protection,6and potential technological applications in racetrack memories,7–9oscillators,10–13detectors,14and logic gates.15The recent experimental evi- dences16–19that N ´eel skyrmions can be stabilized at room temperature in multilayers where a sufficient interfacial Dzyaloshinskii–Moriya interaction (i-DMI)20,21arises, and that they can be manipulated by the spin-transfer torque (STT) from the spin-Hall effect (SHE),22–24have been pushing the research towards the development of a skyrmion based technology.25,26Two key aspects for the skyrmion implementation in devices concern its nucleation and detection. For the former, while the experimen- tal nucleation of skyrmions has been achieved by using external out-of-plane fields,17–19the control of the single skyrmion nucleation at room temperature is still an open challenge, although many numeri- cal works have predicted it.7,27–29For the latter, skyrmions have been detected by different techniques, such as topological Hall effect,6,30,31Lorentz transmission electron microscopy,32,33spin-polarized tunneling microscopy,34–36polar magneto-optical Kerr effect microscope,16scanning transmission X-ray microscopy,17,19and photoemission electron microscopy combined with X-ray magnetic cir- cular dichroism.18However, a suitable method to ensure an easier integration of skyrmionic devices in CMOS systems concerns a fully electrical detection of skyrmions. Recently, the tunneling non- collinear magnetoresistance37has been used, and many works have pointed out the possibility to use the tunnel magnetoresistance (TMR). The key issue is that the skyrmion, once nucleated at room aemail: marco.ricci@unipg.it and gfinocchio@unime.it. 2158-3226/2017/7(5)/056022/7 7, 056022-1 ©Author(s) 2017 056022-2 Tomasello et al. AIP Advances 7, 056022 (2017) temperature, is characterized by a thermal drift38,39because of the translational invariance which impedes the steady presence of the skyrmion below a magnetic tunnel junction (MTJ) read-head. To fix this problem, here we propose a detection protocol based on the TMR and SHE from a microwave current, as well as the combination of SHE and STT, in a three terminal device.23,40–42Such a system is composed of an MTJ with a perpendicular polarizer, built on top of a heavy metal strip, and allows one to control the magnetization dynamics in the MTJ free layer by means of two electrically inde- pendent currents: one flowing through the heavy metal (SHE), and the other one flowing through the MTJ (STT). The application of a microwave current drives continuously the skyrmion motion below the MTJ, leading to a dynamical change of the magnetoresistance signal linked to the out-of-plane component of the magnetization. This signal univocally indicates the presence of the skyrmion, and then, the successfully accomplishment of the nucleation process that can be, for instance, obtained by a dc STT as predicted in.7 MICROMAGNETIC MODEL AND DEVICE The study is performed by means of a state-of-the-art processing tools and micromagnetic solver which numerically integrates the Landau-Lifshitz-Gilbert equation:43–45 (1+2)dmf d=(mfheff)[mf(mfheff)]gBSH 2 0eM2 StFLmf(mf(ˆzjHM)) +gBSH 2 0eM2 StFLmf(ˆzjHM)gBjMTJ 0eM2 StFL"MTJ(mf,mp)f mf mfmp q(V) mfmpg (1) whereis the Gilbert damping, mfis the normalized magnetization of the MTJ free layer, and = 0Mstis the dimensionless time, with 0being the gyromagnetic ratio, and Msthe satura- tion magnetization of the MTJ free layer. heffis the normalized effective field, which includes the exchange, magnetostatic, anisotropy and external fields, as well as the i -DMI46,47and thermal field.48 gis the Land ´e factor,Bis the Bohr magneton, SHis the spin-Hall angle, eis the electron charge, tFLis the thickness of the ferromagnetic free layer, and ˆ zis the unit vector along the out-of-plane direction."MTJ(mf,mp)=2 [1+2(mf
mp)]is the polarization function, where is the spin polarization factor, and mpis the magnetization of the pinned layer, which is considered fixed for all the simula- tions. q(V)(mfmp) is the perpendicular torque term that depends on the voltage applied to the MTJ leads.49 We consider a three-terminal device composed of an ultrathin ferromagnetic layer in contact with a Pt heavy metal (HM), in order to achieve a large enough i-DMI and SHE. The ferromagnet acts as the free layer of the point contact MTJ deposited on top of it, which permits to apply a localized STT (see Fig. 1(a) for a sketch of the device, where a cartesian coordinate system has been introduced). The MTJ free layer has a 400x200 nm2elliptical cross section with thickness of 0.8 nm, and its major axis is oriented along the y-direction, in order to exploit the SHE-driven motion of the skyrmion along the direction perpendicular to the electrical current ( x-axis).8The pinned layer has a fixed out-of-plane easy axis of the magnetization and it is patterned as a nano-contact.12,14 The discretization cell used is 2.5x2.5x0.8 nm3. In our simulations, we consider typical experimental systems (CoNi/Pt),50whose physical parameters are: MS= 600 kA/m, A= 20 pJ/m, perpendicular anisotropy constant ku= 0.60 MJ/m3,D= 3.0 mJ/m2,= 0.1,= 0.66, and SH= 0.1. RESULTS AND DISCUSSION We analyze the skyrmion stability at room temperature T= 300 K in presence of an external normal-to-plane field Hext= 25 mT. The skyrmion core14is along the negative z-direction. Statistical analyses (simulations 500 ns long) of the skyrmion diameter show that its mean value is around 40 nm, therefore we design the nanocontact to have a diameter of 50 nm in order to achieve a sufficiently large TMR signal when the skyrmion goes under it. Once the contact is fixed, we achieve the skyrmion nucleation by injecting a large enough dc current pulse through the MTJ.7After the current jMTJis056022-3 Tomasello et al. AIP Advances 7, 056022 (2017) FIG. 1. (a) Schematic representation of the three-terminal device studied. (b) Trajectory of the skyrmion center at T= 300 K and for Hext= 25 mT, when no microwave current is applied. The solid black line indicates the MTJ free layer under analysis, while the solid green line represents the region under the nanocontact. switched off, the skyrmion does not remain under the nanocontact, because it is subject to thermal fluctuations that give rise to a breathing, shape deformations, as well as a Brownian drift,38,39as we can see in Fig. 1(b), where the trajectory of the skyrmion center is plotted (no microwave current is applied). The thermal drift of the skyrmion, even in presence of a confined ferromagnet, does056022-4 Tomasello et al. AIP Advances 7, 056022 (2017) not allow one to detect it directly with a dc current flowing through a standard TMR read-head, because it is located most of the time out of the nanocontact region. We solve this issue by driving the skyrmion motion via a microwave current. In particular, we can use two independent currents in the three-terminal device, permitting the analysis of three configurations: ( i) only SHE, ( ii) only STT, ( iii) combination of SHE and STT. In the case ( i), we apply an in-plane sinusoidal current jHM(t) =JHMsin(2fHMt) along the x-axis with amplitude JHM= 30 MA/cm2, frequency fHMfrom 0.1 GHz to 1 GHz, and duration of 100 ns. ForfHM<0.3 GHz, the skyrmion is expelled from the free layer, entailing a detection failure. For 0.3 GHz<fHM<0.7 GHz, the skyrmion starts to alternately shift along the y-axis.8In particular, by starting from the region under the nanocontact, the skyrmion moves towards the bottom edge in the semi-period when the current is negative, whereas it goes back towards the nanocontact when the cur- rent is positive. Each time that it passes below the nanocontact induces a variation of the out-of-plane component of the magnetization <mzuc>(see Fig. 2(a)), and, hence, of the TMR/(1hmzuci)2
. By injecting a subcritical dc current via the MTJ, it is possible to convert the oscillating resistance FIG. 2. Temporal evolution of the normalized z-component of the magnetization under the nanocontact (black curve) and the applied normalized microwave current (magenta curve) flowing in the heavy metal only (SHE), for Hext= 25 mT at (a) T= 300 K and (b) T= 0 K. The current amplitude and frequency are 30 MA/cm2and 0.3 GHz, respectively. In (a), <mzuc>exhibits the same frequency 1/ TMof the current 1/ THMbetween 5 ns and 15 ns, and a double frequency between 45 ns and 55 ns, whereas in (b) it oscillates permanently at the same frequency of the current. Inset in (a): spatial distribution of the magnetization at t= 15 ns. The nanocontact and the color scale are also indicated. (Multimedia view) [URL: http://dx.doi.org/10.1063/1.4975998.1] [URL: http://dx.doi.org/10.1063/1.4975998.2] [URL: http://dx.doi.org/10.1063/1.4975998.3]056022-5 Tomasello et al. AIP Advances 7, 056022 (2017) into an output voltage having the same characteristic waveform. This signal identifies the presence of the skyrmion, because, if the skyrmion was not nucleated, the <mzuc>would not show significant dynamical changes. The variation of <mzuc>is characterized by two non-stationary oscillations: one at the same frequency of the current and the other at twice such frequency. This last contribution is linked to a double passage of the skyrmion beneath the nanocontact within a period of the microwave current, and it is due to the stochasticity of thermal fluctuations, which lead the skyrmion to not deterministically cover the same trajectory in each oscillation semi-period (see also MOVIE 1 of Fig. 2 (multimedia view)). To confirm this, we carried out simulations at zero temperature, observing that<mzuc>oscillates only at the same frequency of the current (see Fig. 2(b)). For fHM>0.7 GHz, the oscillatory translation of the skyrmion is too fast to obtain a robust TMR change. As expected, the boundaries of the frequency ranges above described become larger (lower) if the current amplitude is increased (decreased), because of a faster (slower) skyrmion motion. In the case ( ii), we inject a perpendicular microwave current jMTJ(t) =JMTJsin(2fMTJt) in the MTJ with amplitude JMTJ= 3 MA/cm2and same duration and frequency fMTJas for ( i). Although the STT from this current acts as an anti-damping torque, it does not strongly affect the skyrmion dynamics, which, therefore, are mainly due to the thermal fluctuations in all the fMTJrange. The key role is played by the Brownian diffusion that hampers the steady presence of the skyrmion under the nanocontact, resulting in non-stationary jumps of the <mzuc>(see Fig. 3(a)). Therefore, this configuration ( ii) cannot be used for skyrmion detection. FIG. 3. (a) Temporal evolution of the normalized z-component of the magnetization under the nanocontact (black curve) and of the applied normalized microwave current (magenta curve) flowing in the MTJ only (STT), for Hext= 25 mT at T= 300 K. The current amplitude and frequency are 3 MA/cm2and 0.3 GHz, respectively. Three regions are separated by dashed lines and refer to the position of the skyrmion with respect to the nanocontact. (b) Example of time evolution of the normalized z-component of the magnetization under the nanocontact when the SHE and STT act simultaneously, JHM= 30 MA/cm2, JMTJ= 3 MA/cm2and same frequency of 0.3 GHz. Insets: spatial distribution of the magnetization. The nanocontact and a color scale are also indicated.056022-6 Tomasello et al. AIP Advances 7, 056022 (2017) In the configuration ( iii), we simultaneously apply both jHM(t) and jMTJ(t). We fix JHM = 30 MA/cm2and JMTJ= 3 MA/cm2, and consider the same frequency in the range 0.1 GHz -1.0 GHz for both currents. The skyrmion dynamics, and hence, the <mzuc>, exhibits similar features to the case ( i) (frequency range to obtain TMR change, waveform, non-stationary frequency). Fig. 3(b) shows the time evolution of <mzuc>, where the spatial distributions of the free layer magnetization51 are indicated as insets and refer to the time instants pointed by the black arrows. However, we note that the presence of jMTJ(t) decreases the lower frequency of the range where the detection can be performed to 0.2 GHz. This is due to the local STT, which reduces the damping under the nanocontact for a certain semi-period of jMTJ(t), thus behaving as an “attraction pole” for the skyrmion, which is no longer expelled by the SHE. CONCLUSIONS In summary, we have proposed, by means of micromagnetic simulations, a method to electrically detect the presence of a N ´eel skyrmion in a pillar that can be easy generalized for a racetrack memory. We have considered a three-terminal device, where an MTJ is deposited on top, and, by injecting a large enough microwave current through the heavy metal layer (SHE), we are able to continuously drive the skyrmion motion under the point-contact MTJ. In this way, we critically reduce the skyrmion drift induced by thermal fluctuations at room temperature that, in absence of current, provides a stochastic drift of the skyrmion around the nanocontact (the trajectory depends on the initial seed of the thermal fluctuations). The SHE-induced skyrmion motion under the nanocontact leads to a variation of the out-of-plane component of the magnetization below the third terminal, and, consequently, of the TMR. This reading protocol can be very useful for the design of TMR read-out for the electrical detection of skyrmions. 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1.4998244.pdf

Emergence, evolution, and control of multistability in a hybrid topological quantum/ classical system Guanglei Wang , Hongya Xu , and Ying-Cheng Lai Citation: Chaos 28, 033601 (2018); doi: 10.1063/1.4998244 View online: https://doi.org/10.1063/1.4998244 View Table of Contents: http://aip.scitation.org/toc/cha/28/3 Published by the American Institute of Physics Articles you may be interested in Prediction of flow dynamics using point processes Chaos: An Interdisciplinary Journal of Nonlinear Science 28, 011101 (2018); 10.1063/1.5016219 Sensitivity analysis of the noise-induced oscillatory multistability in Higgins model of glycolysis Chaos: An Interdisciplinary Journal of Nonlinear Science 28, 033602 (2018); 10.1063/1.4989982 Introduction to Focus Issue: Time-delay dynamics Chaos: An Interdisciplinary Journal of Nonlinear Science 27, 114201 (2017); 10.1063/1.5011354 Describing chaotic attractors: Regular and perpetual points Chaos: An Interdisciplinary Journal of Nonlinear Science 28, 033604 (2018); 10.1063/1.4991801 Phenomenology of coupled nonlinear oscillators Chaos: An Interdisciplinary Journal of Nonlinear Science 28, 023110 (2018); 10.1063/1.5007747 Asymmetry in electrical coupling between neurons alters multistable firing behavior Chaos: An Interdisciplinary Journal of Nonlinear Science 28, 033605 (2018); 10.1063/1.5003091Emergence, evolution, and control of multistability in a hybrid topological quantum/classical system Guanglei Wang,1Hongya Xu,1and Ying-Cheng Lai1,2,a) 1School of Electrical, Computer, and Energy Engineering, Arizona State University, Tempe, Arizona 85287, USA 2Department of Physics, Arizona State University, Tempe, Arizona 85287, USA (Received 29 July 2017; accepted 13 September 2017; published online 1 March 2018) We present a novel class of nonlinear dynamical systems—a hybrid of relativistic quantum and classical systems and demonstrate that multistability is ubiquitous. A representative setting is coupledsystems of a topological insulator and an insulating ferromagnet, where the former possesses an insulating bulk with topologically protected, dissipationless, and conducting surface electronic states governed by the relativistic quantum Dirac Hamiltonian and the latter is described by the nonlinearclassical evolution of its magnetization vector. The interactions between the two are essentially the spin transfer torque from the topological insulator to the ferromagnet and the local proximity induced exchange coupling in the opposite direction. The hybrid system exhibits a rich variety of nonlineardynamical phenomena besides multistability such as bifurcations, chaos, and phase synchronization. The degree of multistability can be controlled by an external voltage. In the case of two coexisting states, the system is effectively binary, opening a door to exploitation for developing spintronicmemory devices. Because of the dissipationless and spin-momentum locking nature of the surface currents of the topological insulator, little power is needed for generating a significant current, mak- ing the system appealing for potential applications in next generation of low power memory devices.Published by AIP Publishing. https://doi.org/10.1063/1.4998244 Topological quantum materials are a frontier area in con- densed matter physics and material sciences. A represen-tative class of such materials is topological insulators, which have an insulating bulk but possess dissipationless conducting electronic states on the surface. For a three-dimensional topological insulator (3D TI), such as Bi 2Se3, the surface states have a topological origin with a perfectspin-momentum locking, effectively eliminating backscat-tering from non-magnetic impurities and generating elec-tronic “highways” with highly efficient transport. The surface states can generally be described by a two- dimensional Dirac Hamiltonian in relativistic quantummechanics. When a piece of insulating, ferromagneticmaterial is placed on top of a topological insulator, twothings can happen. First, there is a spin-transfer torquefrom the spin-polarized surface current of the topological insulator to the ferromagnet, modulating its magnetization and making it evolve dynamically. Second, the ferromag-net generates an exchange coupling to the Hamiltonian ofthe topological insulator, reducing its quantum transmis-sion from unity and rendering it time dependent. Due tothese two distinct types of interactions in the oppositedirections, the coupled system of topological insulator and ferromagnet constitutes a novel class of nonlinear dynami- cal systems—a hybrid type of systems where a relativisticquantum description of the surface states of the 3D TI anda classical modeling of the ferromagnet based on the LLG(Landau-Lifshitz-Gilbert) equation are necessary. Thehybrid dynamical system can exhibit a rich variety of non- linear dynamical phenomena and has potential applica- tions in spintronics. Here, we review some recent results inthe study of such systems, focusing on multistability. In particular, we demonstrate that multistability can emergein open parameter regions and is therefore ubiquitous inthe hybrid relativistic quantum/classical systems. Thedegree of multistability as characterized by the ratio of the basin volumes of the multiple coexisting states can be externally controlled (e.g., by systematically varying thedriving frequency of the external voltage). The controlledmultistability is effectively switchable binary states thatcan be exploited for developing spintronic memory. Forexample, in spintronics, the multiple stable states of themagnetization are essential for realizing efficient switching and information storage (e.g., magnetoresistive random- access memory—MRAM, with a read-out mechanismbased on the giant magnetoresistance effect). MRAM iswidely considered to be the next generation of the univer-sal memory after the current FLASH memory devices. Akey challenge of the current MRAM technology lies in itssignificant power consumption necessary for writing or changing the direction of the magnetization. The spin- transfer torque based magnetic tunnel junction configura-tion has been developed for power-efficient MRAMs. Forthis type of applications, the coupled TI-ferromagnet sys-tem represents a paradigmatic setting where onlyextremely low driving power is needed for high perfor-mance because of the dissipationless nature of the spin-polarized surface currents of the TI. I. INTRODUCTION The demands for ever increasing processing speed and diminishing power consumption have resulted in thea)Electronic mail: Ying-Cheng.Lai@asu.edu 1054-1500/2018/28(3)/033601/10/$30.00 Published by AIP Publishing. 28, 033601-1CHAOS 28, 033601 (2018) conceptualization and the emergence of physical systems that involve topological quantum states. Such a system can appear in a hybrid form: one component effectively obeying quantum mechanics on a relatively large length scale(/C24lm), while another governed by classical or semiclassical equations of motion, with interactions of distinct physical origin in opposite directions. The topological states basedquantum component acts as an electronic highway, where electrons can sustain a dissipationless, spin-polarized current under a small electrical driving field. The dynamics of the classical component can be nonlinear, leading to a novel class of nonlinear dynamical systems. We believe such sys-tems represent a new paradigm for research on nonlinear dynamics. The purpose of this mini-review article is to intro- duce a class of coupled topological quantum/classical hybridsystems and discuss the dynamical behaviors with a special focus on the issue of multistability. Significant potential applications will also be articulated. The practical motivations for studying the class of topo- logical quantum/classical hybrid systems are as follows. The tremendous advances of information technology relied onthe development of hardwares. Before 2003, the clock speed of the CPU increases exponentially as predicted by Moore’s law and Dennard scaling. 1As we approach the end of Moore’s law, mobile Internet rises, which requires more power efficient and reliable hardwares. Mobile Internet con- nects every user into the network and provides enormousdata, leading to the emergence of today’s most rapidly grow- ing technology—artificial intelligence (AI). 2–4The orders of magnitude increases in abilities to collect, store, and analyzeinformation require new physical principles, designs, and methods—“more is different.” 5In the technological develop- ment, quantum mechanics has become increasingly relevantand critical. In general, when the device size approaches the scale of about 10 nm, quantum effects become important. In the current mesoscopic era, a hybrid systems descriptionincorporating classical and quantum effects is essential. For example, quantum corrections such as energy quantizations and anisotropic mass are necessary in the fabrication ofCMOS, 6the elementary building block of CPU and GPU. In terms of memory devices, the magnetic tunnel junction (MTJ) based spin-transfer torque random access memory(STTRAM) 7,8is becoming a promising complement to solid state drives, whose core is a tunnel junction design with mul- tilayer stacks, where a layer of ferromagnetic materials witha fixed magnetization is used to polarize the spin of the injected current. As a fully quantum phenomenon, the spin polarized current tunnels through a barrier to drive the mag-netization in the magnetically soft ferromagnetic layer—the so-called free layer for information storage. The basic principles of STTRAM was proposed about two decades ago in the context of spintronics before the dis- covery of topological insulators (TIs). 9–11In addition, a current-induced spin-orbit torque mechanism12was proposed as an alternative way to harness the magnetization of con- ducting magnetic or magnetically doped materials with large spin-orbit coupling. With the advance of 3D TIs, much moreefficient operations are anticipated with giant spin-orbit tor- que and spin-transfer torque. Particularly, in comparison with the conventional spin-orbit torque settings in heavymetals with a strong Rashba type of spin-orbit coupling, 13–16 the spin density in 3D topological insulator based systems can be enhanced substantially by the factor /C22hvF=aR/C291, where vFis the Fermi velocity in the TI and aRis the strength of the Rashba spin-orbit coupling in two-dimensional elec- tron gas (2DEG) systems.17The two-layer stack configura- tions of TI/ferromagnet18–20or TI/anti-ferromagnet21,22have recently been articulated, which allow for the magnetization of the depositing magnetic materials to be controlled and the transport of spin-polarized states on the surface of the 3D TIs to be modulated. To be concrete, in this paper, we consider the ferromagnet-TI configuration and focus on the dynamics of the magnetization in the insulating ferromagnet and the two orthogonal current components on the surface of the TI: one along and another perpendicular to the direction of an exter- nally applied electrical field. A schematic illustration of the system is presented in Fig. 1(a), where a rectangular shape of the ferromagnet is deposited on the top of a TI. For the ferro- magnet, the dynamical variable is the magnetization vectorM, whose evolution is governed by the classical Landau- Lifshitz-Gilbert (LLG) equation, 23which is nonlinear. The TI, as will be described in Sec. II, hosts massless spin-1/2 quasiparticles in the low-energy regime ( /C24meV) on its sur- face and generates spin polarized surface currents when a weak electrical field is applied.9–11In fact, the surface states of the TI are described by the Dirac Hamiltonian, rendering it effectively relativistic quantum. Physically, the interac- tions between TI and ferromagnet can be described, as fol- lows. The robust spin polarized current on the surface of the TI generates24a strong spin-transfer torque25to the ferro- magnet, inducing a change in its magnetization vector. The ferromagnet, in turn, generates a proximity induced exchange field in the TI. As a result, there is an exchange coupling term in the surface Dirac Hamiltonian of the TI, FIG. 1. Schematic illustration of a representative relativistic quantum/classi- cal hybrid system, the physical interactions, and the dynamical evolution of the states. (a) A coupled ferromagnet and topological insulator (TI) system, where the ferromagnet is deposited on the top of the TI. (b) The physical interactions: spin-transfer torque (from TI to ferromagnet) and exchange coupling (from ferromagnet to TI). The dynamical evolution of the ferro- magnet is described by the classical, nonlinear LLG equation, and the dissi- pationless, spin-polarized surface states of the TI are determined by theDirac Hamiltonian. Refer to Secs. IIandIIIfor the meanings of the mathe- matical notations and equations.033601-2 Wang, Xu, and Lai Chaos 28, 033601 (2018)which modulates the quantum transmission and leads to a change in the surface current. The two-way interactionsbetween the ferromagnet and TI are schematically illustratedin Fig. 1(b), where the whole coupled system is of a hybrid type: effectively relativistic quantum TI and classical ferro-magnet. The interactions render time dependent dynamicalvariables in both TI and ferromagnet: the surface current for the former and the magnetization vector for the latter. Due to the intrinsic and externally spin-transfer torque induced non-linearity of the LLG equation, the whole configuration repre-sents a nonlinear dynamical system, in which a rich varietyof phenomena such as bifurcations, chaos, synchronization,and multistability can arise. 26 Following the theme of the Focus Issue, in this paper, we focus on the emergence, evolution, and control of multi-stability. When a small external voltage with both a dc andac component is applied to the TI, a robust spin-polarizedsurface current rises, as shown in Fig. 1(a). In certain open parameter regions, the magnetization can exhibit two coex-isting stable states (attractors) with distinct magnetic orienta-tions, each having a basin of attraction. As the phase spacefor the magnetization is a spherical surface, the basin areas of the two attractors are well defined. (This is different from typical dissipative nonlinear dynamical systems in which thebasin of attraction of an attractor has an infinite phase spacevolume. 27) As an external parameter, e.g., the frequency of the ac voltage, is varied, the relative basin areas of the twoattractors can be continuously modulated. In fact, as theparameter is changed systematically, both birth and death ofmultistability can be demonstrated, rendering feasiblemanipulation or control of multistability. While some ofthese results have appeared recently, 26here we focus on those that have not been published. In Sec. II, we provide a concise introduction to the basics of TIs and the Dirac Hamiltonian with a focus on thephysical pictures of the emergence of strong, spin-polarized surface states. In Sec. III, we describe the mechanism of spin transfer torque and the rules of the dynamical evolution ofthe TI-ferromagnet coupled system in terms of the LLGequation and the quantum transmission of the TI. In Sec. IV, we present results of multistability, followed by a discussionof potential applications in Sec. V. II. TOPOLOGICAL INSULATORS One of the most remarkable breakthroughs in condensed matter physics in the last decade is the theoretical predic-tion 9,28,29and the subsequent experimental realization30–32 of TIs.10,11,33TIs are one emergent phase of the material that has a bulk band gap so its interior is an insulator but withgapless surface states within the bulk band gap. The surfacestates are protected by the time-reversal symmetry and there- fore are robust against backscattering from impurities, which are practically appealing to developing dissipationless orlow-power electronics. Moreover, the surface states possessa perfect spin-momentum locking, in which the spin orienta-tion and the direction of the momentum are invariant duringthe propagation. TIs are representative of topologically protected phases of matter, 34,35one theme of last year’s Nobel prize.36Theprediction and experimental realization of TIs benefited from the well-known quantum Hall effect,37also a topological quantum order. The topological phases of matter not only areof fundamental importance but also have potential applica-tions in electronics and spintronics. 38According to the bulk- edge correspondence in topological field theory, gapless edge states exist at the boundary between two materials with different bulk, topologically invariant numbers.10The edge states are protected by the topological properties of the bulkband structures and thus are extremely robust against localperturbations. Depending on the detailed system setting,there are remarkable properties associated with the edgestates such as perfect conductance, uni-directional transpor-tation, and spin-momentum coupling. 9–11 We learned from elementary physics that a perpendicular magnetic field applied to a conductor subject to a longitudinalelectrical field will induce a transverse voltage—the classicHall effect. 39In 1980, it was discovered that, for a 2DEG at low temperatures, under a strong magnetic field, the Hall con- ductivity is quantized exactly at the integer multiple of thefundamental conductivity e 2/h,w h e r e eis the electronic charge and his the Planck constant. This is the integer quan- t u mH a l le f f e c t( u s u a l l yr e f e r r e dt oa sQ H E ) .37Different from the classical Hall effect, the quantized Hall conductivity isfundamentally a quantum phenomenon occurring at the mac-roscopic scale. 40Soon after, new states of matter such as the fractional quantum Hall effect (FQHE),41–44quantum anoma- lous Hall effect (QAHE),35and the quantum spin Hall effect (QSHE)45–48were discovered. Besides their fundamental sig- nificance, such discoveries have led to an unprecedented wayto understand and explore the phases of matter through a con-nection of two seemingly unrelated fields: condensed matterphysics and topology. Heuristically, QHE can be understood in terms of the Landau levels—energy levels formed due to a strong mag-netic field. Classically, an electron will precess under such afield. Quantum mechanically, only the orbits whose circum-ference is an integer multiple of 2 pcan emerge (constructive interference), leading to the Landau levels. QHE can beunderstood as a result of the Fermi level’s crossing throughvarious Landau levels 49as, e.g., the strength of the external magnetic field is increased. At a deeper level, the robustness of the quantized con- ductivity in QHE can be understood as a topological effect. To understand the concept of topology in physical systems,we consider a two-dimensional closed surface characterizedby a Gaussian curvature. According to the Gauss-Bonnet the-orem, the number of holes associated with a compact surfacewithout a boundary can be written as a closed surfaceintegral 4pð1/C0gÞ¼ /H20860ðGaussian curvature Þ/C1da; where we have g¼0 for a spherical surface and g¼1f o ra donut surface (a two-dimensional torus). Different types ofgeometrical surfaces can thus be characterized by a singlenumber. This idea has been extended to physics with aproper definition of geometry in terms of the quantum eigen-states, where a generalized Gauss-Bonnet formula by Chern033601-3 Wang, Xu, and Lai Chaos 28, 033601 (2018)applies. In particular, consider the band structure of a 2D conductor or a 2DEG. Let jumðkÞibe the Bloch wavefunc- tion associated with the mth band. The underlying Berry con- nection is Am¼ihumðkÞjr kjumðkÞi: (1) The Berry phase is given by UB¼þ Am/C1dl¼/H20860ðr /C2 AmÞ/C1ds; (2) where Fm¼r/C2 Amis the Berry curvature. The total Berry flux associated with the mth band in the Brillouin zone is nm¼1 2pðð d2k/C1Fm; (3) which is the Chern number.10,34LetNbe the number of occupied bands. The total Chern number is n¼XN m¼1nm: (4) It was proved34that the Hall conductivity is given by rxy¼ne2 h: (5) As the magnetic field strength is increased, the integer Chern number increases, one at a time, leading to a series of pla- teaus in the conductivity plot. The Chern number is a topo-logical invariant: it cannot change when the underlying Hamiltonian varies smoothly. 34This leads to robust quanti- zation of the Hall conductance. Due to its topological nature and the time-reversal symmetry breaking, dissipationless chiral edge conducting channels emerge at the interface between the integer quantum Hall state and vacuum, which appears to be promising for developing low-power electron- ics but with the requirement of a strong external magnetic field. Nevertheless, the topological ideas developed in the context of QHE turned out to have a far-reaching impact on pursuing distinct topological phases of matter. The QSHE represents a preliminary manifestation of TIs with a time reversal symmetry, as a 2D TI, essentially a quantum spin Hall state, was predicted in the CdTe-HgTe- CdTe quantum well system9and experimentally realized.30 Both CdTe and HgTe have a zinc blende crystal structureand have minimum band gaps about the Cpoint. The origins of the conduction and valance bands are sandpatomic orbi- tals. Compared with CdTe, HgTe can support an inverted band structure about the Cpoint, i.e., CdTe has an s-type conduction band and a p-type valance band, while HgTe has ap-type conduction band and an s-type valance band. Mathematically, the inversion is equivalent to a change in the sign of the effective mass. The difference is in fact a result of the strong spin-orbit coupling in HgTe, which can reduce the band gap and even invert the bandstructure through orbital splitting. (In experiments, the band gap can be widened by increasing the size of the quantum confine- ment.) Intuitively, a CdTe-HgTe-CdTe quantum well can be thought as making a replacement of the layers of Cd atomsby Hg. When the HgTe layer is thin, the properties of CdTe is dominant and the quantum well is in the normal regime. As one increases the thickness of the HgTe layer, eventually the configuration of the conduction and valance bands willbecome similar to that of HgTe so the quantum well is in the inverted regime. The interface joining the two materials with the inverted configuration acts as a domain wall and canpotentially harbor novel electronic states described by a dis- tinct Hamiltonian. In general, regardless of the material parameters, the Hamiltonian can be written in the momentum space asH¼Hðk x;ky;/C0i@zÞ, where zstands for the stacking direction sokzis not a good quantum number. Integrating this Hamiltonian within several dominant zbase states, one can arrive at an effective Hamiltonian for the 2D TI, which is characterized by kxandky. Edge states can be obtained by imposing periodic boundary conditions along one directionand open boundary conditions in another. The inverted band structure, i.e., the basis, and the time-reversal symmetry guarantee the gaplessness of the edge state while the detailedform of the open boundary plays a secondary role only. For apedagogical review of 2D Tis, see Ref. 50. Parallel to the study of 2D TIs, there were efforts in uncovering 3D TIs, for which Bi 1–xSbxwas theoretically pro- posed to be a candidate.28,46The prediction was confirmed experimentally shortly after.31However, since Bi 1–xSbxis an alloy with random substitutional disorders, its surface statesare quite complicated, rendering difficulty in a description based on an effective model. The attention was then turned to find 3D TIs in stoichiometric crystals with simple surfacestates, leading to the discovery 29,32of Bi 2Se3. In particular, it was experimentally observed32that there is a single Dirac cone on the surface of Bi 2Se3. A low-energy effective model was then proposed29for Bi 2Se3as a 3D TI, in which spin- orbit coupling was identified as the mechanism to invert the bands in Bi 2Se3and the four most relevant bands about the C point were used to construct an effective bulk Hamiltonian. Ageneric form of the Hamiltonian was written down in the space constituting the four bases up to the order of Oðk 2Þ, constrained by a number of symmetries: the time-reversal, theinversion, and the three-fold rotational symmetries. The parameter associated with each term was determined by fitting the dispersion relation with the ab initio computational results. The surface states can be obtained by imposing constraintsalong one direction, which mathematically entails replacing k z by/C0i@zwhile keeping the states along the other two direc- tions oscillatory. The effective surface Hamiltonian can becalculated by projecting the bulk Hamiltonian onto the surface states. At present, Bi 2Se3is one of the most commonly studied 3D TIs, which possesses gapless Dirac surface states protectedby the time-reversal symmetry and a bulk band gap up to 0.3 eV (equivalent to 3000 K—far higher than the room temperature). The widely used Hamiltonian for the surface states of an ideal 3D TI is H¼/C22hv Fðr/C2kÞ/C1^z; (6) where vFis the Fermi velocity of the surface states (vF/C256:2/C2105m=s for Bi 2Se3), and r¼ðrx;ry;rzÞare the033601-4 Wang, Xu, and Lai Chaos 28, 033601 (2018)Pauli matrices describing the spin of the surface electron.29 An elementary calculation shows that the dispersion relation of this surface Hamiltonian indeed has the structure of aDirac cone. In proximity to a ferromagnet with a magnetiza-tionm, an extra term m/C1rin the Hamiltonian (6)will be induced. Due to the breaking of the time reversal symmetryby the exchange field, gap opening for the surface states willoccur and backward scatterings will no longer be forbidden. The proximity induced exchange field will thus modulate the charge transport behaviors of the surface states and theunderlying spin density by disturbing the spin texture, whichin turn can be a driving source of the nonlinear dynamicmagnetization in the adjacent ferromagnetic cap layerthrough a spin-transfer torque. III. SPIN-TRANSFER TORQUE Spin-transfer torque25is a major subject of research in spintronics,51a field aiming to understand and exploit the spin degree of freedom of electrons beyond the conventionalcharge degree of freedom. Intuitively, spin-transfer torque isnothing but the exchange interaction between two magneti-zation vectors. In particular, when two magnetization vectorsare brought close to each other, they tend to align or anti-align with each other to evolve into a lower energy state. Inour coupled TI-ferromagnet system, one magnetization vec-tor is the net contribution of the spin-polarized current on thesurface of the TI, and the other comes from the ferromagnet.A setting to generate a spin-transfer torque is schematically illustrated in Fig. 2. The basic physical picture can be described, as follows. When a normal current flows near orwithin a region in which a strong magnetization is present,the spin associated with the current will be partially polar-ized. This implies that, by the law of action and reaction, aspin-polarized current will exert a torque on the magnetiza-tion—the spin-transfer torque. Such a torque will induceoscillations, inversion, and other dynamical behaviors of themagnetization. The ability to manipulate magnetization iscritical to applications, especially in developing memorydevices.Conventionally, a three-layer magnetic tunnel junction (MTJ) is used to study spin-transfer torque, 51–53in which the fixed layer is a ferromagnet with a permanent magnetization. A schematic illustration of an MTJ is presented in Fig. 3. When a normal current is injected into this layer, the spin will be partially polarized due to the individual exchange interaction between each single electron spin and the magne- tization. As a result, there is randomness in the polarization with no definite correlation between the direction of the elec- tron spin and momentum. To separate the fixed from the free layer, a thin insulating separation barrier is needed to form a tunnel junction. The current will travel through the insulating barrier via the mechanism of quantum tunneling, during which the direction of spin will not be altered insofar as the insulating barrier does not have any magnetic impurity. After the tunneling, the spin-polarized current will exert a spin-transfer torque on the magnetization in the free ferro- magnetic layer, modifying the information stored. Readoutof the information, i.e., the direction of the magnetization, can be realized by exploiting the giant magnetoresistance effect. 54–56 An issue with MTJ is that a very large current is needed to reorient the magnetization, motivating the efforts to explore alternative mechanisms with a lower energy require- ment. One mechanism was discovered in ferromagnet/ heavy-metal bilayer heterostructures.14,57The strong Rashba spin-orbit coupling in many heavy metals can be exploited through the mechanism of spin-orbit torque to generate spin-polarized currents via the Edelstein effect, which are gener- ally much stronger than the exchange interaction between the fixed layer and current spin in the conventional MTJ. Moreover, the current in the configuration needs no longer to be restricted to the perpendicular direction, but can have any orientation within the film plane. This new geometrical degree of freedom enables unconventional strategies for manipulating magnetization. For example, it was discov- ered 58that the domain wall motions (essentially the FIG. 2. Schematic illustration of spin-transfer torque. The large green arrow indicates an electron flow under an external longitudinal voltage. The elec- trons are spin polarized (e.g., the surface states of a 3D TI). As the electrons pass through a region in which a magnetization vector is present (e.g., a fer- romagnetic region), it exerts a torque on the magnetization, due to which thespin polarization of the outgoing electrons is changed, henceforth the term “spin-transfer torque.” FIG. 3. Schematic illustration of a magnetic tunnel junction (MTJ). A typi- cal MTJ consists of a fixed layer, a barrier and a free layer. An electrical cur- rent is injected vertically into the fixed layer and comes out from the free layer. The average spin of the current is polarized by the fixed layer, which tunnels through the insulating barrier and drives the magnetization of the free layer. This configuration can be used to modify and detect the magneti-zation direction in the free layer through the giant magnetoresistance (GMR) effect.033601-5 Wang, Xu, and Lai Chaos 28, 033601 (2018)dynamics of magnetization) in the covering magnetic free layer have a sensitive dependence on the spatial distribution of the current generated spin-orbit torque. A disadvantage of the spin-orbit torque configuration is that the heavy metals usually suffer from substantial scatter- ings and the transportation mechanisms are complex. Inaddition, for heavy metals, spin-orbit coupling is essentially a higher-order effect. As a result, the currents are not per- fectly polarized. These difficulties can be overcome byexploiting TIs as a replacement for heavy metals. Mathematically, the Hamiltonian term describing the Rashba spin-orbit coupling has the same form as the effec- tive surface Hamiltonian of a 3D TI, i.e., /C24r/C2k. This is basically the whole Hamiltonian for the surface states of 3DTI under the low energy approximation. The presence of an exchange field from the ferromagnetic cap layer will contrib- ute a Zeeman term to the Hamiltonian. To see this explicitly,we consider the surface Hamiltonian of a 3D TI in the pres- ence of a magnetization /C22hv Fðr/C2kÞ/C1^zþm/C1r¼/C22hvFrx/C1kyþmx /C22hvF/C18/C19/C20 /C0ry/C1kx/C0my /C22hvF/C18/C19 /C21 þmzrz; where the mzterm only induces a band gap due to the breaking of the time-reversal symmetry for the surface states, as shown in Fig. 4(a).T h e mxandmyterms are equivalent to a shift in the center of the Fermi surface, as shown in Fig. 4(b). Consequently, the surface currents and the associated spin den- sities flowing through the magnetization region will be modu- lated. In addition, as the TI contains an insulating bulk with aband gap much larger than the room temperature thermal fluc- tuations, the only conducting modes are those associated with the spin-polarized surface states. The coupled TI/ferromagnetconfiguration has been experimentally realized. 24So far the system provides the strongest spin-transfer torque source—two to three orders of magnitude higher than that in heavy metals. The magnetization dynamics of the ferromagnet depos- ited on the 3D TI can be described by the classical LLGequation,23which captures the essential physical processes such as procession and damping of the magnetization subjectto external torques. The LLG equation is _n¼/C0D /C22hn/C2^xþaGn/C2_nþ1 /C22hT; (7) where the first term represents the procession along the easy axis ^xandDis the anisotropic energy of the ferromagnet. The second term describes the Gilbert damping of strength aG. The last term is the contribution from the external torque, which is the spin-transfer torque T¼hri/C2m¼nhri/C2n; (8) where, for convenience, we use n¼jmj(the magnitude of the magnetization) as a normalizing factor so that n¼m=n becomes a unit vector. IV. EMERGENCE, EVOLUTION, AND CONTROL OF MULTISTABILITY To study the nonlinear dynamics of the coupled TI- ferromagnet system, we couple the transportation of current on the surface of TI with the oscillatory dynamics of the magnetization of the ferromagnet through the mechanisms ofspin-transfer torque and exchange coupling. The variousdirections are defined in Fig. 1(a). The covering ferromag- netic material is insulating so that the conducting current is limited to the 2D surface of the TI: j¼ðj x;jy;0Þ. In addition, the width of the device in the ydirection is assumed to be large, while the system size in the xdirection is on the order of the coherent length so that the transport of the surface cur- rent can be described as a scattering process under a square magnetic potential. The typical time scale of the evolution ofthe magnetization is nanoseconds, which is much slowercompared with the relaxation time of the surface current ofthe TI. We can then use the adiabatic approximation when modeling the dynamics of the surface electrons. Specifically, we solve the time-independent Dirac equation with a con-stant exchange coupling term at a given time and obtain thetransmission coefficient of surface electrons. 59,60The low- FIG. 4. Surface states of topological insulators and the effect of magnetization. (a) Red and blue lines represent the linear dispersion relation of t he surface states of an ideal TI without a gap opening perturbation, with the slope /C22h/C23F. The two states have opposite spin polarizations. The presence of a perpendicular magnetization vector mzleads to a gap opening of size proportional to 2 mz, making the dispersion relation hyperbolic. (b) Magnetization vectors in the plane of the TI, i.e., mxandmy, will shift the central position of the Fermi surface in the wave vector space, inducing an asymmetry with respect to either ky¼0o r kx¼0.033601-6 Wang, Xu, and Lai Chaos 28, 033601 (2018)energy effective surface state Hamiltonian of the TI under a square magnetic potential is given by H¼/C22hvFðr/C2kÞ/C1^zþm/C1rHðxÞHðL/C0xÞ: (9) Compared with Eq. (6), the induced exchange field is modeled by a step function HðxÞin the space to ensure that it only appears within the ferromagnetic region oflength L. To calculate the transmission coefficient through the fer- romagnetic region, we consider the wavefunctions before entering inside and after exiting the ferromagnetic region,and apply the boundary conditions at the interfaces of thethree regions. The result is 59 t¼/C04/C22h/C23F~kxcosh aðAþieihBÞ; (10) where A¼a2ie/C0ih/C22h/C23Fð~kyþi~kxÞ/C0E/C0mzhi /C0a1ie/C0ih/C22h/C23Fð~ky/C0i~kxÞ/C0E/C0mzhi ; B¼a2ie/C0ihðE/C0mzÞ/C0/C22h/C23Fð~ky/C0i~kxÞhi /C0a1ie/C0ihðE/C0mzÞ/C0/C22h/C23Fð~kyþi~kxÞhi ; and E¼/C22h/C23FkF; kx¼kFcosh; ky¼kFsinh; /C22h/C23F~kx¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E2/C0m2 z/C0ð/C22h/C23~kyÞ2q ; /C22h/C23F~ky¼/C22h/C23Fkyþmx; a¼eikFLcosh; a1¼eið~kxþmyÞL; a2¼eið/C0~kxþmyÞL: In these expressions, Eand kFare the Fermi energy and Fermi wave vector of electrons outside the ferromagnetic region, i.e., the linear dispersion region, and his the incident angle of the electron to the ferromagnetic region. Integratingthe transmission coefficient over the incident angle, weget the current densities through the ferromagnetic regionalong the xandydirections as j x¼/C0kF 2pe2 /C22hVðp 2 /C0p 2dhjtj2cosh; jy¼/C0kF 2pe2 /C22hVðp 2 /C0p 2dhjtj2sinh; (11) from which the spin density of the electrons can be calcu- lated as hri¼/C01 e/C23Fj/C2^z¼1 e/C23F/C1ð /C0jy;jx;0Þ¼jx e/C23F/C1g; (12)where V¼VdcþVaccosðXtÞis the driving voltage along thexdirection and g¼ðgx;1;0Þ. A surprising result is that the voltage along the xdirection will induce a current along ydirection, which is a signature of an anomalous Hall effect.26To understand the origin of this current deviation, we examine the integrand of Eq. (11) in the absence of the ferromagnet jtj2¼cos4h cos4hcos2ðkxLÞþð sin2h/C01Þ2sin2ðkxLÞ; (13) which is an even function with respect to the incident angle, leading to a zero jyafter the integration. However, if the effect ofmis taken into account, the quantity jtj2is no longer an even function with respect of h, meaning that the ycomponent of the current contributed by the electrons with incident angles hand/C0hdo not have the same magnitudes, so a net ycompo- nent appears. The quantity jgxj¼jjy=jxj¼jrxy=rxxjis the ratio of the Hall conductance to the channel conductance.60 Dynamical behaviors including chaos, phase synchroni- zation, and multistability in the coupled TI-ferromagnet sys- tem were reported in a previous work.26Here, we present a phenomenon on multistability that was not reported in the previous work: continuous mutual switching of final state through a sequence of multistability transitions. Specifically,we focus on the behavior of the system versus the bifurcation parameter V dc=VacforX=xF¼X=ðD=/C22hÞ¼7:0, as shown in Fig.5(a). Since nis a directional vector of unit length, it con- tains only two independent variables, e.g., nxandny, which we represent using the blue and red colors, respectively. We fix other parameters as n=E¼0:1,kFL¼100, and E2eVac= ð2p/C22h3xF/C232 FÞ¼100. Figure 5(a) shows that there are several critical parameter values about which the system dynamicschange abruptly as reflected by the discontinuous behaviors of the blue and red dots. A detailed investigation in a previous work 26demonstrated that this is a signature of multistability. Because the whole phase space is the surface of a 3D sphere, the relative strength of multistability can be charac- terized by the volumes of basins of attraction of the coexist- ing final states (attractors). For example, for the case of two attractors, the ratio of the volumes of their basins of attrac- tion indicates the relative weight of each state. Figures 5(b) and5(c) show, for X=xF¼10:0, two representative basins forVdc=Vac¼0:5179 and Vdc=Vac¼0:5232, respectively, which are calculated by covering the unit sphere with a100/C2100 grid of initial conditions and determining to which attractor each initial condition leads to. For the two distinct attractors, the values of the dynamical variable n yare differ- ent:ny/C240 and ny/C24/C01, where the sign of the driving volt- age determines the sign of ny. From an applied standpoint, the two stable states are effectively binary, which can be detected through the GMR mechanism. To label the final states, we color a small region on the sphere with the corre-sponding value of n yin each final state and use Albers equal- area conic projection to map the sphere to a plane, which is area-preserving. As the dc driving voltage is increasedslightly, the basin of the blue state expands while that of the red state shrinks. As can be seen from the bifurcation dia- gram, when all the initial conditions lead to the blue state,033601-7 Wang, Xu, and Lai Chaos 28, 033601 (2018)we have ny/C24/C01. If we continue to increase the dc driving voltage from this point, nywill approach 0 eventually, indi- cating that the red state is the sole attractor of the system. During this process, there is a continuous transition of multi-stability, where there is a single state (blue) at the beginning,followed by the emergence and gradual increase of the basin of the red state, and finally by the disappearance of the entire basin of the blue state. From this point on, another transitionin the opposite order occurs, where the red state eventually disappears and replaced by a blue state, and so on. This phe- nomenon of continuous flipping of the final state through asequence of multistability transitions was not reported in the previous work. 26 Multistability in the coupled TI-ferromagnet system can be controlled through parameter perturbations. There are two approaches to altering the final magnetization state. We first fix a particular value of the driving voltage and choose the ini-tial conditions that lead to one of the two possible final states. Transition to the alternative final state can be triggered by applying a perturbation, such as a voltage pulse. Figure 6 shows the approximate locations of the various multistability regimes along the V dc=Vacaxis for different values of the driv- ing frequency. For example, for X=xF¼7:0, there are four multistability regimes located approximately at the four points on the line X=xF¼7:0. We then examine the bifurcations for different values of the driving frequency and mark the corre-sponding transition points in Fig. 6. Since the transition points depend on the initial condition used in calculating the bifurca- tions, the results are only an approximate indicator of the mul-tistability regimes of finite width. In spite of the uncertainties,we obtain a linear behavior of the multistability regimes in the parameter plane of the driving frequency and voltage. We note that there are irregularities associated with the case ofX=x F¼2:0. This is because, in this case, the multistability regimes are too close to each other, rendering indistinguish- able to certain extent the transition regimes.The dependence of the multistability regimes on the driving frequency suggests ease to control the final state. For example, if multistability is undesired, we can choose a rela- tively high value of the driving frequency, e.g., X=xF ¼10:0. In this case, the multistability regimes occupy only a small part of the Vdc=Vacinterval. That is, for most parameter values, there is only a single final state in the system, regard-less of the initial conditions. A proper dc voltage can then be chosen to guarantee that the system approaches the desired final state. If the task is to optimize the flexibility for the FIG. 5. Bifurcations and multistability in the coupled TI-ferromagnet system.(a) A bifurcation diagram for X=x F ¼7:0, where the parameter ratio Vdc=Vacis swept from –1.0 to 1.0. The system exhibits rich dynamics. In sev- eral parameter regimes, there are abrupt changes in the final states. A systematic computation with differentinitial conditions reveals multistability. (b) and (c) For X=x F¼10:0, typical examples of multistability. The phase space of the normalized magnetization nof the ferromagnet is the surface of a unit sphere. All possible initial condi- tions from the spherical surface aredistinguished by different colors. The Albers equal-area conic projection is used to map the initial conditions from the spherical surface to a plane while preserving the area of each final state. The standard parallels of the Albers projection are 2 9p;7 18p. The basins of two final states are shown. FIG. 6. Dependence of multistability regimes on the driving voltage and fre-quency. The blue circles indicate the approximate positions of the multi- stability regimes in the parameter plane of driving voltage and frequency. The positions are those at which discontinuous bifurcations occur with respect to the driving frequency for the specific initial condition nðt¼0Þ ¼ð1;0;0Þ. The width of each regime depends on the condition. The red dashed lines are included for eye guidance. For relatively small values of X=x F(e.g., 2.0), there are more multistability regimes than those for higher values of X=xF. The results indicate that the emergence and evolution of multistability can be controlled by the driving frequency.033601-8 Wang, Xu, and Lai Chaos 28, 033601 (2018)system to switch between distinct final states, we can set a relatively low value of the frequency, e.g., X=xF¼2:0. In this case, the system exhibits a large number of multistabilityregimes. Switching between the final states can be readilyachieved by using a voltage pulse. V. DISCUSSION Multistability is a ubiquitous phenomenon in nonlinear dynamical systems,61–72and also in physical systems such as driven nanowire73,74and semiconductor superlattice.75 Indeed, it is common for a nonlinear system to exhibit mul- tiple coexisting attractors, each with its own basin of attrac-tion. 61,62The boundaries among the distinct basins can be fractal61,62or even riddled,76–87and there is transient chaos88on the basin boundaries. In applications of nonlin- ear dynamics, it is thus natural to anticipate multistability.In a specific physical system, to understand the origin ofmultistability can be beneficial to its prediction and control.Alternatively, it may be possible to exploit multistabilityfor technological systems, such as the development ofmemory devices. The purposes of this mini-review article are twofold. First, we introduce topologically protected phases of matter,a frontier field in condensed matter physics and materials sci- ence, to the nonlinear dynamics community. For this pur- pose, we provide an elementary description of a number ofbasic concepts such as Berry phase and Chern number in thecontext of the celebrated QHE with an emphasis on the topo-logical nature, and topological insulators with dissipation-less, spin-momentum locking surface states that are aremarkable source of the spin-transfer torque for nonlineardynamical magnetization. As a concrete example of a hybridtopological quantum/classical system, we discuss the config-uration of coupled TI and ferromagnet, where the former is arelativistic quantum system and the latter is classical. Thephysical interactions between the two types of systems arediscussed: the spin polarized electron flows on the surface of the TI delivers a spin-transfer torque to the magnetization of the ferromagnet, while the latter modifies the DiracHamiltonian of the former through an exchange coupling.The nonlinear dynamics of this hybrid system has been stud-ied previously, 26including a brief account of multistability. The second purpose is then to present results pertinent tomultistability, which were not reported in the previousworks. Through a detailed parameter space mapping of theregions of multistability, we uncover the phenomenon ofalternating multistability, in which the final states of the sys-tem emerge and disappear alternatively as some parametersare continuously changed. For example, by changing the fre-quency of the driving voltage, one can tune the percentage ofthe multistability regimes in the parameter space. The phe- nomenon provides a mechanism to harness multistability through experimentally realizable means, such as the deliv-ery of small voltage pulses to the TI. The system of coupled TI-ferromagnet is a promising prototype of the building blocks for the next generation ofuniversal memory device. The multistable states can poten-tially be exploited for a binary state operation to store andprocess information.ACKNOWLEDGMENTS We would like to acknowledge the support from the Vannevar Bush Faculty Fellowship program sponsored by the Basic Research Office of the Assistant Secretary ofDefense for Research and Engineering, and funded by theOffice of Naval Research through Grant No. N00014-16-1-2828. 1H. Sutter, “The free lunch is over: A fundamental turn toward concurrency in software,” Dr. Dobb J. 30, 202–210 (2005). 2T. Chouard and L. 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1.1559973.pdf

Thermally excited Trivelpiece–Gould modes as a pure electron plasma temperature diagnostic F. Anderegg, N. Shiga, D. H. E. Dubin, C. F. Driscoll, and R. W. Gould Citation: Physics of Plasmas (1994-present) 10, 1556 (2003); doi: 10.1063/1.1559973 View online: http://dx.doi.org/10.1063/1.1559973 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/10/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Behaviour and stability of Trivelpiece-Gould modes in non-neutral plasma containing small density fraction of background gas ions AIP Conf. Proc. 1521, 89 (2013); 10.1063/1.4796065 Experimental study of parametric dependence of electron-scale turbulence in a spherical tokamaka) Phys. Plasmas 19, 056125 (2012); 10.1063/1.4719689 Thermally excited fluctuations as a pure electron plasma temperature diagnostic Phys. Plasmas 13, 022109 (2006); 10.1063/1.2172928 Thermal excitation of Trivelpiece-Gould modes in a pure electron plasma AIP Conf. Proc. 606, 253 (2002); 10.1063/1.1454291 Power coupling to helicon and Trivelpiece–Gould modes in helicon sources Phys. Plasmas 5, 564 (1998); 10.1063/1.872748 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.193.164.203 On: Tue, 23 Dec 2014 05:22:47Thermally excited Trivelpiece–Gould modes as a pure electron plasma temperature diagnostica F. Anderegg,b)N. Shiga, D. H. E. Dubin, and C. F. Driscoll Institute for Pure and Applied Physical Sciences and Department of Physics, University of California at San Diego, La Jolla, California 92093 R. W. Gould California Institute of Technology, Mail Stop 128-95, Pasadena, California 91103 ~Received 11 November 2002; accepted 19 December 2002 ! Thermally excited plasma modes are observed in trapped, near-thermal-equilibrium pure electron plasmas over a temperature range of 0.05 ,kT,5 eV. The modes are excited and damped by thermal fluctuations in both the plasma and the receiver electronics. The thermal emission spectratogether with a plasma-antenna coupling coefficient calibration uniquely determine the plasma ~and load!temperature. This calibration is obtained from the mode spectra themselves when the receiver-generated noise absorption is measurable; or from separate wave reflection/absorptionmeasurements; or from kinetic theory. This nondestructive temperature diagnostic agrees well withstandard diagnostics, and may be useful for expensive species such as antimatter.©2003 American Institute of Physics. @DOI: 10.1063/1.1559973 # I. INTRODUCTION Un-neutralized plasmas are unique in that they can be trapped in a rotating thermal equilibrium state by static elec-tric and magnetic fields. Steady-state confinement of N 510 3–109electrons, ions, or antimatter particles1,2is rou- tinely used in plasma experiments, atomic physics,3and spectroscopy.4The thermal equilibrium characteristics be- come most evident with the formation of Coulomb crystals5 when pure ion plasmas are cooled to the liquid and solidregimes at sub-Kelvin temperatures. The higher temperatureplasma regime studied here is also well-described by near-equilibrium statistical mechanics, 6with kinetic theories of waves and transport coefficients7amenable to experimental tests. These stable thermal equilibrium plasmas exhibit fluctu- ating electric fields due to the random motions of the par-ticles. The weakly damped plasma waves represent normalmodes of the isolated plasma, with expected average electro-static potential energy of 1 2kTpwhereTpis the plasma tem- perature. These waves are excited by particles as they moverandomly in the plasma; and if a receiver is connected to theplasma, the waves will also be excited by uncorrelated ther-mal fluctuations in the receiver circuit. Conversely, thewaves are absorbed ~e.g., Landau-damped !by the particles and by the real part of the load impedance. In traps withfinite length and radius the modes are discrete, and the Fou-rier spectra of the electric fields exhibit well-separated peaksat the discrete Trivelpiece–Gould ~TG!standing mode frequencies. 8 Somewhat simpler center of mass modes are commonly observed in the ‘‘single particle regime’’ with highly tunedresonant circuits in hyperbolic traps. 9The center of massmode is a well-established diagnostic for particle number; but these oscillations are only weakly coupled to the randomparticle motion through the anharmonicity of the trap, andlittle temperature information can be obtained. Thermal ex-citation of the drift ~diocotron !modes at lower frequencies is more difficult to measure; whereas thermal cyclotron modesat higher frequencies are readily observed in warmnon-neutral 10and hot fusion plasmas;11upper hybrid modes have also been used as a thermal diagnostic.12In space plas- mas, thermal noise is also used as a diagnostic;13there the absence of boundary conditions significantly changes the im-pedance of a mode near a resonance. In the crystallized re-gime, suprathermal equipartition of mode energy has re-cently been observed in dusty plasmas. 14In mirrors of a laser resonator,15thermally excited vibrations are carefully ana- lyzed assuming that each mode has energy kT. Historically, the ionospheric microwave back scattering is probably one of the first observations of radiation scatter-ing by a collective effect, i.e., not by individual particles. 16,17 Similarly, the signals detected in the present experiment arethe result of collective plasma modes and not the result ofuncorrelated charged particles moving randomly. In this paper, the spectrum of thermally excited TG standing modes is measured in pure electron plasmas over atemperature range of 0.05 ,kT p,5 eV, using a room- temperature receiver ~with an effective temperature ;0.05 eV–0.1 eV !. We consider the azimuthally symmetric ( mu 50) modes only. The received spectrum consists of a Lorentzian due to the plasma mode, plus an uncorrelatedbroad spectrum due to the receiver ~load!noise. Each of these components is ‘‘filtered’’ by the load impedance Z , and the mode impedance Zm. When the receiver impedance is non-negligible compared to the mode impedance, a non-Lorentzian spectrum occurs: resonant absorption of the re- a!Paper QI2 2, Bull. Am. Phys. Soc. 47, 249 ~2002!. b!Invited speaker. Electronic mail: fanderegg@ucsd.eduPHYSICS OF PLASMAS VOLUME 10, NUMBER 5 MAY 2003 1556 1070-664X/2003/10(5)/1556/7/$20.00 © 2003 American Institute of Physics This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.193.164.203 On: Tue, 23 Dec 2014 05:22:47ceiver noise by the plasma mode creates a characteristic ‘‘dip and peak’’ in the noise spectrum. By Nyquist’s theorem,10thermal plasmas generate noise proportional to kTptimes the real part of the plasma mode impedance, ZmRe. Conversely, the load-generated noise is pro- portional to the real part of the load impedance Z,Retimes kT,. The area of the received spectral peak is proportional to the average mode energy Wm, normalized by the geometri- cal antenna plasma coupling. The average mode energy is inequilibrium with both the plasma temperature and the loadtemperature; more precisely, we will see that W mis coupled tokTpat the mode damping rate gmand is coupled to kT,at a load damping rate g,. The plasma impedance Zmcan be obtained directly from the received noise spectra when thereceiver impedance and noise are significant; or it can beobtained by a separate wave reflection/absorption measure-ment; or it can be calculated from a kinetic theory of randomtest particles. Overall, the technique allows a rapid non-destructive diagnostic of the plasma temperature with 625% accuracy. II. EXPERIMENTAL APPARATUSES Fluctuation measurements were obtained from pure elec- tron plasmas contained in two similar Penning–Malmbergtraps ~‘‘IV’’ and ‘‘EV’’ !shown schematically in Fig. 1 ~a!. These two traps differ mainly in plasma diameter and mag-netic field strength. The IV trap consists of a series of hollowconducting cylinders of radius r w52.86 cm contained in ul- trahigh vacuum at P’10210Torr with a uniform axial mag- netic field of B530 kG. Electrons are injected from a hot tungsten filament, and contained axially by voltages Vconf ’2200 V on end electrodes. Typical plasmas have N’109electrons in a column length Lp’41 cm, with a plasma ra- diusrp’0.2 cm and a central density n0’107cm23.~For EV, the parameters are B50.375 kG, rp51.7 cm,rw53.8 cm, andLp524 cm. ! The plasma density profile n(r) and the temperature Tp are obtained by dumping the plasma axially and measuring the total charge passing through a hole in a scanning colli-mator plate. Both measurements require shot-to-shot repro-ducibility of the injected plasma, and we typically observevariability dn/n<1%. On IV, a weak ‘‘rotating wall’’ per- turbation at fRW;0.5 MHz is used to obtain steady-state confinement of the electron column.18The EV plasmas ex- pand radially towards the wall with a characteristic ‘‘mobil-ity’’ time of tm’100 s, so the electrons are repetitively in- jected and dumped. The spectrum analyzer scans analyzedhere typically require about 0.5 s to complete. The parallel temperature T iof the electrons can be mea- sured by slowly lowering the confinement voltage and mea-suring the escaping charge. 19On EV, the perpendicular tem- peratureT’can also be measured using a ‘‘magnetic beach’’ analyzer. In general, we find Ti’T’[Tp, since the electron–electron collision rate n’i’100 s21is relatively rapid.20On EV, the electrons equilibrate to kTp’1 eV soon after injection, whereas the electrons in IV cool to kTp ’0.05 eVdue to cyclotron radiation.To control the tempera- ture, we apply auxiliary ‘‘wiggle’’heating by modulating oneelectrode voltage at a frequency f h50.8–1.0 MHz, where fh is adjusted so that all harmonics are distinct from the TG mode of interest. On the EV apparatus, a ‘‘heating burst’’ isapplied before the measurement; in contrast, on the IV appa-ratus, the heating is applied continuously to balance the cool-ing due to the cyclotron radiation. III. MODE SPECTRA We perform plasma wave transmission experiments by applying an rf voltage of amplitude Vexcto a cylindrical elec- trode at one end of the column. Vexcexcites density pertur- bations dnin the column which propagate and induce the measured voltages Vaon a distant receiving cylindrical an- tenna with finite load impedance. Here, the wave transmis-sion cylinder has length L exc55.8 cm, and the cylinder used as a reception antenna has La511.7 cm.The load impedance on IV ~or EV !isR,5750 V~or 50 V) in parallel with C,5440 pF ~or 165 pF !. Figures 2 ~a!and 2 ~b!show the spectrum of standing mu50 Trivelpiece–Gould modes excited by wall excitations ofVexc5280 dbm ~22mV!and 2100 dBm ~2.2mV!at frequencies f50.01–10 MHz. The frequencies of the Trivelpiece–Gould mode can be approximated as f’fpSrp rwDSrwpmz LpDF1 2lnrw rpG1/2F113 2Sv¯ vfD2G. ~1! The wave frequencies scale with the plasma frequency fp [28 MHz ( n/107cm23)1/2, reduced by the fill ratio rp/rw and by the trap radius compared to the axial wavelength.21 The axial wave number is given by kz5pmz/Lp, where axial mode number is mz51 , 2 ,...,5 .W e have also included thermal corrections, which depend on the ratio of v¯ FIG. 1. ~a!Schematic diagram of a cylindrical Penning–Malmberg trap with the receiving circuit. ~b!Electrical circuit analogue to the plasma mode and receiver.1557 Phys. Plasmas, Vol. 10, No. 5, May 2003 Thermally excited Trivelpiece –Gould mode s... This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.193.164.203 On: Tue, 23 Dec 2014 05:22:47[(kT/m)1/2to the wave phase velocity vf. These modes ex- hibit exponential damping at a rate gtotwhich will be seen to be the sum of gmfrom inherent plasma mode effects such as Landau damping, and g,due to dissipative loading by the receiver. The antenna/mode coupling is not strongly frequency sensitive, but does show the expected sin( kzz) dependence: the lesser sensitivity for mz52 apparent in Fig. 2 is due to the location and length Laof the detection cylindrical an- tenna. The peak amplitudes for the continuously driven sinu-soidal modes are independent of the bandwidth ~BW53 kHz!of the spectrum analyzer; whereas the spectral ampli- tude of the intermode noise decreases as (BW) 21/2as ex- pected.At Vexc52100 dBm, the mode fluctuations have am- plitude dn/n;1025. The peak labeled RW is the nonresonant rotating wall drive; the mode measurements pre-sented here have also been obtained with the drive off, andthere appears to be no significant coupling between the weakRW drive and the weak TG at incommensurate frequencies. IV. EMISSION MEASUREMENTS Small peaks representing thermally excited modes are still visible in Fig. 2 ~c!~‘‘no drive’’ !when the transmitter electrode is grounded ( Vexc50). These peaks have ampli- tudes on the receiving antenna of Va52124 dBm with a receiver bandwidth of 3 kHz, representing voltage fluctua- tions on the electrode with spectral intensity Va/ABW ’2.6nV/AHz. Here, the mode amplitudes scale as BW21/2 as expected for BW,g, since the measured power ( }Va2)i s a fraction BW/gof the total mode power in the antenna circuit. Figure 3 shows received spectra of the thermally excited mz51 mode for four different plasma temperatures. These analogue analyzer scans over Df;100 kHz required about10 seconds to complete on a steady-state plasma confined with a ‘‘rotating wall;’’alternately, digital fast Fourier trans-form ~FFT!spectra could be obtained from 0.1 s of the an- tenna voltage sampled at 10 7samples/s.The mode frequency fmincreases slightly with temperature, as expected from Eq. ~1!. The width of the spectral peak represents the total damp- ing,gtot. This total damping consists of the internal ‘‘mode’’ damping gmand the external ‘‘load’’damping g,: gtot[gm1g,. ~2! Here the internal damping is predominantly Landau damp- ing, but any other internal damping ~e.g., collisional !is also included in gm. The width of the peaks in Fig. 3 increases substantially as Landau damping becomes significant for kTp>0.5 eV, i.e., for vf/v¯&5. For the lowest temperature shown in Fig. 3, a substantial distortion of the Lorentzianplasma mode is observed; this will be seen to represent re-ceiver noise reflected by the plasma. Figure 1 ~b!shows a circuit modeling the reception of thermal noise from the plasma. Near a plasma mode at fre-quency vm52pfmwith intrinsic damping gm, the ratio of antenna current to voltage ~i.e., the admittance !Zm21is given by a simple pole, as Zm215Gvm2 i~v2vm!1gm. ~3! Gis the geometric ~capacitive !coupling coefficient between the plasma mode and the receiving electrode; here G’0.5 pF ~or 0.5 cm in CGS units !. On resonance, the mode imped- ance is real with magnitude Rm[ZmRe~vm!5gm/Gvm2, ~4! whereZRe[Re$Z%andZIm[Im$Z%. The voltage Vmrepre- sents ‘‘white’’ noise from the plasma particles. Nyquist’s theorem predicts Vm5A4kTpRe(Zm)df. The currents flow through a resonant mode impedance Zm, and then through a load impedance Z,with its inevitable white noise. FIG. 2. Spectrum of mu50,mz51,2,...,5T rivelpiece–Gould modes for three drive amplitudes including no drive, i.e., thermally excited. FIG. 3. Spectra of the thermally excited mu50,mz51, mode for different plasma temperatures; the solid lines are fits to Eq. ~7!.The temperature Tpis from emission/reflection measurement.1558 Phys. Plasmas, Vol. 10, No. 5, May 2003 Anderegg et al. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.193.164.203 On: Tue, 23 Dec 2014 05:22:47The external electronics ~load!has measured ~input!re- sistanceR,and capacitance C,, or a total load impedance Z,given by Z,215R,211ivC,, ~5! which is essentially constant over the mode resonance. The load impedance is easily measured with a vector impedancemeter. This almost constant Z ,differs from the resonant- circuit loads commonly used in harmonic traps with a smallnumber of particles, 9simplifying the spectral interpretation somewhat. Nyquist’s theorem says that the spectral density of the square of the noise voltage is proportional to kTtimes the real part of the impedance, for both the plasma and the loadnoise sources.Avoltage-divider fraction Z ,/(Zm1Z,) of the thermal plasma-mode voltage Vmwill be measured on the electrode as Va, together with an analogous fraction Zm/(Zm1Z,) of the ~uncorrelated !load noise V,. This gives Va2~f! df54kTpZmReUZ, Zm1Z,U2 14kT,Z,ReUZm Zm1Z,U2 .~6! The voltage Vmfrom electron thermal motion in the plasma is uncorrelated with the Johnson noise of the load V,. Using Eqs.~2!,~3!, and ~5!, Eq. ~6!can be explicitly written as Va2~f! df54kTpRmuZ,u2 uRm1Z,Reu2gtot2 gtot21~v2vm8!2 14kT,Z,Re 3H122~v2vm8!dvm1~gtot22gm22dvm2! gtot21~v2vm8!2J, ~7! where gtot[gm1g,5S11Z,Re RmDgm, dvm[vm2vm8[Z,Imvm2G. The first term of Eq. ~7!describes the Lorentzian ‘‘plasma’’ emission spectrum centered at vm8of width gtot, with amplitude proportional to kTpRm. Thus, the emission spectrum alone does not determine kTpunless prior knowl- edge of the coupling coefficient GallowsRmto be obtained from Eq. ~4!. The second term describes the ‘‘load’’noise as a uniform background, plus a ‘‘dip and peak’’ from the ( v 2vm8) term, plus a Lorentzian absorption, with all three spectral components proportional to kT,Z,Re. In practice, making the load about as ‘‘hot’’ as the plasma produces theoptimal spectra, allowing a calibrated determination of kT p in one measurement. Figure 4 ~a!shows the received spectrum of the mu50, mz51 mode in the EV apparatus, when the load is much colder than the plasma. The spectrum is completely de-scribed by Eq. ~7!. SinceZ ,is known, the spectrum is pa- rametrized by the plasma temperature Tp, mode frequency vm, mode damping gm, coupling coefficient Gand loadtemperature T,. However, the ‘‘load’’ component is too small for the characteristic dip and peak to give an accuratecalibration of the coupling coefficient G. Using G50.43 pF ~0.39 cm in CGS !from a separate measurement described below, we obtain kT p51.89 eV, fm54.063 MHz, gm/vm 52.131023andkT,50.35 eV. In contrast in Fig. 4 ~b!, noise has been deliberately added to the receiver, corresponding to an effectively higherload temperature. The received spectrum has the sameLorentzian ‘‘plasma’’component, but the dip and peak fromthe plasma ‘‘shorting’’ the load noise is more pronounced.This phase-sensitive reflection and absorption of the loadnoise by the plasma determines the antenna coupling coeffi-cientG. Afive-parameter fit to the received spectrum of Fig. 4~b!then gives kT p51.84 eV, fm54.067 MHz, gm/vm 51.631023,G50.43 pF ~0.39 cm in CGS !, andkT,52.5 eV. The standard dump diagnostic gives kTpdump51.9 eV, with no measurable difference for Figs. 4 ~a!and 4 ~b!. Figure 4 ~b!demonstrates that the plasma temperature can be obtained in one measurement if the load is ‘‘noisyenough.’’ Of course, if the emission from the load were todominate the spectrum, the plasma component proportionaltoT pmight be obscured. For Eq. ~6!to be valid, the load must be uncorrelated with the plasma mode. Using a sweptanalogue analyzer, a sine wave of constant amplitude whichtracks the frequency of the receiver ~from a tracking genera- tor!satisfies the criterion. When digitizing the wave form for FFT analysis, a ‘‘random’’ load current should be added tothe antenna/receiver junction. The damping rate gmis mea- FIG. 4. ~a!Spectra of the thermally excited mu50,mz51 mode for kTp 51.9 eV and kT,50.3 eV. ~b!Same with noise added to the receiver (kT,52.5 eV !. The long-dashed line is Eq. ~7!fitted to the data, the solid line is the plasma component, and the short-dashed line is the load noisefiltered by the plasma.1559 Phys. Plasmas, Vol. 10, No. 5, May 2003 Thermally excited Trivelpiece –Gould mode s... This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.193.164.203 On: Tue, 23 Dec 2014 05:22:47surably smaller when the noise added to the load drives the mode to large amplitude, because the wave traps particles atthe phase velocity, thereby reducing the Landau damping.The thermal component @solid curve in Fig. 4 ~b!#can be viewed as a small test wave in the presence of a larger am-plitude wave excited by the load noise. To establish the load ‘‘temperature’’ range over which the five-parameter fit determines T pcorrectly, we varied T, over the range 0.35 ,T,/Tp,200 with a fixed kTp50.9 eV. Figure 5 indicates that kTpis obtained satisfactorily in one measurement if T,/Tp&20. Thermally excited modes are in equilibrium with both the plasma at temperature Tpand the load at temperature T,. Theory suggests that the plasma mode reaches an equilib-rium average energy W mgiven by 05d dtWm5gm~kTp2Wm!1g,~kT,2Wm!. ~8! If the mode were subject to other energy couplings, these couplings would also appear on the right-hand side of Eq.~8!. The average equilibrium mode energy, given by W m5gmkTp1g,kT, gm1g,, ~9! increases as T,/Tpincreases, i.e., the density fluctuations associated with the mode are increasing in the plasma.Clearly the mode is in thermal equilibrium with the plasmaand with the load. Therefore a high temperature load in-creases the average energy of the mode W m; conversely a well coupled ~i.e., large g,) cold load will reduce Wm. Note that the plasma temperature Tp~i.e., the temperature of the particles !remains unchanged because one mode carries1 2kT of electrostatic energy while the plasma has NtotkTof energy whereNtot.109electrons. Similarly, a feedback circuit can present a ‘‘negative im- pedance’’ coupling to the mode. We have actively dampedthermally excited modes, reducing W mby a factor of 25, and observe no measurable change in Tp.V. THE GEOMETRICAL COUPLING COEFFICIENT One can alternately calculate the geometrical coupling coefficient G, or determine it with a separate reflection/ absorption measurement. As we have seen, the emissionspectrum with a noisy receiver effectively incorporates areflection/absorption measurement. We calculate the coupling coefficient Ganalytically us- ing kinetic theory.Analysis of a uniform density collisionlessplasma of radius r pwithz-periodic boundaries of period Lp reproduces the impedance of Eq. ~3!for frequencies near a plasma resonance. In the limit of T!0, assuming that lD !rpand thatkzrw!1, we find that G54p«0LpFm2 11x2ln2~rv/rp!. ~10! Here Fm[(mzp)21@sin(mzpz2/Lp)2sin(mzpz1/Lp)#, withz1andz2representing the left and right ends of the antenna cylinder; xis a dimensionless quantity that satisfies the equation xJ1(x)ln(rw/rp)5J0(x), and is related to the fre- quency of the plasma mode by x5kzrp(vp2/vm221)1/2. For rv/rp@1, one sees x’A2/ln(rw/rp), which implies G .4p«0LpFm2/@112 ln(rw/rp)#. All equations except Eq. ~10!are valid in CGS or SI; in CGS Eq. ~10!would have no 4p«0. Equation ~10!presumes that the plasma column is pen- etrating the cylindrical antenna completely. When the an-tenna is at the end of the plasma and the plasma only par-tially penetrates the electrode, the plasma end point z pwould replace the electrode end z2in estimating Fm. A more com- plete kinetic analysis of a Maxwellian distribution of ‘‘fullydressed’’ test particles including the plasma dielectricproperties 22reproduces the Nyquist theorem of Eq. ~7!, and also gives the nonresonant Debye-shielded fluctuations. Forexample, each ~axial!half of the plasma has total ~frequency- integrated !nonresonant fluctuations ( dN)25(dq)2/e2 ;O(0.1)( lD/rp)3NforlD&rp, showing a strong reduc- tion below the ( dN)2}Nfluctuations expected for fully un- correlated particles.23 A completely experimental determination of Gcan be obtained with a reflection/absorption measurement,10as shown in Fig. 6 ~a!. The direct measurement of Zmuses a directional coupler and lock-in detector to determine thecomplex reflection coefficient r(f) for a weak wave at fre- quencyfincident on the receiving antenna and plasma. This reflection coefficient is defined as the voltage fraction ~and phase !which is reflected by the plasma-loaded antenna com- pared to that reflected by an open circuit without antenna orplasma, i.e., r[V refl~plasma ! Vrefl~open!. ~11! This reflection coefficient depends on the impedance Ztot connected to the directional coupler compared to the imped- anceZ0550Vof the coupler itself, and is given implicitly by Ztot5Z0~11r! ~12r!. ~12! FIG. 5. Plasma temperature Tpas obtained by emission and average mode energyWmobtained from Eq. ~9!for a plasma temperature of 0.92 eV versus normalized load temperature.1560 Phys. Plasmas, Vol. 10, No. 5, May 2003 Anderegg et al. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.193.164.203 On: Tue, 23 Dec 2014 05:22:47Here,Ztot215Zm211ivC,is the total impedance of the plasma-loaded antenna, given by the plasma impedance Zm of Eq. ~3!in parallel with the capacitance C,of the electrode and connecting cable. Figure 6 ~b!shows an example of the measured ampli- tude and phase of the reflected wave as the frequency isscanned across the m z51 mode.The entire signal is reflected at frequencies ffar from the mode frequency fm, because the plasma impedance @Eq.~3!#is large off-resonance, i.e., uZm(f)u@Z0. On resonance, about 25% of the incident wave is absorbed by the plasma, and 75% is reflected. In essence,the depth of the absorption dip indicates how close R mis to the 50 Vof the directional coupler, since ZmIm(fm)50. A fit ~solid line !to ther(f) data using Eqs. ~3!and~12!gives the parameters of ZtotasRm5329 V(5366310212s/cm in CGS!,fm54.458 MHz, gm/v054.631023, andC,5158 pF, resulting in G5g/(v2R)50.49 pF ~or 0.44 cm in CGS !. Figure 7 shows the coupling coefficient Gis robust, i.e., it changes by less than a factor of 2 when the plasma tem-perature changes from 0.7 eV to 3.5 eV. The simple lowtemperature limit of Eq. ~10!, for EV plasma parameters, givesG50.42 pF, shown with a dashed line. In contrast, the mode impedance R mvaries from 30 Vto 2000 Vin that tem- perature range, as Landau damping increases the modedamping gm. Since Gisalmostconstant,Eq. ~4!predictsthat Rmwill increase as gm/vm2. The dashed line represents Rm 5gLandau/vm2G, where gLandauis calculated from plasma pa- rameters. The small discrepancies may be due to a 20% errorin the temperature calibration of the EV apparatus, or possi-bly finite length plasmas may require a correction to Landaudamping. Figure 8 illustrates that R mis directly related to the mode damping gm/vm; here again the dashed line is the simple theory prediction from G50.42 pF.VI. TEMPERATURE DIAGNOSTIC Figure 9 displays the plasma temperature Tpemissionob- tained from the emission spectra versus the plasma tempera- tureTddumpmeasured by dumping the plasma. Data were taken for plasmas with a range of ‘‘geometric’’ parameters(n,r p,Lp) on both EV ~circles !and IV ~triangles !, with varied amounts of plasma heating. Most of the values of Tpemissionwere obtained from four-parameter fits to the emis- sion spectra, together with a separate measurement of Gfor each (n,rp,L). On IV, the value of Gwas determined from a single five-parameter fit to the non-Lorentzian ‘‘ kT50.06 eV’’ spectrum of Fig. 3, giving G50.21 pF ~0.19 cm in CGS !. Implementation of the ‘‘high temperature load’’ techniquehas obviated the directional coupler reflection measurements,and essentially identical values of T pare obtained with a single spectrum. FIG. 6. ~a!Reflection/absorption electronics. ~b!Measured magnitude and phase of the reflection coefficient r. FIG. 7. Coupling coefficient Gand plasma-mode impedance Rmversus plasma temperature. FIG. 8. Normalized plasma mode impedance versus the plasma-modedamping rate.1561 Phys. Plasmas, Vol. 10, No. 5, May 2003 Thermally excited Trivelpiece –Gould mode s... This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.193.164.203 On: Tue, 23 Dec 2014 05:22:47VII. DISCUSSION It should be emphasized that steady-state plasmas often exhibit spectral peaks which are much larger than thermal, asshown in Figs. 2 ~a!and 2 ~b!, because the mode is being externally driven. More subtly, noise on the ostensibly steadyconfinement voltage V confor other rf signals may stimulate particular plasma modes without proportionately increasingthe plasma temperature. In practice, the standard ~destruc- tive!dump diagnostic is valuable in establishing that the ap- paratus is ‘‘quiet,’’and that the mode couples only to T pand T,. Note that under ‘‘noisy’’ confining voltage conditions, Tpemissiondoes not scale linearly with Tpdump. After carefully eliminating unwanted external damping, the measured total mode damping gtotis consistent with the theoretical perspective of inherent plasma mode damping gm and the receiver load-induced damping g,calculated from the known Z,Re. The analysis presumes that gtot5gm1g,; any unforeseen external damping would be included errone-ously into gm. The smallest mode damping obtained on EV in regimes where Landau damping should be negligiblysmall is gm/vm.731024; presuming this residual damping arises from unknown couplings to external resistances, Eq.~4!sets a limit to the ‘‘unknown’’ coupling coefficient and impedance connected to G uRu,28@pFV#. This represents all electrodes other than the antenna/receiver circuit. In summary, we observe thermally excited plasma modes capacitively coupled to an antenna with a room tem-perature amplifier. Using a generalized version of Nyquist’stheorem, we have shown that one single nondestructive mea-surement can determine the coupling coefficient G, theplasma temperature T pand also the load temperature T,, damping rate gmand mode frequency vm/2p. This single measurement technique is possible because the frequency de-pendences of Z m(v) andZ,(v) are distinguishable. The thermally excited plasma modes are coupled to the theorist’sthermal bath 6only at rate gm, which depends sensitively on temperature; and they are also coupled to the load at tem-peratureT ,at a rate g,, which depends on the antenna cou- pling and receiver parameters. This new technique is now aworking diagnostic that may be useful for ‘‘expensive’’ par-ticles as, for example, antimatter. ACKNOWLEDGMENTS We thank Dr. R. E. Pollock for his seminal contributions and Dr. T. M. O’Neil for many fruitful discussions. This work is supported by Office of Naval Research Grant No. N00014-96-1-0239 and National Science Founda-tion Grant No. PHY-9876999. 1R.G. Greaves and C.M. Surko, Phys. Plasmas 4,1 5 2 8 ~1997!. 2M. Amoretti, C. Amsler, G. Bonomi et al., Nature ~London !419,4 5 6 ~2002!; G. Gabrielse, N.S. Bowden, P. Oxley et al., Phys. Rev. Lett. 89, 213401 ~2002!. 3J.P. Sullivan, S.J. Gilbert, and C.M. Surko, Phys. Rev. Lett. 86, 1494 ~2001!. 4K.H. Knoll, G. Marx, K. Hubner, F. Schweikert, S. Stahl, C.Weber, and G. Werth, Phys. Rev. A 54, 1199 ~1996!. 5T.B. Mitchell, J.J. Bollinger, D.H.E. Dubin, X.-P. Huang, W.M. Itano, and R.H. Baughman, Science 282, 1290 ~1998!. 6T.M. O’Neil and D.H.E. Dubin, Phys. Plasmas 5, 2163 ~1998!. 7D.H.E. Dubin, Phys. Plasmas 5, 1688 ~1998!; F. Anderegg, X.-P. Huang, C.F. Driscoll, E.M. Hollmann, T.M. O’Neil, and D.H.E. Dubin, Phys. Rev.Lett.78,2 1 2 8 ~1997!; D.H.E. Dubin, ibid.79, 2678 ~1997!; E.M. Holl- mann, F. Anderegg, and C.F. Driscoll, ibid.82, 4839 ~1999!. 8S.A. Prasad and T.M. O’Neil, Phys. Fluids 26, 665 ~1983!; A.W. Trivel- piece and R.W. Gould, J. Appl. Phys. 30, 1784 ~1959!. 9D.J. Wineland and H.G. Dehmelt, J. Appl. Phys. 46, 919 ~1975!. 10R.W. Gould, Phys. Plasmas 2, 2151 ~1995!. 11I. Fidone, G. Giruzzi, and G. Taylor, Phys. Plasmas 3, 2331 ~1996!. 12R.L. Stenzel and R.W. Gould, J. Appl. Phys. 42, 4225 ~1971!. 13N. Meyer-Vernet and C. Perche, J. Geophys. Res., @Atmos. #94,2 4 0 5 ~1989!. 14S. Nunomura, J. Goree, S. Hu, X. Wang, and A. Bhattacharjee, Bull. Am. Phys. Soc. 46,2 2~2001!. 15A. Gillespie and F. Raab, Phys. Rev. D 52, 577 ~1995!. 16K.L. Bowles, G.R. Ochs and J.L. Green, J. Res. Natl. Bur. Stand., Sect. D 66D, 395 ~1962!. 17N. Rostoker and M.N. Rosenbluth, Phys. Fluids 3,1~1960!. 18F.Anderegg, E.M. Hollmann, and C.F. Driscoll, Phys. Rev. Lett. 81,4 8 7 5 ~1998!; E.M. Hollmann, F. Anderegg, and C.F. Driscoll, Phys. Plasmas 7, 2776 ~2000!. 19D.L. Eggleston, C.F. Driscoll, B.R. Beck,A.W. Hyatt, and J.H. Malmberg, Phys. Fluids B 4,3 4 3 2 ~1992!. 20B.R. Beck, J. Fajans, and J.H. Malmberg, Phys. Plasmas 3, 1250 ~1996!. 21R.C. Davidson, Physics of Nonneutral Plasmas ~Addison-Wesley, Read- ing, MA, 1990 !, Sec. 5.5.2. 22N. Krall and A.W. Trivelpiece, Principles of Plasma Physics ~McGraw- Hill, New York, 1973 !, Chap. 11. 23N.T. Nakata, G.W. Hart, and B.G. Peterson, in Non-Neutral Plasma Phys- ics IV, edited by F. Anderegg, L. Schweikhard, and C.F. Driscoll ~Ameri- can Institute of Physics, New York, 2002 !. FIG. 9. Plasma temperature measured by emission technique, compared to the standard dump temperature measurement. The triangles are from the IVapparatus and the diamonds are from the EV apparatus.1562 Phys. Plasmas, Vol. 10, No. 5, May 2003 Anderegg et al. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.193.164.203 On: Tue, 23 Dec 2014 05:22:47

1.1639948.pdf

Model of thermal erasure in neighboring tracks during thermomagnetic writing A. Lyberatos and J. Hohlfeld Citation: Journal of Applied Physics 95, 1949 (2004); doi: 10.1063/1.1639948 View online: http://dx.doi.org/10.1063/1.1639948 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/95/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Analysis of write-head synchronization and adjacent track erasure in bit patterned media using a statistical model J. Appl. Phys. 109, 07B755 (2011); 10.1063/1.3562869 Characterization of coercivity at a recording speed of granular media for thermally assisted recording J. Appl. Phys. 109, 07B727 (2011); 10.1063/1.3556698 L10-ordered FePtAg–C granular thin film for thermally assisted magnetic recording media (invited) J. Appl. Phys. 109, 07B703 (2011); 10.1063/1.3536794 Thermal stability of the magnetization following thermomagnetic writing in perpendicular media J. Appl. Phys. 94, 1119 (2003); 10.1063/1.1585118 Influence of write conditions on thermal stability J. Appl. Phys. 91, 7086 (2002); 10.1063/1.1447526 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.105.215.146 On: Thu, 18 Dec 2014 04:14:43Model of thermal erasure in neighboring tracks during thermomagnetic writing A. Lyberatosa)and J. Hohlfeld Seagate Research, 1251 Waterfront Place, Pittsburgh, Pennsylvania 15222 ~Received 14 August 2003; accepted 17 November 2003 ! A method is presented for the calculation of the thermal erasure of the recorded information in neighboring tracks and implemented for heat-assisted magnetic recording on FePt granular thinfilms with high perpendicular anisotropy. The thermal erasure in neighboring tracks decaysexponentially with distance from the recording head. It is strongly dependent on the heating, grainsize, and anisotropy but the write field has a relatively small effect. The track width is dependentprimarilyonthetemperatureprofileofthelaserbeam.Secondaryeffectsarisingfromthewritefield,anisotropy, intergranular magnetostatic, and exchange coupling are also considered.Amore squareshaped temperature profile of the laser beam results in substantial improvement in recordingproperties; enhancement of read signal and sharpness of the magnetic transition and reduction oftransition jitter and lubricant loss. © 2004 American Institute of Physics. @DOI: 10.1063/1.1639948 # I. INTRODUCTION The thermal stability of the magnetization imposes a fundamental limit in current ultrahigh density recordingmedia. 1High areal densities are achieved by scaling to smaller bit and ferromagnetic grain sizes. Using small grains,however, results in signal loss from the thermally inducedtransitions of the magnetization over the energy barriers ofanisotropy. For a single grain in a steady field the switchingprobability is p sw512e2rt, ~1! where the rate rof thermal activation is given by the Arrhenius–Ne ´el law:2 r5t215foe2Eb/kT, ~2! where,fo5109–1011s21is the material specific attempt fre- quency,Ebis the energy barrier, and tis the relaxation time. For 1 Tb/in2recording the grain size is sufficiently small D 54–6 nm to be within the range where coherent magnetic reversal ~strictly valid for ferromagnetic ellipsoids3!is thought to be a good approximation. The intrinsic barrier forcoherent switching in one grain is E b5KuV, whereKuis the constant of the uniaxial anisotropy and Vis the volume of the grain. To increase the areal density, materials with higher an- isotropy are required to maintain adequate thermal stabilityfor a period of 10 years ( K uV/kBT.40).1The available field from recording heads is too small to write information onmedia that support recording densities of several Tbit/in 2. Heat-assisted magnetic recording ~HAMR !has been pro- posed as a solution to the writability problem. By heating themedium with a laser beam, the anisotropy field is reduced tovalues well below the head field. HAMR has the potential ofimproving the areal density by about an order of magnitude larger than conventional magnetic recording. 4 In the presence of a field, the energy barrier of activation may be reduced resulting to accelerated thermal decay of themagnetization.The barrier is also modified by the intergranu-lar magnetostatic interactions. 1,5The thermal erasure of neighboring tracks from the side-fringing field of a recordinghead is of increasing importance.Although the write field onthe adjacent track is small, however, it increases with trackdensity and may result in significant information loss overthe 10 9write cycles that typically occur during the lifetime of a hard disk. An accurate and efficient method to determine the ther- mal erasure in neighboring tracks is desirable. The micro-magnetic method using Langevin dynamics to account forthe thermal agitation of the magnetization 6is a powerful method but cannot be applied since the simulation time in-creases exponentially with barrier height and system size.Furthermore, the switching probability for a single passthrough the fringe field is very small and must be accuratelydetermined by repeating the simulation many times. A simplified approach is to consider the coherent mag- netic switching of a single grain in the polycrystalline re-cording medium. 7,8The effective time of thermal decay in Eq.~1!is usually taken as7–9 t5p/v, ~3! wherepis the downtrack thickness of the write pole for perpendicular writing and vis the relative head-medium ve- locity. This approach has verified the reduction of thermalerasure in the presence of intergranular exchange couplingand has demonstrated a superior thermal stability for tilted,compared with perpendicular media. 8The validity of Eq. ~3!, however, has not been demonstrated and the model also ig-nores the distribution in grain size and anisotropy that is ofimportance in the thermal erasure. a!Electronic mail: andreas.lyberatos@seagate.comJOURNAL OF APPLIED PHYSICS VOLUME 95, NUMBER 4 15 FEBRUARY 2004 1949 0021-8979/2004/95(4)/1949/9/$22.00 © 2004 American Institute of Physics [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.105.215.146 On: Thu, 18 Dec 2014 04:14:43A more accurate calculation considers the variation of the relaxation time twith the position of a grain during medium motion.10In the present study, this method is ex- tended to consider the natural dispersion in magnetic prop-erties and is applied to the thermal erasure of neighboringtracks occurring during thermomagnetic writing on mediawith high perpendicular anisotropy. It is shown that the era-sure is primarily determined by the temperature profile of thelaser beam and not by the write field, as in conventionalmagnetic recording. II. MODEL A. Recording system and medium properties A schematic view of the thermomagnetic recording sys- tem is shown in Fig. 1. The recording head is integrated witha planar light guide on a slider. The perpendicular recordingmedium is first subject to the field from a single pole headand then heated by the laser beam. The thick ~;200 nm ! metallic underlayer for heat sinking is assumed to be an in-finitely permeable soft underlayer ~SUL!. The recording medium is a granular thin film with high perpendicular anisotropy. The temperature dependence of themedium saturation magnetization M sis evaluated using the classical mean field theory.11For FePt media with effective atomic spin S.3/2,12the reduced saturation magnetization of the medium is given by13 m~T!5Ms~T! Ms~0!55 6T TCX51 3F2 tanh ~X!1tanhSX 2DG, ~4! whereTCis the Curie temperature and the parameter Xis proportional to the temperature averaged magnetization. Fora givenT/T C, the value of Xis evaluated from Eq. ~4!and used to deduce m(T). The temperature dependence of the first-order anisotropy constant K1is assumed to be given by K1~T! K1~0!5SMs~T! Ms~0!D2 , ~5! which is consistent with the experimental evidence on epi- taxial Fe 552xNixPt45thin films14for the temperature range of interest @Tamb,TC#, whereTambis the operating ~ambient ! temperature of the disk drive. A theoretical treatment of thetemperature dependence of the magnetic anisotropy, on thebasis of the single-ion model is given in Ref. 13, however, Eq.~5!provides a satisfactory description of the sharp rise of the anisotropy below the Curie temperature, shown in Fig. 2.The second-order anisotropy constant and the shape anisot-ropy of the grains can be ignored 13and we set Ku5K1to simplify the calculation. The granular thin film is modeled as an ensemble of Stoner–Wohlfarth particles of uniform magnetization anduniaxial anisotropy with easy axes aligned in the perpendicu-lar~y!direction. The distribution of particles volumes is as- sumed to be lognormal f ~y!51 sVyA2pe21/2~lny/sV!2, ~6! wherey5V/Vm,Vmis the median volume and sVis the width of the distribution. The distribution function of grain anisotropy fields is assumed to be Gaussian g~hK!51 sKA2pe21/2~12hK/sK!2, ~7! where sKandhK5HK/^HK&are independent of tempera- ture, ^HK&52^Ku&/Msbis the mean anisotropy field and Msbis the ~bulk!saturation magnetization of a grain. The laser beam is assumed to be continuously on, repre- senting the worst case scenario for signal decay on the adja-cent track. The temperature profile is assumed to be of theform T ~x,z!5Tamb1DTmaxe21/2@~x2L!21z2#h/s2h, ~8! whereL5(p1c)/2 is the displacement of the laser beam with respect to the head field ~Fig. 1 !. In most calculations we assume a Gaussian beam ( h51) although a comparison with a super-Gaussian beam ( h52)15will also be made. B. Magnetic reversal probability on the adjacent track We consider a single grain moving through the side fringing field of a recording head. The switching rate rmay be regarded to be constant for a sufficiently small timestepDt. The probability that magnetic reversal does not occur in that timestep is FIG. 1. Schematic representation of the thermomagnetic recording system. FIG. 2. Reduced anisotropy field HK(T)/HK(0)@or medium magnetization Ms(T)/Ms(0)] and gradient dHK/dT@scaled by HK(0)/Tc] as a function of temperature T~scaled by the Curie temperature !.1950 J. Appl. Phys., Vol. 95, No. 4, 15 February 2004 A. Lyberatos and J. Hohlfeld [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.105.215.146 On: Thu, 18 Dec 2014 04:14:43pnsw5e2rDt. ~9! The nonswitching probability following a pass through the fringe field is the product of the nonswitching probabilitiesleading to p nsw5e2*r~t!dt. ~10! The rate ris given by Eq. ~2!if the switching against the field direction is negligible. The variation of the rate withtime arises from the change in temperature and vector fieldon the grain in the adjacent track, as it passes close to thewrite pole. The switching probability can be expressed as p sw512e2vo/ve2jmin, ~11! where j5Eb kBT, ~12! jminis the minimum value of jthrough the fringe field and vo5E 2xsxsfo~t!e2@j~t!2jmin#dx ~13! has the dimensions of velocity and is dependent on the field and temperature profiles. The dependence of voandjon time, arises from the change in the temperature, the magneticproperties M s,HK, and the field during the motion of the grain. The integration is restricted to the range ( 2xs,xs) where the field and temperature profiles enhance the switch-ing probability. The probability of thermally activatedswitching during storage is otherwise neglected. If the switching probability is small p sw!1, the approxi- mation psw.vo ve2jmin ~14! indicates that the switching probability is determined prima- rily by the minimum barrier to thermal energy ratio. Crossing the fringe field is equivalent to applying a steady head for an effective time, determined using Eqs. ~1! and~11!. teff5vo vfo8, ~15! wherefo8is the attempt frequency when j5jmin. The calculation of the barrier will be discussed in detail. The attempt frequency is evaluated assuming intermediate-to-high damping ~IHD!so that the damping constant ain the Gilbert equation of motion of the magnetization satisfies thecondition: 16 aEb kBT.1. ~16! The energy dissipated in one precessional cycle is greater than the thermal energy. The condition ensures a Maxwell–Boltzmann distribution of orientations during motion awayfrom the saddle point. It is satisfied for the neighboring tracksince the switching probability is very small ( E B@kBT). The attempt frequency is then given by16,17fo.Vo 2pvov1, ~17! where v125g2 Msb2c1~1!c2~1!, ~18! vo252g2 Msb2c1~0!c2~0!, ~19! are the squares of the well and saddle angular frequencies, v1is also the ferromagnetic resonance frequency, g5 21.76 rad/Oes is the gyromagnetic ratio and Vo5ag 2Msb~11a2!~2c1~0!2c2~0! 1A~c2~0!2c1~0!!224a22c1~0!c2~0!! ~20! is the saddle angular frequency associated with the hyper- bolic paraboloid approximation at the top of the barrier. The coefficients care determined from the Gibbs free energy approximation about the stationary points: G5Gi11 2@c1~i!~a1~i!!21c2~i!~a2~i!!2#, ~21! wherei50, and 1 denote the saddle and energy minimum, respectively, and a1ianda2iare the direction cosines of the magnetization along unit vectors perpendicular and parallelto the plane defined by the magnetization direction at thestationary points. For a single domain grain, the coefficients care given by 16 c1~i!52Ku@cos2ui1hcos~ui2c!#, ~22! c2~i!52Ku@cos22ui1hcos~ui2c!#, ~23! whereh5Ht/HKis the ~reduced !total field on the grain at an angle cto the easy axis and u0,u1are the orientations of the magnetization to the easy axis at the saddle point andminimum of the energy potential respectively. Equation ~17! is valid if the field angle is in the range 10° , c,90°. If the field is parallel to the easy axis ( c50) the attempt frequency is16,17 fo5agHK 11a2AKuV pkBT~12h2!~12h!. ~24! For intermediate angles 0 ,c,10°, a linear interpolation approximation is used for the relaxation time: t~c!5t~0!1@t~10!2t~0!#c 10. ~25! C. Magnetization of the adjacent track The reduced magnetization I5M/Msat an offtrack po- sitionzon the adjacent track is evaluated as the average magnetization over the volume and anisotropy field distribu-tions1951 J. Appl. Phys., Vol. 95, No. 4, 15 February 2004 A. Lyberatos and J. Hohlfeld [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.105.215.146 On: Thu, 18 Dec 2014 04:14:43I~z,N!E 0‘E 0‘ I~y,hK,z,N!w~y!g~hK!dydhK, ~26! where w~y!5e2sV2/2yf~y!, ~27! andI(y,hK,z,N) is the mean magnetization of grains with volumey, anisotropy hKat distance zafterNcrossings. The magnetization is related to the switching probability: I~y,hK,z,N!5122psw~y,hK,z,N!. ~28! The switching probability after a single crossing for all po- sitionszon the neighboring track is small: psw~y,hK!!1 N~29! and we use the approximation pnsw~y,hK,N!5pnswN~y,hK!. ~30! Using Eqs. ~11!,~26!,~28!, and ~30!, the mean magneti- zation is evaluated numerically by approximating the integralover the distributions f(y),g(h K) as a discrete sum: I~z,N!5( i( lw~yi!g~hK,l! 3DyiDhK~2e2Nvoil/ve2jminil 21!. ~31! The mean magnetization of the neighboring track is av- eraged over the track width w ^I~N!&51 wE w/21g3w/21g I~z,N!dz, ~32! wheregis the guard band. D. Calculation of the barrier height The barrier to magnetic reversal Ebis evaluated using the Pfeiffer approximation for coherent switching:18 Eb~t!>KuVS12uHt~t!u HKs~c!D0.8611.14s~c! , ~33! wheres(c)5(sin2/3c1cos2/3c)23/2andc(t) is the field angle to the easy axis. The total field on the grain is the sumof the head, demagnetizing and exchange fields: H t5Hh1Hd1Hexch. ~34! The field of the write pole with SULwas evaluated using the method of superposition of 3D Lindholm head fields.19The error does not exceed 8% provided that ux2p/2u.S,uz 2b/2u.S, whereS5d1d1sis the spacing between the head and the SUL. The approximation to the field on neigh-boring tracks is therefore satisfactory. The demagnetizing field is evaluated assuming the mag- netization in all tracks is saturated and that the transition atthe track edges is infinitely sharp. The demagnetizing field atany point ~x,z!is evaluated by defining a N3Narray of cells centered as that point with cell size equal to the median grainsize. The size of the array is assumed to be sufficiently largeso that the temperature in the exterior is to a good approxi-mation equal to the ambient temperature. The demagnetizing field on the central ( i th) cell at ~x,z!is perpendicular and can be approximated as Hd,iy5SHp2( jÞiDij@Ms~Tamb!2Ms~Tj!#hj 2DiiMs~Tamb!hiDe, ~35! where the sum runs over the cells jin the array, e51,21i f the magnetization is saturated in the positive or negative ~y! direction, respectively, hjis the fraction of the jthcell within a track ~i.e., not in a guard band !,Dijare the elements of the magnetostatic interaction matrix and Hpis the demagnetiz- ing field averaged through the film thickness arising from aperiodic array of stripe domains parallel to the downtrack ~x! direction that is evaluated using the formalism of Ref. 20: H p524pMs~Tamb!~12a! 18Ms~Tamb!( m.0G~dm/P! msin~pma! 3cosS2pmz8 P2pmaD, ~36! where G~x!512e22px 2px, ~37! P5g1wis the track pitch, a5g/P, andz85z2w/2. The image contribution arising from the presence of the soft un-derlayer and the pole head is ignored to simplify the calcu-lation. The strength of the demagnetizing field is thereforeoverestimated in our model and our calculations represent anupper limit to the thermal decay. The mutual exchange energy of neighbor grains iandj, assuming uniform magnetization within each grain is: 21 Eijexch522C*lijdui.uj, ~38! whereC*is related to the stiffness coefficient across the boundary, lijis the contact length, dis the thickness of the recording layer and ui,ujare unit vectors along the direction of the magnetization of the two grains.The exchange field ongrainifrom its neighbor at site jis H exch,ij521 MsbVi]Eijexch ]ui52dC* MsbVilijuj, ~39! whereViis the volume of the grain at site i. Summing over all neighbor grains with magnetization assumed parallel, thenet exchange field at magnetic saturation is H exch,i54^HK&heAVm Viy, ~40! wherehe5C*/@^Ku&Dm#, andDmis the median grain di- ameter. The exchange field is inversely proportional to thegrain size D. We assume that C *}Ms2, i.e., a similar variation to the exchange stiffness coefficient within the grain.11Our model assumption that Ku}Ms2then implies that the exchange pa-1952 J. Appl. Phys., Vol. 95, No. 4, 15 February 2004 A. Lyberatos and J. Hohlfeld [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.105.215.146 On: Thu, 18 Dec 2014 04:14:43rameterheis independent of temperature. The intergranular exchange field is dependent on the temperature averaged di-rection of the atomic spins at the boundary and is thereforeproportional to M s. E. Model parameters The nominal model parameters, unless specified, are as follows. For the geometry of the recording system: polethickness p5150 nm, pole width b535 nm, length of planar light guide c530 nm, head–medium spacing d55 nm, re- cording layer thickness d512 nm, interlayer thickness s 510 nm, and track pitch P560 nm. The gap field is Ho 510 kOe and the relative head–medium velocity v 510 m/s. We consider the motion of the medium within dis- tancexs55pfrom the center of the write pole. For the re- cording medium magnetic properties at the ambient tempera-ture we use: bulk saturation magnetization of the grains M sb51000 emu/cc, mean anisotropy field ^HK&550 kOe, median grain diameter Dm55 nm, width of volume distribu- tionsV50.3 and width of anisotropy distribution sK50.1. The assumption that each grain reverses magnetically whilethe magnetization of other grains is fixed is justified, if theexchange coupling is weak h e,0.05.21For this reason a valuehe50.01 is used in all calculations. If the grains are cylinders, the stability ratio for the median grain size is ^Ku&Vm/kBT.122. For the temperature profile we use: op- erating temperature of the disk drive Tamb5350 K. FWHM 565 nm and DTmax5600 K. In calculating the switching probability the damping constant is taken as a50.1. III. RESULTS A. Calculation of the track width The calculation of signal decay on neighboring tracks requires some knowledge of the width wof the written track. This allows a more accurate calculation of the demagnetizingfield, which is dependent on the guard band between tracks.The guard band is normally demagnetized since the disk isac erased. A small error in the estimate of the track width,however, is not important for the purpose of our calculation,so a simplified method of calculation is adopted as describedbelow. We first note that a lower bound is imposed by the con- dition that the maximum temperature at the track edgeshould not exceed the Curie temperature. Using Eq. ~8!leads to the following inequality: w.2 sF2l nSDTmax Tc2TambDG1/~2h! . ~41! The track edge is defined as the maximum offtrack spacing z where the lowest value of the barrier Ebo5min@Eb(t)#,(2‘ ,t,‘), during the motion of a grain through the fringe field is zero. The anisotropy of the grain is taken as the meananisotropy. This method assumes spontaneous switching andneglects thermally activated and precessional reversal. The barrierE bois shown in Fig. 3 as a function of offtrack spacing z. The polarity of the demagnetizing field, which in our model is assumed to be of maximum strength, modifies theposition of the track edge. The shift Dz5z22z1of the track edge is the maximum displacement arising from randomvariations in the magnetostatic field from previously writtentransitions. Calculation of the track edge using the Langevinmethod 6to incorporate the superparamagnetic behavior close to the Curie temperature, results only in small increase of thetrack width and requires extensive computer effort. In Fig. 4, we study the factors that determine the width of the track. For a grain at the track edge z5w/2, Fig. 4 ~a! shows the variation of temperature T, stability ratio K uV/kT and barrier height to thermal energy ratio Eb/kTas a func- tion of intrack ~x!position. Figure 4 ~b!shows the respective variation of the head field components and effective strengthof the total field, which is derived from Eq. ~33!as H s5@~Hh,y1Hd,y1Hexch,y!2/31~Hh,x21Hh,z2!1/3#3/2. ~42! Spontaneous switching at zero barrier occurs when tempera- ture attains a maximum value at x.L590 nm. The field Hs is strongest close to the edge of the write pole x.70 nm <p/2, however, the anisotropy at that point is still large. The switching occurs when the anisotropy is minimized and thehead field simply determines the final magnetization direc-tion. Figure 4 ~c!shows the variation of the perpendicular demagnetizing and exchange field in relation to the headfield. The intergranular exchange coupling is significantly re-duced at the switching point since it is dependent on the localsaturation magnetization M sand switching occurs close to the Curie temperature. The demagnetizing field far from thewrite pole is smaller than 4 pMs.12.56 kOe because of the guard band. It is further reduced at the switching point by thethermal reduction of the saturation magnetization, however,since it is a nonlocal field, it does not decrease as sharply asthe exchange field. The maximum jitter Dzin the position of the track edge that may arise from the variation of the magnetostatic field isstrongly dependent on the anisotropy field as shown in Fig. 5~a!. and appears to vary as Dz}H K22. The jitter increases with gap field Ho@Fig. 5 ~b!#. It is not sensitive to the form of FIG. 3. Minimum barrier height to thermal energy ratio Eb/kBTa saf u n c - tion of off-track spacing zfor a single grain crossing the fringe field. The dotted line indicates the position where the maximum temperature duringmotion of the grain is equal to the Curie temperature. The demagnetizingfield, of maximum strength may assist or oppose the head field.1953 J. Appl. Phys., Vol. 95, No. 4, 15 February 2004 A. Lyberatos and J. Hohlfeld [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.105.215.146 On: Thu, 18 Dec 2014 04:14:43the Gaussian beam ( DTmax,FWHM) and the pole dimen- sions. The effect of the jitter on the demagnetizing field andsignal decay is small and can be neglected. The track width is determined primarily by the tempera- ture profile of the laser beam.An explanation is provided byFig. 6. The minimum anisotropy H Kat the switching point rises fast with offtrack distance. Since the track edge is de-termined, to first order, by the condition H K5Ht,y, it occurs close to the point where Tmax5Tc. The sharp rise of the anisotropy of FePt media occurs since the gradient dHK dz5dHK dTdT dz, ~43! is dominated by the temperature gradient dHK/dTclose tothe Curie point. The maximum in dT/dzinstead occurs at a more distant point, at z5s.27.6 nm for the Gaussian tem- perature profile. The variation of the track width with the standard devia- tion of the Gaussian profile is shown in Fig. 7. The trackwidth is primarily determined by the temperature profile andincreases with gap field. The increase is small for high an-isotropy media, as discussed above. The write field in heat-assisted recording must be suffi- ciently strong to overcome any variations in the demagnetiz-ing field during the cooling process. It is sufficient to ensurethe validity of this condition at the track edge. Figure 8shows the variation of the demagnetizing and exchange FIG. 4. For a grain at the track edge ( z524.75 nm), we plot as a function of the position xin the down-track direction: ~a!temperature ~T!, stability ratioKuV/kBTand barrier height to thermal energy ratio Eb/kBT;~b!com- ponents of the head field Hhand the strength of the total field ( Hs);~c! perpendicular components of the head, demagnetizing, and exchange field. FIG. 5. Maximum jitter of the track edge Dz~nm!as a function of ~a! anisotropy field HK~kOe!at the ambient temperature and ~b!the gap field Ho. FIG. 6. Temperature ~T!, temperature gradient ( dT/dz), anisotropy field (HK), and gradients of the anisotropy ( dHK/dT,dHK/dz) as a function of the off-track position z, for the in-track position xwhen the barrier Ebto magnetic reversal attains a minimum value. The dotted line marks the posi-tion where Tis the Curie temperature.1954 J. Appl. Phys., Vol. 95, No. 4, 15 February 2004 A. Lyberatos and J. Hohlfeld [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.105.215.146 On: Thu, 18 Dec 2014 04:14:43fields at the switching point ( x.L,z5w/2) as a function of track width, which varies by changing the FWHM of theGaussian beam.An increase in track width reduces the guardband so the strength of the demagnetizing field is increased.For fixed track width, a reduction in temperature excursionDT maxrequires the use of a wider beam ~FWHM !@for in- stance assume that Tmax5Tcand consider Eq. ~41!#. The heating in the vicinity of the switching point then extendsover a wider area and the reduction of the demagnetizingfield is more pronounced. For our choice of model param-eters, the head field at the switching point must exceed 3.5kOe corresponding to a gap field H o.8.3 kOe. This is a lower limit to the field required to write on the media, sincea stronger field may be required to achieve saturation.AgapfieldH o510 kOe was adopted in most calculations. B. Calculation of the signal decay on the adjacent track To determine the thermal decay of the magnetization us- ing Eq. ~31!we calculate vofor grains of different volume and anisotropy. The variation appears to be approximately of the form vo}HK/AVas is indicated by Fig. 9. The thermal decay of the magnetization in the adjacent track, expressed as a percentage of grains that are magneti-cally switched, is approximately logarithmically dependenton the offtrack distance from the write pole ~Fig. 10 !.W e assumeN510 9crossings through the fringe field. The loga-rithmic variation arises since the minimum in the barrier to thermal energy ratio increases approximately linearly withofftrack spacing. The signal decay is relatively weaker at theedge of the track furthest from the write pole, since the de-magnetizing field is reduced in the vicinity of the guardband. A reduction of the width of the beam reduces heatingon the adjacent track and signal decay.The relative reductionin temperature DT/Tdecreases far from the write pole so the difference in signal decay is smaller.Areduction in the writefield similarily reduces the thermal decay. Figure 11 shows the signal decay in the adjacent track, averaged over the track width, as a function of the grainanisotropy field H Kat the ambient temperature, the FWHM of the Gaussian beam and DTmax. A better thermal stability is achieved by increasing the anisotropy and reducing the FIG. 7. Width ~w!of the written track as a function of the standard deviation of the Gaussian laser beam, for different choice of gap field Ho. The dotted line is obtained assuming the maximum temperature reached is Tcand using Eq.~41!. FIG. 8. Sum of the demagnetizing and exchange fields at the switching point on the track edge ~where the barrier to magnetic reversal vanishes !as a function of track width ~w!. Results are presented for different maximum heating temperatures. FIG. 9. Velocity voevaluated using @Eq.~13!#as a function of ~a!anisot- ropy field HKat ambient temperature scaled by the mean anisotropy and ~b! grain volume ( V) scaled by the median. FIG. 10. Fraction of the magnetization switched in the adjacent track after N5109passes through the fringe field, as a function of the spacing zin the off-track direction. Results are presented for different gap field Hoand width of the Gaussian laser beam.The dotted lines mark the boundary of thetrack.1955 J. Appl. Phys., Vol. 95, No. 4, 15 February 2004 A. Lyberatos and J. Hohlfeld [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.105.215.146 On: Thu, 18 Dec 2014 04:14:43heating. For some fixed offtrack position z, the fraction switched Fis related to minimum barrier height approxi- mately as ln F}2(Eb/kBT)min, and the minimum barrier is approximately proportional to HK(Tamb)@Eq.~33!#. The ob- servation in Fig. 11 that ln F}2HK(Tamb) in the limit of strongHKwhere the approximation Ebo.MsHKV/2 is more appropriate, is therefore not surprising. A logarithmic varia-tion of thermal decay with the FWHM of the Gaussian beamis observed in Fig. 11 ~b!. The deviation observed for a wide beam occurs since the thermal decay is strong and results incomplete demagnetization of the track edge closest to thepole which cannot further contribute to the decay. The signal decay increases as expected with the tempera- ture excursion DT maxas shown in Fig. 11 ~c!. The signal de- cay becomes pronounced when DTmax5525 K. At that point an increase in demagnetizing strength occurs from a ratherabrupt change in track width.This effect appears to be absentwhen the FWHM is varied @Fig. 11 ~b!#. Next we consider the problem of optimization of the temperature profile in thermomagnetic writing. The basic re-quirements are that the written track width is large to in-crease the read signal and that DT maxis small to avoid ex- cessive lubricant loss, subject to the condition that thethermal decay of the magnetization in the adjacent track is tolerable. The retained magnetization after a typical N5109 crossings of the fringe field, over the lifetime of the disk drive must be M>0.9Ms. A larger track width is attained by increasing DTmaxas shown in Fig. 12. Hence, the optimal value DTmaxois deter- mined by the requirement that the retained average magneti-zation in the adjacent track, following N510 9crossings is M50.9Ms. The optimal temperature excursion DTmaxocan be re- duced by increasing the width of the Gaussian beam, asshown in Fig. 13, however, the track width is also reduced.Asignificant improvement in recording properties is attainedby modification of the temperature profile, for instance usinga super-Gaussian beam as shown in Fig. 14. For the same track width, DT maxois substantially reduced ~Fig. 13 !. It is also desirable to reduce transition jitter arising from laser power fluctuations, which normally result in modula-tion of DT max. The requirement is that the temperature gra- dientdT/dzis large at the magnetic transition. Here, we consider the transition at the track edge. The gradient dT/dz for optimal thermal decay ( DTmaxo) is increased when a nar- row beam is used, as shown in Fig. 15. Transition sharpness in conventional magnetic recording is inversely proportional to write field gradient.22In thermo- magnetic writing the effective write field gradient incorpo-rates a term from the temperature variation in anisotropy: 23 FIG. 11. Fraction of the magnetization in the adjacent track after N5109 passes through the fringe field and track width ~w!as a function ~a!the anisotropy field ( HK),~b!the FWHM of the Gaussian beam using DTmax 5500 K, and ~c!the maximum temperature excursion DTmax. FIG. 12. Mean magnetization of the adjacent track after N5109passes through the fringe field and track width ~w!as a function of the temperature excursion DTmax. FIG. 13. Maximum temperature excursion DTmaxofor tolerable signal decay in the adjacent track and track width ~w!as a function of the FWHM of the laser beam. The full/dotted curves are obtained with a Gaussian/super-Gaussian temperature profile.1956 J. Appl. Phys., Vol. 95, No. 4, 15 February 2004 A. Lyberatos and J. Hohlfeld [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.105.215.146 On: Thu, 18 Dec 2014 04:14:43dHheff dx5dHh dx2dHK dTdT dx, ~44! which is normally the dominant term, so transition sharpness is very large, as is confirmed by micromagneticsimulations. 24A measure of the improvement in transition sharpness at the track edge is provided by the ratio(dH K/dz)/(dHh,y/dz). The variation with FWHM is shown in Fig. 15.Transition sharpness is reduced when a low powerwide beam is used.Amore square shaped temperature profilesuch as super-Gaussian results in substantial improvement inboth transition jitter and sharpness. IV. CONCLUSIONS A method to determine signal decay in neighboring tracks from repeated crossing through the side-fringing fieldof a recording head was developed. This is applied to heat-assisted magnetic recording using a single pole head andFePt polycrystalline thin films with high perpendicular an-isotropy. For FePt media the width of the thermomagneti-cally written track is primarily determined by the tempera-ture profile induced by continuous application of the laserbeam, since the magnetic anisotropy rises sharply below the Curie temperature.Arelative small enhancement of the trackwidth occurs by the write field which is inversely propor-tional to the square of the magnetic anisotropy. The probability of thermal magnetic reversal for a single grain on a neighboring track is determined primarily by theminimum value of the barrier to thermal energy ratio incrossing the fringe field and decays exponentially with off-track distance from the write pole, increasing magnetic an-isotropy and decreasing width of the laser beam. For tolerable signal decay in the neighboring track, an increase in maximum heating temperature to improve readsignal, transition jitter, and sharpness is possible if a narrowbeam is used, however, lubricant loss may be unacceptableand a stronger write field must also be applied. A moresquare-shaped temperature profile results in substantial im-provement in recording properties and may allow the use ofa wider beam. The confinement of the beam to the dimen-sions required for 1 Tb/in 2recording is in practice difficult. ACKNOWLEDGMENTS The authors are grateful to K. Yu Guslienko, T. McDaniel, and T. Rausch for helpful discussions. 1D. Weller, A. Moser, L. Folks, M. E. Best, W. Lee, M. F. Toney, M. Schwickert, J.-U. Thiele, and M. F. Doerner, IEEE Trans. Magn. 36,1 0 ~2000!. 2L. Ne´el, Ann. Geophys. ~C.N.R.S. !5,9 9~1949!. 3E. C. Stoner and E. P. Wohlfarth, Philos. Trans. R. Soc. London, Ser. A 240, 599 ~1948!. 4S. Cumpson, P. Hidding, and R. Coehoorn, IEEE Trans. Magn. 36,2 2 7 1 ~2000!. 5T. Shimatsu, H. Uwazumi, Y. Sakai, A. Otsuki, I. Watanabe, H. Muraoka, and Y. Nakamura, IEEE Trans. Magn. 37, 1567 ~2001!. 6E. D. Boerner and H. N. Bertram, IEEE Trans. Magn. 34, 1678 ~1998!. 7K. Z. Gao and H. N. Bertram, IEEE Trans. Magn. 38, 3675 ~2002!. 8K. Z. Gao and H. N. Bertram, IEEE Trans. Magn. 39, 704 ~2003!. 9H. J. Richter, S. Z. Wu, and R. Malmhall, IEEE Trans. Magn. 34, 1540 ~1998!. 10A. Lyberatos, S. Khizroev, and D. Litvinov, Jpn. J.Appl. Phys., Part 1 42, 1598 ~2003!. 11A. H. Morrish, The Physical Principles of Magnetism ~Krieger, Malabar, FL, 1965 !. 12G. H. O. Daalderop, P. J. Kelly, and M. F. H. Schuurmans, Phys. Rev. B 44, 12054 ~1991!. 13A. Lyberatos and K. Yu Guslienko, J. Appl. Phys. 94, 1119 ~2003!. 14J.-U.Thiele, K. R. Coffey, M. F.Toney, J.A. Hedstrom, andA. J. Kellock, J. Appl. Phys. 91, 6595 ~2002!. 15M. Santarsiero and R. Borghi, J. Opt. Soc. Am. A 16,1 8 8 ~1999!. 16W.T. Coffey, D. S. F. Crothers, J. L. Dormann, L. J. Geoghegan, and E. C. Kennedy, Phys. Rev. B 58, 3249 ~1998!. 17W. F. Brown, Jr., Phys. Rev. B 130, 1677 ~1963!. 18H. Pfeiffer, Phys. Status Solidi 118, 295 ~1990!. 19K. Okuda, K. Sueoka, and K. G. Ashar, IEEE Trans. Magn. 24,2 4 7 9 ~1988!. 20M. Mansuripur, in The Physical Principles of Magneto-Optical Recording ~Cambridge University Press, Cambridge, 1995 !, Chap. 13, p. 756. 21H. Zhou, H. N. Bertram, and M. E. Schabes, IEEE Trans. Magn. 38, 1422 ~2002!. 22H. N. Bertram, Theory of Magnetic Recording ~Cambridge University Press, New York, 1994 !. 23T. Rausch, Ph.D. thesis, Carnegie Mellon University, January, 2003. 24J. R. Hoinville, IEEE Trans. Magn. 37,1 2 5 4 ~2001!. FIG. 14. Temperature variation at center of written track ( z50) for a Gaussian ~a!and super-Gaussian ~b!profile, with maximum temperature excursion DTmax5600 K and FWHM 565 nm. FIG. 15. Temperature gradient ( dT/dz) and anisotropy gradient to head field gradient ratio, at the track edge, for optimal temperature excursion (DTmaxo), as a function of FWHM of a Gaussian ~full curves !and super- Gaussian ~dotted curves !beam.1957 J. Appl. Phys., Vol. 95, No. 4, 15 February 2004 A. Lyberatos and J. Hohlfeld [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.105.215.146 On: Thu, 18 Dec 2014 04:14:43

1.4891502.pdf

Remote Walker breakdown and coupling breaking in parallel nanowire systems S. Krishnia, I. Purnama, and W. S. Lew Citation: Applied Physics Letters 105, 042404 (2014); doi: 10.1063/1.4891502 View online: http://dx.doi.org/10.1063/1.4891502 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/105/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Effects of notch shape on the magnetic domain wall motion in nanowires with in-plane or perpendicular magnetic anisotropy J. Appl. Phys. 111, 07D123 (2012); 10.1063/1.3677340 Currentinduced coupled domain wall motions in a twonanowire system Appl. Phys. Lett. 99, 152501 (2011); 10.1063/1.3650706 Relation between critical current of domain wall motion and wire dimension in perpendicularly magnetized Co/Ni nanowires Appl. Phys. Lett. 95, 232504 (2009); 10.1063/1.3271827 Magnetic domain wall collision around the Walker breakdown in ferromagnetic nanowires J. Appl. Phys. 106, 103926 (2009); 10.1063/1.3264642 Spin-current-induced magnetization reversal in magnetic nanowires with constrictions J. Appl. Phys. 97, 10C705 (2005); 10.1063/1.1851434 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 157.211.3.38 On: Sat, 29 Nov 2014 08:47:06Remote Walker breakdown and coupling breaking in parallel nanowire systems S. Krishnia, I. Purnama, and W. S. Lewa) School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371 (Received 21 March 2014; accepted 16 July 2014; published online 29 July 2014) In a multiple nanowire system, we show by micromagnetic simulations that a transverse domain wall in a current-free nanowire can undergo a remote Walker breakdown when it is coupled to a nearby current-driven domain wall. Moreover, for chirality combination with the highest coupling strength, the remote Walker breakdown preceded the current-induced Walker breakdown. TheWalker breakdown limit of such coupled systems has also been shifted towards higher current densities, where beyond these, the coupling is shown to be broken. VC2014 AIP Publishing LLC . [http://dx.doi.org/10.1063/1.4891502 ] The dynamic behavior of a magnetic domain wall (DW) under the influence of electrical current, which was first pro-posed in the mid-90s, 1has been extensively studied in recent years for potential application in non-volatile magnetic solid state memory.2In the design of such memory devices, the magnetic data stored in the nanowires can be shifted to the position of the reading sensor by applying a spin polarized current. Such movement is initiated primarily by the spin-transfer torque phenomenon on the DW. Most studies, henceforth, have been focused on the effort to understand the dynamics of current-driven DW. For instance, it was foundthat the DW propagation velocity can be increased by using modulated nanowire structures 3,4or by injecting high- frequency pulsed current.5,6Nevertheless, the design of high-density DW-based memory devices implies that the nanowires should eventually be placed as close as possible to each other. Therefore, it is important to investigate howthe interaction between DWs in closely spaced nanowires affects the current-driven DW motions. 7,14These DWs have been shown to carry intrinsic magnetic charges based ontheir chiralities and shapes, 8–11and it is possible for them to interact with each other. We found that the interaction between two neighboring DWs that have the opposite mag-netic charges is oscillatory in nature. 12,13It is also possible to make use of the interaction to remotely drive a DW with a fixed chirality in the neighboring nanowire, without thedirect application of current. 7We showed that the remote drive technique can be extended by making use of the inter- nal DW compression force to drive multiple DWs in thecurrent-free nanowire. 7,24However, the understanding of the effect of the chirality combinations of the two DWs on the remote-driving phenomenon is still illusive. In this Letter, we show that it is possible to induce a remote-driving and also a remote-Walker breakdown phe- nomenon in a current-free nanowire by making use of themagnetostatic coupling between DWs in a system of closely spaced ( /C2450 nm) nanowires. Such DWs are also found to be able to retain their fidelity at a higher current density limit.Depending on the chirality combinations of the coupled DWs, it is possible to induce a remote DW structural break-down in the current-free nanowire. The DW dynamics in the closely spaced nanowires are investigated by using the OOMMF 15micromagnetic simula- tion program, with the addition of the spin-transfer torque term to the Landau-Lifshitz-Gilbert (LLG) equation. The chosen Ni 80Fe20material parameters were initially set to: saturation magnetization ( Ms)¼8.6/C2105A/m, exchange stiffness constant ( A)¼1.3/C210/C011J/m, damping constant (a)¼0.01, non-adiabatic spin-torque constant ( b)¼0.04,21,25 and zero magnetocrystalline anisotropy. Each nanowire has a length of 10 lm, width of 100 nm, and thickness of 6 nm. A mesh size of 5 nm /C25n m/C23 nm was used throughout this work. First, the interaction between two head-to-head (HH) transverse DWs in a two-nanowire structure was investi-gated. The DW is classified as a HH when the surrounding magnetic domains points towards it. Fig. 1(a) shows the schematic diagram of the simulation model of the two-nanowire structure (top view), in which a HH DW is nucleated at x¼1lm in the bottom nanowire, and at x¼2lm in the upper nanowire. The separation between the nanowires was maintained at 50 nm. Fig. 1(b) shows all pos- sible four chirality combinations: Up-Down (UD), Up-Up (UU), Down-Down (DD) and Down-Up (DU), with the chir-ality of the DWs in the bottom and the upper nanowires indi- cated by the first and the second letters, respectively. Spin polarized current is then applied to the bottom nanowire todrive the bottom DW (current-driven DW). However, the simulations also show that the upper DW is driven in the same direction as the bottom DW, even though there is nocurrent that is applied to the upper nanowire (remote-driven DW). The motion of the upper DW in the absence of direct current can be attributed to the stray magnetic field interac-tions between the upper and the bottom DWs. 16 The calculated average velocities ( v) of the coupled do- main walls (CDWs) for all possible four chirality combina-tions are shown in Fig. 1(c). The average velocity is calculated by taking the average of the velocity of the remote-driven and current-driven DWs, using the formulaa)Author to whom correspondence should be addressed. Electronic mail: wensiang@ntu.edu.sg 0003-6951/2014/105(4)/042404/4/$30.00 VC2014 AIP Publishing LLC 105, 042404-1APPLIED PHYSICS LETTERS 105, 042404 (2014) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 157.211.3.38 On: Sat, 29 Nov 2014 08:47:06v¼L 2Dmx Dt.22The velocity of CDWs is found to increase line- arly with respect to the applied current density for all chiral-ity combinations. For a fixed current density, the average velocity of the CDWs in the two-nanowire system is approx- imately half of the velocity of a single DW system. The DWs can be depicted as two magnetic charges that are repelling each other but are confined along the length of the nanowires. For a HH transverse DW, the bulk of themagnetic charge is concentrated at the base of its triangular shape. Hence, the strength of the interaction between the two DWs is different depending on the chirality combination.The difference in the interaction strength does not affect the average velocity of the CDWs; however, it directly deter- mines the maximum current density that can be applied tothe system while still maintaining the coupling between the two DWs. As shown in Fig. 1(c), the coupling between the current-driven and remote-driven DWs are broken after acertain threshold current density ( J th), whose value depends on the chirality combination of the two DWs. The threshold current densities ( Jth) for the UD, UU, DD, and DU chirality combinations are 6.73 /C21012, 5.30/C21012, 4.89/C21012, and 3.06/C21012A/m2, respectively. The maximum velocities (vmax) of the coupled DWs for UD, UU, DD, and DU chiral- ity combinations are 604, 495, 459, and 294 m/s, respec- tively. UD has the strongest interaction as well as the highest Jthandvmaxbecause the magnetic charges of the two DWsare located the closest as compared to the other chirality combinations. The coupling strength of UU and DD combi- nations are about equal because the distance between the magnetic charges in both combinations is the same. DU hasthe weakest interaction between the two DWs because the magnetic charges are located the furthest away from each other as shown in the Supplementary material. 23 In general, for all chirality combinations, the coupling between the current-driven DW and the remote-driven DW is broken when the applied current density is increased beyond their respective Jth. However, the coupling breaking process is different depending on the chirality combination.Figs. 2(a)and2(b) show the position of each DW in the sys- tem with respect to the simulation time for J>J th,f o rU U and UD combinations, respectively. The shapes and relativepositions of both the current-driven and remote-driven DWs at different stages of the simulation are also shown. For both UU and DD chirality combinations, the coupling breakingphenomenon is preceded by a change in the shape of the current-driven DW, which is also known as the Walker breakdown phenomenon. 17,18For a single nanowire with a width of 100 nm and thickness of 6 nm, the Walker break- down limit is JWB¼4.28/C21012A/m2. Hence, our results have shown that it is possible to suppress and shift theWalker breakdown limit of a current-driven DW by utilizing FIG. 1. (a) Schematic diagram (top view) of a two-nanowire system employed in our micromagnetic model. The current is applied to the bottom nanowire, while no current is applied to the upper nanowire. (b) Four possi- ble domain wall chirality combinations: UD, UU, DD, and DU. (c) Plot ofvelocities of four possible domain wall chirality combinations as a function of applied current density. Dotted lines represent the threshold current den- sities for all four chirality combinations. FIG. 2. (a) and (b) The position of the domain walls in two different systemsas a function of simulation time for current densities higher than threshold. The domain wall chiralities and relative positions are also shown at different times of simulation. Dotted and solid lines represent the position of remote-driven domain wall and current-driven domain wall, respectively. (a) UU (b) UD, the inset represents the chirality change of DWs with time.042404-2 Krishnia, Purnama, and Lew Appl. Phys. Lett. 105, 042404 (2014) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 157.211.3.38 On: Sat, 29 Nov 2014 08:47:06the two-nanowire system, as the Jthof the UU, DD, and UD chirality combinations are well above JWB[Fig. 1(c)]. For UD chirality combination, the coupling breaking is preceded by two shape-change phenomena, as shown byFig. 2(b). Inset shows how the chiralities of both the current-driven and remote-driven DWs change with respect to the simulation time. The first shape-change occurs at theremote-driven DW (point A in the inset), as the transverse component ( M y) of the remote-driven DW changes from negative to positive. The second shape-change occurs at the current-driven DW (point B in the inset). The shape-change phenomenon at the remote-driven DW is unexpected asthere is no current being applied to the corresponding nano- wire. To substantiate the result, separate simulations on a single nanowire were performed. We found that for a singlenanowire with a width of 100 nm and thickness of 6 nm, Walker breakdown can be achieved by driving the DW up tov WB/C25620 m/s with the application of an external field (HWB)o f1 8G a u s s .I nat w o - n a n o w i r es y s t e m ,t h es t r a y magnetic field from the current-driven DW is responsible for the remote driving of the DW in the current-free nano-wire. It is possible for the stray field from the current- driven DW to exceed H WB19and drive the remote DWs with the speed of >620 m/s, which results in the remote Walker breakdown in the current-free nanowire. The detailed explanation of remote Walker breakdown is described in supplementary material.23 The distance between the current-driven lower DW and the remote-driven upper DW along the x-axis for various applied current densities for UD chirality combination isshown in Fig. 3(a). The interwire separation here is 50 nm, and the applied current density is below the Walker break- down limit. The distance between the two DWs is shown todecrease as the applied current density is increased, which shows that the coupling between the two DWs is Columbic in nature. Fig. 3(b)shows the change in the distance between the two DWs for various interwire separations. The results show that the distance between the two DWs decreases as the interwire separation is increased. However, two differenttrends are observed depending on the proximity between the nanowires. When the two nanowires are placed very close to each other ( d<30 nm), the distance between the two DWs decreases abruptly with applied current density. The differ- ent trends can be attributed to the charge distribution within the DWs. Petit et al. 20have shown that the magnetic charge distribution of a DW has a Gaussian shape: For the HH DW, a strong positive charge is spread out along the base of the triangular shape of the DW, while a weak negative charge isspread out along the apex of the triangular shape. The weak negative charge at the apex of the DW affects the coupling in the two-nanowire system when the interwire spacing issmall ( d<30 nm). Each of the DW then forms a dipole, and the interaction between the two DWs becomes dipole-to- dipole. However, when the interwire spacing between thetwo nanowires is large ( d>30 nm), the presence of the weak negative charge is suppressed, and thus, the interaction between the two DWs becomes charge-to-charge. Thecharge-to-charge interaction is depicted by the gradual change in the DW distance in Fig. 3(b) for interwire separa- tion ( d)>30 nm. The threshold current density is also foundto decrease with increasing nanowire spacing ( d), as shown in Fig. 3(c). Fig. 4shows the comparison between the two- and three-nanowire systems of the strongest coupling strength (UD and UDD, respectively) with a single nanowire system FIG. 3. (a) Distance between adjacent DWs as a function of applied current density showing Columbic force nature of magnetostatic forces. (b) Distance between adjacent DWs against separation between nanowires for different current densities. (c) Threshold current densities as a function of separation between nanowires.042404-3 Krishnia, Purnama, and Lew Appl. Phys. Lett. 105, 042404 (2014) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 157.211.3.38 On: Sat, 29 Nov 2014 08:47:06in terms of the average velocity as a function of the applied current density. For interwire spacing ¼50 nm, the average velocity of the CDWs is found to decrease with the increaseof the total number of DWs in the system. For a particular current density, the velocity of the CDWs in the three- and two-nanowire systems is reduced to about 1/3 and 1/2 ofthe single nanowire system, respectively. The linear reduc- tion in the velocity with the increase in the DWs in the sys- tem is due to an increase in the inertia of the system. 13 However, the maximum threshold current density for the CDWs is found to be increased with the number of the DWs in the system. The threshold current density ofthe three-nanowire system is twice as compared to the sin- gle nanowire system. The interaction between DWs in three-nanowire system and the comparison of it with single-and two-nanowire system are detailed in the supplementary material. 23 In conclusion, we have shown that it is possible to induce a remote Walker breakdown in the current-free DW by exploiting the magnetostatic interaction between DWs in a closely spaced nanowire system. The remoteWalker breakdown phenomenon strongly depends on the chirality combination of both current-driven and remote- driven DWs. The Walker breakdown limit of suchcoupled systems has also been shifted towards higher cur- rent density values. Increasi ng the current density beyond the Walker breakdown limit causes the breaking of themagnetostatic coupling. This is important in the design of high-density memory devices , where the nanowires shall be placed very close to each other. 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Kinane, T. R. Charlton, S. Langridge, A. Potenza, S. S. Dhesi, P. S. Keatley, R. J. Hicken, B. J. Hickey, and C. H. Marrows, Phys. Rev. B 81, 020413(R) (2010). 22D. G. Porter and M. J. Donahue, J. Appl. Phys. 95, 6729 (2004). 23See supplementary material at http://dx.doi.org/10.1063/1.4891502 for: (1) Coupling strength for DU chirality combination. (2) Remote Walker breakdown in current-free nanowire. (3) Interaction between three HH transverse DWs in three-nanowire system. 24I. Purnama, M. C. Sekhar, S. Goolaup, and W. S. Lew, IEEE Trans. Magn. 47, 3081 (2011). 25M. C. Sekhar, S. Goolaup, I. Purnama, and W. S. Lew, J. Appl. Phys. 115, 083913 (2014). FIG. 4. Plot showing the velocity variation of the CDWs in different nano-wire systems as a function of applied current densities.042404-4 Krishnia, Purnama, and Lew Appl. Phys. Lett. 105, 042404 (2014) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 157.211.3.38 On: Sat, 29 Nov 2014 08:47:06

1.4975996.pdf

Rate-dependent extensions of the parametric magneto-dynamic model with magnetic hysteresis S. Steentjes , M. Petrun , G. Glehn , D. Dolinar , and K. Hameyer Citation: AIP Advances 7, 056021 (2017); doi: 10.1063/1.4975996 View online: http://dx.doi.org/10.1063/1.4975996 View Table of Contents: http://aip.scitation.org/toc/adv/7/5 Published by the American Institute of PhysicsAIP ADV ANCES 7, 056021 (2017) Rate-dependent extensions of the parametric magneto-dynamic model with magnetic hysteresis S. Steentjes,1M. Petrun,2G. Glehn,1D. Dolinar,2and K. Hameyer1 1Institute of Electrical Machines, RWTH Aachen University, D-52062 Aachen, Germany 2Institute of Power Engineering, FERI, University of Maribor, SI-2000 Maribor, Slovenia (Presented 3 November 2016; received 23 September 2016; accepted 11 November 2016; published online 8 February 2017) This paper extends the parametric magneto-dynamic model of soft magnetic steel sheets to account for the phase shift between local magnetic flux den- sity and magnetic field strength. This phase shift originates from the damped motion of domain walls and is strongly dependent on the microstructure of the material. In this regard, two different approaches to include the rate-dependent effects are investigated: a purely phenomenological, mathematical approach and a physical-based one. © 2017 Author(s). All article content, except where other- wise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). [http://dx.doi.org/10.1063/1.4975996] I. INTRODUCTION Soft magnetic steel sheets (SMSSs) are due to their technical as well as economical properties indispensable in many contemporary electrical devices. Their widespread use requires adequate modelling of magnetization processes inside SMSSs. Especially for the use in applied engineering, such models should be adequately simple, whereas the magnetization processes should be described as accurate as possible. These two goals are in general very difficult to realize in the modelling process due to complex magnetization processes inside SMSSs. Such processes include hysteresis and non-linear skin effect due to macro- and microscopic eddy currents. Therefore, the accurate description of magnetization processes in SMSSs remains a largely unsolved physical and engineering problem.1 Contemporary models for applied engineering are mostly based on a simplified one-dimensional description that takes into account the macroscopic eddy currents. Such description is suitable for thin and long SMSSs, where the simulation of the magnetization process is reduced to the solution of the well-known diffusion equation. This description links the magnitudes of the magnetic field strength Hand magnetic flux density Bin a material with conductivity . Due to highly non-linear and hysteretic relation between HandB, the discussed description can be solved by applying various approaches, whereas most require spatial discretization of the observed SMSS.2One of the recent approaches is represented by the parametric magneto-dynamic (PMD) model. Using the PMD model in combination with a static, rate-independent hysteresis model the diffusion problem can be solved effectively, whereas the model is based on sound physical background.2–4The PMD is especially convenient when the lamination model has to be incorporated into an electric circuit such as, e.g., for the simulation of dc-dc converters. Both field- and flux-driven versions exist.2–4 However, all approaches that solve the discussed diffusion problem underestimate the magne- tization dynamics and consequently the total power loss, especially when modeling materials with a coarser-grained structure. This underestimation originates from not considering microscopic eddy currents around moving domain walls in the original problem description. These eddy currents can become unacceptably large and lead to a lag in the flux density Bbehind the applied field H.5This phenomenon can be taken into account by extending the PMD model using two different approaches. As the discussed process resembles a viscous-like friction, it can be accounted for in the PMD model by introducing the notion of the “fast” magnetic viscosity similar to the Landau-Lifshitz-Gilbert equation for magnetic viscosity.6,7Alternatively also the rate-dependent model8can be applied. 2158-3226/2017/7(5)/056021/4 7, 056021-1 ©Author(s) 2017 056021-2 Steentjes et al. AIP Advances 7, 056021 (2017) The aim of this paper is to present and analyze both versions of the upgraded PMD model that appear to be more versatile than existing models. Both discussed extended model versions significantly increase the prediction of the dynamic magnetization as well as total losses under arbitrary magnetizing conditions. The model parameters are thought to be material dependent so it is reasoned that the model can be used to accurately predict losses in a wide range of materials magnetized under sinusoidal as well as non-sinusoidal flux waveforms. II. THEORETICAL BACKGROUND The PMD model bases on the average values of magnetic variables inside individual slices (flux tubes) of the SMSS, which allows taking into account the distribution of the induced eddy currents inside all the slices and their influence on the excitation of magnetic field inside the SMSS. The PMD is expressed in the form of a matrix differential equation (1), where represents a vector of the magneto-motive forces generated by the applied current ipin the excitation winding, ¯H¯ is a vector of average magnetic field strengths as hysteretic functions of the average magnetic fluxes in the slices and Nis a vector with the number of turns npof the excitation winding. =Nip=¯H(¯)lm+Lmd¯ dt. (1) The matrix of magnetic inductance Lmdepends only on the geometric, material properties and on the discretization of the observed SMSS, i.e., the number of slices N. Magnetic hysteresis enters into the PMD in the constitutive relation. In this paper the static hysteresis is considered using the Tellinen hysteresis model. The microscopic eddy currents are, however, generated by movement of domain walls when SMSSs are exposed to dynamic magnetizing fields. These currents generate additional losses as well as influence the dynamic magnetization processes inside SMSSs. Additionally, there are a number of other mechanisms responsible for the excess loss. As a result, the observed dynamic hysteresis loops are additionally inflated. Consequently all adequate classical eddy current models typically underestimate the dynamic loops as well as total power loss inside SMSSs. This deficit is therefore addressed using one of the so called excess field extension of the discussed classical eddy current models. A. Magnetic viscosity The lag in the flux density Bbehind the applied field Hcaused by the discussed microscopic phenomena can be effectively solved by adding the so-called magnetic viscosity, described by (2), H(t)=Hh(B)+ Rm 1+B2 B2s!!1dB(t) dt1= (2) where Hh(B) represents the magnetic field strength due to the static hysteresis, d B/dtis the change rate of the magnetic flux density B,is a directional variable, whereas Rm,Bsandare model param- eters.6,7The presented description is similar to the Landau-Lifshitz-Gilbert equation for magnetic viscosity.6,7Main advantages of the proposed model are that it provides an integral description of the complex phenomena of underlying excess loss and does not contradict their complex underlying physics. Furthermore, it is also very flexible and can adequately describe the excess component in non-oriented as well as grain oriented SMSSs. B. Rate-dependent extension The rate-dependent extension was originally developed for homogenized and comprehensive description of all dynamic effects inside SMSSs.9In this way, impacts of both micro- and macroscopic eddy currents as well as other effects were taken into account simultaneously. The dynamic effects are described based on an intuitive differential equation (3), which delays its input (supplementary) variable Hhwith respect to the actual field strength H. The discussed extension depends additionally056021-3 Steentjes et al. AIP Advances 7, 056021 (2017) also on the change rate of the magnetic flux density d B/dt, whereas a,bandcare the model parameters. dHh(t) dt=a(H(t)Hh(t))bdB(t) dt+cdH(t) dt. (3) Regardless the original intention of the presented model, such model can be also used to extend classical eddy current models like the PMD. In this way, the discussed extension is used to describe only the excess component of the field produced, whereas the macroscopic eddy currents are calculated with the PMD. Such an approach is also more consistent with underlying physics and with the commonly accepted loss separation theory of SMSSs. III. RESULTS For this comparison the original voltage-driven PMD model is extended by the two rate-dependent extensions (2) and (3) and implemented using MATLAB/Simulink software. Different versions of the PMD model by applying the discussed rate-dependent extensions are evaluated and compared, where experimental data (measured voltages) are used directly as the PMD model input. The data of the evaluated NO soft magnetic steel sheets M400-50A, the experimental setup and the PMD model are presented in Refs. 3, 4, 9. The classical model without rate-dependent effects is abbreviated in the figures as “cl.”, whereas the viscosity based extension is abbreviated as “cl. + visc.” and the mathematic approach as “cl.+ r.d.” The parameters of the viscous-like friction term in (2) Rmand are identified from the frequency dependence of the excess loss whereas Bsis given by the saturation magnetic polarization of the material. More complicated expressions for (2) can be found in Ref. 6. The parameters of the mathematic model are identified by matching the ascending branch of the major hysteresis loop measured at 50 Hz, i.e., minimizing the least square error. The obtained parameter sets are depicted in Table I. The obtained PMD models are tested for different sinusoidal excitation waveforms for frequen- cies up to f= 1000 Hz and magnetic flux densities up to Bmax= 1:5T. Different rate-dependent models are evaluated by comparing the calculated and measured major and minor dynamic hysteresis loops for the NO steel grade M400-50A. In order to provide a comprehensive analysis, in addition the classical model without inclusion of rate-dependent effects is evaluated. At 50 Hz (Fig. 1 (left)) the effect of magnetic viscosity is not significant. This can be observed comparing the three models. The viscosity-based extension overestimates the loop width, whereas the rate-dependent matches the TABLE I. Parameters of the Rate-Dependent Extensions. Rm  Bsin T Visc. (2) 1 2 1.98 a b c R.d. (3) 5500 64 0.81 FIG. 1. Comparison of measured and modelled hysteresis loops at 50 Hz (left) and 100 Hz (right) for magnetic flux densities of 0.5 T, 1.0 T and 1.5 T in M400-50A.056021-4 Steentjes et al. AIP Advances 7, 056021 (2017) FIG. 2. Comparison of measured and modelled hysteresis loops at 400 Hz (left) and 1000 Hz (right) for magnetic flux densities of 0.5 T, 1.0 T and 1.5 T in M400-50A. measured curve very well with small deviations approaching material saturation. Increasing the frequency to 100 Hz increases the importance of rate-dependent effects. The classical solution under- estimates the loop width, i.e., the energy loss. Again the rate-dependent model (3) describes the hysteresis loop shape accurately. A further increase in frequency leads to an improved accuracy of the viscosity-based model (Fig. 2). In contrast, the mathematical model, which was identified at 50 Hz, obeys a completely different shape at 400 and 1000 Hz near the boots of the hysteresis loops. In general, by introducing a rate-dependent term in the constitutive law of the PMD it is possible to improve the prediction of the loop shape, i.e., energy loss significantly, in particular at higher frequencies. The viscosity-based model allows a good estimation of the loop shape without any additional parameter identification procedure just using the parameters obtained from the classical excess loss theory in combination with the saturation polarization. In contrast, the mathematical model performs better at those frequencies where it was identified. Therefore, it has a smaller predictive value and needs some re-parameterization or dedicated identification scheme. IV. CONCLUSION This paper compares and analyzes two rate-dependent extensions of the PMD model under sinusoidal magnetization waveforms. The models are compared in terms of identification procedure and accuracy of the hysteresis loop shape prediction at frequencies from 50 Hz to 1000 Hz and different magnetic flux densities. The application of the coupled approach (lamination model plus rate-dependent hysteresis model) allows improving the loss calculation as well as the prediction of magnetization dynamics without increasing the computational burden or the need for any additional measurements for parameter identification purposes. 1S. Zirka, Y . Moroz, S. Steentjes, K. Hameyer, K. Chwastek, S. Zurek, and R. Harrison, J. Mag. Mag. Mat. 394(2015). 2M. Petrun, S. Steentjes, K. Hameyer, and D. Dolinar, Magn., IEEE Trans. 52(3) (2016). 3M. Petrun, V . Podlogar, S. Steentjes, K. Hameyer, and D. Dolinar, Magn., IEEE Trans. 50(4) (2014). 4M. Petrun, V . Podlogar, S. Steentjes, K. Hameyer, and D. Dolinar, Magn., IEEE Trans. 50(11) (2014). 5M. Celasco, A. Masoero, P. Mazzetti, and A. Stepanescu, J. Appl. Phys. 57(1985). 6S. E. Zirka, Y . I. Moroz, P. Marketos, and A. J. Moses, Magn., IEEE Trans. 42(9) (2006). 7S. E. Zirka, Y . I. Moroz, P. Marketos, and A. J. Moses, Physica B 343(14) (2004). 8J. F¨uzi, COMPEL 18(3) (1999). 9M. Petrun, S. Steentjes, K. Hameyer, and D. Dolinar, Magn., IEEE Trans. 51(1) (2015).

1.3310017.pdf

Out-of-plane current controlled switching of the fourfold degenerate state of a magnetic vortex in soft magnetic nanodots Youn-Seok Choi, Myoung-Woo Yoo, Ki-Suk Lee, Young-Sang Yu, Hyunsung Jung et al. Citation: Appl. Phys. Lett. 96, 072507 (2010); doi: 10.1063/1.3310017 View online: http://dx.doi.org/10.1063/1.3310017 View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v96/i7 Published by the American Institute of Physics. Additional information on Appl. Phys. Lett. Journal Homepage: http://apl.aip.org/ Journal Information: http://apl.aip.org/about/about_the_journal Top downloads: http://apl.aip.org/features/most_downloaded Information for Authors: http://apl.aip.org/authors Downloaded 20 Mar 2013 to 140.254.87.103. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsOut-of-plane current controlled switching of the fourfold degenerate state of a magnetic vortex in soft magnetic nanodots Youn-Seok Choi, Myoung-Woo Yoo, Ki-Suk Lee, Young-Sang Yu, Hyunsung Jung, and Sang-Koog Kima/H20850 Research Center for Spin Dynamics & Spin-Wave Devices, Nanospinics Laboratory, Research Institute of Advanced Materials, Department of Materials Science and Engineering, Seoul National University,Seoul 151-744, Republic of Korea /H20849Received 10 November 2009; accepted 16 January 2010; published online 17 February 2010 /H20850 We report on an observation of transitions of the fourfold degenerate state of a magnetic vortex in soft magnetic nanodots by micromagnetic numerical calculations. The quaternary vortex states inpatterned magnetic dots were found to be controllable by changing the density of out-of-plane dc orpulse currents applied to the dots. Each vortex state can be switched to any of the other states byapplying different sequence combinations of individual single-step pulse currents. Each step pulse has a characteristic threshold current density and direction. This work offers a promising way formanipulating both the polarization and chirality of magnetic vortices. © 2010 American Institute of Physics ./H20851doi:10.1063/1.3310017 /H20852 The unique magnetization configurations of magnetic vortices in the restricted geometries of magnetic nanodotsof various shapes 1,2are of a growing interest. The vortex states are characterized by the combinations of an in-planecurling magnetization of either counter-clockwise /H20849c=+1 /H20850or clockwise /H20849c=−1 /H20850orientation /H20849denoted by chirality c /H20850along with an out-of-plane core magnetization of either upward /H20849p=+1 /H20850or downward /H20849p=−1 /H20850orientation /H20849represented by polarization p /H20850. In recent years, a dynamic phenomenon of ultrafast switching between the binary pstates has been observed by applications of not only linearly oscillating, 3,4circularly rotating5–7in-plane fields but also their pulse forms8with lower-power consumption. In addition to such field-driven p switching, in-plane ac /H20849Refs. 9and10/H20850and out-of-plane dc currents11,12have been reported to allow for the pswitching, as well. Although there have been extensive studies on only p switching,3–14only c-switching.15,16or switching of flux- closure orientation in nanorings, the reliable control of tran-sitions between the quaternary vortex states /H20849fourfold degen- erate states /H20850has not been explored so far. In this letter, we present the results of micromagnetic numerical simulationsof out-of-plane current driven switching of the fourfold de-generate vortex state. We also report a promising means ofmanipulating individual switching from each of the fourstates to any of the other states and back again, using differ-ent sequence combinations of characteristic single-step pulsecurrents, as found from this study. In the present study, we used Permalloy /H20849Py, Ni 80Fe20/H20850 cylindrical dots of a radius R=100 nm and different thick- nesses, L=10 and 17 nm, as shown in Fig. 1/H20849a/H20850. The initial ground state was either /H20849p,c/H20850=/H20849+1,+1 /H20850or/H20849p,c/H20850=/H20849+1,−1 /H20850. We numerically calculated the magnetization dynamic mo- tions of individual unit cells /H20849size: 2/H110032/H11003Lnm3Ref. 17/H20850in the Py dots using the OOMMF code18that employs the Landau–Liftshitz–Gilbert equation,19including the so-calledSlonczewski spin-transfer torque /H20849STT /H20850.20The STT term is expressed as TSTT=/H20849aSTT /Ms/H20850M/H11003/H20849M/H11003mˆP/H20850with aSTT =1 2/H9266h/H9253Pj0//H20849/H926202eM sL/H20850, where mˆPis the unit vector of spin polarization direction, hthe Plank’s constant, /H9253the gyromag- netic ratio, Pthe degree of spin polarization, j0the current density, /H92620the vacuum permeability, ethe electron charge, andMsthe saturation magnetization. In order to conduct the model study of spin-polarized out-of-plane current-drivenvortex excitations in the free-standing dots, we assumedthat the spin polarization points in the − zdirection, /H20849i.e., S pol=−1 /H20850along with P=0.7 /H20851see Fig. 1/H20849a/H20850/H20852. The current flow was in the + zdirection /H20849denoted as ip=+1 /H20850. The circumfer- ential Oersted fields /H20849OHs /H20850due to the current flow were taken into account using Biot–Savart’s formulation.21In such a free standing model of metallic materials, we did not con- a/H20850Author to whom correspondence should be addressed. Electronic mail: sangkoog@snu.ac.kr. (a) jospol=- 1 2RL xyz 1 1p c=+ =± (b) (c) (d) 02 0 00.4 0.2 0υ(km/s) t(ns) 02 0 4 00.4 0.2 0υ(km/s) t(ns) FIG. 1. /H20849Color online /H20850/H20849a/H20850Schematic illustration of the model geometry. The initial vortex states used are /H20849p,c/H20850=/H20849+1,+1 /H20850and /H20849+1,/H110021/H20850. The directions of the out-of-plane current and the spin polarization of electrons are denoted asi p=+1 and Spol=−1, respectively. The colors and the height of the surface correspond to the in-plane and out-of-plane components of the local mag-netizations, respectively. /H20849b/H20850,/H20849c/H20850, and /H20849d/H20850represent simulation results on a vortex gyrotropic motion, only “ p” switching, and “ p-PLUS- c” switching, respectively. The current densities used are j 0=7.0/H11003106, 2.0/H11003107, and 6.0/H11003107A/cm2for/H20849b/H20850,/H20849c/H20850, and /H20849d/H20850, respectively. In /H20849b/H20850and /H20849c/H20850are also shown the orbital trajectories of motions of the up core /H20849solid line /H20850and the down core /H20849dashed line /H20850and their instantaneous speeds.APPLIED PHYSICS LETTERS 96, 072507 /H208492010 /H20850 0003-6951/2010/96 /H208497/H20850/072507/3/$30.00 © 2010 American Institute of Physics 96, 072507-1 Downloaded 20 Mar 2013 to 140.254.87.103. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionssider thermal heating by current flow because the tempera- ture increase caused by the interface between an electrodeand the nano-pillar is negligible. 22In real systems, however, the other interfaces of sufficiently large contact areas could yield considerable thermal heating in the cases of high cur-rent densities. From simulation results on specific dot dimensions, e.g., /H20849R,L/H20850=/H20849100 nm, 17 nm /H20850and with the indicated initial state, /H20849p,c/H20850=/H20849+1,−1 /H20850, three different characteristic dynamic behaviors are observed: a vortex gyrotropic motion, the switching of palone and the simultaneous switching of both pandc, as represented in Figs. 1/H20849b/H20850–1/H20849d/H20850, respectively. The dynamic behaviors from the /H20849p,c/H20850=/H20849+1,−1 /H20850state contrasted with different values of the current density j 0applied, as shown in Fig. 1. Note that, with this initial vortex state, the circumferential OH orientation is antiparallel with the orien-tation of c=−1, and the orientation of p=+1 is parallel with the current direction, i p=+1. Figure 2diagrams the distinct dynamic behaviors in different regimes of j0. In regime I 0/H20849j0 below a certain critical density, jcri/H20850there is no further vortex excitation. In regime I /H20849j0cri/H11021j0/H11021j0I-II/H20850, we observed a typical low-frequency translation mode, the so-called vortex gyro- tropic motion.21,23–25In this regime, the vortex core exhibits a spirally rotating motion with a continuously increasing or-bital radius, finally attaining the stationary motion of a con-stant orbital radius along with the eigenfrequency, as illus-trated by the orbital trajectory of the vortex core motion andits speed /H20851see Fig. 1/H20849b/H20850/H20852. More detailed descriptions of re- gimes I 0and I are given in Ref. 24. In contrast to the earlier I 0and I regimes, in regime II /H20849j0I-II/H11021j0/H11021j0II-III/H20850, only pswitching occurs, maintaining the initial state of c=−1 /H20849without c-switching /H20850, as shown in Fig.1/H20849c/H20850, after the application of the dc current during the re- quired p-switching time. The switching time, indicated by the solid circles and line in the diagrams, decreases withincreasing j 0. For the lower j0values in regime II, the p-switching takes place via the creation and annihilation of a vortex-antivortex pair inside a given dot, according to thesame mechanism as described in earlier studies. 3,4,8The ve- locity of the up core just before its switching was observed tobe 330 /H1100637 m /s/H20851see Fig. 1/H20849c/H20850/H20852, which is the same as the critical velocity driven by oscillating in-plane and circularrotating fields /H20849or currents /H20850. 4After the core orientation is re- versed, the velocity of the down core decreases rapidly, andthen there is no motion any more. Such out-of-plane dc cur-rent driven pswitching dynamics is analogous to a circular- rotating-field driven core switching reported in Refs. 5–7.O n the contrary, for the higher j 0values in regime II, the p-switching mechanism differed from that observed in the lower j0region. Further investigation of the mechanisms is required.26 With further increases of j0across j0II-III, an additional vortex dynamic was found that represents the switching ofboth pandcwith certain time intervals of the order of ap- proximately nanoseconds /H20849hereafter denoted as “ p-PLUS- c” switching /H20850. In this regime /H20849j 0/H11022j0II-III/H20850, compared with the ear- lier regimes, the j0values are large, producing relatively high-strength OHs, e.g., 1.25 kOe at j0=2.0/H11003108A/cm2 forR=100 nm and L=17 nm. Thus, the high-strength OH assists such “ p-PLUS- c” switching via the reduction of the Zeeman energy through the additional c-switching from the antiparallel to the parallel orientation between the c-state and the given OH. Note that such additional cswitching never happens for the initial state of c·ip=+1 /H20851see Fig. 2/H20849b/H20850/H20852. How- ever, the switching mechanism is not simple but differentfrom the field driven magnetization reversals typically ob-served in ring-type elements. 16Further simulations reveal that the cswitching by only the OH /H20849excluding STT effect /H20850 occurs above j0=1.8/H11003108A/cm2, which value is six times greater than the threshold value, j0II-III=0.3/H11003108A/cm2ob- tained considering both the STT and OH effects.26 Figure 2/H20849b/H20850shows additional switching diagrams for the different initial vortex states and dot size, as noted. Generaltrends of those switching diagrams are the same for the caseofc=−1, but there is no region III in the diagrams for the other case of c=+1. This is because the orientation of the c-state is parallel with that of the given OH, as mentioned before. On the basis of the results above, we propose a promis- ing means by which each of the fourfold ground vortex statescan be simply but reliably manipulated. In Fig. 3, we sche- matically illustrated ten different pulse sequences composedof single-, double- or triple-step pulses. Each current pulse is of either density value, j 0I-II/H11021j0/H11021j0II-IIIorj0/H11022j0II-III, and ei- ther current direction, + zor −z. The lengths of the step pulses are supposed to be at least longer than the switching times,t p/H2084910/H11011250 ns /H20850and tp+c/H20849/H1102130 ns /H20850, required for “ p” and “p-PLUS- c” switchings, respectively. For example, for p switching, single-step pulses with tp,/H20849pulse types ¬and −/H20850, are necessary. Here, the current density and length of thestep pulses were determined by the initial vortex state andthe dot dimensions. Contrastingly, c-state switching can be achieved only by combining the two different processes of“p-PLUS- c” and “ p” switching, in order to return the p-state FIG. 2. /H20849Color online /H20850Switching diagrams of no excitation, vortex gyration, “p” and “ p-PLUS- c” switching with respect to j0, as indicated by regimes, I0, I, II, and III, respectively. The closed symbols indicate the time required for each switching at a given value of j0. The vertical lines represent the boundaries between the different vortex excitations, corresponding to the threshold j0values /H20849j0cri,j0I-II,a n d j0II-III/H20850that distinguish each switching re- gime. The dot size and initial vortex state used are noted in each diagram.The hatched area corresponds to the initial ground state before “ p” and “p-PLUS- c” switching.072507-2 Choi et al. Appl. Phys. Lett. 96, 072507 /H208492010 /H20850 Downloaded 20 Mar 2013 to 140.254.87.103. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsto its original state, since c-switching never occurs alone, but occurs along with pswitching. The pulse types, °,±,², and ³effect c-switching. The pulse sequences for “p-PLUS- c” switching are relatively complex: the single- step pulses of ®and ¯are required for switching between /H20849p,c/H20850=/H20849+1,−1 /H20850, and /H20849/H110021, +1 /H20850, but for switching between /H20849p,c/H20850=/H20849+1,+1 /H20850and /H20849/H110021,/H110021/H20850, the triple-step pulses of ´ and µare necessary. The reason is that the latter switching can occur only through the serial processes of switchingfrom the given /H20849/H110021,/H110021/H20850state to /H20849+1,/H110021/H20850and /H20849/H110021, +1 /H20850, and then to /H20849+1,+1 /H20850, or vice versa. Thus, the pulse sequences ´ and µ, are the results of the combinations of ¬+¯+¬and −+®+ −, respectively. Consequently, each vortex state can be switched to any of the other states directly through twodifferent processes of the “ p” and “ p-PLUS- c” switching or their combinations, by one of the pulse sequences 27indicated by the arrows and the numbers noted in Fig. 3. In summary, we have studied transitions of the fourfold degenerate state /H20849both polarization and chirality /H20850of a vortex in soft magnetic dots, manipulated by changing the densityand direction of out-of-plane dc or pulse currents applied tothe dots. It is proposed that individual switchings from eachvortex state to any of the other states are controllable withthe different sequences of four characteristic single-steppulses found from this study. This work provides a means ofcontrolling the full degree of freedom of the fourfold degen-erate state of a magnetic vortex in confined magnetic dots.This work was supported by Basic Science Research Program through the National Research Foundation of Korea/H20849NRF /H20850funded by the Ministry of Education, Science and Technology /H20849Grant No. 20090063589 /H20850. 1T. Shinjo, T. Okuno, R. Hassdorf, K. Shigeto, and T. Ono, Science 289, 930 /H208492000 /H20850; A. Wachowiak, J. Wiebe, M. Bode, O. Pietzsch, M. Morgen- stern, and R. Wiesendanger, ibid. 298,5 7 7 /H208492002 /H20850. 2L. Thomas, C. Rettner, M. Hayashi, M. G. Samant, S. S. P. Parkin, A. Doran, and A. Scholl, Appl. Phys. Lett. 87, 262501 /H208492005 /H20850; J.-Y. Lee, K.-S. Lee, S. Choi, K. Y. Guslienko, and S.-K. Kim, Phys. Rev. B 76, 184408 /H208492007 /H20850. 3B. Van Waeyenberge, A. Puzic, H. Stoll, K. W. Chou, T. Tyliszczak, R. Hertel, M. Fähnle, H. Brückl, K. Rott, G. Reiss, I. Neudecker, D. Weiss,C. H. Back, and G. Schütz, Nature /H20849London /H20850444,4 6 1 /H208492006 /H20850. 4K.-S. Lee, K. Y. Guslienko, J.-Y. Lee, and S.-K. Kim, Phys. Rev. B 76, 174410 /H208492007 /H20850; K. Y. Guslienko, K.-S. Lee, and S.-K. Kim, Phys. Rev. Lett. 100, 027203 /H208492008 /H20850. 5S.-K. Kim, K.-S. Lee, Y.-S. Yu, and Y.-S. Choi, Appl. Phys. Lett. 92, 022509 /H208492008 /H20850. 6M. Curcic, B. Van Waeyenberge, A. Vansteenkiste, M. Weigand, V. Sack- mann, H. Stoll, M. Fahnle, T. Tyliszczak, G. Woltersdorf, C. H. Back, andG. Schutz, Phys. Rev. Lett. 101, 197204 /H208492008 /H20850. 7K.-S. Lee, S.-K. Kim, Y.-S. Yu, Y.-S. Choi, K. Y., Guslienko, H. Jung, and P. Fischer, Phys. Rev. Lett. 101, 267206 /H208492008 /H20850. 8R. Hertel, S. Gliga, M. Fähnle, and C. M. Schneider, Phys. Rev. Lett. 98, 117201 /H208492007 /H20850. 9K. Yamada, S. Kasai, Y. Nakatani, K. Kobayashi, H. Kohno, A. Thiaville, and T. Ono, Nature Mater. 6, 270 /H208492007 /H20850. 10S.-K. Kim, Y.-S. Choi, K.-S. Lee, K. Y. Guslienko, and D.-E. Jeong, Appl. Phys. Lett. 91, 082506 /H208492007 /H20850. 11J.-G. Caputo, Y. Gaididei, F. G. Mertens, and D. D. Sheka, Phys. Rev. Lett. 98, 056604 /H208492007 /H20850. 12D. D. Sheka, Y. Gaididei, and F. G. Mertens, Appl. Phys. Lett. 91, 082509 /H208492007 /H20850; Y. Liu, H. He, and Z. Zhang, ibid. 91, 242501 /H208492007 /H20850. 13S.-K. Kim, K.-S. Lee, Y.-S. Choi, and Y.-S. Yu, IEEE Trans. Magn. 44, 3071 /H208492008 /H20850. 14S. Bohlens, B. Kruger, A. Drews, M. Bolte, G. Meier, and D. Pfannkuche, Appl. Phys. Lett. 93, 142508 /H208492008 /H20850. 15B. C. Choi, J. Rudge, E. Girgis, J. Kolthammer, Y. K. Hong, and A. Lyle, Appl. Phys. Lett. 91, 022501 /H208492007 /H20850. 16T. Yang, M. Hara, A. Hirohata, T. Kimura, and Y. Otani, Appl. Phys. Lett. 90, 022504 /H208492007 /H20850; M. T. Moneck and J.-G. Zhu, IEEE Trans. Magn. 44, 2500 /H208492008 /H20850. 17We used the dot thickness Lfor the cell dimension along the zdirection. The material parameters corresponding to Py were as follows: the satura-tion magnetization M s=8.6/H11003105A/m, the exchange stiffness Aex=1.3 /H1100310−11J/m, the damping constant /H9251=0.01, and the gyromagnetic ratio /H9253=2.21/H11003105m/As, with a zero magnetocrystalline anisotropy. 18The version of the OOMMF code used is 1.2a4. See http://math.nist.gov/ oommf . 19L. D. Landau and E. M. Lifshitz, Phys. Z. Sowjetunion 8,1 5 3 /H208491935 /H20850;T . L. Gilbert, Phys. Rev. 100, 1243 /H208491955 /H20850. 20J. C. Slonczewski, J. Magn. Magn. Mater. 159,L 1 /H208491996 /H20850. 21Y.-S. Choi, S.-K. Kim, K.-S. Lee, and Y.-S. Yu, Appl. Phys. Lett. 93, 182508 /H208492008 /H20850. 22C.-Y. You, S.-S. Ha, and H.-W. Lee, J. Magn. Magn. Mater. 321,3 5 8 9 /H208492009 /H20850; S.-S. Ha and C.-Y. You, Phys. Status Solidi A 204,3 9 6 6 /H208492007 /H20850. 23B. A. Ivanov and C. E. Zaspel, Phys. Rev. Lett. 99, 247208 /H208492007 /H20850. 24Y.-S. Choi, K.-S. Lee, and S.-K. Kim, Phys. Rev. B 79, 184424 /H208492009 /H20850. 25K. Yu. Guslienko, B. A. Ivanov, V. Novosad, Y. Otani, H. Shima, and K. Fukamichi, J. Appl. Phys. 91, 8037 /H208492002 /H20850. 26K.-S. Lee, M.-W. Yoo, and S.-K. Kim /H20849unpublished /H20850. 27All of the simulation results shown in this paper are specific to the simu- lation conditions described in Fig. 1/H20849a/H20850and the text. The ten types of pulse sequences shown in Fig. 3were also demonstrated for the “ p”a n d “p-PLUS- c” switchings for another conditions while holding p•Spol=−1. With the other conditions for p•Spol=+1, there were neither gyrotropic motion and nor any switching. FIG. 3. /H20849Color online /H20850/H20849a/H20850Current pulse sequences /H20849comprised of single, double, or triple pulses /H20850required for individual switching between each pair of the quadruple vortex states shown in /H20849b/H20850.072507-3 Choi et al. Appl. Phys. Lett. 96, 072507 /H208492010 /H20850 Downloaded 20 Mar 2013 to 140.254.87.103. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissions

1.1421211.pdf

Picosecond large angle reorientation of the magnetization in Ni 81 Fe 19 circular thin- film elements J. Wu, D. S. Schmool, N. D. Hughes, J. R. Moore, and R. J. Hicken Citation: Journal of Applied Physics 91, 278 (2002); doi: 10.1063/1.1421211 View online: http://dx.doi.org/10.1063/1.1421211 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/91/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Influence of a Dy overlayer on the precessional dynamics of a ferromagnetic thin film Appl. Phys. Lett. 102, 062418 (2013); 10.1063/1.4792740 Ferromagnetic microstructured thin films with high complex permeability for microwave applications J. Appl. Phys. 109, 07A323 (2011); 10.1063/1.3560035 Dependence of anisotropy and damping on shape and aspect ratio in micron sized Ni 81 Fe 19 elements J. Appl. Phys. 95, 6998 (2004); 10.1063/1.1687273 Thickness effects on magnetic properties and ferromagnetic resonance in Co–Ni–Fe–N soft magnetic thin films J. Appl. Phys. 91, 8462 (2002); 10.1063/1.1453950 Studies of coupled metallic magnetic thin-film trilayers J. Appl. Phys. 84, 958 (1998); 10.1063/1.368161 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.248.155.225 On: Sun, 23 Nov 2014 05:35:37Picosecond large angle reorientation of the magnetization in Ni 81Fe19 circular thin-film elements J. Wu,a)D. S. Schmool,b)N. D. Hughes, J. R. Moore, and R. J. Hickenc) School of Physics, University of Exeter, Stocker Road, Exeter EX4 4QL, United Kingdom ~Received 28 March 2001; accepted for publication 1 October 2001 ! Large angle picosecond reorientation of the magnetization has been studied in circular Ni 81Fe19 thin-film elements of 30 mm diameter and 500 Å thickness by means of an optical pump–probe technique. The sample was pumped by an optically triggered magnetic field pulse and probed by atimeresolvedmagneto-opticalKerreffectmeasurement.Thetemporalprofileofthepulsedfieldandthe in-plane uniaxial anisotropy of the element were first determined from measurements made inlarge static fields where the magnetization exhibited small amplitude ferromagnetic resonanceoscillations. Measurements of large amplitude oscillations were then made in a smaller static fieldthat was still larger than the in-plane uniaxial anisotropy field and sufficient to saturate the sample.Using the measured temporal profile of the pulsed field, the Landau–Lifshitz–Gilbert equation wasused to model the motion of the magnetization as a coherent rotation process. The same values ofthe anisotropy and damping constants provided an adequate simulation of both the high and lowfield data. The magnetization was found to move through an angle of up to about 30° onsubnanosecond time scales. The dependence of the reorientation upon the direction of the staticapplied field and observed deviations from the coherent precession model are discussed. © 2002 American Institute of Physics. @DOI: 10.1063/1.1421211 # I. INTRODUCTION Fast magnetic reorientation is being intensively studied as higher bit rates are sought in magnetic recording systems.In longitudinal magnetic recording, the alignment of the stor-age medium has relied upon the thermally activated reversalof an ensemble of single domain particles, while the polepieces of the write head and the soft layer of the magnetore-sistive read sensor have been realigned by a combination ofdomain wall motion and domain rotation. However, as theswitching times of these small magnetic structures move intothe picosecond regime, it becomes necessary to rotate all thespins coherently. In this study, we show firstly that thesample anisotropy and the temporal profile of the pulsed fieldmay be characterized by optical pump probe measurements,and secondly, that the same technique may be used to inves-tigate large angle processional motion of the magnetization. In studies of high speed magnetic reorientation, a mag- netic field pulse with a short rise time must be applied to theelement under investigation. This may be achieved by dis-charging a transmission line with a reed switch, 1the pulse duration being controlled by the length of the transmissionline. The sample is placed close to the transmission line andexperiences the magnetic field associated with the transientcurrent. Nanosecond pulses of hundreds of kilo-Oersted canbe generated in microcoils 2while kilo-Oersted pulses a few picoseconds in duration can be obtained from the electronbeam at a synchrotron source. 3Much information about thereorientation process may be obtained by examining the rem- anent state of the sample. However, for a full understanding,it is necessary to observe the full trajectory of the magneti-zation during the reorientation process. The past decade hasseen the development of a number of experimental tech-niques that allow dynamics to be studied in the time domainwith subnanosecond resolution. These include real timeinductive 4and magnetoresistive5measurements. With femto- second pulsed laser sources now commonly available, theoptimum temporal resolution has been obtained frommagneto-optical pump-probe techniques. The optical pumppulse can be used to trigger a high voltage pulse generatorconnected to a transmission line, but this technique generallysuffers from electronic trigger jitter. This can be eliminatedby using a photoconductive switch to directly gate a chargedtransmission line. 6Pulsed fields of the order of 1 kOe may be obtained with a bias voltage of the order of 10 V if the widthof the transmission line tracks is reduced to just a few mi-crons. Both the linear 7and nonlinear8magneto-optical Kerr effect ~MOKE !have been used to probe the magnetization dynamics and information about the motion of the vectormagnetization has been obtained. 9–11The optical technique offers excellent spatial resolution, particularly with the recentdevelopment of near field magneto-optical techniques. 12 The magnetic reorientation process depends firstly upon the strength, shape, and orientation of the field pulse, andsecondly upon magnetic and structural properties of the sample such as shape, magnetic anisotropy, the presence ofexchange bias fields, and the nature of the magnetic damp-ing. Precessional switching was inferred from experiments inwhich pulsed fields as small as 2 kOe and of a few picosec-onds duration were applied within the plane of a thin mag- a!Present address: Department of Physics, University of York, Heslington, York YO10 5DD, United Kingdom. b!Present address: Departamento de Fisica, Faculdade de Cie ˆncias, Univer- sidade do Porto, R. Campo Alegre 687, P-4169-007 Porto, Portugal. c!Electronic mail: r.j.hicken@exeter.ac.ukJOURNAL OF APPLIED PHYSICS VOLUME 91, NUMBER 1 1 JANUARY 2002 278 0021-8979/2002/91(1)/278/9/$19.00 © 2002 American Institute of Physics [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.248.155.225 On: Sun, 23 Nov 2014 05:35:37netic film.13It was found that the pulsed field should be applied orthogonal rather than antiparallel to the initial mag-netization direction so as to exert a larger initial torque per-pendicular to the film plane.As the magnetization tips out ofthe film plane, the thin-film demagnetizing field exerts astrong in-plane torque that reduces the switching time. How-ever device applications require fast reorientation of mag-netic elements in pulsed fields of more modest strength, anda number of studies have been reported in which time re-solved measurements of reorientation have been made after afield pulse has been supplied by a transmission line. 8,11,14 Domain wall processes occur on nanosecond timescales11but these may be suppressed by the application of a bias field.Picosecond switching times may then be observed althoughthe switching may still be incoherent. The rise time of thepulsed field within the magnetic element should be as shortas possible but may be increased by eddy currentshielding 10,15,16within the element that depends upon the di- mensions of the element. The size of the magnetic reorienta-tion is also effected by damping. Ferromagnetic resonance~FMR !experiments made on samples of different thicknesses 17and on individual samples at different frequencies,18have shown that the damping cannot be de- scribed by a single phenomenological constant. Simulationssuggest that a large amplitude uniform precession decaysthrough the generation of spin waves. 19Finally, we note that fast switching requires that the magnetization settles quicklyinto its final orientation. This requires either critical dampingof the precessional motion or careful tailoring of the shape ofthe pulsed field to match the period of precession. 14,20 It is clear that fast magnetic reorientation is a compli- cated process and experimental studies are invaluable in ad-vancing our understanding of this phenomenon. In thepresent work, we study the reorientation of the magnetizationin 30 mm diameter Ni 81Fe19elements that are expected to exhibit qualitatively similar dynamics to the pole pieces in athin-film write head structure, or to the free layer in a spin-valve sensor. Our aim has been first to show that all param- eters affecting the switching can be deduced from pump–probe measurements.These parameter include the anisotropyconstants and damping parameter of the sample, and the tem-poral profile of the pulsed field. We have then used thepump–probe technique to investigate the reorientation be-havior as the relative orientation of a small pulsed field, witha duration of the order of 1 ns, and a static bias field is variedthrough 360°. II. EXPERIMENT The sample chosen for the present study was a circular Ni81Fe19thin-film element of 30 mm diameter and 500 Å thickness produced by a standard photolithographic lift-offtechnique. A row of identical dots was defined on a glasssubstratewithacentertocenterseparationof100 mmchosen to avoid significant dipolar coupling between dots. Photore-sist was first spun onto the substrate and patterned, before theNi 81Fe19was sputtered at an Ar pressure of 5 mTorr in a vacuum chamber with a base pressure of 1 31027Torr. A protective capping layer of 130 Å of Al 2O3was deposited without breaking the vacuum. The sputtering was performedin a bias field of 156 Oe produced by a pair of SmCo 5per- manent magnets placed in close proximity to the substratewhich was at ambient temperature throughout. Finally, theremaining photoresist was removed leaving the row ofNi 81Fe19dots. The basic layout of our optical pump–probe apparatus has been described previously.9The experiment is built around a mode-locked Ti:Sapphire laser that produces 100 fspulses at a repetition rate of 82 MHz. The laser was tuned toa wavelength of 750 nm for the present study. Each pulse issplit into a pump and probe component. The pump is used totrigger a current pulse in the device shown in Fig. 1. Thedevice consists of an interdigitated photoconductive switch,grown on a semi-insulating intrinsic GaAs substrate, that isconnected to a coplanar stripline.The stripline consists ofAustrips of 30 mm width and separation, and is terminated by a 1.5Vsurface mount resistor and a 47 nF surface mount capacitor before being connected to a 20 V power supply.The temporal profile of the current pulse in the transmissionline was monitored with a 500 MHz oscilloscope connectedin parallel with the surface mount resistor. The sample wasplaced face down on the transmission line where it experi-enced the magnetic field associated with the current. Theglass substrate was oriented so that the row of dots made asmall angle with the transmission line ~this angle has been exaggerated in Fig. 1 !. This ensured that one dot lay directly above a coplanar strip, and experienced an in-plane pulsedfield, while another ~not adjacent to the first !lay between the strips and experienced an out-of-plane pulsed field. A time delay was introduced between the pump and probe pulses by reflecting the pump from a retroreflector ona stepper motor driven translation stage. Both beams wereexpanded by a factor of 10 and then focused with achromatsof 15 cm focal length. The pump spot was defocused to adiameter of about 1 mm on the interdigitated structure. Theprobe spot was focused to a diameter of less than 20 mm and FIG. 1. The measurement geometry and an expanded view of the sample position are shown in schematic form.279 J. Appl. Phys., Vol. 91, No. 1, 1 January 2002 Wuet al. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.248.155.225 On: Sun, 23 Nov 2014 05:35:37was incident upon the back of the glass substrate at an angle of about 43°. The spot positions were monitored continu-ously with a charge coupled device camera equipped with azoom lens that gave a 360 on-screen magnification factor. The probe spot was positioned on the center of the chosenelement. Clearly, the large spot size does not allow us toobtain information about the spatial variation of the dynami-cal magnetization. Measurements were made stroboscopi-cally at a fixed time delay and then the delay was changed torecord the magnetic response as a function of the time delay.The probe beam was initially s-polarized and acquired asmall rotation and ellipticity due to the MOKE when re-flected from the sample. An optical bridge was used to mea-sure the Kerr rotation. By chopping the pump beam and us-ing phase sensitive detection, a resolution of close to 1 m degree may be achieved. The device shown in Fig. 1 wasplaced between the pole pieces of an electromagnet so thatthe transmission line was parallel to the plane of incidence.The electromagnet allowed wide field optical access andcould be freely rotated about an axis normal to the sampleplane. III. THEORY The response of the magnetization of the magnetic ele- ment to the pulsed field may be describe by the Landau–Lifshitz–Gilbert equation ]M ]t52uguM3Heff1a MSM3]M ]tD, ~1! in which ais the Gilbert damping constant, Heffis a total effective field acting upon the magnetization and g5g3p 3(2.80) MHz/Oe where gis the spectroscopic splitting fac- tor. In the most general case, the dynamical magnetization isnonuniform and it is necessary to solve Eq. ~1!simulta- neously with the Maxwell’s equations by means of a finiteelement analysis. However, we will use a simpler model inwhich the magnetization of the element is assumed to beuniform and rotates coherently under the influence of thepulsed field. In this case, H effis conveniently evaluated from Heff521 M„uEtot ~2! Etot52M@H1h~t!#2Ku~ukˆ!212pM2uz2, ~3! whereu˜M/Mand the total energy density Etotincludes the Zeeman energy of the magnetization in the static field Hand the pulsed field h(t), a uniaxial anisotropy with in-plane easy axis in the direction parallel to the unit vector kˆ, and the thin-film demagnetizing energy. For a small amplitude pre-cession of the magnetization, we may derive some simplealgebraic results for the precession frequency that we willrequire later in the analysis of our experimental results. Letus first assume that uHuis sufficiently large that h(t) may be neglected when calculating the precession frequency. Byminimizing the free energy in Eq. ~3!, the static orientation ofMis given byMHsin u2Kusin@2~uk2u!#50, ~4! where ukis the angle between Hand the in-plane easy axis, anduis the angle between MandHas shown in Fig. 1. The precession frequency is then given by Sv gD2 5SHcosu12Ku Mcos@2~uk2u!#D 3SHcosu12Ku Mcos2~uk2u!14pMD. ~5! Let us now retain h(t) but instead assume that Mis quasi- aligned with H85H1h(t). In this case, the precession fre- quency is given by Sv gD2 5SH812Ku Mcos2ukDSH812Ku Mcos2uk14pMD ~6! H85~H2cos2wH1~HsinwH2h!2!1/2, ~7! where wHis the angle between Hand the optical plane of incidence as shown in Fig. 1. When the amplitude of preces-sion is not small then Eq. ~1!may be solved numerically if the temporal profile of the pulsed field is known. Since theelement is thicker than the optical skin depth, generalizedFresnel reflection coefficients for the interface between a di-electric and ferromagnetic metal may be used to calculate theinstantaneous Kerr rotation of the probe beam due to a mix-ture of the longitudinal and polar Kerr effects as we havediscussed previously. 15 IV. CHARACTERIZATION OF THE PULSED FIELD It is essential to determine the temporal profile of the pulsed magnetic field before attempting to interpret the re-sults of the pump–probe experiments. The trace obtainedfrom the oscilloscope, which has been plotted in Figs. 2 ~a! and 2 ~b!, gives a general impression of the pulse shape that is useful when aligning the pump beam. The trace showsoscillations with a period of the order of 1 ns that are asso-ciated with reflections of the pulse from the power supply.However, the trace does not reveal the finer features of thepulse shape due to the finite bandwidth ~500 MHz !of the oscilloscope. Also, the height of the trace is not a reliablemeasure of the current amplitude because the small valuesurface mount resistor has considerable inductive impedanceat high frequencies 21that is difficult to characterize. In order to estimate the amplitude of the current, we instead attachedthe oscilloscope across the interdigitated switch and mea-sured the drop in voltage that occurs when the switch isgated.Taking into account the bandwidth of the oscilloscope,we estimate a peak voltage change of about 30% of the biasvoltage. We calculate 22that our transmission line has a char- acteristic impedance of approximately 86 V, which implies a peak current of 70 mA in the transmission line. By integra-tion of the Biot–Savart law we calculate a peak pulsed fieldof about 15 Oe immediately above the tracks of the trans-mission line.280 J. Appl. Phys., Vol. 91, No. 1, 1 January 2002 Wuet al. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.248.155.225 On: Sun, 23 Nov 2014 05:35:37High field FMR measurements can be used as a high bandwidth probe of the temporal profile of pulsed magneticfields. 23When FMR oscillations are induced by a pulsed field applied in the plane of a thin-film sample, the largethin-film demagnetizing field causes the magnetization to behighly elliptical with the long axis of the ellipse lying in thesample plane. However, if the pulsed field is applied perpen-dicular to the film plane, then the trajectory has a more cir-cular shape. 9,16In this case, the measured Kerr rotation is dominated by the polar MOKE signal from the out-of-planecomponent of the dynamical magnetization. The shape of theenvelope of the oscillatory Kerr signal then follows the tem-poral profile of the pulsed field and is relatively insensitive tothe values of the optical constants of the sample. With this inmind, the probe beam was focused on the element betweenthe tracks of the transmission line, where the pulsed field liesperpendicular to the plane of the sample, and relatively largestatic fields were applied in the plane of the sample andperpendicular to the plane of incidence. The Kerr rotationsrecorded for three different static field values are shown inFig. 2 ~c!. The flat trace at negative time delays suggests that resonant pumping from successive laser pulses can be ig-nored.Simulations were performed by solving the Landau– Lifshitz–Gilbert equation numerically and calculating theexpected Kerr rotation. The oscilloscope trace was used as afirst guess of the true pump field profile. The simulations areinsensitive to the in-plane anisotropy field due to the largevalue of the static field, but the values of the spontaneousmagnetization M, thegfactor, and the damping constant a must be adjusted to reproduce the shape of the oscillations. The simulations for the three static field values were com-pared with the experimental traces and the pump field profilewas gradually adjusted until the simulations were in satisfac-tory agreement with the data. The final profile, shown in Fig.2~d!, is qualitatively similar to the original oscilloscope trace but the rise time is now much shorter and there is evidenceof short time scale reflections that arise from impedance mis-matches on the transmission line structure itself. This tempo-ral profile was used in all subsequent calculations. Finally,we note that provisional values of the sample parameters aresufficient for the initial characterization of the pulsed field.These will be discussed again in the next section where thefinal values are determined more accurately. The final valuesforM,K u, andghave been used in the simulations of Fig. 2~c!to ensure self-consistency in the fitting, but a somewhat larger value of a50.02 was required, which we will discuss in the final section of this article. V. SAMPLE CHARACTERIZATION The in-plane anisotropy of the sample was initially stud- ied by longitudinal MOKE measurements at a wavelength of633 nm with a focused spot size of about 20 mm. Hysteresis loops were acquired by probing through the glass substratewith the magnetic field applied at various angles in the planeof the sample. Measurements were first made with an unfo- FIG. 2. ~a!and~b!The pulse shape obtained from the oscilloscope is shown on two different time scales. ~c!Measured ~fine line !and simulated ~bold line!high field FMR oscillations are shown. ~d!The pulsed field profile used to produce the simulations in ~c!is shown. FIG. 3. ~a!Hard and easy axis loops are plotted for the continuous Ni81Fe19 film. ~b!Hard and easy axis loops are plotted for the circular Ni81Fe19 thin-film element.281 J. Appl. Phys., Vol. 91, No. 1, 1 January 2002 Wuet al. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.248.155.225 On: Sun, 23 Nov 2014 05:35:37cused beam on a continuous film of similar thickness. The results are shown in Fig. 3 ~a!and reveal a clear uniaxial character and a hard axis saturation field of about 7 Oe. Theeasy axis direction corresponds to that of the magnetic fieldapplied during the film growth. The loops obtained from thecircular element were more distorted, suggesting that the sig-nal obtained with the focused laser spot is an ill-defined av-erage from a few domains within the element. The domainstructure may also be different to that in the continuous filmsdue to the modified dipolar energy and the introduction ofadditional pinning sites at the edge of the element. In Fig.3~b!, we show loops corresponding to the hard and easy di- rections as determined from pump–probe measurements thatwe discuss in the next paragraph. Clearly, the anisotropy field cannot be easily determined from these hysteresis loops. Pump–probe measurements were next made on the ele- ment centered on the lower track of the transmission linewhere the pulsed field lay in the sample plane, pointing ver-tically downward as shown in Fig. 1.Astatic field of 216 Oewas applied in different directions in the plane of the sample.Measurements were made as the field was rotated through360° in steps of 10°. The measured Kerr rotation has beenplotted in Fig. 4. The static field is sufficiently large that themagnetization is quasi-aligned with the static field. We seethat the amplitude of the Kerr signal is a minimum when thepulsed and static fields are parallel. This is easily understoodsince the pulsed field exerts no torque upon the magnetiza-tion in this case. Fast Fourier transforms were taken of theexperimental data. The power spectra showed single peakswhose positions have been plotted in Fig. 5. Although thereis some scatter in these measured frequencies, it is immedi-ately obvious that the frequencies do not have the 180° pe-riod expected for a sample with uniaxial anisotropy. Thesymmetry is reduced by the presence of the pulsed fieldwhich remains fixed as the static field is rotated and whichhas magnitude of similar order to that of the in-plane anisot-ropy field. Equations ~6!and~7!may be used to describe this variation if some further approximations are made to accountfor the time dependence of the pulsed field. First, we assumethat the effective static orientation of Mis parallel to H rather than the instantaneous total field H 85H1h(t). Sec- ondly, since the magnitude of the pulsed field is time depen-dent, we use an average value of 6.8 Oe determined fromFig. 2.Assuming also a value of g52, Eqs. ~6!and~7!were fitted to the measured frequency values. The fit yielded val-ues ofM5720 emu/cm 3and 2Ku/M512 Oe. The orienta- tion of the easy axis was found to be 20° from the opticalplane of incidence and hence also 20° from the direction ofthe magnetic field applied during the film growth. These pa-rameters values and a damping constant of a50.012 were then used in all subsequent calculations. Simulated Kerr ro-tation traces have been plotted in Fig. 4 and are in reasonablequalitative agreement with the experimental data. We do notexpect to reproduce the amplitude of oscillation correctly because the optical constants of the film are not sufficientlywell known, indeed, we use values reported for pure nickel. 16 The complex optical constants influence the way in whichthe longitudinal and polar Kerr effects combine to give thetotal rotation signal. The longitudinal and polar contributionschange with field azimuth due to changes in both the trajec-tory of the magnetization precession and the optical geom-etry. An incorrect choice of optical constants can lead tosome variation in the relative amplitudes of the simulationsand experimental data for different azimuths in Fig. 4. Fre-quencies obtained from Fourier transforms of the simulationshave also been plotted in Fig. 5 and lie close to the curvecalculated from Eqs. ~4!and~7!. The good agreement be- tween experiment and simulation in Fig. 5 gives additionalcredence to the pulsed field values shown in Fig. 2. Measurements were next made as a function of the static field strength with the static field applied in the plane ofincidence at an angle of wH5180°. The results are shown in FIG. 4. Time dependent Kerr rotation is plotted for measurements in which a static field of 216 Oe was applied in different directions in the sampleplane. The fine and bold curves represent the measured and simulated rota-tion, respectively.282 J. Appl. Phys., Vol. 91, No. 1, 1 January 2002 Wu et al. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.248.155.225 On: Sun, 23 Nov 2014 05:35:37Fig. 6 ~a!. Simulated curves are also shown that are in good qualitative agreement with the experimental data. In eachcase, the equilibrium orientation of the static magnetizationwas calculated from Eq. ~4!and used as the starting point for the time dependent calculation. The frequencies of oscilla-tion of the measured curves were again determined from theFourier power spectra and are plotted in Fig. 6 ~b!with a curve calculated from Eqs. ~4!and~5!. The parameter values assumed in Fig. 5 were used again in Fig. 6 ~b!, and their validity is confirmed by the good agreement between the curve and the measured frequencies. VI. MEASUREMENTS OF LARGE ANGLE REORIENTATION Having characterized the pulsed field and the magnetic properties of the Ni 81Fe19dot immediately above the lower track of the transmission line, the rotation scan of Fig. 4 wasrepeated in a smaller static field of 54 Oe. The measured andsimulated Kerr rotations have been plotted in Fig. 7. Thegeneral trends are the same in the two sets of curves in Fig. FIG. 5. The frequencies of oscillation of the curves in Fig. 4 are plotted as a function of wH. The frequencies of the measured and simulated curves are represented by diamond and circular symbols, respectively. The curve wascalculated from Eqs. ~6!and~7!. FIG. 6. ~a!The measured ~fine line !and simulated ~bold line !Kerr rotation is plotted for different values of the static field for the case that wH5180°. ~b!The FMR frequencies deduced from the experimental curves in ~a!are plotted against the value of the static field. The curve was calculated fromEqs.~4!and~5!. FIG. 7. Time dependent Kerr rotation is plotted for measurements in which a static field of 54 Oe was applied in different directions in the sample plane.The fine and bold curves represent the measured and simulated rotation,respectively.283 J. Appl. Phys., Vol. 91, No. 1, 1 January 2002 Wu et al. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.248.155.225 On: Sun, 23 Nov 2014 05:35:377 but the agreement between experiment and simulation is generally poorer than in Fig. 4. Fourier transforms wereagain taken of both the experimental and simulated Kerr ro-tation curves and the peak frequencies are plotted in Fig. 8.The frequencies of the simulated curves are seen to varythrough a greater range than those of the experimentalcurves. However, it is debatable whether the Fourier trans-form of so few periods of oscillation is really a reliable mea-sure of the average precession frequency. No curve appearsin Fig. 8 because Eqs. ~6!and~7!are no longer valid in the low field limit. If we assume that the simulations do provide an ad- equate description of the dynamics, then we may considerthe trajectory of the magnetization corresponding to eachtrace in Fig. 7. As we mentioned previously, the magnetiza-tion exhibits a very flat precession ( M z!M) due to the in- fluence of the thin-film demagnetizing field and the fact thatthe pulsed field lies in the plane of the sample. We define thein-plane reorientation angle wRas the maximum angular de- viation of the projection of the magnetization upon the x–y plane. The dependence of wRupon wHhas been plotted in Fig. 9 ~a!. The scatter in the graph results from the form of h(t) used in the simulation. The motion is sensitive to the relative phase of the peaks in h(t) and the precession of the magnetization,14,20and this varies in a complicated way as the orientation of the static field is changed.As expected themagnitude of wRis a minimum when wHis close to 90° and 270° because Mandh(t) are almost parallel and so little torque is applied to M. The Kerr rotation and the time de- pendence of the magnetization have been plotted in Figs.9~b!and 9 ~c!, respectively, for the wH540° azimuth ~Hand hare 130° apart !which lies close to the maximum in the wR curve. From Fig. 7, it can also be seen that the simulated and experimental Kerr rotations agree particularly well for thisazimuth. Figure 9 ~c!suggests that a reorientation angle of close to 30° occurs about 350 ps after the application of thepulsed field. VII. DISCUSSION The results presented in the preceding sections demon- strate the utility of the optical pump–probe technique bothfor the investigation of picosecond magnetic processes andthe characterization of basic material properties. High fieldmeasurements can probe the pulsed field profile with a band- width limited by the FMR frequency that is in turn limited bythe magnitude of the static field that may be applied.Aband-width of about 20 GHz was adequate for the present studysince the aim was to characterize the pulse used in measure-ments at much lower static fields. Since the precession fre-quency is only about 2 GHz in these latter measurements, themotion of the magnetization is relatively insensitive to Fou-rier components of the pulsed field with frequency greaterthan 20 GHz.We expect the pulsed field within the sample torise more slowly when the pulsed field is applied perpendicu-lar rather than parallel to the sample plane due to eddy cur-rent shielding. 15,16The rise time of the pulse used in the low field reorientation measurements may be some tens of pico-seconds shorter than that determined in Sec. IV. This mayaffect the phase of the oscillations shown in Figs. 4 and 7,but, due to the complicated shape of the pulse profile, wehave not attempted to account for this effect. We would ex-pect the out-of-plane component of the magnetization M Zto generate eddy currents at the circumference of the circularelement.We have already noted that the precession trajectoryhas a more circular shape and hence a larger M zcomponent when the pulsed field is perpendicular to the sample plane.This may explain why a larger damping constant was re-quired in the characterization of the pulsed field. The in-plane magnetic anisotropy of the Ni 81Fe19ele- ment could not be easily determined from the MOKE hys- FIG. 8. The frequencies of oscillation of the curves in Fig. 8 are plotted as a function of wH. The diamond and circle symbols correspond to the ex- perimental and simulated curves, respectively. FIG. 9. ~a!The maximum angular deviation of the projection of the mag- netization upon the x–yplane, wR, is plotted against the angle wHdefined in Fig. 1. ~b!The experimental ~fine line !and simulated ~bold line !Kerr rotations are plotted for wH540°.~c!The three components of the magne- tization are plotted as a function of time for the wH540° simulation.284 J. Appl. Phys., Vol. 91, No. 1, 1 January 2002 Wuet al. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.248.155.225 On: Sun, 23 Nov 2014 05:35:37teresis loops because the detailed quasi-static micromagnetic reversal process was not known. MOKE microscopy wouldprovide more information but domain images generally re-quire further interpretation before the anisotropy strength canbe deduced. Pump–probe measurements of the frequency ofprecession in a large static applied field are more reliablebecause the sample is always saturated. We note that hyster-esis loops from continuous films grown simultaneously withthe dots may be misleading. We suggest that the photolitho-graphic lift-off technique may lead to strain relief in the cir-cular element that affects a change in the magnetic anisot-ropy. This might explain the observation that the uniaxialanisotropy axis was misaligned from the direction of themagnetic field applied during the sample growth. More de-tailed structural studies are required to confirm this sugges-tion.The inclusion of the average pulsed field in Eqs. ~6!and ~7!for the precession frequency is a simple but crude ap- proximation. In reality, the instantaneous precession fre-quency changes continuously as the magnitude of the pulsedfield changes. However, we have demonstrated that this ap-proach provides a useful check on the calculated pulsed fieldstrength in the present case and we expect that the pulsedfield strength could be determined more accurately for fieldswith simpler temporal profiles such as the step fields that wehave used previously. 9 We have assumed that the pulsed field is uniform across the area of the circular element and that its value is equal tothat calculated at the center of the element. In fact, there issome variation in the magnitude and direction of the pulsedfield across the width of the coplanar stiplines as we havediscussed previously. 16Nevertheless, the coherent rotation model appears to work well for large static fields and goodqualitative agreement is obtained between experiment andsimulation in Fig. 4. While better agreement could be ob-tained through further variation of the sample parameters foreach azimuth, our aim has been to be obtain the best overallagreement with a single set of parameters.Adjustment of theoptical constants might further improve the simulations butwe have not attempted to do this due to the computation timerequired. We also note that any misalignment of thes-polarized beam gives rise to a small p-polarized compo- nent, the intensity of which may change due to the transverseKerr effect, leading to an effective rotation of the total elec-tric field. We would expect this effect to be most obviousaround wH50 and 180° in Fig. 4 where there is a large time dependent component of the magnetization transverse to theplane of incidence. Since the agreement between experimentand simulation is no worse for these azimuths than for othersin Fig. 4, we conclude that this misalignment effect is prob-ably not significant in the present study. We expect the magnetization to reorient by a rotation process, even in the smaller static field used in Fig. 7, be-cause the sum of the static and pulsed fields H 8is always larger than the anisotropy field. The static field is essentialfor our stroboscopic measurement technique since it returnsthe system to its initial state after each pulse has been ap-plied. However, this case is also of technological interestsince a bias field may be built into a thin-film structure bymeans of interlayer exchange coupling to a high coercivityunderlayer or by direct exchange coupling to an antiferro- magnet. The disagreement between simulation and experi-ment for some values of wHmay indicate that the rotation process is becoming incoherent. The good agreement be-tween experiment and simulation in Fig. 9 ~b!suggests that the magnetization can be switched through an angle of up to30° by means of a rotation process. The maximum switchingangle occurred when wH550° so that handHwere 140° apart. An angle of 90° is expected to be best for very shortpulsed fields since then the maximum torque is applied to themagnetization before it reorients significantly. In the presentcase, the pulsed field has longer duration and a somewhatdifferent orientation of the pulsed field is required so that themaximum integrated torque is obtained during the reorienta-tion process. The damping constant awas found to be simi- lar for both the small and large amplitude motion of Figs. 4and 7, suggesting that the uniform precession is not stronglydamped by spin wave generation under the present condi-tions. In future studies, it would be interesting to explorehow well the coherent model applies and whether the valueof the damping parameter remains unchanged as the value ofthe static field is reduced further. In conclusion, optical pump–probe experiments have shown that reorientation through angles of up to 30° may beachieved in times as short as 350 ps in Ni 81Fe19thin-film elements. The motion is well described by a coherent rota-tion process. However, to understand this behavior, it is nec-essary to perform a thorough characterization of the mag-netic properties of the element and determine the temporalprofile of the pulsed magnetic field.We have shown how thismay be achieved with pump–probe measurements made inlarger static applied fields on the same sample. Larger pulsedmagnetic field are now required to extend these measure-ments to a wider range of magnetic materials and improvedspatial resolution is required so that the coherence of theswitching may be explored in greater detail. ACKNOWLEDGMENT The authors gratefully acknowledge the financial support of the United Kingdom Engineering and Physical SciencesResearch Council ~EPSRC !. 1R. P. Cowburn, J. Ferre ´, S. J. Gray, and J. A. C. Bland, Phys. Rev. B 58, 11507 ~1998!. 2K. Mackay, M. Bonfim, D. Givord, and A. Fontaine, J. Appl. Phys. 87, 1996 ~2000!. 3C. H. Back, D.Weller, J. Heidmann, D. Mauri, D. Guarisco, E. L. Garwin, and H. C. Siegmann, Phys. Rev. Lett. 81, 3251 ~1998!. 4T. J. Silva, C. S. Lee,T. M. Crawford, and C.T. Rogers, J.Appl. Phys. 85, 7849 ~1999!. 5R. H. Koch, J. G. Deak, D. W. Abraham, P. L. Trouilloud, R. A. Altman, Y. Lu, W. J. Gallagher, R. E. Scheuerlein, K. P. Roche, and S. S. P. Parkin,Phys. Rev. Lett. 81, 4512 ~1998!. 6M. R. Freeman, R. R. Ruf, and R. J. Gambino, IEEE Trans. Magn. 27, 4840 ~1991!. 7W. K. Hiebert, A. Stankiewicz, and M. R. Freeman, Phys. Rev. Lett. 79, 1134 ~1997!. 8T. M. Crawford, T. J. Silva, C. W. Teplin, and C. T. Rogers, Appl. Phys. Lett.74, 3386 ~1999!. 9R. J. Hicken and J. Wu, J. Appl. Phys. 85, 4580 ~1999!. 10Y.Acreman, C. H. Back, M. Buess, O. Portmann,A. Vaterlaus, D. Pescia, and H. Melchior, Science 290, 492 ~2000!.285 J. Appl. Phys., Vol. 91, No. 1, 1 January 2002 Wuet al. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.248.155.225 On: Sun, 23 Nov 2014 05:35:3711B. C. Choi, M. Belov, W. K. Hiebert, G. E. Ballentine, and M. R. Free- man, Phys. Rev. Lett. 86, 728 ~2001!. 12M. R. Freeman,A.Y. Elezzabi, and J.A. H. Stotz, J.Appl. Phys. 81, 4516 ~1997!. 13C. H. Back, R. Allenspach, W. Weber, S. S. P. Parkin, D. Weller, E. L. Garwin, and H. C. Siegmann, Science 285, 864 ~1999!. 14M. Bauer, R. Lopusnik, J. Fassbender, and B. Hillebrands, Appl. Phys. Lett.76,2 7 5 8 ~2000!. 15J. Wu, J. R. Moore, and R. J. Hicken, J. Magn. Magn. Mater. 222, 189 ~2000!. 16J. Wu, N. D. Hughes, J. R. Moore, and R. J. Hicken, J. Magn. Magn. Mater. ~in press !.17A. Azevedo, A. B. Oliveira, F. M. de Aguiar, and S. M. Rezende, Phys. Rev. B62, 5331 ~2000!. 18J. F. Cochran, R. W. Qiao, and B. Heinrich, Phys. Rev. B 39, 4399 ~1989!. 19E. D. Boerner, H. N. Bertram, and H. Suhl, J.Appl. Phys. 87, 5389 ~2000!. 20T. M. Crawford, P. Kabos, and T. J. Silva, Appl. Phys. Lett. 76,2 1 1 3 ~2000!. 21Philips Components Application Note, ‘‘Space-saving 4-resistor arrays in 0804 package.’’ 22K. C. Gupta, R. Garg, and I. J. Bahl, Microstrip Lines and Slotlines ~Artech House, Boston, 1979 !. 23A. Y. Elezzabi and M. R. Freeman, Appl. Phys. Lett. 68, 3546 ~1996!.286 J. Appl. Phys., Vol. 91, No. 1, 1 January 2002 Wuet al. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.248.155.225 On: Sun, 23 Nov 2014 05:35:37

1.5143382.pdf

AIP Advances 10, 025116 (2020); https://doi.org/10.1063/1.5143382 10, 025116 © 2020 Author(s).Recurrent neural networks made of magnetic tunnel junctions Cite as: AIP Advances 10, 025116 (2020); https://doi.org/10.1063/1.5143382 Submitted: 23 December 2019 . Accepted: 25 January 2020 . Published Online: 11 February 2020 Qi Zheng , Xiaorui Zhu , Yuanyuan Mi , Zhe Yuan , and Ke Xia AIP Advances ARTICLE scitation.org/journal/adv Recurrent neural networks made of magnetic tunnel junctions Cite as: AIP Advances 10, 025116 (2020); doi: 10.1063/1.5143382 Submitted: 23 December 2019 •Accepted: 25 January 2020 • Published Online: 11 February 2020 Qi Zheng,1Xiaorui Zhu,1Yuanyuan Mi,2Zhe Yuan,1,3,a) and Ke Xia1,3,4 AFFILIATIONS 1Center for Advanced Quantum Studies and Department of Physics, Beijing Normal University, Beijing 100875, China 2Center for Neurointelligence, Chongqing University, Chongqing 400044, China 3Center for Quantum Computing, Peng Cheng Laboratory, Shenzhen 518005, China 4Shenzhen Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China a)Author to whom correspondence should be addressed: zyuan@bnu.edu.cn ABSTRACT Artificial intelligence based on artificial neural networks, which are originally inspired by the biological architectures of the human brain, has mostly been realized using software but executed on conventional von Neumann computers, where the so-called von Neumann bottle- neck essentially limits the executive efficiency due to the separate computing and storage units. Therefore, a suitable hardware platform that can exploit all the advantages of brain-inspired computing is highly desirable. Based upon micromagnetic simulation of the magnetization dynamics, we demonstrate theoretically and numerically that recurrent neural networks consisting of as few as 40 magnetic tunnel junctions can generate and recognize periodic time series after they are trained with an efficient algorithm. ©2020 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/1.5143382 .,s In the past decade, significant progress has been made in arti- ficial intelligence, where advanced algorithms using artificial neural networks (ANNs) have been successfully applied in image recogni- tion, data classification, and other areas.1,2As an impressive exam- ple, the deep learning technique has shown an overwhelming advan- tage in the confrontation between a human and computer in the game of go.3–5ANNs resulting from simulating biological archi- tectures of the human brain possess the intrinsic advantages of the brain including parallel computation, distributed storage, low energy consumption, etc. Nevertheless, these advanced algorithms are mostly implemented using software and are still executed on conventional computers with the von Neumann architecture, where the advantages of brain-inspired computing are unfortunately not fully exploited.5 There have been many attempts to design and fabricate neuro- morphic hardware devices,6–9which are not limited by the von Neu- mann bottleneck and intrinsically possess all the aforementioned advantages. Neuromorphic chips using the standard CMOS circuits, such as the IBM TrueNorth, consisting of billions of transistors canperform brain-inspired computing with remarkably low power.10,11 Magnetic materials, however, have the potential to further increase the energy efficiency and areal density of devices by several orders of magnitude. An example has been shown in the devices of random number generation, where the most energy-efficient implementa- tion of the CMOS circuit consumes 2.9 pJ/bit and the circuit area of 4004μm2.12The device based on magnetic tunnel junctions (MTJs) only costs 20 fJ/bit and 2 μm2in area.13In addition, memristors made of resistive and phase-change materials have attracted much attention in the realization of ANNs.14–18Compared to resistive and phase-change memristors, magnetic materials have faster dynamics at a time scale of nanoseconds and high endurance of more than 1015 cycles for magnetization switching.19–21More importantly, the mag- netization dynamics can be well described by the phenomenological Landau–Lifshitz–Gilbert equation,22,23which has been examined in the past half century in the research communities of magnetism and spintronics. Recently, spintronics-based brain-inspired computing was used to realize the Hopfield model of memory.24The sound recognition could be significantly improved under the help of spin AIP Advances 10, 025116 (2020); doi: 10.1063/1.5143382 10, 025116-1 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv torque nano-oscillators,25,26whose nonlinear magnetization dynam- ics with memory is essential to capture the distinct acoustic features encoded in frequencies. A voltage-controlled stochastic spintronic device is implemented in experiment, where the stochastic behav- ior of magnetic switching is incorporated in an ANN to recognize the handwritten digits.27To date, most of the spintronic devices of brain-inspired computing have been applied in the recognition of static images or patterns, and little is known about their capability of temporal signal processing. Reservoir computing is particularly suitable for encoding time series,28,29in which the reservoir is physically a recurrent neu- ral network (RNN).30The sparse and usually random connections among the neurons in the RNN ensure the capability to describe sufficiently complex functions.31The relatively simple structure is another advantage of the RNN in the hardware implementation.32,33 In this paper, we report a spintronic realization of RNNs with MTJs, which were used as the basic units of spin-transfer torque magnetic random access memory. The nonlinear magnetization dynamics of an MTJ driven by an electrical current allows us to replace one neu- ron in the RNN by a single MTJ. By performing a micromagnetic simulation, we demonstrate that an RNN consisting of as few as 40 MTJs can generate and recognize sequential signals after an efficient training process. The capability of the network can be significantly improved by increasing the number of MTJs. We consider 40 MTJs as artificial neurons, which are ran- domly and sparsely connected with one another via either positive or negative unidirectional synapses, as schematically illustrated in Fig. 1(a). An MTJ consists of two thin magnetic layers separated by an insulating layer, as plotted in Fig. 1(b). The bottom layer has a fixed in-plane magnetization, usually pinned by an antiferromag- netic material via exchange bias.34The magnetization of the top (free) layer in the MTJ can be excited by an injected electrical cur- rent following the Landau–Lifshitz–Gilbert equation in the presenceof current-induced spin-transfer torques35–37 ˙m=−γm×Heff+αm×˙m +τϵm×mp×m−βτm×mp. (1) Here, mis the magnetization direction of the free layer and Heffis the effective magnetic field, including the exchange, anisotropy, and demagnetization fields. The gyromagnetic ratio γand Gilbert damp- ingαare both material parameters. The last two terms in Eq. (1) are the adiabatic and nonadiabatic spin-transfer torques, respectively, where mpdenotes the magnetization direction of the fixed magnetic layer and the magnitude of the torque τ= (γ̵hP/μ0eM st)jdepends on the current polarization P, the current density j, the saturation magnetization Ms, and the free layer thickness t. The Slonczewski parameterϵ=Λ2/[(Λ2+ 1) + ( Λ2−1)m⋅mp] characterizes the angu- lar dependence of the torque with the dimensionless parameter Λ ∈[0, 1].βis the nonadiabaticity of the spin-transfer torque and is usually much smaller than one. In this work, the dynamic equation is solved numerically using the micromagnetic simulation program MuMax3.38 The magnetization of the free layer, which is perpendicular to the fixed layer at equilibrium, starts to precess about its easy axis with an external current. Therefore, the total resistance Rof the MTJ, which depends on the relative magnetization orientation of the two magnetic layers, exhibits an oscillation as a function of time. In the regime of a small current density, the amplitude of the oscillation decays gradually [see Fig. 1(c)], and the output signal of the artificial neuron is quantitatively defined by the difference between the last maximum and minimum values of the resistance ( ΔR) within 1.5 ns since the electrical current is injected. In this way, the driving force of the magnetization precession, i.e., the injected electrical current density, can be defined as the input of the artificial neuron, while the resulting oscillatory resistance ΔR FIG. 1 . (a) Sketch of an RNN made of MTJs. The black arrows denote the synaptic connections between MTJs. The red lines show the connections from every MTJ to the output node with adjustable weights. The black lines at the bottom represent the feedback connec- tions that transport the output signal to every MTJ in the RNN. (b) The structure of an MTJ. (c) Damped oscillation in the resistance of the MTJ as a function of time due to the precessional motion of the free magnetic layer under the current density j= 0.9 ×1010A/m2. The dif- ference between the last maximum and minimum resistances ΔRwithin 1.5 ns after injecting the current is defined as the output. (d) The output of the artificial neuron as a function of current density. The shaded range of the current den- sity is used to excite the magnetization dynamics. Inset: Calculated ΔR/R0for a large range of j, where R0denotes the resistance of the MTJ at equilibrium. AIP Advances 10, 025116 (2020); doi: 10.1063/1.5143382 10, 025116-2 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv corresponds to the response or output. If one increases the current density, ΔRincreases monotonically and nonlinearly. Using micro- magnetic simulation, we can determine the nonlinear response func- tion of the artificial neuron, which is plotted in Fig. 1(d). We choose an electrical current density in the range of [0.8, 1.0] ×1010A/m2 and a corresponding ΔR∈[114, 145] Ω in this work. Both the input and output are normalized to be in the range of [0, 1] when we consider the signal transfer among MTJs (see the supplementary material). Owing to the small range of jthat we choose in this work, the resistance change ΔRis very small compared with the value at equilibrium R0= 11.05 kΩ. In practice, ΔR/R0can be increased up to 20% by applying larger current density, as shown in the inset of Fig. 1(d). Every MTJ is connected to the “output neuron” of the RNN via two synapses: one transfers the output signal of every MTJ to the “output neuron” ( wout) and the other provides feedback from the “output neuron” to the MTJ ( wf). Here, only the weights wout are varied during the “learning” process, while the synapses within the RNN and the feedback synapses wfare all fixed. Such an RNN can maintain time-dependent activation by the mutual interactions of neurons even without an external input. The detailed parame- ters and learning algorithm of this network can be found in the supplementary material. As illustrated in Fig. 1(a), the weighted summation over the output signals of all the MTJs is defined as the output of the RNN. We first let the RNN generate a target sinusoidal function f(t) = Asin(2πt/T) with A= 0.87 and T= 45 ns. A very efficient algorithm called “force-learning scheme”39is applied, in which the weights woutare tuned by comparing the error between the RNN output and the desired target function. In practice, the RNN output follows thetarget function very quickly. As shown in Fig. 2(a), after 3000 ns, the output is already in perfect agreement with the target function. After learning for 4500 ns, we no longer vary wout, and the network sustains the generation of the same function as its output. This sug- gests the success of the learning scheme in this artificial RNN, which merely consists of 40 MTJs. Att= 5700 ns, we abruptly change the amplitude and period of the target sinusoidal function with A= 0.65 and T= 80 ns. At the same time, the force-learning algorithm is launched again to train the network via tuning the weights wout. The output significantly deviates from the new target function immediately after 5700 ns, but they superpose each other after 3000 ns of learning. We turn off the learning process after t= 8700 ns, and the RNN steadily generates the new sinusoidal function afterwards. More complex time series can be learned using an RNN with more MTJs. For instance, by defining two-dimensional coordinates xand y, which are both time-dependent functions, we can repro- duce handwritten Chinese characters. As schematically illustrated in Fig. 3(a), we construct an RNN with 800 MTJs and two output nodes for xandy. Instead of using feedback from the output nodes, we introduce two input nodes, where the ideal target functions are imported to the RNN to increase the learning efficiency. Moreover, we allow tunability of the random and sparse connections in the RNN to improve its flexibility and transferability, because a single RNN is used to generate the two coordinates simultaneously. We choose the Chinese character meaning “teacher” written with a writing brush, as shown in Fig. 3(b). Two functions of time x(t) and y(t) are defined in a two-dimensional coordinate system to follow the stroke order of this character. Since both the connec- tions in the RNN and the output weights are adjusted in the learning FIG. 2 . (a) RNN output as a function of time (the red line). The target function is shown as the black line for compar- ison. Learning is performed in the first 4500 ns. (b) RNN output with a different target function after t= 5700 ns. AIP Advances 10, 025116 (2020); doi: 10.1063/1.5143382 10, 025116-3 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv FIG. 3 . (a) Sketch of the RNN used for writing a Chinese character with two input and two output nodes. The nodes output1 and output2 are used to generate the xand ycoor- dinates, respectively, as a function of time. The connection weights among the MTJs and the output weights wout1and wout2are adjusted during the learning, which are illustrated by red lines. (b) The Chinese character meaning “teacher” written with a Chinese writing brush. (c) The output by the network in (a) reproducing the character in (b). Time is rep- resented by colored dots with a uniform time interval of 1.5 ns. process, we employed the so-called innate training algorithm.40Such an algorithm is more robust and efficient for convergence. In addi- tion, the innate training algorithm is practically highly resistant to noise or perturbations. The specific implementation of the innate training algorithm has two steps. In the first step, the connection weights inside the RNN are tuned to allow every MTJ to have its own sustained response to a pulse input. The success of this train- ing is achieved when this sustained response becomes invariant for different initial conditions of the MTJs. This step is essential to improve the robustness of the network and produce insensitivity to noise. Having adjusted the connection weights in the RNN, next we apply the force-learning algorithm to tune the output weights wout1 andwout2. In this step, the target periodic functions x(t) and y(t), which are implemented with the period of 170 ns, are imported from the two input nodes. After training for 10 periods, the output is plotted in Fig. 3(c), which successfully reproduces the handwritten Chinese character. In addition to the generation of periodic functions, an RNN made of MTJs can also be applied to the recognition of time series. The structure of the network is shown in Fig. 4(a), where a time- dependent function is imported to the RNN from the input node. After adjusting the output weights woutof the network, the output can have a different response to the corresponding input functions. To demonstrate recognition by the RNN, we input two simple func- tions into the RNN: a square wave 2 sgn[sin( ω1t)] and a sinusoidal function sin( ω2t) withω1= 0.16 GHz and ω2= 0.21 GHz. The designed target functions for the sinusoid and the square wave are +1 and −1, respectively. For every 60 ns, we input one type of function, either the sinu- soid or the square wave, as plotted in Fig. 4(b). Here, random noise is superposed on the function, which is approximately 5% of the mag- nitude of the function. Then, we adjust the output weights woutto let the output of the RNN match the required target function [Fig. 4(c)]. Such learning is performed 60 times in the first 3600 ns and thenthe weights are fixed in later recognition. To avoid the influence of the previous recognition, we deliberately reset all the MTJs at the beginning of every recognition process, i.e., the initial output is always 0. The real RNN output is plotted in Fig. 4(d) as a func- tion of time. Depending on the averaged output value, the RNN can recognize all the input waves 100 times (from 3600 ns to 9600 ns) successfully. The numerical simulation we have done so far is a proof of concept and hence MTJs with fast precessions are chosen in sim- ulation to reduce computational cost. In experiment, lower fre- quency precessions may be preferred, which can be done by using the vortex magnetization in the free layer.26Moreover, the high frequency resistance is technically difficult to measure, so the mea- surable voltage of the MTJs can be used as the neuron output, which is just another nonlinear function of the input current. One also needs to consider two possible difficulties in experiment, i.e., the non-identical MTJs and phase noise, while the details are pro- vided in the supplementary material. We show that a RNN made of 40 MTJs with 25 different sizes works as well as the RNN with identical MTJs. Phase noise is one of the key issues limiting the functionality and performance of MTJ-based dynamical devices.41 The RNN output is indeed affected by the phase noise, but can be systematically improved by increasing the number of MTJs in the RNN. The synapses here are not realized using magnetic devices. Instead, we merely consider a hybrid system with the artificial neu- rons modeled by MTJs and an external storage for the synaptic weights. The implementation is analogous to the present neuromor- phic chip, where static random access memory is employed to store the adjustable synaptic weights.8–11There are several proposals of trainable artificial synapses made of magnetic and resistive materials in the literature,20,42such as the Hall bars consisting of perpendic- ular magnetic multilayers,24the spintronics memristors based on domain walls,43–46and the MTJ-based devices with multiple elec- trical resistances.19,47,48The technical challenge for applying these AIP Advances 10, 025116 (2020); doi: 10.1063/1.5143382 10, 025116-4 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv FIG. 4 . (a) The structure of the RNN used for time series recognition. (b) The input data, (c) target function, and (d) output of the RNN as a function of time. Learn- ing is performed for the first 3600 ns and the recognition is carried out in the next 6000 ns. All input functions have been successfully recognized. proposals is the effective and precise adjustment of the synaptic weights in the training process. The imperfection in synapses is explicitly examined in the supplementary material including the fluctuation of the synaptic weights, signal delay in the RNN, and failure to update part of the output synapses during the training process. Nevertheless, the RNN can still learn to generate the tar- get function, indicating a high tolerance of the RNN for imperfect synapses. We have demonstrated that the generation and recognition of time series can be achieved by a MTJ-based recurrent neu- ral network. Using micromagnetics to simulate the magnetization dynamics of the MTJs, we have shown that the RNN can learn to generate an arbitrary periodic function. With enough MTJs, an RNN can even be trained to simulate a handwritten character of the Chinese writing system. The recognition of different time- dependent functions has also been successfully performed using such a network. Moreover, this MTJ-based RNN is found to have a high tolerance to size dispersion of the MTJs. In time series recognition, such an RNN is resistant to the noise of the input signals. The demonstration of this spintronic implementation of neu- romorphic computing suggests that MTJs are very promising can- didates for artificial neurons. Owing to the low energy cost and small geometric size, magnetic devices are expected to significantlyimprove the energy efficiency and integration density of neuro- morphic devices. MTJs have ultrafast dynamics in the nanosecond regime and high endurance of more than 1015cycles because mag- netization dynamics does not involve any atomic motion as in the diffusive memristors. In addition, MTJs can be naturally integrated with the artificial synapses made of non-volatile magnetic memories such that all magnetic/spintronic neuromorphic chips can eventu- ally be achieved. The proposed magnetic synapses also attract great attention in research,19,20,24,42–48where the synaptic weight needs to be precisely adjusted during learning. See the supplementary material for the theoretical methods and numerical details, the effects of non-identical MTJs, phase noise of MTJs, and the imperfect synapses. This work was financially supported by the National Key Research and Development Program of China (Grant No. 2017YFA0303300), the National Natural Science Foundation of China (Grant Nos. 11734004, 61774018, 61604013, 61774017, and 31771146), the Recruitment Program of Global Youth Experts, and the Fundamental Research Funds for the Central Universities (Grant Nos. 2018EYT03 and 2018STUD03). Y.M. acknowledges the financial support of Beijing Municipal Science and Technol- ogy Commission (Grant No. Z171100000117007) and Beijing Nova Program (Grant No. Z181100006218118). AIP Advances 10, 025116 (2020); doi: 10.1063/1.5143382 10, 025116-5 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv REFERENCES 1S. Lawrence, C. L. Giles, A. Chung Tsoi, and A. D. Back, IEEE Trans. Neural Networks 8, 98 (1997). 2M. M. Najafabadi, F. Villanustre, T. M. Khoshgoftaar, N. Seliya, R. Wald, and E. Muharemagic, J. Big Data 2, 1 (2015). 3Y. LeCun, Y. Bengio, and G. Hinton, Nature 521, 436 (2015). 4J. Schmidhuber, Neural Networks 61, 85 (2015). 5D. Silver, A. Huang, C. J. Maddison et al. , Nature 529, 484 (2016). 6B. V. Benjamin, P. Gao, E. McQuinn, S. Choudhary, A. R. Chandrasekaran, J.-M. Bussat, R. Alvarez-Icaza, J. V. Arthur, P. A. Merolla, and K. Boahen, Proc. IEEE 102, 699 (2014). 7E. Neftci, J. Binas, U. Rutishauser, E. Chicca, G. Indiveri, and R. J. Douglas, Proc. Natl. Acad. 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1.3558986.pdf

Influences of film microstructure and defects on magnetization reversal in bit patterned Co/Pt multilayer thin film media Vickie W. Guo, Hwan-Soo Lee, and Jian-Gang Zhu Citation: J. Appl. Phys. 109, 093908 (2011); doi: 10.1063/1.3558986 View online: http://dx.doi.org/10.1063/1.3558986 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v109/i9 Published by the American Institute of Physics. Related Articles Magnetic anisotropy and anomalous Hall effect of ultrathin Co/Pd bilayers J. Appl. Phys. 112, 093915 (2012) Nano-faceting of Cu capping layer grown on Fe/Si (111) and its effect on magnetic anisotropy J. Appl. Phys. 112, 093916 (2012) Magnetic properties of the magnetophotonic crystal based on bismuth iron garnet J. Appl. Phys. 112, 093910 (2012) Design of remnant magnetization FeCoV films as compact, heatless neutron spin rotators Appl. Phys. Lett. 101, 182404 (2012) Observation of rotatable stripe domain in permalloy films with oblique sputtering J. Appl. Phys. 112, 093907 (2012) Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 08 Nov 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsInfluences of film microstructure and defects on magnetization reversal in bit patterned Co/Pt multilayer thin film media Vickie W. Guo,1,a)Hwan-Soo Lee,2,b)and Jian-Gang Zhu1 1Data Storage Systems Center, Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA 2Advanced Materials & Device Lab, Samsung Electro-Mechanics, Suwon, Gyunggi-Do 443-743, South Korea (Received 23 September 2010; accepted 22 January 2011; published online 9 May 2011) Quasisingle crystalline and polycrystalline Co/Pt multilayered films were prepared via sputtering technique. The polycrystalline Co/Pt multilayers exhibited an appreciable number of planar defects such as twin boundaries and stacking faults whereas few defects were present for the quasisingle crystalline films. The polycrystalline films had smoother surface, and as patterned into arrays ofsmall islands, a smaller critical size for single domain was unexpectedly observed. The corresponding magnetic domain images revealed that nucleation interestingly occurred at any locations of a patterned element, which was attributed to the observed defects. Moreover,micromagnetic modeling was utilized to further quantitatively study influences of an anisotropically soft region (which can represent existing defects) in the patterned element on nucleation field in terms of exchange coupling strength. VC2011 American Institute of Physics . [doi:10.1063/1.3558986 ] I. INTRODUCTION Bit patterned media (BPM) is a promising strategy for increasing magnetic recording density beyond 1Tb/in2.1–3In BPM, grains are strongly exchange coupled, and magneticisolation between bits is achieved by patterning the media into the isolated elements. Typically, grains in each bit switch simultaneously, exhibiting a Stoner-Wolfarth 4type reversal. Understanding of the magnetization reversal for patterned elements, which are likely to be influenced by fun- damental magnetic properties such as anisotropy, magnetiza-tion, exchange coupling, and magnetostatic interaction, is a key to designing desirable BPM media. Extrinsic parameters, e.g., lithographic variations 5,6or film microstructure7–10controlled by seed layers, defects, grain boundary, and grain orientation have also been known to be critical. Previously, the authors investigated the effects of me- dium microstructure on the switching field (SF) and its distri- bution (SFD) in bit patterned media.11Two types of film, namely, quasisingle crystalline (QSC) and polycrystalline (PC) Co/Pt based multilayers were investigated. The two types of samples showed similar anisotropy field and coer- civity in the as-deposited. However, when patterned, PC exhibited a lower SF in any pattern size of comparison thanQSC. PC had a notably smaller critical size below which pat- terned islands were uniformly magnetized. In this study, the microstructure and roughness for the two types of film were examined using transmission electron microscopy (TEM) and low angle x-ray reflectivity (XRR), respectively. We also correlate magnetization reversal andexchange coupling in the patterned media with the observed microstructural and morphological features, in conjunctionwith a micromagnetic study. In the simulation, a soft region (that presents a defect region) with a low anisotropy (H soft) was advertently set at a certain location of the entire regionof a patterned bit. The two locations chosen for the investiga- tion were the center and the corner of the bit. A comparison between the two cases was made in terms of nucleation fieldas a function of exchange coupling constant for different H softvalues. II. EXPERIMENTAL The QSC and the PC Co/Pt multilayer films were fabri- cated via sputtering technique, and the films were subse- quently patterned into a periodic array of dot or square shaped features, using electron-beam lithography. The pat-terns were ranged from 1000 nm ( ¼1lm) to 100 nm in size. Other details of patterning procedures can be found in elsewhere. 12 In fabricating the QSC films, prior to a 6nm thick Pt layer, a 15 nm thick Ag seed layer was deposited onto a sin- gle crystal Si (111) substrate to reduce the lattice mismatchbetween the Pt (111) and the Si (111). The Co/Pt multilayers were composed of 10 repetitions of bi-layers Co(6A ˚)/Pt(18 A ˚). A 3 nm thick Pt layer was deposited on top as a capping layer.For the fabrication of the PC film, a 3 nm thick Ta seed layer was deposited onto a Si (100) substrate prior to the deposition of the Pt layer. No Ag underlayer was used. See Fig. 1.T h e other deposition conditions were identical. Due to the absence of the Ag underlayer, grains of both the Pt and the Co were randomly oriented in the film pl ane. Details of the preparation of Pt/Co multilayers were described in the previous publication. 11 The magnetic properties of the samples were measured by an alternating gradient magnetometer (AGM). Surface roughness was measured using an atomic force microscope (AFM) and low angle x-ray reflectivity (XRR). Magnetica)Electronic mail: vickie.guo@gmail.com b)Electronic mail: jisoo725@naver.com 0021-8979/2011/109(9)/093908/5/$30.00 VC2011 American Institute of Physics 109, 093908-1JOURNAL OF APPLIED PHYSICS 109, 093908 (2011) Downloaded 08 Nov 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsdomain imaging was performed using a magnetic force microscope (MFM). Film textures and microstructures were characterized by an x-ray diffractometer (Philips X’pert Prowith x-ray lens) using Cu K aradiation and by a transmission electron microscope (TEM) operating at 200 kV. III. MICROMAGNETIC SIMULATION Patterned bits based on Co/Pt multilayers are structured utilizing a three-dimensional array of multilayered cubes shown in Fig. 2. The cubic mesh represented the face-cen- tered cubic (fcc) crystal microstructure of both the Co and the Pt. In the mesh, the thicknesses of individual Co and Pt layers can be set differently prior to computation, however,the ratio of the Co and the Pt was fixed to 1:3, which was shown to exhibit strong perpendicular anisotropy. 13Since Pt atoms adjacent to the Co layer can be polarized and exhibit anet increase in magnetic moment, a Pt monolayer next to the Co layer was taken into account for the energy calculation. A single Co monolayer and the adjacent two polarized Ptmonolayers were combined into a single magnetic identity to reduce the size of calculated data array and computation time. The Pt polarization was assumed to be one atomic layerof Pt. The magnetic anisotropy (K Co-Pt) and saturation mag- netization (M Co-Pt) of the combined layer are determined by the following relationships: KCo/C0Pt¼KCo/C1tCoþ2KPt/C1tPt ðÞ tCo/C0Pt(1) Ms;Co/C0Pt¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi M2 s;Co/C1tCoþ2M2 s;Pt/C1tPt tCo/C0Pts (2)where t Co,tPt, and t Co-Pt are the thicknesses of a Co layer, the polarized Pt monolayer, and the combined Co-Pt layer, respectively. K Coand K Ptare the crystalline anisotropies of the Co and the polarized Pt, and M s,Coand M s,Ptare the satu- ration magnetizations for the Co and the polarized Pt, respectively. The critical energy terms were taken into consideration for each mesh cell, and the spin dynamics was based on the phenomenological Landau-Lifshitz-Gilbert (LLG) equationand Brown’s micromagnetic theory. 14,15The simulated grid size of Co/Pt multilayer was 50 nm. The thicknesses of Co and Pt were 0.6 nm and 1.8 nm, respectively. Values of mag-netic properties were chosen to be close to the fabricated film. Magnetizations for the Co and the polarized Pt were set to 1200 emu/cc and 100 emu/cc, respectively. The saturationmagnetization of the sputtered Co/Pt film was /C24650 emu/cc. The magnetization of the Pt was set to attain the observed magnetization of the sputtered film. The perpendicular ani-sotropy was 4 /C210 6erg/cc. The lateral exchange coupling (A1) within a layer was 1 /C210/C06erg/cm, and the exchange coupling (A 2) between the two Co layers was 1 /C210/C07erg/ cm. The Gilbert damping constant, awas 0.1.14 IV. RESULTS AND DISCUSSION A key microstructural distinction comes from compar- ing selected area electron diffraction of plan view samplesfor PC and QSC. In Fig. 3(a), the selected area electron dif- fraction (SAED) taken along a zone axis (111) consisted of concentric rings, which are a characteristic of polycrystallinegrowth. In Fig. 3(b), a plan-view image of the sample [Si/ Ag/Pt/(Co/Pt) 10] is shown. No clear grain boundaries were seen. The corresponding diffraction pattern shows both Coand Pt have been epitaxially grown. The diffracted spots from the Ag and the Pt are both sharp, indicating good epi- taxial growth and few defects. FIG. 2. (Color online) Schematic of three-dimensionally meshed Co/Pt mul- tilayer structure. A 1and A 2are the lateral exchange coupling constant and the exchange coupling constant between two adjacent layers, respectively. ‘a’ is the lattice constant. FIG. 3. Plan-view TEM images and the corresponding electron diffraction patterns: (a) polycrystalline films and (b) quasisingle crystalline films. FIG. 1. (Color online) Schematic of two types of film: (a) Polycrystalline (PC) films and (b) quasisingle crystalline (QSC) films.093908-2 Guo, Lee, and Zhu J. Appl. Phys. 109, 093908 (2011) Downloaded 08 Nov 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsIn Fig. 4, a cross-sectional micrograph of an unpatterned PC Co/Pt multilayer is shown. The most common defect we observed within a columnar grain was twinning. An appreci- able number of twin boundaries (TBs) and stacking faults(SFs) through thickness direction were identified for the PC Co/Pt multilayer samples whereas notable twinning and few stacking faults were present for QSC. The presence of twin-ning was consistent with the /scan result for QSC where six evenly spaced diffracted peaks for the Ag and the Pt (200) planes were observed. There should be three when antici-pated from the stereographic projection of single crystal face centered cubic (111) lattice. For face-centered cubic (fcc) crystals, the stacking of (111) plane is ABCABC… whereas as twin boundaries are present, the stacking sequence can be ABCBA… Stacking faults of Co that grows on Pt (111) yields a stack of ABAB.Co based multilayers have been shown to have a large perpendicular magnetic anisotropy (PMA). 16,17This was achieved when the layers were grown with (111) textured. The corresponding inset of the cross-sectional image at a lower magnification reveals that the PC film is quite smooth in terms of surface roughness. No physical voids andseparation at grain boundaries were observed, suggesting that the film be strongly exchange coupled. An apparent difference in roughness between the two types of film was probed as the low angle x-ray reflectivity (XRR) measurement was carried out (see Fig. 5). The pres- ence of surface roughness leads to a progressive decrease inthe specular intensity of an entire reflectivity curve. Rough- ness for QSC is apparently greater. The Kiessig fringes which attenuate faster for QSC also supports that the inter-face of QSC is rougher. The AFM measurements also confirmed that the average roughness (R a) for QSC was rougher. The R afor QSC was 0.50 nm where that for PC was 0.22 nm. The measurement area was 5 lm/C25lm. The smoother surface for PC may lead to a less variation in ferromagnetic properties throughintergrains. Less variation of intergranular exchange cou- pling for PC would help to form single domain as patterned. In Figs. 6(a)–6(c), various in-plane ac fields were applied for a pattern size of 500nm. The remanent magnet- ization state was obtained through demagnetization using a damped alternating in-plane field. The ranges of ac field arenoted on top of the each sub-figure. The in-plane field was utilized for the investigation since field perpendicular to the film plane failed to produce multi-domain states for pat-terned islands. In the Co/Pt system studied here, domain wall pinning field appeared to be much smaller that nucleation field. There are a couple of notable traits for the observed re- manent states provided by varying ac field. First, not all the elements can nucleate reverse domains. A few remainunchanged as indicated by the arrows in white, suggesting their anisotropy field (H k) is higher than that for the rest of the patterned elements. Secondly, on a close look at the do-main configurations, it was found that initial nucleation sites FIG. 4. Cross-sectional TEM of unpatterned polycrystalline Co/Pt multi- layer. The arrows in black indicate twin boundaries (TBs) and stacking faults (SFs) were shown as well in through thickness direction. FIG. 5. (Color online) Low angle x-ray reflectivity curves for PC and QSC(as unpatterned). The specular intensity for QSC attenuates faster, indicating that the surface roughness is greater than that of PC. FIG. 6. (Color online) MFM images for different in-plane ac fields: (a) þ//C0 5kOe, (b) þ//C08kOe, and (c) þ//C012kOe. The pattern size for (a), (b), and (c) was 500 nm in length. FIG. 7. (Color online) MFM images [4] of the PC films for different patternsizes: (a) 500 nm, (b) 200 nm, and (c) 100 nm. A damped ac in-plane field with a peak amplitude of 12kOe was applied. The insets are shown for com- parison. Figs. 7(a)and7(b)are taken from Fig. 6 of Ref. 11. (Reprinted with permission from V. W. Guo, H.-S. Lee, Y. Luo, M. T. Moneck, and J.-G. Zhu, IEEE Tran. Magn. 45, 2686 (2009), copyright 2009 IEEE).093908-3 Guo, Lee, and Zhu J. Appl. Phys. 109, 093908 (2011) Downloaded 08 Nov 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionswere not necessarily located at the corner, which we pre- sumed was likely to be weaker in terms of anisotropy. In fact, nucleation occurred at any locations of an element, suchas border, corner or even center of the element, suggesting that another factor other than the pattern shape play a more important role in magnetization reversal. In Fig. 7, MFM images of the PC for three different pat- tern sizes of 500, 200, and 100 nm were shown. In-plane field with a peak amplitude of 12 kOe was used. The patternsizes of 500 and 200 nm displayed multi-domain configura- tions. The critical size for single domain state appears to be 100 nm whereas most of islands with a size of 200 nm exhib- ited single domain for the QSC as shown in Fig. 7(b). The idea of a larger variation in principal magnetic properties such as magnetization, anisotropy, or exchange coupling due to roughness for QSC does not account for the observations. The QSC revealed a higher switching field(SF) than that for PC as patterned down to 1 lm or smaller as well. 11The QSC consistently displayed a larger SF by about 2000 Oe than that of PC. We take into account a possibility that defects in film would play a crucial role in magnetization reversal, in partic- ular, for PC where population of defects in the film is signifi-cant. Under this situation, locations where defects reside within a patterned element can act as a preferable site for nucleation during magnetization reversal. Magnetization re-versal for QSC appears to be much more uniform than that of PC. Typically pack-man type domain was observed as shown in the inset of Fig. 7(a). In Fig. 8, in order to quantitatively describe the effects of anisotropy and exchange coupling on a localized volume, a soft region with a low anisotropy has been advertently setat a certain location of the entire region of a patterned bit. In a real patterned structure, the soft region can represent either local variation in intrinsic properties or local defectsoccurred during fabrication process. The two locations chosen for this investigation were the center and the corner of the bit. A comparison between thetwo cases was made in terms of nucleation field as a function of exchange coupling constant. Calculations were carried out for different H softvalues, ranging from 0.2 H kto 0.8 H k where H kis the anisotropy field for the rest of the area in the calculated bit.The nucleation field for the center case monotonically increased with respect to exchange coupling strength. The nucleation field becomes less sensitive to exchange couplingstrength when the H softis comparable to the H k. Of particular note is that there is a minimum nucleation field observed for the case where the soft region is at the cor-ner of a bit. See Fig. 7(b). As exchange coupling increased from 2 /C210 /C07to 4/C210/C07erg/cm for H soft¼0.4 H k, the nucleation field decreased from 3300 to 2400 Oe. Under thiscircumstance, a certain exchange coupling can keep the mag- netization of the soft region tilted from the easy axis. This results in a lower field to switch the soft region due to the angle dependence of the switching field whereas, in the cen- tered case, exchange coupling experienced by the soft regionis balanced out and does not provide a lower switching field. The figure points that when H softis sufficiently smaller than H k, the nucleation field in a patterned bit can significantly vary according to defect location and exchange coupling strength. V. CONCLUSIONS Influences of film microstructure and defects on magnet- ization reversal process for patterned Co/Pt perpendicular multilayer films were studied. Currently, there is no directevidence, clarifying which types of defect, TBs or SFs, dom- inate magnetization reversal. Nonetheless, the location of defects residing in the film appeared to be important in deter-mining characteristics of the patterned elements. Micromag- netic study quantitatively described the effects of a soft region in a patterned element on nucleation field as functionsof anisotropy and exchange coupling. ACKNOWLEDGMENTS This work was supported by the Data Storage Systems Center of Carnegie Mellon University. The authors would like to thank Drs. Yueling Qin and Anup Roy for their helpon the TEM work. 1T. Thomson, G. Hu, and B. D. Terris, Phys. Rev. Lett. 96, 257204 (2006). 2E. A. Dobisz, Z. Z. Bandia ´c, T.-S. Wu, and T. Albrecht, Proc. IEEE 96, 1836 (2008). FIG. 8. (Color online) Comparison between the two cases: (a) center of a bit and (b) corner of the bit. The nucleation field versus exchange coupling constant for different H softvalues.093908-4 Guo, Lee, and Zhu J. Appl. Phys. 109, 093908 (2011) Downloaded 08 Nov 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions3H. J. Richter, A. Y. Dobin, R. T. Lynch, D. Weller, R. M. Brockie, O. Heinonen, K. Z. Gao, J. Xue, R. J. M. v. d. Veerdonk, P. Asselin, and M. F. Erden, Appl. Phys. Lett. 88, 222512 (2006). 4E. C. Stoner and E. P. Wohlfarth, Phil. Trans. R. Soc. London A 240, 599 (1948). 5Y. Kitade, H. Komoriya, and T. Maruyama, IEEE Tran. Magn. 40, 2516 (2004). 6J. Kalezhi, J. J. Miles, and B. D. Belle, IEEE Tran. Magn. 45, 3531 (2009). 7J. M. Shaw, W. H. Rippard, S. E. Russek, T. Reith, and C. M. Falco, J. Appl. Phys. 101, 023909 (2007). 8R. Sbiaa, C. Z. Hua, S. N. Piramanayagam, R. Law, K. O. Aung, and N. Thiyagarajah, J. Appl. Phys. 106, 023906 (2009). 9B. D. Terris, M. Albrecht, G. Hu, T. Thomson, and C. T. Rettner, IEEE Tran. Magn. 41, 2822 (2005).10J. W. Lau, R. D. McMichael, S. H. Chung, J. O. Rantschler, V. Parekh, and D. Litvinov, Appl. Phys. Lett. 92, 012506 (2008). 11V. W. Guo, H.-S. Lee, Y. Luo, M. T. Moneck, and J.-G. Zhu, IEEE Tran. Magn. 45, 2686 (2009). 12M. T. Moneck, J.-G. Zhu, X. Che, Y. Tang, H. J. Lee, S. Zhang, K.-S. Moon, and N. Takahashi, IEEE Tran. Magn. 43, 2127 (2007). 13A. S. H. Rozatian, B. D. Fulthorpe, T. P. A. Hase, D. E. Read, G. Ashcroft, D. E. Joyce, P. J. Grundy, J. Amighian, and B. K. Tanner, J. Magn. Magn. Mater. 256, 365 (2003). 14J.-G. Zhu, Ph.D. thesis (University of California, 1989). 15C. Kittel, Rev. Mod. Phys. 21, 541 (1949). 16C. Chappert and P. Bruno, J. Appl. Phys. 64, 5736 (1988). 17D. Weller, L. Folks, M. Best, E. E. Fullerton, B. D. Terris, G. J. Kusinski, K. M. Krishnan, and G. Thomas, J. Appl. Phys. 89, 7525 (2001).093908-5 Guo, Lee, and Zhu J. Appl. Phys. 109, 093908 (2011) Downloaded 08 Nov 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

1.4922084.pdf

Enhancement of the anti-damping spin torque efficacy of platinum by interface modification Minh-Hai Nguyen,1Chi-Feng Pai (白奇峰),1,a)Kayla X. Nguyen,1David A. Muller,1,2 D. C. Ralph,1,2and R. A. Buhrman1,b) 1Cornell University, Ithaca, New York 14853, USA 2Kavli Institute at Cornell, Ithaca, New York 14853, USA (Received 24 March 2015; accepted 23 May 2015; published online 2 June 2015) We report a strong enhancement of the efficacy of the spin Hall effect (SHE) of Pt for exerting anti-damping spin torque on an adjacent ferromagnetic layer by the insertion of /C250:5 nm layer of Hf between a Pt film and a thin, /C202n m , F e 60Co20B20ferromagnetic layer. This enhancement is quantified by measurement of the switching current density when the ferromagnetic layer is thefree electrode in a magnetic tunnel junction. The results are explained as the suppression of spin pumping through a substantial decrease in the effective spin-mixing conductance of the interface, but without a concomitant reduction of the ferromagnet’s absorption of the SHE generated spincurrent. VC2015 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4922084 ] The experimental determination that a current density Je flowing through certain high-atomic-number metals can gen- erate a quite substantial transverse spin current density Js through the spin Hall effect (SHE)1–3has been a major factor in the recent focus on the study of spin-orbit interaction effects in heavy metal - ferromagnet (HM jFM) thin film mul- tilayer systems. The fraction of this spin current that is absorbed by the ferromagnetic film generates a spin transfer torque on the FM, characterized by a spin-torque (ST) effi- ciency nSH/C17ð2e=/C22hÞJabsorbed s =Je/C20hSH/C17ð2e=/C22hÞJs=Je, where hSHis the “internal” spin Hall angle. For magnetic excitation by the anti-damping ST effect, which is the mechanism by which the spin Hall torque can achieve magnetic manipula- tion using the least possible current, the critical current density scales /a=nSH, and the write energy /ða=nSHÞ2q, where ais the Gilbert damping of the FM jHM bilayer and q is the electrical resistivity of the HM. Large ST efficiencies have been measured for Pt, beta-phase or amorphous Ta, and beta-phase W films: nPt SH¼0:04/C00:09,4–6nb/C0Ta SH/C250:12,7 andnb/C0W SH/C250:3.8The large values of nSHforb-Ta and b-W, together with the relatively small values of damping for thin b-Ta and b-WjFM bilayers, have enabled low-current ST switching and ST microwave excitation of the free electrode of nanoscale magnetic tunnel junctions (MTJs),8demonstrat- ing the feasibility of the SHE for three-terminal memory de- vice and ST nano-oscillator applications9–11as well as new classes of spin logic circuits.12–17However, the high resistivity ofb-Ta and b-W,/C21180lX/C1cm, can be problematic when the write energy and device heating are important consi- derations. While the lower resistivity of Pt films, qPt /C2520lX/C1cm for isolated films (which can be different from the averaged resistivity of thin Pt having adjoining metal layers of high resistivity),18makes Pt seemingly more attrac- tive for energy-efficient ST devices, the smaller value of nSHfor Pt and a much higher damping for FM jPt bilayers4,7 greatly diminish the effectiveness of anti-damping ST for Pt devices. Recent works19–21have shown strong and complicated effects of a thin insertion layer on the spin orbit torques inHMjFM systems. Here, we report that a thin, /C250:5n m , H f layer inserted between a Pt film and a thin Fe 60Co20B20 (FeCoB) layer causes large reductions in both the currentdensity and write energy needed for ST switching. The pres-ence of the Hf reduces the Gilbert damping aby more than a factor of 2 by suppressing spin pumping and at the same time results in n PtjHf SH/C250:12, approximately 2 times higher than the spin torque efficiency reported with Pt jNi81Fe19 bilayers. Pt jHfjFeCoB is therefore a preferred SHE structure for use in anti-damping ST applications. Our work suggeststhat there may be additional opportunities for the enhance-ment of spin Hall torque effects through the further opti- mized modification of HM jFM interfaces. Understanding the consequences of the Hf insertion layer requires an analysis of the processes contributing tomagnetic damping and spin transmission at an HM jFM interface. The phenomenon of spin pumping, which is typi-cally analyzed via use of the drift-diffusion equation, 22 increases the magnetic damping ain HM jFM structures compared to a0, the intrinsic damping parameter in the ab- sence of the HM, because the precession of the FM magnet-ization leads to a loss of spin angular momentum in theHM 23resulting in a¼a0þc/C22h2 2e2MstFMG"# eff: (1) Here c¼1:76/C21011Hz T/C01is the gyromagnetic ratio, Msis the saturation magnetization of the FM, tFMis the thickness of t h eF Ml a y e r ,a n d G"# effis the “effective spin-mixing con- ductance” of the HM jFM interface. G"# effc a nb ee x p r e s s e di n terms of the bare spin mixing conductance of the interface G"# (here, we are assuming that jReG"#j/C29j ImG"#j24), and the spin conductance of the HM layer, Gext/C17tanhðtHM=ksÞ=ð2qHMksÞ, as24–27a)Present address: Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA. b)Author to whom correspondence should be addressed. Electronic mail:buhrman@cornell.edu 0003-6951/2015/106(22)/222402/5/$30.00 VC2015 AIP Publishing LLC 106, 222402-1APPLIED PHYSICS LETTERS 106, 222402 (2015) G"# eff¼G"# 1þG"#=Gext: (2) Pt has a relatively large value of G"# effand is therefore a good “spin sink,” since the typical resistivity of Pt films, qPt/C2520/C025lX/C1cm in combination with a spin attenua- tion length kPt s/C251:2/C01:4n m ,28,29results in Gext/C25 1:8/C21015X/C01m/C02(assuming tHM/C29ks), while the bare mixing conductance of common Pt jFM interfaces is usually of a similar value, e.g., G"#/C251:2/C21015X/C01m/C02has been reported29for PtjPy (Py ¼Ni81Fe19). Consequently, G"# efffor PtjFM bilayers is considerably higher, /C210:7/C21015X/C01m/C02, than that found, for example, for the b-WjFeCoB system,30 0:16/C21015X/C01m/C02. When tFMis small, as it must be in ST devices, the large value of G"# effcauses a large increase in a above a0for PtjFM bilayers (e.g., more than a factor of 4 for a 1.8 nm FeCoB layer, see below). This greatly reduces the efficacy of Pt for anti-damping ST applications, although theSHE of Pt is still quite effective for driving the displacement of domain walls in perpendicularly magnetized free layers where the increased damping is not an issue. 31,32 Turning to the spin torque efficiency nSH, this is of course affected by the net interfacial spin transmissivity in the opposite direction, that is, by the extent to which spincurrents generated by the SHE in the HM are transmitted through the HM jFM interface to exert a spin torque on the FM. The drift-diffusion analysis for this situation 24,25indi- cates that unless Re G"#/C29Gextthere will be substantial spin back-flow from the interface, which reduces the efficiency of the SHE (relative to the internal spin Hall angle, hSH)i n exerting a damping-like spin torque on the FM nSH¼hSH/C2G"#tanhtHM 2ks/C18/C19 tanhtHM ks/C18/C19 G"#þGext ¼hSH/C2G"# eff GexttanhtHM 2ks/C18/C19 tanhtHM ks/C18/C19 : (3) This reduction can be quite significant. For example, applying the analysis above to the Pt jPy interface yields nSH /C250:25hPt SHindicating that the lower bound of nSH/C250:05 as established by the ST ferromagnetic resonance (ST-FMR) study of Pt jPy bilayers of Liu et al.4,28is considerably lower than the actual internal spin Hall angle of the Pt film hPt SH/C250:20.33This is similar to the result of the same analy- sis applied to Pt jCo and Pt jCoFe interfaces.34This analysis suggests that by an appropriate choice of materials and con- trol of the interface structure, we could possibly achieve higher ST efficiencies, and indeed recent studies using differ-ent Pt jFM combinations have reported n SH/C250:1 in some cases.33,34 With the objective of investigating means to suppress spin-pumping and to enhance, or at least not degrade, the spin torque efficiency of Pt for MTJ switching applications, we produced jjTa(1)jPt(4)jHf(tHf)jFeCoB( tFeCoB )jMgO(1.6) j Ru(2) and jjTa(1)jPt(4)jHf(tHf)jFeCoB( tFeCoB )jMgO(1.6) jFeCoB (4)jHf(5)jRu(5) multilayer films18(here, jjrepresents the ther- mally oxidized Si substrate and the numbers in parentheses arethicknesses in nm). The high resistivity Ta was used for ad- hesion and smoothing purposes, while Hf was chosen for thisinvestigation because initial FMR studies indicated a lowG "# efffor Hf jFeCoB, and the previous work has demonstrated that there are negligible current-induced spin-orbit torquesproduced at Hf jFM interfaces. 35The Hf thickness tHfwas varied in fine steps from 0.33 to 0.76 nm with a relativeuncertainty of 5%, while the FeCoB layer thicknesses t FeCoB were 1.6 and 1.8 nm. We also fabricated and measured con- trol samples with tHf¼0 (i.e., no Hf spacer). The samples were annealed at 300/C14C for 30 min in a background pressure of<10/C07Torr. Since certain transition metal elements when incorpo- rated into magnetic tunnel junction structures can be quitemobile, either during deposition and subsequent annealingsteps, we investigated this possibility with respect to the Hfinsertion layer by using electron energy loss spectroscopy(EELS) 36to study the spatial-dependent composition of some of our samples in a 100 keV Nion UltraSTEM. Fig. 1 shows the EELS data for a tFeCoB¼1:6n m , tHf¼0:5n m sample. Since the obtained intensity is proportional to therelative distribution of the element, it is readily seen that aportion of Hf has diffused through the FeCoB layer into theMgO layer, where it is now oxidized. By integrating over theintensity, the amount of Hf in between the Pt and FeCoBlayers is estimated to be /C2470% of the total amount of Hf de- posited, which corresponds to a thickness of /C240.35 nm for this nominal 0.5 nm Hf sample. This indicates that a thin,conformal, and continuous Hf spacer of approximately twoatomic layers or so in thickness is formed on the surface ofPt layer, which is consistent with the high (negative) forma-tion enthalpy of HfPt compounds. 37,38 The magnetic properties of the FeCoB layer in the first set of multilayers were ch aracterized by SQUID magnetometry and anomalous Hall measurements18which indicated a satura- tion magnetization Ms¼ð1:5660:06Þ/C2106A=m, and also an apparent “magnetic dead layer” thickness of td¼0:760:1n m . FIG. 1. ABF-STEM (annular bright field - scanning transmission electron microscopy) image of jjTa(1)jPt(4)jHf(0.5) jFeCoB(1.6) jMgO(1.6) jRu(2) sample and the corresponding EELS line profile that shows Hf diffusion through the FeCoB into the MgO.222402-2 Nguyen et al. Appl. Phys. Lett. 106, 222402 (2015)By fitting the data from measurem ent of the effective magnetic anisotropy energy Keffteff FeCoB as a function of FeCoB effective thickness teff FeCoB¼tFeCoB/C0tdto the standard model for the thickness dependence of the magnetic anisotropy39 Keffteff FeCoB¼ðKV/C0ð1=2Þl0M2 sÞteff FeCoBþKS; (4) the interface and bulk anisotropy energy densities are estimated to be KS¼0:4560:03 mJ =m2andKV¼0:6060:03 MJ =m3, respectively. This value of KSis smaller than typical for TajFeCoB jMgO multilayers, while KVis similar to a recent report.40 Finally, measurements of the in-plane effective demag- netization field l0MefffortFeCoB¼1:6 nm and tFeCoB¼ 1:8n m jjTa(1)jPt(4)jHf(tHf)jFeCoB( tFeCoB )jMgO(1.6) jRu(2) samples indicated that the insertion of a thin layer Hf at the interface of Pt and FeCoB has a significant effect on l0Meff, with a local minimum attHf¼0:5 nm for both series.18We tentatively attribute this behavior to the role of the Hf inser-tion layer in both reducing the positive volume anisotropy effect from elastic strain from the underlying Pt and in enhancing the surface anisotropy energy through reductionof strain at the FeCoB jMgO interface. In Fig. 2(a),w es h o w aðt HfÞ, determined by frequency- dependent ST-FMR measurements,18for the two different FeCoB thicknesses, 1.6 nm and 1.8 nm, with the results clearlydemonstrating that a deposited Hf layer as thin as 0.35 nm, or even less, is effective in greatly reducing the spin-pumping- induced increase in a. All of the samples with the Hf insertion layer exhibit a decrease in aby a factor of 2 or more com- pared to Pt jFeCoB(1.8 nm) with no insertion layer (also shown in Fig. 2(a)). We quantified the effect of the 0.5 nm Hf inser- tion layer on G "# effof a series of tHf¼0:5 nm samples as the function of teff FeCoB .F i g . 2(b) shows the best fit to the damping coefficient aðtef f FeCoBÞdata (solid line in Fig. 2(b))t oE q . (1) which yields a0/C250:006 (broken line) and G"# eff/C250:24 /C21015X/C01m/C02.T h i s G"# effvalue is nearly as low as the value of 0:16/C21015X/C01m/C02observed in the b-WjFeCoB system.30 Similar measurements made on a series of tHf¼0 controlsamples18yielded G"# eff/C25ð1:160:1Þ/C21015X/C01m/C02,s i m i l a r , although somewhat higher, than the previous results for PtjNi81Fe19, confirming the strong effectiveness of the inser- tion of a nominal 0.5 nm Hf layer in suppressing spin pump- ing, as reflected by the large reduction of Da¼a/C0a0shown in Fig. 2(a). To determine nSHfor the Pt jHfjFeCoB trilayers, we measured the ST switching current of a FeCoB free layer in a MTJ, which is the application for which we seek to improve the spin Hall efficacy.8To accomplish these switch- ing current measurements, we patterned the second set of multilayers by electron beam lithography and ion milling(described in supplementary material 18) into three terminal SHE-MTJ devices which consisted of elliptical FeCoB j MgOjFeCoB MTJs, typically with lateral dimensions of /C2550/C2180 nm2, on top of a Ta jPtjHf microstrip approxi- mately 1 :2lm wide as shown schematically in Fig. 3(a). The magnetization of the free FM layer could be controlled either by an in-plane external field along the major axis of the tun-nel junction or by a direct current through Pt layer, and the orientation of the magnetic free layer can be determined by the differential resistance of the MTJ. Figures 3(b)–3(d) show the results for t FeCoB¼1:6n m andtHf¼0:5 nm devices. From the field switching behavior, the coercivity l0Hcis determined to be about 4.5 mT and the tunneling magnetoresistance (TMR) is 80%. Figure 3(b) shows the current-switching behavior for a 50 /C2180 nm2 MTJ, 1.2 lm channel device at a ramp rate of 0.0013 mA/s, for which the switching occurs at average critical currents Ic¼60:4 mA. The switching currents at different ramp rates are shown in Fig. 3(c). By fitting the data to the thermally assisted spin torque switching model,41we find that the zero- thermal-fluctuation switching current is I0¼0:7160:08 mA, which is, considering the geometry of the device and assum- ing that all current flows through the comparatively low-resistivity 4 nm Pt layer, equivalent to a current density of FIG. 2. Gilbert damping parameter aofjjTa(1)jPt(4)jHf(tHf)jFeCoB( tFeCoB )j MgO(1.6) samples measured by frequency-dependent ST-FMR. (a) Damping parameter versus Hf thickness tHffor the tFeCoB¼1:6 nm (circles) andtFeCoB¼1:8 nm (squares) samples. The horizontal broken line indicates the fitted damping parameter (0.006) for an isolated FeCoB layer. (b) Damping parameter versus FeCoB effective thickness of the tHf¼0:5n m samples. The solid line shows the fitting result from which the magneticdamping parameter of isolated FeCoB film is estimated. The ellipses in (a) and (b) indicate the same data points. FIG. 3. Current-induced switching behavior of jjTa(1)jPt(4)jHf(0.5) jFeCoB (1.6)jMgO(1.6) jFeCoB(4) three-terminal devices. (a) Schematic structure of jjTajPtjHfjFeCoB jMgOjFeCoB three-terminal SHE-MTJ devices. (b) Differential resistance versus total current Iapplied to the channel at a ramp rate of 0.0013 mA/s for 50 /C2180 nm2MTJ with a 1.2 lm channel. The switching currents are determined to be Ic/C2560:4 mA. Broken lines connect the data points, indicating the magnetic switching events. (c) Plot of switch- ing currents at different ramp rates of 0.0013 mA/s for a 50 /C2180 nm2MTJ with a 1.2 lm channel. Solid lines show fitted results. (d) I0versus channel width wof 70/C2240 nm2devices. The linear fit (line) gives the average cur- rent density J0¼ð1:5560:12Þ/C21011A=m2.222402-3 Nguyen et al. Appl. Phys. Lett. 106, 222402 (2015)J0¼ð1:560:2Þ/C21011A=m2. The same measurement and analysis were also performed for tFeCoB¼1:6n m , tHf¼0:5n m , 7 0 /C2240 nm2devices with different channel widths. As shown in Fig. 3(d), we confirmed that the switch- ing current I0varies linearly with the channel width w,a s expected, and that the average zero-fluctuation switching current density J0¼ð1:660:1Þ/C21011A=m2for that series of devices is consistent with that of the 50 /C2180 nm2MTJ, 1.2lm-wide channel device. We used the results of the measurements of the switch- ing current density J0as plotted in Fig. 4(a) as a function of tHfto calculate nSH, using the measured values for Meffanda mentioned above and the formula8,42 nSH¼2e /C22hl0Msteff FeCoB aHcþMeff 2/C18/C19 =J0: (5) Those latter results are plotted in Fig. 4(b). For 0 /C20tHf /C200:6 nm, the spin torque efficiency fluctuates about the av- erage value nSH¼0:10, with a peak value nSH¼0:1260:02 at both tHf¼0:0 nm and tHf¼0:5 nm. While a quantitative analysis using the drift-diffusion model of these results for the spin torque efficiency of the Pt jHfjFeCoB trilayer struc- tures is conceptually challenging in the tHf/C250:5 nm ultra- thin limit, this is less of an obvious concern for the Pt jFeCoB bilayer samples. If we use qPt¼24lX/C1cm as determined for our samples18andkPt s¼1:2 nm (Ref. 29) (determined for samples having the same electrical resistivity), we have that for the Pt layer Gext¼1:7/C21015X/C01m/C02. Equations (2)and(3) then yield G"#¼GextG"# eff=ðGext/C0G"# effÞ/C253:1/C21015X/C01m/C02 andnPtjFeCoB SH =hPt SH¼0:65, where the latter is a considerably higher ratio than reported for a Pt jPy bilayer nPtjPy SH=hPt SH ¼0:25,33which signifies that our Pt jFeCoB interface has a sig- nificantly higher spin curre nt transmissivity. With nPtjFeCoB SH =hPt SH ¼0:65, the high spin torque efficiency nPtjFeCoB SH ¼0:1260:02 obtained from the switching measurements indicates that hPt SH¼0:1860:03, quite consistent with the spin Hall angle val- ues recently reported from analyzes of experiments on Pt jPy, PtjCo, and Pt jCoFe systems.33,34 Returning to the results from the devices with the Hf insertion layer, while the substantial decrease in nSHwhen tHfis increased from 0.6 nm to 0.76 nm is perhaps qualita- tively consistent with an increased attenuation of the spincurrent by a thicker Hf layer,35the lack of significant varia- tion of nSHfor 0/C20tHf/C200:6 nm is quite surprising in light of the strong suppression of spin pumping, which represents a factor of 4 reduction of G"# effby a Hf insertion only one to two atomic layers thick. Within the drift-diffusion analysis, the most straightforward explanation for this spin pumping reduction is that the spin mixing conductance G"# HfjFeCoB /C28G"# PtjFeCoB, with the alternative being that the Hf layer has a very low spin conductance, 1 =ð2qHfkHf sÞ, together with tHf/C21kHf s. In light of the measured nSHðtHfÞresults (Fig. 4 (b)), there are fundamental challenges for both explanations. A low G"#will enhance the back flow of the spin current from the Hf jFeCoB interface, lowering nSHas implied by Eq.(3). This could be counteracted if Gextis also lowered by a similar degree by the Hf insertion, but since the experimen- tal evidence is that Hf has no significant SHE a low Gext;Hf would also result in a strong attenuation of the spin current from the Pt before it reaches the Hf jFeCoB interface.18 Another possible explanation could be that the Hf insertion results in a decreased Gext;Pt, or an enhanced hPt SH, through intermixing, but the similarity of the averaged resistivity of the Pt layer with and without the Hf insertion, together withthe results of experiments with PtHf alloys that will be reported elsewhere, appears to make this alternative explana- tion unlikely. Given these contradictions between the spinpumping and spin backflow (spin accumulation) predictions of the drift-diffusion equation and the results reported here, we tentatively conclude that with the very thin layers that areemployed in this system, where interfacial scattering is a dominant factor, drift-diffusion simply does not provide an adequate understanding of the essential spin transport details.To achieve that a Boltzmann equation analysis of the interfa- cial spin transmissivity and a more detailed description of the electronic structure of the interface are likely to berequired. In summary, we have maintained a high spin torque effi- ciency n SH¼0:1260:02 in Pt jFeCoB based three terminal SHE-MTJ devices, while substantially reducing the effect of spin pumping in increasing the damping of the thin FeCoB free layer. We have achieved this by introducing a thin, nomi-nally 0.5 nm, Hf layer between the Pt SHE layer and theFeCoB. This reduced the magnetic damping to a/C250:012 without significantly changing the spin torque efficiency and thus subsequently lowered the SHE switching current densityto/C251:6/C210 11A=m2. This value is approximately a factor of 2 lower than achieved previously in similar Ta jCo40Fe40B20 SHE-MTJ devices having much higher resistivity. The decrease in the damping can be attributed to a suppression of spin pumping brought about by a large reduction of the effec- tive spin mixing conductance G"# effof PtjHfjFeCoB compared to PtjFM but in a way that does not reduce the absorption of spin current at the FM interface. Although further theoretical investigation is necessary for a complete explanation and opti-mally to guide further improvements, the experimental deter- mination that Pt jHfjFeCoB samples can provide high spin torque efficiency, together with an electrical resistivity muchless than for b-Ta and b-W, demonstrates clearly that Pt jHf provides an attractive alternative to those materials for anti- damping SHE torque logic devices for which impedance andlow excitation power are important criteria.FIG. 4. (a) Switching current density J0and (b) spin Hall torque efficiency nSHversus Hf thickness for tFeCoB¼1:6 nm (circles) and tFeCoB¼1:8n m (squares) samples. J0achieves a minimum at tHf¼0:5 nm. Within the uncer- tainty, nSH/C250:10 for tHf<0:6 nm with local maxima /C250:12 at tHf¼0a n d 0.5 nm but then decreases for thicker Hf spacer. Dashed lines connect the data points to guide the eye.222402-4 Nguyen et al. Appl. Phys. Lett. 106, 222402 (2015)The authors thank J. Park and Y. Ou of Cornell University; C. T. Boone of NIST; and L. H. V. Lea ˜o of Federal University of Pernambuco (Brazil) for fruitfuldiscussions. We thank S. Parkin of IBM Almaden ResearchCenter for sharing a manuscript reporting a related studyprior to publication. This work was supported in part by theNSF/MRSEC program (DMR-1120296) through the CornellCenter for Materials Research, ONR, and the SamsungElectronics Corporation. We also acknowledge support fromthe NSF (Grant No. ECCS-0335765) through use of theCornell Nanofabrication Facility/National NanofabricationInfrastructure Network. A patent application has been filedon behalf of the authors regarding technology applications ofsome of the findings reported here. 1M. I. Dyakonov and V. I. Perel, Phys. Lett. A 35, 459 (1971). 2J. Hirsch, Phys. Rev. Lett. 83, 1834 (1999). 3A. Hoffmann, IEEE Trans. Magn. 49, 5172 (2013). 4L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett. 106, 036601 (2011). 5A. Ganguly, K. Kondou, H. 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1.347853.pdf

Domainwall dynamic instability Samuel W. Yuan and H. Neal Bertram Citation: Journal of Applied Physics 69, 5874 (1991); doi: 10.1063/1.347853 View online: http://dx.doi.org/10.1063/1.347853 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/69/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Novel Studies of Ferroelectric DomainWall Dynamics AIP Conf. Proc. 1349, 1235 (2011); 10.1063/1.3606312 Low-temperature domain-wall dynamics in weak ferromagnets Low Temp. Phys. 28, 337 (2002); 10.1063/1.1480240 Domainwall dynamics in thick Permalloy films J. Appl. Phys. 73, 5992 (1993); 10.1063/1.353497 Domainwall dynamics in garnet films with orthorhombic anisotropy J. Appl. Phys. 54, 6577 (1983); 10.1063/1.331892 DomainWall Dynamics in Fe from Pulsed NMR Experiments J. Appl. Phys. 39, 440 (1968); 10.1063/1.2163469 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.189.205.30 On: Wed, 10 Dec 2014 19:06:38Domain-wall dynamic instability Samuel W. Yuan and H. Neal Bertram Center for Magnetic Recording Research, University of California, San Diego, La Jolla, California 92093 Computer simulation is utilized to follow the dynamic evolution of a one-dimensional Bloch domain wall. The Landau-Lifshitz equation with Gilbert damping is solved with the I applied field along the domain magnetization. Walker’s solution is verified for the case where the anisotropy field is much larger than the Walker critical field. When the critical field is comparable to the anisotropy field, the wall motion exhibits instability. For-an external field lower than the Walker critical field, after some transient behavior, the steady-state Walker-type wall configuration is reached; when the applied field is above the Walker limit, the initial single wall gradually evolves into an odd number of Rloch walls with consecutively opposite senses. Wall motions for soft ferrite material with different dampings are also investigated. For very small damping the stability holds well. However, for medium damping, it is found that below some critical field, smaller than Walker’s, the steady-state solution is stable. Above that field, the single wall first develops a frontal wake which enlarges until instability when an odd number of walls form. ,_ ,.. I. INTRODUCTION The study bf high-frequency effectsin magnetic heads is of increasing importance since”data rates in magnetic storage technology are reaching 100-200 MHz. Recording heads are made of soft magnetic material and consist in general of a-dense collection of domains in zero external field. The analysis of the dynamic behavior is complicated; however, the study of a simplified 180” Bloch domain wall can serve as a starting point to more complex systems. There has been significant study of the static structure and stable motion of a single domain wall erformed first by Walker”” and subsequent researchers. ‘VgA uniform dc ,.i magnetic field is applied along the easy axis of an infinite uniaxial medium containing a single 180” domain wall. Un- der the simplifying assumption that the magnetizations are in a plane whose normal direction has a fixed angle 4 (see Fig. 1) with the direction of wall motion, the equation of motion can be solved analytically.’ There is a critical field (henceforth called the Walker field) Hc = 27rcNc (where a is the reduced phenomenological damping constant as defined in Sec. II, and Me is the saturation magnetization) beyond which no self-consistent exact solution is available. In an applied field Ho < Hc, the wall plane aquires a fixed azimuthal angle 4 upon application of an external field. The wall moves uniformly in steady state with contracted width compared to a static wall. When Ho > Ho Schryer and Walker2 found approximately that the magnetization NG. 1. Cartesian coordi- * ~~~~ precesses about the field and a periodic component appears in the forward motion of the wall. Here we examine the motion of a Bloch wall by nu- merically solving the Landau-Lifshitz equation with Gilbert damping.’ Since a one-dimensional wall is consid- ered, discretization is only required along the wall motion direction. Without any a priori assumptions and con- straints introduced by the aforementioned analytical ap- proach, we are able to investigate the validity and stability of the Walker classical wall motion,’ and seek a more gen- eral solution. II. MODEL AND COMPUTATIONAL SCHEME _ A detailed study of wall motion involves discretization of a continuous medium and solution of the Landau- Lifshitz-Gilbert equation of motion at each space-time mesh point: dM ~==?+W%xd - j& Mx WxJ&), where the effective field is (1) (2) Using the Cartesian coordinate system depicted in Fig. 1, the total energy density of the domain wall can be writ- ten as follows, consisting of contributions from exchange, uniaxial anisotropy, magnetostatic, and Zeeman energy, re- spectively, with applied field pointing in the easy axis: (3) Note that the normally long-range magnetostatic interac- tion reduces to a local point field in the case of a one- dimensional variation, which greatly simplifies the compu- tation. A computer simulation adopting the Adams algo- rithm’ is used to follow the detailed dynamic evolution of 5874 J. Appl. Phys. 69 (8), 15 April 1991 0021-8979/91/085874-03$03.00 @ 1991 American institute of Physics 5874 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.189.205.30 On: Wed, 10 Dec 2014 19:06:38the domain-wall motion. An infinite domain wall is. ap- proximated by a chain of spins with finite length (about 30 times the domain-wall thickness’) with its end spins pinned at 8 = 0 and P. The mesh discretization follows the wall motion, so that excessive discretization is not re- quired. The spatial and time measurements are scaled by the conventional domain wall thickness m (Ref. 9) and gyromagnetic ratio 7, respectively. Therefore the equation of motion can be rewritten as dm x=mXh - amX CmXh), , where m = M/M,, a = A./y, and we re$ace A by K wher- ever the former occurs in the (scaled) effective field h. We have used 300 as well as 600 spatial mesh points; with essentially the same results. The time step has been selected (0.01 or 0.1 in scaled units) so as not to cause a serious stiffness problem” of Eq. (4)) and to permit a rea- sonable computing time. Numerical error has been typi- cally controlled to within 5.0 X 10 - ‘. Ill. RESULTS AND DISCUSSION A. YIG material (4rM,, z Hk) Starting from a static Bloch wall without external field, the wall behavior after applying a uniform dc field is fol- lowed. We have used the same parameters for yttrium iron garnet material as those used in a related paper by Schryer and Walker,’ and are able to reproduce their results. The time dependence of magnetization, domain-wall width, and velocity is examined. For example, for 4?rM, = -1700 Oe (MO = 135 emu/cm3), anisotropy field Hk = 2K/Mo = 80 Oe, and damping constant a = 0.01, the Walker field is Hc = 8.5 Oe. In this case Hk $ Hc. The numerical results are in excellent agreement with Ref. 2: The wall keeps its identity throughout all ranges of applied magnetic field. When the applied field is below the Walker field, the wall motion tends exponentially to the steady-state Walker so- lution. Above the Walker field, however, the domain wall undergoes periodic oscillation superimposed on a net for- ward motion. 1 i For damping constant a = 0.1, Hk then becomes com- parable with He For external field Ho < Hc, a transient bow wave oscillation first appears, which makes the mag- netization vectors close to the wall region precess at local sites around the applied field (Fig. 2). A steady-state Walker wall is formed soon after this transient. When the applied field Ho is large compared to HQ a transient rear wake appears first. This wake is a spatial oscillation of the magnetization vectors following the wall motion. Subse- quently .a large frontal part of the wall rotates beyond 180”. As this trend increases, the wake remains in the rear part of the wall. The frontal magnetizations eventually attain rotations that are beyond 360” with the z-axis destroying the single Bloch-wall identity. This is equivalent to having two head-to-tail 180” walls with the same winding sense, with a separate wall in the front rotating in an opposite .-. FIG. 2. Typical transient bow waves encountered after an ex- ternal field is applied to an ini- tially static wall. Notice the rear wakes following the waI1 center. The arrow below shows the wall motion direction (different per- spective) . direction. (Figure 4 should be referred to for a graphic illustration of this process.) B. Ferrites (4dlI,,g Hk) Ferrite materials have been studied with typical pa- rameters: 4?rMo = 5000 Oe, anisotropy field H,+ = 25 Oe. For small damping a = 0.001, the dynamic evolution yields the subcritical and supercriticai wall velocity versus time relations as shown in Fig. 3. For Ho < Hc, a steady- state Walker configuration is obtained, with the velocity reaching a slight maximum and then declining to its equi- librium value [Fig. 3 (a)]; when Ho > Hc, the same charac- teristic oscillation [Fig. 3(b)] occurs as that shown in Ref. The steady-state Walker configuration is used as an initial condition for solving the equation of motion in order to study stability.” A linear stability study by Magyari and Thomas” yields a Walker solution linearly stable up to a critical value less than the Walker limit. This critical field corresponds to the domain-wall maximum velocity, be- yond which the Daring mass13 changes from a positive to a negative value:?& this critical field, the Doring mass is infinite, which physically indicates the onset of instability. However, our results show that linear stability analysis is not, adequate for the purpose of describing general stability properties to this highly nonlinear problem. For a damping constant a = 0.1, the Walker field is 250 Oe, and the crit- ical field corresponding to the maximum wall velocity12 is 124 Oe. Numerical results show that instability occurs when H o z 70 Oe, which is much less than that predicted by Magyari and Thomas. The single wall [Fig: 4(a)] first develops a frontal wake, which rotates more than 180” . . i .t. t FIG. 3. Velocity vs time relation for ferrites, 47-&f; = 560COe, a = 0.001, the Walker field Hem= 2.5 06~ (a) subcritiial, Ho= 2.0 Oe, (b) supercrit- ical, Ho = 3.0 Oe. 5875 J. Appl. Phys., Vol. 69, No. 8,15 April 1991 S. W. Yuan and H. N. Bet-tram 5875 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.189.205.30 On: Wed, 10 Dec 2014 19:06:38(4 3.2r r- 2.4 I i 4 0 1.6 t = o,6 0.8 (b) :::E r- e 1.6 1 I t = 1.3 0.8 0.0- 0 10 20 30 x (Cl 3.2 L I/ -- 1 2.4 0 1.6 0.8 (d) x FIG. 4. WaII configurations indicating instability of the Walker solution. Figures to the left show the spatial variations of the 0 tingle (defined in Fig. 1) of the wall. 4?rM, = 5000 Oe, a = 0.1, Hk = 25 Oe, Ho = 75 Oe. (a) The Walker steady-state wall profile as initial condition, (b) frontal wake appears as time progresses, (c) enlarges to render the original single wall unstable, (d) at last generates multiwails. The arrow below (a) is the wall motion direction. The vertical arrow above each vector plot indicates the wall center in a co-moving coordinate. , from the favored domain magnetization direction [Fig. 4(b)]. The magnetization vectors within this wake con- tinue to rotate around the wall motion direction [Fig. 4(c)], extending beyond 360” [Fig. 4(d)]. Thus the frontal wake enlarges until it generates two extra walls similar to the case for YIG materials when Hk z HC The time re- quired for the original wall to become unstable decreases as the applied field is increased. For even higher external fields, instability sets in with an avalanche generation of extra walls. For Ho > 200 Oe, the initially single wall evolves into three walls. As the frontal wall moves further ahead, an extra pair of head-to- tail walls soon appear in the rear. This is followed by a further increase in the number of walls. As many as nine walls have been seen in this case. There have been intensive studies of similar type of domain-wall dynamics in the language of nonlinear soliton problems. 14-16 Using inverse scattering methods,17 an ana- lytic solution of the general form of the Landau-Lifshitz equation without damping has been achieved, which allows soliton, bion, and many-soliton solutions.‘4 These results show trends that are in good agreement with the results presented here. IV. CONCLUSION The Walker solution to the one-dimensional domain wall motion is but one of many possible solutions to a coupled nonlinear problem. Its stability range is limited by material parameters. Walker’s solution’,’ is verified for the case where the anisotropy field is much larger than the Walker field: When the applied field is below the Walker field, wall motion settles to the steady-state Walker solu- tion, either exponentially or after transients; above the crit- ical field, however, the domain wall has a periodic compo- nent in its forward motion. When the Walker field is comparable to the anisotropy field, wall motion exhibits instability. It is found that above a certain threshold of external field, the initially single wall breaks down to mul- tiwalls, indicating the onset of chaotic or turbulent behav- ior 18,19 ACKNOWLEDGMENTS The authors would like to thank Drs. John Sloncze- wski, Neil Smith, and Jian-Gang Zhu for useful discus- sions. This research was supported by the National Science Foundation under Grant No. NMR-87-0742 1. ‘J. F. Dillon, in Magnetism, edited by G. T. Rado and H. 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Pokrovsky, S’olitons (North-Holland, Amsterdam, 1986). I7 M J Ablowitz and H. Segur, Solitons and the Inverse Scattering Trans- . . form (SIAM, Philadelphia, 198 1). ‘sF. Waldner, J. Magn. Magn. Mater. 31-34, 1015 (1983). “H. Suhl and X. Y. Zhang, J. Appl. Phys, 61, 4216 (1987). 5876 J. Appl. Phys., Vol. 69, No. 8,15 April 1991 S. W. Yuan and H. N. Bertram 5876 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.189.205.30 On: Wed, 10 Dec 2014 19:06:38

1.3561753.pdf

Micromagnetic study of switching boundary of a spin torque nanodevice Yan Zhou,1,a/H20850Johan Åkerman,1,2and Jonathan Z. Sun3 1Department of Microelectronics and Applied Physics, Royal Institute of Technology, Electrum 229, 164 40 Kista, Sweden 2Department of Physics, University of Gothenburg, 412 96 Gothenburg, Sweden 3IBM T. J. Watson Research Center, P.O. Box 218, Yorktown Heights, New York 10598, USA /H20849Received 11 August 2010; accepted 12 February 2011; published online 7 March 2011 /H20850 We report on a numerical study of the micromagnetic switching process of a nanostructured spin torque device. We show that incoherent spin waves can be excited over a wide range of current andfield even at zero temperature. These large amplitude, incoherent, and nonzero kspin wave modes are shown to alter the switching phase boundary from that calculated within a macrospin model. Thepresence of telegraphic transitions between different spin wave modes may also contribute to theso-called back-hopping phenomenon where the switching probability varies nonmonotonically withincreasing bias current. © 2011 American Institute of Physics ./H20851doi:10.1063/1.3561753 /H20852 Spin transfer torque /H20849STT /H20850provides an effective means to switch the magnetization of a free layer in a nanostruc-tured spin-valve /H20849SV /H20850or magnetic tunnel junction /H20849MTJ /H20850. STT is under intensive investigation for possible applicationas random access memory 1–13and microwave signal generators.8,13–21To properly study the switching perfor- mance of such nanodevices, micromagnetic simulation be-comes a necessary and powerful tool, especially for samplesizes /H20849typically above 30 nm /H20850where higher order /H20849k/H110220/H20850spin wave modes become important. 18,22–28Recent experimental work also suggests that, unlike in all-metal SVs, STT switch-ing in MgO-barrier-based MTJs involves higher energy mag-netic excitations, and thus, full-scale micromagnetic simula-tions are required for a more complete understanding of theswitching processes. 29–32 In some STT experiments using a /H20841Co /H20841Cu /H20841Co /H20841current- perpendicular-to-plane /H20849CPP /H20850SV nanopillar,33it was found that the measured switching phase boundary in the current-magnetic field /H20849I;H/H20850space gives an intercept with the ap- plied field axis of only 430 Oe, which is well below the value of 2 /H9266Msof Co /H208499 kOe /H20850expected from a single domain model. The effect of finite temperature alone is insufficient toaccount for this discrepancy. In this work, we perform a full-scale micromagnetic analysis including the current-generatedoersted field and find that the higher order spin wave modescan be excited in the case under study. The calculated inter-cept agrees with the experimental value. Our simulations aredone at zero kelvin. Even so, there appears to be large re-gions in parameter space where STT can generate a largeamount of incoherent spin waves, effectively softening the material magnetically, leading to a distributed, probabilisticswitching behavior. The free layer of Co material has the shape of thin rect- angular box with lateral dimensions of 50 nm /H1100310 nm and a thickness of 3 nm. The saturation magnetization is1400 emu /cm 3. The other material and testing parameters are the same as in Ref. 33. To emulate the macrospin limit in our micromagnetic simulation, we first use an unrealisticallylarge exchange constant A=30 /H9262erg /cm, which is one order of magnitude larger than typical exchange constant for CoA=3/H9262erg /cm. The resulting switching phase diagram is shown in Fig. 1. The switching threshold positions in /H20849I,H/H20850 phase-space can be mainly divided into three regions, P, AP, and O, which is a transitional region separating P and APregions. In this region, the free layer magnetization is driveninto a coherent precession with spin wave vector value k =0. The three insets in Fig. 1show typical time-domain traces as examples of the system’s response to a small dis-turbance away from the easy-axis. For very small current,and when the applied field is smaller than the anisotropyfield, only the P state is stable. When increasing the currentup to the onset of instability boundary, the cone angle of themagnetization starts to open up. We also found the existenceof coherent oscillations /H20849gray region labeled “O” in Fig. 1/H20850, where the nonlinearity can stabilize a limit cycle withoutnecessitating a final global magnetization reversal. We haveconfirmed that such coherent precession persists throughoutthe entire transition region /H20849O/H20850from the instability line to the switching line. beyond the switching line, we get completemagnetization reversal /H20849AP /H20850. We can now compare our large- Amicromagnetic simulation /H20849solid line /H20850with a small- angle linearized analytical solution within a macrospin a/H20850Currently with Hong Kong Institute of Technology, Hong Kong; electronic mail: yanzhouy@hotmail.com.500600 Micromagnetic 1.0)Macrospin 300400 0 1 02 03 04 00.5 Time(ns)Mx(ave )ngtyOe) 200300 01ave)1.00x(ave) Swi tchinInstabili tHapp(O P AP O 0 1 2 3 4 5 6 70100 02468-1Mx(a Time (ns)0123450.99Mx Time(ns) 0 1 2 3 4 5 6 7 I( m A ) FIG. 1. /H20849Color online /H20850Phase boundaries in /H20849I;H/H20850calculated using an ex- change constant of A=30/H9262erg /cm. Blue /H20849dark gray /H20850: parallel state /H20849P/H20850; yellow /H20849light gray /H20850: antiparallel state /H20849AP /H20850; and gray: oscillatory dynamic states /H20849O/H20850. Inset: the temporal evolution of the average magnetization along the easy-axis /H20849x/H20850for different phase states P, O, and AP. The two dotted lines represent the instability and switching boundaries in the figure, the details ofwhich is explained in the main text.APPLIED PHYSICS LETTERS 98, 102501 /H208492011 /H20850 0003-6951/2011/98 /H2084910/H20850/102501/3/$30.00 © 2011 American Institute of Physics 98, 102501-1model /H20849dotted line /H20850. The two approaches give approximately the same switching phase boundary, indicating that the largerexchange constant substantially suppresses any inhomoge-neous excitations of the magnetization. The remaining smalldifference between a pure macrospin calculation and our mi-cromagnetic calculation is due to the nonuniform magneto-static field at the sample edges since the sample shape isassumed to be a simple rectangular box and not an oblateellipsoid. We now turn to a more realistic situation, where the correct exchange constant of A=3 /H9262erg /cm is used in the micromagnetic simulation; the temperature is still set to ab-solute zero. The resulting phase boundaries in /H20849I,H/H20850are plot- ted in Fig. 2. To highlight the behavior of the different phases, we make a cut at an applied field of 300 Oe andstudy the system as a function of increasing drive current.For small current /H20849e.g., I=3.8 mA /H20850, below the instability threshold, we find the occurrence of static nonuniform cant-ing of the spins in the free layer /H20849see the first inset in Fig. 2/H20850. In this mode, the magnetization is slightly tilted away fromthe equilibrium parallel state. If the drive current is in-creased, different chaotic dynamic phases appear, represent-ing incoherent spin wave excitations and even switching, asshown by the solid red circle /H208492/H20850and solid gray circle /H208493/H20850in region C, respectively. In this chaotic dynamic region /H20849and at zero temperature /H20850incoherent spin waves lose or regain their stability over very small current or field intervals. Furtherincreasing the current leads to a complete and deterministicreversal of the free layer magnetization /H20849see the last inset where I=5.5 mA /H20850. From our simulations, we define an “in- stability line” separating P and C regions and a “determinis-tic switching line” demarcating the boundary between C andAP regions. Even at zero kelvin, our simulations show the presence of such chaotic switching regime /H20849C/H20850which corresponds to a distributed and probabilistic switching region for realistic ex-change constant values. 30The chaotic dynamics observed for realistic exchange constant values is partially due to the com-petition of exchange-stiffness energy and the energy result-ing from nonuniform magnetostatic interactions. In addition,the current-induced oersted field may also play an important role in initiating such high order incoherent spin wave modesdue to its highly inhomogeneous spatial distribution. The ex-citation of incoherent spin waves can be reduced by a de-crease in the magnetostatic-to-exchange energy ratio such asin the case of an increased value of exchange-stiffness con-stant Aas shown above. We have also checked several inter- mediate exchange constant values, and have found that the/H20849lower /H20850onset current of instabilities does not change much with Abut the upper boundary of the current region where the chaotic excitations exist increases with decreasing A.W e also checked that the critical thickness of the free layer co-balt, for the transition of the system from single domain tochaotic behavior, decreases with decreasing A, consistent with the findings by other groups. 26 The time-evolution of the unit vector of the free layer magnetization mˆis found from the Landau–Lifshitz–Gilbert– Slonczewski equation1,2 dmˆ dt=− /H20841/H9253/H20841mˆ/H11003Heff+/H9251mˆ/H11003dmˆ dt+aJmˆ/H11003 /H20849mˆ/H11003Mˆ/H20850, /H208491/H20850 and the last term is Slonczewski spin torque with the mag- nitude aJ=/H20849/H20841/H9253/H20841/H6036/H9257I/2eV/H92620Ms/H20850.Vis the volume of the free layer. Msis its saturation magnetization and /H9257is the spin polarization of the current. Equation /H208491/H20850can be transformed into the following set of differential equations in a spherical coordinate system in thesmall cone angle limit,34 d/H9258 d/H9270=−/H9258/H20853/H9251/H208491+h/H20850+hp/H20851sin /H20849/H9278/H20850+/H9251cos /H20849/H9278/H20850/H20852cos /H20849/H9278/H20850+hs/H20854, d/H9278 d/H9270=−1− hp/H20851cos /H20849/H9278/H20850−/H9251sin /H20849/H9278/H20850−h+hs/H9251/H20852. /H208492/H20850 The dimensionless units are: /H9270=/H9253Hkt//H208491+/H92512/H20850,h=H/Hk, where His the applied field in the same direction as the easy-axis direction defined by Hk,hp=Kp/K=4/H9266Ms/Kis the reduced easy-plane anisotropy field, with K=1 /2MsHkbeing the uniaxial anisotropy energy. hs=/H20849/H6036/2e/H20850/H9257I/VM sHkis the dimensionless spin current density. By treating the damping and spin current as a perturbation, the average rate of energychange dU /d /H9270of the free layer can be obtained as34 1 K/H20883dU d/H9270/H20884=−2 /H208491+h/H20850/H20875/H208731+h+1 2hp/H20874/H9251+hs/H20876/H925802, /H208493/H20850 where /H92580is the initial value of the polar angle. The above equation gives the on-axis stability threshold for spin-current-driven motion at the small cone-angle limit.Thereby, we have the threshold for an instability toward anincreasing cone angle within the framework of macrospinmodel 33,34 Ic/H20849H/H20850=/H208492e//H6036/H20850/H20849/H9251//H9257/H20850/H20849VM S/H20850/H20849H+Hk+2/H9266Ms/H20850. /H208494/H20850 From Eq. /H208494/H20850, we get the straightforward solution for critical switching current without applied magnetic field Ic/H20849H=0 /H20850 =/H208492e//H6036/H20850/H20849/H9251//H9257/H20850/H20849VM S/H20850/H20849Hk+2/H9266Ms/H20850and the slope of the switching boundary of the phase diagram as a function of I andH,dIdH =/H208492e//H6036/H20850/H20849/H9251//H9257/H20850/H20849VM S/H20850. These values are known to differ from experimental observations by more than a fac- tor of 5.33To address this issue, we seek to examine the corresponding slope and intercept as discussed above but500600 43 2 1 01x(ave) 300400 012 Time (ns)02 0 4 0 Time (ns)02 0 4 0-1Mx Time (ns)Oe) 200300 3 4 32 1erm inisticwitching C PHapp(O AP tability 0 1 2 3 4 5 6 70100 DeterSw Inst 0 1 2 3 4 5 6 7 I (mA) FIG. 2. /H20849Color online /H20850Phase boundaries in /H20849I,H/H20850. Blue /H20849gray /H20850: parallel state /H20849P/H20850; yellow /H20849light gray /H20850: antiparallel state /H20849AP /H20850; and gray /H20849black /H20850: chaotic dynamics state /H20849c/H20850. Inset: the magnetization trajectories for different currents when Happ=300 Oe /H208491/H20850I=3.25 mA, /H208492/H20850I=4.01 mA, /H208493/H20850I=4.21 mA, and /H208494/H20850I=5.34 mA. The solid red circle /H208492/H20850and solid gray circle /H208493/H20850in region C represent a typical incoherent spin wave excitation and a complete reversalof the free layer magnetization, respectively. The instability line separates Pand C states and the deterministic switching line gives the boundary of Cand AP states.102501-2 Zhou, Åkerman, and Sun Appl. Phys. Lett. 98, 102501 /H208492011 /H20850using full micromagnetics simulation. This is shown in Fig. 2. Obviously, such comparison will depends on the coeffi- cient of /H208492e//H6036/H20850/H20849/H9251//H9257/H20850/H20849VM S/H20850, some of which are not directly determined by experiments for a given device. By comparing the ratio of the intercept and slope, however, one eliminate toa large degree the uncertainties due to these parameters /H20849they cancel out in the ratio /H20850, and thus give a more reliable com- parison of the model with experiment. The initial comparisonbetween macrospin model and experiments were also donethis way, as discussed in Refs. 33and35. The ratio of the intercept and slope can be readily extracted from the abovefigures and is highlighted in Fig. 3with the red /H20849black /H20850circle. As shown in Fig. 3, the micromagnetic simulations give good agreement with the interception value obtained in ex-periments, while the single-mode spin wave model gives amuch higher value of the ratio. This suggests that the modi-fication of the switching boundary involves a collection ofspin wave modes, whose systematic dependence in terms ofthreshold values as well as occupation numbers on spin-torque-providing current Ias well as the applied field H, give rise to an additional field-current dependence, resulting in adifferent global switching threshold from those predicted bysingle-mode models. It has recently been experimentally demonstrated that the spin torque driven switching of nanosized MTJ involvesa significant amount of backhopping behavior, 30,36where the switching probability varies nonmonotonically upon increas-ing the bias voltage. This can be explained, at least partially,by the higher order incoherent spin wave excitations ob-served in our micromagnetics simulations /H20849see C region of Fig. 2/H20850. The detailed micromagnetics study of the back- hopping behavior needs future investigation. In conclusion, we have carried out a full-scale micro- magnetic analysis of STT switching in /H20841Co /H20841Cu /H20841Co /H20841CPP SV nanopillars. We find that STT can generate a large amount ofincoherent spin waves, even at zero temperature, which leadsto an effective softening of the magnetic properties. The re-sulting phase diagrams show regions of both deterministicswitching and chaotic behavior. A comparison with STT ex-periments show good agreement of the switching boundarylines. Since the strong effective softening is not captured inmacrospin calculations, our results seem to explain the typi-cal large discrepancy between experimentally measured STT switching phase diagrams, and calculations based on a mac-rospin approximation. Support from The Swedish Foundation for strategic Re- search /H20849SSF /H20850, The Swedish Research Council /H20849VR /H20850, and the Göran Gustafsson Foundation is gratefully acknowledged.Johan Åkerman is a Royal Swedish Academy of SciencesResearch Fellow supported by a grant from the Knut andAlice Wallenberg Foundation. The micromagnetic results areobtained using the LLG simulator developed by Mike Schei-nfein. We thank R. K. Dumas and S. Bonetti for criticalreading of the manuscript. 1L. Berger, Phys. Rev. B 54, 9353 /H208491996 /H20850. 2J. C. Slonczewski, J. Magn. Magn. Mater. 195,2 6 1 /H208491999 /H20850. 3J. Z. Sun, J. Magn. Magn. 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The dashed dotted line is the extended deterministic switching line given by micromagnetics simulationsin Fig. 2. The solid line represents the extended experimental switching line /H20849Ref. 33/H20850. The red /H20849black /H20850circle highlights the comparison of the intercep- tion values of the boundary line with H appaxis between micromagnetic simulation and experiment /H20849Ref. 33/H20850.102501-3 Zhou, Åkerman, and Sun Appl. Phys. Lett. 98, 102501 /H208492011 /H20850Applied Physics Letters is copyrighted by the American Institute of Physics (AIP). Redistribution of journal material is subject to the AIP online journal license and/or AIP copyright. For more information, see http://ojps.aip.org/aplo/aplcr.jsp

1.1908755.pdf

Fifty-Fourth Meeting of the Acoustical Society of America Citation: The Journal of the Acoustical Society of America 29, 1248 (1957); doi: 10.1121/1.1908755 View online: https://doi.org/10.1121/1.1908755 View Table of Contents: http://asa.scitation.org/toc/jas/29/11 Published by the Acoustical Society of AmericaTHE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA VOLUME 29, NUMBER 11 NOVEMBER, 1957 Program of the Fifty-Fourth Meeting of the Acoustical Society of America UNIVERSITY OF MICHIGAN, ANN ARBOR, MICHIGAN, OCTOBER 24, 25, ANt) 26, 1957 Session A. Acoustics, Microwaves, and the Solid State WINSTON E. KOCIC, Chairman Invited Papers A1. MicrowavemAcoustic Analog Experiments at Giittingen. ERWIN MEYER, III. Physikalisches Institut der Universitiit, GSttingen, Germany. A2. Acoustically Induced Transition between Nuclear Spin Levels. W. H. TANTILLA, Michigan State University, East Lansing, Michigan. A3. Interactions of Acoustic and Electromagnetic Waves in Solid-State Materials. G. WEINREICa, Bell Telephone Laboratories, Murray Hill, New Jersey. A4. Some Low-Temperature Acoustic Properties of Solid-State Materials. ROBERT W. MORSE, Brown University, Providence, Rhode Island. A5. Acoustic Research in England with Particular Reference to the Physics Department, Imperial College. R. W. B. STEPIES, Imperial College, University of London. Session B. Hearing I. J. c. R. LICKLIDER, Chairman Invited Papers B1. Adaptation in Sensory Systems. J.P. EGAN' AND J. D. MILLER, Hearing and Communication Laboratory, University of Indiana, Bloomington, Indiana. B2. Measurement of Auditory Adaptation. J. Doa.Lr) HARRIS, Medical Research Laboratory, Sub- marine Base, New London, Connecticut. Contributed Papers B3. On the Relationship of the Intensity and Duration of Noise Exposure to Temporary Threshold Shift. WALTF. R SPIETH, Operational Applications Laboratory, Air Force Cambridge Research Center, Bolling Air Force Base, D. C.-- Evidence from industrial hearing surveys suggests that a hazardous noise exposure criterion can be established on the basis of an 8-hr exposure. It would be desirable to extend this criterion to shorter durations. For example, in many military situations, personnel may be exposed to noises of extremely high intensities but for relatively short durations, as in the operation of jet afterburners. To the extent that the temporary threshold shift TTS is related to more permanent hearing loss, this study attempts to explore the extension of the hazardous criterion to shorter durations. Specifically, the exchange relations between intensity and duration of a white noise, required for equal TTS, was examined. For TTS measured at 4 kc following white noise exposure, the exchange relation is, approximately, --6 db of noise level per doubling of exposure-duration. This finding is in agreement with previously reported research from this, and other laboratories. In the present experiment, the range of conditions over which this relation holds is 4 min exposure at 118 db to 32 min at 100 db. However, 130 db for ! min to 124 db for 2 min of exposure produce definitely less TTS than the range of condi- tions cited above. B4. Intensity of Noise Exposure and the Time-Course of Recovery from Temporary Hearing Loss. WILLIAM J. TRIT- TIPOE AND WALTER SPIETH, Operational Applications Labora- tory, AFCRC, Bolling Air Force Base, D. C.--It would be reasonable to expect that temporary threshold shift (TTS) following exposure to higher sound levels would be an increas- ing function of the exposure level. However, numerous in- vestigators have observed that when duration of exposure is held constant, a greater TTS may occur after exposure to a less intense level than after a more intense level. This is frequently observed when the TTS is measured soon after exposure, but less frequently observed when the TTS is measured 3 min or longer after exposure. Thus, Miller found that 3 min of white noise at 115 db produced more TTS than at 120 db (measured 6 min later at 4000 cps). In the present study TTS was measured by the Bdkdsy methodat 4000 and 6000 cy for 10 min following exposure to 3 min durations of thermal noise. Five different noise levels were used. They ranged, in 5 db steps, from 108 to 128 db. Each of 9 ears was run twice on each noise condition. Only ! ear showed a definite pattern of decreasing TTS with increasing intensity. B5. Binaural versus Monaural Auditory Fatigue. IRA J. HIRSH, Central Institute for the Deaf, St. Louis, Missouri.m Theoretical considerations regarding the interaction between 1248 FIFTY-FOURTH MEETING 1249 the two ears suggest the possibility that the amount of auditory fatigue (threshold elevation) in one ear following exposure to sound will depend, in part, upon whether the exposure was directed at that ear only or at both ears. Ex- posures to noise and to tone were directed at either ear and at both ears of five observers, and continuous threshold recording of the absolute threshold for one ear was made following each exposure. The results indicate no significant difference in the fatiguing effect in one ear between exposures on that ear and exposures to both ears. (This research was supported in whole or in part by the U.S. Air Force under Contract No. AF33(616)-3505, monitored by Aero Medical Laboratory.) B6. Electrical Responses of the Auditory System to Differ- ent Types of Transient Stimuli. E. DE BOER, Research Labora- tory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts.--In physiological experiments on the cat's auditory system clicks have often been used because of their ability to produce synchronized neural firings. In this study the class of impulsive stimuli has been extended. The electrical signals fed to the earphone are close approxima- tions to members of the family of singularity functions. In the present experiments doublets, impulses, step, and ramp functions have been used. Each of these functions is the time integral of the previous one, with a conversion factor involving the time dimension. The actual wave form of the acoustic stimulus at the animal's eardrum could not be observed. The above-mentioned integral relation, however, holds for what may be thought of as the linear part of the system. As a typical result, it has been found that the peripheral neural response, N, to the step is equal to that of the pulse pro- vided the pulse has the same height as the step, and a length of 20 sec. This holds over a fairly large range of amplitudes (20-40 db). Similar results have been found for the other signals. Preliminary results will also be reported with respect to both microphonic and cortical responses to the above- mentioned stimuli. [This work was supported in part by the U.S. Army (Signal Corps), the U.S. Air Force (Office of Scientific Research, Air Research and Development Com- mand), and the U.S. Navy (Office of Naval Research).-I no off-responses in the anesthetized animal (burst durations shorter than 60 msec and fall times longer than 10 msec). These findings suggest that these off-responses in unanesthe- tized animals may involve neurons other than those which give rise to on-responses' the possibility that inhibitory neuronal links at levels below the auditory cortex are involved will be discussed. [This work was supported in part by the U.S. Army (Signal Corps), the U.S. Air Force (Office of Scientific Research, Air Research and Development Com- mand), ard the U.S. Navy (Office of Naval Research).] B8. Harmonic Distortion in Cochlear Models. JUERGEN TONNDORF, University Hospitals, Iowa City, Iowa.--The en- velope over the train of waves traveling along the cochlear partition is asymmetrical: the distal slope being steeper than the proximal one [-G. yon Bdkdsy, J. Acoust. Soc. Am. 19,452 (1947)]. In experiments on models, this asymmetry was seen to increase with intensity. Since the mean revolving velocity of Bdkdsy's "eddies" also varies with intensity, artificial "eddies" were introduced into the model while the intensity remained constant. Thereby the asymmetry of the envelope was also increased. Propulsion of the "perilymphatic" fluids by the "eddies" converts particle motion (ordinarily in closed orbits) [-Juergen Tonndorf, J. Acoust. Soc. Am. 5, 558 (1957)-1 into cycloids. As the eddies accelerate along the partition (see the paper by Tonndorf), the cycloids expand with distance, its velocity attaining a sawtooth-like pattern. The asymmetry of the envelope can be explained satisfactorily by the combined effect upon the partition of the distorted cycloids in both "perilymphatic scalae." However, the same theoretical considerations lead to asymmetry of the displace- ment pattern of the partition and to an effect similar to peak clipping, both of which increase with distance. This distorted displacement pattern in the distal region of the partition in turn affects particle motion in the adjacent fluids, whence the distortion is propagated proximally by the eddies to be resolved in the usual way (see the paper by Tonndorf) along the cochlear partition. [This research was supported in whole or in part by the U.S. Air Force, under Contract No. AF-41 (657) 148, monitored by the School of Aviation Medicine, UASF, Randolph Air Force Base, Texas.-I B7. Off-Responses to Noise Bursts at the Auditory Cortex of the Cat. T. T. SANDEL AND N. Y-S. KIANG, Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts.--In a previous report (November, 1956, 52nd meeting of the Acoustical Society) we concluded that cortical off-responses observed in cats under barbiturate anesthesia are in fact responses to acoustic transients that appear in the signal as it is turned off. We have since found in unanesthetized cats with high spinal sections off-responses that seem to occur in the absence of such acoustic transients. This conclusion is based on the following considerations; first, these off-responses are observed when bursts of noise are turned off and are abolished by injections of barbiturates: secondly, the latencies of these off-responses are 6-8 msec longer than the latencies of the previously reported off- responses to acoustic transients: and thirdly, these off- responses occur also under stimulus conditions which produce B9. Endolymphatic Oxygen Tension in the Cochlea of the Guinea Pig. GEORGE MISRAHY, K. HILDRETH, E. W. SHINABARGER, L. CLARK, AND E. RICE, Wright Air Develop- ment Center, Wright-Patterson Air Force Base, Ohio.--Oxygen tension of the endolymph in the scala media of the guinea pig cochlea was measured using a micropolarographic technique. Near the stria vascularis, the oxygen tension was 55-75 mm Hg. Deeper in the scala media it decreased gradually to !6-25 mm Hg. Following sound the oxygen tension fell below normal with very gradual recovery. Breathing pure nitrogen reduced the oxygen tension rapidly. On rebreathing air it usually returned to normal as rapidly overshooting for one or two minutes. If recovery from anoxia was slow no overshoot occurred. Breathing 10% carbon dioxide in air increased the oxygen tension more than pure oxygen. On return to normal air the oxygen tension fell back to normal or slightly below if the animal had previously breathed pure oxygen. 1250 ACOUSTICAL SOCIETY OF AMERICA Session C. Shock and Vibration FRF. D MINTZ, Chairman Invited Papers C1. Displacement Measurement by Electronic Counting. SEYMOUR EnELMAN, ERNF. ST R. SMITH, ANn EARLE JONES, National Bureau of Standards, Washington, D. C.--This paper reports further progress in our development of methods for calibrating vibration pickups using the wavelength of light as the standard length. Methods described previously J. Acoust. Soc. Am. 27, 728 (1955);28, 780 (1956) and to be published are fatiguing, time consuming, and require trained personnel. Use of a photomultiplier and electronic counter to count interference fringes passing a slit makes it possible to measure fairly large as well as small amplitudes over a wide frequency range while the use of an averaging technique improves the sensitivity of the counting method to a small fraction of a wave- length of light. Measurement by counting is rapid and easy and can be performed by untrained personnel after the equipment is set up working. C2. Instrumentation of a Random Vibration System to Meet Current Test Specifications. DORMAN E. PRIEST, The Calldyne Company, Winchester, Massachusetts.--Common random test specifications include a shaped random vibration level in g per cps, a load in pounds, and a frequency region in which equalization for i, rregularities of response are specified. By a straightforward process the size and characteristics of the shaker and other system components can be determined which will satisfy most of the important factors involved. Important limitations imposed by the nonlinear resonant load reactions, the high-frequency resonances of the shaker armature structures, and the peak voltage limit of the power amplifier make a new approach to part of the specifications desirable. A revision of the usual test method is suggested which avoids impractical attempts to equalize for these load reac- tions and deletes certain high-frequency regions. In these, a complex pattern of standing waves in the shaker table, fixture, and specimen does not allow the usual definition of the acceleration level. C3. Role of Electrodynamic Shaker System in Simulating Vibrational Environments. KENNETH J. METZCAR, The Calidyne Company, Winchester, Massachusetts.--The basic philosophies of simulating vibrational environments are examined in the light of practical electrodynamic shaker systems. Two basically different approaches to simulation are discussed, both of which assume that a component attached to a structure is the subject for test. A distinction is made between the field measurements used as a basis for simulation which represent the responses of particular structures to an environment and the actual environmental forces. The first approach to simulation replaces the structure on which the component is mounted by a shaker. The motion of the shaker is then controlled in such a manner as to reproduce in a statistical sense the response of the structure to the actual environment. A second approach replaces the environmental forces which excite the structure and component by a shaker. The force output of the shaker is then controlled to reproduce the response of the structure to the actual environment. The two approaches are compared pointing out the relative advantages and disadvantages between the two with emphasis on the shaker system requirements. Contributed Papers C4. Measurement of Mechanical Impedance and Its Applications. E. L. HIXSON ANn A. F. WITTENBORN, Defense Research Laboratory, The University of Texas, Austin, Texas.- The impedance concept for mechanical elements is briefly developed through the analogy to electric circuits. This concept then makes the whole field of circuit analysis with its highly developed set of rules, theorems, and mathematics available for the solution of mechanical problems. Many devices are available for the application of forces and the measurement of motion and force. Some of these are com- bined in several devices designed specifically to measure the driving-point impedance of soils. An analog computer is pre- sented that takes force and velocity information from the impedance devices, computes their complex ratio and plots impedance as the measurement is being made. Examples of the impedance of soils, sponge rubber, and a simple beam are presented. Impedance information is used to predict the motion of the bar to a sinusoidal driving force and Norton's equivalent circuit is used to predict the motion of a mass attached to the bar. Finally, the transient response of a mass dropped on sponge rubber is predicted from the equivalent circuit as determined from impedance measurements. Each of these cases is verified experimentally. C5. Dynamic Force Measuring System for Machinery Vibration Measurements. T. W. BARRACLOUGH AND R. D. COLLIER, Electric Boat Division, General Dynamics Corpora- tion, Groton, Connecticut.--The system consists of four piezo- electric force gauges, which are installed under machinery feet and are terminated in a large siesmic mass. Individual force gauges are designed to measure the vibratory forces applied normal to the piezoelectric elements. The instan- taneous voltage developed is directly proportional to the instantaneous applied vibratory force. The outputs from the (four) force gauges are fed through a set of four cathode followers where the outputs are combined in two ways: (1) as a phasor sum to provide a single vertical force measure- ment, (2) adding the outputs of two gauges, subtracting the outputs of the other two gauges and introducing electrical equivalent moment arms, to provide torque measurements about the axis through the center of gravity of the machine. These fundamental measurements are based on force and torque equations developed from the six motions of an assumed rigid body. At the fundamental frequency the vertical force measurement, Fy, is a measure of static unbalance and the torque measurement, Tz, is primarily that of dynamic unbalance. Confirming experiments have been performed. FIFTY-FOURTH MEETING 1251 The response of the system is 10-2000 cps calibrated within -I-5%. The system is used to balance assembled machinery and to select quiet machines based on discrete frequency analyses. C6. Methods of Generating High-Intensity Sound with Loudspeakers for Environmental Testing of Electronic Components Subjected to Jet and Missile Engine Noise. JoHN K. HILLIARD AND WALTER T. FIALA, Altec Lansing Corporation.--Unsymmetrical reverberant sound chambers having cubical contents ranging from 30 cubic inches to 20 cubic feet and plane wave tubes two to four inches in diameter will be described. These chambers and tubes are energized by various types of loudspeakers to produce the frequency range required for the environmental testing of electronic components which may be subjected to high sound fields in airframe and missile application. C7. Shake Table for Calibration of Velocity Pickups. J. W. WEscoTT, J. H. PRou% AD W. H. FOLLETT, Engineering Research Institute, University of Michigan, Ann Arbor, Michi- gan.--A Shake Table was developed and built by the Engineer- ing Research Institute to calibrate low-frequency (0 to 200 cps) velocity pickups. The platform that supports the pickup to be tested is 6 in. in diameter and will support a load of approximately 30 lb. This makes the use of a table limited by force it can deliver except at very low frequencies. The table will operate with a 10 lb load to a frequency of 150 cps. The platform displacement is 0.125 in. peak-to-peak. The plat- form and its load are supported by air bellows. This is an improvement over a spring support due to the fact that it has greater damping and it is more easily adjusted to different loads. The adjustment consists of just putting more air in the bellows. The table is driven by a dc push-pull power amplifier. This delivers a current to a tapped coil on the Shake Table that is located in a magnetic field. The field is set up by a coil energized by 24 volts. The power amplifier can be driven by any convenient source delivering about 1 volt. (Parts of this research were supported by Tri-service Contract No. DA-36- 039-sc-52654.) C8. Preliminary Investigation of Some Blanket Damping Treatment Variables. N. E. BxR XND S.S. Kusxx, Engineering Research Institute, University of Michigan, Ann Arbor, Michigan.--An investigation of the dependence of vibration damping capacity on frequency, septum weight, blanket density, and blanket thickness was carried out by means of thick plate decay rate apparatus for several fibrous- glass-blanket-septum configurations. Correlations between these parameters will be presented. Session D. Noise ROBERT M. SERWOOD, Chairman Invited Paper D 1. Sound Propagation through the Atmosphere. M.D. BuaraAaD, Armour Research Foundation of Illinois Institute of Technology, Chicago 16, Illinois.--The propagation of sound from an airborne source to the ground has been studied. The source was a propeller type airplane flown at altitudes up to 4800 ft and distances up to 9600 ft. Angles of elevation of the source with respect to the earth's surface were 2 ø , 5 ø , 15 ø , 30 ø , and 90 ø . The weather varied to include typical year-round conditions in the Chicago, Illinois, area and winter in Phoenix, Arizona. The noise propagated to the earth was analyzed to determine the variation in attenuation as a function of frequency and the relative position of the noise source. A statistical analysis was made to determine the effects of various weather param- eters on propagation of the sound. Average values of sound attenuation were obtained. In addition, the effects of temperature, temperature gradient, humidity, wind, and wind gradient are given in an empirical formula. Propagation does not vary significantly for elevation angles between 5 ø and 90 ø , but for the 2 ø angle there is evidence of ground surface effects. The most important factor for mini- mizing the noise on the ground on a particular day was wind direction. Other meteorological condi- tions do not appear to be significant for controlling noise from flight operations around an airport. (Sponsored by the Aero Medical Laboratory, WCRDB, Wright-Patterson Air Force Base, Ohio.) Contributed Papers D2. Near Field Jet Engine Noise Contours. E. J. KIRCI-I- MAN AN A. L. Owv., The Martin Company, Baltimore, Maryland.--Using the method of near field jet noise prediction developed by Bolt Beranek and Newman, a dual pod installa- tion of after burning engines has been investigated for the following cases: (1) inboard 100% power, (2) inboard with afterburner, (3) outboard 100% power, (4) outboard with afterburner, (5) inboard 100% and outboard 100%, (6) inboard afterburner and outboard 100%, (7) inboard 100% and outboard afterburner, and (8) inboard afterburner and outboard afterburner. A comparison is given between the pre- dicted data and measured data. In the case of single engines, the experimental results indicate that, in the very high inten- sity areas at least, sufficiently close predictions can be made for design purposes. A modified method has been developed to approximate the experimental results for multiengine con- figurations. D3. Noise Control Treatment for a 15-Passenger Com- mercial Helicopter. LA¾oN N. Mm,E ANt) LEo L. BF. RANEIC, Bolt Beranek and Newman Inc., Cambridge, Massachusetts, A HAY SZENFL, Vertol Aircraft Corporation, Morton, Pennsylvania.--In 1955 a noise measurement and noise re- duction study was carried out on a military model H-21 helicopter. Based on this study, two commercial versions of this helicopter were constructed early in 1957 by the Vertol Aircraft Corporation for the French Government for use in the recent International Air Exposition in Paris. This paper presents the noise control treatments contained in the com- mercial helicopters and compares inside noise levels before 1252 ACOUSTICAL SOCIETY OF AMERICA and after the treatment. The noise reduction achieved by the treatment in the passenger cabin averages approximately 5 db in the low-frequency octave bands and 20-25 db in the high-frequency octave bands. The resulting noise levels in- side the completed helicopters are comparable to those of commercial fixed-wing airliners. D4. Noise Levels Inside the Prototype Caravelle Twin-Jet Airliner During Flight. LAYMON N. MILLER AND LEO L. BERANEK, Bolt Beranek and Newman Inc., Cambridge, Massachusetts.--During recent tests of the external noise of the French commercial jet airliner "Caravelle," noise level measurements were also made inside the aircraft during flight. Although the acoustic treatment inside the passenger cabin was incomplete at the time of the measurements, the inside noise levels were significantly lower than those of con- ventional propeller driven aircraft in present-day use. This is attributed largely to the location of the two jet engines at the rear of the fuselage. Noise levels are presented for a number of flight conditions. As an example, during cruise at 7100 rpm at 15 000-ft altitude, the noise levels in the forward part of the cabin were found to be in the range of 75 to 85 db re 0.0002 microbar in the low-frequency octave bands and averaged approximately 60 db in the speech interference bands of 600-4800 cps. D5. Circurnaural Earphone Coupling with Improved Per- formance as an Ear Defender. E. A. G. SHAW AND G. J. THIESSEN, Division of Applied Physics, National Research Council, Ottawa, Canada.--It is customary to use a relatively small coupling volume (typically 15-50 cm a) to provide maximum earphone sensitivity at speech frequencies, al- though a much greater volume is desirable to minimize the penetration of extraneous noise, particularly at low fre- quencies. This dilemma is resolved by making the effective coupling volume frequency dependent. For example the cavity may be divided into two parts coupled by an acoustic resistance, both parts then being effective at low frequencies but only one above a transition frequency (300 cps in a typical case). Alternatively, the two cavities may be coupled by an acoustic mass. When a large total coupling volume is used, it is desirable to damp out high-frequency standing waves with suitable porous material. The introduction of such material is also beneficial at low frequencies where the wave motion tends to be isothermal thus further increasing the effective cavity size. D6. Component Characteristics for an Active Ear Defender. WILLARD F. MEEKER, Radio Corporation of America, Camden, New Jersey.--Proposed methods of electronic noise reduction at a listerers ears have been described previously [J. Acoust. Soc. Am. 28,773 (A) 1956)']. A headset incorporating electronic noise reduction has been termed an active ear defender. One method proposed uses a microphone under each earphone in a negative feedback arrangement. Another method makes use of a microphone located outside the earphone cushion to provide a canceling signal. The canceling arrangement is very sensitive to amplitude and phase errors. Examination of its electrical analog reveals that a surprisingly large band width is necessary because of the critical phase requirements. The band width required for the negative feedback arrange- ment is also large but amplitude and phase characteristics within the useful band are less critical. In this paper the microphone and earphone characteristics required for effective noise reduction are discussed. A negative feedback system is described which provides approximately 15 db of noise reduction from 100 to 200 cps. Component characteristics which will make possible greater amounts of noise reduction are discussed. (Work supported by U.S. Air Force and U.S. Army Signal Corps.) D7. Theory of Pulse Jet Noise Generation. PETER J. WESTERVELT AND RICHARD J. MCQUILLIN, Department of Physics, Brown University, Providence, Rhode Island.--Pulse jet noise arises from three separate and distinct mechanisms. The pulsating air flow at the inlet and exhaust constitute simple monopole souces whose source strengths can be related to the instantaneous volume flow. The fact that the exhaust flow is directed causes a large periodic flux of momen- tum to be directed along the axis of the jet giving rise to a dipole or force source. The strength of this force source is clearly related to the thrust of the jet. Finally, the jet, mixing with the surrounding air, will produce noise of the Lighthill variety stemming from the convected quadripole sources. One problem consists in determing the proper admixture of the three fundamental sources necessary to reproduce the measured far-field directivity. Another problem consists in relating the three source strengths to the measured flow and thrust characteristics of the jet. Both of these problems are currently being studied. Preliminary analysis indicates that the fundamental frequency is generated by a monopole and that the higher harmonics contain monopole, dipole, and quadripole radiation. (Work supported by the Wright Air Development Command.) D8. Correlations and Spectra of Pressure Fluctuations on the Wall Adjacent to a Turbulent Boundary Layer. MARK HARRISON, David Taylor Model Basin, Navy Department, Washington, D. C.--A fully developed turbulent boundary layer with a zero pressure gradient has been studied. A subsonic wind tunnel was used. The pressure fluctuations on the wall of the wind tunnel were measured by small flush mounted microphones about of an inch in radius. Data for the dimensionless spectral density of the pressure fluctua- tions, P (d e)/0 s U0a/ *, as a function of the frequency parameter f*/Uo, will be presented; P), p, U0,/5', andfare, respectively the spectral density, fluid density, free stream velocity, boundary layer displacement thickness, and frequency. Data for the coefficient (p(t))/pUo4 will be presented. The longitudinal and transverse cross correlations for the pressure fluctuation were studied by using two flush mounted micro- phones. These correlations were measured for the total pressure fluctuation and as a function of frequency. Data for the latter will be presented as a function of fox/Uo and x/*; x is the distance between the two microphones and f0 is the midfrequency of a sufficiently narrow tunable filter. Based upon these measurements, an evaluation of the concept of frozen lamellae of turbulence convected at a local mean flow velocity will be presented D9. Hydrodynamic Stability of the Jet-Edge Whistle. RICHARD H. LYON, Department of Electrical Engineering, University of Minnesota, Minneapolis, Minnesota.--This paper deals with the ability of present day knowledge of hydro- dynamic stability to predict the onset of oscillation in a jet-edge whistle. Neutral stability curves for the free jet and the thin plate will be studied for their relation to this problem One finds that no immediate or obvious connection exists between the frequencies produced in these systems and the frequencies produced in the whistle when the jet and plate are combined. A series of experiments are described which lead to an experimentally determined neutral stability curve for the jet-edge whistle. These experiments indicate a strong destabilizing influence of the edge on the jet flow. A discussion of the physical basis of this destabilizing effect follows. (This work has been sponsored by a grant-in-aid from the Graduate School of the University of Minnesota.) FIFTY-FOURTH MEETING 1253 Session E. Propagation and Seismics UNO INGARD, Chairman Contributed Papers El. Energy Balance in Acoustic Streaming. PETER J. WESTERVELT, Department of Physics, Brown University, Providence, Rhode Island.--The power required to maintain the Eckart type of acoustic streaming is calculated and found to be of sufficient magnitude to contribute measureably to the attenuation of large amplitude sound waves. A second problem has also been solved. When sound interacts with objects, streaming occurs due to the vorticity sources localized near the surface of the objects. An energy theorem is proved which is valid for objects small compared with the wave- length of sound and which enables the energy dissipated by the streaming motion to be expressed entirely in terms of the first-order oscillatory motion thus obviating the need for calculating the second-order streaming motion. [Work sup- ported jointly by the U.S. Air Force (Wright Air Develop- ment Command) and the Office of Naval Research.-] E2. On the Prediction of the Attenuation of Sound Propa- gated Over Ground. DAVID N. I<EAST AND FRANCIS M. WIENER, Bolt Beranek and Newman Inc., Cambridge, Massachusetts.--The attenuation of sound propagated out of doors can be separated into attenuation due to spherical divergence and attenuation due to atmospheric effects. The latter, which shall be referred to as excess attenuation, appears to be principally caused by molecular absorption, temperature and wind gradients, and turbulence. Recently, extensive field data have been obtained which have led to a series of empirical functions which enable one to predict the excess attenuation as a function of frequency and range for a given set of micrometeorological and terrain conditions. Although limited to those cases where both the sound source and the receiver are near the ground, this procedure is of great utility in estimating the noise exposure at some distance from a sound source. (This work was supported by the U.S. Army Signal Corps under Contract No. DA-36-039SC-64503.) E3. Sound Transmission through Thin Plates at Oblique Incidence. E.G. EICm. ER AND R. F. LAMBERT, Department of Electrical Engineering, University of Minnesota, Minneapolis, Minnesota.Transmission loss measurements of thin, rec- tangular aluminum plates employing the (0.1) acoustic mode of a wave guide are described. Theoretical transmission loss formulas are derived for the limiting cases of small and large ratios of plate span to plate thickness. In the case of large ratios, a "mass law" formulation is obtained which exhibits an angular dependence characteristic of the (0.1) mode. Almost complete sound transmission predicted for angles of incidence near grazing is verified experimentally for all plates studied. Some discrepancies between theory and experiment for small ratios are discussed and an attempt is made to empirically correct the calculations. These discrepancies are here attributed to imprefect boundary clamping and internal plate dissipation. propagate without attenuation. The present paper gives solutions for higher phase velocities, showing that no attenua- tion in the direction of the axis is required. Special attention is given to the limiting case of phase velocity equal to the speed of compressional waves in the solid, which yields the coupling of a fluid-filled borehole to a plane compressional wave in the surrounding solid. In the low-frequency limit, this coupling agrees with an approximate expression pub- lished earlier [J. Acoust. Soc. Am. 25, 906-915 (1953)-]. E5. Elastic Waves in Lake Ice Produced by Impact of Falling Weights. DONALD H. CLEMENTS, DAVID E. WILLIS, AND J. T. WILSON, Engineering Research Institute, University of Michigan, Ann Arbor, Michigan.--Experimental studies of the propagation of elastic waves in lake ice were made at Lake Margrethe, Michigan, and Garrison Dam, North Dakota. These studies were designed to investigate, in particu- lar, flexural waves in the ice-water system produced by hammer drops, cylinder drops, and sledge hammer impacts on lake ice and nearby land. A method similar to the standard seismic-refraction techniques was employed for the wave studies. Flexural wave dispersion curves obtained from records of mechanical impulses on ice are, in general, con- sistent with theoretical curves based on the "layer" theory of Press and Ewing (1951) for the observed ice thickness and elastic parameters. Three groups of points on the observed dispersion curves obtained from Lake Margrethe data are flexural waves with frequencies from 2 to 8 cps, 13 to 25 cps, and 40 to 140 cps. The first and third groups exhibit dis- persion consistent with the combined thickness of a two- layer floating ice sheet. The second group follows dispersion for a wave path in the upper thin layer only. Hammer drops on land generate flexural waves in the layer of frozen ground in the frequency range 15 to 40 cps. These waves are trans- mitted across the lake-shore boundary and follow a wave path in the upper thin layer of lake ice. A continuous train of flexural waves in the frequency range 4 to 140 cps was observed on records of cylinder drops taken at Garrison Dam, North Dakota. (Parts of this work were supported by Tri-service Contract No. DA-36-039-sc-52654.) E6. Plant Resonance of Seismometers. DAVID E. WILLIS AND J. T. WILSON, Engineering Research Institute, University of Michigan, Ann Arbor, Michigan.--A seismometer resting on an elastic medium can be regarded as a simple damped oscillator. Methods for measuring the "plant" resonance of seismometers are discussed along with the various parameters affecting the resonance. Laboratory measurements on sand and rubber, and field measurements on different types of soils were made using an electromechanical oscillator to generate elastic waves. The results are tabulated and the correlations with resonances observed in machine foundations are discussed. E4. Elastic Waves along a Cylindrical Bore. J. E. WHITE, Research Center, The Ohio Oil Company, Littleton, Colorado.-- This paper discusses axially symmetric solutions for elastic waves propagating along an infinite cylinder, namely those which can be expressed as F(r) e k e -iw. M. A. Biot [J. Appl. Phys. 23, 997-1005 (1952)-] presented results for phase velo- cities along the axis less than shear velocity in the solid and indicated that for higher phase velocities, the waves cannot E7. Seismic Level Recorder. H. J. BUGAJSKI AND J. C. JOHNSON, Engineering Research Institute, University of Michigan, Ann Arbor, Michigan.The seismic level recorder is a sensitive recording ac vacuum tube voltmeter. The prime intention in the design of this recorder was to provide a portable level data system which could be left operating unattended for long periods of time. The seismic level recorder incorporates a high gain seismic amplifier connected to a 1254 ACOUSTICAL SOCIETY OF AMERICA peak-to-peak rectifier whose output is fed through a logarith- mic driver to an Esterline-Angus ink recorder. The frequency response is 1 to 100 cps at the 6 db points. Provisions have been made for plugin 12 db octave band-pass filters. The calibrated scale covers approximately a 50-db range of signals allowing a wide variety of detectors and signals to be explored. (Parts of this work were supported by Tri-service Contract No. DA-36-039-sc-52654.) Session F. Speech KV. NNV. Trt N. STv. VV. NS, Chairman Contributed Papers Fl. Relative Intensities of Sounds at Various Anatomical Locations of the Head and Neck during Phonation of the Vowels. HV. RBV. RT J. OYv. R, The Ohio State University Research Foundation, Columbus, Ohio.--Speech signals from 16 different anatomical locations were recorded as subjects intoned different vowels at a constant level. Power level analyses were made to determine the relative intensity of the signals. It was found that significant difference in intensity existed among the anatomical locations. It was found that the composite intensities of the test vowels were statistically equivalent at the larynx, the sides of the neck, and at the angles of the rami. The next rank of intensity was noted at the back of the neck, back of the head, chin, and nose. The next most intense location was the top of the head, with least intensity measured at the six positions of the two tem- ples, midback of the head, forehead, and the sides of the head above the ears. In general, previous observations of "the nearer the larynx, the more intense the sound" is borne out by this study. Some locations of lesser intensity were sub- jectively evaluated as providing more faithful signals. The consensus was that most faithful reproduction resulted in the vowels recorded from the forehead, with the most intense reproduction afforded by the larynx and neck areas. [Done in connection with contract AF 19(604)-1757, monitored by the Operational Applications Laboratory, Air Research and Development Command, Bolling Air Force Base, Washington 25, D. C.J F2. Masking of English Words by Prolonged Vowel Sounds. HENRY M. MOSER, JOHN J. DREHER, AND JOHN J. O'NEILL, The Ohio State University Research Foundation, Columbus, Ohio.--One-hundred and ten monosyllabic words selected from the Thorndike list of 1000 most frequently occurring words in English to represent equally each of 10 vowels were presented to 300 American listeners in an articulation test. Also tested were 72 spondee words, half selected from those in use in audiological tests and half from those in frequent use in air traffic control, further to represent the same vowel sounds. Masking of the stimuli was accomplished by separately recording each of nine prolonged vowels intoned by a trio of male voices. Results indicate that vowels of equal sound pressure levels differ considerably in masking effectiveness, that words containing a specific vowel are not masked optimal- ly by the same vowel, and that spondees are masked by prolonged vowels in the same rank order as are the mono- syllables. Prolonged vowel sounds with relatively high con- centrations of energy between 700 and 1000 cps are most effective as masking agents. Rank order correlation of observed masking effectiveness with masking effectiveness predicted by the Strassberg method is 0.52, the Beranek method is 0.59, and by the Pickett-Kryter method, 0.69. Some obser- vations on resistance of words to masking are made in relation to phonemic transition areas within words. [Donein connection with contract AF 19(604)-1577, monitored by the Operational Applications Laboratory, Air Research and Development Command, Bolling Air Force Base, Washington 25, D. C.-I F3. Peak Factors in Subdivisions of the Speech-Frequency Band. WEAN'r WATItEN-DUNN AND DAWD W. LPlCE, Electronics Research Directorate, Air Force Cambridge Research Center, Air Research and Development Command, Bedford, Massachusetts.--The relative powers and the corresponding peak factors in octave or half-octave divisions of the speech- frequency band take on renewed interest in the light of a recent report by Schneider [Frequenz 10, 97-106 and 152-161 (1956)-] on the intelligibility of clipped speech where the speech is divided into sub-bands and clipped. Examination of the available power and peak-factor data leads to the conclusion that the latter has not been measured with an accuracy sufficient for the present purpose. Measurements of the peak factors in subdivisions of the speech-frequency band, using modern techniques, will be presented and related to previous power and peak-factor measurements. F4. Some Cues for the Distinction between Voiced and Voiceless Stops in Initial Position. A.M. LInERMAN,* P. C. DELATTRE, AND F. S. COOPER, Haskins Laboratories, New York, New York.--A series of experiments with synthetic speech indicated that each of the voiced stops could be made to sound like its voiceless counterpart by eliminating the beginning of the first-formant transition. The amount of first-formant "cutback" required to produce a voiceless stop varied according to the stop, the vowel which followed it, and the listener, but was in no case greater than 40 msec. It was also found that filling the first part of all formant transi- tions with noise (i.e., aspiration) contributed to the impression of voicelessness, but the importance of aspiration as a cue appeared to be considerably less than that of the first-formant cutback. Many listeners showed great consistency in differ- entially indentifying the patterns as voiced or voiceless even when the stimulus difference was seemingly very small. In one experiment, for example, several listeners sorted the patterns perfectly on the basis of a 10-msec difference in the starting time of the first formant. [This work was supported in part by the Carnegie Corporation of New York, and in part by the Department of Defense in connection with Contract DA 49-170-sc-2159.-I * Also, University of Connecticut, Storrs, Connecticut. ? Also, University of Colorado, Boulder, Colorado. F5. Clipping and Transmission Methods. FREDmC}t Vmm AND KVRt H. HAASE, Communications Laboratory, Electronics Research Directorate, Air Force Cambridge Research Center, Bedford, Massachusetts.--Methods and techniques of (!) audioband (AB) clipping, (2) single side band (SSB) clipping, (3) double side band (DSB) clipping, and (4) partial band (PB) clipping both with and without control, will be explained and discussed in relation to distortion, intelligibility, and power gain. Assuming constant power gain, the above methods are arranged in order of increasing understandability and decreasing distortion, provided they are not further distorted during transmission. If the signal is influenced by fading, Doppler effects, or atmospheric disturbances during trans- FIFTY-FOURTH MEETING 1255 mission, the modulation envelope becomes the important factor. For a rectangular modulating signal the resulting envelope will also be rectangular in AB and DSB. It is related to the shape of the modulating signal in SSB and PB but it is no longer rectangular. To suppress transmission distortions, clipping at the receiver can only be used in AB and DSB, for it would introduce additional distortion in SSB and PB. When distortion in transmission is considered, the various methods of clipping will be rank-ordered in a different sequence. F6. Identification of Turbulent Speech Sounds. JACOB WIREN, Northeastern University, Boston, Massachusetts.--In the study of turbulent signals there seems to occur two classes, one which is stationary in nature and the other possessing transient characteristics. The former occurs in speech as fricatives (such as s) and the latter as stop sounds (such as t). It is often desirable to devise methods for separating them automatically. These will find their particular application directly in an electronic binary selection system for phoneme classification or indirectly for a speech analysis synthesis scheme using continuous parameters. Three methods have been considered for this task. They involve the use of (a) the duration of the sounds, (b) the total number of zero crossings of the wave form of the sound, and (c) the change of energy or spectrum during the onset of the sound with respect to a fixed time interval or a fixed number of zero crossings of the wave form. The relative merits of each method will be dis- cussed, and a more detailed presentation will be made for the results of the third method. [-The research in this paper has been made possible through support and sponsorship extended by the Electronics Research Directorate of the Air Force Cambridge Research Center, under Contracts AF19 (604)-1039, Item I, and AF19(604)-2198. F7. Mathematical Expressions for Speech Sound Spectra. KURT HAASE, FRIEDRICH VILBIG, AND ALVIN DRAKE, Com- munications Laboratory Electronics Research Directorate, Air Force Cambridge Research Center, Bedford, Massachusetts.- Mathematical expressions, developed for the harmonic line spectrum of vowels and for the noiselike spectrum of con- sonants, will be shown. In both cases the spectrum content is encircled by the envelope. This is characterized by several peaks which identify the formant or quasi-formant frequencies. The spectrum can be seen as a modulation of these formant frequencies by a properly chosen pitch or noise pulse. The distortions which appear if this approximate sound representa- tion is transformed by a nonlinear device producing square and higher order terms will be investigated and discussed. The distortions can be measured by the harmonic distortion factor. For the same distortion factor the influence of the distortion frequencies may be quite different depending on the energy distribution in the sound spectrum and the co- efficients of the nonlinear device. Taking these characteristics into account provides a method for correlating the harmonic distortion factor with understandibility. F8. Speech Synthesis by Rule. FRANCES INGEMANN,* Haskins Laboratories, New York, New York.--It has seemed useful for both theoretical and practical reasons to attempt to write rules for synthesizing speech on the basis of the experiments performed by colleagues at the Haskins Labora- tories. Because the discrete nature of phonemes as perceptual units is not reflected on the acoustic level, it is not feasible to synthesize speech by putting together prefabricated phonemes. On the other hand, synthesis of syllabic units need not require an inventory of rules comparable with the number of syllables in the language, since the rules can be phrased in terms of the constituent phonemes and combinatorial opera- tions. The rules can be further simplified by writing them in terms of the acoustic features that correspond to the standard phonetic dimensions of place and manner; syllables and words are then synthesized by combining the rules for the classes to which the constituent phonemes belong. An example of speech synthesized by rule will be presented. [This work was supported by the Carnegie Corporation of New York, the Veterans Administration, and the Department of Defense in connection with Contract DA 49-170-sc-2159.-] * Now at the University of Kansas, Lawrence, Kansas. F9. Pitch Discrimination for Synthetic Vowels. JAMES L. FLANAGAN, Bell Telephone Laboratories, Inc., Murray Hill, New Jersey, AND MICHAEL G. SASLOW, Acoustics Laboratory, Massachusetts Institute of Technology, Cambridge, Massachu- setts.--A psychoacoustic experiment to determine the just- discriminable changes in the fundamental frequency of synthetic vowels is described. The experimental variables investigated are: three synthetic vowels (/i, a, ae/), two sound pressure levels (60 and 80 db re 0.0002 dyne/cm'), and two fundamental frequencies (80 and 120 cps). A method of constant stimuli is employed in which subjects judge the pitch of the vowel as "higher" or "lower" in comparison to a fixed standard. The data indicate that the changes in fundamental frequency discriminated correctly 75% of the time are of the order of 0.3 to 0.5 cps. Application of the results to speech compression is suggested. F10. Word Intelligibility of Time Compressed Speech. R. G. KLUMPP AND J. C. WEBSTER, Human Factors Division, U.S. Navy Electronics Laboratory, San Diego 2, California.By sampling, compressing, abutting, storing, and speed reduction, Fairbanks and associates have shown that comprehension is good for: (1) connected speech compressed 50% (0.5) in time, and (2) a single PB list compressed up to 0.15. In 1929, Fletcher and associates using turntable speedup with its attendant frequency distortion found that CV, VC, and CVC syllables dropped rapidly in intelligibility at speedup cor- responding to time reductions of 0.8 (down about 30% at 0.7). In one case a complex technique using simple language showed great advantage. In the other case a simple technique showed limited advantage using difficult speech material. In our work on message (simple language) storage schemes, the desirability of obtaining time compressed speech with simple modifications of existing equipment has become evident. Preliminary experimental data show that by using speedup of tape-recorded speech, numbers, and flight phrases can tolerate time compressions of 0.6 without serious loss in intelligibility, and PB words lose only 5% in intelligibility with a time compression of 0.7. Fll. Continuous Analysis Speech Band-Width Compres- sion System. T. E. BAYSTON AND S. J. CAMPANELLA, Melpar, Inc., Falls Church, Virginia.An investigative program conducted at Melpar under U.S. Air Force sponsorship has resulted in the development of a speech band-width com- pression system capable of transmitting speech within a total band width of 150 cps (as compared with 3000 cps for speech transmission by conventional means). This system consists of an analyzer which extracts from the acoustical speech signal six vocal analog control signals, a suitable transmission link, and a synthesizer which employs the control signals to reproduce the essential features of the original speech. The vocal analog control signals vary slowly in time because of natural physical constraints, and thus require considerable less transmission band width than the original acoustical signal. Articulation tests conducted on 1256 ACOUSTICAL SOCIETY OF AMERICA the system have resulted in scores of 67 % for PB (phonetically balanced) word lists and 88% on Spondee word lists. A breadboard model of this system will be described and tape recordings of processed speech will be presented. [This work was supported by Contract AF33 (600)-3.-1 F12. Present Status of Formoder. S. H. CrAc, R. BAca, JR., W. F. KIc, AD R. Smc¾s, Northeastern University, Boston, Massachusetts.--Formoder, short for Formant- Moment Coder, is a system which utilizes slowly varying parameters and discrete linguistic features extracted from the speech spectrum for speech band-width compression. Formoder offers a different approach than Vocoder for the approximation of the speech spectrum by using FM as well as AM parameters. In addition, the use of certain linguistic features limits the dynamic ranges needed in the analyzer and synthesizer components. The system falls short of the ideal representation of a time varying dynamic model of the speech mechanism, but it holds promise for an improvement in compression ratio over that of Vocoder. This is perhaps the reason for continued study of essentially similar systems in various laboratories. This paper is intended to discuss further the above-mentioned points and will describe the present status of Formoder together with system evaluation and articulation tests. In a typical operation of the system using two formants and one moment, articulation tests show the Harvard PB word lists score over 40% and sentence score over 80%. I-The research in this paper has been made possible through support and sponsorship extended by the. Electronics Research Directorate of the Air Force Cambridge Research Center, under Contract AF19(604)-1039, Item I, and AF19 (604)-2198.-] Session G. Symposium on Unsolved Problems in Acoustics R. BRucv. LDsAY, Chairman Invited Papers G1. Electroacoustics and Transducers. FREt)ERICh: V. HvT, Harvard University, Cambridge, Massachusetts. G2. Sonic Engineering. TaEOt)OR HVETER, Raytheon Manufacturing Company, Wayland, Massa- chusetts. G3. Properties of Matter. Solids. ROBERt W. MORSE, Physics Department, Brown University, Providence, Rhode Island. G4. Properties of Matter. Liquids. TaEOr)ORE LIXOVIXZ, Catholic University of America, Washing- ton, D.C., and Naval Ordnance Laboratory, White Oak, Maryland. G5. Noise, Shock, and Vibration. ROBERt O. FE}tR, General Engineering Laboratory, General Electric Company, Schenectady, New York. G6. Medical Acoustics. WILLIAt J. FRY, Bioacoustics Laboratory, Department of Electrical Engineer- ing, University of Illinois, Urbana, Illinois. G7. Physiological and Psychological Acoustics. MERLE LAWRENCE, Department of Otolaryngology, University of Michigan, Ann Arbor, Michigan. G8. Speech and Communication. GEORGE A. MILLER, Harvard University, Cambridge, Massachusetts, Session H. Hearing II J,us P. EGAN, Chairman Contributed Papers H1. Difference Limen for Pure Tone Diminution. ROGER E. KIRC, The Baldwin Piano Company, Cincinnati, Ohio.- The difference limen for linear pure tone diminution was investigated with diminution rate and fundamental frequency as parameters. Electronic means provided two tone diminu- tion rates which were independently adjustable. The two tones were automatically recycled five times each. The initial loudness level of the tones was 80 phons. The terminal loudness level of the tones was randomized. This was ac- complished by randomizing the duration of the tones. The tones were presented to subjects over PDR-8 earphones. The method of constant stimuli was used to determine the difference limen. An analysis of the results indicates that the difference limen for tone diminution is of the order of 4« to 5øfo over the range of diminution rates investigated. The difference limen is relatively independent of the fundamental frequency of the tone. The relationship between the difference limen and diminution rate is described by an equation of the form Y=0.048X where Y and X are measured in db/sec. H2. Explanation of Limens of Loudness. J. R. PIERCE, Bell Telephone Laboratories, Inc., Murray Hill, New Jersey.- It is proposed that the limen of loudness /xS is equal or proportional to the root mean square deviation in the number of pulses produced by the sound in the time T of observation. By making a plausible assumption concerning the deviation and by assuming that loudness S is proportional to the pulse rate one obtains/xS = A-mT-/2S m. Here A is a constant which must be a function of frequency if the equation is to fit experimental data. The fit is encouraging but not com- pletely convincing. If valid, the hypothesis might be used to explain other acoustic phenomena. H3. Some Factors Governing the Lateralization of High- Frequency Complex Wave Forms. E. E. DAVID, JR., NEWMAN GUTTUrAl, AD W. A. vA BEREIJC, Bell Telephone Labora- tories, Inc., Murray Hill, New Jersey.The experiments described here were designed to investigate the reciprocity of interaural intensity and arrival-time differences for complex FIFTY-FOURTH MEETING 1257 signals above 1500 cps. When clicks are brought to the ears at nearly the same times and intensities and at a typical repetition rate of 20 per second, single clicks are perceived as originating inside the head. As had previously been shown, place of lateralization depends upon interaural arrival-time and intensity differences. Centering may be achieved over a range of these differences. Our results show that with 2000 cps high-pass clicks a very nearly linear trading relationship exists between the time and intensity differences (in decibels) when the product of the two intensities is constant, that is, when the sum of the binarual intensities (in decibels) is constant. The slope of the function depends upon sensation level and is independent of repetition rate at moderate levels and rates up to at least 140 clicks per second. Lateralization may also be controlled by masking the clicks with noise. Similar experiments with other wave forms will be reported also. H4. Comparison of Threshold in Air and in Water. WALTER N. WAINWRIGHT, U.S. Navy Underwater Sound Laboratory, Fort Trumbull, New London, Connecticut.--Hearing thresholds were measured for two subjects over a frequency range of 250 to 4000 cps. These measurements were made both in air and in water. When threshold intensity levels are compared it is shown that the greatest loss in hearing acuity occurred over the frequency band from 500 to 2000 cps and amounted to about 20 db, while below 300 cps the threshold intensity in water was lower than that in air. Measured intensity levels for an underwater swimmer speaking into a face mask are also presented. H$. Aural Azimuth Discrimination Ability of a Human Observer. F. FRAZIER, J. C. JOHNSON, AND R. C. FITZPATRICK, Engineering Research Institute, University of Michigan, Ann Arbor, Michigan.--A preliminary study has been made for the purpose of determining the ability of an individual to accurately estimate the direction of sound arrival. The observers were shielded such that no visual aids could be used. A limited statistical analysis has been applied to the data from this study, and significant though not conclusive, results were obtained. [Parts of this research were supported by Tri-service Contract No. DA-36-039-sc-52654.1 H6. Detection of Beats in Repeated Bursts of Tone. M. H. GOLDSTEIN, JR., AND T. T. SANDEL, Research Labora- tory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts.--Listeners report hearing beats in repeated bursts of tone of low intensity provided the burst rate stands in a nearly harmonic relation to the frequency of the tone and provided that certain conditions of rise time of the bursts and frequency of the tone (carrier) are met. A typical signal that will give rise to beats at a rate of 2/sec might be one in which bursts of a tone of 2001 cps are re- peated at a rate of 200 per second; the bursts of tone are presented at a level of 35 db above threshold with a rise time of 0.25 msec for each burst, and a sound-time fraction of 0.5. In these experiments, listeners were asked to report the presence or absence of beats as the frequency of the carrier was slowly changed. For a given test, repetition rate, sound- time fraction, and rise time were held constant. For a nearly harmonic situation the probability of detection of beats is greater for short rise times, and carriers of relatively low frequency. Data on the masking of the beats with wide- band noise, high-pass filtered noise, and low-pass filtered noise are also presented. These experiments permit com- parison of various mathematical representations of these stimuli with respect to their ability to predict the listeners responses. [This work was supported by the U.S. Army (Signal Corps), The U.S. Air Force (Office of Scientific Research, Air Research and Development Command), and the U.S. Navy (Office of Naval Research).i H7. Detection of Multicomponent Auditory Signals in Noise. DAVID M. GREEN, Massachusetts Institute of Tech- nology, Cambridge, Massachusetts.--How can the detection of a multicomponent signal be predicted if the detection of each single component of the complex signal is known? In the detection of a single sinusoidal signal, Fletcher's critical-band hypothesis asserts that all but a narrow band of frequencies near the signal to be detected may be ignored. What will happen if two or more components, widely separated in frequency, are simultaneously presented is uncertain. Complex signals using various durations, frequency spacings, and am- plitudes were generated. Both monaural and binaural observa- tions were made. In each experiment the detection of each individual component was first determined. Then a complex signal was generated by adding two or more of these com- ponents together. The detection of the complex signal was determined. Three models are considered: (1) a model which asserts there is no summation, (2) a statistically independent threshold model, and (3) a statistical decision model. The results support the third model. If two equal-amplitude sine waves have equal detectability and are separated by several critical bands, the same detection for the combination is obtained if each individual component is reduced 1.5 db. Thus, the total energy in the complex is greater than either single component for a given detectability. [This research was supported by U.S. Air Force Contract AF 18(600)-1219. The author held a National Science Foundation Fellowship.l H8. Effect of Multiple Observations on the Detectability of Signals in Noise. JOHN A. SWETS, ELIZABETH F. SHIPLEY, AND DAVID M. GREEN, Massachusetts Institute of Technology, Cambridge, Massachusetts.--The use of repeated presentations of a given signal event as an experimental technique in psychoacoustic studies permits the inference of several general properties of the hearing process. In addition to the immediate datum, i.e., the amount and rate of gain in de- tectability th. at results from repeated observations, this method provides information concerning the observer's ability to integrate over successive observations, the amount of noise generated within the auditory system, some character- istics of the frequency-selectivity process, and the effects of uncertainty about signal frequency. Data are presented from experiments that permitted five observations of each signal where the signal consisted of a pulsed tone in noise. Both independent noise, i.e., noise that was statistically independent from one presentation to another, and identical noise, i.e., noise that was exactly the same on each of the five presenta- tions, were used. Using the independent noise, the detect- ability index d improved according to the square root of the number of observations. Using identical noise, an inference concerning the portion or total noise affecting the detection process that is of internal origin was made. Experiments designed to distinguish between two opposing models of the process of frequency analysis and to show the effects of uncertainty as to signal frequency are described. [This research was supported by funds provided under U.S. Air Force Contract AF 19(604)-1738 with the Operational Applications Laboratory, Air Force Cambridge Research Center, Bolling Air Force Base, Washington, D.C.] H9. Measurements of Hearing Threshold by Bone Con- duction. EDITH L. R. CORLISS, ERNEST L. SMITH, JUNE O. EANET, MAHLON D. BURKHARD,* AND WALTER KOIDAN, National Bureau of Standards, Washington, D. C.Measure- ments of threshold of hearing by bone conduction have been made for a group of young normal subjects. For each thres- hold, measurements of the mechanical impedance of the 1258 ACOUSTICAL SOCIETY OF AMERICA subjects' foreheads and mastolds were taken. By use of the impedance data, thresholds obtained in terms of displacement amplitude also were converted into threshold force deter- minations. The results provide additional data for design of an artificial mastold and serve as a source of tentative thres- hold values. Threshold data to be reported are for a driving tip of 2 cm diam applied with a static coupling force of 1 kg. * Now at Armour Research Foundation, Chicago, Illinois. Session I. Architectural Acoustics WILLIAM A. JACK, Chairman Contributed Papers I1. Sound Absorption Coefficients of Acoustical Materials Backed by Large Depth Air Spaces. W. A. JACK, Johns- Manville Research Center, Manville, New Jersey.--The Johns- Manville reverberation chamber technique uses a sample area 8 by 8 ft. A deep pit, beneath the sample, fillable with water to the height desired, provides a ready means for investigating any air space up to 46 in. in depth. Below 600 cps strong depth dependencies were found. The results on a variety of materials are presented. 12. Example of the Dependence of Sound Absorption on the Area of Sample. ROBERT W. YOUNG, If. S. Navy Elec- tronics Laboratory, San Diego, California.--The sound absorp- tion in a solid concrete room was measured by the reverbera- tion method, first with the room entirely bare, then with a "border" of l-in. Acousti-Celotex (46 fff' in area) around the ceiling, then with a "border" added around the top of the walls (total acoustical tile 112 fff-), and finally with the entire ceiling and one-third of the side walls covered (265 of acoustical tile). The test sound, generated by mechanical impact, was filtered on reception into third-octave and half- octave bands. The volume of the rectangular room was 1350 ft a and its area 793 fff'. With the border of 46 ft 2 the co- efficients determined in the customary fashion by the Sabine reverberation equation were near those published for 1¬ in. Acousti-Celotex. At 125 and 8000 cps the coefficients changed little with the area of sample. At 1000 cps, however, the measured coefficient dropped from 0.93 to 0.44, to 0.25 as the sample area was increased. This is possible evidence for a change in effective mean free path and for the need of an appropriate adjustment in the reverberation formula. The name Sabine absorption coefficient is suggested for the usual coefficient computed by the reverberation technique, to distinguish it from the energy absorption coefficient that does not exceed unity. 13. Definition Measurement Apparatus. J. L. COLLINS AND C. D. ANDERSON, Defense Research Laboratory, The University of Texas, Austin, Texas.--A laboratory apparatus for the measurement of acoustic "definition" or "clearness" as pro- posed by Professor Erwin Meyer [J. Acoust. Soc. Am. 26, 630-636 (1954) has been built and used to determine the definition of several school classrooms. This paper will describe the instrumentation used for making these measure- ments and discuss a few of the typical readings. Further results of the definition measurements will be presented in the paper "Acoustic Measurements in School Classrooms" by Mr. R. N. Lane et al. 14. Acoustic Measurements in School Classrooms. R. N. LANE, E. E. MIKESKA, AND C. D. ANDERSON, Defense Research Laboratory, The University of Texas, Austin, Texas.--Acoustic measurements were made in the classrooms of twenty-two different school buildings in a study sponsored by the Acousti- cal Materials Association. Interior acoustical treatments in the rooms included full ceiling treatment, peripheral treat- ment, and no treatment. Measurements included reverbera- tion time, definition, percentage articulation, transmission loss and noise reduction between rooms, followed by a questionnaire survey of the teachers who use the classrooms. The questionnaire returns indicated some interesting teacher impressions which will be reported, although not enough teachers were polled to treat their answers in a statistical manner. 15. On the Steady-State Transmission in Rooms. M. R. SC}ROEDER, Bell Telephone Laboratories, Inc., Murray Hill, New Jersey.--In 1935, E. C. Wente [J. Acoust. Soc. Am. 7, 123 (1935)] suggested an approach to the problem of finding a quantitative measure other than reverberation time for the acoustical quality of a room. The effect of reverberation time on acoustical quality was fairly well known at that time, and it had become apparent that reverberation time alone, though the most important room-acoustical parameter, was not sufficient to describe the acoustical behavior of an en- closure. Wente's approach consisted of measuring the steady- state transmission level from one point within a room to another as a function of frequency. He and other workers in the field suggested a variety of statistical quantities obtainable from such transmission curves as quality measures to supple- ment reverberation time. In this paper it will be shown that for large rooms these new parameters do not contain new information. In fact, the transmission curves can be proved to be members of an ensemble of statistical functions with a given amplitude distribution and a spectrum which depends only on reverberation time. Large rooms in the above sense are characterized by the fact that for each frequency in the range considered many normal modes are excited simul- taneously. Assuming random distributions of the eigen- frequencies and the excitation coefficients [M. R. Schroeder, Acustica 4, Beih. 1,456 (1954)] of the normal modes an in- equality for the volume of the room can be derived which defines the range of validity of the statistical theory [M. R. Schroeder, Acustica 4, Beih. 2, 594 (1954)]. This theory has been confirmed by measurements in many concert halls, theaters, and broadcast studios [H. Kuttruff and R. Thiele, Acustica 4, Beih. 2, 614 (1954)]. 16. Development of Acoustic Attenuators for Air-Condi- tioning Ducts. CAR. BECKER, Weathertron Department, General Electric Company, Bloomfield, New Jersey.--The air-borne noise path along ducts is important for frequencies above approximately 150 cps. While long, tortuous ducts often provide inherent attenuation, special treatment can be necessary for short ducts. Treatment effectiveness is judged in terms of air flow resistance in relation to minimum acoustic attenuation which will be provided for a range of installation conditions. Design of acoustic duct attenuators involves selec- tion of suitable means for providing sound absorptive surfaces, effective over the wide frequency range involved, and their FIFTY-FOURTH MEETING 1259 arrangement in the air stream. Methods are discussed for measuring acoustic attenuation in relation to air-flow re- sistance. Attenuation measurements without air flow permit detailed examination of acoustic effectiveness of attenuator components and their arrangement. Subsequent attenuation measurements under rated air-flow conditions are used to determine any potential source of self-noise from turbulence. 17. Measurement of Noise Inside Ducts. IRA DYER, Bolt Beranek and Newman Inc., Cambridge, Massachusetts.--Re- cently Kerka, and earlier Beranek, Reynolds, and Wilson, discussed the measurement of sound power radiated by a fan in a straight duct, by measurements of the sound pressure inside the duct. One of the difficulties associated with this technique is that above the first cutoff frequency the sound pressure varies markedly across a transverse section of the duct and the relation between mean square pressure and power is not simple. The transverse variation can be under- stood by application of mode theory to propagation of noise in the duct; this theory is in good agreement with measure- ments reported by others. Furthermore, in given frequency ranges, it is possible to select a measurement position inside the duct such that the mean square pressure is related to the power flow by the usual free-field equation. 18. Some Experiments on Sound Propagation in Damped Air Ducts with Flow Velocities up to 80 m/sec. ERWIN MEYER, F. MECHEL, AND G. KURTZE, III. Physikalisches Institut der Universitiit G6ttingen, G6ttingen, Germany.--In a rectangular duct with the cross section 35X100 mm one larger side is lined with sound absorbing devices. The used absorbers are rock wool (layer-thickness 6 cm) with and without perforated coverings, and undamped as well as damped resonators of the Helmholtz type. The flow velocities have been varied between 0 and 80 m/sec. The microphone used in the measure- ments has a probe tube with a front opening in the form of a slot around the tube in a distance of 5 tube diameters from the front end. The opening is covered by copper gauze. Measurements have been made with respect to sound attenua- tion and sound velocity as functions of frequency for different flow velocities, the signal to noise ratio being about 30 db. In general the sound attenuation in the duct decreases with increasing flow velocity. For absorbers without resonance character this is observed over a comparatively large frequency range. In the case of resonance absorbers the resonant fre- quency is shifted towards higher frequencies with increasing flow velocity. The attenuation decreases for the resonant frequency but above this frequency reaches higher values than without flow. Between these frequencies we have a range with largely decreased attenuation which latter even may reach negative values so that an amplification is observed. 19. Attenuation of Sound in Lined Air Ducts. A. J. KING, Metropolitan Vickers Electrical Co. Ltd., Manchester, England.-- Theoretical predictions of the performance of silencers are apt to be difficult and unreliable so experimental results are often very valuable. A previous paper contained curves giving the attenuation of sound with frequency in lined ducts of various widths and thicknesses of lining of rock wool. Subsequent experience has shown the need for an extension of these curves to cover thicker linings and splitters and advantage has been taken of an opportunity to obtain further data on 16-in. splitters and air channels. New curves including these data are given for 50% space factor, together with an extra- polation to 32-in. splitters. Thicknesses greater than 16 in. are unlikely to be used unless frequencies lower than 100 cps are important. For covering very wide frequency bands two stage filters are recommended. Session J. Electroacoustics RICHARD K. COOK, Chairman Invited Papers J1. Some Observations on Reproduced Sound in an Automobile. B. A. SCHWARZ ANt) D. E. BRINKERHOFF, Delco Radio Division, General Motors Corporation, Kokomo, Indiana.---Some of the physical and psychological factors of reproduced sound in an automobile, are discussed. The environ- mental conditions, such as: ambient noise as it affects audible frequency and dynamic range; "listening room" size, audience seating arrangement, and acoustical properties, are compared with these condi- tions in a living room or concert hall. The transducer efficiency, size, loadings, required power handling, frequency balance, frequency and dynamic range, and the effects of its location on the distribution and delivery to the listeners are considered. Psychological factors affecting the desirability of many experimental loudspeaker locations tested, are presented. Contributed Papers J2. Stereophonic Sound Reproduction in the Home. HARRY F. OLSON, R CA Laboratories, Princeton, New Jersey.--To achieve realism in a sound reproducing system three funda- mental conditions must be satisfied, namely' first, the fre- quency range must be such as to include all the audible components of the various sounds to be reproduced' second, the volume range must be such as to permit noiseless and distortionless reproduction of the entire range of intensity associated with the sounds; third, the reverberation and spatial characteristic of the original sound must be preserved. This paper describes experiments and tests conducted over a period of a decade with the object of determining the factors which play a major role in establishing realism in reproduced sound under practical operating conditions in the home. The subjects investigated include the following: acoustic fre- quency preference; nonlinear distortion; stereophonic fre- quency preference; noise; localization (lateral and depth); reverberation. J3. Bigradient Uniaxial Microphone. HARRY F. OLSON, Joan PR.STON, AND Joan C. BL.AZ.Y, RCA Laboratories, Princeton, New Jersey.--A second-order unidirectional micro- phone has been developed consisting of two uniaxial micro- phones connected in series opposition. This system has been termed the bigradient uniaxial microphone. The directional efficiency, that is, the energy response to random sounds is one-ninth. The high discrimination which this microphone exhibits to sounds which originate from the sides and rear makes it particularly suitable for long distance sound pickup in radio, television, sound motion pictures, phonograph re- cording, and sound reinforcing systems. 1260 ACOUSTICAL SOCIETY OF AMERICA J4. Miniature Loudspeakers for Personal Radio Receivers. JOHN C. BLEAZE¾, JOHN PRESTON, AND EVERETT G. MAY, R CA Laboratories, Princeton, New Jersey.--This paper presents two small experimental loudspeakers of the same general design, suitable for use in personal radio receivers. The design of the magnetic structure is unconventional in that the loudspeaker cone housing and the magnet occupy the same space, and the over-all loudspeaker depth is thereby reduced. The vibrating system of the loudspeaker is con- ventional so that the directional characteristics, distortion, and frequency response are, in general, similar to a completely conventional loudspeaker of the same diameter. Some general design considerations applicable to this type loudspeaker are discussed, together with a chart summarizing these design considerations. J5. Lumped-Parameter Design Formulas for Electrically Driven Tonpilz Strongly Loaded at One End. FRITZ PORDES, U. $. Navy Underwater Sound Laboratory, Fort Trumbull, New London, Connecticut.--The Tonpilz (two lumped masses linked by a lumped compliance) is loaded only on one end and me- chanical losses are attributed to the loaded end. The compliant member as a whole, or a fraction of it, is either piezoelectric, electrostrictive, or magnetostrictive, and its electromechani- cally active portion is electrically lossless. Formulas developed in terms of dimensionless variables and their graphical plots allow one to arrive quickly at the optimum design(s) with given material constants for stated performance requirements. Considerable saving in time spent in computations is achieved. J6. Dynamic Mechanical Stability in the Variable-Reluc- tance and Electrostatic Transducers. CHARLES H. SHERMAN, U. $. Navy Underwater Sound Laboratory, Fort Trumbull, New London, Connecticut.--Large displacement behavior of the variable-reluctance, and the mathematically equivalent electrostatic, transducing mechanism was analyzed in order to evaluate the importance of mechanical instability as a limiting factor in high-power sound projectors. Previous work by Hunt et al. had shown the existence of static instability. This paper first considers the effect of transients on static stability and then shows that instability can also occur under steady-state dynamic conditions, even when' the system is statically stable. When the excitation is increased in order to increase the amplitude of the oscillations, the average displacement also increases, which leads to collapse of the gap. This interdependence of the oscillatory motion and the average displacement, intentionally excluded from linearized treatments, is the cause of dynamic instability. Detailed calculations show the conditions for instability for typical values of mechanical Q, polarization, and frequency. Under steady-state dynamic conditions, the average displacement can exceed the static limit of « the gap length, and the relative oscillatory displacement amplitude can reach « or more without collapse. Although such instability renders the transducer inoperative, the calculations indicate that it will seldom be a serious limitation. J7. Intracardiac Acoustics. JOHN D. WALLACE, U.S. Naval Air Development Center, Johnsville, Pennsylvania and Division of Cardiology, Philadelphia General Hospital, Phila- delphia, Pennsylvania, JAMES R. BROWN, JR., U.S. Naval Air Development Center, Johnsville, Pennsylvania, DAVID H. LEWIS AND GEOROE W. DEITZ, Division of Cardiology, Phila- delphia General Hospital, Philadelphia, Pennsylvania, AND ALI ERTUORUL, Cocuk Saglik Institusu, Ankara, Turkey.-- The authors have previously described the development of the intracardiac phonocatheter. These phonocatheters have been used routinely for the past year at the Heart Station of the Philadelphia General Hospital. This paper reports the findings of studies on dogs as well as one hundred human patients including children as young as eleven weeks. Pre- liminary measurements have been made of the transmission loss from chamber to chamber in the heart as well as from the various chambers to the chest wall. Spectrograms have been used in the studies of the several heart sounds, and frequency- time-intensity contours have been derived. The nature and origin of the sounds have been investigated in respect to the diagnosis of specific heart defects which in several cases have been subsequently confirmed by surgery. J8. Acoustic Direction Determination. H. J. BUAJSKI AND J. H. PROUT, Engineering Research Institute, University of Michigan, Ann Arbor, Michigan.--A method has been devised to automatically compute and display the azimuth of an acoustic signal generating source. Three sensors are arranged in a given pattern so that a signal wave front traversing the pattern from any direction becomes part of a mathematical function with respect to time. The signals as obtained from the sensors are converted to time pulses and fed to three respective delay lines to which are connected many coincidence channels. The output of each coincidence channel is fed to a display board which designates the given azimuth of the signal wave front. Limited experimentation has shown that a narrow beam system, less than five degrees, is feasible. The system is not frequency sensitive below a given upper limit determined by the spacing of the sensors. The proposed method suggests many possible uses from a single directional unit to an elaborate omnidirectional system. Where a passive omnidirectional detection unit is desired, the time pulses from three sensors could be transmitted by wire or radio to the master computing unit and display some distance away. (Parts of this research were supported by Tri-service Contract No. DA-36-039-sc-52654.) J9. Transient Tilt" Distortion in Hearing Aids. EDWIN D. BURNETT, THELMA H. JENSEN, AND EDITH L. R. CORLISS, National Bureau of Standards, Washington, D. C.--When some hearing aids are exposed to acoustic signal trains with sharp onset and cutoff, the envelope of the acoustic output is tilted. Upon frequency analysis, this is found to correspond to generation of the gating frequency, which is actually below the normal pass band of the instrument. Subjectively, instru- ments exhibiting this "tilt" in response envelope sound noisy and distorted, although their frequency response may be fairly flat and their steady-state nonlinear distortion may be measured to be low. This appears to be a type of inter- modulation test which is not critical with respect to pulse- repetition rate and is fairly insensitive to frequency. This work was supported by the Department of Medicine and Surgery of the Veterans Administration. JlO. Precise Automatic Method for Measuring Hearing Aid Response. THELMA H. JENSEN, EDWIN D. BURNETT, EDITH L. R. CORLISS, AND MAHLON D. BURKHARD,* National Bureau of Standards, Washington, P. C.--This paper describes equipment used at the National Bureau of Standards for rapid, precise measurement of hearing aid properties. By using an insert technique, allowance can be made for varia- tions in the measuring system. A compression circuit which is relatively fast acting serves to stabilize the source. Logarith- mic amplifiers and a differencing circuit make possible the direct recording of gain. A Panoramic analyzer serves as a swept wave analyzer for distortion measurements. The system has proved satisfactory for free-field measurements as well as cavity pressure measurements. This work was sup- ported by the Department of Medicine and Surgery of the Veterans Administration. * Now at Armour Research Foundation, Chicago, Illinois. FIFTY-FOURTH MEETING 1261 Session K. Ultrasonics ROBERT T. BEYER, Chairman Contributed Papers K1. Frequency and Temperature Dependence of Internal Friction in Pure Copper. W. P. MASON AND H. B6mEL, Bell Telephone Laboratories, Inc., Murray Hill, New Jersey.-- Measurements have been made at low strain amplitudes, of the internal friction, Q-i, for 99.999% pure copper for fre- quencies from 100 cy to 14 mc and from 4.2øK to room temperature. If one plots the Q- values against the log of the frequency for a given temperature, a linearly decreasing relation is obtained. At low temperatures the internal friction becomes very small. Although the loop length dependence agrees with the Koehler pinned loop model of internal friction, the present data shows that the frequency dependence does not. It is suggested that another source of dissipation is needed. The present data suggest that this is connected with the displacement occuring as the loop moves and is of a hysteresis nature. The frequency and temperature data show that the hysteresis lag in the motion cannot follow the applied stress at high frequencies or low temperatures. K2. Acoustical Techniques for the Study of Electrochemical Processes. ERNEST YEAGER AND FRANK HOVORKA, Ultrasonics Research Laboratory, Western Reserve University, Cleveland, Ohio.--At relatively low frequencies, sound waves produce an alternating component in the potential of a reversible electrode in accord with the thermodynamic dependence of the potential on pressure and temperature. With increasing frequency, the ac response of the electrode decreases until at sufficiently high frequencies only the alternating components associated with the variation of the electrode-solution interface capacity and the ionic vibration potentials remain. The departure of the response from thermodynamic expectations occurs because the electrode processes responsible for the establishment of the potential are too slow for instantaneous equilibrium to be maintained in the sound field. The situation is analogous to the relaxation effects associated with the excess sound absorp- tion in polyvalent electrolytes such as magnesium sulfate. From the frequency dependence of the alternating potentials at various electrolyte concentrations and temperature, infor- mation can be obtained as to the orders, rate constants, and energy barriers for various electrode processes, e.g., electron transfer from a metallic electrode to an ion. Since most electrochemical systems involve several consecutive steps, a spectrum of relaxation frequencies is anticipated. The measure- ment of these acoustically produced alternating potentials offers considerable promise for the study of electrode processes which are too rapid for conventional electrochemical tech- niques. (This paper is based on research sponsored by the Office of Naval Research.) K3. Measurement of Dynamic Shear Impedance of Liquids at High Ultrasonic Frequencies. H. J. McSKImN, Bell Tele- phone Laboratories, Inc., Murray Hill, New Jersey.--Further development of a shear reflectance technique reported on previously is here described [Phys. Rev. 75, 936 (1949)-]. Shear waves at 30 mcps are reflected at a low grazing angle from a polished surface on a fused silica rod. Simultaneously waves are transmitted through a second path designed to provide a precisely matched reference channel with high relative stability. The specimen liquid when placed on the polished surface produces a change of amplitude and phase for the reflected wave from which the complex shear imped- ance can be calculated. Illustrative data is presented for a series of Dow Corning DC-200 silicone fluids having steady flow viscosities from 1.5 to 200 000 centistokes. At 39 mcps and 25øC relaxation effects are present for viscosity grades as low as 5 centistokes. The limiting stiffness (Voigt) is -2 X 107 dynes/cm'--in the range for rubber-like materials. K4. Recording Ultrasonic Interferometer for Liquids. JOSEPH L. HUNTER, MICHAEL J. MARKOWSK, JOSEPH J. KLo?ovtc, AND LEONARD F. BRUENtNC, John Carroll Univer- sity, Cleveland, Ohio.--An interferometer has been constructed for the measurement of velocity and absorption in liquids. The design includes features for achieving and maintaining close parallelism between emitter face and reflector face over a large range of movement of the reflector. The reflector distance may be changed either manually or automatically. If driven automatically, the drive is taken from the gear box of a Bruel and I<jaer graphic level recorder through a flexible shaft and a reducing gear, so that the motion of the reflector is synchronized with the motion of the recorder paper. A precision gauge with a full range of six inches is firmly attached to the reflector shaft for a check on the travel of the reflector. If greater precision of distance measurements is required, provision is made for the insertion of gauge blocks. For extremely minute motions a differential drive may be inserted. The instrument is constructed in subassemblies, so as to be adaptable over a wide range of absorption coefficients. Convenience features include a self-contained temperature bath and accessibility of the crystal holder for ease of change of emitters. K5. Absorption and Velocity Measurements in Castor Oil. JOSEPH L. HUNTER AND LEONARD F. BRUENING, John Carroll University, Cleveland, Ohio.--Velocity and absorption in castor oil have been measured over a frequency range from 3 to 15 mc, and over a temperature range from 0 ø to 90øC. The velocity measurements indicate that dispersion, if present at all, is extremely small, and probably beyond the experimental accuracy. On the other hand, the change of velocity with temperature is extremely large, varying from 1570 msec - at 0øC, to 1320 msec - at 90øC. The measured absorption coefficient is definitely below the Stokes low- frequency value in all cases where the absorption coefficient is large, and definitely above where the absorption coefficient is small. Although the temperature at which the crossover point occurs varies greatly with frequency, (from 25øC at 3 mc to 90 ø at 15 mc) an interesting feature of the results is that the value of the coefficient at the crossover point is always approximately the same. This fact makes it appear that absorption in castor oil is entirely a viscosity phenomenon, in spite of the fact that there are marked departures from the simple frequency-squared law. K6. Velocity of Sound in a Liquid Containing Gas Bubbles. H. B. KAR?LUS, Armour Research Foundation of Illinois Institute of Technology, Chicago, Illinois.--At low frequencies consideration of the quasi-static elastic moduli and densities of the gas and liquid parts of the mixture yields a value of sound velocity which is much lower than the velocity in either constituent. The computation has been confirmed experimentally in the concentration regions of 1 to 60% air by volume. The sound velocity was found from the change of phase in a plane progressive wave as a function of distance. Concentrations were determined from a measure of the hydrostatic pressure of the mixture. The "froth" was made by pumping the water rapidly past a porous glass filter through which the air was forced. A trace of detergent re- 1262 ACOUSTICAL SOCIETY OF AMERICA tarded the coalescence of bubbles. Minimum sound velocity in a water air mixture occurs at 50% concentration where the velocity is approximately 22 mps. K7. Radio-Frequency Bridge Methods in Ultrasonic Inter- ferometry. E. F. CAROME, F. A. GUTOWSKI, AND D. E. SCHUELE, John Carroll University, Cleveland, Ohio.--The sound absorption coefficient in liquids may be shown to be simply related to the acoustic resistance of a liquid column [-J. Acoust. Soc. Am. 29, 769 (A) (1957). It is possible to determine accurately the variations of this resistance by employing radio-frequency bridge techniques. Such a method has been used to measure the temperature dependence of the absorption coefficient of castor oil in the frequency range from 2 to 10 M c. The experimental accuracy and consistency of the results indicate that this method may aid in removing discrepancies in the existing data. In addition, because of its extreme sensitivity the bridge-type detection system appears to be ideal for velocity and dispersion measurements. K8. Effects of Ultrasound on Aqueous Methyl Iodide. EDWIN C. STEINER AND JOHN KARPOVICH, Physical Research Laboratory, The Dow Chemical Company, Midland, Michigan.q The irradiation of a heterogeneous system of methyl iodide and water with high-intensity ultrasound causes a free radical reaction to occur. The reaction was followed to its completion by measuring the volume of the gases liberated and the conductivity of the solution. The gaseous, liquid, and solid products were collected and identified. The gases liberated were principally methane along with lesser amounts of ethane, ethylene, propane, propylene, acetylene, butane, etc. The main liquid product was methylene iodide, whereas ethylene iodide was the main solid product. Hydrogen iodide and iodine were also formed. Methylene iodide, which has a low vapor pressure, was not decomposed by ultrasound. This observation supports the hypothesis that the aqueous methyl iodide reaction takes place in the vapor phase. K9. The Effects of Ultrasound on Growing Bone. JUSTINE L. VAUGHEN, AND LEONARD F. BENDER, Physical Medicine and Rehabilitation, University of Michigan Medical School, Ann Arbor, Michigan.--The epiphyseal area of growing bone is currently considered a contra-indicated site for ultrasonic therapy. Previous studies indicate that damage occurs to bone cortex and bone marrow with high intensities of ultra- sound. The effect of clinical doses of ultrasound upon the epiphyses of growing rabbits was studied. The left knee area of 20 animals was treated underwater with 1 w/cm ' from a 12 cm - sound head for five minutes daily from the age of three months until there was x-ray evidence of epiphyseal closure (6 to 8 months). The untreated hind leg served as control. Comparison of the data showed no significant differ- ence in bone length, microscopic appearance, or rate or manner of epiphyseal closure between the treated and control leg. Session L. Intelligibility KARL KRYTER, Chairman Contributed Papers L1. Cocktail Party Effect. IRWIN POLLACK AND J. M. PICKETT, Operational Applications Laboratory, Air Force Cambridge Research Center, Bolling Air Force Base, Washington, D. C.--Acousticians have long recognized that the cocktail party is a stimulating environment for inquiry into the basic problem of acoustics. The cocktail party provides a critical test of listening to a talker against extraneous sound sources. The "cocktail party effect" hence has been identified with situations which require selective directional listening in a multitalker environment. After appropriate operational re- search of cocktail parties, we brought the problem to the laboratory. Unfortunately, we were forced to reduce the problem to more sober proportions. We tested the intelligi- bility of recorded monosyllabic PB words against a babble of 1, 2, 4, or 7 independent talkers reading newspaper text. Two test conditions were examined: (1) test words were introduced stereophonically with one set of background talkers presented over one earphone and another set of talkers presented over the other earphone; and (2) test words were introduced with a single set of background talkers presented over a single earphone. The cocktail party effect was identified with the difference between the speech-to- background ratios required for 50% intelligibility under the two conditions. The effect ranged from 12 db with one back- ground voice to 6 db with seven background voices (in each ear). L2. Effects of High Pass, Low Pass, and Band Rejection Filtering upon Articulation Test Scores. IRL D. KRYTER, Operational Applications Laboratory, Air Force Cambridge Research Center, Bolling Air Force Base, Washington, D.C.-- Articulation tests (CVC nonsense syllables) were conducted with a wide variety of HP and LP filter and gain conditions in an attempt to determine bands of speech frequencies that con- tribute equally to the intelligibility of speech. Comparison is made between our results and those reported by French and Steinberg (1947). In addition, tests with HP and LP filtering were run in which several portions of the range of speech fre- quencies were sharply rejected. It was found that with certain of these latter filter arrangements the articulation scores that were obtained greatly exceeded those predicted by any "arti- culation index" (A1) based on the simple addition of the con- tributions of "equal articulation bands." There will be a brief discussion of the theory underlying different methods used for dividing the frequency range into bands supposedly con- tributing equally to speech intelligibility. L3. Prediction of Speech Intelligibility at High Noise Levels. J. M. PICKETT AND IRWIN POLLACK, Operational Applications Laboratory, Air Force Cambridge Research Center, Bolling Air Force Base, Washington, D. C.--Word articulation tests were conducted with three noise spectra (spectral slopes of - 12, 0, and +6 db per octave) and with two speech spectra (spectral tilt of 0 to +6 db per octave) with noise levels up to 130 db SPL. Loss of intelligibility was observed at higher noise levels for all spectral conditions examined. The effective change in the speech-to-noise ratio (S/N), defines the effect of high noise levels upon intelligibility. This quantity is that change in the S/N ratio which, at moderate listening levels, produces the equivalent change in intelligibility observed at high noise levels. To a first approximation, (S/N), is primarily dependent upon the over-all noise level. For example, as the over-all FIFTY-FOURTH MEETING 1263 noise level is increased from 90 to 125 db SPL, (S/N)e increases to 4.5 db (averaged over spectra and averaged over S/N ratio). However, (S/N), is not invariant with spectra and S/N ratio. (S/N), is generally higher at more favorable S/N ratios, and it is also somewhat lower when speech and noise spectra are parallel than when divergent. The use of (S/N), for predicting intelligibility at high noise levels is compared with procedures suggested by French and Steinberg and by Beranek. L4. Relationship of Speaker Intelligibility to the Sound Pressure Level of Continuous Noise Environments of Various Spectra and Octave-Band Widths. GILBERT C. TOLHURST, Acoustic Laboratory, U.S. Naval School of Aviation Medicine, N. A. S., Pensacola, FIorida.--The effects of six tilted and six octave-band ambient noise spectra upon speaker intelligibility were studied as each spectrum was presented at six different sound pressure levels through 125 db. Constant level record- ings were made of 48 speakers talking in the noises, and these were subsequently played back to panels of listeners. The mean speaker intelligibility data were subjected to analyses testing speech score differences attributable to spectra and sound pressure level. The results indicate that, generally, as the ambient noise surrounding talkers is increased in sound pressure level there is a decrement in speaker intelligibility whether the noise is of a tilted or an octave-band spectrum. The differences were highly significant statistically. The -3 db per octave tilted spectrum yielded an average decrease in speaker scores of 2.3 percentage points per db increase of noise between 105-125 db of noise. The most detrimental octave-band noise, 1200-2400 cps, averaged 0.75 percentage points lower scores per db noise level increase. Statistically significant differences were found among each of the two general spectra types affecting speaker intelligibility. Greater differences were exhibited among the tilted spectra than among the octave-band noises. L5. One-Syllable Words. HENRY M. MOSER, The Ohio State University Research Foundation, Columbus, Ohio.--A systematic listing of the monosyllabic words in American English is made according to sound. An empirical approach is followed in which sounds are produced alone or in combina- tions. If the auditory signal is recognized as a word, it is listed in conventional spelling. A number of words not included in dictionaries are identified by this method. While the list was prepared to supply additional single syllable words to supple- ment those already used in radiotelephone communication, it can be utilized in language studies to show the frequency of occurrence of vowels, consonants, and consonant combina- tions at the beginning and endings of words. Frequency of occurrence of the speech sounds in various combinations are presented in tabular form. [Done in connection with contract AF 19(604)-1577, Operational Applications Laboratory, AFCC.3 L6. Relationships among Threshold and Intelligibility Word Lists. BRUCe. M. SIEa.NTHXL.R AND EDWARD HARDICK, Speech and Hearing Clinic, The Pennsylvania State Univer- sity.--Since development of the PB-50 word lists and their use for measuring discrimination loss, the PB-familiar (Hudgins), PB-kindergarden (Haskins), and the W-22 lists (Hirsh et al.) have been assembled. The relationship between P.A.L. test No. 9 and the PB-50 articulation curve has been described, as has the relationship between C.I.D. tests W-2 and W-22. Although the PB-F and PB-K lists are potentially useful in audiologic practice, their relationships to other P B lists and threshold tests have not been reported. Three series of observation were done: (a) Twenty-two normal hearing adults were given tests No. 9 and W-2, and articulation curves for PB-50, PB-F, PB-K, and W-22 lists. (b) Thirty- four children, ages eight to eighteen years, were given tests W-2 or the Picture Identification Test for threshold, and tests PB-50, PB-F, and PB-K at 40 db H.L. (c) Seventeen normal hearing and twelve hypacusic adults were given P.A.L. tests No. 9, and PB-F articulation curves to deter- mine 50% intelligibility levels. The relative db levels for the threshold tests and various articulation curves are presented. L7. Word Intelligibility and Position in Sentence. HERBERT RUBENSTEIN AND J. M. PICKETT, Operational Applications Laboratory, Air Force Cambridge Research Center, Bolling Air Force Base, Washington, D. C.--There is evidence that the manner in which context is distributed affects its constraint on verbal responses. For example, it has been demonstrated that a bilaterally distributed context is most effective in reducing the number of acceptable meanings for a given word. Another investigation has shown that nouns are most predictable at the end of a sentence and least predictable at the beginning of a sentence. In these investigations the contexts were pre- sented visually; the present experiment was undertaken to ascertain whether similar results would be obtained with speech presented against a background of noise. Complete declarative sentences, all of the same length, were constructed in pairs; e.g., for each sentence containing the test noun in initial position, there was a semantically equivalent sentence made up of the same words but with the test noun in medial or final position. The members of a pair were presented at different experimental sessions in such a way that each experi- mental condition was equally represented at each session. The results indicate that, when the signal is compressed so that all words are approximately equal in intensity, intelligi- bility is lowest in initial position and highest in final position. L8. Confidence Ratings, Message-Reception, and the Receiver Operator Characteristic. Louis R. DECKER AND IRWIN POLLACK, Operational Applications Laboratory, Air Force Cambridge Research Center, Bolling Air Force Base, D.C. m Can an observer rate the accuracy of his message reception? Apparently, yes. Listeners were presented with spondee words against a white noise background. For each word presented, the listener recorded his received message. Im- mediately thereafter, he assigned a confidence rating with respect to his accuracy of message reception. We find that the assignment of confidence ratings does not interfere with the accuracy of message reception. More important, the rela- tive accuracy of message reception is directly related to the confidence rating. This relationship is relatively invariant over a range of speech-to-noise ratios. It may be noted that the rating procedure directly yields the entire trading relation between correct confirmations and false alarms--the receiver operating characteristic (ROC). The form of the ROC curves obtained by the rating procedure is independent of average word difficulty and is similar to that obtained by successive experiments with fixed acceptance criteria. 1264 ACOUSTICAL SOCIETY OF AMERICA Session M. Underwater Sound JAMES L. STEWART, Chairman Invited Papers MI. Sounds Produced by the Unsteady Flow of Water. M. STRASBERG, David Taylor Model Basin, Washington, D. C.--The production of sound by the unsteady flow of a homogeneous fluid is dis- cussed, with special attention given to those phenomena of practical concern as sources of underwater noise. Brief reviews are given of recent experimental and theoretical investigations of the sound associated with the following flow phenomena: (1) unsteady flow within an unbounded fluid, related to the noise from turbulent jets and wakes; (2) fluctuating hydrodynamic forces due to unsteady flow about a rigid body, related to "aeolian tones" and the rotation sound from a propeller; and (3) excita- tion of the vibration of an elastic body by the flow. A short motion picture produced by L. Prandtl will be shown to illustrate several kinds of unsteady flow. M2. Long Wavelengths in Stratified Media: Comparisons between Theory and Experiment. I. TOLSTOY, Columbia University, Hudson Laboratories, Dobbs Ferry, New York.--A principal char- acteristic of propagation in a medium of rapidly varying properties is its strong dispersion. For a transient source this means a rapidly changing form of the traveling wave. For simple harmonic exci- tation it results in different rates of energy transport for different frequencies and thus in resonant and antiresonant excitation frequencies. Both types of phenomena have been studied experimentally, although most of the past work has been limited to the dispersion of pulses, and in particular to the concept of "arrival time" of the various Fourier components. This concept, although not rigorous, has been very useful and has found many applications and verifications in geophysics and acoustics. In the last five years successful efforts have been made to obtain more detailed agreement between theory and experiment, with particular reference to the quantitative prediction of sound fields and wave forms. Several such experiments will be discussed here. One of these is a model experiment with a tran- sient source, designed around Pekeris' treatment of the two-layered half-space. Another is a simple harmonic experiment carried out in shallow water at sea. In both cases detailed and quantitative agreement has been secured within quite acceptable limits of error. Contributed Papers M3. Sound Propagation in Dissipative Layered Media. R. P. RYAN,* R. K. EB¾, AND A. O. WILLIAMS, JR., Department of Physics, Brown University, Providence, Rhode Island.-- When sound is propagated from a point source in a layer of fluid, comparable in depth to the acoustic wavelength and overlying a much thicker layer of fluid with greater density and sound velocity, "wave-guide" or normal-mode effects and cylindrical spreading are to be expected, according to well- known theory. If the thicker layer absorbs sound, total internal reflection is destroyed, and the sound in the thin layer should be attenuated according to relations derived by Kornhauser and Raney [J. Acoust. Soc. Am. 27, 689 (1955). If the thicker layer is also shear-elastic, this attenuation should increase, through conversion of compressional into shear waves [-K. Ergin, Bull. Seismological Soc. Am. 42, 349 (1952). These predictions have been confirmed by measure- ments in a water layer 0.5-2 cm thick, over a slab of rubber for which the density, absorption, and sound velocities were known. The frequency range was 55 to 600 kc. Vertical modal patterns of sound intensity and interactions of two modes were also found to agree with theory. (Supported in part by the Office of Naval Research.) * On leave for graduate study from the U.S. Naval Mine Defense Lab- oratory, Panama City, Florida. M4. Sound Propagation in Layered Media with a Rough Interface. R. K. EBY AND A. O. WILLIAMS, JR., Department of Physics, Brown University, Providence, Rhode Island. m When sound propagates in a fluid layer of suitable thickness, overlying a much thicker layer of greater density and sound velocity, the normal-mode propagation expected ideally [-C. L. Pekeris, "Theory of propagation of explosive sound in the sea," in Memoir 27 of the Geological Society of America (1948) is a result of total internal reflection. Anything pre- venting such reflection, such as a rough interface, should cause attenuation of the sound. A qualitative explanation is that some diffusely reflected sound strikes the interface at less than critical angles and escapes. Such attenuations have been studied with the apparatus of the accompanying paper by Ryan, Eby, and Williams. Small solid spheres were spread on the rubber-water interface; the extra attenuation caused by them was measured in the water layer as a function of frequency, water depth, sphere size, and area density of spheres. This attenuation is roughly proportional to area density of spheres, and to their acoustic scattering cross sections, and varies additionally with sphere size, water depth, and frequency, Frequencies were 100-250 kc; sphere radii 0.07-0.23 cm; water depths 0.5-2.0 cm. Glass, lead, and steel spheres behaved alike. (Supported in part by the Office of Naval Research.) M5. Noise Produced by Boundary Layer Turbulence of a Rotating Cylinder. EUGEN SKUDRZYK AND GILLIAN P. HADDLE, Ordnance Research Laboratory, The Pennsylvania State Univer- sity, University Park, Pennsylvania.--A rotating cylinder was equipped with hydrophones and immersed in water. Measure- ments of the velocity distribution and the noise level were performed. The results have been used as a guide line to derive a general theory of the noise production by plane or curved, smooth, and rough surfaces and to estimate the admissible size of surface roughnesses under given conditions. M6. Arrays for the Investigation of Ambient Noise in the Ocean. V. C. ANDERSON, University of California Marine Physical Laboratory of the Scripps Institution of Oceanography, San Diego, California.--A discussion of the important param- eters to be considered in the use of directional arrays for the study of noise fields in the ocean leads to specialized array designs. Some properties of circular and spherical arrays composed of omnidirectional elements are presented. The AUTHOR INDEX 1265 calculation of directivity index for a spherical array is carried out by the use of the space-correlation functions of an isotropic noise field. The measured, broad-band directivity pattern of a circular plane array as a function of frequency is also shown. (This work represents one of the results of research carried out under contract NObsr 72512, NE 120221-4-5-6 with the Bureau of Ships, Navy Department.) M7. Evaluation of Surface Reverberation in Sonar. J. W. HORTON, U. $. Navy Underwater Sound Laboratory, New London, Connecticut.--Several measures have been devised for evaluating those characteristics of an acoustic medium, or of its boundaries, which determine the level of reverberation following the transmission of a pulse of acoustic energy. Of those measures which are applicable to surface reverberation, the coefficient giving the rate per unit area at which acoustic energy reaching a scattering surface along a given ray path is returned toward the source is particularly useful in computing the equivalent plane wave level of reverberation as a function of time. The present paper discusses the relations between this coefficient and other factors which must be taken into account in computing the total transmission loss suffered by acoustic energy returned in the form of reverberation. The significance of these relations with respect to practical prob- lems is discussed briefly. A simple and accurate method for measuring the value of the backscattering coefficient in any given situation is described. M8. Alteration of the Bubble Pulse Amplitude of an Underwater Explosion. OTTO H. HILL, Defense Research Laboratory, The University of Texas, Austin, Texas.--The underwater detonation of a conventional high explosive is characterized by the production of a shock wave and an oscillating gas globe, containing primarily the reaction pro- ducts, which in certain circumstances may result in the almost periodic generation of an acoustic pulse. If one postulates a scheme for altering the internal energy of the gas globe which will not affect the kinetic energy associated with the water, and which provides for internal pressure equilibration in a time sufficiently short that the actual radius of the bubble does not appreciably alter, then it is possible to obtain relations which yield the effective alteration of the bubble pulse amplitude as a function of the change in total energy produced in the bubble system, the time at which the alteration is made, and the initial history of the bubble. M9. Several General Purpose Transducer and Reflector Combinations for Underwater Sound Work.* M. V. MECH- LER,i C. M. MCIINNEY, C. D. ANDERSON, AND F. A. COLLINS, Defense Research Laboratory, The University of Texas, Austin, Texas.--Several transducer and reflector combinations are described which are useful over a fairly wide range of fre- quencies. One form consists of a right-angle conical reflector with a line transducer mounted on the principle axis of the cone. The units described have aperatures ranging from 6 in. to 48 in. Frequencies covered are from 25 kc to 0.5 mc. One arrangement allows simultaneous operation of the unit as a projector and as a hydrophone. Another transducer, designed to be used in the fractional megacycle region, uses a con- ventional parabolic reflector with a probe at the focus. Means are shown for rapidly changing the beam width, shading, and focus. Several other experimental transducers are described briefly. Calibration data in the form of direc- tivity patterns, sensitivity, and impedance are shown. * This work was sponsored by the Bureau of Ships. I Now with Collins Radio Company, Dallas, Texas. THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA VOLUME 29, NUMBER 11 NOVEMBER, 1957 Author Index to Abstracts of Ann Arbor Meeting Anderson, C. D.--I3, 14, M9 Anderson, V. C.--M6 Bach, R., Jr.--F12 Barnett, N. E.C8 Barraclough, T. W.C5 Bayston, T. E.--Fll Becker, CarlmI6 Bender, Leonard F.K9 Beranek, Leo L.--D3, D4 Bleazey, John C.--J3, J4 Bõmmel, H.--K1 Brinkerhoff, D. E.J 1 Brown, James R., Jr.--J7 Bruening, Leonard F.K4, K5 Bugajski, H. J.--E 7, J8 Burkhard, Mahlon D.--D1, H9, J 10 Burnett, Edwin D.J 9, J 10 Campanella, S. J.--Fll Carome, E. F.--K7 Chang, S. H.--F 12 Clark, L.--B9 Clements, Donald H.--E5 Collier, R. D.C5 Collins, F. A.M9 Collins, J. L.I3 Cooper, F. S.--F4 Corliss, Edith L. R.H9, J9, J 10 David, E. E., Jr.--H3 de Boer, E.--B6 Decker, Louis R.--L8 Deitz, George W.--J7 Delattre, P. C.F4 Drake, Alvin--F7 Dreher, John J.mF2 Dyer, Iraqi 7 Eanet, June O.H9 Eby, R. K.M3, M4 Edelman, Seymour--C1 Egan, J. P.--B 1 Eichler, E. G.E3 Ertugrul, AliJ 7 Fehr, Robert O.--G5 Fiala, Walter T.--C6 Fitzpatrick, R. C.--H5 Flanagan, James L.--F9 Follett, H.C7 Frazier, F.--H5 Fry, William J.--G6 Goldstein, M. H., Jr.--H6 Green, David M.--H7, H8 Gutowski, F. A.--K7 Guttman, Newman--H3 Haase, Kurt H.--F5, F7 Haddle, Gillian P.--M5 Hardick, Edward--L6 Harris, J. Donald--B2

1.1851671.pdf

Micromagnetic studies of vortices leaving and entering square nanoboxes Daniel Dotse and Anthony S. Arrott Citation: Journal of Applied Physics 97, 10E307 (2005); doi: 10.1063/1.1851671 View online: http://dx.doi.org/10.1063/1.1851671 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/97/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Micromagnetic calculation of dynamic susceptibility in ferromagnetic nanorings J. Appl. Phys. 105, 083908 (2009); 10.1063/1.3108537 Three-dimensional micromagnetic finite element simulations including eddy currents J. Appl. Phys. 97, 10E311 (2005); 10.1063/1.1852211 Magnetic normal modes of nanoelements J. Appl. Phys. 97, 10J901 (2005); 10.1063/1.1852191 Micromagnetic simulation studies of ferromagnetic part spheres J. Appl. Phys. 97, 10E305 (2005); 10.1063/1.1850073 Spin-polarized current-driven switching in permalloy nanostructures J. Appl. Phys. 97, 10E302 (2005); 10.1063/1.1847292 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 157.211.3.38 On: Sun, 30 Nov 2014 12:00:51Micromagnetic studies of vortices leaving and entering square nanoboxes Daniel Dotse and Anthony S. Arrotta! Center for Interactive Micromagnetics, Department of Physics and Chemistry, Virginia State University, Petersburg, Virginia 23806 sPresented on 10 November 2004; published online 6 May 2005 d The passage of the virtual vortex of the Cstate into a square cross-section nanobox proceeds by the development of a direction of polarization to break the symmetry. This happens in a dynamictransition for nanoboxes below a certain height, while for higher nanoboxes there exists a stablepolarized Cstate over a narrow range of applied fields that increases as the square of the excess of the height above the critical height. © 2005 American Institute of Physics . fDOI: 10.1063/1.1851671 g Adecade ago Dalhberg and Zhu 1popularized the experi- mental and theoretical synergism between magnetic micros-copy and micromagnetics. Microscopy glimpses the behaviorof patterned materials useful in information technology. Mi-cromagnetics, though simplified by approximations and ne-glect of some details, provides complete descriptions of themagnetic configurations and what might happen in transi-tions between states. Great progress has occurred in bothexperiment and theory in recent years. 2Yet, because this is a complex field, there continue to be subtleties that have es-caped attention. One of these is the nature of the transitionsbetween two well-known states of a magnetic nanobox ofsquare cross section with the height less than the width; theconfigurations are the simple vortex state and the Cstate. The subtle effects reported here are more of a computationalcuriosity than of practical importance. Experimentally thesewould be difficult to observe because they involve symmetrybreaking that is sensitive to defects and barriers that aresmall compared to room-temperature thermal activation en-ergies. The simplest vortex state has the magnetization of the core of the vortex and of all four vertical edges in the samesense along the short zaxis, about which the magnetization circulates.When a field is applied perpendicular to the zaxis, e.g., along the xaxis, the vortex center moves away from the zaxis in the ydirection. At a critical exit field the vortex becomes unstable and exits through the yface. The exiting vortex leaves behind a virtual vortex state, usually called theCstate, with its center outside of the box. There is a range of heights of the nanobox for which the Cstate is stable in the exiting field. The field can then be lowered to observe thevirtual vortex reentering the box. Simple micromagnetic cal-culation of the exit field and the field for reentry led to someerratic results for which there is a simple explanation and alesson for practitioners of the art of micromagnetics. This kind of effect is well established as it has to do with the relative stability of three states, which could be desig-nated as 1, 0, and −1, or up, down, and in the plane. Newelland Merill have referred to this as a three-tined pitchforkbifurcation. 3Here the problem is that the real vortex is po-larized, either up or down, but the virtual vortex can be po- larized or unpolarized because it does not have a real core.For the virtual vortex to reenter the box, symmetry must bebroken, for example, by a small applied field along the z axis. When the vortex leaves, the field is sufficiently highthat the virtual vortex is unpolarized. On lowering the x-axis field theCstate becomes unstable with respect to the vortex reentering the box at a field that depends on the bias appliedalong the zdirection.The polarization increases in a dynamic jump as the virtual vortex passes in through the yface. Above a critical height d c, the transition is not directly from the unpolarized Cstate to the vortex state, but rather through a polarized Cstate that is stable even if the z-axis bias field is removed.The vortex state, the unpolarized Cstate, and the polarized Cstate are compared in Fig. 1. The energy drop between the Cstate and the vortex state at reentry is much larger than the energy barrier for reentry.The latter is so small that it would not be observable at roomtemperature. The smallness of the barriers and the very weaktorques associated with them present difficulties for micro-magnetic calculations, necessitating extrapolation and inter-polation techniques 4to determine the phase boundaries of the polarized Cstate in a diagram of applied x-axis field and heightdin thezdirection. We demonstrate this with square boxes of permalloy, Py. The square sides are 54 354 nm2, which is large compared adElectronic mail: aarrott@vsu.edu FIG. 1. The vortex state sleftd, unpolarized Cstate scenter d, and the polar- izedCstate sright d,asviewedfromthetop supper dandfromtheside slower d from which the vortex leaves and enters as the Hxfield is first increased and then decreased.JOURNAL OF APPLIED PHYSICS 97, 10E307 s2005 d 0021-8979/2005/97 ~10!/10E307/3/$22.50 © 2005 American Institute of Physics 97, 10E307-1 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 157.211.3.38 On: Sun, 30 Nov 2014 12:00:51to the characteristic length of the competition between ex- change and magnetostatic demagnetization energies of Py,for which l ex=5 nm. The critical height above which the polarized Cstate is stable is dc=32 nm. Below this the un- polarized virtual vortex enters without first achieving thestable polarized state. The upper limit of consideration is d =38 nm. At this height the vortex state on leaving in the critical exit field passes through the Cstate to form a flower state from which reentry of the vortex is more complicated. The vortex exits for H x=485 Oe independent of din the range of interest here. The direction of the net magnetizationkM xlof theCstate is in the same direction as the field. The field for reentry depends strongly on d.A td=36 nm reentry occurs through the polarized Cstate which forms for Hx ,360 Oe. The calculated reentry field at Hx=222 Oe is sen- sitive to the choice of grid size in the micromagnetic calcu-lations, which were done using Scheinfein’s Landau–Lifshitz–Gilbert sLLG dmicromagnetics simulator. 5The effect of grid size is shown in Fig. 2, where kMzlis shown as a function of Hx. If the grid size is too large, the vortex itself, as calculated, is unpolarized, because the most energetic por-tion of the vortex hides at the intersection of four computa-tional cells. Because we restrict the calculation to cubic grid cells, the choices of grid size are given by 54/ nand the heights for calculation limited to 54 m/n, wherenandmare integers.At d=36 nm one has the range of grid sizes shown in Fig. 2. This calculation is performed by first finding a region ofstability of the polarized grid state in the presence of an H z =1 Oe bias field. The grid size must be substantially less than lex=5 nm to properly assess the range of stability of the polarized Cstate. For a 9-nm grid a vortex is hysteretic onmoving between cells. For the 18-nm grid the calculation gives a vortex with or without polarization. Without the bias field, the field of reentry of the vortex depends on waiting for an instability at the level of theround-off error of the computer to grow by over ten orders ofmagnitude.Aconventional hysteresis loop calculation wouldfollow the central tine of the pitchfork without falling off theridge of instability. Indeed, it was the erratic behavior of asimple exercise in computing the properties of the box as afunction of the height that led to the discovery of the role ofthe stable polarized Cstate. Once a stable polarized Cstate is found, it is used as a starting point for calculations to higher and lower H xfields to find the limits of stability. The transition from the polarizedCstate to the unpolarized Cstate is continuous with the z component of the magnetization going to zero as sqrt sH −H cd. The vortex enters as a first-order jump. The range of stability of the polarized Cstate is shown in the phase diagram of Fig. 3. Note that it takes a negativefield to drive the vortex in for d z,32.5 nm. For higher boxes a positive field is needed to keep the vortex out.The width ofthe region of stability increases as the square of the increasein height above the critical height that is found by curvefitting to be d c=32.0 nm. Because d=32 nm was so close to the critical height for the formation of the polarized Cstate, the torques were exceedingly small. The path method6is used to find the dependence of kMzl onHzat each of several choices of Hx. The basic assumption of the path method is that the equilibrium configurations arespecified with considerable accuracy by just stating kM zl, almost independent of the fields used to produce the configu- ration. For each configuration all the terms in the energy areevaluated. The sum of the demagnetization energy and theexchange energy is called the internal energy of the configu-ration. sThe anisotropy energy is also included, even though it is negligible. dThe derivative of the internal energy with respect to the kM zlVis the field in the zdirection, which would hold the configuration in equilibrium sstable or not d. Unstable equilibrium solutions are investigated with micro- FIG. 2. Calculations with different grid sizes s18–1 nm cubes dof the net fractional zcomponent of the magnetization for the polarized Cstate be- tween the onset of polarization at high fields and the transition at low fieldsto the vortex state with larger polarization. The calculation for the large18-nm grid size produces a vortex that appears to have no polarization eventhough it comes from a polarized Cstate. FIG. 3. Phase diagram for the stability of the polarized Cstate, shown as the dashed region between the calculated transition fields Hxfor the unpolarized to polarized Cstate transition sdiamonds dand the polarized Cstate to vortex state transition ssquares dfor the indicated heights dzof the square box of permalloy with 54-nm sides. The lower curve striangles dis the difference in transition fields fit to sdz−dcd2withdc=32.0 as the best fit.10E307-2 D. Dotse and A. S. Arrott J. Appl. Phys. 97, 10E307 ~2005 ! [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 157.211.3.38 On: Sun, 30 Nov 2014 12:00:51magnetics by readjusting the applied field from time to time to maintain the equilibrium solution for a given kMzl. The torques are so low that the approach to equilibrium is ex- tremely slow. The equilibrium values of kMzlare found by extrapolation over many time constants. The result is judged to have converged when the extrapolated values are nolonger changing with calculation time. The magnetizationcomponents and the energy are becoming completely expo-nential in approach to equilibrium after times of the order ofone time constant. All of the short wavelength perturbationsdamp out quickly, while the system as a whole moves toequilibrium at glacial speeds. The above calculations were carried out for a box rather than a circular cylinder to avoid problems with serratededges and their effect on reentry of vortices. But this workwas stimulated by the excellent studies of vortices in cylin-ders by the Regensburg group. 7The Center for Interactive Micromagnetics under the di- rection of Professor Carey E. Stronach is supported by theAFOSR. This work constitutes part of a Masters Thesis byone of the authors sD.D. din the Department of Chemistry and Physics at Virginia State University. 1E. D. Dahlberg and J. G. Zhu, Phys. Today 48s4d,3 4 s1995 d. 2R. Wiesendanger, M. Bode, A. Kubetzka, O. Pietzsch, M. Morgenstern, A. Wachowiak, and J. Wiebe, J. Magn. Magn. Mater. 272–276,2 1 1 5 s2004 d. 3A. J. Newell and R. T. Merrill, J. Appl. Phys. 84, 4394 s1998 d; G. Iooss and D. D. Joseph, Elementary Stability and Bifurcation Theory , 2nd ed. sSpringer, New York, 1990 d. 4A. S. Arrott, Introduction to Micromagnetics inUltrathin Magnetic Struc- tures, edited by B. Heinrich and J. A. C. Bland sSpringer, Berlin, 2004 d, Vol. 4. 5M. R. Scheinfein and A. S. Arrott, J. Appl. Phys. 93, 6802 s2003 d. 6http://llgmicro.home.mindspring.com/ 7M. Rahm, M. Schneider, J. Biberger, R. Pulwey, J. Zweck, D. Weiss, and V. Umansky, Appl. Phys. Lett. 82,4 1 1 0 s2003 d.10E307-3 D. Dotse and A. S. Arrott J. Appl. Phys. 97, 10E307 ~2005 ! [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 157.211.3.38 On: Sun, 30 Nov 2014 12:00:51

1.4975132.pdf

Finite-dimensional colored fluctuation-dissipation theorem for spin systems Stam Nicolis , Pascal Thibaudeau , and Julien Tranchida Citation: AIP Advances 7, 056012 (2017); doi: 10.1063/1.4975132 View online: http://dx.doi.org/10.1063/1.4975132 View Table of Contents: http://aip.scitation.org/toc/adv/7/5 Published by the American Institute of PhysicsAIP ADV ANCES 7, 056012 (2017) Finite-dimensional colored fluctuation-dissipation theorem for spin systems Stam Nicolis,1,aPascal Thibaudeau,2,band Julien Tranchida1,2,c 1CNRS-Laboratoire de Math ´ematiques et Physique Th ´eorique (UMR 7350), F ´ed´eration de Recherche “Denis Poisson” (FR2964), D ´epartement de Physique, Universit ´e de Tours, Parc de Grandmont, F-37200 Tours, France 2CEA DAM/Le Ripault, BP 16, F-37260 Monts, France (Presented 2 November 2016; received 23 September 2016; accepted 4 November 2016; published online 26 January 2017) When nano-magnets are coupled to random external sources, their magnetization becomes a random variable, whose properties are defined by an induced probability density, that can be reconstructed from its moments, using the Langevin equation, for mapping the noise to the dynamical degrees of freedom. When the spin dynamics is discretized in time, a general fluctuation-dissipation theorem, valid for non-Markovian noise, can be established, even when zero modes are present. We discuss the subtleties that arise, when Gilbert damping is present and the mapping between noise and spin degrees of freedom is non–linear. © 2017 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). [http://dx.doi.org/10.1063/1.4975132] I. INTRODUCTION For any system, in equilibrium with a bath, the fluctuation-dissipation relation (FDR) plays an important role in defining consistently its closure, since it relates the fluctuations of the subsystem of the dynamical degrees of freedom, that one is, by definition, interested in, with the fluctua- tions of the degrees of freedom that are defined as uninteresting and are lumped under the term “dissipation”. The essential reason behind this relation is that, for equilibrium situations, it is possible to define a probability measure on the space of states, with respect to which the average values, that enter in the FDR, can be unambiguously computed. So this can be modified, if the dynamical degrees of freedom are so affected by the immersion in the bath, that they must be replaced by others–the interaction with the bath leads to a phase transition and the equilibrium measure is not unitarily equivalent to the measure of the dynamical degrees of freedom, in the absence of the bath. While it is possible to address these questions by numerical simulations, and reconstruct the density that way, what has, really, changed in the last years is that experiments of great precision, that probe both issues, have become possible, particularly in magnetic systems.1It is in such a context that the FDR has become of topical interest.2–4 In such systems, since the noise affects the magnetic field, that makes the spin precess, it is not additive, but multiplicative. While, already, for additive noise, the issue of the “backreaction” of the dynamical degrees of freedom on the bath is quite delicate, for multiplicative noise it becomes even more difficult to evade and must be addressed. Further complications arise when the fluctuations are colored, namely posses finite intrinsic correlation time.5,6In such a situation, no FDR has been unequivocally obtained, that relates the intensity of the fluctuations to the damping constant.7 aElectronic mail: stam.nicolis@lmpt.univ-tours.fr bElectronic mail: pascal.thibaudeau@cea.fr cElectronic mail: julien.tranchida@cea.fr 2158-3226/2017/7(5)/056012/4 7, 056012-1 ©Author(s) 2017 056012-2 Nicolis, Thibaudeau, and Tranchida AIP Advances 7, 056012 (2017) In this note, we wish to study these issues in the context of magnetic systems placed in random magnetic fields, whose distribution can have an auto–correlation time comparable to the time scale defined by the precession frequency. The aim of this communication is to sketch out a route for establishing a FDR in a quite general setting,8that will be shown to be consistent to previous results for magnetic systems, obtained in the limit of white-noise fluctuations, and can be readily adapted beyond this context, especially for explicit calculations. A remaining challenge is to obtain the stochastic equation, that defines the mapping between noise and the dynamical degrees of freedom, that are identified with the spin components of a nanomagnet, and whose solution does, indeed, describe a normalizable density for the spin configurations. II. GAUSSIAN APPROXIMATION In order to better grasp the issues at stake, we shall start with a finite number of dynamical degrees of freedom, sA n. The time index nruns from 0 to N 1 and will be identified with the evolution time instant, in the continuum limit; the flavor index Aruns from 1 to Nfand labels “internal” degrees of freedom–it will label the components of the spin. The summation convention on repeated indices is assumed. We assume that these dynamical degrees of freedom are immersed in a bath. The bath is described by variables A nand is defined by the partition function Z= NfY A=1N1Y n=0dA ne1 2A nFABDnmB m (1) The matrix Facts on the flavor indices and the matrix Don the “target space” indices–that describe the instants in time. The white noise case corresponds to taking Dnm=nm=2. The simplest colored noise case corresponds to taking Dnm=nm=2 n, with not all the nequal. Furthermore, if it cannot be put in diagonal form at all, then it describes higher derivative effects. The average of a functional Fof the variables A nis then well defined as hFi=1 Z NfY A=1N1Y n=0dA nF[]e1 2A nFABDnmB m (2) From this expression we may deduce the moments of the degrees of freedom of the bath: D A nE =0 D A nB mE =f F1gABf D1g nm(3) with the others deduced from Wick’s theorem. What we notice here is that, for non–diagonal matrices, FandD, the degrees of freedom of the bath that have well–defined properties, i.e. the degrees of freedom that are eigenstates of these matrices, are linear combinations of the A n. So it makes sense to work in that basis. In this context, the white noise limit corresponds to the case in which Dis the identity matrix–all components have the same relaxation time. The colored noise case, then can be identified as that, where Dis not the identity matrix. When we immerse a physical system in such a bath it can happen that the eigen-bases of the system and of the bath do not match. The map between the degrees of freedom of the bath and the dynamical degrees of freedom is provided by a stochastic equation. For instance, one consider the Landau-Lifshitz-Gilbert equation ˙s=!s+s˙s+E(s), where the vielbein Econtains both an antisymmetric part sand at least an additional non-zero diagonal element. Because this vielbein is invertible, we can express as a function of s. To illustrate the procedure, we start with the case of linear equations: A n=fA BCm nsB m (4)056012-3 Nicolis, Thibaudeau, and Tranchida AIP Advances 7, 056012 (2017) Assuming that the matrices are invertible, we obtain the change of variables (we shall study presently what happens when the matrices have zero modes) sA n=f f1gA Bf C1gm nB m (5) The Jacobian is a constant that can be absorbed in the normalization of the partition function,9so we obtain the partition function for the dynamical degrees of freedom, Z= NfY A=1N1Y n=0dsA ne1 2sA0 n0fB B0FABfA A0Cn0 nDnmCm0 msB0 m0 (6) that defines the correlation functions–for the finite-dimensional case the moments–of the dynamical degrees of freedom. The 1–point function vanishes, hsA ni=0, while the 2–point function is given by the expressionD sA nsB mE =f [f1Ff]1gABf [C1DC]1g nm(7) This is the FDR for the present case, that relates the parameters, fA BandCm n, of the spin dynamics, with the parameters, FABandDnm, of the bath. III. WHEN ZERO MODES ARE RELEVANT Let us now consider the case when the matrices fA Band/or Cm nhave zero modes, a case that is relevant for the physical system studied in this paper. The zero modes imply, quite simply, that we cannot replace all of the A nby the sA n, since we cannot invert eq. (4); we can, only, replace the non–zero modes. The matrices fand/or Care not of full rank–but they surely have positive rank, otherwise the stochastic map does not make sense. When we replace the non–zero modes, we shall generate quadratic terms in the sA n–but, since we do not replace all of the A n, on the one hand there will be mixed terms, while there will remain the terms quadratic in the A n, that correspond to the zero modes. When we integrate over the zero modes, the A nthat we could not express directly as linear combinations of the sA n, we shall encounter Gaussian integrals over them that contain terms linear in the zero modes and the sA nalready replaced. The result of these Gaussian integrations will be quadratic contributions to the already present sA n, that enter in the action with the opposite sign to their coefficients. The system will be stable, if these contributions do not completely cancel the existing ones and will lead to a modification of the FDR. Let us see this in action. We shall take Nf= 3 and fA B="A BC!C, with!a fixed vector in flavor space. In the magnetic case it will correspond to the fixed part of the precession frequency. We immediately remark that fA Bhas one zeromode, along the vector !. Since this vector is fixed, without loss of generality, we may take it to lie along the z–axis:!=(0, 0,!3). The stochastic equation, eq. (4), takes the form 1 n=!3Cm ns2 m 2 n=!3Cm ns1 m(8) We may replace these in the partition function for the noise; but we must integrate over 3 nseparately. We remark that they do not involve s3 n, the component of the dynamical degrees of freedom, parallel to the precession vector. IfFAB=AB, i.e. the spherical symmetry is imposed, we immediately deduce that the integration over3 ndecouples from the rest and just gives a contribution to the normalization. The partition function for the dynamical degrees of freedom, s1 nands2 n, is given by the expression Z= 2Y A0=1N1Y n=0dsA ne(!3)2 2sA0 m0[CDC]n0m0sA0 m0. (9) There is a subtle point here: the motion of the A0= 1, 2 flavor components is a rotation, with precession frequency!3, about the z–axis, so the combination, ( s1 m)2+(s2 m)2should appear–and it does. Therefore we deduce the FDR for this case, that corresponds to Larmor precession: D sA nsB mE = !32f C1DCg1 nm(10)056012-4 Nicolis, Thibaudeau, and Tranchida AIP Advances 7, 056012 (2017) If the spherical symmetry is not imposed, in flavor space, e.g. FAB=AB+AB(1AB), we would have had terms linear in 3 n, along with the quadratic terms and additional contributions when we would have integrated over the 3 n. IV. BEYOND THE GAUSSIAN APPROXIMATION Now let us address the issue of non–linear stochastic maps, also relevant for the Landau–Lifshitz– Gilbert equation. Let us replace eq. (4) by A n=fA (1)BC(1)m nsB m+fA (2)BCC(2)ml nsB msC l(11) In this case the Jacobian of the transformation, between the degrees of freedom of the bath and the degrees of freedom that describe the “interesting” dynamics, is not a constant: JAB nk(s)A n sB k=fA (1)BC(1)k n +f fA (2)BCC(2)kl n+fA (2)CBC(2)lk ng sC l(12) This means that, if it is possible to neglect the zero modes and the concomitant fluctuations in the sign of the determinant, which is true in perturbation theory, the partition function for the spin degrees of freedom is given by the expression Z=f dsA ng detJAB nk(s)e1 2A n(s)FABDnmB m(s)(13) where the(s) are defined by eq. (11). The expression in the exponent contains terms that are quadratic and quartic in the spin variables. The fluctuation–dissipation relation can then be deduced from the Schwinger–Dyson equations9, Z1f dsA ng@ @sL k sA1n1


sAInIdetJAB nk(s)e1 2A n(s)FABDnmB m(s) =0 (14) These relations can be used to generalize eqs. (10) and express the fact that the spin degrees of freedom are in equilibrium with the bath. The determinant can be introduced into the exponent using anti-commuting variables, that describe the dynamics of the bath9. It should be stressed that, since the (s) are polynomials in the spin degrees of freedom, once the determinant has been expressed in terms of anti-commuting fields, there is a finite number of parameters that define the dynamics and, thus, enter in the fluctuation–dissipation relation. Indeed, if D nmis not the identity matrix, which means that the dynamics is not ultra–local in time, tunneling between configurations implies that the effects of the determinant and it sign will be, inevitably and, thus, implicitly, be generated by the dynamics, therefore it suffices to sample the correlation functions by the action of the spin degrees of freedom. The subtleties of the dynamics are encoded in the relation between the noise fields and the spins, so it is at that point that the zero modes need to be taken into account. There are not any issues of principle, involved, however, precisely because the system is consistently closed. How to sample the correlation functions will be reported in detail in future work. 1M. G. M ¨unzenberg, Nature Materials 9, 184 (2010). 2A. Mitra and A. J. Millis, Physical Review B 72, 121102 (2005). 3V . L. Safonov and H. N. Bertram, Physical Review B 71, 224402 (2005). 4W. T. Coffey and Y . P. Kalmykov, Journal of Applied Physics 112, 121301 (2012). 5R. Kupferman, G. A. Pavliotis, and A. M. Stuart, Physical Review E 70, 036120 (2004). 6M. Nishino and S. Miyashita, Physical Review B 91, 134411 (2015). 7M. Baiesi, C. Maes, and B. Wynants, Physical Review Letters 103, 010602 (2009). 8C. Aron, G. Biroli, and L. F. Cugliandolo, Journal of Statistical Mechanics: Theory and Experiment 2010 , P11018 (2010). 9J. Zinn-Justin, Quantum Field Theory and Critical Phenomena , 4th ed., International series of monographs on physics No. 113 (Clarendon Press, Oxford, 2011) oCLC: 767915024.

1.4822058.pdf

Thermally activated in-plane magnetization rotation induced by spin torque L. Chotorlishvili, Z. Toklikishvili, A. Sukhov, P. P. Horley, V. K. Dugaev et al. Citation: J. Appl. Phys. 114, 123906 (2013); doi: 10.1063/1.4822058 View online: http://dx.doi.org/10.1063/1.4822058 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v114/i12 Published by the AIP Publishing LLC. Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 27 Sep 2013 to 128.143.23.241. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsThermally activated in-plane magnetization rotation induced by spin torque L. Chotorlishvili,1Z. Toklikishvili,2A. Sukhov,1P . P . Horley,3V. K. Dugaev,1,4,5V. R. Vieira,5 S. Trimper,1and J. Berakdar1 1Institut f €ur Physik, Martin-Luther-Universit €at Halle-Wittenberg, Heinrich-Damerow-Str. 4, 06120 Halle, Germany 2Physics Department of the Tbilisi State University, Chavchavadze Ave. 3, 0128 Tbilisi, Georgia 3Centro de Investigaci /C19on en Materiales Avanzados (CIMAV S.C.), Chihuahua/Monterrey, 31109 Chihuahua, Mexico 4Department of Physics, Rzesz /C19ow University of Technology Al. Powstanc /C19ow Warszawy 6, 35-959 Rzesz /C19ow, Poland 5Department of Physics and CFIF, Instituto Superior T /C19ecnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisbon, Portugal (Received 18 May 2013; accepted 9 September 2013; published online 25 September 2013) We study the role of thermal fluctuations on the spin dynamics of a thin permalloy film with a focus on the behavior of spin torque and find that the thermally assisted spin torque results in new aspects of the magnetization dynamics. In particular, we uncover the formation of a finite, spin torque-induced, in-plane magnetization component. The orientation of the in-plane magnetizationvector depends on the temperature and the spin-torque coupling. We investigate and illustrate that the variation of the temperature leads to a thermally induced rotation of the in-plane magnetization. VC2013 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4822058 ] I. INTRODUCTION We are witnessing a growing body of research on various phenomena related to the transfer of angular momentum by means of an electric current.1The fact that the electron current carries/transfers spin is well-known, the interest to thisphenomenon was fueled however by the experimental demon- strations that the electric current can strongly affect the mag- netization dynamics in nanostructures 6–9with important consequences for technological applications, such as steering magnetic domain walls2and vortices, conception of high- frequency electrical oscillators,3,5and the magnetization reversal in magnetic layers via exerting spin torque.4The latter is achieved by a spin-polarized charge current and has been demonstrated for various magnetic nanostructures10–13,15and magnetic tunnel junctions.16While several microscopic mech- anisms relevant for various nanosystems have been discussed; on the macroscopic level, the effect of the spin-polarizedcurrent can also be described by the well-established macro- scopic Landau-Lifshitz-Gilbert (LLG) equation 17upon inclu- ding the appropriate spin-torque terms. Another important factor, which can influence substan- tially the magnetization switching in nanostructures, is the effect of thermal fluctuations18(here we refer to the very comprehensive recent overview17and references therein for the details of the well-studied finite-temperature spin dynam- ics). This effect can be captured by including fluctuatingLangevin fields into the LLG equation. Following the stand- ard protocol, 18the magnetization trajectory can be identified as the average over the ensemble of noninteracting nanopar-ticles and described by the Fokker-Planck (FP) equation. In this paper, we consider the combined influence of thermal fluctuations and the spin torque terms previouslyderived for the case of the current-induced motion of a mag- netic domain wall in a quasi-one-dimensional ferromagnet with easy-axis and easy-plane anisotropies. 14We show thatsuch a torque term leads to an interesting physical phenom- enon of thermally activated in-plane magnetization rotation. We will show that in case of spin torque exerted by spin polarized current, orientation of the in-plane magnetizationcan be easily switched and controlled by thermal heating or thermal cooling of the system. Discovered effect may have promising applications based on controlling magnetizationdynamics in nanostructures. A key issue in our result is the ratio between the thermal activation energy and the Zeeman energy of the magnetization vector in the external drivingmagnetic field. This means from the experimental viewpoint that our theoretical proposal can be easily implemented by tuning the amplitude of the external driving magnetic field.Thus, the in-plane magnetization vector can be controlled and switched by out-of-plane external magnetic field. To obtain analytical solutions of the FP equation, we de- velop a perturbation approach which substantially differs from the previously discussed methods. 19–25The advantage of our approach is that it allows for obtaining some analyti-cal solutions with high accuracy in arbitrary order of the per- turbation theory. II. MODEL The finite temperature magnetization dynamics in a thin ferromagnetic layer in the presence of a spin torque and an external magnetic field can be described by the following stochastic LLG equation:14,18,26 dM dt¼ceM/C2ðHef fþhðtÞÞ /C0cekM/C2½M/C2ðHef fþhðtÞÞ/C138 þbM/C2sþaM/C2ðM/C2sÞ: (1) Here, Heffis the effective magnetic field and h(t)i st h er a n d o m Langevin magnetic field related to the thermal fluctuations, a and bare the Slonczewski spin torque constants, ceis the 0021-8979/2013/114(12)/123906/9/$30.00 VC2013 AIP Publishing LLC 114, 123906-1JOURNAL OF APPLIED PHYSICS 114, 123906 (2013) Downloaded 27 Sep 2013 to 128.143.23.241. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsgyromagnetic ratio for electrons, and kis a phenomenological (Gilbert) damping constant. For convenience, we introduce dimensionless quantities. Thus, we deal with a normalizedmagnetization vector jMj¼1, a dimensionless (rescaled) damping k!k=jMj, the torque constants a=x 0!eaand b=x0!eb. The dimensionless time t!x0tis defined through the frequency of the Larmor precession in the effec- tive field x0¼cjHef fj. The anisotropy field for the ferromag- netic system can be evaluated for thin film alloys of thepermalloy class (Fe-Ni, Fe-Co-Ni) by using the formula 27,28 bA¼2K1=Ms,w h e r e K1is the anisotropy coefficient and Ms is the saturation magnetization. In particular, for a thin film26,28,29Fe25Co25-Ni50, the saturation magnetization of the film is of the order of Ms/C251025 G, the anisotropy constant K1/C254/C2103erg=cm3, the anisotropy field bA/C257:8O e , and the anisotropy field in units of the frequency is xp ¼cebA/C250:138/C2109Hz ( ce¼1:755/C2107Oe/C01c/C01), while the Zeeman frequency in the reasonable strong externalmagnetic field is x 0¼cejH0j/C2517/C2ð102/C4104ÞMHzðjH0j /C25102/C4104OeÞ:Since x0/C29xp, we conclude that for the Fe-Co-Ni alloy, the dominating factor is the external magneticfieldH ef f¼ð0;0;H0Þ. The components of the Langevin field ha;a¼x;y;z obey the following correlation relations: hhðtÞi ¼ 0; hhaðtÞhbðt0Þi ¼ 2kTdabdðt/C0t0Þ;(2) where the averaging ( h/C1/C1 /C1i) is performed over all possible realizations of the random field hðtÞ. For the derivation of the stochastic Fokker-Planck equation, we follow Ref. 30and use the functional integration method in order to average the dy-namics over all possible realizations of random noise field. As shown in Ref. 30, this method is quite general and straightfor- ward, and for the case of a small coupling between the system and the bath, it recovers previously obtained results. 18 We define the distribution function in the following form: fðN;tÞ¼h pðt;½h/C138Þih;pðt;½h/C138Þ ¼dðN/C0MðtÞÞ:(3) Here, Nis the unit vector on the sphere, and we assume that the random field hðtÞstands for a Gaussian noise with the associated functional F½hðtÞ/C138 ¼1 Zhexp/C01 2gðþ1 /C01dsh2ðsÞ2 643 75: (4) Here, Zh¼ÐDhFis the normalization factor andÐDh denotes the functional integration over all possible realiza- tions of the random field hðtÞandg¼2kT. Note that for convenience we measure the temperature in units of theLarmor frequency x 0¼cjHef fj. Therefore, the dimension- less temperature is defined via the expression T!kBT=x0/C22h. Taking into account the relations30 dhaðsÞ dhbðtÞ¼dabdðs/C0tÞ;ð DhdnF½h/C138 dha1ðt1Þdha2ðt2Þ…dhanðtnÞ¼0; dp dt¼/C0@p @NdM dt; (5)and following the standard procedure,30we deduce from Eq. (1)the following FP equation: @f @t¼/C0@ @N/C26 /C0N/C2HeffþkN/C2N/C2Heffþ þebN/C2sþeaN/C2N/C2s/C0kTN/C2N/C2@ @N/C27 f: (6) Solving for such a time-dependent FP equation is a difficult problem even in the absence of spin torque terms.22–25In the presence of spin torque, the analytical consideration of thenon-stationary FP equation becomes intractable. To proceed further, we consider a particular configuration of the spin tor- que s¼ðs;0;0Þ 14,15and the driving external field Hef f ¼ð0;0;H0Þterms. Here, for convenience, the amplitude of the renormalized magnetic field is set to one. We will look for the perturbation solution of Eq. (6)and consider the case e¼1=x0/C281(x0is the Larmor precession frequency in the external constant magnetic field) as a small parameter of the theory and look for a stationary solution of Eq. (6)in the form f¼Cexp1 TðN/C1HeffÞþewðNÞ…/C20/C21 : (7) Here, wðNÞis a function of the vector N. A zeroth-order solu- tionf0¼Cexp½1 TðN/C1HeffÞ/C138corresponds to the solution in the absence of the spin torque. In the stationary case, inserting Eq. (7)in Eq. (6)and after straightforward calculations, we obtain f¼Cexp/C201 TðN/C1HeffÞþea kTðN/C1sÞ þeb 2kT2N/C1ðs/C2HeffÞþ…Þþ…/C21 :(8) Here, in Eq. (8), we assume a high temperature limit, b¼1=T¼kB=ceH0/C22h/C281, and therefore, we can neglect higher order terms in the inverse temperature. For conven-ience, in the intermediate equations in what follows we set x 0¼ceH0¼1;kB¼1;/C22h¼1. As we show below, the val- ues of the temperature defines the limits of the application ofthe perturbation theory. We also neglect the higher order terms that are proportional to the small parameter e. III. OBSERVABLE QUANTITIES Using the distribution function (8), we can evaluate the mean values of the components of the magnetization vector using the following parameterization: Mx¼sinhcosu;My ¼sinhsinu;Mz¼cosh;0/C20h/C20p;0/C20u/C202p. In the ab- sence of the spin torque, the distribution function takes on the following form: dwðh;uÞ¼fðhÞdX¼Z/C01ðbH0ÞexpðbH0coshÞdX:(9) Here, ZðbH0Þ¼ð expðbH0coshÞdX¼4p bH0sinhðbH0Þ (10) is the partition function and dwðh;uÞdefines the probability that the magnetization vector ~Mis oriented within a solid123906-2 Chotorlishvili et al. J. Appl. Phys. 114, 123906 (2013) Downloaded 27 Sep 2013 to 128.143.23.241. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsangle of the width dX¼sinhdhdu. Taking into account Eqs. (9)and(10), we find Mx¼My¼0 and Mz¼LðH0=TÞ; (11) where LðxÞ¼cothðxÞ/C01 xis the Langevin function. In the case of the high temperature limit H0=T<1, that means for T>ceH0/C22h=kB, we have Mz/C25ceH0/C22h=3kBT. In the case of low temperatures, T<ceH0/C22h=kB, we have Mz¼M ¼1. For the square components of the magnetization, we have M2 x¼M2 y¼LðbH0Þ bH0;M2 z¼1/C02LðbH0Þ bH0: (12) We see that Eq. (12) conserves the magnetization vector M2 xþM2 yþM2 z¼1. For the dispersion, we have ðDMiÞ2¼ðMi/C0MiÞ2;i¼x;y;z; ðDMxÞ2¼ðDMyÞ2¼T H0LðH0=TÞ; ðDMxÞ2¼1/C02T H0LðH0=TÞ/C0L2ðH0=TÞ:(13) By using the explicit form of solutions (11)and partition func- tion(10), we can evaluate the mean energy of the system U¼/C0@ @blnðZðbHÞ Þ¼/C0 ceH0/C22hLceH0/C22h kBT/C18/C19 ; (14) and the heat capacity CV¼@U @T/C18/C19 V¼kB1/C0ð4pÞ2 Z2ceH0/C22h kBT/C16/C172 43 5: (15) If the spin torque terms are taken into account, the results are changed. The distribution function takes the form f¼Cexp½acoshþedsinhcosu/C0egsinhsinu/C138; a¼bH0;d¼as kT;g¼bs 2kT2:(16) The expressions for magnetization components in this case are quite involved and are presented in the Appendix. For the particular case of e2ðd2þg2Þ=2/C281, i.e., for 1 T/C18/C192 <2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a b/C16/C174 þ2x2 0k2 b2s2s /C02a b/C16/C172 ; (17) the expressions for the mean components of the magnetiza- tion vector simplifies to MxðH0;TÞ/C25/C0as kH0x0LðH0=TÞ; MyðH0;TÞ/C25þbs 2kx01 TH0LðH0=TÞ; MzðH0;TÞ/C25LðH0=TÞ/C01 2x2 0a2s2 k2H2 0þb2s3 4k21 H2 0T2 ! /C23LðH0=TÞþH0 TL2ðH0=TÞ/C0H0 T/C18/C19 ;(18)and M2 xðx0;H0;TÞ/C25T H0LðH0=TÞ /C01 2x2 0a2s2 k2H2 0þb2s2 4k21 H2 0T2 ! L2ðH0=TÞ þ1 2x2 03a2s2 k2H2 0þb2s2 4k21 T2H2 0 ! /C21/C03T H0LðH0=TÞ/C18/C19 ; M2 yðx0;H0;TÞ/C25T H0LðH0=TÞ /C01 2x2 0a2s2 k2H2 0þb2s2 4k21 H2 0T2 ! L2ðH0=TÞ þ1 2x2 0a2s2 k2H2 0þ3b2s2 4k21 H2 0T2 ! /C21/C03T H0LðH0=TÞ/C18/C19 ; M2 zðx0;H0;TÞ/C25 1/C02T H0LðH0=TÞ/C18/C19 þ1 x2 0a2s2 k2H2 0þb2s2 4k21 H2 0T2 ! L2ðH0=TÞ /C02 x2 0a2s2 k2H2 0þb2s2 4k21 H2 0T2 ! /C21/C03T H0LðH0=TÞ/C18/C19 : (19) From Eq. (19), it is easy to see that the normalization condi- tion holds, M2¼1:Equation (17) defines the minimum val- ues of the temperature, for which the solutions (18) and(19) are still valid. In particular, taking into account that x0=bs/C291, from Eq. (17), we obtain T>Tcr;Tcr/C25/C22h kBffiffiffiffiffiffiffiffiffiffi x0bs 3kr : (20) Equation (20) shows that the temperature, above which our approach is valid, increases with the amplitudes of external field x0¼ceHor of the torque bs. The meaning of Eq. (18) is straightforward. The torque leads to a formation of trans-versal components Mx;yðx0;H0;TÞ, while the external field tries to align the magnetization along the zaxis. Taking typical values of the parameters for the thin film Fe25-Co 25-Ni50, such as x0¼cejH0j/C2517/C2104MHz ;jH0j ¼104Oe;k¼10/C04for the maximal value of the critical threshold temperature Tcrwe have Tcr<70 K. In the limit of a strong field and low temperatures, T>ceH0/C22h=kB, we obtain Mzðx0;TÞ/C251. From Eq. (18) we also see that Mxðx0;TÞ Myðx0;TÞ/C25/C02a bkBT ceH0/C22h: (21)123906-3 Chotorlishvili et al. J. Appl. Phys. 114, 123906 (2013) Downloaded 27 Sep 2013 to 128.143.23.241. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsThe meaning of Eq. (21) is that we can rotate the magnetization’s transversal component in the plane via (cf. Fig. 1). Using Eq. (A1), we find U¼/C0@ @blnsinðbH0Þ bH0/C18/C19 /C0@ @bln 1þe2 2ðd2þg2ÞLðbH0Þ bH0 ! (22) and for the mean energy, we obtain U¼/C0ceH0/C22hLceH0/C22h kBT/C18/C19 /C0/C22h 2x0a2s2 k2þ3 4b2s2 k2ceH0/C22h kBT/C18/C192 ! LceH0/C22h kBT/C18/C19 /C0kBT 2x2 0a2s2 k2þ1 4b2s2 k2ceH0/C22h kBT/C18/C192 ! /C21/C0ceH0/C22h kBT/C18/C192 sinh2ceH0/C22h kBT/C18/C190 BBB@1 CCCA: (23) We can evaluate now the heat capacity C V¼@U @T¼kB 1/C0ceH0/C22h kBT/C18/C192 sinh2ceH0/C22h kBT/C18/C190 BBB@1 CCCAþ3 4b2s2 k2x0/C22h kBT/C18/C193 LceH0/C22h kBT/C18/C19 þ1 2x2 0b2s2 k2ceH0/C22h kBT/C18/C192 1/C0ceH0/C22h kBT/C18/C192 sinh2ceH0/C22h kBT/C18/C190 BBB@1 CCCA8 >>>>< >>>>: þ1 x2 0a2s2 k2þb2s2 k2ceH0/C22h kBT/C18/C192 ! /C2ceH0/C22h kBT/C18/C192 sinh2ceH0/C22h kBT/C18/C19 LceH0/C22h kBT/C18/C199 >>>>= >>>>;(24) and the change of the heat capacity due to the spin torque dC V¼kB3 4b2s2 k2x0/C22h kBT/C18/C193 LceH0/C22h kBT/C18/C19 þ1 2x2 0b2s2 k2ceH0/C22h kBT/C18/C192 1/C0ceH0/C22h kBT/C18/C192 sinh2ceH0/C22h kBT/C18/C190 BBB@1 CCCA8 >>>>< >>>>: þ1 x2 0a2s2 k2þb2s2 k2ceH0/C22h kBT/C18/C192 !ceH0/C22h kBT/C18/C192 sinh2ceH0/C22h kBT/C18/C19 LceH0/C22h kBT/C18/C199 >>>>= >>>>;: (25) IV. NUMERICAL RESULTS Let us inspect the temperature dependence of the mean values of the magnetization components using the analytical results derived in Sec. III. We note again that the analytical solutions [Eqs. (18) and(19)] contain first and second order terms. First order terms correspond to the solution in the ab- sence of spin torque and are valid for arbitrary values of tem- perature, while second order terms are defined fortemperatures T>T cr, see Eq. (20). Since we measure tem- perature in units of Tcr, the solution obtained using perturba- tion theory is not well-defined in the vicinity of T/C251. Therefore, we expect to see a slight loss of smoothness of the magnetization curves in the vicinity around this area. However, our main finding of thermally activated in-planemagnetization rotation (see Eq. (20)) is well defined for arbi- trary values of the temperature. Fig. 1shows the rotation ofFIG. 1. Rotation of the magnetization in the xyplane with a varying ratio between the spin torque parameters aandb. The angle of the rotation is defined via hðTÞ¼tan/C01MxðTÞ MyðTÞ/C16/C17 . Plotted using Eq. (18). The temperature unit is defined viax0/C22h kB, here x0¼ceH0is the Larmor precession frequency x0¼1;k¼1. The maximal rotation angle of the magnetization Dh/C25p=2 is reached for the temperature DT/C2510x0/C22h kB.123906-4 Chotorlishvili et al. J. Appl. Phys. 114, 123906 (2013) Downloaded 27 Sep 2013 to 128.143.23.241. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsthe in-plane component of the magnetization induced by the change of the temperature and is plotted using Eq. (18). Note that the expression for the MzðH0;TÞin Eq. (18) contains two terms. The first term recovers the result obtained without the spin torque Eq. (11) and is defined for arbitrary temperatures. While the second term in Eq. (18) is the contribution of the perturbation theory and therefore according to Eq. (20) is defined only for temperatures above Tcr. We should take this into account when plotting MzðH0;TÞusing Eq. (20). We see that the rotation amplitude depends on the ratio between the spin torque constants a/b and for a=b>1 has a maximum. The temperature depend- ence of the mean values of the in-plane magnetization com- ponents Mx;Myis shown in Figs. 2–4. We see that themaximal values of Mzdecreases with the increase of the spin torque component a(see Fig. 4). Now we present square components of the magnetiza- tion plotted using Eq. (19), see Figs. 5–7. The general conclusion is that the asymmetry between the spin torque coefficients aandbhas more important con- sequences for the thermal rotation of the magnetization inthexyplane and the mean values of the magnetization com- ponents Mx;y;z; however, it is less evident for the mean values of the square of the components M2 x;y;z. Finally, we show the temperature dependence of the dispersion for the Mzcompo- nent of the magnetization (Fig. 8). We see that for different ratios between the spin torque constants a/b, the values of the dispersion are different. At higher temperatures, all these different values merge together. Additionally, we perform full numerical finite- temperature calculations based on the solution of theFIG. 2. Dependence of the magnetization component MxðTÞon the tempera- ture for different values of the spin torque constant a, plotted using Eq. (18). The temperature unit is defined viax0/C22h kB, here x0¼ceH0is the Larmor pre- cession frequency x0¼1;k¼1. FIG. 3. Dependence of the magnetization component MyðTÞon the tempera- ture for different values of the spin torque constant b. Temperature unit is defined viax0/C22h kB;here x0¼ceH0is the Larmor precession frequency x0¼1;k¼1. FIG. 4. Same as Fig. 3for the magnetization component MzðTÞ. Slight loss of the smoothness of the magnetization curve in the vicinity T/C241 is con- nected to the fact that perturbation solution is not well-defined in the area which is marked out by dashed lines.FIG. 5. Dependence of the magnetization component M2 zðTÞon the tempera- ture for different values of the spin torque constant aandb, plotted using Eq.(18). Area in which perturbation theory is not well defined is marked out by two dashed lines. Temperature unit is given byx0/C22h kB, with x0¼ceH0;x0¼1, and k¼1. FIG. 6. The same as Fig. 5but for M2 xðTÞ. Area in which perturbation theory is not well defined is marked out by two dashed lines. FIG. 7. The same as Fig. 5but for M2 yðTÞ.123906-5 Chotorlishvili et al. J. Appl. Phys. 114, 123906 (2013) Downloaded 27 Sep 2013 to 128.143.23.241. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsstochastic LLG equation by means of the Heun method which converges in quadratic mean to the solution inter- preted in the sense of Stratonovich.31Exact numerical solu- tion of the stochastic LLG equation is important since analytical results are obtained in the framework of perturba- tion theory and therefore are valid for the temperatures abovecritical temperature T>T cronly. In order to observe de- pendence of the magnetization components on the tempera- ture and spin torque constants, we numerically solvestochastic LLG equation (1)and generate random trajecto- ries on the sufficiently large time interval until magnetization components after relaxation process reaches stationary re-gime. In the stationary regime, values of the magnetization components are time independent and depend on the temper- ature and spin torque parameters only. Therefore, after aver-aging results over the ensemble of random trajectories for the magnetization components, we obtain mean values which we can compare to the mean values of the magnetizationcomponents obtained via the solution of stationary Fokker- Plank equation Eq. (8).I nF i g . 9, the rotation of the magnet- ization denoted by the angle hreproduces the analytical results of Eq. (21). As we see, depending on the ratiobetween the spin torque constants a/bmaximal values of the observed rotational angle ðDhÞ max/C25p=2 is in a good quanti- tative and qualitative agreement with the analytical results presented in Fig. 1. Fig. 10shows all three magnetization components for the chosen values of the spin current. Theseresults are in full agreement with the analytical results depicted in Figs. 2–4, which predict a decay of the magnet- ization with increasing temperatures. We have deviationbetween analytical and numerical results only in the area below critical temperature T<T cr/C251 where perturbation theory used in analytical calculations is not defined. Our fullnumerical results supplement for the low temperature case, i.e., for T<1 in dimensionless units, or in real (non-scaled) units T<70½K/C138. The numerics can go beyond the range of validity of the perturbation theory. The numerically accurateFIG. 9. Demonstration of the magnetization rotation based on the numerical solution of the stochastic LLG. The as-obtained trajectories of the magnet- ization for the parameters related to Fe-Ni or Fe-Co-Ni (saturation magnet- ization MS¼1025½G/C138, spin-torque s¼ðs;0;0Þ, external magnetic field H0z¼104[Oe] and rescaled damping constant k¼1). Ensemble- averaged over 100 realizations for each time step after assuring that the mag- netization reached the quasi-equilibrium scaling is done. The definition ofangle hðTÞis given in the caption of Fig. 1. We see that numerical result fits qualitatively and quantitatively with the analytical results obtained in the framework of the perturbation theory.FIG. 8. The dispersion for the zcomponent of the magnetization, without the spin torque DMzðTÞ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðDMzÞ2q ¼1/C02T H0LðH0=TÞ/C0L2ðH0=TÞhi1=2 and in the case of a spin torque DMST zðTÞ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðDMzÞ2q ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M2 z/C0ðMzÞ2q . The mean values for the case in the presence of the torque are defined via Eqs. (18)and(19). The temperature unit is the same as Fig. 1. FIG. 10. Ensemble-averaged (over 100 realizations) magnetization components calculated for the parameters listed in the caption of Fig. 9. Good agreement between numerical and analytical results is evident. We have deviation between analytical and numerical results for the component Myonly in the area below critical temperature T<Tcr/C251, where perturbation theory used in analytical calculations is not defined. Our full numerical results supplement for the low temperature case, i.e., for T<1 in dimensionless units, or in real (non-scaled) units T<70½K/C138.123906-6 Chotorlishvili et al. J. Appl. Phys. 114, 123906 (2013) Downloaded 27 Sep 2013 to 128.143.23.241. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsresults for the magnetization are smooth. Fig. 11additionally presents the zero temperature equilibrium from which for certain value of the ratio a/bthe non-zero temperature calcu- lations start. In particular, Fig. 11defines equilibrium ground state of the system for the zero temperature. This zero tem- perature ground state depends on the torque parameters.Finally, in Fig. 12, we show the effect of the magnetization rotation calculated for each time step for the averaged values of the squared projections of the magnetization, i.e., M2 x;y;z. V. CONCLUSION In this work, we studied the thermally assisted spin- torque and its influence on the magnetization dynamics in athin permalloy film and presented results for Fe 25Co25Ni50. We found that the spin torque term leads to nontrivial dy- namical effects in the finite temperature magnetization dy-namics. Assuming the spin torque terms to be small compared with the Larmor precessional term, we developed a perturbational approach to the Fokker-Planck equation andobtained analytical expression for the distribution function including the spin torque terms. In particular, we proved that the spin torque term leads to the formation of a non-vanishing in-plane magnetization component. The ratio between the mean values of the components Mxand My defines the orientation of the in-plane magnetization vector Mxðx0;TÞ Myðx0;TÞ/C25/C02a bkBT ceH0/C22h. We find that the orientation of the in- plane magnetization depends on the ratio between the spin torque constants a/band between the temperature and the amplitude of the external magnetic field T=H0. Therefore, changing the temperature leads to a thermally induced rota- tion of the in-plane magnetization vector. We name this as“thermally activated in-plane magnetization rotation.” We found that if from the two spin torque terms eb½~M;~s/C138; ea½~M½~M;~s/C138/C138, the last term is the dominant one a>bthe effect of the thermally activated in-plane magnetization rota- tion is enhanced (cf. Fig. 1). ACKNOWLEDGMENTS Financial support by the Deutsche Forschungsgemeinschaft (DFG) through SFB 762 and SU 690/1-1 and Contract BE2161/5-1 is gratefully acknowledged. This work was supported by the National Science Center in Poland as a research project FIG. 11. Illustration of the relaxation of the magnetization from the initially chosen arbitrary states to the zero temperature equilibrium state. I n case of zero temperature, relaxation of the magnetization vector is connected to the phenomenological damping constant k. As we see due to the spin torque terms transver- sal components of the magnetization vector are different from the zero in the equilibrium. Initial state is chosen as Mzðt¼0Þ¼/C0 1. (a) a=b¼0:1, (b) a=b¼0:2, (c) a=b¼1, and (d) a=b¼10; other parameters are as those listed in the caption of Fig. 9. FIG. 12. Rotation of the magnetization as a function of temperature (T2½0:7/C138) calculated from trajectories averaged for each temperature at quasi-equilibrium after relaxation time srel/C25sprecess =k. Number of averag- ing is 100, a=b¼1. Initial magnetization state is fMx;My;Mzgðt¼0Þ ¼f0;1;0g.123906-7 Chotorlishvili et al. J. Appl. Phys. 114, 123906 (2013) Downloaded 27 Sep 2013 to 128.143.23.241. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsin years 2011–2014, and CONACYT of Mexico (Basic Science Project No. 129269). APPENDIX: EXPLICIT EXPRESSIONS FOR THE MAGNETIZATION COMPONENTS We use the partition functionZðaÞ¼4psinha a1þe2ðd2þg2Þ 2LðaÞ a/C18/C19 : (A1) Here, we give some expressions for the components of the magnetization Mx¼edLðbH0Þ bH01 1þe2ðd2þg2Þ 2LðbH0Þ bH0 ! /C25edLðbH0Þ bH01/C0e2ðd2þg2Þ 2LðbH0Þ bH0 ! ; My¼/C0egLðbH0Þ bH01 1þe2ðd2þg2Þ 2LðbH0Þ bH0 ! /C25/C0egLðbH0Þ bH01/C0e2ðd2þg2Þ 2LðbH0Þ bH0 ! ; Mz¼LðbH0Þ 1þe2ðd2þg2Þ 2LðbH0Þ bH0 ! þe2ðd2þg2Þ 21 bH01/C03LðbH0Þ bH0/C18/C19 1þe2ðd2þg2Þ 2LðbH0Þ bH0 ! /C25LðbH0Þ1/C0e2ðd2þg2Þ 2LðbH0Þ bH0 ! þe2ðd2þg2Þ 2bH01/C03LðbH0Þ bH0/C18/C19 /C21/C0e2ðd2þg2Þ 2LðbH0Þ bH0 ! : (A2) For the modulus squares of the magnetization components, we find M2 x¼LðbH0Þ bH0 1þe2ðd2þg2Þ 2LðbHÞ bH0/C18/C19 þe2ð3d2þg2Þ 2ðbHÞ21/C03LðbH0Þ bH0/C18/C19 1þe2ðd2þg2Þ 2LðbH0Þ bH0 ! M2 y¼LðbH0Þ bH0 1þe2ðd2þg2Þ 2LðbH0Þ bH0/C18/C19 þe2ðd2þ3g2Þ 2ðbH0Þ21/C03LðbH0Þ bH0/C18/C19 1þg2ðd2þg2Þ 2LðbH0Þ bH0 ! ; M2 z¼1/C02LðbHÞ bH0 1þe2ðd2þg2Þ 2LðbH0Þ bH0/C18/C19 þe2ðd2þg2Þ 24 ðbH0Þ23LðbH0Þ bH0/C01/C18/C19 þLðbH0Þ bH0 ! 1þe2ðd2þg2Þ 2LðbH0Þ bH0/C18/C19 ; (A3) where a¼bH0¼ceH0/C22h=kBT.1J. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996); L. Berger, Phys. Rev. B 54, 9353 (1996); M. Stiles and J. Miltat, “Spin-transfer torque and dynamics,” in Spin Dynamics in Confined Magnetic Structures III (Springer, Berlin/Heidelberg, 2006), pp. 225–308; D. Ralph and M. Stiles, J. Magn. Magn. Mater. 320, 1190 (2008); A. Brataas, A. D. Kent, and H. Ohno, Nature Mater. 11, 372 (2012). 2S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320, 190 (2008). 3S. I. Kiselev et al.,Nature 425, 380 (2003). 4E. B. Myers, D. C. Ralph, J. A. Katine, R. N. Louie, and R. A. Buhrman, Science 285, 867 (1999). 5M. Tsoi et al.,Nature 406, 46 (2000). 6A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu, and T. 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1.3689011.pdf

Multi-frequency magnonic logic circuits for parallel data processing Alexander Khitun Citation: Journal of Applied Physics 111, 054307 (2012); doi: 10.1063/1.3689011 View online: http://dx.doi.org/10.1063/1.3689011 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/111/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Magnonic beam splitter: The building block of parallel magnonic circuitry Appl. Phys. Lett. 106, 192406 (2015); 10.1063/1.4921206 Spin wave based parallel logic operations for binary data coded with domain walls J. Appl. Phys. 115, 17D505 (2014); 10.1063/1.4863936 Non-volatile magnonic logic circuits engineering J. Appl. Phys. 110, 034306 (2011); 10.1063/1.3609062 Magnetoelectric spin wave amplifier for spin wave logic circuits J. Appl. Phys. 106, 123909 (2009); 10.1063/1.3267152 Parallel processing and analysis of thermographic data AIP Conf. Proc. 615, 558 (2002); 10.1063/1.1472847 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.117.125.76 On: Tue, 15 Sep 2015 13:12:09Multi-frequency magnonic logic circuits for parallel data processing Alexander Khitun Electrical Engineering Department, University of California Riverside, California, 92521, USA (Received 8 November 2011; accepted 24 January 2012; published online 7 March 2012) We describe and analyze magnonic logic circuits enabling parallel data processing on multiple frequencies. The circuits combine bi-stable (digital) input/output elements and an analog core. Thedata transmission and processing within the analog part is accomplished by the spin waves, where logic 0 and 1 are encoded into the phase of the propagating wave. The latter makes it possible to utilize a number of bit carrying waves on different frequencies for parallel data processing. Theoperation of the magnonic logic circuits is illustrated by numerical modeling. We also present the estimates on the potential functional throughput enhancement and compare it with scaled CMOS. The described multi-frequency approach offers a fundamental advantage over the transistor-basedcircuitry and may provide an extra dimension for the functional throughput enhancement. The shortcoming and potentials issues are also discussed. VC2012 American Institute of Physics . [http://dx.doi.org/10.1063/1.3689011 ] I. INTRODUCTION Modern logic circuits consist of a large number of tran- sistors fabricated on a surface of a silicon wafer and intercon- nected by metallic wires. The transistors are arranged toperform Boolean operations (e.g. NOT, AND logic gates). Within this approach, the computational power is measured in the number of operations per time per area. In the past sixtyyears, a straightforward approach to functional throughput enhancement was associated with the increase of the number of transistors, which is well known as the Moore’s law. 1 Decades of transistor-based circuitry perfection resulted in the Complementary Metal–Oxide–Semiconductor (CMOS) technology, which is the basis for the current semiconductorindustry. Unfortunately, CMOS technology is close to the fundamental limits mainly due to the power dissipation prob- lems. 2The latter stimulates a big deal of interest to the post- CMOS technologies able to overcome the current constrains. The most of the “beyond CMOS” proposals are focused on the development of a new switch – a more efficient tran-sistor. 3There is still some room for the semiconductor transistors improvement by implementing novel materials (e.g. nano-tubes,4graphene,5tunneling-based transistors,6 etc.). However, it is difficult to expect that the introduction of a new material may extend the Moore’s law for multiple generations as it used to work for CMOS. At this moment, itis important to identify possible routes to alternative (possi- bly transistor-less) logic devices which may lead to a more powerful logic circuitry and offer a pathway for long-termdevelopment. There are several fundamental constrains in- herent to transistor-based logic devices: (i) logic variable is a scalar value (voltage), (ii) metallic interconnects do not per-form any functional work on data processing, (iii) the transistor-based approach is volatile, requiring a permanent power supply even though no computation is performed, and(iv) one transistor can process only one bit at a time . Addressing these fundamental issues is the key leading to a more efficient logic circuitry.Magnonic logic circuits exploiting magnetization as a state variable and spin waves for information transmission and processing is one of the possible solutions. The basicconcept of the spin wave logic circuits have been described in the preceding works. 7–10The utilization of waves for data transmission makes it possible to code logic 0 and 1 in thephase of the propagating wave, make use of waveguides as passive logic elements for phase modulation and exploit wave interference for achieving logic functionality. In thiswork, we describe and analyze the possibility of multi- frequency operation, where each frequency can serve as an independent information channel for information transmis-sion and processing. The rest of the paper is organized as follows. In Sec. II, we describe the basic elements and the principle of operation of a single-frequency magnonic logiccircuit. Next, we extend consideration to the multi-frequency logic circuits. The operation of the multi-frequency circuits is illustrated by numerical modeling in Sec. III. The advan- tages and the challenges of the multi-frequencies approach are discussed in the Secs. IVandV, respectively. II. PRINCIPLE OF OPERATION AND BASIC ELEMENTS We start with the description of the single-frequency operating magnonic logic circuit schematically shown inFig. 1. The circuit comprises the following elements: (i) magneto-electric cells, (ii) magnetic waveguides – spin wave buses, and (iii) a phase shifter. The magneto-electric cell(ME cell) is a multi-functional element aimed to serve as a memory and data processing unit. We assume the ME cells to have the two key properties: (i) magnetic bi-stability (twostable states of magnetization at zero applied voltage), and (ii) electric control of magnetization (voltage applied to the ME cell affects its magnetic state). The ME cell can be real-ized by utilizing multiferroics – materials possessing mag- netic and electric polarization (e.g. synthetic multiferroics combining piezoelectric and magnetoelastic materials 11). 0021-8979/2012/111(5)/054307/9/$30.00 VC2012 American Institute of Physics 111, 054307-1JOURNAL OF APPLIED PHYSICS 111, 054307 (2012) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.117.125.76 On: Tue, 15 Sep 2015 13:12:09The electric and magnetic polarizations of the ME cell are connected via the magneto-electric coupling: the change ofthe electric polarization affects the magnetic polarization, and vice versa, the change of magnetization affects the elec- tric polarization. The latter makes it possible to control themagnetic state of the ME cell by the applying an electric field. 12For simplicity, we assume that ME cell is designed in such a way as to have two stable states of magnetization atzero applied voltage. And the applied voltage affects the direction of the anisotropy field (easy-axis rotation 12). Each ME cell is placed in the direct contact with the spin wave bus (spins of the ME cells are coupled to the spins of the spin wave bus via exchange interaction). The buses are simply the strips of ferromagnetic material (e.g. NiFe)aimed to transmit the spin wave signals from the input to the output ME cells. Spin wave propagation (phase velocity, attenuation) depends on the waveguide material and shape.The phase of the propagating spin wave is the most impor- tant parameter in our consideration. For simplicity, we assume that the spin wave propagating through the spinwave buses preserve the initial phase while the change of the phase occurs only via passing the phase shifter. The phase shifter is a passive element providing the certain phasechange to the propagating spin wave. Spin wave dispersion depends on many parameters including the material proper- ties and the shape of the spin wave bus, and the presence ofthe external magnetic field. For example, an additional phase shift to the propagating spin wave may be provided bymodifying the width/thickness of the waveguide. In this case, the phase shifter may be simply a part of the spin wave bus. It is also possible to achieve a phase shift by introducingan additional layer of magnetic material attached to the spin wave bus providing an additional bias field. In general, the phase shifters may be constructed to provide any desirablephase shift from 0 to 2 p. In this work, we consider circuits with a p-phase shifter, which is sufficient for logic gate construction. The principle of operation is the following. Initial infor- mation is received in the form of voltage pulses. Input 0 and 1 are encoded in the polarity of the voltage applied to theinput ME cells (e.g. þ10 mV correspond to logic state 0, and /C010 mV correspond to logic 1). The applied voltage disturbs the magnetization of the ME cell and results in the spin wavesignal (spin wave packet) excitation in the spin wave buses. The polarity of the applied voltage defines the direction of the magnetization change in the ME cell, and, in turn, theinitial phase of the excited spin wave. The waves excited by positive and negative voltages have the same amplitude but a p-phase difference. Thus, the input information is translated into the phase of the excited wave (e.g. initial phase 0 corre- sponds to logic state 0, and initial phase pcorresponds to logic 1). Then, the waves propagate through the magneticwaveguides and interfere at the point of waveguide junction. For any junction with an odd number of interfering waves, there is a transmitted wave with a non-zero amplitude. Inthis case, the phase of the transmitted wave always corre- sponds to the majority of the phases of the interfering waves (for example, the transmitted wave will have phase 0, if thereare two or three waves with initial phase 0; the wave will have a p-phase otherwise). The transmitted wave passes the phase shifter and accumulates an additional p-phase shift (e.g. phase 0 !p, and phase p!0). Finally, the spin wave signal reaches the output ME cell. The switching of the ME cell is achieved as a combined effect of two: interaction withthe incoming spin wave and magneto-electric coupling. The voltage is applied to the output ME cell prior to computation. The direction of the easy-axis of the output ME cell underthe applied voltage is assumed to have 90 /C14with respect to the steady state position at zero applied voltage. At the moment of spin wave arrival, the voltage applied to theoutput cell is turned off. It is assumed that the applied volt- age is turning off much faster than the time required for mag- netization to relax. In this scenario, the output ME cell isplaced in the metastable state (magnetization is along the hard axis perpendicular to the two stable states). The incom- ing spin wave defines the direction of the magnetizationrelaxation by providing a “seed magnetization change” to- ward one or another steady state. For instance, the ME cell relaxes to one of the steady states if the phase of the incom-ing spin wave is in the range from /C0p/2 to p/2, and it relaxes to the other state if the phase of the incoming spin wave is in the range from p/2 to 3 p/2. The amplitude of the incoming spin wave is of minor importance for the electric-field assisted switching. The only requirement for the spin wave amplitude is to be above the thermal noise level to ensure thereliable switching. The correlation between the magnetiza- tion of the input and the output ME cells is defined entirely FIG. 1. (Color online) Schematic view of the single-frequency operating magnonic logic circuit. There are three inputs (A,B,C) and the output D. The inputs and the output are the ME cells connected via the ferromagnetic waveguides – spin wave buses. The input cells generate spin waves of the same amplitude with initial phase 0 or p, corresponding to logic 0 and 1, respectively. The phase is controlled by the polarity of the input voltagepulse (e.g. 610 mv). The waves propagate through the waveguides and interfere at the point of junction. The phase of the wave passed the junction corresponds to the majority of the interfering waves. The phase of the trans- mitted wave is inverted by passing the phase shifter. The phase of the trans- mitted wave defines the magnetization of the output ME cell D. The Table illustrates the data processing in the phase space. The circuit can operate as NAND or NOR gate for inputs A and B depending the third input C (NOR ifC¼1, NAND if C ¼0).054307-2 Alexander Khitun J. Appl. Phys. 111, 054307 (2012) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.117.125.76 On: Tue, 15 Sep 2015 13:12:09by the phases of the spin waves transmitted form the input to the output. For example, the magnetization of the output ME cell shown in Fig. 1becomes oriented in the opposite direc- tion to the magnetization of the majority of the input ME cells. The result of computation is preserved in the magnetiza- tion state of the output ME cell. The retention time depends on the materials properties and the shape of the ME cell. There are two possible mechanisms for read-out. The outputME cell may be integrated with a magnetic tunneling junc- tion (MTJ) allowing for read-out via the magnetoresistance measurements. The read-out can be also accomplished bydetecting the voltage across the ME cell. The internal cou- pling between the magnetic and electric polarizations in mul- tiferroic materials makes it possible to detect the change ofelectric field (voltage) caused by the change of the magnetic state. 13The later offers a convenient way of magnetic-to- electric domain conversion (e.g. spin wave with phase 0 toproduce þ10 mV output corresponding to logic state 0, and spin wave with phase pto produce-10 mV corresponding to logic 1). In this work, we focus our consideration on the possibil- ity of building multi-frequency magnonic logic devices. One of the specific features of magnonic circuits is the combina-tion of the digital (bi-stable) inputs/outputs and the analog core. Information processing within the analog core is associated with the manipulation of the phases of the propa-gating waves. The Truth Table inserted in Fig. 1shows the input/output phase correlation. The waveguide junction works as a Majority logic gate. The amplitude of the trans-mitted wave depends on the number of the in-phase waves, while the phase of the transmitted wave always corresponds to the majority of the phase inputs. The p-phase shifter works as an Inverter in the phase space. As a result of this combination, the three-input one-output gate in Fig. 1can operate as a NAND or a NOR gate for inputs A and Bdepending on the third input C (NOR if C ¼1, NAND if C¼0). Such a gate can be a universal building block for any Boolean logic gate construction. Most importantly is that theanalog core can operate on a number of frequencies at the same time by exploiting wave superposition. The general view of the multi-frequency magnonic circuit is shown in Fig. 2. The structure and the principle of operation are similar to the above described example except there are multiple ME cells on each of the input and outputnodes. These cells are aimed to operate (excite and detect) spin waves on different frequencies (e.g. f 1,f2,...fn). The operational frequency of the ME cell can be tuned by the cellsize/shape/composition. In order to avoid the cross talk among the cells operating on different frequencies, the cells are connected with the spin wave buses via the magnoniccrystals 14serving as frequency filters. Each of these crystals allows spin wave transport within a certain frequency range enabling the ME cell isolation. Within the spin wave buses,spin waves of different frequencies superpose, propagate, and receive a p-phase shift independently of each other. Logic 0 and 1 are encoded into the phases of the propagatingspin waves on each frequency. The output ME cells are con- nected to the spin wave buses via the magnonic crystals aswell in order to receive spin wave signal on the specific frequency. The Truth Table shown in Fig. 1can be applied for the each of the operating frequencies. Thus, the consid- ered circuit can perform NAND or NOR operations on the number of bits at the same time. III. NUMERICAL SIMULATIONS In this section, we present the results of numerical simu- lations aimed to illustrate the operation of magnonic logic circuits. The operation procedure includes several majorsteps: spin wave generation by the input ME cell, spin wave propagation and interference, obtaining a p-phase shift by passing the phase-shifter, and the output ME cell switchingdepending the phase of the incoming spin wave. The process of the spin wave excitation by the ME cell is modeled by using the Landau-Lifshitz equation: d~m dt¼/C0c 1þa2~m/C2~Heffþa~m/C2~Heff/C2/C3 ; (1) where ~m¼~M=Msis the unit magnetization vector, Msis the saturation magnetization, cis the gyro-magnetic ratio, and a is the phenomenological Gilbert damping coefficient. The effective magnetic field ~Heffis the sum of the following: ~Heff¼~Hdþ~Hexþ~Haþ~Hb; (2) where ~Hdis the magnetostatic field, ~Hexis the exchange field, ~Hais the anisotropy field ( ~Ha¼ð2K=MSÞð~m/C1~cÞc*,Kis the uniaxial anisotropy constant, and ~cis the unit vector along the uniaxial direction), ~Hbis the external bias magnetic field. It is assumed that the application of the bias voltage V to the ME cell results in the easy-axis rotation ( ~cvector rota- tion) is as follows: cx¼0, cy¼cosh,cz¼c0sinh, FIG. 2. (Color online) Schematic view of the multi-frequency magnonic cir- cuit. There are multiple ME cells on each of the input and output node aimed to excite and detect spin waves on the specific frequency (e.g. f1,f2,...fn). The cells are connected to the spin wave buses via the magnonic crystals serving as the frequency filters. Within the spin wave buses, spin waves of different frequencies superpose, propagate, and receive a p-phase shift inde- pendently of each other. Logic 0 and 1 are encoded into the phases of thepropagating spin waves on each frequency. The output ME cells recognize the result of computation (the phase of the transmitted wave) on one of the operating frequency.054307-3 Alexander Khitun J. Appl. Phys. 111, 054307 (2012) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.117.125.76 On: Tue, 15 Sep 2015 13:12:09h¼p 2V VG/C18/C19 ; (3) where VGis the voltage resulting in a 90 degree easy axis rotation from the Y axis toward the Z axis. The material pa-rameters taken in simulations are typical for permalloy films c¼2.0/C210 7rad/s/Oe, 4 pMs¼10 kG, 2 K/ Ms¼4 Oe.15,16 Figure 3illustrates the process of spin wave excitation by the ME cell. In our simulations, we considered a strip of permalloy uniformly magnetized along the Y axis. The ME cell is considered as a part of the strip, where the local ani-sotropy field depends on the applied voltage according to Eq. (3). A dome-shaped voltage pulses of 300 ps is applied to the ME cell. The local change of the anisotropy field results inthe spin wave excitation (spin wave is propagating along the X axis). The amplitude of the spin wave is calculated by Eqs. (1)–(2). The graph in Fig. 3shows the magnetization of the ME cell (black curve) and the amplitude of the excited spin wave (red curve) as a function of time. The plot illus- trates two possible magnetization responses to the positiveand negative input voltages. The polarity of the applied volt- age defines the direction of the ME cell’s magnetization change (e.g. the initial phase 0 or pof the excited spin wave). The propagation, interference and phase accumulation are simulated by the analytical model described in Ref. 17. This simple model is found to be in strong agreement with experimental data on spin wave propagation in permalloy. 17We assume that each of the input ME cell generates a spinwave packet propagating along the X axis. The wave packet consists of a Gaussian distribution of wave vectors that is 2 /d in width and centered about k 0. The magnetization compo- nents in the Cartesian coordinates are given as follows: Mx¼Cexpð/C0t=sÞ d4þb2t2exp/C0d2ðx/C0/C23tÞ2 4ðd4þb2t2Þ"# sinðk0x/C0xtþ/Þ; My¼Cexpð/C0t=sÞ d4þb2t2exp/C0d2ðx/C0/C23tÞ2 4ðd4þb2t2Þ"# cosðk0x/C0xtþ/Þ; Mz¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M2 s/C0M2 x/C0M2 yq ; ð4Þ where Cis a constant proportional to the amplitude, sis the decay time, /is the initial phase, /C23andbare the coefficients of the first and second order terms, respectively, in the Taylor expansion of the nonlinear dispersion xðkÞ. In this case, the propagation of the spin wave packet can bedescribed by just one magnetization component ( M xorMy). Hereafter, we present the results of numerical modeling for just one magnetization component perpendicular to the spinwave propagation direction - M y. In our simulations, we take s¼1.0 ns, d¼1.0lm,/C23=1 04m/s, and neglect the second order term b= 0 for simplicity. First, we simulate the operation of the single-frequency circuit ( f¼0.4 GHz, k 0¼0.25 lm/C01). In Fig. 4, there are several plots showing spin wave signals excited by the inputME cells (on the left), signal after the point of waveguide junction (in the center), and the spin wave signal after the phase shifter (on the right). The amplitudes of the spin wavesare normalized to the saturation magnetization M s.Three input ME cells generate the three spin wave packets of the same shape and distribution except the initial phase /C0/.A s an example, we took the initial phase /=0for inputs A and C, and the opposite phase /¼pfor input B. We intention- ally restrict our consideration by the small amplitudes My /C28Msto consider only the linear regime. Thus, at the point of waveguide junction, the total magnetization can be found as a sum of the three superposing packets form inputs A, B,and C. The amplitude of the transmitted signal may vary depending the number of in-phase waves (two or three) while the phase of the propagating wave ( /=0) corresponds to the majority of the phases (e.g. MAJ(0, p,0 )¼0). The wave propagating through the phase shifter signal accumu- lates an additional p-phase shift (the length of the inverter is taken to be 100 nm). The process of the output ME cell switching is illus- trated in Fig. 5. At the moment of switching, the output ME cell is polarized along the Z axis (the V Gvoltage is applied to the output ME cell). As the spin wave packet reaches the output ME cell, the bias voltage is turned off, and the mag-netization starts to relax toward the stable state along or op- posite to the Y axis. The phase of the incoming wave defines the way of relaxation and the final magnetization state of theoutput ME cell (e.g. /=0will lead to the positive M y, and /=pwill lead to the negative My). There are two curves in Fig. 5showing two possible relaxation trajectories FIG. 3. (Color online) (A) Schematics on the spin wave excitation by the ME cell. The ME cell is a multiferroic structure enabling magnetization con- trol by the electric field. The applied voltage rotates the direction of the ani-sotropy field from the Y axis toward the Z axis and results in the spin wave excitation. (B) Results of numerical modeling showing the magnetization change of the ME cell (black curve) and the excited spin wave in the spin wave bus (red curve). The plot shows the direct correspondence between the polarity of the applied voltage and the direction of magnetization change (initial phase) of the excited spin wave.054307-4 Alexander Khitun J. Appl. Phys. 111, 054307 (2012) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.117.125.76 On: Tue, 15 Sep 2015 13:12:09depending on the phase of the incoming wave. As we stated before, the final state of the ME cell is determined by the phase not the amplitude of the spin wave signal. The results of numerical modeling shown in Figs. 3–5are aimed to illustrate the main idea of spin wave circuit – logic functionality by manipulating the phases of the propagating waves. There are no non-linear devices (switches) in the ana- log core but only passive elements: waveguides, junction and phase shifter. The latter let us extend this approach to multiplewaves simultaneously propagating and interfering in one struc- ture. Next, we simulate the operation of the multi-frequency circuit shown in Fig. 2. For simplicity, we consider three waves for each input A, B and C. We assume three ME cells on each input to generate spin wave packets centered about the three different wave vectors k 1,k2,a n d k3, corresponding to the three operating frequencies f1,f2,a n d f3, respectively. As an example, we take k1¼0.02 lm/C01,k2¼0.25 lm/C01,a n d k3¼2.0lm/C01corresponding to 0.032 GHz, 0.4 GHz, and 3.2 GHz, respectively. The phase velocity /C23, damping con- stants,and the packet width dare taken the same as in the pre- vious example of the single-frequency circuit. In Fig. 6,t h e r e are plots showing spin wave signals excited by the ME cells (on the left), signal after the point of waveguide junction FIG. 4. Results of numerical simulations illustrating the operation of the single-frequency logic circuit shown in Fig. 1. The plots on the right show the My component of the spin wave signal generated by the input ME cell. The plots in the center and on the left show the signal after the point of junction and the p-phase shifter, respectively. The spin wave signals are approximated by the wave packets with Gaussian distribution (width d¼1lm and centered about k0¼0.25 lm/C01. FIG. 5. (Color online) Output ME cell switching as a function of the phase of the incoming spin wave The ME cell is polarized along the Z axis (the V G voltage applied to the output ME cell) prior to the switching. The bias volt- age is turned off at the moment of the spin wave arrival. The magnetization starts to relax toward the stable state along or opposite to the Y axis. The relaxation trajectory is defined by the phase of the incoming wave. The blue curve and the red curves show the two possible trajectories corresponding to 0 and pphase of the incoming wave, respectively.054307-5 Alexander Khitun J. Appl. Phys. 111, 054307 (2012) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.117.125.76 On: Tue, 15 Sep 2015 13:12:09(in the center), and the signal after the phase shifter (on the right). The spin wave signal on each input is the sum of three packets centered about three different wave vectors. Each of the plots on the left side of Fig. 6shows three curves corre- sponding to the three packets, and the insert on the plot depicts the sum of three. As in the previous example, the information is encoded in the initial phase for each packet (23possible phase combinations for each input, 83possible combinations for 3 inputs). The resultant magnetization at the point of wave- guides junction is calculated as a sum of the nine input packets(shown in the inserts in Fig. 6). The wave passing through the junction comprises three wave packets with 2 3possible phase combinations. It is assumed that each of the packets accumu-lates a p-phase shift independently of the others. There are three output ME cells receiving signals on one of the informa- tion carrying frequencies ( f 1,f2,f3)to recognize and store one of the 23final states. Theoretically, it is possible to build a classical wave-based core, which can be in the superposition of 2Npossible states, where Nis the number of the input ME cells. The Nbits can divided between nffrequencies depending on the number of inputs per logic gate a,N¼nf/C2a. (e.g. threeinput bits per frequency for the universal gate shown in Fig.1). IV. DISCUSSION The principle of operation of the multi-frequency magnonic circuit is fundamentally different from the conven-tional transistor-based circuits and combines a number of novel ideas on information transmission and processing. The major critical concerns can be divided into the following: (i)practical feasibility of the basic components (e.g. ME cells), (ii) the ability to excite and recognize multiple spin waves on different frequencies, (iii) possible power dissipationissues due to the number of operating channels, and (iv) fault tolerance of the phase-based circuits. Some of the circuit components have been experimen- tally realized (e.g. spin wave interferometer 18) and some components (e.g. ME cells) are currently under study. Spin wave transport in nanometer scale magnetic waveguides hasbeen intensively studied during the past decade. 16,17,19,20The coherence length of spin waves in ferromagnetic materials FIG. 6. (Color online) Results of numerical modeling illustrating the operation of the triple-frequencies logic circuit shown in Fig. 2. The operating wave vec- tors are k1¼0.02 lm/C01,k2¼0.25 lm/C01, and k3¼2.0lm/C01. The plots on the right show the Mycomponent of the spin wave signal generated by the input ME cell. The plots in the center and on the left show the signal after the point of junction and the p-phase shifter, respectively. The inserts show the total magnet- ization as a sum of all superposed packets.054307-6 Alexander Khitun J. Appl. Phys. 111, 054307 (2012) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.117.125.76 On: Tue, 15 Sep 2015 13:12:09(e.g. NiFe) exceeds tens of microns at room temperature,16,17 which allows us to utilize spin wave interference at the micrometer scale. The typical spin wave group velocity is10 4m/s and the relaxation time is about 0.8 ns for NiFe at room temperature. This short attenuation time and relatively slow propagation are not critical for the sub-micrometerscale logic devices. ME cell is the key element in the described approach. The operation of the ME cell is based on the effect ofmagneto-electric coupling enabling magnetization control by applying an electric field and vice versa. For a long time, the most of interest on magneto-electric coupling has been asso-ciated with the single-phase multiferroics – a unique class of materials inherently possessing electric and magnetic polar- izations (e.g. BiFeO 3).21However, the limited number of room temperature single-phase multiferroics as well as the relatively low magnetic polarization makes difficult their practical utilization in magnonic circuits. On the other hand,the combination of piezoelectric and magnetostrictive mate- rials (so called synthetic multiferroics) may produce much bigger effects. An electric field applied across the piezoelec-tric produces stress, which, in turn, affects the magnetic polarization of the magnetostrictive material. As a result of the electro-mechanical-magnetic coupling the magnetizationof the magnetostrictive materials can be controlled by the electric field. A number of piezo-magnetostrictive pairs have been studied during the past decade and magneto-electriccoupling coefficients have been tabulated. 11In the specific case of magnonic circuitry, it is important to produce signifi- cant change of magnetization in order to excite spin waves.Recent experimental data on PZT/Ni pair have shown the 90 /C14magnetization rotation in Ni as a function of the electric field applied across the PZT layer.22The utilization of the synthetic multiferroics has a big advantage for the ME cell engineering as the frequency response of the cell can be tuned by the thickness of the piezoelectric/magnetostrictivematerials, which makes it possible to realize a frequency- selective input/output elements based on the same materials. Reliable operation of the output ME cells is another important challenge. The output cell relaxation is triggered by the seed magnetization change provided by the all incom- ing spin waves. As one can see from the results of numericalmodeling in Fig. 6, the widths of the output signal increase, which makes difficult the precise filtering. The error immu- nity of the read-out is directly related to the quality of themagnonic crystals 14serving as frequency filters. Magnonic crystals can be fabricated as a composition of two materials with different magnetic properties or as a single materialwaveguide with periodically varying dimensions. Frequency band gaps have been experimentally observed in a grating- like structure comprising shallow grooves etched intothe surface of an yttrium-iron-garnet film, 23in a one- dimensional arrays of permalloy nanostripes separated by the air gaps,24and in a synthetic nanostructure composed of two different magnetic materials.25The obtained data show the feasibility of using magonic crystals as the frequency filters with frequency band gaps of several GHz, which canbe tuned by the bias magnetic field. 25According to the theo- retical study,26the efficiency of the magnonic crystals can befurther improved by using the frequency-depending damp- ing. It would be of benefit to exploit this physical phenom- enon for spin wave signal filtering and cross-talk prevention. The operation of the multi-frequency magnonic circuit requires a frequency-independent phase shifter providing the same phase shift to the waves of different frequencies. Intheory, such a device can be realized by using the domain wall of a special shape as described in Ref. 27. However, practical realization may not require any sophisticated struc-ture as there is no need in the precise control for the phase shift of the frequency-independent shifter. The mechanism of the voltage-assisted switching allows for a quite big phasemargins (e.g. þ//C0p/2 phase variation of the incoming spin wave would lead to the same final state of the ME cell). The phase shift accumulated by the spin waves on differentfrequencies may vary within þ//C0p/4 range without affecting the logic functionality. These big phase margins can be translated in the permissible frequency range, which dependson the spin wave dispersion in the given material/structure. For example, the phase shift difference between spin waves of 4 GHz and 5 GHz frequency would be less than pi/8assuming linear dispersion. The energy per operation in the magnonic logic circuits depends on the number of ME cells and the energy requiredfor magnetization rotation in each cell. It is important to note that the electric field required for magnetization rotation in Ni/PZT synthetic multiferroic is about 1.2 MV/m. 22The latter promises a very low, order of aJ, energy per switch achievable in nanometer scale ME cells (e.g. 24 aJ for 100 nm /C2100 nm ME cell with 0.8 lm PZT). Thus, the max- imum power dissipation density per 1 lm2area circuit oper- ating at 1 GHz frequency can be estimated as 7.2 W/cm2 (three input cells per one frequency). An addition of an extra operating frequency would linearly increase the power dissi- pation in the circuit. The upper limit for the power dissipa- tion in magnonic circuits may be higher than the one ofthe silicon counterparts, as the metallic waveguides can be placed on the non-magnetic metallic base (e.g. Cu) to enhance the thermal transport. Optimistically, it may bepossible to build magnonic devices with hundreds of simul- taneously operating channels before power dissipation will appear as the critical issue. There are certain concerns associated with the reliability of information encoding in the phase of the propagating wave. As noticed in Ref. 28, the phase of the information carrying spin wave can be disrupted by scattering on the waveguide imperfections. Dispersion may be another serious problem as the velocity of the spin wave is frequency-dependent. Someof these critical comments have been clarified by the recent experimental data on three-wave interference in 20 nm thick permalloy film presented in Ref. 29. Three spin waves were excited by microwave (3 GHz) signals produced by the AC electric currents. The initial phases (0 or p) were controlled by the direction of the excitation current. All 2 3phase combi- nation have been tested and reliable output detected.29This experiment validated the feasibility of using spin wave inter- ference and demonstrated robust operation at room tempera-ture. In general, the fault tolerance of the wave-based devices is defined by the operating wavelength k. A propagating wave054307-7 Alexander Khitun J. Appl. Phys. 111, 054307 (2012) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.117.125.76 On: Tue, 15 Sep 2015 13:12:09is not affected by the structure imperfection which size is much less than the wavelength. So far, all of the experimen- tally demonstrated spin wave prototypes operate with themicrometer-scale waves. 29–31It is not clear if the wavelength scaling down below 100 nm will lead to any significant issues. Multi-frequency operation implies the simultaneous transmission of a number of spin waves through the ferro- magnetic waveguides. The inevitable frequency mixing dueto the inherent magnetic non-linearity will result in the appearance of spurs in the frequency domain. The latter may be a serious issue limiting the bandwidth and the number ofoperating frequencies. However, it may be possible to keep the amplitude of the spur signals much below the main signal level. We would like to refer to the experimental data on thesimultaneous spin wave excitation by two microwave signals on different frequencies reported in. 31Indeed, there were observed output signals on the mixed frequencies (e.g.f 1þf2,f1/C0f2,2f1/C0f2, etc.) However, the amplitude of the mix frequency signals was about 20 dB less than the ampli- tude of the main signals. A multi-frequency spin wave trans-port is an unexplored field, which requires a more detailed experimental study. To the best of our knowledge, there is no systematic study on the bandwidth and the frequencymixing in the ferromagnetic waveguides, which is the subject for the further investigations. Finally, we would like to estimate the potential func- tional throughput enhancement due to the use of multiple frequencies and compare it with the conventional CMOS. In Fig.7(a), we present the estimates on the functional through- put in [Ops/ns cm 2] for the Full Adder Circuit built of scaled CMOS (blue markers) and magnonic multi-frequency circuit (red markers). The estimates for the CMOS Full Adder cir-cuit are based on the data for the 32 nm CMOS technology (area¼3.2lm 2, time delay ¼10 ps from Ref. 32). The esti- mates for further generations are extrapolated by using thefollowing empirical rule: the area per circuit scales as /C20.5 per generation, and the time delay scales as /C20.7 per genera- tion. The estimates for the magnonic circuit are based on themodel for the single frequency operating circuit presented in Ref. 10. The area per circuit scales as 25/C2k, where kis the wavelength. The time delay t delay is the sum of the following: the time required to excite spin wave, the propagation time, and the time required for the output ME cell switching: tdelay¼textþtpropþtrelax. The propagation time is easy to estimate by dividing the circuit length by the spin wave group velocity tprop¼3k/t(e.g. 100 nm per 10 ps). Less reliable are the estimates on the ME cell excitation and relax-ation times due to the lack of experimental data. As a con- servative estimate, we take t ext¼trelax¼100 ps, which is experimentally observed in magnetic memory devices.33The graph in Fig. 7(a) shows significant (two orders of magni- tude) functional throughput enhancement over the scaled CMOS. The overall advantage is due to the two parts (i)magnonic logic can outperform the transistor-based approach even for the single frequency circuits, as the wave-based logic circuits requires a fewer number of elements; 10(ii) an additional advantage comes from the use of multiple fre- quencies. The plot in Fig. 7(b) shows the relative functionalthroughput enhancement FNas a function of the number of frequencies N(independent information channels). The enhancement is estimated by the following formula: FN¼F1þN 1þDs sN/C18/C19 /C11þDt tN/C18/C19 ; (5) where F1is the functional throughput for a single frequency, Ds/sis the relative area increase associated with the addition of one extra frequency, the Dt/tis the relative time delay increase associated with the addition of one extra frequency. In numerical estimates, we assumed Ds/s¼5%, and Dt/t =1%. As one can see from Fig. 5(b), the relative enhance- ment of using multiple frequencies estimated by Eq. (2)has a maximum and starts to decrease beyond the optimum chan-nel number. The decrease at the large number of channels is due to the additional area and time delay introduced by each new input/output port. The optimum number of the opera-tional channels varies for different logic circuits. The lower is the area/time delay per an additional input/output port the FIG. 7. (Color online) (a) Numerical estimates on the functional throughput for the Full Adder Circuit built of CMOS (blue markers) and magnonic multi-frequency circuit (red markers). The estimates for the CMOS circuit are based on the 32 nm CMOS technology. The estimates for smaller feature size are extrapolated: the area per circuit scales as /C20.5 per generation, and the time delay scales as /C20.7 per generation. The estimates for the mag- nonic circuit are based on the model for the single frequency operating cir- cuit reported in Ref. 10and Eq. (2). (b) Numerical estimates on the functional throughput as a function of the number of frequencies (independ- ent information channels) by Eq. (2).054307-8 Alexander Khitun J. Appl. Phys. 111, 054307 (2012) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.117.125.76 On: Tue, 15 Sep 2015 13:12:09more prominent is the advantage of the multi-frequency circuits. V. CONCLUSIONS We have described the concept of multi-frequency mag- nonic logic circuits taking the advantage of wave superposi- tion for parallel data processing. The operation is illustrated by numerical modeling. According to the presented esti-mates, mutli-frequency magnonic logic circuits may provide an orders of magnitude functional throughput enhancement over the scaled CMOS. There is a number of questionsregarding the operation of the circuit components and overall system stability, which require further study. In summary, we want to emphasize the key point of this work. Wave-based magnonic logic devices offer a fundamental advantage over the CMOS technology. The ability to use multiple fre- quencies as independent information channels opens a newdimension for functional throughput enhancement and may provide a route to a long-term development. ACKNOWLEDGMENTS The work was supported by the DARPA program on Non-volatile Logic (program manager Dr. Devanand K. She- noy) and by the Nanoelectronics Research Initiative (NRI) (Dr. Jeffrey J. Welser, NRI Director) via the Western Insti-tute of Nanoelectronics (WIN, director Dr. Kang L Wang). 1G. E. Moore, Electronics 38, 114 (1965). 2V. A. Sverdlov, T. J. Walls, and K. K. Likharev, IEEE Trans. Electron Devices 50, 1926 (2003). 3K. Bernstein, R. K. Cavin, W. Porod, A. Seabaugh, and J. Welser, Proc. 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Crawford, and C. T. Rogers, J. Appl. Phys. 85, 7849 (1999). 17M. Covington, T. M. Crawford, and G. J. Parker, Phys. Rev. Lett. 89, 237202 (2002). 18Y. Wu, M. Bao, A. Khitun, J.-Y. Kim, A. Hong, and K. L. Wang, J. Nano- electron. Optoelectron. 4, 394 (2009). 19M. Bailleul, D. Olligs, C. Fermon, and S. Demokritov, Europhys. Lett. 56, 741 (2001). 20V. E. Demidov, J. Jersch, K. Rott, P. Krzysteczko, G. Reiss, and S. O.Demokritov, Phys. Rev. Lett. 102, 177207 (2009). 21S.-W. Cheong and M. Mostovoy, Nat. Mater. 6, 13 (2007). 22T. K. Chung, S. Keller, and G. P. Carman, Appl. Phys. Lett. 94, 132501 (2009). 23A. V. Chumak, A. A. Serga, B. Hillebrands, and M. P. Kostylev, Appl. Phys. Lett. 93, 022508 (2008). 24M. P. Kostylev, G. Gubbiotti, J.-G. Hu, G. Garlotti, T. Ono, and R. L. Stamps, Phys. Rev. B 76, 054422 (2007). 25Z. K. Wang, V. L. Zhang, H. S. Lim, S. C. Ng, M. H. Kuok, S. Jain, and A. O. Adeyeye, Appl. Phys. Lett. 94, 083112 (2009). 26V. V. Kruglyak and A. N. Kuchko, J. Magn. Magn. Mater. 272, 302 (2004). 27I. V. Ovchinnikov and K. L. Wang, Phys. Rev. B 82, 024410 (2010). 28S. Bandyopadhyay and M. Cahay, Nanotechnology 20, 412001 (2009). 29P. Shabadi, A. Khitun, P. Narayanan, M. Bao, I. Koren, K. L. Wang, and C. A. Moritz, Proceedings of the Nanoscale Architectures (NANOARCH), 2010 IEEE/ACM International Symposium (2010), p. 11. 30M. P. Kostylev, A. A. Serga, T. Schneider, B. Leven, and B. Hillebrands, Appl. Phys. Lett. 87, 153501 (2005). 31M. Bao, A. Khitun, Y. Wu, J.-Y. Lee, K. L. Wang, and A. P. Jacob, Appl. Phys. Lett. 93, 072509 (2008). 32A. Chen, private communication (2010). 33G. E. Rowlands, T. Rahman, J. A. Katine, J. Langer, A. Lyle, H. Zhao, J. G. Alzate, A. A. Kovalev, Y. Tserkovnyak, Z. M. Zeng, H. W. Jiang, K. Galatsis, Y. M. Huai, P. K. Amiri, K. L. Wang, I. N. Krivorotov, and J.-P. Wang, Appl. Phys. Lett. 98, 102509 (2011).054307-9 Alexander Khitun J. Appl. Phys. 111, 054307 (2012) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.117.125.76 On: Tue, 15 Sep 2015 13:12:09

1.5099913.pdf

J. Appl. Phys. 126, 043902 (2019); https://doi.org/10.1063/1.5099913 126, 043902 © 2019 Author(s).Measurement of spin mixing conductance in Ni 81Fe19/α-W and Ni 81Fe19/α-W heterostructures via ferromagnetic resonance Cite as: J. Appl. Phys. 126, 043902 (2019); https://doi.org/10.1063/1.5099913 Submitted: 12 April 2019 . Accepted: 30 June 2019 . Published Online: 23 July 2019 W. Cao , J. Liu , A. Zangiabadi , K. Barmak , and W. E. Bailey ARTICLES YOU MAY BE INTERESTED IN Organic photodetectors with frustrated charge transport for small-pitch image sensors Journal of Applied Physics 126, 045501 (2019); https://doi.org/10.1063/1.5102179 Characterization of the electromechanical properties of YCa 4O(BO 3)3 single crystals up to 800 °C Journal of Applied Physics 126, 045104 (2019); https://doi.org/10.1063/1.5093102 Photoluminescence-based detection of mechanical defects in multijunction solar cells Journal of Applied Physics 126, 044503 (2019); https://doi.org/10.1063/1.5106414Measurement of spin mixing conductance in Ni81Fe19/α-W and Ni 81Fe19/β-W heterostructures via ferromagnetic resonance Cite as: J. Appl. Phys. 126, 043902 (2019); doi: 10.1063/1.5099913 View Online Export Citation CrossMar k Submitted: 12 April 2019 · Accepted: 30 June 2019 · Published Online: 23 July 2019 W. Cao,a) J. Liu, A. Zangiabadi, K. Barmak, and W. E. Baileyb) AFFILIATIONS Materials Science and Engineering, Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York 10027, USA a)Electronic mail: wc2476@columbia.edu b)Electronic mail: web54@columbia.edu ABSTRACT We present measurements of interfacial Gilbert damping due to the spin pumping e ffect in Ni 81Fe19=W heterostructures. Measurements were compared for heterostructures in which the crystallographic phase of W, either α(bcc)-W or β(A15)-W, was enriched through deposi- tion conditions and characterized using X-ray di ffraction and high-resolution cross-sectional transmission electron microscopy. Single-phase Ni81Fe19=α-W heterostructures could be realized, but heterostructures with β-W were realized as a mixed α-βphase. The spin mixing conductances for W at interfaces with Ni 81Fe19were found to be signi ficantly lower than those for similar heavy metals such as Pd and Pt, but comparable to those for Ta, and independent of enrichment in the βphase. Published under license by AIP Publishing. https://doi.org/10.1063/1.5099913 I. INTRODUCTION The heavy metals Ta, W, and Pt have drawn attention as charge- to-spin-current-converters using spin Hall and related e ffects.1–4Beta phase W, β-W, with the topologically close-packed A15 structure,5 possesses a “giant ”spin Hall angle of θSH/C250:3–0:4.3,6The spin transport properties of β-W, such as the spin Hall angle θSHand spin di ffusion length λSD, have been characterized by di fferent methods.3,6–8In these studies, the metastable β-W layers were depos- ited directly on the substrate, were only stable for small W thickness,and were presumably stabilized through residual water vapor oroxygen on the substrate surface; thicker W films typically revert to the stable (bcc) αphase. Recently, some of us 9–11have optimized a di fferent method to stabilize the metastable- β-phase, using the introduction of N 2gas12 while sputtering at low power. Relatively thick (over 100 nm) monophase β-Wfilms could be stabilized this way, when deposited on glass substrates. This technique has allowed deposition ofmajority βphase W for 14 nm W films on CoFeB, as CoFeB/W (14 nm), and of minority βphase for 14 nm W films on Ni and Ni 81Fe19(“Py”), as Ni/W(14 nm) and Py/W(14 nm). In the presentwork, we have prepared both monophase Py/ α-W (here Py/ “α”-W) and mixed phase Py =(αþβ)-W (here Py/ “β”-W) heterostructures using our optimized sputtering technique to enrich the fraction of β-W. Crystallographic phases of W were characterized by X-ray diffraction (XRD) and high-resolution cross-sectional transmission electron microscopy (HR-XTEM); secondary structural informationwas provided by electrical resistivity measurements at room tem- perature. We note that our measurements cannot distinguish between purely metallic, A15 β-W, and A15 W oxide or nitride (e.g., W 3O); the identity of β-W as a purely metallic phase or a compound is a longstanding controversy.12,13 In ferromagnet (FM)/normal metal (NM) heterostructures, pure (chargeless) spin currents can be injected from the FM into the NM by exciting ferromagnetic resonance (FMR) in the FMlayer, “pumping ”out the spin current. 14,15If the spin current is absorbed in the NM layer, the in fluence of “spin pumping ”can be observed through the increase in the linewidth of the resonance, proportional to frequency ωas Gilbert damping, due to the loss of angular momentum from the precessing spin system.14,15The efficiency of the spin pumping e ffect for a given interface is charac- terized through the spin mixing conductance (SMC) g"# FM=NM.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 126, 043902 (2019); doi: 10.1063/1.5099913 126, 043902-1 Published under license by AIP Publishing.The SMC is also an important parameter for the interpretation of inverse spin Hall e ffect (ISHE) measurements,4,16in which the spin Hall angle θSHis measured by pumping chargeless spin current into the NM by FMR and measuring spin-to-charge current con-version through the generated charge current. Measurements ofspin mixing conductance for Py =α-W and Py =β-W have not been reported previously, although some measurements have been reported for W oxide. 8For these measurements, the simplest way to isolate the contribution of the FM/NM interface to the damping,and thus the spin pumping e ffect and spin mixing conductance g "# FM=NM, is to deposit the FM on the bottom and the NM on top, so that comparison structures without the NM layer have nearly identical microstructure. The ability to deposit enriched β-W on Py rather than on an insulating substrate is thus important for themeasurement of spin mixing conductance of Py =β-W. In this paper, we report measurements of spin mixing conductances for Py/“α”-W and Py/ “β”-W interfaces using variable-frequency, swept- field FMR, as in our previous work. 17–19 II. SAMPLE PREPARATION Ultrahigh vacuum (UHV) magnetron sputtering was used to deposit substrate =Ta(5 nm) =Cu(5 nm) =Ni81Fe19(Py)/W/Cu(5 nm)/ Ta(5 nm) heterostructures on both oxidized Si and glass substrates at room temperature, with base pressure better than 2 /C210/C08Torr. The samples consist of two thickness series in “α”-W and “β”-W for a total of four series. In the first thickness series, the thickness of Py (tPy¼5n m ) w a s fixed and the thickness of W was varied, with tW¼2, 5, 10, and 30 nm, for both “α”-optimized and “β”-optimized conditions. This thickness series was used for resistivity measure- ments, X-ray di ffraction (XRD) ( tW¼10, 30 nm), high-resolution cross-sectional transmission electron microscope (HR-XTEM)(t W¼30 nm), and FMR characterization. In the second thickness series, the thickness of W ( tW¼10 nm) was fixed and the thickness of Py was varied, with tPy¼3, 5, 10, and 20 nm, also for both “α” and “β”conditions. This thickness series was used only for FMR characterization. The same stacks without W layers, Py(3, 5, 10, and20 nm), were deposited as reference samples for FMR measurements. One heterostructure with reverse deposition order, “α”-W(10 nm)/Py (5 nm), was deposited in the absence of N 2gas and characterized by XRD and FMR; this was not possible for “β”-W because the βphase cannot be stabilized on Cu underlayers.10 The W layers in all samples were deposited with 10 W power, nearly constant deposition rate ( ,0:1A/C14=s) and an Ar pressure of 3/C210/C03Torr. Nitrogen gas, with 1 :2/C210/C05Torr pressure mea- sured by a residual gas analyzer, was introduced to promote thegrowth of βphase W. 10 III. STRUCTURAL CHARACTERIZATION Crystalline phases of W in the Py/W heterostructures were characterized primarily by XRD (Sec. III A ), with supporting mea- surements by HR-XTEM (Sec. III B), and finally with some indirect evidence in room-temperature electrical resistivity measurements(Sec. III C ). Our basic findings are that films deposited without N 2, optimized for “α”-W, are nearly single-phase αin Py/ “α”-W, while in the Py/ “β”-W optimized heterostructures, deposited in the pres- ence of N 2, the W layers are mixed αþβphase, with a roughly50% –50% mixture of α-W and β-W averaged over a 10 nm film. The phase composition within the first 5 nm of the interface may have a slightly greater fraction of α-W, but β-W could be positively identi fied here as well. A. X-ray diffraction Both symmetric ( θ/C02θ) and grazing-incidence, fixed sample angle X-ray di ffraction (XRD) scans were carried out on Py(5 nm)/ W(10 nm) and Py(5 nm)/W(30 nm) heterostructures deposited onglass substrates. The scans are compared for “α”-W and “β”-W depositions. Scans were recorded using Cu K αradiation and a com- mercial di ffractometer. The symmetric ( θ/C02θ) scans, with scattering vector perpen- dicular to film planes, are presented first. We point out some obvious features of the symmetric XRD spectra, shown in Figs. 1(a) and1(b). For the Py/ “α”-W(30 nm) film in Fig. 1(a) , all peaks can be indexed to the close-packed planes, Cu(111)/Py(111) (fcc) and α-W(110) (bcc). The small peak at 2 θ¼36/C14can be indexed to the reflection of a small amount of Cu Kβradiation from α-W(110). Moving to the thinner αphase film in Fig. 1(b) , Py/ “α”-W(10 nm), it is still the case that all re flections can be indexed to the close- packed Cu(111)/Py(111) and α-W(110) planes. However, there is greater structure in these re flections, presumably due to finite-size oscillations (Laue satellites), expected to be more evident in thinnerfilms. Nearly identical spectra are recorded for the 10 nm “α”-Wfilms regardless of deposition order: Py(5 nm)/ “α”-W(10 nm) and “α”-W ( 1 0n m ) / P y ( 5n m ) fil m ss c a t t e rX - r a y sv e r ys i m i l a r l y ,a ss h o w ni n Fig. 1(b) . We should note that Cu deposited on Ta has strong {111} texture in our films. Py (Ni 81Fe19) deposited on Cu also has strong {111} texture; growth of Py on Cu and vice versa is found to be largelycoherent within grains. Both layers are fcc with similar lattice parame- ters: a Cu/C253:61 A/C14for Cu10,20andaPy/C253:55 A/C14for Py,10,21with a small mis fits t r a i no f ϵ¼jaCu/C0aPyj=aCu/C252%. The XRD peaks for (111)-re flections in bulk phases, broadened by finite-size e ffects (FWHM /C251:7/C14for 5 nm films, using the Scherrer equation22,23), are very close to each other, at 44 :2/C14(Py) and 43 :4/C14(Cu), respectively, so we expect (and have observed) one averaged peak for Cu and Py. The nominal “β”-Wfilms (red lines) clearly show the presence of the βphase through the unique β-W(200) re flection at 2 θ≃36/C14. This unique re flection is very strong in the “β”-W(30 nm) hetero- structure [ Fig. 1(a) ] but weaker as a proportion of the total intensity in the thinner “β”-W(10 nm) heterostructure [ Fig. 1(b) ]. In Fig. 1(a) , experimental β-W(200) and β-W(210) re flections have intensities in a ratio similar to the theoretical scattering intensity ratios forrandomly-oriented βgrains. This is not the case for the thinner “β”-W(10 nm) heterostructure in Fig. 1(b) ; here, the unique β-W (200) peak is less intense than expected. We interpret the relativeweakness of β(200) as the presence of a large fraction of αgrains in the nominal Py/ “β”-W(10 nm) heterostructure. In order to quantify the amount of α-W in the nominal “β”-Wfilm, we have carried out grazing-incidence measurements of Py(5 nm)/W(10 nm) samples (20 /C14/C202θ/C20100/C14) on the same diffractometer, as illustrated in Fig. 1(c) . The samples were mea- sured at a fixed source position of 5/C14with 0 :1/C14step size, 0 :25/C14fixed slit, and the 15 mm beam mask. From the TEM measurements in Fig. 2(b) ,w e find that the deposited “α”-Wfilms have {110}Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 126, 043902 (2019); doi: 10.1063/1.5099913 126, 043902-2 Published under license by AIP Publishing.texture, i.e., the hexagonal arrangement (60/C14angles) of the {011} α-W re flections away from the surface normal. Thus, with the grazing-incidence geometry, in which the scattering vector does not remain perpendicular to the film plane, the relative intensities of the peaks will not match theoretical calculations (vertical lines)based on randomly-oriented, untextured films. For example, the α-W(200) peak (blue, /difference58/C14in 2θ) almost vanishes in the XRD scan here, due to the {011} α-W texture. Here, we focus on the α-W(211) peaks near 2 θ¼72/C14, observed in both Py/ “α”-W and Py/ “β”-W samples. As shown in FIG. 1. X-ray diffraction (XRD) measurements for Py(5 nm)/ “α”-W(tW) (blue) and Py(5 nm)/ “β”-W(tW) (red) deposited on glass substrates. (a) tW¼30 nm and (b) tW¼10 nm. Solid vertical lines show the calculated re flections and intensities for α-W and β-W peaks. (c) Grazing-incidence XRD measurements for Py(5 nm)/ “α”-W(10 nm) and Py(5 nm)/ “β”-W(10 nm) samples. The inset shows the α(211) re flections observed in both samples. The blue and the red dashed lines refer to thefits for Py/ “α”-W and Py/ “β”-W, respectively. The black dashed line refers to the identical quadratic background.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 126, 043902 (2019); doi: 10.1063/1.5099913 126, 043902-3 Published under license by AIP Publishing.the inset of Fig. 1(c) , the α-W(211) peaks (60/C14/C202θ/C2085/C14) were fitted as the sum of the Lorentzian peak and identical background, assumed quadratic in 2 θ, for both Py/ “α”-W and Py/ “β”-W samples. First, we fit the α-W(211) peak (blue) in the Py/ “α”-W sample to the summed function to determine the Lorentzian peakand quadratic background parameters. Next, we use this fitted background in the fit to the α-W(211) peak (red) in the Py/ “β”-W sample. The two fitted α-W(211) peaks are shown as blue (for Py/“α”-W) and red (for Py/ “β”-W) dashed lines in the inset of Fig. 1(c) . The fits reproduce the experimental data well in the fitted region. The integrated α-W(211) peak (i.e., the 2 θ-integrated area between the measured data and the fitted background) for the Py/“β”-W sample has roughly half the intensity of the integrated peak for the Py/ “α”-W sample. Assuming that the nominal α-W is 100% αphase and that the αgrains in mixed phase “β”-W have similar {110} texture, as is supported by the HR-XTEM measure-ments in Figs. 2 and3, we conclude that the Py/ “β”-W(10 nm) film is roughly 50% α-W and 50% β-W. B. Transmission electron microscopy The phases of the nominal Py(5 nm)/ “α”-W(30 nm) and the nominal Py(5 nm)/ “β”-W(30 nm) samples deposited on oxidized Si substrates were characterized in high-resolution cross-sectional imaging, selected-area di ffraction, and focused-beam nanodi ffrac- tion, by transmission electron microscopy (for details, see Ref. 32). Figure 2 shows a cross-sectional image and di ffraction pattern for the nominal Py/ “α”-W(30 nm) heterostructure. First, one can see from the mass contrast between W and the 3d transition metal elements (Ni, Fe, Cu) that the Py/W and W/Cu interfaces are rela-tively flat and sharp on the scale of the image resolution of /difference3 nm, presumably broadened by topographic variation through thethickness of the TEM foil. Second, based on (less pronounced) diffraction contrast parallel to the interface, the grains appear to be columnar, in many cases extending through the film thickness, with an average (lateral) grain diameter of 10 –20 nm. The selected-area di ffraction (SAD) pattern can be indexed according to unique (111)Py//(011) α-Wfiber texture, as shown by the hexagonal arrangement (60 /C14angles) of the {011} re flections in α-W, and thearrangement of {111} re flections in Py, /difference70:5/C14away from the (vertical) fiber axis. The calculated di ffraction spots based on {111} Py//{011} α-Wfiber texture with 1-fold rotational symmetry about thefilm-normal axis are shown in the inset of Fig. 2(b) ; good agreement is found. Cross-sectional images and di ffraction patterns for the Py/ “β”-W(30 nm) heterostructure are shown in Fig. 3 . Here again, in Fig. 3(a) , the mass contrast shows similarly well-de fined interfaces, but the topographic variations have a shorter wavelength, due pre- sumably to smaller, more equiaxed grains in the mixed-phase“β”-W. Circles indicate areas where convergent nanobeam electron diffraction (CBED) patterns were taken. The di ffraction patterns over these small regions can be indexed to single phases: fcc Ni 81Fe19(Py) in green, bcc α-W in blue, and A15 β-W in red. The CBED patterns in Fig. 3(a) confirm that the nominal “β”-Wfilm is mixed-phase α-W and β-W. The critical question for distinguishing the spin mixing conductances of α-W and β-W in Py/W is the identity of the W phase located within the first several nanometers of the interface with Py: the pumped spin current isejected through the interface and absorbed over this region; see thex-axis of Fig. 6 . We have addressed this question locally using high- resolution imaging [see Fig. 3(b) ] and over a larger area using fre- quency analysis [see Fig. 3(c) ] of the image, roughly equivalent to SAD. In Fig. 3(b) , a 10 nm area (red box) shows what appears to be a single-crystal region with (1 /C2211)[110]Py//(011)[1 /C2211]α-W//(002) [200] β-W, indicating that the βcrystals may nucleate on top of the αcrystals; however, this is contrary to our previous observations 10 and not distinguishable in the image from the superposition of grains through the foil, with nucleation of βat the Py/W interface. The discrete spatial Fourier transform (FT) of this region showsthat the four vertically/horizontally circled β-W{002} spots are similar in intensity to the six α-W{011} spots, supporting a similar β-W content in this region. Carrying out a spatial FT of the full selected region within 5 nm of the interface (dotted box) inFig. 3(a) , we can con firm that β-W is indeed present adjacent to the interface, as indicated by the β-W{002} FT spots in Fig. 3(c) , although these appear to be somewhat less intense than the α-W {011} spots. FIG. 2. (a) High-resolution cross- sectional transmission electron micros-copy (HR-XTEM) image of SiO 2/Ta (5 nm)/Cu(5 nm)/Py(5 nm)/ “α”-W(30 nm)/ Cu(5 nm)/Ta(5 nm) heterostructure. Theα-W grains are columnar with a lateral radius of 10 –20 nm, with larger grain size in the growth direction. (b) Selected-area diffraction (SAD) pattern ofthe heterostructure, showing the pre-ferred texture of α-W grains on the Py layer, {111}Py/ /{011} α-W (see the calcu- lated pattern in the inset). No sign ofβ-W was detected in this heterostructure.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 126, 043902 (2019); doi: 10.1063/1.5099913 126, 043902-4 Published under license by AIP Publishing.C. Resistivity Four-point probe van der Pauw resistivity measurements were performed at room temperature on the first thickness series of samples ( tPy¼5n m fixed, variable tW) deposited on 25 /C225 mm square glass substrates, i.e., glass substrate/Ta(5 nm)/Cu(5 nm)/Py(5 nm)/W( t W)/Cu(5 nm)/Ta(5 nm). Two point probes for current and two point probes for voltage were placed at the four corners of the square coupons. For square samples, the voltage-to-current ratios were converted to resistance per square using the known geometricalfactor π=ln 2/C254:53. 24To isolate the W resistances, we plot the thickness-dependent sheet conductance and fit according to 1 Rtotal¼Gtotal¼G0þ4:53 ρWtW, (1) where Rtotal (Gtotal) is the total resistance (conductance) of the sample, ρWandtWare the resistivity and the thickness of the W layer, and G0is the parallel conductance of other layers in the stack. We have veri fied Ohmic response by fitting the proportional dependence of voltage Von current Iover the range 2 mA /C20I/C20 10 mA for each sample. Figure 4 summarizes the total conductance Gtotal¼1=Rtotal as a function of W thickness tWfor all Py(5 nm) =W(tW) heterostructures. Solid lines represent linear fits for the W resistivity ρW, assumed constant as a function of Wthickness for “α”-W and “β”-W samples. The extracted resistivity for“α”phase W ραis found to be /difference35μΩcm and for “β”phase W ρβ/difference148μΩcm. The resistivity for “β”-W more than four times greater than that for “α”-W is due in large part to the much smaller grain size for β-W and is typically observed in prior studies.25Here, the resistivity for “α-W”is larger by a factor of 2 –3 thanfilms deposited at room temperature and postannealed in pre- vious work,26also attributable to a smaller grain size in these films deposited at ambient temperature. The resistivity measurements for these thin films might be taken as indirect evidence for the pres- ence of the βphase in the nominal “β”-W layers. IV. FERROMAGNETIC RESONANCE MEASUREMENTS The four thickness series of Py( tPy)=W(tW)films, for “α”-W and “β”-W, as described in Sec. IIwere characterized using variable-frequency field-swept FMR using a coplanar waveguide (CPW) with a center conductor width of 300 μm. The bias mag- neticfield was applied in the film plane ( pc-FMR or parallel condi- tion). For details, see, e.g., our prior work in Ref. 20. Figure 5 summarizes half-power FMR linewidth ΔH1=2as a function of frequency ω=2πfor Py(5 nm), Py(5 nm)/ “α”-W (10 nm), and Py(5 nm)/ “β”-W(10 nm) samples. The measurements were taken at frequencies from 3 GHz to above 20 GHz. Solid lines are linear regression of the variable-frequency FMR linewidth FIG. 3. (a) HR-XTEM image of SiO 2/Ta(5 nm)/Cu(5 nm)/Py(5 nm)/ “β”-W(30 nm)/Cu(5 nm)/T a(5 nm), showing mixed-phase α-W and β-W. Convergent nanobeam electron diffraction (CBED) patterns, bottom, reveal the coexistence of separated α-W,β-W, and fcc Py. (b) Close-up of one region near the Py/W interface in (a), with discrete spatial Fourier Transform (FT). The FT is consistent with a single-crystal pattern of (1 /C2211)[110]Py/ /(011)[1 /C2211]α-W//(002)[200] β-W, as shown in the calculated pattern (bottom right). (c) FT of the interface region (dotted box), showing coexistence of α-W and β-W in the first 5 nm W adjacent to the Py/W interface.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 126, 043902 (2019); doi: 10.1063/1.5099913 126, 043902-5 Published under license by AIP Publishing.ΔH1=2¼ΔH0þ2αω=γ, where ΔH1=2is the full-width at half- maximum, ΔH0is the inhomogeneous broadening, αis the Gilbert damping, ωis the resonance frequency, and γis the gyromagnetic ratio. Good linear fits were obtained with resonance frequency ω=2πfor experimental linewidths ΔH1=2(ω) of all the samples measured. For the first sample thickness series Py(5 nm) =W(tW), we plot damping parameters αextracted from the linear fits, as a function of W thickness in Fig. 6 . Standard deviation errors in the fit for αare/difference2/C210/C04. The Gilbert damping αsaturates quickly as a func- tion of tWfor both “α”-W and “β”-W, with almost all of the e ffect realized with the first 2 nm of W. Loosely speaking, this fast satura- tion implies a short spin di ffusion length λSD/C202 nm, so the iden- tity of the W phase ( αorβ) over this length scale near the interface is most relevant. The averaged damping, αPy=“α”/C0Wand αPy=“β”/C0W, is shown as horizontal dashed lines in the figure. αPy=“α”/C0Wis slightly smaller than αPy=“β”/C0W, but this may be within experimental error. Due to spin pumping, the damping is enhanced with the addition of W layers Δα¼αPy=W/C0αPy, normalized to the Gilbert damping αPyof the reference sample without W layers. The e ffective SMC g"# effat the Py/W interfaces can be calculated as follows: Δα¼γ/C22hg"# eff (4πMS)tPy, (2) where γis the gyromagnetic ratio, /C22his the reduced Planck constant, and 4 πMS/C2510 kG is the saturation inductance of Py. In this series of samples, the e ffective SMC at the Py/ “α”-W inter- face g"# Py=“α”/C0W/C257:2+0:3n m/C02and the e ffective SMC at the Py/“β”-W interface g"# Py=“β”/C0W/C257:4+0:2n m/C02. These values are significantly lower than those reported in Ref. 8for CoFeB/W (20–30 nm/C02), as measured by spin-torque FMR. For the second sample thickness series Py( tPy)/W(10 nm), we plot the extracted Gilbert damping αand damping enhancement Δα¼αPy=W/C0αPyas a function of Py thickness in Fig. 7 . The enhanced damping is normalized to the Gilbert damping αPyof reference samples with the same Py thickness tPy. The result is in good agreement with the inverse thickness dependence of contributed damping predicted from Eq. (2). The experimental FIG. 4. The total conductance Gtotal¼1=Rtotalas a function of W thickness. Blue dots refer to Py(5 nm)/ “α”-W(tW) samples, and red dots refer to Py(5 nm)/ “β”-W(tW) samples. The solid lines are linear fits. FIG. 5. Half-power FMR linewidth ΔH1=2spectra of reference samples: Py(5 nm) (black), Py(5 nm)/ “α”-W(10 nm) (blue), and Py(5 nm)/ “β”-W(10 nm) (red). The solid lines are linear fits. FIG. 6. Gilbert damping αof the reference samples: Py(5 nm) (black), Py(5 nm)/ “α”-W(tW) (blue), and Py(5 nm)/ “β”-W(tW) (red). The blue and red dashed lines refer to averaged enhanced damping for Py(5 nm)/ “α”-W(tW) and Py(5 nm)/ “β”-W(tW), respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 126, 043902 (2019); doi: 10.1063/1.5099913 126, 043902-6 Published under license by AIP Publishing.data are fitted with Eq. (2)to extract the e ffective SMC. In this series of samples, the e ffective SMC at the Py/ “α”-W interface g"# Py=“α”/C0W/C256:7+0:1n m/C02and the e ffective SMC at the Py/- “β”-W interface g"# Py=“β”/C0W/C257:4+0:3n m/C02. Previous studies on W have shown that the formation of α-W is preferred, for thicker W layers (e.g., 10 nm).3,26We also prepared the sample “α”-W(10 nm)/Py(5 nm) with reverse deposition order, with the same seed and cap layers, on an oxidized Si substrate.Here, the top surface of the 10 nm thick α-W layer is pure αphase, as shown by XRD in Fig. 1(a) . We performed the same FMR mea- surement on the reverse-order sample; its Gilbert damping enhancement Δαis plotted as the green dot in Fig. 7 . This point almost overlaps with the measurement for the normal order samplePy(5 nm)/ “α”-W(10 nm), indirectly supporting the conclusion that the phase of the Py/ “α”-W interface is similar to the phase of the “α”-W/Py interface, i.e., almost 100% αphase W. Note that it was not possible to deposit a reverse-order βphase sample because no βphase W could be stabilized on Cu using our technique. 10 The FMR measurements of spin mixing conductance g"#for Py/“α”-W and Py/ “β”-W are new in this study. We find that the value is similar to that measured for Ta27(g"#/difference10 nm/C02) regard- less of the enriched phase. First-principles-based calculationsincluding relativistic e ffects 28forg"#at Py/NM interfaces have shown that Ta, next to W in the periodic table, is a good spin sink due to its large spin-orbit coupling (SOC) but has a relatively smallg "#/difference8–9n m/C02. The e fficient absorption of the spin current can be connected with a large SOC from the large atomic number, andthe low SMC can be connected to relatively poor band matching across the Py/W interface, compared with that for Py/Cu or Py/Pt. 28The conclusion for Ta is consistent with our experimental results for the Py/W system, i.e., the rapid saturation of Gilbertdamping within the first 2 nm of W, indicating that W is also a good spin sink, with a similarly low g "#/difference7n m/C02.V. DISCUSSION We have found very little di fference between the spin scatter- ing properties (spin mixing conductance and spin di ffusion length) ofα-W and mixed phase ( αþβ)-W. The simplest interpretation is that both spin mixing conductances and spin di ffusion lengths are nearly equal for the two phases. However, despite our development of an optimized technique9–11to stabilize the βphase, our control over the amounts of deposited αandβphases is less than com- plete, particularly near the Py/W interface. The “α”-structure we deposited, Py/ α-W, is nearly /difference100% α phase. We observed no strong β-W peaks in the XRD scans and neither crystalline structure nor di ffraction patterns for the β phase in HR-XTEM characterization. According to our previous work,10,26,29we know that ionically and covalently bonded sub- strates/underlayers are favorable for the formation of some β-W, whereas metallic underlayers promote α, so on Py even at a thick- ness of 2 nm, the nominally α-Wfilm is fully αif deposited in the absence of nitrogen. In the thinnest “β”-structure which we can characterize by XRD, Py/ “β”-W(10 nm), we identify a roughly 50% –50% mixture ofαandβphases. If this balance persists at the interface as well, the SMC cannot di ffer by more than 10 –20% for the two phases. While the measurement of the 5 nm region near the interface seems to show somewhat less than 50% βphase, there is still a sub- stantial population of β-W in this region, and it would seem that a strong di fference in SMC for α-W and β-W should be resolvable if present. Given that the measured values are very similar, we con- clude that the αandβphases do not di ffer strongly in this spin transport study. One might ask why the spin mixing conductance, in contrast to the spin Hall angle,3does not di ffer much for the two phases of W. The spin mixing conductance (SMC) g"# FM=NMis a property of FIG. 7. Damping enhancement Δα¼αPy=W/C0αPyof Py(tPy)/“α”-W(10 nm) (blue), Py( tPy)/“β”-W(10 nm) (red), and “α”-W(10 nm)/Py(5 nm) (green) samples, normalized to the Gilbert damping of reference samples αPywith the same Py thickness. Solid lines refer to fitting with Eq. (2). The inset shows the Gilbert damping αof the reference samples: Py( tPy) (black), Py( tPy)/“α”-W(10 nm) (blue), Py (tPy)/“β”-W(10 nm) (red), and “α”-W(10 nm)/Py(5 nm) (green).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 126, 043902 (2019); doi: 10.1063/1.5099913 126, 043902-7 Published under license by AIP Publishing.the FM/NM interface, rather than a bulk property of the NM layer. The SMC may be approximated (in a single-band, free-electron model) as g"#/C25κk2 FA=4π2, where kFis the Fermi wave number for the NM, κrepresents the number of scattering channels in units of one channel per interface atom, and Ais the total surface area of the interface.30Despite the possibility that bulk β-W has a stronger effective spin-orbit coupling and spin Hall e ffect due to its A15 structure, β-W could have similar numbers of conducting channels per atom at the FM/NM interface as α-W, which could lead to similar values of SMC measured here. Another possibility is that the spin di ffusion length λSDmay vary along the W layer thickness, due to nonuniformly distributed α-W and β-W phases in “β”-W samples. If this is true, fitting a single spin di ffusion length for spin pumping into very thin W layers will be problematic.31However, because we have observed a very rapid saturation of Gilbert damping over the first 2 nm of W for both “α”-W (almost pure αphase) and “β”-W (mixed phase) in Fig. 6 , we can only assign an upper bound for λSD, similarly short in the two phases. VI. CONCLUSIONS In summary, we report measurements of spin mixing conduc- tances of Py/W films with controlled amounts of αandβphase W, measured by Gilbert damping through ferromagnetic resonance(FMR). We find no strong di fferences in the spin mixing conduc- tances of Py/ α-W and Py/ β-W, measured as g "#¼6:7–7:4n m/C02, although control of the βphase is seen to be more di fficult near the interface with Py. Our experimental results also indicate that W, nomatter of which phase, is a good spin sink, but with relatively smallspin mixing conductance in Ni 81Fe19(Py)/W, similar to Ta in Py/Ta. ACKNOWLEDGMENTS The authors thank Daniel Paley of Columbia Nano Initiative for the grazing-incidence XRD scans and Kadir Sentosun ofColumbia University for the satellite peak calculations. This work was supported by the U.S. NSF-DMR-1411160. REFERENCES 1E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Lett. 88, 182509 (2006). 2L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett. 106, 036601 (2011). 3C.-F. Pai, L. Liu, Y. Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, Appl. Phys. Lett. 101, 122404 (2012). 4H. L. Wang, C. H. Du, Y. Pu, R. Adur, P. C. Hammel, and F. Y. Yang, Phys. Rev. Lett. 112, 197201 (2014). 5H. Hartmann, F. Ebert, and O. Bretschneider, Z. Anorg. Allg. Chem. 198, 116 (1931). 6Q. Hao and G. Xiao, Phys. Rev. Appl. 3, 034009 (2015).7J. Liu, T. Ohkubo, S. Mitani, K. Hono, and M. Hayashi, Appl. Phys. Lett. 107, 232408 (2015). 8K.-U. Demasius, T. Phung, W. Zhang, B. P. Hughes, S.-H. Yang, A. Kellock, W. Han, A. Pushp, and S. S. P. Parkin, Nat. 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Omelchenko, C. Coutts, N. R. Lee-Hone, R. Hübner, D. Broun, B. Heinrich, and E. Girt, Phys. Rev. B 94, 054416 (2016). 32Focused ion-beam (FIB) and FEI Helios NanoLab 660 were used to prepare foils for TEM studies. To protect the heterostructures against the ion-beam damage during sample preparation, amorphous platinum (1 :5μm thick) was sputtered on the surface of the wafers by electron and ion beam. TEM and high- resolution cross-sectional TEM (HR-XTEM) analyses were performed by image Cs-corrected FEI Titan Themis 200 at an accelerating voltage of 200 kV. The nanobeam electron di ffraction pattern (DP) technique and the Fourier transform (FT) analysis of the HRTEM have been utilized to identify the nature of each phase at a scale of 1 –2 nm wide. The nanobeam DPs were obtained by FEI Talos TEM operating at 200 kV. The second condenser aperture was set to 50 μmt o obtain a small beam-convergence angle. In the di ffraction mode, the beam was condensed to a spot ( /difference1–2 nm), and a convergent electron beam di ffraction (in this case, known as the Kossel-Möllenstedt pattern) was acquired at di fferent locations on the sample.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 126, 043902 (2019); doi: 10.1063/1.5099913 126, 043902-8 Published under license by AIP Publishing.

1.3504356.pdf

Tilt-over mode in a precessing triaxial ellipsoid D. Cébron,a/H20850M. Le Bars, and P . Meunier Institut de Recherche sur les Phénomènes Hors Equilibre, UMR 6594, CNRS et Aix-Marseille Universités, 49 rue F . Joliot-Curie, BP146, 13384 Marseille Cédex 13, France /H20849Received 8 April 2010; accepted 14 September 2010; published online 4 November 2010 /H20850 The tilt-over mode in a precessing triaxial ellipsoid is studied theoretically and numerically. Inviscid and viscous analytical models previously developed for the spheroidal geometry by Poincaré /H20851Bull. Astron. 27, 321 /H208491910 /H20850/H20852and Busse /H20851J. Fluid Mech. 33, 739 /H208491968 /H20850/H20852are extended to this more complex geometry, which corresponds to a tidally deformed spinning astrophysical body. Asconfirmed by three-dimensional numerical simulations, the proposed analytical model provides anaccurate description of the stationary flow in an arbitrary triaxial ellipsoid, until the appearance atmore vigorous forcing of time dependent flows driven by tidal and/or precessional instabilities.©2010 American Institute of Physics ./H20851doi:10.1063/1.3504356 /H20852 I. INTRODUCTION The flow of a rotating viscous incompressible homoge- neous fluid in a precessing container has been studied forover one century because of its multiple applications, such asthe motions in planetary liquid cores and the generation ofplanetary magnetic fields /H20849e.g., Refs. 1and2/H20850. In the sphe- roidal geometry, the early work of Poincaré 3demonstrated that the flow of an inviscid fluid has a uniform vorticity.Later, viscous effects have been taken into account as a cor-rection to the inviscid modes in considering carefully thecritical regions of the Ekman layer. 4–6Indeed, the Poincaré solution is modified by the apparition of boundary layers andsome strong internal shear layers are also created in the bulkof the flow, which do not disappear in the limit of vanishingviscosity. 5Besides, at high enough precession rates, these shear layers may become unstable7–9and in a second transi- tion, the entire flow becomes turbulent: this is the precessioninstability. These experimental works have been completedby numerical studies in cylindrical, spherical, and spheroidalgeometries, 10–14in particular for studying kinematic dynamo models in a spheroidal galaxy15or for geophysical applications.16–20 However, in natural systems, both the rotation and the gravitational tides deform the celestial body into a triaxialellipsoid, where the so-called elliptical /H20849or tidal /H20850instability may take place /H20849see Refs. 21–25for details on this instability and its geophysical and astrophysical applications /H20850. The el- liptical instability takes place in any rotating fluid whosestreamlines are elliptically deformed /H20849see, e.g., Ref. 26or Ref. 27/H20850. It comes from a parametric resonance of two iner- tial waves of the rotating fluid with the tidal /H20849or elliptical /H20850 deformation of azimuthal wave number m=2. 25Similarly, it has been suggested that the precession instability comesfrom the parametric resonance of two inertial waves with theforcing related to the precession of azimuthal wave numberm=1, which comes from the deviations of the laminar tilt- over base flow from a pure solid body rotation. 28–30How-ever, it has also been suggested that the precession instability is related to a shear instability of the zonal flows that appearin a precessing container /H20849e.g., Ref. 18/H20850. Clearly, the precise origin of the precession instability is still under debate and isbeyond the point of the present work. But since tides andprecession are simultaneously present in natural systems, itseems necessary to study their reciprocal influence, in pres-ence or not of instabilities. The full problem is rather com-plex and involves three different rotating frames: the precess-ing frame, with a period T p/H1101526 000 yr for the Earth, the frame of the tidal bulge, with a period around Td/H1101527 days for the Earth, and the container or “mantle” frame, with aperiod T s/H1101523.93 h for the Earth. As a first step toward the full study of the interaction between the elliptical instabilityand the precession, we consider the particular case where thetriaxial ellipsoid is fixed in the precessing frame /H20849T d=Tp/H20850, which allows the theoretical approach to be analytically trac- table. We show in Fig. 1a sketch of this configuration. The paper is organized as follow. In Sec. II, the Poincaré and Busse analytical models are extended to precessing tri-axial ellipsoids. Then in Sec. III, our analysis is validated bycomparison with a numerical simulation. II. ANALYTICAL SOLUTION OF THE FLOW IN A PRECESSING TRIAXIAL ELLIPSOID In this section, we consider first the case of an inviscid fluid and extend the Poincaré model to a precessing triaxialellipsoid, which allows us to obtain explicit analytical solu-tions. We then tackle the viscous case in extending the Bussemodel, following the method of Noir et al. 9As sketched in Fig. 1, we consider the rotating flow inside a precessing tri- axial ellipsoidal container of principal axes /H20849a1,a2,a3/H20850.W e define /H20849Ox1,Ox2,Ox3/H20850in the frame of the tidal bulge which is also the precessing frame, such that Oxiis along the prin- cipal axis aiof the ellipsoidal container and /H20849Ox3/H20850is the mantle rotation axis. We note /H9024the imposed mantle angular velocity and use /H9024−1as a timescale. We also introduce the mean equatorial radius Req=/H20849a1+a2/H20850/2, which is used as a length scale. Consequently, the problem is fully described bya/H20850Electronic mail: cebron@irphe.univ-mrs.fr.PHYSICS OF FLUIDS 22, 116601 /H208492010 /H20850 1070-6631/2010/22 /H2084911/H20850/116601/8/$30.00 © 2010 American Institute of Physics 22, 116601-1six dimensionless numbers: the ellipticity /H9280=/H20841a12−a22/H20841//H20849a12 +a22/H20850, the aspect ratio a3/a1, the Ekman number E=/H9263//H9024Req2, where /H9263is the kinematic viscosity of the fluid, and the three components of the dimensionless precession vector /H9024pin the inertial frame of reference, i.e., the angle /H9258between /H9024pand ex3, the angle /H92582between /H9024pandex1, and the angular pre- cession rate /H9024p, which is positive for prograde precession and negative for retrograde precession. A. Inviscid Poincaré tilt-over mode for a triaxial ellipsoid We look for a base flow solution of the incompressible Euler equations, i.e., /H11509u /H11509t+u·/H11612u=−/H11612p−2/H9024p/H11003u, /H208491/H20850 /H11612·u=0 , /H208492/H20850 inside the triaxial ellipsoid in the precessing frame. In this frame, using the ellipsoid equation x12 a12+x22 a22+x32 a32=1 , /H208493/H20850 we transform the ellipsoidal geometry into a sphere with the transformation /H20849xk/H11032/H20850=/H20849xk/ak/H20850k/H33528/H208491,2,3 /H20850and we write the velocity field in this sphere as /H20849Uk/H11032/H20850=/H20873Uk ak/H20874 k/H33528/H208491,2,3 /H20850. /H208494/H20850 As suggested in Ref. 3, we focus on the so-called “simple motions” such that the velocity U/H20849U1,U2,U3/H20850is de- scribed as linear combinations of the coordinates. This hy- pothesis leads to a solid body rotation in the sphere, i.e.,U /H11032=/H9275/H11032/H11003r/H11032, where /H9275/H11032/H20849/H92751/H11032,/H92752/H11032,/H92753/H11032/H20850depends on time a priori .Equation /H208494/H20850then gives the following velocity field in the ellipsoid: U=/H20849Uk/H20850k/H33528/H208491,2,3 /H20850 =/H20873ak ak−1/H9275k+1/H11032xk−1−ak ak+1/H9275k−1/H11032xk+1/H20874 k/H33528/H208491,2,3 /H20850, /H208495/H20850 where permutations k/H33528/H208491,2,3 /H20850are used. Now we have to find the simple motions solution of the Euler equations for the rotational part of the flow, taking intoaccount the no-penetration boundary conditions for its irro-tational part, which leads to the so-called Poincaré flow. Wefollow the method of Noir 31rather than the Lagrangian method of Poincaré which is more laborious. We deduce therotation rate vector /H9275from the velocity field /H208495/H20850 /H9275=1 2/H11612/H11003U=/H20849/H9275k/H20850k/H33528/H208491,2,3 /H20850 =1 2/H20875/H20873ak−1 ak+1+ak+1 ak−1/H20874/H9275k/H11032/H20876 k/H33528/H208491,2,3 /H20850. /H208496/H20850 Note that /H9275/H11032is independent of the space coordinates so that /H9275is uniform. Taking the rotational of Eq. /H208491/H20850, the inviscid equation for /H9275gives the following three scalar equations: d/H9275k dt+/H20849/H9251k+1,k−/H9251k−1,k/H20850/H9275k−1/H9275k+1 =/H9251k−1,k/H9024p,k−1/H9275k+1−/H9251k+1,k/H9024p,k+1/H9275k−1 /H208497/H20850 for permutations k/H33528/H208491,2,3 /H20850, with the coefficients /H9251i,j =2 //H20849/H9257ij+2/H20850=2/H20849/H9257ji+1/H20850//H20849/H9257ji+2/H20850and the different ellipticities /H9257ij=ai2/aj2−1 of the container. The equations for a spheroidal geometry are recovered with a1=a2. We focus on the stationary solutions of the problem. In this particular case, we solve the system /H208497/H20850analytically, which gives /H92751=/H92753 /H925212a32+a22 /H92531/H9024p,1, /H208498/H20850 /H92752=/H92753 /H925212a32+a12 /H92532/H9024p,2, /H208499/H20850 where /H9252ij=/H20849ai2+aj2/H20850//H208492aiaj/H20850=/H9252jiand /H9253i=/H92753/H20849ai2−a32/H20850//H925212 +2/H9024p,3a1a2. The corresponding velocity field writes U1=a1/H92753 /H925212/H20873−x2 a2+2a1/H9024p,2x3 /H92532/H20874, /H2084910/H20850 U2=a2/H92753 /H925212/H20873x1 a1−2a2/H9024p,1x3 /H92531/H20874, /H2084911/H20850 U3=a3/H92753 /H925212/H208732a3/H9024p,1x2 /H92531−2a3/H9024p,2x1 /H92532/H20874. /H2084912/H20850 Note that the choice of /H92753is arbitrary here. Actually, /H92753is determined by the boundary /H20849Ekman /H20850layer and thus has to be determined by a viscous study, following for instance themethod of Busse, 5as described in Sec. II B. Note also that this velocity field is divergent for /H92531=0 or /H92532=0: the inviscid FIG. 1. /H20849Color online /H20850Sketch of the problem under consideration. A hollow solid but deformable spheroid /H20849i.e., the mantle /H20850is filled with liquid and set in rotation at a constant angular velocity /H9024along its axis /H20849Ox3/H20850. The spheroid axis is tilted at the precession angle /H9258and fixed on a rotating table, which rotates at the precession rate /H9024p. Two fixed rollers aligned with /H20849Ox3/H20850allow to transform the spheroid into a triaxial ellipsoid by compression along theaxis /H20849Ox 2/H20850perpendicular to the rotation axis.116601-2 Cébron, Le Bars, and Meunier Phys. Fluids 22, 116601 /H208492010 /H20850study gives two resonances, which correspond to resonance between the frequencies of, respectively, the precessionalforcing and the tilt-over /H20849see Ref. 9for details /H20850. These linear resonances are reached for two specific precession rates /H20849de- pending on the aspect ratio /H20850which are /H9024 p,3=a32−ai2 a12+a22/H92753 /H2084913/H20850 fori/H33528/H208491,2 /H20850. It is clear that oblate ellipsoids /H20849a1,a2/H11022a3/H20850 have their resonance in the retrograde regime /H20849i.e., in the range /H9024p,3/H110210/H20850, whereas prolate ellipsoids /H20849a1,a2/H11021a3/H20850have their resonance in the prograde regime. Finally, compared to the spheroidal case, an important result here is the apparitionof the second resonance created by the equatorial ellipticityof the container. B. Viscous study of the tilt-over mode in a triaxial ellipsoid Following Ref. 5, it is possible to take into account the viscosity in the study of the flow in a precessing triaxialellipsoid. Here, we focus on the equivalent method of Noiret al. 9based on the equilibrium between the inertial torque /H9003i, the pressure torque /H9003p, and the viscous torque /H9003v. Keep- ing the leading terms for these three torques, the generaltorque balance given in Ref. 5for a steady rotating flow q= /H9275/H11003rin the precessing frame within a volume Vwith a surface /H9018writes 2/H20885 Vr/H11003/H20849/H9024p/H11003q/H20850dV/H9003i =−/H20886 /H9018pr/H11003nd/H9018/H9003p +E/H20886 Vr/H11003/H116122qdV/H9003v , /H2084914/H20850 where ris the position vector and nis the unit vector normal to/H9018pointing outward /H20849see Ref. 9/H20850. Now, we have to calcu- late these terms for a triaxial ellipsoid. At first order, the pressure gradient equilibrates the cen- trifugal force, which gives p=1 2/H20858 i=13 /H20900/H20849/H9275i+12+/H9275i−12/H20850xi2−/H20858 j=1 j/HS11005i3 /H9275i/H9275jxixj/H20901. /H2084915/H20850 Hence in the limit of small ellipticities, i.e., at first order in /H9257=1− a3/a1and/H92572=1− a2/a1, the pressure torque writes /H9003p=−/H20886pr/H11003nd/H9018=/H20849/H9003p,k/H20850k/H33528/H208491,2,3 /H20850=I/H20900/H20849/H9257−/H92572/H20850/H92752/H92753 −/H9257/H92751/H92753 /H92572/H92751/H92752/H20901, /H2084916/H20850 where Iis the moment of inertia in the spherical approxima- tion. Note that the expression of Noir et al.9is recovered for the spheroid /H20849i.e.,a1=a2hence /H92572=0/H20850. After little algebra, the precessional torque simply writes /H9003i=I/H9024p/H11003/H9275. /H2084917/H20850Finally, Eqs. /H2084916/H20850and /H2084917/H20850give/H9003p·/H9275=0 and /H9003i·/H9275=0 and thus with Eq. /H2084914/H20850,/H9003v·/H9275=0. Consequently, the fluid being in a stationary state, there is no differential rotationalong /H9275 /H9275·e3=/H92752. /H2084918/H20850 Indeed, in the rotating frame of the fluid, the angular rate of the container along /H9275is null: the only differential rotation between the fluid and the container is in the equatorial planesuch that no spin-up process occurs. Note that this can alsobe recovered with a boundary layer analysis, in the same wayas Busse, 5which shows that the volumic Ekman pumping is solution of the inviscid bulk equations provided that the so-called solvability condition /H2084918/H20850is verified. According to this equation, also named the no spin-up condition in Ref. 9, the only relative motion between the interior and the boundary isa rotation given by /H9275eq=/H9275−/H9024. Then, we need to calculate the viscous torque due this equatorial differential rotation.This calculation relies on the fact that in order to maintainthe basic stationary state, the torque supplied to the fluid tocounterbalance the viscous friction is given by the decay ratethat would occur if, at a given time, the precession is turnedoff. This decay rate is simply given by the rate at whichenergy is dissipated by the Poincaré mode in a free system att=0. This rate is given by the Greenspan’s theory, 6valid in the frame rotating with the fluid. Thus, this linear solution forthe viscous decay of the spin-over mode in a rotating fluidleads to introduce a new Ekman number E f=E//H9275and a new unit of time t˜=t/H9275, scaled with the fluid rotation rate /H9275. Ac- cording to Greenspan,6the time evolution of /H9275eqin the non- rotating frame is /H9275eq/H20849t˜/H20850=e/H9261rt˜/H20881Ef/H20875cos/H20849/H9261it˜/H20881Ef/H20850/H9275eq/H208490/H20850 − sin /H20849/H9261it˜/H20881Ef/H20850/H9275/H11003/H9275eq/H208490/H20850 /H9275/H20876 /H2084919/H20850 with/H9261r=−2.62 and /H9261i=0.259. Note that strictly speaking in triaxial ellipsoids, the ellipticity modifies the growth rate andeigenfrequency of inertial modes. However, in the limit ofsmall ellipticities we consider here, this modification can beneglected. Using this equation, reintroducing the variables Eandt, the equatorial viscous torque is /H9003 v=I/H20873d/H9275eq dt/H20874 t=0=I/H20881/H9275E/H20900/H9261r/H92751+/H9261i/H92752//H9275 /H9261r/H92752−/H9261i/H92751//H9275 /H9261r/H20849/H92753−1/H20850/H20901. /H2084920/H20850 Then, the torque balance given by Eq. /H2084914/H20850projected onto the rotation axis of the fluid /H9275/H20851the no spin-up condition /H2084918/H20850/H20852, as well as onto the principal axes ex1andex3, yields the following system of equations: /H927512+/H927522=/H92753/H208491−/H92753/H20850, /H2084921/H20850 /H9024p,2/H92753−/H9024p,3/H92752 =/H20849/H9257−/H92572/H20850/H92752/H92753+/H20873/H9261r/H92751/H927531/4+/H9261i/H92752 /H927531/4/H20874/H20881E, /H2084922/H20850116601-3 Tilt-over mode in a precessing triaxial ellipsoid Phys. Fluids 22, 116601 /H208492010 /H20850/H9024p,1/H92752−/H9024p,2/H92751=/H92572/H92751/H92752−/H9261r/H927531/4/H208491−/H92753/H20850/H20881E. /H2084923/H20850 The supplementary terms compared to Ref. 5or Ref. 9 do not allow simplifying the system of equations into onlyone and the full system has to be solved numerically to ob-tain the rotation axis components of the fluid. This nonlinearsystem can be solved in an efficient way with a continuationmethod /H20849successive perturbations on the a 2axis /H20850starting from Busse’s solution in a spheroid. An example is shown inFig. 2where the solution in the spheroidal geometry /H20849the case /H9263=10−5m2/s in Fig. 3 of Ref. 9, which gives an Ekman number of E=3/H1100310−5/H20850is compared to a slightly deformed triaxial ellipsoid /H20849/H9280=0.03 /H20850with the same ratio a3/a1.I tc a n be noticed that even a very small tidal deformation /H9280radi- cally changes the obtained solution. III. NUMERICAL AND EXPERIMENTAL VALIDATION Our purpose here is to validate and test the range of validity of our analytical solution by comparison with nu-merical simulations of the full nonlinear Navier–Stokesequations in a precessing ellipsoid. A. Numerical resolution We consider the rotating flow inside a triaxial ellipsoidal container of principal axes /H20849a1,a2,a3/H20850, as sketched in Fig. 1. We work in the precessing frame of reference. Starting from rest, a constant tangential velocity U/H208811−/H20849x3/a3/H208502is imposed from time t=0 all along the outer boundary in each plane of coordinate x3perpendicular to the rotation axis /H20849Ox3/H20850, where Uis the imposed boundary velocity at the equator. We intro- duce the timescale /H9024−1by writing the tangential velocity along the deformed outer boundary at the equator U=/H9024Req. We then solve the Navier–Stokes equations with no-slipboundary conditions, taking into account a Coriolis force as-sociated with the precession /H9024 p, i.e., in the frame /H20849Ox1,Ox2,Ox3/H20850of the tidal bulge, which is also the precess- ing frame, we solve/H11509u /H11509t+u·/H11612u=−/H11612p+E/H9004u−2/H9024p/H11003u, /H2084924/H20850 /H11612·u=0 . /H2084925/H20850 In this work, the range of parameters studied is E/H1135010−3and/H9280/H113490.32. Once a stationary or periodic state is reached, we determine the rotation rate in the bulk of thefluid, i.e., outside the viscous boundary layer. To do so, weintroduce an interior homothetic ellipsoid, in a ratio /H9260, and we define the bulk rotation rate /H9275as the mean value of rotation rate over this homothetic ellipsoid. FollowingRef. 32, we consider a dimensionless viscous layer thick- ness of /H9254/H9263/H110155/H20881Eand thus choose for the homothetic ratio /H9260/H110151−/H9254/H9263/H110151−5/H20881E. Note that the spheroidal case can be ef- ficiently solved by spectral methods /H20849see, e.g., Ref. 19or Ref. 20/H20850. But for the triaxial ellipsoids we are interested in, there is no simple symmetry. Our computations are thus per-formed with a finite element method, which allows us tocorrectly reproduce the geometry and to simply impose theboundary conditions. The solver and the numerical methodare described in details in Ref. 24. B. Experimental setup The experimental setup has been described in Refs. 30 and33for a precessing cylinder and readers should refer to these papers for more details. For this paper, the experimenthas been slightly modified in order to study the precession ofa spheroid. This allows validating the flow at small Ekmannumbers, which is impossible numerically. Unfortunately,this setup is limited to a spheroidal geometry /H20849a 1=a2/H20850and is not able to validate the theory for a triaxial ellipsoid. Further modifications to the setup, such as two rollers compressingthe spheroid, would be needed to make the equatorial planeelliptical. The homemade spheroid has been obtained by assem- bling two half-spheroidal cavities drilled in solid Plexiglascylinders. The accuracy of the machines /H2084910 /H9262m/H20850ensured that the step between the two parts would be smaller than the Ekman layer /H20849of the order of 300 /H9262ma t E=10−5/H20850. This spheroid of equatorial diameter 17 cm and aspect ratioa 3/a1=0.85 is filled with water and mounted on a motor which is itself located on a rotating platform. The angularvelocities of the spheroid and the platform are stable within0.1% and the precessing angle /H9258between the two axes was varied from 5° to 15° with an accuracy of 0.1°. Pulsed yttrium aluminum garnet /H20849YAG /H20850lasers are used to create a luminous sheet perpendicular to the axis of therotating platform. A particle image velocimetry /H20849PIV /H20850camera is located on the rotating platform, aligned with the axis ofrotation of the spheroid. This allowed to obtain PIV measure-ments in a plane almost parallel to the equatorial plane/H20849x 1,x2/H20850and located at a distance x3=5 cm above it. Since the YAG lasers are not located on the rotating platform, the mea- surement plane is tilted with an angle /H9258with respect to the equatorial plane, which introduces an error of up to 2 cm /H20849for /H9258=15° /H20850in the axial location of the velocity vectors. How- ever, the measured velocity components exactly correspond−10 −8 −6 −4 −2 00.80.850.90.951 Ωp(rpm)ω ε=0 ε= 0.03 FIG. 2. /H20849Color online /H20850Theoretical amplitude of tilt-over mode angular ro- tation rate /H9275for a spheroid /H20849/H9280=0/H20850and for a slightly deformed triaxial ellip- soid /H20849/H9280=0.03 /H20850. The other parameters used in this figure are those of Ref. 9: the rotation rate of the container is /H9024=207 rpm and a1=0.125 m /H20849such that the Ekman number is E=3/H1100310−5/H20850,/H9258=9°, a3/a1=0.96, and /H92582=0°. The spheroidal case corresponds to the plot /H9263=10−5m2/si nF i g .3o fR e f . 9. Note that even a very small tidal deformation significantly changes theobtained solution.116601-4 Cébron, Le Bars, and Meunier Phys. Fluids 22, 116601 /H208492010 /H20850to the x1,x2components of the velocity because the camera is aligned with the axis of the spheroid. There is no distortionof the images at the air-Plexiglas interface /H20849because it is a plane /H20850but there are distortions of the images due to the Plexiglas-water spheroidal interface. These deformations aresmall because the refractive index of the Plexiglas and thewater are close. They were calculated analytically andchecked experimentally using a grid. They are locatedmostly at the boundary of the spheroid and they introduce amaximum error of 15% on the radial displacement of theparticles. This does not bias the measurements because onlythe central region of the velocity field was used for the treat-ment of the data. The PIV images were rotated numericallyin order to remove the background rotation of the flow be-fore being treated by a homemade cross-correlation PIV al-gorithm. In the frame of reference of the spheroid, the two- dimensional velocity field was found to be a nearly uniformtranslation flow whose direction and amplitude vary with theprecession frequency /H9024 p. This mean flow is linked to the equatorial component of the angular rotation rate /H20849/H92751,/H92752/H20850, which creates a uniform translation flow /H20849/H925131x3/H92752, −/H925132x3/H92751/H20850in the plane x3=5 cm. This flow corresponds to the last terms found in Eqs. /H2084910/H20850and /H2084911/H20850in the case a1=a2. The first terms of these equations correspond to the axial angular rotation /H20849/H92753/H20850, which in the frame of reference of the spheroid is equal to /H92753−1 in dimensionless form and is thus small outside of the resonances. The measurements ofthe mean velocity and of the mean vorticity thus give in asimple manner the three components of the rotation vector /H9275. Such measurements have been done for the first time in a precessing spheroid and will be compared to theoretical andnumerical results in the following. C. Validation in the spheroidal case Before dealing with the problem of precessing triaxial ellipsoids, a first step of this work has been to simultaneouslycheck the validity of our numerical tool and the validity ofthe theoretical development of Busse 5over an extended range in Ekman number. To do so, we combine numericaland experimental approaches. In Fig. 3, the theory proposed in Ref. 5is validated over more than three decades of Ekman numbers for two differentangles of precession. The interest of the presented results istwofold. First, they validate our numerical model, which willnow be used to study triaxial ellipsoids. Second, this com-pletes the previous validations of Busse’s theory /H20849e.g., Refs. 9and13/H20850, in particular for rather large Ekman numbers. Note that, as already noted in Ref. 13, the theoretical analysis requires that the angle between the rotation vector of thecontainer and the rotation vector of the fluid is small sinceotherwise the Ekman boundary layer analysis is no longervalid. This could explain the differences between the experi-mental determinations and the theoretical predictions at thehighest precessional forcing angle /H20849 /H9258=15° /H20850. A supplementary confirmation of both the numerical model and Busse’s theory at such Ekman numbers is given in Fig. 4/H20849a/H20850, where the com- ponents of the tilt-over rotation axis are in an excellentagreement for a large range of precession rate. At large pre- cession rates, the inviscid results of Poincaré3are recovered. D. Numerical simulations of precessing triaxial ellipsoids A first series of simulations have been performed for a3/a1=0.86, /H9258=10°, /H92582=45°, E=1 /600,/H9280=0.1, and various precession rates. Results are presented in Fig. 4/H20849b/H20850. An ex- cellent agreement is found between the numerical simula-tions and the analytical viscous solution all along the ex-plored range, which demonstrates the validity of ourextended viscous theory. Note also that the triaxial Poincaréinviscid flow is recovered far from the resonances. As al-ready noted above, an important feature of the triaxial geom-etry is the apparition of a second resonance. As already notedin Sec. II A, according to the Eq. /H2084913/H20850, the two resonances are in the retrograde regime here because a 1,a2/H11022a3. Note also that as already described in the literature /H20849see, e.g., Ref. 9/H20850, the viscosity naturally smoothes the inviscid resonance peaks but also modifies their position. The tilt-over mode is thus well described by our analyti- cal model at relatively large Ekman number and relativelylow precession rate and tidal eccentricity. However, two10−610−510−410−310−2−0.4−0.200.20.40.60.81 Eωiω1(expe.) ω2(expe.) ω3(expe.) ω1(num.) ω2(num.) ω3(num.) (a) 10−610−510−410−310−2−0.500.51 Eωiω1(expe.) ω2(expe.) ω3(expe.) ω1(num.) ω2(num.) ω3(num.) (b) FIG. 3. /H20849Color online /H20850Comparison of the theoretical rotation rate compo- nents of Busse’s solution for a spheroid /H20849continuous lines /H20850with experimental PIV measurements and numerical simulations performed over a large rangeof Ekman numbers for /H9024 p=−0.14 and a3/a1=0.85. /H20849a/H20850For a precession angle of 5°. /H20849b/H20850For a precession angle of 15°.116601-5 Tilt-over mode in a precessing triaxial ellipsoid Phys. Fluids 22, 116601 /H208492010 /H20850types of instabilities can be expected for more vigorous flows: the tidal or elliptical instability at relatively smallEkman and/or large eccentricity /H9280and the precession insta- bility at relatively small Ekman and/or large precession rate.Focusing on the tidal instability, we keep the Ekman numberabove the precession instability threshold and we increasethe eccentricity above the elliptical instability threshold. The appearance and form of the tidal instability in the presence of a rotating tidal deformation aligned with the ro- tation axis have been studied elsewhere. 23,24This case corre- sponds in our notations to the specific configuration /H9258=0. The most well-known mode is the so-called spin-over mode,which appears for low values of tidal bulge rotation. Asshown in the Appendix, a theoretical stability analysis dem-onstrates that this mode may grow on the Poincaré flow.Note that the spin-over mode, which is an unstable mode ofthe tidal instability, and the tilt-over mode, which is the baseflow of precession, are actually identical and hence shouldboth be described by our stationary analytical solution. Thisis indeed the case, as shown in Fig. 5/H20849a/H20850around /H9024 p=0 where numerical and analytical solutions are in good agreement.Moreover, by comparison with the known results of the el-liptical instability in the special case /H9258=0 /H20849see Refs. 23and24for details /H20850, we expect the elliptical instability to lead to an unstationary mode for /H9024pbelow the resonance band of the spin-over and to no elliptical instability for /H9024pabove the resonance band of the spin-over. These results are also re-covered in Fig. 5/H20849a/H20850, where oscillatory motions are indeed observed for /H9024 p/H33528/H20851−0.5;−0.2 /H20852, whereas good agreement is found for /H9024p/H110220 between analytical and numerical results. Finally, a systematic study of the excited mode frequency asa function of the Ekman number for /H9024 p=−0.4 is shown in Fig.5/H20849b/H20850. As expected from the stability analysis presented in the Appendix, the transition from a stationary flow to anunstationary one is found below E/H110151/300. IV. CONCLUSION This paper presents the first analytical solution and the first numerical simulations of the precessing flow inside atriaxial ellipsoid. The extension of the Busse viscous solutionto this more complex geometry is shown to provide an accu-rate description of the stationary flow, which may be desta-bilized by oscillating modes of the tidal and/or precessioninstabilities for more vigorous forcing. Clearly, the completestudy of instabilities in such a system deserves more workand will be the subject of future studies with increased nu-merical power and a new experimental setup. But the study−1 −0.5 0 0.5 1−1−0.500.51 Ωpωi ω1(num.) ω2(num.) ω3(num.) (a) −1 −0.5 0 0.5 1−1−0.500.51 Ωpωi ω1(num.) ω2(num.) ω3(num.) (b) FIG. 4. /H20849Color online /H20850The rotation rate components of the flow in a spher- oid, shown in /H20849a/H20850, are compared in /H20849b/H20850with those in the ellipsoid with the same aspect ratio a3/a1=0.86 /H20849/H9258=10°, /H92582=45°, and E=1 /600 /H20850. The theo- retical rotation rate components of the inviscid Poincaré solution and of itsextension to triaxial ellipsoids are given by the black dashed and dotted lines, assuming that /H92753=/H925212=a12+a22/2a1a2/H20849see the Appendix /H20850. The theoret- ical rotation rate components of the Busse viscous solution and of its exten-sion to triaxial ellipsoids are represented by the continuous lines. The twoinviscid resonances /H92531and/H92532are represented by the vertical black lines. /H20849a/H20850 In the spheroid, the validity of our numerical model is confirmed over alarge range of precession rate by its good agreement with the Busse solution./H20849b/H20850In the ellipsoid with the same aspect ratio but with /H9280=0.1, our analytical solution is in good agreement with the numerical results.−0.5 0 0.5−1−0.500.51 Ωpωi ω1(num.) ω2(num.) ω3(num.) 0 100 200 300 400 500 600 70000.10.20.30.40.50.60.7 1/Ef (b)(a) FIG. 5. /H20849Color online /H20850We consider the case /H9258=10°, a3/a1=0.86, /H92582=0, and /H9280=0.317 /H20849corresponding to /H20849a1+a2/H20850/2=a3/H20850./H20849a/H20850Evolution of the rotation rate components of the flow with the amplitude of precession for a fixed value ofthe Ekman number E=1 /600. The legend is the same as in Fig. 4/H20849b/H20850. Filled symbols correspond to cases where the flow is oscillatory, whereas emptysymbols correspond to a stationary flow. Note that the sign reversal at /H9024 p =0 is due to the two possible symmetric orientations of the spin-over mode of the tidal instability. /H20849b/H20850We now focus on the /H9024p=−0.4 case, where the flow is oscillatory at E=1 /600 as shown in /H20849a/H20850and we study the evolution of the pulsation fof the flow with the Ekman number. The flow is stationary for an Ekman number larger than E=1 /300 and then oscillates at a fixed pulsation f/H110150.67.116601-6 Cébron, Le Bars, and Meunier Phys. Fluids 22, 116601 /H208492010 /H20850of the stationary flow performed here already highlights the fact that even a very small tidal deformation significantlymodifies the resulting flow. This conclusion is directly rel-evant to the dynamics of planetary cores and atmospheres,where precession and tidal deformation simultaneously takeplace. APPENDIX: INSTABILITY OF THE INVISCID POINCARÉ FLOW IN A TRIAXIAL ELLIPSOID The work in Ref. 34has recently studied the stability of a rotating flow with elliptical streamlines in the particularcase where the rotation axis executes a constant precessionalmotion about a perpendicular axis. Using the Wentzel–Kramers–Brillouin /H20849WKB /H20850method, 34they achieve to quan- tify both the influence of this particular precession on theelliptical instability and the influence of the strain on theCoriolis instability for such an unbounded cylindrical baseflow. In this appendix, we consider the stability of the invis-cid Poincaré flow in triaxial ellipsoids against small inviscidperturbative rotations. The theoretical expression of thegrowth rate for perturbations that are linear in space vari-ables /H20849i.e., corresponding to the classical spin-over mode /H20850 can be readily obtained with the same method as in Ref. 35, but with the base flow Uobtained in Sec. II A, using an arbitrary value of /H92753. As reminded in Ref. 25, the most gen- eral perturbation-flow linear in the spatial coordinates is uk−Uk=Ki+1/H20849t/H20850ai ai−1xi−1−Ki−1/H20849t/H20850ai ai+1xi+1, /H20849A1/H20850 where the scalar amplitude Kiof these small inviscid pertur- bative rotations can be written Ki=/H9255kie/H9268t. The total flow u have to satisfy the inviscid vorticity equation. Then, Ubeing the flow at order 0, the flow at first order in /H9280is solution of Mk=0 , /H20849A2/H20850 where k=/H20849ki/H20850i/H33528/H208491,2,3 /H20850andMi sa3/H110033 matrix given by M=/H20900A12 −A21−B12/H9268 C1 B13/H9268 D12 −B23/H9268 C2 D21/H20901, /H20849A3/H20850 where Aij=2/H9024p,ja3 aj/H9253i /H9253j, Bij=ai2+aj2 aiaj, Ci=/H9253i a3ai, Dij=−4/H9024p,i/H9024p,3aj2 /H9253i. The solution flow at first order is nontrivial if det /H20849M/H20850 =0 and the growth rate /H9268is obtained in solving this equation. Actually, the third degree equation det /H20849M/H20850=0 writes in the simple form /H92683+p/H9268=0 and the growth rate is given by/H9268=/H20881−p=/H20881A12B13D21+A21B23D12−C1C2B12 B13B23B12./H20849A4/H20850 Note that the dimensionless base flow of Gledzer and Ponomarev35is recovered for the particular value /H92753=/H925212.I t can also be noticed that the extension of the result of Gledzerand Ponomarev 35to the particular case of a rotating tidal bulge at /H9024p=/H208490,0,/H9024br/H20850given in Ref. 24is recovered: /H9268=/H20881−C1C2 B13B23 =/H20881−/H20849a12−a32+2a1a2/H9024br/H20850/H20849a22−a32+2a1a2/H9024br/H20850 /H20849a12+a32/H20850/H20849a22+a32/H20850. /H20849A5/H20850 Note finally that this method gives the theoretical inviscid growth rate, which has then to be corrected by a viscous surfacic damping term /H9273/H20881E,36where /H9273is a constant /H9273/H110112.62. The threshold is then given explicitly by Ec=1 /H92732A12B13D21+A21B23D12−C1C2B12 B13B23B12. /H20849A6/H20850 For the case presented in Fig. 5, i.e., /H9258=10°, a3 =/H20849a1+a2/H20850/2,/H92582=0, and /H9280=0.317, we predict a critical Ekman number for instability around Ec/H110151/300, and for E=1 /600, we predict an unstable spin-over mode in the range /H9024p/H33528/H20851−0.13;0.12 /H20852, both in good agreement with the numerical results. 1W. Malkus, in Lectures on Solar and Planetary Dynamos (Energy Sources for Planetary Dynamos) , edited by M. R. E. Proctor and A. D. Gilbert /H20849Cambridge University Press, London, 1993 /H20850. 2A. Tilgner, “Precession driven dynamo,” Phys. Fluids 17, 034104 /H208492005 /H20850. 3R. Poincaré, “Sur la précession des corps déformables,” Bull. Astron. 27, 321 /H208491910 /H20850. 4K. Stewartson and P. Roberts, “On the motion of a liquid in a spheroidal cavity of precessing rigid body,” J. Fluid Mech. 17,1 /H208492004 /H20850. 5F. Busse, “Steady fluid flow in a precessing spheroidal shell,” J. Fluid Mech. 33, 739 /H208491968 /H20850. 6H. Greenspan, The Theory of Rotating Fluids /H20849Cambridge University Press, Cambridge, 1968 /H20850. 7W. Malkus, “Precession of the Earth as the cause of geomagnetism,” Sci- ence 160, 259 /H208491968 /H20850. 8J. P. Vanyo, P. Wilde, P. Cardin, and P. Olson, “Experiments on precessing flows in the Earth’s liquid core,” Geophys. J. Int. 121, 136 /H208491995 /H20850. 9J. Noir, P. Cardin, D. Jault, and J. P. Masson, “Experimental evidence of nonlinear resonance effects between retrograde precession and the tilt-overmode within a spheroid,” Geophys. J. Int. 154,4 0 7 /H208492003 /H20850. 10R. Hollerbach and R. R. Kerswell, “Oscillatory internal shear layers in rotating and precessing flows,” J. Fluid Mech. 298, 327 /H208491995 /H20850. 11L. Quartapelle and M. Verri, “On the spectral solutions of the 3-dimensional Navier-Stokes equations in spherical and cylindrical re-gions,” Bull. Astron. 90,1 /H208491995 /H20850. 12A. Tilgner, “Magnetohydrodynamic flow in precessing spherical shells,” Geophys. J. Int. 379, 303 /H208491999 /H20850. 13A. Tilgner and F. H. Busse, “Fluid flows in precessing spherical shells,” J. Fluid Mech. 426, 387 /H208492001 /H20850. 14J. Noir, D. Jault, and P. Cardin, “Numerical study of the motions within a slowly precessing sphere at low Ekman number,” J. Fluid Mech. 437,2 8 3 /H208492001 /H20850. 15M. R. Walker and C. F. Barenghi, “High resolution numerical dynamos in the limit of a thin disk galaxy,” Geophys. Astrophys. Fluid Dyn. 76,2 6 5 /H208491994 /H20850.116601-7 Tilt-over mode in a precessing triaxial ellipsoid Phys. Fluids 22, 116601 /H208492010 /H2085016A. Tilgner, “Non-axisymmetric shear layers in precessing fluid ellipsoidal shells,” J. Fluid Mech. 136, 629 /H208491999 /H20850. 17S. Lorenzani and A. Tilgner, “Fluid instabilities in precessing spheroidal cavities,” J. Fluid Mech. 447, 111 /H208492001 /H20850. 18S. Lorenzani and A. Tilgner, “Inertial instabilities of fluid flow in precess- ing spheroidal shells,” J. Fluid Mech. 492, 363 /H208492003 /H20850. 19D. Schmitt and D. Jault, “Numerical study of a rotating fluid in a spheroi- dal container,” J. Comput. Phys. 197, 671 /H208492004 /H20850. 20C.-C. Wu and P. H. Roberts, “On a dynamo driven by topographic preces- sion,” Geophys. Astrophys. Fluid Dyn. 103, 467 /H208492009 /H20850. 21R. Kerswell and W. V . R. Malkus, “Tidal instability as the source for Io’s magnetic signature,” Geophys. Res. Lett. 25, 603 /H208491998 /H20850. 22L. Lacaze, P. Le Gal, and S. Le Dizès, “Elliptical instability in a rotating spheroid,” J. Fluid Mech. 505,1 /H208492004 /H20850. 23M. Le Bars, L. Lacaze, S. Le Dizès, P. Le Gal, and M. Rieutord, “Tidal instability in stellar and planetary binary system,” Phys. Earth Planet. Inter. 178,4 8 /H208492010 /H20850. 24D. Cébron, M. Le Bars, J. Leontini, P. Maubert, and P. Le Gal, “A sys- tematic numerical study of the tidal instability in a rotating triaxial ellip-soid,” Phys. Earth Planet. Inter. 182,1 1 9 /H208492010 /H20850. 25R. Kerswell, “Elliptical instability,” Annu. Rev. Fluid Mech. 34,8 3 /H208492002 /H20850. 26F. A. Waleffe, “On the three-dimensional instability of strained vortices,” Phys. Fluids 2,7 6 /H208491990 /H20850.27M. Le Bars, S. Le Dizès, and P. Le Gal, “Coriolis effects on the elliptical instability in cylindrical and spherical rotating containers,” J. Fluid Mech. 585, 323 /H208492007 /H20850. 28R. Kerswell, “The instability of precessing flow,” Geophys. Astrophys. Fluid Dyn. 72,1 0 7 /H208491993 /H20850. 29R. Kerswell, “On the internal shear layers spawned by the critical regions in oscillatory Ekman boundary layers,” J. Fluid Mech. 298,3 1 1 /H208491995 /H20850. 30R. Lagrange, P. Meunier, C. Eloy, and F. Nadal, “Instability of a fluid inside a precessing cylinder,” Phys. Fluids 20, 081701 /H208492008 /H20850. 31J. Noir, “Ecoulement d’un fluide dans une cavité en précession: Approches numériques et expérimentales,” Ph.D. thesis, Université Joseph-Fourier,Grenoble 1, 2000. 32J. M. Owen and R. H. Rogers, Flow and Heat Transfer in Rotating Disc Systems, Volume 1: Rotor-Stator Systems /H20849Research Studies/Wiley, Taunton/New York, 1989 /H20850. 33P. Meunier, C. Eloy, R. Lagrange, and F. Nadal, “A rotating fluid cylinder subject to weak precession,” J. Fluid Mech. 599, 405 /H208492008 /H20850. 34M. M. Naing and Y . Fukumoto, “Local instability of an elliptical flow subjected to a Coriolis force,” J. Phys. Soc. Jpn. 78, 124401 /H208492009 /H20850. 35E. B. Gledzer and V . M. Ponomarev, “Instability of bounded flows with elliptical streamlines,” J. Fluid Mech. 240,1 /H208491992 /H20850. 36M. Kudlick, “On the transient motions in a contained rotating fluid,” Ph.D. thesis, Massachusetts Institute of Technology, 1966.116601-8 Cébron, Le Bars, and Meunier Phys. Fluids 22, 116601 /H208492010 /H20850Physics of Fluids is copyrighted by the American Institute of Physics (AIP). Redistribution of journal material is subject to the AIP online journal license and/or AIP copyright. For more information, see http://ojps.aip.org/phf/phfcr.jsp

1.1690151.pdf

Generalized magnetostatic modes around large magnetization motions Giorgio Bertotti, Roberto Bonin, Isaak D. Mayergoyz, and Claudio Serpico Citation: J. Appl. Phys. 95, 7046 (2004); doi: 10.1063/1.1690151 View online: http://dx.doi.org/10.1063/1.1690151 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v95/i11 Published by the AIP Publishing LLC. Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 13 Jul 2013 to 129.174.21.5. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsGeneralized magnetostatic modes around large magnetization motions Giorgio Bertottia) Istituto Elettrotecnico Nazionale Galileo Ferraris, I-10135 Torino, Italy Roberto Bonin Dipartimento di Fisica, Politecnico di Torino, I-10129 Torino, Italy Isaak D. Mayergoyz Department of Electrical and Computer Engineering, University of Maryland, College Park, Maryland 20742 Claudio Serpico Dipartimento di Ingegneria Elettrica, Universita `di Napoli ‘‘Federico II,’’I-80125 Napoli, Italy ~Presented on 8 January 2004 ! The notion of magnetostatic mode is extended to far-from-equilibrium conditions. A general formulation for the calculation of mode frequencies and associated magnetization patterns isobtained by linearizing the Landau–Lifshitz–Gilbert equation around previously derived analyticalsolutions for spatially uniform driven magnetization motions. Analogies and differences withrespect to Walker modes are discussed. © 2004 American Institute of Physics. @DOI: 10.1063/1.1690151 # I. INTRODUCTION A saturated ferromagnetic body can spontaneously devi- ate from uniform magnetization. The spectrum of these per-turbation modes has been thoroughly investigated under vari- ous approximations. 1Short-wavelength spin-wave modes are studied by neglecting magnetostatic boundary conditions atthe body surface. However, these boundary conditions be-come dominant when the mode wavelength is comparable tothe body linear dimensions. Under these conditions, analyti-cal results can still be obtained by neglecting exchangeforces, which is a good approximation for bodies of lineardimensions much larger than the exchange length. The fre-quency spectrum and the space-time structure of the result-ing magnetostatic modes have been accurately studied. 2–4 Amuch more complex situation is encountered when the magnetization of the body is not in the equilibrium saturationstate, but executes large motions driven by a forcing externalfield. Two are then the basic questions to be answered: ~i! under what conditions the driven motion of the magnetiza-tion is uniform in space; ~ii!how one can characterize the spontaneous deviations of the magnetization from the mainuniform motion. The answer to the first question was givenin Ref. 5, where it was shown that under particular symmetryconditions a ferromagnetic body will respond to circularlypolarized radio-frequency fields by spatially uniform, dy-namically stable magnetization modes, termed Pmodes.As a first, partial answer to the second question, it was shown inRef. 6 that generalized spin-wave modes do in fact exist evenfar from equilibrium, as quasiperiodic perturbations of P modes. In this article we discuss in what terms the notion ofmagnetostatic mode can be extended to far-from-equilibriumconditions as well, in the form of spontaneous deviationsfromP-mode motions under conditions where exchange forces play a negligible role. II. EQUATION FOR GENERALIZED MAGNETOSTATIC MODES We consider a nonconducting ferromagnetic particle of spheroidal shape, characterized by uniaxial crystal anisot-ropy along the symmetry axis e i. The particle is subjected to the external field ha(t)5ha’(t)1haiei, whereha’(t)i sa circularly polarized component lying in the ( x,y) plane per- pendicular to ei:ha’(t)5ha’(cosvtex1sinvtey). The mag- netization M(r,t) inside the particle obeys the Landau- Lifshitz-Gilbert ~LLG!equation, with zero normal derivative at the body surface. In terms of the normalized magnetiza-tionm5M/M s(Msis the saturation magnetization !, this equation takes the dimensionless form ]m ]t2am3]m ]t52mˆheff, ~1! where time is measured in units of ( gMs)21~gis the abso- lute value of the gyromagnetic ratio !andais the damping constant. The dimensionless effective field heff~normalized toMs) is given by heff5ha’~t!1hM1~hai1kmi!ei1„2m. ~2! According to our previous assumptions the anisotropy field is equal to kmiei, wheremiis the magnetization component alongeiandk52K1/m0Ms2is the normalized crystal anisot- ropy constant. The exchange field is represented by the term„ 2m, where spatial coordinates in the Laplacian are mea- sured in units of the exchange length lEX5(2A/m0Ms2)1/2. The magnetostatic field hMmust be obtained by solving magnetostatic Maxwell equations with the usual interfaceconditions at the body surface. It was shown in previous articles 5,6that Eqs. ~1!and~2! always admit spatially uniform time-harmonic solutions fora!Author to whom correspondence should be addressed; electronic mail: bertotti@ien.itJOURNAL OF APPLIED PHYSICS VOLUME 95, NUMBER 11 1 JUNE 2004 7046 0021-8979/2004/95(11)/7046/3/$22.00 © 2004 American Institute of Physics Downloaded 13 Jul 2013 to 129.174.21.5. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsarbitrary values of the control parameters ( hai,ha’,v). These magnetization modes, termed Pmodes, represent mo- tions where the magnetization precesses around the particlesymmetry axis in synchronism with the rf field. Therefore,theP-mode magnetization m 0(t) is fully described by its tilt angle u0with respect to eiand by the angle f0by which it lagsha’(t). The angles ( u0,f0) are solutions of the pair of equations: n05(hai2v)/cos u01keff,n025ha’2/sin2u0 2a2v2, where n05avcotf0andkeff5k1N’2Ni(Niand N’are the particle demagnetizing factors !. In addition, one finds that the number of Pmodes associated with given val- ues of the field amplitude and frequency can only be two orfour. Knowledge of explicit analytical formulas for m 0(t) per- mits one to study spontaneous deviations from the P-mode motion in rigorous terms, by linearizing the LLG equationaroundm 0(t). To this end, let us consider the slightly per- turbedPmodem0(t)1dm(r,t), with udm(r,t)u!1. The perturbation must preserve the magnetization magnitude,which means that m 0dm50. With this in mind, let us con- sider the two-dimensional, time-dependent vector basis(e 1(t),e2(t)), where e1points along ( ei3m0(t))3m0(t) ande2alongei3m0(t). One finds: e1(t)5cosu0cos(vt 2f0)ex1cosu0sin(vt2f0)ey2sinu0ei,e2(t)52sin(vt 2f0)ex1cos(vt2f0)ey. This basis is perpendicular to m0 by definition, so it is the natural basis for the representation of the perturbation: dm5dm1(r,t)e1(t)1dm2(r,t)e2(t). By inserting m0(t)1dm(r,t) into Eq. ~1!, by expressing m0in terms of ( u0,f0) and by using the previous equations for e1(t) ande2(t) one obtains the following linearized equation fordm1anddm2: S1a 2a1D] ]tSdm1 dm2D5SdhM2 2dhM1D 1S2avcosu0 2n01N’1„2 n02N’2ksin2u02„22avcosu0DSdm1 dm2D, ~3! where dhM15dhM"e1(t) and dhM25dhM"e2(t) represent thee1ande2components of the magnetostatic field created by the perturbation dm. Equation ~3!is valid for any arbitrary perturbation. In Ref. 6 it was used to extend the notion of spin wave tofar-from-equilibrium conditions and to make a number ofpredictions about the onset of spin-wave instabilities indriven systems. Those spin-wave instabilities exist also forthe systems considered in the present article and shouldeventually be included in the analysis that follows, devotedto the study of magnetostatic effects. Similarly to what occurs at equilibrium, magnetostatic effects will be dominant when we consider perturbationswith wavelength on the order of the linear dimensions of theparticle, in cases where these dimensions are much largerthan the exchange length. Exchange forces can then be ne-glected, i.e., the Laplacian terms in Eq. ~3!can be dropped. In addition, although the properties of Pmodes crucially depend on the value of the damping constant aeven if a!1, P-mode perturbations can be safely studied under the ap-proximation a.0, as is usually done for normal modes in linear systems. Finally, we limit our analysis to the particularcase of a spherical particle with vanishing crystal anisotropy,i.e., k50, which, as we will see, greatly simplifies the treat- ment of boundary conditions. Let us assume that all timedependencies are harmonic with angular frequency vM, i.e., dm5am(r)exp(ivMt) and dhM5ah(r)exp(ivMt). Under these assumptions and approximations, Eq. ~3!is reduced to the much simpler form SivM vH 2vHivMDSdm1 dm2D5SdhM2 2dhM1D, ~4! where vH[n02N’5(hai2v)/cos u02Ni. The formal similarity of Eq. ~4!to the law governing the usual Walker modes is striking. Yet, there is a substantial difference: thislaw holds now in the time-dependent basis ( e 1(t),e2(t)). Therefore it describes an anisotropic constitutive law withtime-dependent anisotropy directions controlled by theP-mode orientation m 0(t). The correct way to solve magne- tostatic Maxwell equations for this medium with time-dependent properties is by passing to a new reference frame,rotated at the angular frequency varoundei, in which both theP-mode direction m0and the ( e1,e2) basis become sta- tionary. We take the new rotating Cartesian axes along(e x8(t),ey8(t),ei), withey8(t)[e2(t) andex8(t)5e2(t)3ei. The basis ( e1,e2,m0) is obtained from ( ex8,ey8,ei) by a ro- tation of amplitude u0around the e2axis. We will denote by (x8,y8,z) the Cartesian coordinates along the ( ex8,ey8,ei) axes, respectively. In the new reference frame, the aniso-tropic properties of the medium become time independent.However, there is a price to pay for this: the original LLGequation has to be modified in order to introduce the appro-priate convection terms associated with the change of refer-ence frame. More precisely, in the rotating frame the simpli-fied equation previously derived from Eq. ~3!, i.e., Eq. ~4!, takes the form SivM vH 2vHivMDSdm1 dm2D2v] ]f8Sdm1 dm2D5SdhM2 2dhM1D,~5! where f8is the azimuthal angle around eimeasured with respect to ex8. The connection between f8and the azimuthal angle fdefined in the laboratory frame with respect to exis f5f81vt2f0. The study of generalized magnetostatic modes is reduced to the joint solution, in the ( x8,y8,z) frame, of Eq. ~5!and of the equation for the magnetostatic potential associated with the perturbation magnetostatic field,i.e., dhM5„cM. III. GENERALIZED WALKER MODES We will discuss one particular family of Walker-type nonuniform solutions. The structure of Eq. ~5!prompts us to look for solutions of the form dm1,25a1,2~r,u!exp~2ipf8!exp~ivMt!, ~6! where (r,u,f8) are the spherical coordinates associated with (x8,y8,z) andpis an integer.There is no a priorireason why solutions with this symmetry should exist. We will verify a7047 J. Appl. Phys., Vol. 95, No. 11, Part 2, 1 June 2004 Bertotti et al. Downloaded 13 Jul 2013 to 129.174.21.5. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsposteriori when this is indeed the case. By substituting Eq. ~6!into Eq. ~5!and by inverting the ensuing matrix relation one finds Sdm1 dm2D5SxHixM8 2ixM8xHDSdhM1 dhM2D, ~7! where xH5vH vH22vM82, ~8! xM85vM8 vH22vM82 andvM85vM1pv. Equation ~7!describes a vector relation in the (e1,e2,m0) basis, written in two-dimensional form thanks to the fact that the m0component of dmis zero by definition. Let us transform Eq. ~7!to the (ex8,ey8,ei) basis by making use of the known relationship between the(e 1,e2,m0) and (ex8,ey8,ei) bases. To first order in u0, which is a good approximation whenever the P-mode motion is not too large, the result reads Sdmx8 dmy8 dmiD5SxH ixM8 2xHsinu0 2ixM8 xHixM8sinu0 2xHsinu02ixM8sinu00D 3SdhMx8 dhMy8 dhMiD. ~9! Equation ~9!provides the desired anisotropic constitutive law which can be used in magnetostatic Maxwell equationsto obtain the equation obeyed by the magnetostatic potential cM. One finds that the potential satisfies the equation ~11xH!S]2cM ]x821]2cM ]y82D1]2cM ]z222xHsinu0]2cM ]x8]z50 ~10! inside the particle and Laplace equation outside of it. Equa- tion ~10!is similar to the equation for ordinary Walker modes, except for the additional anisotropic term propor-tional to sin u0. Also the interface conditions at the particle surface are the same as the ones for Walker modes, despitethe fact that we are solving the problem in the rotating frame(e x8,ey8,ei). This is because the frame rotation has no effect on the spherical shape of the particle. One can verify that theparticular family of Walker-type potentials, given inside theparticle by the expression cM5Ap~x82iy8!p11, ~11!p51,2,... is a solution of the problem we are considering @in Walker’s notation,2these would be modes ( m,m,0), with m[p11]. For this family of solutions hMx85~p11!Ap~x82iy8!p,hMy852i~p11!Ap~x82iy8!p, ~12! hMi50. By substituting these expressions into Eq. ~9!one obtains the corresponding components of dmand eventually dm1,dm1 by passing to the basis ( e1,e2,m0). All magnetization com- ponents are proportional to ( x82iy8)p5(rsinu)pexp (2ipf8), in agreement with Eq. ~6!. This confirms that we obtained indeed admissible solutions to the problem. The eigenfrequencies associated with the modes de- scribed by Eqs. ~12!are obtained by noting that vM85vM 1pvis the quantity playing the role of Walker eigenfre- quency in Eqs. ~7!and~8!. Therefore, Walker’s results for (m,m,0) modes2directly apply to vM8, that is, vM82vH 5(p11)/@2(p11)11#. This is equivalent to vM5hai2v cosu02Ni2pv1p11 2~p11!11. ~13! It is interesting to see if the generalized magnetostatic modes are indeed reduced to the corresponding ordinary Walkermodes in the limit of vanishing P-mode motions, i.e., cos u0!1. In this limit, the Pmode is reduced to static satu- ration.We see from Eq. ~6!that the mode time dependence in the laboratory frame is of the form exp @i(vM1pv)t#, because f85f2vt1f0. According to Eq. ~13!, this is the correct Walker-mode frequency apart from a shift by the amount2 v. This shift is due to the fact that the dm1,2components are time dependent in the laboratory frame. In the limit ofvanishing P-mode motion, the basis ( e 1,e2,m0) is reduced to a basis which is rotated at the angular frequency varound theeiaxis. Therefore, when the perturbation is expressed in terms of its time-independent components dmx,yan addi- tional vtterm appears in the phase of the perturbation which cancels the previously mentioned shift. One can also verifythat the space dependence is reduced to that of ordinaryWalker modes. This close correspondence between ordinary and gener- alized Walker modes can be pictured as follows. When theP-mode motion becomes increasingly large, the spatial pat- tern of (m,m,0) Walker modes is slightly affected. However, the plane in which the mode lives is no longer the ( e x,ey) plane but the time-dependent ( e1,e2) plane, whose normal m0executes a precession of angular frequency varoundei under the driving action of the rotating field. ACKNOWLEDGMENTS This work was partially supported by the U.S. Depart- ment of Energy and by MIUR-FIRB contract No.RBAU01B2T8. 1A. Gurevich and G. Melkov, Magnetization Oscillations and Waves ~Chemical Rubber Corp., Boca Raton, FL, 1996 !. 2L. Walker, Phys. Rev. 105, 390 ~1957!. 3L. Walker, J. Appl. Phys. 29, 318 ~1958!. 4P. Fletcher and R. Bell, J. Appl. Phys. 30,6 8 7 ~1959!. 5G. Bertotti, I. Mayergoyz, and C. Serpico, Phys. Rev. Lett. 86,7 2 4 ~2001!. 6G. Bertotti, I. Mayergoyz, and C. Serpico, Phys. Rev. Lett. 87, 217203 ~2001!.7048 J. Appl. Phys., Vol. 95, No. 11, Part 2, 1 June 2004 Bertotti et al. Downloaded 13 Jul 2013 to 129.174.21.5. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions

1.3067416.pdf

Keeping track of the Earth's magnetic field L. R. Alldredge Citation: Physics Today 5, 11, 8 (1952); doi: 10.1063/1.3067416 View online: http://dx.doi.org/10.1063/1.3067416 View Table of Contents: http://physicstoday.scitation.org/toc/pto/5/11 Published by the American Institute of PhysicsKEEPING TRACK OF THE EARTH'S MAGNETIC FIELDFig. 1 By L. R. Alldredge MAN has been studying the magnetic field of the earth for centuries. The existence of this field of force must be deduced by the use of scientific instru- ments since, unlike gravity, it is not detectable by our human senses. Measurements of the earth's magnetic field have been made from deep down in mines up to nearly one hundred miles above the surface. The discovery of the magnetic ore magnetite, often called lodestone, paved the way for the development of the magnetic compass, which has been used by navi- gators since the eleventh century. It was discovered that a piece of lodestone mounted on a block of wood and floated in water always indicated a particular direction at a given location. It was later found that such a compass pointed true north only in certain places. At other localities it would point a little to the east or west of true north. This error with respect to true north has become known as the magnetic declination or varia- tion of the compass. It was not discovered until the 16th century that magnetic forces tended to tip a mag- netized needle from the horizontal. The angle between the final position of a magnetized needle free in space and the horizontal plane is called the dip angle or angle of inclination. The general characteristics of this invisible field of force gradually became known and in the year 1600William Gilbert published his famous volume De Magnete pointing out that the earth itself acts as though it were a great spherical magnet as shown schematically in Fig. 1. At any point on or near the earth and at a given in- stant of time the magnetic field has a definite direction and magnitude. The field points in the direction in which an isolated north magnetic pole would tend to move, and the magnitude of the field is measured by the strength of its force on such a pole. At each point, the magnetic field can be completely described by giving the magnitude, declination, and the inclination, or by giving the vertical and horizontal component of the field and the declination, or by giving the vertical, true north, and true east components. For the past three hundred years, geomagneticians have been mapping the characteristics of this field with great success. During the same period many investiga- tors have also considered the problem of the origin of the earth's magnetic field, and while several new theo- ries have been proposed which may ultimately lead to a satisfactory explanation the answer still remains ob- scure. L. R. Alldredge, chief of the electricity and magnetic research division of the U. S. Naval Ordnance Laboratory, Silver Spring, Maryland, has served NOL as a physicist since 1945. PHYSICS TODAYCharacteristics of the Field If the earth's magnetic field could be mapped once and these values were to remain unchanged for all fu- ture time, the problem would indeed be a rather simple one. A complete, detailed declination survey would re- sult in magnetic navigation maps which could be as ap- propriately used in the year 2000 as today. Unfortunately this is not the case. The sparse data which are available near the magnetic poles, where the horizontal component of field is small, indicate that the places where the horizontal component of field is zero, known as the magnetic dip poles, shift slowly in time. This means that in the polar regions large annual changes in declination will occur. It would be quite possible for a compass to reverse its direction during one year at a point near a magnetic dip pole. In some locations quite far removed from the magnetic poles, where more detailed and accurate data are available, the direction in which the compass points changes as much as one quarter of a degree per year. Offhand this may appear to be a rather small rate of change, but it requires that surveys be made regularly in order to maintain the required accuracy of the magnetic naviga- tion charts. These long-period changes are called secular changes. In addition to the slow change with time there are much faster changes which add to the difficulty of making a coherent survey, but which do not result in permanent changes tending to make magnetic charts obsolete. There are more or less regular variations as- sociated with the positions of the sun and moon. There are also irregular variations which occur during mag-netic storms. Except for very severe storms, these rela- tively short period variations have amplitudes which arc too small to disturb a navigator using a magnetic compass, but they are large enough to cause difficulty to the geomagnetician gathering data for the construc- tion of magnetic secular change charts. The irregular variations which occur during magnetic storms may continue for several days. These variations contain energy distributed over a broad frequency range, with the amplitude decreasing at the higher frequencies. Most measurements of this type have failed to record variations which occur faster than one cycle per second. Below this frequency limit the amplitude of disturb- ances increases with latitude, indicating a dependence upon ionospheric and auroral activity. As an example of magnetic activity of this type, a pulse having a dura- tion of 1 second and an amplitude of 0.1 gamma (1 gamma = 10~5 oersteds) occasionally appears superim- posed upon the ambient vertical component of approxi- mately 44,000 gamma at Tucson, Arizona. If sufficiently sensitive detectors are used, it is found that magnetic field variations occur throughout the audio range and up through the radio frequency spec- trum. These have been studied extensively as unwanted radio disturbances above 10,000 cycles per second. Very little is known about the natural magnetic field varia- tions in the range from 1 to 10,000 cycles per second. There are several research teams engaged in exploring this region at the present time, so conditions may soon be clarified. It appears that most of the energy in all but the lowest part of this frequency band originates in atmospheric storm centers. Since these centers are con- centrated in the equatorial areas, it is expected that Fit- - ADAK. ALASKA 48— -MAGNITUDE OF EARTH'S TOTAL MAGNETIC FIELD ADAK. ALASKA TO KWAJALEIN ATOLL -'•--, « 40-—— MAGNITUDE OF EARTH'S- TOTAL MAGNETIC FIELDDATA TAKEN FROM >945 HYDROGRAPHIC CHARTS 32 28 24 NORTH LATITUDE (DEGREES)FLIGHT LINE 1v KWAKE AJALEHAD IDW INK AY NOVEMBER 195210 intensities will fall off rapidly with increased latitude. It seems certain that the major causes of variations in the magnetic field other than the long period secular changes are external to the earth's crust, whereas the sources of the main part of the earth's field and the causes of the secular changes are deeply rooted within the earth. In searching for the origin of the main part of the earth's field, much can be learned from the experimen- tal approach of carefully charting the field components over the surface of the earth and studying the secular changes of these components. Magnetic charts of this type showing the declination of the compass are of most interest to navigators. A knowledge of the secular change pattern will permit the extrapolation of present day charts for use in subsequent years. When gathering data to be used in constructing magnetic charts, the short period field variations must be understood so their effects can be eliminated from the resulting charts. The short period fluctuations play only a nuisance role in chart making. The utility of magnetic declination maps in the sci- ence of navigation is obvious, but the practical value of the other elements of the earth's magnetic field (such as its vertical and horizontal components) is not so evident. For a long time, these components were only of academic interest. Later, component measurements were used in geophysical prospecting in the search for magnetic ores or possible oil-bearing substructures. More recently, with the advent of airborne total field magnetometers, total field intensity maps have been used in airborne geophysical surveys. It is conceivable that in the future several of the magnetic field com- ponents will be used in routine navigation procedures. When the amplitudes of any of the magnetic elements are plotted for measurements made on the surface of the earth or at a low altitude, many localized irregu- larities appear which may be much larger than the time variations mentioned earlier. The amplitude and extent of such space perturbations ("anomalies) depend on the magnetic characteristics of the surrounding geology. The degree to which these anomalies are preserved in chart making depends upon the scale and desired use of the resulting chart. In most geophysical surveys where detailed geology is of interest the anomalies are the im- portant parts of the resulting map. Data taken for a small scale navigation chart are purposely smoothed, leaving in only broad regional effects. Fig. 2 shows an example of the degree of smoothing which may occur in compiling world-wide charts. The dotted line was taken from a recent magnetic chart. The solid line is the actual field which was measured during a flight from Adak, Alaska, to Kwajalein using an airborne magne- tometer. This flight was made at an average altitude of 1000 feet and the ocean depths ranged from 2000 to 3000 fathoms. The many anomalies which appear would be reduced in amplitude if the flight had been made at a higher altitude. The general divergence between the dotted and solid curves in the vicinity of Adak indicates a need for new data to improve the charts.Existing Instruments Most of the instruments that have been used to measure the relatively higher frequency time variations are of the induction coil type. Such instruments indi- cate the rate of change of field and their response must be integrated if the field is desired. Various filter and recording circuits are used and no knowledge of the absolute value of the field can be obtained. They are straightforward and are based on well known principles and will not be discussed further here. The instruments used to obtain suitable data for chart making are of more interest and are therefore discussed in more detail. Nearly all magnetic field measurements before World War II were made using either rotating coils or the de- flection and oscillations of small magnets. Many of these instruments are still best for certain types of field measurements. All magnetic components can be measured very ac- curately on land with these instruments, and many of them have been adapted for less accurate work at sea in nonmagnetic ships. Ordinary floating and gimbaled compass needles are, of course, in this class and are used in aircraft as well as at sea for very rough meas- urements. Component-measuring instruments of this type have not been adapted very successfully for use in aircraft, although a few insensitive surveys have been made with their help. During World War II a new type of magnetometer was developed. The sensitive mechanism in this magne- tometer is a thin core of high-permeability material which is driven to saturation by an applied alternating magnetic field. Harmonics of the applied field are de- veloped which are proportional to the magnetic field to be measured along the core axis. Three of these detecting elements are mounted mutu- ally perpendicular. Two of them (called orienting induc- tors) control servo motors so that the third one is al- ways maintained with its axis parallel to the direction of the earth's field. Since the magnetometer is stabilized by the earth's magnetic field itself, it is practically in- dependent of the motion of the vehicle carrying it and is readily usable in aircraft. This type of magnetometer has become known as the magnetic airborne detector and is commonly called the MAD. Several different in- struments which use this self-orienting saturable re- actor principle have been developed by government and private groups. During the war the MAD was used to locate sub- marines by detecting their magnetic fields. Near the close of hostilities the MAD took its place as an im- portant geophysical survey instrument. Since the war, private industry and government agencies have flown millions of miles to obtain total field intensity contour maps of areas which were of interest because of oil or mineral possibilities. The advent of these successful airborne magnetome- ters brought clearly into focus the desirability of mak- ing world-wide magnetic surveys from the air rather than in slow-moving surface craft. This need was fur- PHYSICS TODAYFig. j. Universal magne- tometer orienting detector mechanism Fig. 4. Orienting detector mechanism pendulously sus- pended ther emphasized by the fact that long-range planes could quickly cover large ocean areas which had never been covered before or at best were last surveyed be- fore 1929 when the last nonmagnetic ship "Carnegie" was destroyed by explosion in Samoa. The MAD measured only the magnitude of the earth's total field. This single bit of information about the earth's field is valuable and has been used exten- sively for geophysical prospecting. It is not, however, adequate for making world magnetic charts. The way in which the data are taken does not permit the calcula- tion of inclination, declination, or any of the component magnetic intensities. Improved Instruments Recently a new instrument has been completed which will permit the continuous recording during flight of all the data required for complete magnetic mapping. Basi- cally an orienting saturable reactor magnetometer like the MAD instruments, it is built in such a way that the orientation angles of the detector coil, when it is slaved to point along the field vector, can be recorded auto- matically. In this way the entire field vector, amplitude and direction, is determined so that all of the desired components can be computed. This instrument is called a universal airborne magnetometer. Fig. 3 is a photograph of the orienting detector mechanism. The three mutually perpendicular saturable reactor units are contained in the roughly cylindrically shaped part at the very bottom of the picture. This cylindrical part is gimbally mounted so that it can berotated about its own axis and about a second axis parallel to the center line of the entire mechanism. Servo motors located at the top portion of the mecha- nism are connected through reduction gearing to con- centric shafts on the center line of the mechanism to turn the cylinder about these gimbal axes. The servo motors are controlled by the outputs of the two orient- ing inductors in the cylindrical part to orient the third inductor parallel with the earth's field. Synchro trans- mitters located adjacent to the servo motors are geared to the gimbal axes to permit measuring the instantane- ous direction of the earth's field with respect to the mechanism. The entire orienting detector mechanism is pendu- lously suspended, as shown in Fig. 4, to provide its own vertical reference. The pendulum is damped by means of a baffle immersed in a viscous silicone oil. Orienta- tion of the mechanism with respect to geographic me- ridians is determined by means of astral observations. The three-dimensional Helmholtz coil system which is centered on the saturable inductor elements is used to compensate for permanent magnetism in the aircraft. The position of the aircraft over the surface of the earth is determined by various combinations of dead reckoning, celestial observations, Radar, Loran, Shoran, visual observation, and terrestrial photography. The instrument just described is now being flight tested and there are already assurances that the result- ing airborne measurements will attain approximately the same accuracy obtainable by the older point-by- point methods used in nonmagnetic ships. Components NOVEMBER 1952 n12 should be accurate to within 100 gamma except in the general vicinity of the earth's magnetic poles. In order to make magnetic measurements from ships, the initial expenditure for equipment is large. Special nonmagnetic ships, such as the "Carnegie" and the "Research", which has been constructed in England but is not yet fitted out, must be built. Fortunately, mod- ern airplanes are constructed principally of aluminum, which is nonmagnetic, although there are many parts of an airplane such as motors and steel torque tubes which can cause difficulty with magnetic measurements. Many modern military planes with their armor and dc-oper- ated motors with ground return are unfit to carry a sensitive magnetometer. Some aircraft can, however, without great difficulty, be modified into suitable ve- hicles for magnetic work. In a few cases it has been possible to install the magnetometer detector in a ply- wood extension of the tail-cone section of the aircraft or on a wing-tip. An early MAD tail cone installation is shown in Fig. 5. The new instrument is, however, much larger than the wartime MAD and because of its pendulum mounting has required a carefully compen- sated inboard installation. The new instrument has re- cently been installed in a Navy P2V air craft in which it is now being tested. An earlier model was installed in a B-29 aircraft and has been used successfully for nearlv 3 years.Comparison of Survey Methods A nonmagnetic ship using conventional magnetic in- struments normally obtains data at stations separated by 200 or 300 miles. The "Carnegie" occupied 6000 declination stations during seventeen years of service. Because of the secular changes occurring during this period the problem of reducing all of the data to a common time base was an extremely burdensome task. Several cooperating airplanes could complete a world- wide survey rapidly enough to avoid this time-consum- ing problem. The airborne techniques which are being developed will give continuous data along a flight line. Furthermore, airborne measurements can be made over any part of the earth and at various altitudes. Because of the differences in the type of data ob- tained by nonmagnetic ships and by proposed airborne techniques, it is impossible to make an accurate com- parison of costs for the two methods. Taking into account the difference in the speeds of the ship and plane and the time required for mainte- nance and bad weather delays, assuming that each is equipped with the same instruments, it is estimated that from ten to fifteen ships would be required to ob- tain as much data as a single plane. Cost estimates for the two methods would also greatly favor the airborne measurements. Fig. 5. PBY with tail cone housing for magnetometer {all figures courtesy Naval Ordnance Laboratory)

1.4884419.pdf

Voltage induced magnetostrictive switching of nanomagnets: Strain assisted strain transfer torque random access memory Asif Khan, Dmitri E. Nikonov, Sasikanth Manipatruni, Tahir Ghani, and Ian A. Young Citation: Applied Physics Letters 104, 262407 (2014); doi: 10.1063/1.4884419 View online: http://dx.doi.org/10.1063/1.4884419 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/104/26?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Acoustically assisted spin-transfer-torque switching of nanomagnets: An energy-efficient hybrid writing scheme for non-volatile memory Appl. Phys. Lett. 103, 232401 (2013); 10.1063/1.4838661 Field assisted spin switching in magnetic random access memory J. Appl. Phys. 99, 08H708 (2006); 10.1063/1.2172578 Critical-field curves for switching toggle mode magnetoresistance random access memory devices (invited) J. Appl. Phys. 97, 10P507 (2005); 10.1063/1.1857753 Estimation of thermal durability and intrinsic critical currents of magnetization switching for spin-transfer based magnetic random access memory J. Appl. Phys. 97, 10C707 (2005); 10.1063/1.1851912 Novel magnetostrictive memory device J. Appl. Phys. 87, 6400 (2000); 10.1063/1.372719 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 131.156.59.191 On: Thu, 11 Sep 2014 12:15:24Voltage induced magnetostrictive switching of nanomagnets: Strain assisted strain transfer torque random access memory Asif Khan,a)Dmitri E. Nikonov, Sasikanth Manipatruni, Tahir Ghani, and Ian A. Y oung Components Research and Portland Technology Development, Intel Corp., Hillsboro, Oregon 97124, USA (Received 7 April 2014; accepted 11 May 2014; published online 2 July 2014) A spintronic device, called the “strain assisted spin transfer torque (STT) random access memory (RAM),” is proposed by combining the magnetostriction effect and the spin transfer torque effectwhich can result in a dramatic improvement in the energy dissipation relative to a conventional STT-RAM. Magnetization switching in the device which is a piezoelectric-ferromagnetic heterostructure via the combined magnetostriction and STT effect is simulated by solving theLandau-Lifshitz-Gilbert equation incorporating the influence of thermal noise. The simulations show that, in such a device, each of these two mechanisms (magnetostriction and spin transfer torque) provides in a 90 /C14rotation of the magnetization leading a deterministic 180/C14switching with a critical current significantly smaller than that required for spin torque alone. Such a scheme is an attractive option for writing magnetic RAM cells. VC2014 AIP Publishing LLC . [http://dx.doi.org/10.1063/1.4884419 ] Scaling of complementary metal-oxide-semiconductor (CMOS) electronics proceeded according to Moore’s law over the last four decades.1The International Technology Roadmap for Semiconductors projects the development of CMOS for the next 13 yr, it also lists possible alternative devices going beyond it. Currently, many beyond CMOSoptions are under active research, 2,3and spintronic devices are promising options.4,5While the control of the magnetiza- tion through a magnetic field in magnetic random-access-memories (RAMs) or through the spin polarized current in spin transfer torque (STT) random-access-memories 6,7has been the main-stay in the field of spintronics, of late, electricfield control of the spintronic device functionalities via dif- ferent magnetoelectric effects has received significant inter- ests. Benchmarking of spintronic devices in different logicarchitectures shows that magnetoelectric spintronics can be much more energy efficient than its STT or magnetic field counterparts. 4This is primarily due to the fact that in an STT-RAM, the critical current density for switching is large (/C241 MA/cm2) resulting in significant dissipation of the order of/C24106kBT,kB, and Tbeing the Boltzmann constant and the room temperature. To date, several mechanisms for the coupling between the electric field and the magnetic order in multiferroicmagnetic-ferro/piezoelectric heterostructures have been reported, 8–13,15–23namely, charge modulation (such as in Fe-BaTiO 3),14–16exchange interactions (for example, in CoFeB-BiFeO 3),17,18and magnetostriction (for example, in La0.7Sr0.3MnO 3/PMN-PT,19CoFe-BaTiO 3,20,21or FeRh- BaTiO 322). Logic and memory applications require an 180/C14 reversal of the magnetization.18,23Furthermore, for the readout of the magnetic state, it is required to build a mag- netic tunnel junction (MTJ) on top of the nanomagnetof the multiferroic heterostr ucture. Hence, from a fabrica- tion point of view, it is desirable to achieve the 180 /C14switching by applying an out-plane electric field on theferro/piezo-electric layer. While such 180 /C14reversal of the magnetization by an applied electric field has been pre-dicted theoretically based on multiferroic interface cou- pling and inter-layer magnetic exchange coupling in heterostructures, such as Fe/Au/Fe/PbTiO 3,24experimen- tally 180/C14reversal has been reported only for in-plane elec- tric fields.12On the other hand, a robust 90/C14switching of the magnetization in magnetoelectric heterostructures atroom temperature compatible with the MTJ structure has been reported experimentally in recent publications, such as in CoFe-BaTiO 3via magnetostriction.15,16 For magnetization reversal, the dynamics of magnetiza- tion rotation, which is governed by the non-linear Landau- Lifshitz-Gilbert (LLG) equation, requires a minimum currentof certain spin polarization to be flowed through the nano- magnet, which dislodges the magnetization from its initial easy axis direction and rolls it over the energy barrier alongthe hard axis overcoming the Gilbert damping effect, thereby switching the direction of magnetization by 180 /C14with respect to its initial direction. The value of the critical cur-rent depends on the initial angle of the magnetization with respect to the easy axis h 0,which typically depends on the thermal fluctuations. If h0(<90/C14) is large, the critical current for magnetization reversal could be small. Hence, if the mag- netization of the free layer in an STT-RAM is already rotated by an angle close to 90/C14by an external mechanism, then a spin polarized current significantly smaller than the critical current can switch the magnetization by another 90/C14leading to a complete magnetization reversal, thereby, reducing theenergy dissipation. Based on this, one can imagine a spin- tronic device combining magnetostrictive and spin transfer torque based switching, where each of the two mechanismsprovides in a 90 /C14switching of the magnetization leading a complete 180/C14switching. In this paper, we analyze such a strain assisted STT-RAM using the Landau-Lifshitz-Gilbertequation incorporating the effect of stochastic thermal noise and show that such a scheme does produce deterministic switching at a very low switching current.a)Present address: Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, California 94720. Electronic mail: asif@eecs.berkeley.edu. 0003-6951/2014/104(26)/262407/5/$30.00 VC2014 AIP Publishing LLC 104, 262407-1APPLIED PHYSICS LETTERS 104, 262407 (2014) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 131.156.59.191 On: Thu, 11 Sep 2014 12:15:24We start by considering a hybrid piezoelectric- ferromagnetic heterostructure shown in Fig. 1. We take a tet- ragonal mono-domain ferroelectric, Pb(Zr 0.2Ti0.8)O3(PZT) film grown on a conducting bottom electrode as the piezo-electric layer on top of which a nanomagnet is lithographi- cally patterned. The material parameters and the thickness of the PZT layer are listed in Table I. The application of voltage between the nanomagnet and the bottom electrode generates an in-plane bi-axial strain in the piezoelectric layer, which is transferred to the magnet. Fig. 1shows the strain generated along x- and y-axis in the piezoelectric ( e xxandeyy, respec- tively). For PZT film with (001) surface orientation (ferro-electric polarization along the 6z-axis), the bi-axial strain is equal in magnitude and sign along the in-plane crystallo- graphic directions, h010iandh100i. The strain components along the h010iandh100idirections are given by e xx¼eyy ¼d31Ez,d31being the piezoelectric coefficient of the PZT layer and Ezbeing the out-of-plane electric field. On the other hand, for PZT with (110) surface orientation, the appli- cation of an out-of-plane electric field creates two different strains along the two in-plane crystallographic directions,h001iandh101i. In such a case, it can be shown that the strain components along the h001iandh101idirections are given by e xx¼(d31þd33)Ez/2ffiffiffi 2p and eyy¼d31Ez/ffiffiffi 2p . spectively. We will later show that the aforementioned dif- ference in the strain distribution results in an interesting dif- ference in the magnetization dynamics of the nanomagnetson PZT films with (001) and (110) surface orientation. Next, we analyze the response of the nanomagnet to a stress pulse. In the presence of a bi-axial stress, the magneticanisotropy of the nanomagnet changes due to the inverse magnetostriction effect. The energy contribution due to the stresses, { r i}, (i/C17xx;yy) making angles, { di} with the unit vector along the direction of the magnetization m!is given by Estrain¼3 2kX iricos2di; (1) where kis the magnetostrictive coefficient of the magnetic material. We take Co 0.6Fe0.4as the nanomagnet material, which has been demonstrated to have a large magnetostric- tive coefficient.25Since kis positive for Co 0.6Fe0.4, a tensile stress favors the alignment of the magnetization along theaxis of stress. Assuming the complete transfer of strain from the piezoelectric layer to the nanomagnet, the stress and thestrain in the nanomagnet are related by r i¼Yei(i/C17xx;yy), Ybeing the Young’s modulus of the magnetic material. The total energy of magnet with a perpendicular magnetic anisot- ropy, Hkupon the application of a biaxial stress is given by Etotal¼EPMAþEstrain¼1 2l0MsHksin2hþEstrain;(2) where Ms,l0, and hare the saturation magnetization of the magnet, the vacuum permeability, and the angle of the mag-netization with respect to the þz axis, respectively. The ani- sotropy field due to the stress is calculated using the following relation: H r/C131!¼@Estrain @m!: (3) The dynamics of the nanomagnet is described by the modi- fied Landau-Lifshitz-Gilbert equation, which is as follows:26 @m! @t¼/C0cl0m!/C2Hef f/C131/C131!hi þam!/C2@m! @t/C20/C21 þIp! eNs;(4) where cis the electron gyromagnetic ratio, ais the Gilbert damping coefficient, Ip!is the component of the vector spin current perpendicular to the magnetization, m!entering the nanomagnet and Nsis the total number of Bohr magnetons per magnet. Hef f/C131/C131!¼Hr/C131!þHPMA/C131/C131/C131!þHN/C131!is the effective mag- netic field and HPMA/C131/C131/C131!andHN/C131!are the fields due to perpendic- ular magnetic anisotropy and stochastic noise, respectively. The noise field, HN/C131!¼Hi^xþHj^yþHk^zacts isotropically on the magnet and hence can be described as27 hHlðtÞi ¼ 0; (5) hHlðtÞHkðt0Þi ¼2akBT l2 0cMsVdðt/C0t0Þdlk; (6) where kBis the Boltzmann constant, Tis the temperature, and Vis the volume of the nanomagnet. In order for the initial conditions of the magnets to be randomized, the initial angle of the magnets follows the relationship: hh2i¼kBT MsVl0HPMA. First, we simulate the magnetization dy- namics of the nanomagnet in the presence of a uniform bi- axial tensile stress, rð¼rxx¼ryyÞ. In order to understand the steady state condition of the magnet under the effect of the uniform bi-axial stress, we note in Eq. (2)that without an applied stress (i.e., r¼0Þ, the total energy is the minimum along6z-axis ( h¼0/C14and 180/C14). For a uniform bi-axial stress, r, the stress energy can be written as Estrain¼3 2krcos2h. Hence, with the increase of the bi-axial tensile stress, the anisotropy energy along the 6z-axis FIG. 1. Hybrid ferromagnetic-piezoelectric structure. Co 0.6Fe0.4and PZT are taken as the ferromagnet and the piezoelectric material, respectively. The material parameters are listed in Tables IandIIrespectively. Upon the application of a positive voltage at the top electric, the strain generated along x- and y-axis in the piezoelectric are exxandeyy, respectively.TABLE I. Material parameters and thickness of the PZT layer. Parameter Value Piezoelectric coefficient d31 180 pm/V [Ref. 31] Piezoelectric coefficient d33 60 pm/V [Ref. 32] Dielectric constant er 500 Thickness dPE 30 nm262407-2 Khan et al. Appl. Phys. Lett. 104, 262407 (2014) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 131.156.59.191 On: Thu, 11 Sep 2014 12:15:24increases and above a critical stress, rc¼l0MsHk 3k,h¼90/C14 (the xy-plane) becomes the minimum energy plane. Assuming the continuity of strain at the interface between the nanomagnet and the underlying piezoelectric, the voltagerequired across the piezoelectric to generate the stress is given by V PE¼rc Yd31dPE;where dPEis the thickness of the piezoelectric layer. The critical bi-axial stress, rcfor this system is calculated to be /C0132 MPa which corresponds to a voltage of 110 mV across the piezoelectric. Figs. 2(a) and 2(b) show the dynamics of the magnetization upon the appli- cation of a uniform bi-axial stress, r¼/C0200 MPa which cor- responds to a VPE¼7 mV. The initial direction of the magnetization is taken along the /C0z axis. We observe in Fig. 2(a) that the time required for the 90/C14rotation of the magnetization is /C2410 ns and after 10 ns, the magnetization moves stochastically in the xy-plane. Figs. 2(c) and 2(d) show the time required for the 90/C14rotation of the magnetiza- tion as functions of randVPE, respectively. Next, we consider the case for non-uniform bi-axial strain. In this case, the magnetization dynamics is simulated with the stress energy given by Eq. (1)forrxx6¼ryy. For a non-uniform bi-axial stress with jrxxj>jryyj, it can be shown that, for rxx>rc¼l0MsHk 3k,6x-axis is the minimum energy direction along which the magnetization will rest at thesteady state. Figs. 3(a) and 3(b) show the dynamics of the magnetization upon the application of 167 mV across the piezoelectric, which creates a non-uniform stress ofr xx¼/C0200 MPa and ryy¼/C066 MPa. We note in Fig. 3(b) that, at the steady state, the magnetization wiggles along the x-direction. This is in contrast with the case for uniform biax-ial strain (Fig. 2(b)), where the steady state magnetization does not have any preferential direction in the xy-plane. Figs. 3(c)and3(d) show the time required for the 90 /C14rotationof the magnetization as functions of rxxand VPE, respec- tively. We note in Fig. 3(d) that under non-uniform biaxial stress, the time required for 90/C14rotation is significantly less than that for uniform biaxial stress for a given VPE. It is interesting to note that once a biaxial strain (uni- form or non-uniform) has rotated the magnetization by 90/C14, upon the removal of the stress, the magnetization has equalpossibility to revert back to the original state and to switch by 180 /C14with respect to the original state due to the effect of stochastic noise. Hence, magnetostriction effects alone can- not deterministically switch the magnetization by 180/C14. However, if a small spin current can be injected through themagnet after the magnetization has rotated by 90 /C14, it can deterministically switch the magnetization by 180/C14. Fig. 4(a) shows the schematic diagram of an STT-RAM built on topof a strain element. For the nanomagnet considered in this work (see Table II), the critical current is calculated to be 27lA, for which the switching time is /C2415 ns. In order to simulate combined strain-STT scheme for magnetization re- versal, we apply a 291 mV across the piezoelectric for 10 ns, which creates a biaxial strain r xx¼ryy¼r¼/C0350 MPa. Afterwards, we apply a 1 ns current pulse of 15 lA. Fig. 4(b) shows the piezoelectric voltage and current pulse sequence and the corresponding spin dynamics. We note in Fig. 4(b) that the magnetization switches by 90/C14in/C245 ns upon the application of the strain pulse. Upon the application of the current pulse, it takes another /C246 ns to completely switch by 180/C14with respect to the initial direction. Fig. 4(c) shows the spin dynamics for the magnet when a 291 mV voltage pulse has been applied for 10 ns (which creates a non-uniformbiaxial stress r xx¼/C0350 MPa, ryy¼/C0116 MPa) followed by a 1 ns current pulse of 15 lA. From Fig. 4(c), the time required for the strain driven 90/C14switching and the FIG. 2. The effect of a uniform bi-axial stress. (a) Projections of magnetization on the axes vs. time upon the applica-tion of a bi-axial stress r¼/C0200 MPa. (b) The trajectory of magnetization. (c) and (d) Time required for the a 90 /C14 rotation of magnetization as functionsof the stress (c) and the voltage across the piezoelectric (d).262407-3 Khan et al. Appl. Phys. Lett. 104, 262407 (2014) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 131.156.59.191 On: Thu, 11 Sep 2014 12:15:24subsequent spin transfer torque driven 180/C14switching are /C243 ns and /C244 ns, respectively. It is important to compare the energy dissipated in a conventional STT-RAM and a strain assisted STT-RAM.For the conventional STT-RAM, the energy dissipated is given by ESTT¼I2Rtcurrent (I¼the magnitude of the charge current, R¼the resistant of the tunnel junction, and tcurrent ¼the duration of the spin current pulse) which equals FIG. 3. The effect of a non-uniform bi- axial stress. (a) Projections of magnet- ization on axes vs. time upon the appli- cation of a non-equal bi-axial stress(r xx¼/C0200 MPa and ryy¼/C046 MPa) corresponding to voltage of /C0167 mV across the piezoelectric layer. (b) The trajectory of magnetization. (c) and (d) Time required for the a 90/C14rotation of the magnetization as functions of the stress (c) and the voltage across thepiezoelectric (d). FIG. 4. Strain assisted STT-RAM (a) The schematic diagram of a strain assisted STT-RAM. (b) Projections of magnetization in the axes vs. time upon the application of a uniform bi-axial stress ( r xx¼/C0350 MPa and ryy¼/C0350 MPa) and then applying a 15lA current for 1 ns. (c) Projections of magnetization on the axes vs. time upon the application of a uniform bi- axial stress ( rxx¼/C0350 MPa and ryy¼/C0116 MPa) and then applying a 15lA current for 1 ns. Both cases show deterministic switching by 180/C14.262407-4 Khan et al. Appl. Phys. Lett. 104, 262407 (2014) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 131.156.59.191 On: Thu, 11 Sep 2014 12:15:241.6/C2107kBT for the simulated values I¼27lA, R¼6kX (Ref. 27), and tcurrent ¼15 ns. On the other hand, for the strain assisted STT-RAM scheme, the total energy dissipated with strain assisted switching is EStrain /C0STT¼Epiezo þEReduced /C0BarrierSTT . It leads to a shorter required current pulse tcurrent as well as smaller switching current due to an angle deflection obtained by strain. The energy dissipated for generating the stress is Epiezo¼1 2CV2, where Cis the capaci- tance of the piezoelectric capacitor given bye0erA dpiezo. For r¼/C0350 MPa, Epiezo is calculated to be 5.1 /C2103kBT. For strain assisted switching, our simulation results in I¼15lA and tcurrent ¼1 ns, EStrain-STT ¼3.3/C2105kBT. Hence the energy dissipation in a strain assisted STT-RAM is /C2450 times smaller than its conventional counterpart. LLG dynamics of nanomagnet-piezoelectric heterostruc- ture devices due to magnetostriction were previously studied considering only uniaxial strain28–30and stochastically unstable cases.29,30However, in an out-of-plane electric field geometry compatible with an MTJ readout circuit, the piezo- electric material generates a bi-axial strain, which we studyin this paper. In addition, any other mechanism of magneto- electric coupling that can lead to a 90 /C14rotation of the mag- netization (such as the exchange bias effect), can lead to similar decrease in the energy dissipation in an STT-RAM. In summary, we investigated the magnetostrictive switching of nanomagnets in the presence of stochastic noise. We show that a deterministic switching of magnetiza- tion by 90/C14, from in-plane to out-of-plane directions and vice versa, is enabled by this mechanism. However, a determinis- tic 180/C14switching of the magnetization fails due to thermal noise, resulting in random outcomes. A device structure isproposed that combines the magnetostrictive and the spin transfer torque effects. It is shown that in such a device, the energy dissipation is nearly 50 times lower than that in aconventional STT-RAM of equal dimensions. 1G. E. Moore, Electronics 19, 114 (1965). 2D. Nikonov and I. Young, Proc. Int. Electron Dev. Meet. 2012 , 25.4.1. 3D. Nikonov and I. A. Young, Proc. IEEE 101, 2498 (2013). 4I. Zutic, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323–410 (2004).5D. Nikonov and G. I. Bourianoff, J. Supercond. Novel Magn. 21, 479–493 (2008). 6M. Tsoi, A. G. M. Jansen, J. Bass, W.-C. Chiang, M. Seck, V. Tsoi, and P.Wyder, Phys. Rev. Lett. 80, 4281 (1998). 7E. B. Myers, D. C. Ralph, J. A. Katine, R. N. Louie, and R. A. Buhrmann, Science 285, 867 (1999). 8Y. Chen, T. Fitchorov, C. Vittoria, and V. G. Harris, Appl. Phys. Lett. 97, 052502 (2010). 9T. Brintlinger, S.-H. Lim, K. H. Baloch, P. Alexander, Y. Qi, J. Barry, J.Melngailis, L. Salamanca-Riba, I. 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Ma, F. Zavaliche, L. Chen, J. Ouyang, J. Melngailis, A. L. Roytburd, V. Vaithyanathan, D. G. Schlom, T. Zhao, and R. Ramesh, Appl. Phys. Lett. 87, 072907 (2005).TABLE II. Material parameters of the nanomagnet. Parameter Value Magnetization Ms 8/C2105A/m (1Tesla/ l0) Perpendicular anisotropy HPMA 8/C2104A/m (0.1Tesla/ l0) Barrier height Eb 40 k BT Thickness d 1.2 nm Width a 58 nm Gilbert’s coefficient a 0.027 Magnetostriction coefficient, k þ2/C210/C04[Ref. 25] Young’s modulus, Y 2/C21011Pa262407-5 Khan et al. Appl. Phys. Lett. 104, 262407 (2014) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 131.156.59.191 On: Thu, 11 Sep 2014 12:15:24

1.4861576.pdf

Dynamics of skyrmions in chiral magnets: Dynamic phase transitions and equation of motion Shi-Zeng Lin, Charles Reichhardt, Cristian D. Batista, and Avadh Saxena Citation: Journal of Applied Physics 115, 17D109 (2014); doi: 10.1063/1.4861576 View online: http://dx.doi.org/10.1063/1.4861576 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/115/17?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Phase separation dynamics in a two-dimensional magnetic mixture J. Chem. Phys. 136, 024502 (2012); 10.1063/1.3674270 Dynamic phase transitions for ferromagnetic systems J. Math. Phys. 49, 053506 (2008); 10.1063/1.2913504 XRay Induced Magnetic Phase Transition in CoW Cyanide Probed by XMCD AIP Conf. Proc. 882, 526 (2007); 10.1063/1.2644581 On the theory of magnetic phase transitions in magnets with a large single-ion anisotropy Low Temp. Phys. 28, 883 (2002); 10.1063/1.1531392 Phase transitions in planar magnetic nanostructures Appl. Phys. Lett. 72, 2041 (1998); 10.1063/1.121258 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 213.0.47.10 On: Tue, 01 Apr 2014 08:20:28Dynamics of skyrmions in chiral magnets: Dynamic phase transitions and equation of motion Shi-Zeng Lin,a)Charles Reichhardt, Cristian D. Batista, and Avadh Saxena Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA (Presented 5 November 2013; received 6 September 2013; accepted 15 October 2013; published online 10 January 2014) We study the dynamics of skyrmions in a metallic chiral magnet. First, we show that skyrmions can be created dynamically by destabilizing the ferromagnetic background state through a spin polarized current. We then treat skyrmions as rigid particles and derive the corresponding equationof motion. The dynamics of skyrmions is dominated by the Magnus force, which accounts for the weak pinning of skyrmions observed in experiments. Finally, we discuss the quantum motion of skyrmions. VC2014 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4861576 ] Skyrmions as topological excitations were first proposed as a model for baryons by Skyrme half a century ago.1Later on, skyrmions were realized in many different condensed matter systems, such as quantum Hall devices, multibandsuperconductors, liquid crystals, and chiral magnets. The ob- servation of triangular skyrmion lattices in metallic chiral magnets, like MnSi or Fe 0.5Co0.5Si, has sparked tremendous interest in this topological textures.2–4For the case of mag- nets, spins wrap a sphere once when moving from the center to the outer region of the skyrmion. The typical size of anindividual skyrmion and the skyrmion lattice constant is of order 10–100 nm. The skyrmion lattice was realized in a very small portion of the temperature-magnetic field phasediagram of bulk crystals, while it was found to be more sta- ble for thin films and nanowires. 4–6Indeed, skyrmions are found to be stable up to room temperature in FeGe.4More recently, skyrmions were also discovered in insulating chiral magnets (e.g., Cu 2OSeO 3).7,8These findings suggest that skyrmions could be ubiquitous in magnetic materials withoutinversion symmetry. In metallic magnets, the conduction electrons interact with skyrmions through the Hund’s coupling, which is largerthan the Fermi energy. The spins of the conduction electrons are forced to be aligned with the local spins of the skyrmion, which yields emergent electromagnetic fields arising from theBerry phase that the electrons pick up because of the spin alignment. 9On the other hand, the skyrmion can also be driven by the electrons via the spin transfer torque mechanism. Themeasured threshold current density to make skyrmions mobile against pinning induced by defects is of order 10 6A/m2,i . e . , five orders of magnitude smaller than the depinning currentsof magnetic domain walls. 10–12Consequently, skyrmions are believed to be promising for applications in spintronics, as they can be easily driven by a spin polarized current. For application purposes, it is important to create sky- rmions in a controlled fashion. The generation of skyrmions by electrical means would be advantageous to minimize de-vice sizes. Because a skyrmion is a topological object, it ispossible to treat it as a particle and to derive the correspond- ing equation of motion. The particle-like description of the skyrmion dynamics is more transparent and convenient for numerical simulations. In this paper, we summarize ourrecent study on skyrmion dynamics. First, we demonstrate a novel mechanism to create skyrmions by destabilizing the fer- romagnetic (FM) state with a spin polarized current. Then,we present a particle-like equation of motion for skyrmions and explain the origin of the weak pinning. Finally, we study the quantum motion of skyrmions and discuss the possible ex-perimental signature of the quantum effect of skyrmions. We consider a thin film of a metallic chiral magnet described by the Hamiltonian 13–17 H¼ð dr2Jex 2ðrnÞ2þDn/C1r/C2 n/C0Ha/C1n/C20/C21 ; (1) where Jexis the exchange interaction, Dis the Dzyaloshinskii-Moriya (DM) interaction due to the absenceof inversion symmetry in the system, and the last term is the Zeeman interaction with an external magnetic field H a. Here, nis the unit vector representing the direction of the classical spin, and we have taken the continuum limit because the skyrmion size is much larger than the lattice constant. The system is assumed to be uniform along the z direction. The phase diagram of His known at T¼0.17A single wave-vector magnetic spiral is stable at low fields. For inter-mediate fields, the triangular skyrmion lattice is stabilized in order to lower the Zeeman energy term (the magnetization along the field direction is nonzero for the skyrmion latticesolution). Finally, a fully polarized FM state becomes stable for large enough fields. The total number of skyrmions in the film is Q¼Ðdr 2qðrÞwith the skyrmion density qðrÞ ¼n/C1ð@xn/C2@ynÞ=ð4pÞ.18The electric field generated by the skyrmion motion is E¼n/C1ð rn/C2@tnÞ.9Dimensionless units are used throughout this paper.19 The spin dynamics is described by the Landau-Lifshitz- Gilbert equation20–22 @tn¼ðJ/C1r Þn/C0cn/C2Heff/C0a@tn/C2n; (2)a)Author to whom correspondence should be addressed. Electronic mail: szl@lanl.gov. 0021-8979/2014/115(17)/17D109/3/$30.00 VC2014 AIP Publishing LLC 115, 17D109-1JOURNAL OF APPLIED PHYSICS 115, 17D109 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 213.0.47.10 On: Tue, 01 Apr 2014 08:20:28where the first term is the adiabatic spin transfer torque and the last term is the Gilbert phenomenological damping. We have a/C281 for typical chiral magnets. Without current (J¼0), the magnon dispersion in the FM state is XðJ¼0;kÞ¼cð1þiaÞHaþJexk2/C0/C1 =ða2þ1Þ. The gap is induced by the applied magnetic field. However, Jcan also be finite in presence of conduction electrons. If we take the conduction electrons as a reference frame, there is a Doppler shift of the magnon spectrum XðJ;kÞ¼XðJ¼0;kÞ/C0k/C1v, where v¼/C0Jis the velocity of the conduction electrons in dimensionless units. This Doppler shift was recently observed.23The gap of the magnon spectrum vanishes for large enough current, Jm¼2cffiffiffiffiffiffiffiffiffiffiffiHaJexp=ða2þ1Þ, indicating that the FM state is no longer stable. In our simulations, we find that skyrmions are dynamically created right after thisinstability. 24The reason is that the current density, J, couples to the emergent vector potential A/C17ic/C22hb†rb=egenerated by non-coplanar spin configurations via the Lagrangian termL JA¼J/C1A(where bis the spin coherent state18). Spin states with non-zero Aare favored by this coupling. In the presence of the DM interaction, the lowest energy state with non-zeroAis a state with skyrmions. This mechanism can be used to create skyrmions by injecting current in a nanosized disk made of a chiral magnet. 25 To demonstrate the creation of skyrmions by current, we performed numerical simulations by putting a skyrmion in the FM state as an initial condition. The skyrmion radiatesspin waves when it is driven by a current, as shown in Fig. 1. The spin gap vanishes at a threshold current and more sky- rmions are dynamically created. The number of skyrmionsincreases continuously after the instability and finally satu- rates [see Fig. 2(b)]. Meanwhile, as it is shown in Fig. 2(a), the electric field increases because it is proportional to thedensity of skyrmions. However, the ratio of the Hall electric field E ?to the longitudinal electric field Ekis independent of Jbecause it is an intrinsic property of rigid skyrmions. This threshold current obtained from our simulations is consistent with the analytical result for Jm. For stronger drives, the sky- rmions become strongly distorted (they are no longer circularobjects) because the spin precession cannot follow the fastmoving skyrmion. As a result, skyrmions are destroyed at such high current densities via the softening of some of itsinternal modes. For typical parameters, the current densities for generating and destroying skyrmions are the order of 10 12A/m2.24 The current-induced instability of the FM state also occurs for D¼0. Because the current density is coupled with the vector potential in the Lagrangian, a state with a nonzeroAis created to minimize the energy after the instability, sim- ilar to the case when D6¼0 we explained above. However, there is no preferred chirality for D¼0. Consequently, a chi- ral liquid phase is stabilized according to our simulations. To characterize the chiral liquid phase, we introduce the aver- aged chirality Qand the absolute chirality P/C17Ðdr 2jqðrÞj. As shown in Fig. 3,Q/C250 while P>0 fluctuates strongly over space and time, indicating a chiral liquid phase. A non- zero electric field is also generated by the fluctuations of FIG. 1. Spatial structure of the nxcomponent of the spin wave radiated by the skyrmion motion. The skyrmion location is labelled in the figure and the arrow denotes the direction of skyrmion motion. Here, a¼0.1,J¼1.4 and Ha¼0.6.FIG. 2. (a) Electric field parallel to the current Ekand the electric field per- pendicular to the current E?and (b) total number of skyrmions Qas a func- tion of the spin current J. The arrows in (a) and (b) indicate the direction of current sweep. Here, a¼0.1 and Ha¼0.6. FIG. 3. (a) Electric field parallel to the current Ekand the electric field per- pendicular to the current E?and (b) total number of skyrmions Qand abso- lute chirality Pas a function of the spin current, J, when Jis increased. Here, a¼0.1,D¼0, and Ha¼0.6.17D109-2 Lin et al. J. Appl. Phys. 115, 17D109 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 213.0.47.10 On: Tue, 01 Apr 2014 08:20:28chirality [see Fig. 3(a)]. Note that in a metal, the dominant contribution to the electric field comes from electrons. Skyrmions are topological excitations and can be treated as particles if the internal modes are not excited. In this case,their equation of motion is 19 4pa cvi¼FMþFLþX jFdðrj/C0riÞþX jFssðrj/C0riÞ;(3) which can be derived by using Thiele’s approach.26The term on the left-hand side accounts for the damping of sky- rmion motion, which is produced by the underlying dampingof the spin precession and damping due to conduction elec- trons that are localized around the skyrmions (for metals). F M¼4pc/C01^z/C2viis the Magnus force per unit length, which is perpendicular to the velocity. FL¼2p/C22he/C01^z/C2Jis the Lorentz force due to the external current, which arises from the emergent quantized magnetic flux U0¼hc/ecarried by the skyrmion in the presence of a finite current. Fdis the interaction between skyrmions and quenched disorder and Fssis the short-range pairwise interaction between two sky- rmions. The damping is weak, a/C281, and the Magnus force FMis dominant over the dissipative force 4 pavi=c. Equation (3)also describes the skyrmion motion in insulators, where the dissipation due to conduction electrons is absent and FL¼0. The rather strong Magnus is one reason behind the weak pinning that has been observed for skyrmions. In the pres- ence of a pinning center or an obstacle, skyrmions can easily be scattered with a velocity perpendicular to the pinning orrepulsive force. Thus, as illustrated in Figs. 4(a)and4(b), the influence of the pinning center or obstacle can be minimized by avoiding passing through them. When the dissipativeforce is dominant, a/C291, the pinning force becomes very strong because skyrmions have to pass through the pinning center, as sketched in Fig. 4(c). A similar conclusion was also reached by numerical simulations of the continuum model [Eqs. (1)and(2)]. 27Finally, we discuss the quantum motion of skyrmions by quantizing Eq. (3).28In a clean sample, skyrmions occupy the lowest Landau level with their wave function stronglylocalized due to the strong emergent magnetic field. In the presence of a pinning potential, the lowest Landau level for skyrmions is split into quantized levels. The transitionbetween different levels can be observed experimentally by microwave absorption measurements in the frequency region around x/C25100 GHz and at low temperatures T<0.5 K for typical materials parameters. Computer resources for numerical calculations were provided by the Institutional Computing Program in LANL.This publication was made possible by funding from the Los Alamos Laboratory Directed Research and Development Program, Project No. 20110181ER. 1T. H. R. Skyrme, Proc. R. Soc. London, Ser. A 260, 127 (1961). 2S. M €uhlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii, and P. B €oni,Science 323, 915 (2009). 3X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H. Han, Y. Matsui, N. Nagaosa, and Y. Tokura, Nature 465, 901 (2010). 4X. Z. Yu, N. Kanazawa, Y. Onose, K. Kimoto, W. Z. Zhang, S. Ishiwata, Y. Matsui, and Y. Tokura, Nature Mater. 10, 106 (2011). 5S. Heinze, K. V. Bergmann, M. Menzel, J. Brede, A. Kubetzka, R. Wiesendanger, G. Bihlmayer, and S. Blgel, Nat. Phys. 7, 713 (2011). 6X. Yu, J. P. DeGrave, Y. Hara, T. Hara, S. Jin, and Y. Tokura, Nano Lett. 13, 3755 (2013). 7S. Seki, X. Z. Yu, S. Ishiwata, and Y. Tokura, Science 336, 198 (2012). 8T. Adams, A. Chacon, M. Wagner, A. Bauer, G. Brandl, B. Pedersen, H. Berger, P. Lemmens, and C. Pfleiderer, Phys. Rev. Lett. 108, 237204 (2012). 9J. Zang, M. Mostovoy, J. H. Han, and N. Nagaosa, Phys. Rev. Lett. 107, 136804 (2011). 10F. Jonietz, S. M €uhlbauer, C. Pfleiderer, A. Neubauer, W. M €unzer, A. Bauer, T. Adams, R. Georgii, P. B €oni, R. A. Duine, K. Everschor, M. Garst, and A. Rosch, Science 330, 1648 (2010). 11X. Z. Yu, N. Kanazawa, W. Z. Zhang, T. Nagai, T. Hara, K. Kimoto, Y. Matsui, Y. Onose, and Y. Tokura, Nature Commun. 3, 988 (2012). 12T. Schulz, R. Ritz, A. Bauer, M. Halder, M. Wagner, C. Franz, C. Pfleiderer, K. Everschor, M. Garst, and A. Rosch, Nat. Phys. 8, 301 (2012). 13A. N. Bogdanov and D. A. Yablonskii, Sov. Phys. JETP 68, 101 (1989). 14A. Bogdanov and A. Hubert, J. Magn. Magn. Mater. 138, 255 (1994). 15U. K. R €oßler, A. N. Bogdanov, and C. Pfleiderer, Nature 442, 797 (2006). 16J. H. Han, J. Zang, Z. Yang, J.-H. Park, and N. Nagaosa, Phys. Rev. B 82, 094429 (2010). 17U. K. R €oßler, A. A. Leonov, and A. N. Bogdanov, J. Phys.: Conf. Ser. 303, 012105 (2011). 18A. Altland and B. D. Simons, Condensed Matter Field Theory (Cambridge University Press, Cambridge, 2010). 19S.-Z. Lin, C. Reichhardt, C. D. Batista, and A. Saxena, Phys. Rev. B 87, 214419 (2013). 20Y. B. Bazaliy, B. A. Jones, and S.-C. Zhang, Phys. Rev. B 57, R3213 (1998). 21Z. Li and S. Zhang, Phys. Rev. Lett. 92, 207203 (2004). 22G. Tatara, H. Kohno, and J. Shibata, Phys. Rep. 468, 213 (2008). 23V. Vlaminck and M. Bailleul, Science 322, 410 (2008). 24S.-Z. Lin, C. Reichhardt, C. D. Batista, and A. Saxena, Phys. Rev. Lett. 110, 207202 (2013). 25S.-Z. Lin, C. Reichhardt, and A. Saxena, Appl. Phys. Lett. 102, 222405 (2013). 26A. A. Thiele, Phys. Rev. Lett. 30, 230 (1973). 27J. Iwasaki, M. Mochizuki, and N. Nagaosa, Nature Commun. 4, 1463 (2013). 28S.-Z. Lin and L. N. Bulaevskii, Phys. Rev. B 88, 060404 (2013). FIG. 4. (a) and (b): Schematic view of a skyrmion passing through a pinning center (a) and obstacle (b) when the Magnus force is dominant. When the Magnus force is dominant over the dissipative force, the skyrmion is deflected by the pinning centers or obstacles. (c) Same as (a) and (b) except that the dissipative force is dominant. The skyrmion has to overcome the pinning site or obstacle by passing through it.17D109-3 Lin et al. J. Appl. Phys. 115, 17D109 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 213.0.47.10 On: Tue, 01 Apr 2014 08:20:28

1.881130.pdf

Clouds of Trapped Cooled Ions Condense into Crystals Barbara Goss Levi Citation: 41, (1988); doi: 10.1063/1.881130 View online: http://dx.doi.org/10.1063/1.881130 View Table of Contents: http://physicstoday.scitation.org/toc/pto/41/9 Published by the American Institute of Physics SEARCH & DISCOVERY CLOUDS OF TRAPPED COOLED IONS CONDENSE INTO CRYSTALS Photographs of ions trapped by elec- tric and magnetic fields are revealing ordered structures ranging from a few ions in crystalline arrays to thou- sands of ions arranged on the surfaces of concentric shells. In some experi- ments with a handful of singly charged ions, varying certain param- eters of the trap causes the regular structures to dissolve rapidly into clouds and just as rapidly to recrystal- lize—behavior resembling a phase change or an order-to-chaos transi- tion. All these regular structures occur in systems with well-defined constituents and interactions. As such they can provide useful insights into collective phenomena such as cluster formation, Wigner crystalliza- tion and other strongly coupled plas- mas. Ordered arrays of trapped charged particles are not new. In 1959 Ralph F. Wuerker, Haywood Shelton (both at TRW, Redondo Beach, California ) and Robert V. Langmuir (Caltech) photographed regular arrays of charged aluminum particles that were about 20 microns in diameter, and found that they successively melted and recrystallized. The recent experiments with ions, however , in- volve better-defined systems, whose particles have identical masses and charges. Furthermore, because indi- vidual ions have smaller charges than the aluminum particles, the new ex- periments can use far larger numbers of particles. Cooling in a Paul trap To study clusters with a small num- ber of ions, the experimenters confine them with a configuration of electric fields known as a Paul trap. The effective confining potential results from applying a radiofrequency elec- tric field between the end plates and a ring-shaped electrode in the center of the cylindrical trap. The electric quadrupole field has hyperbolic equi- potentials in which the motion of ions is harmonic to first order. The thermal motion of the ions is- 800 - 400 LASER DETUNING (MHz) damped by laser cooling. (See the article by David J. Wineland and Wayne M. Itano in PHYSICS TODAY, June 1987, page 34.) In this tech- nique, the ions are illuminated by a laser beam at a frequency just below one of the absorption lines of the ions. By the Doppler effect, the laser light appears at a higher frequency when the ions, jostling in thermal motion, move toward the beam direction. Be- cause the radiation appears to these ions to be at resonance, the ions absorb the radiation and are slowed by conservation of momentum. Sub- sequent spontaneous emissio n of the radiation is isotropic and, on average, does not change the ions' momentum. Ions have been cooled below 10 mK by this technique. The ions can beExcitation spectra from ions trapped by rf electric fields and cooled by loser radiation, as measured by the ion fluorescence When the rf voltage is high (570 V), the ions move randomly in a cloud and the spectrum is broadened by the Doppler shifts (red curve) When the rf voltage is lowered below a certain threshold, rhe spectrum shifts suddenly ro a sharp peak characteristic of an ordered stare. In the orange and yellow curves, correspondin g ro rf voltages of 460 V ond 360 V, respectively, arrows denote region of crystalline structure. The horizonral axis shows rhe amount by which the cooling laser is tuned below rhe resonant frequency. (Adapted from ref. 3.) imaged using the fluorescence pro- duced as they re-emit the absorbed radiation. Laser cooling was pro- posed in 1975 by Theodor Hansch (now at the University of Munich and the Max Planck Institute for Quan- tum Optics, Garching, West Ger- many) and Arthur Schawlow (Stan- ford University) and, independently, by Wineland (now at the National Bureau of Standards, Boulder, Colora- do) and Hans Dehmelt (University of Washington). The behavior of a system of trapped, cooled ions depends on the Coulomb coupling constant F, which is the ratio of the Coulomb interaction energy between neighboring, singly charged ions to their mean thermal energy. When F is greater than 1, the PHYSICS TODAY SEPTEMBER 1988 17 system enters the strong-coupling re- gion. For very large values of F, a collection of ions is expected to ar- range itself in a regular array. With the thermal motion so greatly re- duced, each particle is essentially pinned at a point where the trap forces pulling it toward the center just balance the Coulomb repulsion forces from the other ions pushing it away from the center. In 1980 Werner Neuhauser, Martin Hohenstatt and Peter Toschek of the University of Heidelberg, together with Dehmelt, photographed several ions in a Paul trap and found that the size of the two-ion image agreed with the expected equilibrium distance.1 However, they did not have sufficient resolution to detect any possible structure. From clouds to crystals Last year, at the Max Planck Insti- tute for Quantum Optics, a group led by Herbert Walther found evidence of ordered structures in systems con- taining 2 to 50 magnesium ions in a Paul trap.2-3 Walther's fellow exper - imenters are Frank Diedrich, Ekke- hard Peik, Jan Min Chen and Wolf- gang Quint. They studied changes in the intensity of the fluorescence emitted by ions as they varied the laser detuning, that is, the amount by which the frequency of the cool- ing laser beam falls below the ab- sorption line. When a few ions move randomly in a cloud, the fluorescent spectrum is expected to be quite broad because of the Doppler shifts. The experiment- ers find such broad spectra when the rf field applied to the trap has a voltage high enough to cause the ions to have considerable thermal motion. (See the figure on page 17.) However, at a lower value of rf field intensity, the fluorescent spectrum jumps dis- continuously to the sharply peaked shape associated with a single ion. This jump occurs as the magnitude of the laser detuning is decreased below a certain value. The Max Planck group interprets this behavior as a phase transition from a cloud-like state to a crystalline state in which motions are correlated. The narrow fluorescence peak in the crystal im- plies that the ions have assumed relatively fixed positions. At still smaller values of the laser detuning, a second jump in the spectrum suggests a transition back to the ion cloud. The Max Planck group is able to induce transitions between clouds and crystals not only by varying the laser detuning but also by altering either the power of the cooling laser radiation or the magnitude of the rfvoltage. In all of these cases, they find hysteresis effects expected for a phase transition: For example, jumps from the crystalline state to the cloud- like state always occur at higher rf voltages than transitions in the oppo- site direction. Walther and his colleagues con- firmed the transitions between states by observing the system visually. They imaged the fluorescence with a photon-counting system and video- taped the images. At the expected value of laser detuning, they saw the ion cloud suddenly crystallize, with individual ions clearly resolved. The transition occurs in the less than 0.04 sec between successive frames. The photograph on the cover shows one of the ordered structures they observed, a seven-ion configuration in which neighboring ions are about 20 mi- crons apart. Also last year, a group headed by Wineland at the National Bureau of Standards trapped and photographed a number of ions in regular crystal- line arrays such as rings or linear configurations.4 (See the photograph above.) The NBS team calls these structures "pseudomolecules." The separation distances between ions in these pseudomolecules are on the order of several microns—much greater than the spacings between atoms in a real molecule. The NBS group members include James Berg- quist, Itano, John Bollinger and Charles Manney. In their experi- ments on Hg+, they cool the ions to temperatures below 8 mK, corre- sponding to a T value of about 500. The crystalline structures seen at NBS agree with the configurations that would minimize the potentialSix mercury ions align along rhe z axis in a Paul trap for certain values of rhe rrap volrage. Each ion is pinned ar a point where rhe Coulomb forces pushing ir outward ore jusr balanced by rhe rrap forces pulling ir inward. In rhis false-color phorograph ions are preferentially located in rhe red areas, which are rhe regions of most intense fluorescence. Neighboring ions are separared by several microns. (Courtesy of rhe National Bureau of Standards.) energy for a given ring voltage, ac- cording to calculations by Daniel Du- bin and Thomas O'Neil (University of California, San Diego). In these calcu- lations the configurations change ab- ruptly with the ring voltage. Dubin points out, however, that the term "phase transition" is not precisely correct for a finite system at finite temperatures. The NBS experimenters studied in some detail a pseudomolecule consist- ing of pairs of Hg+ ions. They deter- mined a particular absorption line for an individual Hg+ ion and deduced the vibrational frequency of the pseu- domolecule from the measured side- bands, which reflect the Doppler shift in the absorption frequency caused by the ion motion. The value of vibra- tional frequency determined in this way agreed well with theoretical pre- dictions. Toschek and his colleagues at the University of Hamburg—Th. Sauter, H. Gilhaus, Neuhauser and R. Blatt— have found metastable vibrational states of a single barium ion in a Paul trap,5 and have also photographed clusters of two, three and four ions of that same element. They recently reported an observation of two novel cooling schemes for trapped particles. Nature of the transitions John Hoffnagle, Ralph DeVoe and Richard Brewer (all of IBM's Alma- den Research Center in California) together with Luis Reyna (IBM T. J. Watson Research Center, Yorktown Heights, New York) , have analyzed the behavior observed by Walther and his colleagues in terms of transitions from order to chaos.6 The IBM group asserts that the relevant equations of 18 PHYSICS TODAY SEPTFMBFR 19flfl SEARCH 6 DISCOVERY motion contain ingredients that should lead to chaos: a radiofrequency driving term, dissipation due to laser cooling, and a Coulomb term that couples the ions nonlinearly. Brewer argues that the noise from fluctu- ations in ion recoil after spontaneous emission establishes the initial condi- tions that displace the ions from their equilibrium positions and start them on their path to chaos. He believe s the noise term is small enough that it plays no role in the subsequent, deter- ministic development of chaos. Brewer and his colleagues analyzed in two dimensions the simplest possi- ble system—two barium ions in a Paul trap—and traced its evolution as a function of a parameter q, which depends, among other things, on the rf voltage. As q increases, the mo- tions in the radial and axial directions become more strongly coupled until, at a critica l value of q, the amplitudes of both motions suddenly increase and exhibit an erratic temporal and spatia l dependence, characteristic of the chaotic state. The researchers find that the separation between two initially adjacent points in phase space diverges exponentially, as ex- pected in chaotic systems. The IBM team made direct mea- surements and photographed images of the behavior of two Ba+ ions for comparison with their model predic - tions. They found that the transition from an ordered to a disordered state occurs in the real system at the q value predicte d in their model. At a lower value of q the physical system recrystallizes; the model, however, does not exhibit condensation at this point because of a problem in trunca- tion. In related work, the IBM group has identified a heating source they feel is important. They find that Doppler shifts caused by the oscillation of ions at the frequency of the rf trap voltage can change the effect of the laser from cooling to heating. The main heating mechanism previously identified in Paul traps was rf heating. The Max Planck investigators, joined by institute colleagues Rein- hold Blumel and Wolfgang Schleich and by Yuen-Ron Shen of the Univer- sity of California, Berkeley, have used three-dimensional molecular dynam- ics calculations to simulate the mo- tion of any number of ions in a Paul trap.7 From these simulations they can extract the excitation spectra and the jumps in them, and also reproduce the hysteresis loops seen in the flu- orescence. Their calculations predict the value of the control parameter at which condensation from the cloud phase to the crystal phase occurs.The theoretical modeling does not indicate that an adiabatic change in the rf voltage will melt the crystals, although melting is observed in the experiments. However, the theory shows that the crystal may become very sensitive to fluctuations in the laser intensity, which could then trig- ger this transition. This group explore d the system dynamics and studied the relation between the stability of ion clouds and radiofrequency heating in Paul traps. The results suggest that the heating stems from deterministic chaos as ions in the chaotic cloud phase gain kinetic energy from the rf field that drives the ions in the Paul trap. The experimenters use this scenario to offer an explanation for the sharpness of the observed few-body phase transi- tions. The heating rate depends criti- cally on the phase-spac e diameter of the ion configuration. When the ion separation is of the order of a typical ionic lattice constant, no heating oc- curs and the ions perform multiply periodic motion. Strong heating, however , sets in suddenly at a critical size of the ionic array. For very large clouds the heating rate becomes negli- gible, which again permits regular motion of the ions. Pure ion plasmas To study large collections of trapped ions, called pure ion plasmas, experi- menters typically use a different type of trap. The Paul trap can hold only a certain number of ions before rf heating becomes too large for the ions to be successfully cooled. Studies of plasmas with several hundred to a few thousand ions cooled to tempera- tures below 10 mK have therefore been carried out with Penning traps, in which a static magnetic field, rather than the rf field of the Paul trap, helps to confine the ions. A large number of ions stored in a trap is analogous to a one-component plasma, which consists of a single species of charges embedded in a uniform background charge of oppo- site sign. In a particle trap or storage ring, the trapping fields effectively play the role of the neutralizing background charge. A decade ago, O'Neil and John Malmberg of the University of California, San Diego, suggested that a pure electron plasma can be confined in a Penning trap and cooled to the cryogenic temperature range, where one expects to obtain liquid and crystal states. They showed that the thermal equilibrium of such a system is identical to that of a one-component plasma. With a simple sign change, these ideas also apply to a pure ion plasma.For many years theoretical studies of one-component plasmas considered only systems of infinite extent, but the possibility of experimental studies on finite systems in the strong-cou- pling region has recently stimulated theorists to look at the effect of small size and realistic boundary conditions on the resulting structures. Rather than condensing into a body-centered cubic structure, as predicted for un- bounded systems, a system with sev- eral hundred ions is expected to form concentric spheroidal shells. This general behavior was outlined in 1986 by the late Aneesur Rahman (Univer- sity of Minnesota) and John P. Schiffer (Argonne National Lab and the University of Chicago) as part of molecular dynamics calculations they were doing to simulate the behavior of ions in storage rings.8 In 1988 Dubin and O'Neil predicted this spheroidal structure specifically for the Penning trap.9 The ions diffuse rather freely on the surfaces of these shells, but not between them. O'Neil explained to us that the system behaves like a liquid on the surfaces of the shells but like a solid between the shells. Such behav- ior resembles that of smectic liquid crystals. As temperatures are lowered further and the coupling gets stronger, the ions assume a hexagonal lattice structure on the shell surface. Sarah Gilbert (NBS), Bollinger and Wineland have seen the predicted shell structure in experiments with clouds of beryllium ions in a Penning trap.10 This team used a laser probe beam as well as two cooling beams, one perpendicular and one diagonal to the axis of the trap, which parallels the magnetic field. Each laser beam induces fluorescence and enables the experimenters to see sections of the shells that form. (See the photograph on page 20.) The structures they have seen range from a single shell with 20 ions to 16 shells with a total of 15 000 ions. The number of shells corre- sponds to what the theory predicts for the given number of ions. However, some of these shells had cylindrical rather than the predicted spheroidal shapes, a finding that is not yet understood. Gilbert and her colleagues also explored the dynamical behavior of ions on the shells: They switched off the fluorescence from the ions in certain regions of the cloud by tuning the probe laser beam to a frequency corresponding to a nonfluorescing state. By tracing the diffusion of these "tagged" ions, they found that the ions within a shell moved dis- tances of more than 100 microns in 0.1 sec, but that ions took several seconds to diffuse between shells. PHYSICS TODAY SEPTEMBER 19S8 19 Malmberg and his collaborators at San Diego have confined pure elec- tron plasmas in Penning traps and cooled them to the cryogenic tempera- ture range. These plasmas are much larger (10"' electrons) than the ion plasmas but the correlation strength achieved is lower. The San Diego group feels that the electron plasmas are probably large enough to exhibit the bulk properties of an unbounded one-component plasma, but that sub- stantially higher f values must be reached before a crystal can be ob- tained. Strings of beads If ions can condense in electromagnet- ic traps, they might also crystallize in an ion storage ring, where the parti- cles are similarly cooled and confined. In heavy-ion rings, where ionization levels are high, the F value could be orders of magnitude higher than it is for singly charged ions. Measure- ments made in 1980 at the NAP-M proton storage ring in Novosibirsk, USSR, hinted that the beam there had some coherent structure." In their model of ions in a storage ring, Rahman and Schiffer found that when the density of beam particles is very high, the ions will arrange them- selves on the surfaces of concentric cylinders, which wrap around the ring with the axis at their centers. On each cylindrical surface, the ions oc- cupy points on a triangular grid. If the beam density is sufficiently low, the ions space themselves regularly along the axis, like a string of beads. Everyone is now eager to look for these jewels in the heavy-ion storage rings now under construction at such places as GSI in Darmstadt, West Germany; the Max Planck Institute in Heidelberg, West Germany; and the University of Aarhus in Den- mark. All are expected to produc e beams below the temperature of 1 K that precluded further investigation of ion crystals at Novosibirsk. Die- trich Habs (Max Planck Institute for Nuclear Physics, Heidelberg) has dis- cussed several models of the behavior of ions in these rings.12 Schiffer told us that several features of real ion storage rings not included in the calculations he did with Rahman may make it more difficult to see the predicted structures. One is the time variation of the focusing fields. A more serious omission, he felt, was the curvature of the ring. (The model treats the ring as a straight cylinder.) Ions closer to and further from the ring's center have different travel times. This difference may introduce a shear and cause differential cooling across the beam. However, with ap-Arrongemenr in cylindrical shells of 15 000 ions in o Penning trap is evidenr in rhis photograph. The cross section of rhe shells is most visible where rhe laser beams (two for cooling and one for probing) cross rhe ion rrap. The ions ore found ro arrange Themselves on rhe surface of 11 concentric shells and o central column. Ions diffuse rarher freely on rhe shell surfaces bur nor between shells. (From ref. 10.) propriate cooling and some adjust- ments in parameters, the new ion rings may yet produc e ordered beams. —BARBARA GOSS LEVI References 1. W. Neuhauser, M. Hohenstatt, P. Tos- chek, H. Dehmelt, Phys. Rev. A 22, 1137 (19801. 2. F. Diedrich, J. Krause, G. Rempe, M. O. Scully, H. Walther, in Laser Spectroscopy VIII (Springer Series in Optical Sciences 55), S. Svanberg, W. Person, eds., Springer-Verlag, Berlin (1987), p. 133. 3. F. Diedrich, E. Peik, J. M. Chen, W. Quint, H. Walther, Phys. Rev. Lett. 59. 2931 (1987). 4. D J. Wineland, J. C. Bergquist, W. M. Itano. J. J. Bollinger, C. H. Manney, Phys. Rev. Lett. 59, 2935 (1987).5. Th. Sauter, H. Gilhaus, W. Neuhauser, R. Blatt, P. E. Toschek, to be published in Europhys. Lett. 6. J. Hoffnagle. R. G. DeVoe, L. Reyna, R. G. Brewer , Phys. Rev. Lett. 61, 255 (1988). 7. R. Blumel. J. M. Chen, E. Peik, W. Quint, W. Schleich, Y.-R. Shen, H. Walther, Nature 334, 309 (1988). 8. A. Rahman, J. P. Schiffer, Phys. Rev. Lett. 57, 1133 (1986). 9. D. H. E. Dubin, T. M. O'Neil, Phys. Rev. Lett. 60, 511 (1988). 10. S. L. Gilbert, J. J. Bollinger, D. J. Win- eland, Phys. Rev. Lett. 60, 2022 (1988). 11. E. N. Dementiev, N. S. Dikansky, A. S. Medvedko, V. V. Parkhomchuk, D. V. Pestrikov, Sov. Phys. Tech. Phys. 25, 1001 (1980). 12. D. Habs, in Frontiers of Particle Beams, M. Month, S. Turner, eds., Springer-Verlag, Berlin (1986). 20 PHYSICS TODAY SEPTEMBER 1988

1.3646498.pdf

A new parametrizable model of molecular electronic structure Dimitri N. Laikov Citation: J. Chem. Phys. 135, 134120 (2011); doi: 10.1063/1.3646498 View online: http://dx.doi.org/10.1063/1.3646498 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v135/i13 Published by the American Institute of Physics. Additional information on J. Chem. Phys. Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors Downloaded 18 Mar 2013 to 129.89.24.43. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsTHE JOURNAL OF CHEMICAL PHYSICS 135, 134120 (2011) A new parametrizable model of molecular electronic structure Dimitri N. Laikova) Chemistry Department, Moscow State University, 119992 Moscow, Russia (Received 23 June 2011; accepted 14 September 2011; published online 7 October 2011) A new electronic structure model is developed in which the ground state energy of a molecular sys- tem is given by a Hartree-Fock-like expression with parametrized one- and two-electron integrals over an extended (minimal + polarization) set of orthogonalized atom-centered basis functions, thevariational equations being solved formally within the minimal basis but the effect of polarization functions being included in the spirit of second-order perturbation theory. It is designed to yield good dipole polarizabilities and improved intermolecular potentials with dispersion terms. The molecularintegrals include up to three-center one-electron and two-center two-electron terms, all in simple analytical forms. A method to extract the effective one-electron Hamiltonian of nonlocal-exchange Kohn-Sham theory from the coupled-cluster one-electron density matrix is designed and used to getits matrix representation in a molecule-intrinsic minimal basis as an input to the parametrization pro- cedure – making a direct link to the correlated wavefunction theory. The model has been trained for 15 elements (H, Li–F, Na–Cl, 720 parameters) on a set of 5581 molecules (including ions, transition states, and weakly bound complexes) whose first- and second-order properties were computed by the coupled-cluster theory as a reference, and a good agreement is seen. The model looks promis-ing for the study of large molecular systems, it is believed to be an important step forward from the traditional semiempirical models towards higher accuracy at nearly as low a computational cost. © 2011 American Institute of Physics . [doi: 10.1063/1.3646498 ] I. INTRODUCTION Atomistic computer simulations of complex chemical systems and materials at quantitative level are a great challenge for modern science: not only a high accuracy of computed potential energy surfaces is needed for systemswith many atoms but also a higher speed of computation for a thorough sampling of the configurational space. Molecular mechanical force fields pioneered 144 years ago are still almost the only practical method of calculation in many fields thanks to their very high speed and despite their well-known limitations. With their fixed bonding topology,they are designed first of all for conformational studies where nonbonding interactions play the main role, their treatment of electrostatics may be as sophisticated as to account for po-larization effects. 2Extensions to treat one (or few) chemical reaction steps3need careful parameter adjustment for each active center studied. General reactive force fields4,5with geometry-dependent bond orders and atomic charges seem to be a logical next step and already remind of models withinthe Hohenberg-Kohn 6density functional theory (DFT). Our own experience with this kind of models leads us to believe that instead of designing more and more complicatedcharge and bond order functionals it should be much easier to incorporate at least a single matrix diagonalization of an effective one-electron Hamiltonian 7into the model (likely in a linear-scaling fashion8,9), just like the mainstream DFT a)Electronic mail: laikov@rad.chem.msu.ru. URL: http://rad.chem.msu.ru/ ~laikov/ .took the Kohn-Sham path,10and thus we escape the force field and enter the realm of molecular quantum mechanics. At the other extreme, rigorous wavefunction methods give useful results starting with the second-order many- body perturbation theory (MP2),11but much better with the coupled-cluster theory12with single and double substitutions (CCSD)13or – as the golden standard – with the further per- turbative account of triple substitutions (CCSD(T)).14Their fifth, sixth, and seventh power scaling of computational cost with the system size and huge prefactors, as well as the slow convergence of the computed properties with the basis setsize, make them hopelessly slow for large molecules, but they are indispensable for getting small-molecule reference data for training and testing all kinds of models. Kohn-Sham DFT on its way from local 10to generalized- gradient approximations,15,16with further inclusion of a fraction17or, much better, the full long-range part18of the nonlocal exchange and the dispersion tail,19,20has grown into a rather accurate electronic structure model with favor- able system-size scaling properties. In a standard implemen-tation, localized atom-centered basis functions are used to solve the self-consistent field (SCF) equations (plane wave techniques 21are limited to the less accurate pure density func- tionals and cannot work as fast with the nonlocal exchange), the analytical evaluation of two-electron Coulomb integrals and the numerical integration of exchange-correlation terms are the bottleneck for smaller system sizes but can be made linear scaling starting from around 1000 atoms, in that limit,we estimate the computation of DFT energy functional to be about 10 6to 108times slower than the most sophisti- cated polarizable force field energy and gradient evaluation. 0021-9606/2011/135(13)/134120/10/$30.00 © 2011 American Institute of Physics 135 , 134120-1 Downloaded 18 Mar 2013 to 129.89.24.43. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions134120-2 Dimitri N. Laikov J. Chem. Phys. 135 , 134120 (2011) Density-fitting techniques22,23can speed up the calculation of these terms by up to 100 times but only for the pure DFT. Pseudospectral methods24,25can show up to 100 times speed- up also for the nonlocal-exchange DFT. Even if the integrals were for free, there would be another serious bottleneck in the linear algebra of SCF equations. A smallest meaningfulatomic basis set with 5 functions for H and 14 or 18 for Li– Ne would yield little to no sparsity of the density matrix for typical three-dimensional molecules with up to 3000 atoms 26 and around 30 000 basis functions, so about 1013floating- point multiplications and additions would be needed to do one matrix-multiply! In this regime, one SCF energy calculation can hardly take much less than one day even on a modern high-performance parallel computer. The only way for the electronic structure theory to com- pete with the force field methods lies in the use of a finite- dimensional model Hamiltonian defined by its matrix ele-ments in a minimal atomic basis representation. This is an alternative strategy with the Kohn-Sham DFT – instead of defining the universal density functional in some limited (ap-proximate, parametrized) form and then computing the aris- ing integrals rigorously at each molecular geometry, these molecular integrals themselves can be directly modeled asthe functions of atomic coordinates. One may note that the Coulomb potential of the nuclei in molecules is a very spe- cial class of external potentials for a system of electrons, andthe widely used density functionals are not as universal as thought, being often fitted 27to molecular data. The minimal basis representation is not a limitation – not a fixed free-atom but a molecule-adapted set of (deformed) atomic functions is implied. In our earlier work,28we have shown how to extract such a basis set from an accurate (or exact) solution of Kohn- Sham equations at a given molecular geometry in a unique and molecule-intrinsic way. The effective local potential ofKohn-Sham theory can, in turn, be extracted from the corre- lated wavefunction theory by a stable numerical procedure, 29 in Sec. II D of this work we derive its analog for the case of nonlocal exchange. (After we had done ours, we were made aware of a parallel work on the effective valence shell Hamiltonians30–33but for a fully correlated – and not SCF – treatment of the ground and excited states). Now that the ef- fective minimal-basis matrix elements of the model Hamilto- nian have the first-principles foundation and can be computednumerically, they can be used to guide the work on their ap- proximations. Into this category fall, the semiempirical electronic struc- ture models based on the neglect of diatomic differential overlap (NDDO) formalism that have had nearly half a cen- tury of conservative evolution. Conceived as the simplest approximations 34for valence-only minimal-basis SCF calcu- lations with most multicenter integrals set to zero, they be-came a breakthrough in computational chemistry 35when a systematic parametrization36–41to fit experimental molecu- lar data turned them into predictive phenomenological mod-els. As the growing computer power allowed the DFT, MP2, and other rigorous methods to be applied to chemically inter- esting systems and the standards of accuracy tightened, thesemiempirical models began to lose the game. Some limited orthogonalization corrections were studied 42,43but lead onlyto a limited improvement. Poor intermolecular potentials, es- pecially for hydrogen-bonding, were tried to be cured44,45by simple diatomic dispersion corrections46and further by tri- atomic corrections47,48in the spirit of molecular mechanics – by adding the terms that depend on the atomic coordinates only and not on the electronic state. A proper treatment of po-larizability within the minimal-basis formalism is not straight- forward, one solution 49is to add polarization functions into the basis but this slows down the calculations. A more at-tractive way was found 50,51in which a polarization term bor- rowed from molecular mechanics is added on top of the SCF equations. With all these recent developments, the desired im- provement in accuracy of the semiempirical NDDO models for many applications is still not achieved. Here, we report our new parametrizable electronic struc- ture model that evolves along a different line. We consider an extended atomic valence basis set that has radial andangular polarization functions added to the minimal set on each atom, and then we design a new two-layer SCF method (Sec. II A) in which the variational equations are defined within the minimal subspace and the contribution from the polarization subspace is included in the spirit of second-order perturbation theory. It should be noted that our SCF methodis fairly different in many ways from other known dual-basis techniques. 52–54The parametrizable molecular integrals of our model (Sec. II B) have more general functional forms and include new terms not seen in the traditional semiempirical NDDO models. We add the long-range dispersion corrections into the two-electron integrals – a more meaningful treatment of the two-electron correlation effect that depends on the elec- tronic state. Our model also treats the molecular polarizabilityin a natural and self-interaction-free way. The parametriza- tion procedure (Sec. II C) we use to train our model has a number of new terms added into the optimization process:the effective one-electron Hamiltonian matrices extracted (Sec. II D and Ref. 28) from the correlated wavefunction the- ory, the molecular dispersion coefficients (Sec. II E) and the molecular electrostatic potentials on the surfaces (Sec. II F), as well as the second-order response properties (force con- stant matrices, atomic polar tensors, and dipole polarizabili-ties) – all these values coming from the high-level (coupled- cluster) calculations are required to be reproduced (in the least-squares sense) by the parametrized SCF model. All this methodology defines our model up to the val- ues of parameters that have to be optimized on a set of refer- ence molecular data. Here, we report (Sec. III) a preliminary parametrization for 15 elements H, Li–F, Na–Cl on a very di- verse set of molecular structures to assess the worst-case per- formance of the model – it stands the test, seems to be already usable and useful for some applications, and awaits a future extension to heavier elements. II. THEORY A. Energy expression In our model, the total energy of a molecular system with Knuclei at positions {rk},k=1,...,K , in an external field Downloaded 18 Mar 2013 to 129.89.24.43. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions134120-3 Parametrizable electronic structure model J. Chem. Phys. 135 , 134120 (2011) v(r) can be written as a sum of three terms E=E0+E1+E2, (1) the one independent of the valence electronic structure E0=/summationdisplay kek+/summationdisplay k<k/primeqkqk/prime|rk−rk/prime|−1+/summationdisplay kqkv(rk), (2) being the sum of core electron energies ek, the repulsion of atomic cores with charges {qk}and their interaction with the external field; the Hartree-Fock energy within the minimalatomic basis E 1=1 2/summationdisplay μνσ/parenleftbig Hμν+Fμνσ/parenrightbig Dμνσ, (3) Fμνσ=Hμν+/summationdisplay μ/primeν/primeσ/primeRμνσμ/primeν/primeσ/primeDμ/primeν/primeσ/prime, (4) the indices μ,ν=1,...,M running over the minimal set, the spin label σ=±1 2; and the second-order term E2=/summationdisplay μνσF(2) μνσDμνσ, (5) F(2) μνσ=/summationdisplay α(FαμσXανσ¯sνσ+FανσXαμσ¯sμσ), (6) Xαμσ=Fαμσ εμ−εα, (7) Fανσ=Hαν+/summationdisplay μ/primeν/primeσ/primeRανσμ/primeν/primeσ/primeDμ/primeν/primeσ/prime, (8) that accounts for the effect of polarization functions (la- beled by α=M+1,...,N ) in a perturbative way. The one- electron density matrices Dμνσ=Nσ/summationdisplay i=1CμiσCνiσ (9) withNσelectrons for each spin are formed from the orthogo- nal coefficient matrices /summationdisplay μCμiσCμj σ=δij. (10) The orthogonalized atomic valence basis functions φμ=φmμlμnμkμ (11) can also be labeled by their principal n, angular l, and az- imuthal mquantum numbers and atomic centers k, for each l no more than one nvalue is used in either minimal or polar- ization set. The one-electron integral matrix is a sum Hμν=Tμν+/summationdisplay k/primeVμνk/prime+V(v) μν (12) of kinetic energy, effective core potential, and external field terms. The spin-dependent two-electron integrals, Rμνσμ/primeν/primeσ/prime=Rμνμ/primeν/prime−δσσ/primeRμ/primeνμν/prime, (13) are built from the spinless Coulomb repulsion integrals by the proper antisymmetrization. The values εμ<0 and εα>0i nthe denominator of Eq. (7)are taken as atomic constants and play the role of diagonal energies of the zeroth-order prob- lem of perturbation theory, their signs imply that E2≤0. Had we had ¯sμσ≡1i nE q . (6)as we did in an earlier model, the total energy Ewould be a cubic function of the density ma- trix, whereas the Hartree-Fock energy E1alone is quadratic – the perturbative inclusion of higher order basis set effects is leading to the appearance of effective three-electron inte- grals. Moreover, E2would be quadratic in the external field – as needed for the proper account of polarizability within a formally minimal-basis treatment. Our experience has shown, however, that the simplest choice of ¯sμσ≡1 often leads to a collapse of E2when atoms in a molecule get crowded – we have overcome this problem using the damping factors sμσ=/parenleftBigg 1+/summationdisplay αX2 αμσ/parenrightBigg−1 (14) that need to be spherically averaged ¯sμσ=1 1+2lμ/summationdisplay νsνσδlμlνδkμkν (15) before the insertion into Eq. (6). It is noteworthy that our model has the integrals of only two types: either with all functions from the minimal set (all- minimal) as in Eq. (4)or with all but one from the minimal and one from the polarization set (minimal-polarization one-electron and 3-minimal-1-polarization two-electron integrals) as in Eq. (8). The energy expression (1)is minimized under the orthog- onality constraints (10) to get the ground-state solution of the electronic structure problem, the stationary conditions can be given in the form of SCF one-electron equations /summationdisplay ν(Fμνσ−δμν/epsilon1iσ)Cνiσ=0, (16) with the effective Hamiltonian matrix derived as Fμνσ=∂E/∂D μνσ, (17) the explicit expression for the latter is lengthy but straightfor- ward. To compute a one-electron property V, such as dipole moment or molecular electrostatic potential in a consistent way, the energy derivative formalism should be used leading to V=/summationdisplay μνσVμνDμνσ+/summationdisplay ανVανDαν, (18) with the minimal-polarization block of the effective density matrix given by the derivative Dαν=∂E/∂H αν, (19) its explicit form is again lengthy but straightforward. We have written a computer code that solves these SCF equations to get the energy and its first and second derivatives with respect to the atomic coordinates and the applied uni- form electric field, the derivatives being computed in a fully analytic way. Downloaded 18 Mar 2013 to 129.89.24.43. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions134120-4 Dimitri N. Laikov J. Chem. Phys. 135 , 134120 (2011) B. Molecular integrals The one- and two-electron integrals in the energy ex- pression (1)refer to the orthogonalized atom-centered basis functions (11) and as such are nontrivial functions of (in gen- eral) all atomic coordinates of the molecular system. Theycould have been computed from the first principles using a three-level extension of our zero-bond-dipole orthogonaliza- tion scheme 28applied to a fixed all-electron atomic basis – the core set is orthogonalized first, the minimal valence set is orthogonalized to the core and then within itself, and the polarization set is orthogonalized to the core and valence andat last within itself. Such first-principles integrals, however, can give, at best, only a rough approximation to the Hartree- Fock molecular energy within a polarized basis, but we areaiming at a parametrizable model that can yield accurate molecular properties with the electron correlation included in the spirit of Kohn-Sham density-functional theory. Thus, ourmodel uses parametrized explicit analytical formulas for the molecular integrals giving values that differ somewhat from the first-principles ones. The two-electron integrals are split into a sum R μνμ/primeν/prime=R(0) μνμ/primeν/prime+R(6) μνμ/primeν/prime, (20) of “electrostatics” and “dispersion” parts, the former quickly (exponentially) reaching the asymptotics of Coulomb inter- action of the two ( μνandμ/primeν/prime) charge distributions and the latter having a characteristic r−6asymptotic tail. For the inte- grals with one polarization function Rανμ/primeν/prime=R(0) ανμ/primeν/prime, (21) only the electrostatic part is used. The electron correlation can already be modeled by decreasing the magnitude of the two-electron integrals. The one-electron integrals (12)have long-range Coulomb terms V μνk/primeand it is much easier to work with their short- range analogs ¯Fμν=Hμν+/summationdisplay μ/prime/parenleftBig R(0) μνμ/primeμ/prime−1 2R(0) μμ/primeμ/primeν/parenrightBig pμ/prime, (22) and in the same way for ¯Fαν, where a diagonal promolecule density matrix with constant spherically symmetric atomicpopulations p μ=plμkμ (23) is used to add the promolecule Coulomb and exchange terms, so that they fully neutralize the core charges /summationdisplay lplk=qk. (24) It is trivial to rewrite the energy expression in terms of ¯F andRintegrals, working with the short-range integrals not only simplifies the parametrization but is also of great ad- vantage for large-scale calculations. Moreover, some two-electron terms can be accounted for (on the average) only within the ¯Fintegrals and neglected in the Rintegrals, as we do in the following. The (promolecule) one-electron integrals ¯F μνare of two kinds: one-center if kμ=kνand two-center ifkμ/negationslash=kν, but they should also depend on the positions ofthe other atomic centers k/negationslash=kμ,kν. In our model, we use an additive scheme ¯Fμν=¯Fμν,0+/summationdisplay k/negationslash=kμ,kν¯Fμν,k, (25) where the first leading term is either one- or two-center and the sum runs over the two- or three-center corrections, respec- tively. The leading one-center integrals are atomic constants ¯Fμν,0kμ=kν=δmμmνδlμlνFlμkμ, (26) ¯Fαν,0kα=kν=0. (27) The leading two-center integrals ¯Fμν,0kμ/negationslash=kν=/summationdisplay mAlμ mmμ(zμν)Alν mmν(zμν)Fmlμlνkμkν(rμν), (28) as well as ¯Fαν,0can be reduced to functions of interatomic dis- tances and the well-known transformation matrices Al mm/prime(z) of spherical harmonics upon spatial rotation, zμν=rμν/rμν, rμν=rkμ−rkν,r μν=|rμν|. (29) The two-center corrections to the one-center integrals ¯Fμν,kkμ=kν/negationslash=k=/summationdisplay mlAlμ mmμ(zμk)Alν mmν(zμk)Bllμlν 0mmVllμlνkμk(rμk), (30) as well as ¯Fαν,kare done in the same way, making further use of the triple products Bll/primel/prime/prime mm/primem/prime/primeof spherical harmonics. The most complicated are the three-center corrections ¯Fμν,kwith kμ/negationslash=kν/negationslash=kthat can be exactly reduced only down to non- trivial functions of three variables – we have experimented with the triple series in prolate spheroidal coordinates whichcan be made very accurate, but in the end we have chosen the (approximate) factorization of the three-center terms ¯F μν,kkμ/negationslash=kν/negationslash=k=/summationdisplay λδkλkSμλSνλfλ, (31) into the products of two-center terms Sμλkμ/negationslash=kλ=/summationdisplay mAlμ mmμ(zμλ)Alλ mmλ(zμλ)Smlμlλkμkλ(rμλ), (32) where λlabels the functions of some expansion basis on cen- terkin which the underlying operator is diagonal with eigen- values fλ, andSμλare the overlap integrals. The three-center terms with one polarization function have minor effect and we set them to zero, ¯Fαν,kkα/negationslash=kν/negationslash=k= 0. (33) Of the two-electron integrals, only the one- and two- center ones are of fundamental importance and are included in our model, the rest are generally small and are set to zero. The one-center integrals, R(0) μνμ/primeν/primekμ=kν=kμ/prime=kν/prime= R(0) μνμ/primeν/prime,0+/summationdisplay k/negationslash=kμR(0) μνμ/primeν/prime,k, (34) Downloaded 18 Mar 2013 to 129.89.24.43. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions134120-5 Parametrizable electronic structure model J. Chem. Phys. 135 , 134120 (2011) have the leading term of atomic constants R(0) μνμ/primeν/prime,0kμ=kν=kμ/prime=kν/prime=/summationdisplay mlBllμlν mmμmνBllμ/primelν/prime mmμ/primemν/primeGllμlνlμ/primelν/primekμ (35) and additive corrections for the effect of surrounding atoms R(0) μνμ/primeν/prime,kkμ=kν=kμ/prime=kν/prime/negationslash=k= δmμmνδmμ/primemν/primeδlμlνδlμ/primelν/primeVlμlμ/primekμk(rμk), (36) for the latter we have also tested the general expression R(0) μνμ/primeν/prime,kkμ=kν=kμ/prime=kν/prime/negationslash=k=/summationdisplay mm/primell/primeBllμlν mmμmν ×Bl/primelμ/primelν/prime m/primemμ/primemν/primeBl/prime/primell/prime 0mm/primeVl/prime/primell/primelμlνlμ/primelν/primekμk(rμk), (37) but found only the terms with l/prime/prime=l=l/prime=0t ob ei m p o r t a n t , we also find it safe to set R(0) ανμ/primeν/primekα=kν=kμ/prime=kν/prime= 0. (38) Of the two-center two-electron integrals, only the long-range ones R(0) μνμ/primeν/primekμ=kν/negationslash=kμ/prime=kν/prime=/summationdisplay ¯m¯¯m¯m/prime¯¯m/primemll/primeAlμ ¯mmμ(zμμ/prime)Alν ¯¯mmν ×(zμμ/prime)Alμ/prime ¯m/primemμ/prime(zμμ/prime)Alν/prime ¯¯m/primemν/prime(zμμ/prime) ×Bllμlν m¯m¯¯mBl/primelμ/primelν/prime m¯m/prime¯¯m/primeGmll/primelμlνlμ/primelν/primekμkμ/prime(rμμ/prime), (39) and in the same way R(0) ανμ/primeν/prime, are included in the model, we studied the effect of other two-center terms and found it safe to set them to zero. We should stress that it is thanks to the special properties of the underlying zero-bond-dipole28or- thogonalization of the basis functions that all two-electron integrals involving two-center product charge distributions,φ μ(r)φν(r),kμ/negationslash=kν, are small and can be neglected. The two- electron dispersion-model integrals are naturally limited to the isotropic two-center terms R(6) μνμ/primeν/primekμ=kν/negationslash=kμ/prime=kν/prime= δμνδμ/primeν/primeG(6) lμlμ/primekμkμ/prime(rμμ/prime). (40) The zeroth-order eigenvalues in Eq. (7)can be optimized as atomic parameters but we find it enough to set εμ=¯Fμμ,ε α=1. (41) Now that all molecular integrals in our model are de- fined in terms of constants and functions of one variable r, parametrized formulas for the latter are needed. The short- range functions of Eqs. (28),(30),(32), and (36) can be fitted to a high accuracy by the general expansion f(r;a,c 0,...,c n)=exp(−ar)n/summationdisplay κ=0(ar)κcκ, (42) if enough terms are taken, the long-range functions of Eq. (39) can be done likewise with one long-range term added gl(r;q,a,c 0,...,c n)=qr−1−lu2l(ar)+a1+lexp(−ar)n/summationdisplay κ=0(ar)l+κcκ, (43) with un(r)=1−exp(−r)n/summationdisplay m=0rm m!, (44) and the dispersion tail corrections (40) can easily be added in the form g(6)(r;a,c)=cr−6u9(ar). (45) We have experimented with these general expansions within our electronic structure model (working with more than 20 000 atomic and atom-pair parameters for a set of seven chemical elements!) but found later that much more compactexpressions using sum rules with mostly atomic parameters work quite well and yield more meaningful optimized param- eter values. The short-range terms can be given either by asingle exponential ( n=0) term of Eq. (42) or by a three- parameter hyperbolic-secant function f o(r;a,b,c )=cexp(−ar+b)(1+exp(−2ar+2b))−1, (46) withb>0. For the long-range terms, the leading term of Eq.(43) is enough for all multipole-multipole and charge- multipole interactions and only a single exponential termneeds to be added for the charge-charge interactions. The sum-rule formulas used in this work are as follows. In Eq. (28),w eh a v e F mlμlνkμkν(r)=f(r;√aμaνamlμlνξμξν,cμcνcmlμlνξμξν), (47) with the shorthand notation for atomic parameters aμ≡alμkμ, cμ≡clμkμ, andξμbeing atom-group labels – in particular, ξμ=1i fμis on H, ξμ=2i fμis on a second-row atom Li– F, and ξμ=3f o rμon Na–Cl. Thus, we work with atomic parameters and diatomic atom-group parameters amlμlνξμξν, cmlμlνξμξνin Eq. (47), and this is also done in the formulas be- low. For minimal-polarization analog of Eq. (28), we choose Fmlαlνkαkν(r)=fo/parenleftbigg r;2aαaν aα+aνamlαlνξαξν,bξμξν,cαcνcmlαlνξαξν/parenrightbigg (48) and it should be understood that, here and below, each integral class has its own set of parameters, for example, aνin Eq. (47) is not the same as aνin Eq. (48) – the same notation is used to save space. In Eq. (30), we choose the form Vllμlνkμk(r) =fo/parenleftbigg r;2(aμ+aν)ak aμ+aν+2akallμlνξμξk,bξμξk,1 2(cμ+cν)ckcllμlνξμξk/parenrightbigg (49) for both minimal-minimal and minimal-polarization integrals. In Eq. (32),w eh a v e Smlμlλkμkλ(r)=fo/parenleftbigg r;2aμaλ aμ+aλ,bξμξλ,cμcλcmlμlλξμξλ/parenrightbigg . (50) Downloaded 18 Mar 2013 to 129.89.24.43. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions134120-6 Dimitri N. Laikov J. Chem. Phys. 135 , 134120 (2011) In Eq. (36),w eh a v e Vlμlμ/primekμk(r) =fo/parenleftbigg r;2(aμ+aμ/prime)ak aμ+aμ/prime+2ak,bξμξk,1 2(cμ+cμ/prime)ckclμlμ/primeξμξk/parenrightbigg . (51) In Eq. (39),t h ef o r m Gmll/primelμlνlμ/primelν/primekμkμ/prime(r)=gl+l/prime/parenleftbigg r;qllμlνql/primelμ/primelν/primeqmll/prime, 2(aμ+aν)(aμ/prime+aν/prime) aμ+aν+aμ/prime+aν/prime,−1 2δ0lδ0l/prime/parenrightbigg (52) is used for both 4-minimal and 1-polarization-3-minimal two-electron integrals, with the atomic multipoles qllμlνand the fundamental constants of multipole-multipole interaction qmll/prime, in the charge-charge 4-minimal case one exponential term is added. In Eq. (40),w eh a v e G(6) lμlμ/primekμkμ/prime(r)=g(6)/parenleftbigg r;2aμaμ/prime aμ+aμ/prime,cμcμ/prime/parenrightbigg . (53) As can be seen, the atomic a-parameters in all these for- mulas play the role of radial scale factors, the atomic c- parameters are multiplicative prefactors, and the atom-group b-parameters control the shape of functions at shorter dis- tances r. C. Parametrization procedure We parametrize our model by minimizing the function E=EE+EF+ED+Eg+EH+Ed +Ea+Eq+EU+E6+Ec, (54) that measures the deviation of molecular properties predicted by the model from the reference values computed by the higher level theory on a set of molecules (molecular geome-tries). The energy term E E=/summationdisplay nwE n/parenleftBigg/summationdisplay mcmn(˜Em−Em)/parenrightBigg2 , (55) withm=1,...,M labeling each molecule, can be used not only to fit each predicted energy ˜Emto the reference value Em in the trivial case cmn=δmnbut also for giving higher weights wE nto some chemically meaningful energy differences, such as conformational energy changes, reaction energies, and ac- tivation barriers. The effective one-electron Hamiltonian terms EF=/summationdisplay mwF mtr{(˜Fm−Fm)2} (56) and the density-matrix terms ED=/summationdisplay mwD mtr{(˜Dm−Dm)2} (57) for each molecule take ˜Fmfrom Eq. (17) andFmfrom the theory of Sec. II D below, the density matrices ˜DmandDm come from the diagonalization of ˜FmandFm. One may putallwD m=0 and work with only wF m/negationslash=0, assuming that an accurate F-matrix should also yield an accurate D-matrix, but in practice we have not-so-small errors in F-matrices and find it a better compromise to balance between FandDterms, thus giving more importance to the occupied-occupied and occupied-virtual blocks of F. The energy first gmand second Hmderivatives with re- spect to atomic coordinates for each molecule mare gathered into the terms Eg=/summationdisplay mwg m(˜gm−gm)TW2 m(˜gm−gm), (58) EH=/summationdisplay mwH mtr{((˜Hm−Hm)Wm)2}, (59) where the weight matrix W=/parenleftbig H2+h2 01/parenrightbig−1/2(60) is a well-behaved positive-definite replacement for the inverse H−1. This weighting scheme emphasizes the weak modes and, in our experience, it drives the model towards a balanced reproduction of potential energy surfaces. We set h0=2−10 in this work, close to the lowest eigenvalue of Hfor the water dimer (H 2O)2used as a prototype. For reference geometries at stationary points ( gm=0), as is normally the case, Eq. (58) can be seen as a weighted sum of squared distances betweenthe stationary points estimated by the model and the refer- ence, whereas Eq. (59) is a weighted sum of squared dimen- sionless relative deviations of force constant values. Electric dipole moments and polarizabilities E d=/summationdisplay mwd m|˜dm−dm|2, (61) Ea=/summationdisplay mwa mtr{(˜am−am)2}, (62) as well as atomic polar tensors Eq=/summationdisplay mwq mtr{(˜qm−qm)T(˜qm−qm)}, (63) are included into the optimization in a straightforward way as are the molecular electrostatic potentials EU=/summationdisplay mwU m/summationdisplay κ(˜U(rκ)−U(rκ))2sκ, (64) on the grids of points rκat discretized surfaces with elements sκ(see also Sec. II F). Molecular dispersion coefficients (discussed in Sec. II E below) are treated in the form of square roots E6=/summationdisplay mw(6) m/parenleftBig ˜c1/2 6,m−c1/2 6,m/parenrightBig2 , (65) as these have a more natural system-size scaling. The last (optional) term is a sum of soft constraints Ec=/summationdisplay κwc κs/parenleftBigg/summationdisplay pcpκxp,¯cκ,¯¯cκ/parenrightBigg , (66) Downloaded 18 Mar 2013 to 129.89.24.43. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions134120-7 Parametrizable electronic structure model J. Chem. Phys. 135 , 134120 (2011) s(x,a,b )=⎧ ⎪⎨ ⎪⎩(x−a)2,x < a , 0, a<x<b , (x−b)2,b < x ,(67) on linear combinations of optimization parameters {xp}. D. Effective Hartree-Fock-like Hamiltonian from correlated wavefunction theory Given the one-electron nonidempotent density matrices Dμνσfrom a correlated wavefunction calculation, with μ,ν labeling (in this section only) the functions of an extended atomic basis, we set up our Hartree-Fock-like self-consistent field procedure based on the minimization of the energy ex-pression E (w)=/summationdisplay μνσD(w) μνσHμν+1 2/summationdisplay μνσμ/primeν/primeσ/primeD(w) μνσRμνσμ/primeν/primeσ/primeD(w) μ/primeν/primeσ/prime +w 2/summationdisplay μνμ/primeν/primeσ/parenleftbig D(w) μνσ−Dμνσ/parenrightbig Rμνμ/primeν/prime/parenleftBig D(w) μ/primeν/primeσ−Dμ/primeν/primeσ/parenrightBig (68) with respect to the idempotent density matrices Dμνσ. Equa- tion(68) is a sum of the Hartree-Fock energy and a quadratic density penalty function weighted by w, the associated effec- tive Hamiltonian matrix F(w) μνσ=Hμν+/summationdisplay μ/primeν/primeσ/primeRμνσμ/primeν/primeσ/primeD(w) μ/primeν/primeσ/prime +w/summationdisplay μ/primeν/primeRμνμ/primeν/prime/parenleftBig D(w) μ/primeν/primeσ−Dμ/primeν/primeσ/parenrightBig (69) is a sum of the Fock matrix and a local spin-dependent corre- lation potential matrix, the latter arises as the scaled Coulomb potential of the difference spin density. If a complete basis were used, the limit w→∞ , if exists, should yield the ex- act effective Hamiltonian of Sec. II B of Kohn and Sham’s work.10In practice, we work with a finite incomplete basis and a reasonable finite value of wshould be chosen. Our pro- cedure can be seen as a nonlocal-exchange-local-correlationanalog of the local-exchange-correlation procedure of Zhao et al. 29 E. Molecular dispersion coefficients For two molecules at a large distance rbetween their cen- ters, the leading asymptotic term of the dispersion part of the intermolecular potential is c6r−6withc6being a function of the relative orientation. The orientational average of c6is a bimolecular constant that can be most easily computed within MP2 as c6=2 3mol.1/summationdisplay aiσmol.2/summationdisplay a/primei/primeσ/prime|/angbracketleftφaσ|r|φiσ/angbracketright|2|/angbracketleftφa/primeσ/prime|r|φi/primeσ/prime/angbracketright|2 /epsilon1iσ−/epsilon1aσ+/epsilon1i/primeσ/prime−/epsilon1a/primeσ/prime(70) from the dipole moment integrals over one-electron wave- functions φand the one-electron energies /epsilon1,i’s label the oc- cupied and a’s the virtual states each localized on one of themolecules. Much more complicated expressions can be de- rived for higher order correlation methods, but we will use Eq.(70) in this work as it gives accurate enough values for our purpose. We take only the case of two identical molecules and such c6becomes a molecular constant of interest; more- over, we have seen that a heteromolecular c6is quite close to the geometric mean of two molecular constants. Within our parametrizable model the value to put into Eq. (65) is simply ˜c1/2 6=/summationdisplay μσDμμσcμ (71) withcμas in Eq. (53). By construction, Eq. (53) always gives isotropic c6values that follow the rule of geometric mean. F. Surfaces for sampling molecular electrostatic potentials A smooth surface is preferable for sampling the molecu- lar electrostatic potential, so we have tried the isodensity sur- faceρ(r)=ρ0and found that ρ0≈10−6should be used to enclose most of the density as needed. While this may work well for the exact density, the one calculated within a lim-ited finite basis approximation may not be accurate enough at such small values in the tail region; indeed, we have seen some rather weird shapes for molecules as simple as LiF. In this work, we set a new definition of molecular surface p(r)=p 0with p(r)=/integraldisplay s(r/prime−r)ρ1(r/prime,r/prime/prime)s(r/prime/prime−r)d3r/primed3r/prime/prime(72) in terms of the one-electron density matrix ρ1(r,r), the den- sity being its diagonal part ρ(r)=ρ1(r,r), as inspired by the analysis of the exchange repulsion effects. For the broadening function s(r) localized around r=0,we make the simplest choice s(r)=cexp(a|r|2), (73) in the limit a→∞ withc=(a/π)3/2Eqs. (72)and(73) yieldp(r)→ρ(r). With c=1,a=1/16, and p0=1/4w e get good surfaces for the molecules studied in this work. Thesurface discretization is best done with the spherical quadra- ture rules. 55,56 III. CALCULATIONS The training set of 5581 molecules used in this work cov- ers a broad range of structures built up from 15 elements H, TABLE I. Atomic basis set sizes for CCSD calculations.a Atom Set Core Contracted Primitive H L1 2,1 8,4 Li, Be L11 4,3,1 12,8,4 B, C L1 1 3,2,1 12,8,4 N, O, F L1+1 1 4,3,1 14,10,4 Na, Mg L11 1 5,4,2 18,13,8 Al, Si, P L1 2,1 4,3,1 18,13,5 S, Cl L1+1 2,1 5,4,1 20,15,5 aNote: Number of radial functions for each angular quantum number is given. Downloaded 18 Mar 2013 to 129.89.24.43. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions134120-8 Dimitri N. Laikov J. Chem. Phys. 135 , 134120 (2011) TABLE II. Atomic basis set sizes in the parametrized model.a Atom lmax μ lmax α lmax λ H 011 L i , B e 112 B, C, N, O, F 1 2 2 N a , M g 112Al, Si, P, S, Cl 1 2 2 aNote: Maximum angular quantum number in the minimal lmax μ, polarization lmax α,a n d projection (31)lmax λbasis. Li–F, Na–Cl, there are both energy-minimum and transition- state geometries of neutral, cationic, and anionic species with closed-shell and (high-spin) open-shell electronic configura- tions dominated by a single determinant. We have spent a lotof time setting up this database guided by our chemical intu- ition, finding meaningful structures of (almost) all chemical compositions with up to three nonhydrogen and any numberof hydrogen atoms, and less thoroughly for up to eight nonhy- drogen atoms. We tried to sample all kinds of chemical bonds – from metallic through covalent to ionic, hydrogen bondeddimers and clusters (for example, neutral, protonated, and de- protonated water clusters with up to six O atoms) as well as weaker intermolecular complexes are also carefully chosen. The reference data were generated by the CCSD method 13with nonrelativistic Hamiltonian and correlation- consistent atomic basis sets57of sizes shown in Table I.T h eL1set has one set of valence polarization functions, the L11 set is for a correlated treatment of (outermost) core shells, and theL1+1 set has diffuse functions added as Rydberg shells – this is the entry-level basis for quantitative CCSD calcu- lations, we would have preferred the next level ( L2,L22, L2+1 ) had we had enough computer power. An archive of all molecular data files is available for download.58 The model is defined by the energy expression of Sec. II A, the molecular integrals of Sec. II B, and the atomic basis set sizes given in Table IIwith one radial function for each angular symmetry. Besides the full model, eight simpli- fied models with some classes of integrals set to zero are also analyzed. All parameters of each model have been simulta- neously optimized on the full reference data set, the weightsassigned to each molecular property and the values of each error term in Eq. (54) at convergence are given in Table III, full details of the settings along with the optimized parametervalues can be found in the attached files. 58Our optimization algorithm computes numerically all first derivatives of molec- ular properties with respect to the parameters and uses linearsearches along the optimal direction to find the nearest lo- cal minimum, the choice of the starting guess is so far from straightforward that it cannot be documented here. It is not trivial to judge the quality of such a model by the net error terms, but still some insight into the role of the whole classes of parametrized molecular integrals can be gotas they are removed from the model in the order of increasing TABLE III. Errors in mo lecular properties from the parametrized models. Integral termaEqs. Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7 Model 7/primeModel 8 Model 9 ¯F(kμ=kν/negationslash=k) μν,k(30) +++++++++ ¯F(kμ/negationslash=kν/negationslash=k) μν,k(31) ++++++++ G(l>0,kμ=kν=kμ/prime=kν/prime) llμlνlμ/primelν/primekμ(35) ++++++ q(l>0) llμlν(52) +++++ R(0)(kμ=kν=kμ/prime=kν/prime/negationslash=k) μνμ/primeν/prime,k(36) ++++ ¯F(kα/negationslash=kν) αν,0(28) ++ ¯F(kα=kν/negationslash=k) αν,k(30) ++ R(0)(kα=kν/negationslash=kμ/prime=kν/prime) ανμ/primeν/prime,0(39) +++ R(6) μνμ/primeν/prime (40) ++ + Number of parametersb720 690 524 513 434 406 378 378 264 185 Error termcEqs. w EEtotal (55) 213975.8 1013.0 1718.5 2206.1 3307.7 5006.1 5940.6 4498.5 8176.7 12026.6 EEreact (55) 222228.1 255.1 497.8 562.6 1022.6 1459.8 1662.8 591.4 3010.9 4251.5 EF (56) 2−2998.3 989.6 1131.8 1190.0 1013.9 1107.5 1266.4 902.5 1684.5 1811.7 ED (57) 1 267.2 283.1 522.7 552.3 675.7 771.5 1042.8 720.9 1318.9 1115.3 Eg (58) 253194.3 3255.2 5067.6 5978.6 9712.6 12905.1 13887.4 3432.9 5439.0 8111.3 EH (59) 1 1961.9 1987.2 3398.4 3957.8 6313.1 7774.1 9329.4 1872.7 2990.5 4356.6 Ed (61) 2−37.5 7.6 11.7 22.6 31.5 56.9 63.6 47.2 86.7 119.7 Ea (62) 2−1062.6 58.0 70.0 ... ... ... ... ... ... ... Eq (63) 1 196.1 199.9 289.3 477.6 615.7 661.4 782.9 669.1 1315.5 1258.3 EU (64) 23324.3 324.5 332.2 331.6 305.9 262.3 255.6 278.5 263.5 232.2 E6 (65) 2−385.4 ... 200.4 284.2 ... ... ... ... ... ... (60) h0 2−102−102−102−102−102−102−102−52−52−5 aThe integral terms of the equations given are included in the model if marked by the “ +” sign, otherwise set to zero. bThe number of independent parameters optimized for each model. cThe error terms of the equations given with “ w” the assigned weights and the “ h0” values of Eq. (60) are listed (in a.u.) for each model if included in the optimization, otherwise assigned zero weight and marked by “–.” Downloaded 18 Mar 2013 to 129.89.24.43. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions134120-9 Parametrizable electronic structure model J. Chem. Phys. 135 , 134120 (2011) importance. As expected, the dispersion terms (40) have the smallest impact as shown by Model 2. The role of the polarization basis functions is dramatic – Model 4 that hasthem all removed nearly doubles the most important error terms, and about 2/3 of this effect come from the one-electron integrals (28)and(30), Model 3 with only the two-electron integrals (39) over the polarization functions included is the simplest one to properly account for the molecular dipole polarizability but this does also improve the other molecularproperties. Starting from Model 4, we can study the role of the integrals over the minimal basis that are new to our model. The two-center corrections to one-center two-electron integrals (36) are the least important, though Model 5 without them shows already about 1.5 greater values of most errorfunctions. Further degradation is caused by zeroing out the multipole moments in the two-center two-electron integrals (52) as in Model 6 and, further on, in the one-center two- electron integrals (35) as in Model 7. At this point, Model 7 is already too simplified to support the burden of the first (58) and second (59) derivative terms as they were before, so we need to relieve it by setting a larger value of h 0in Model 7/prime to get ready for the last strike. A dramatic degradation is seen as the three-center corrections to two-center one-electronintegrals (31) are zeroed out in Model 8, and yet again as the two-center corrections to one-center one-electron integrals (30) are dropped in Model 9. With this overview, it should be clear that these integral terms have not only the quantitative but also the qualitative impact on the performance of the model – the simplified models may even fail to reproduce the existence of some stationary point on molecular potential energy surfaces, for example, Model 6 predicts the symmetryof the hydrogen peroxide molecule H 2O2to be C 2hinstead of C 2, and Model 8 gives the cyclobutane molecule C 4H8 the D 4hsymmetry instead of the right D 2d– the reliable prediction of intra- and intermolecular conformations is a challenge that only our full model seems to meet. A binary executable code of our program for LINUX/x86_64 architecture is made available58to the interested researchers worldwide for further testing and some preliminary applications of the new model. IV. CONCLUSIONS We are pleased to have developed the new parametriz- able electronic structure model that both has a number ofgood formal properties and performs well in numerical tests. It evolved through a trial-and-error process until it has reached a satisfactory level of maturity. With this parameter set for 15elements, many interesting systems can already be studied, we are going to see how it works for the structure predic- tion of biomolecules and the chemical processes in condensedphase. If some weaknesses will be found, the model can be reparametrized on a more representative training set. Future work may include: building a comprehensive database of prototype molecular structure for further training and testing of the model, its extension to heavier elements, a reparametrization based on more accurate reference data from CCSD or CCSD(T) calculations with a next-level basis, the investigation of linear-scaling techniques 59–64for solving theSCF equations of our model to find a both accurate and fast algorithm for studies of large systems. A formalism to treat excited electronic states dominated by single excitations froma single determinant can also be worked out, either within the configuration interaction with single substitutions (CIS) or within the linear response theory. ACKNOWLEDGMENTS It took us five years of hard work to come up with this new electronic structure model. 1S. Lifson and A. Warshel, J. Chem. Phys. 49, 5116 (1968). 2A. Warshel and M. Levitt, J. Mol. Biol. 103, 227 (1976). 3A. Warshel and R. M. Weiss, J. Am. Chem. Soc. 102, 6218 (1980). 4D. W. Brenner, Phys. Rev. B 42, 9458 (1990). 5A. C. T. van Duin, S. Dasgupta, F. Lorant, and W. A. Goddard III, J. Phys. Chem. A 105, 9396 (2001). 6P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964). 7R. Hoffmann, J. Chem. Phys. 39, 1397 (1963). 8X.-P. Li, R. W. Nunes, and D. Vanderbilt, P h y s .R e v .B 47, 10891 (1993). 9A. H. R. Palser and D. E. Manolopoulos, Phys. Rev. B 58, 12704 (1998). 10W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965). 11C. 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Chem. 10, 209 (1989). 40M. Y . Filatov, O. V . Gritsenko, and G. M. Zhidomirov, Theor. Chim. Acta 72, 211 (1987). 41J. J. P. Stewart, J. Mol. Model. 13, 1173 (2007). 42M. Kolb and W. Thiel, J. Comput. Chem. 14, 775 (1993). 43W. Weber and W. Thiel, Theor. Chem. Acc. 103, 495 (2000). 44J. P. McNamara and I. H. Hillier, Phys. Chem. Chem. Phys. 9, 2362 (2007). 45T. Tuttle and W. Thiel, Phys. Chem. Chem. Phys. 10, 2159 (2008). 46R. Ahlrichs, R. Penco, and G. Scoles, Chem. Phys. 19, 119 (1977). 47J.ˇRezá ˇc, J. Fanfrlík, D. Salahub, and P. Hobza, J. Chem. Theory Comput. 5, 1749 (2009). Downloaded 18 Mar 2013 to 129.89.24.43. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions134120-10 Dimitri N. Laikov J. Chem. Phys. 135 , 134120 (2011) 48M. Korth, J. Chem. Theory Comput. 6, 3808 (2010). 49C. H. Rhee, R. M. Metzger, and F. M. Wiygul, J. Chem. Phys. 77, 899 (1982). 50T. J. Giese and D. M. York, J. Chem. Phys. 123, 164108 (2005). 51D. T. 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1.364646.pdf

Effect of the large magnetostriction of Terfenol-D on microwave transmission G. Dewar Citation: Journal of Applied Physics 81, 5713 (1997); doi: 10.1063/1.364646 View online: http://dx.doi.org/10.1063/1.364646 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/81/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Enhanced microwave absorption in columnar structured magnetic materials J. Appl. Phys. 112, 083908 (2012); 10.1063/1.4758384 Effects of magnetocrystalline anisotropy constant K 2 on magnetization and magnetostriction of Terfenol-D Appl. Phys. Lett. 98, 012503 (2011); 10.1063/1.3533910 Cluster glass induced exchange biaslike effect in the perovskite cobaltites Appl. Phys. Lett. 90, 162515 (2007); 10.1063/1.2730737 Interface roughness effects on coercivity and exchange bias J. Appl. Phys. 97, 10K105 (2005); 10.1063/1.1847931 A nonlinear constitutive model for Terfenol-D rods J. Appl. Phys. 97, 053901 (2005); 10.1063/1.1850618 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.237.29.138 On: Thu, 27 Aug 2015 03:41:53Effect of the large magnetostriction of Terfenol-D on microwave transmission G. Dewar Physics Department, University of North Dakota, Grand Forks, North Dakota 58202 The calculated transmission of microwave power at GHz frequencies through Terfenol-D (Tb0.27Dy0.73Fe2) is strongly influenced by the enormous magnetostriction parameter of this material. For conditions satisfying ferromagnetic resonance there is a peak in the transmissioncorresponding to sound and spin waves transporting energy through a slab-shaped sample. Themagnetostriction mixes the character of these waves; the calculated transmission also depends onthe exchange stiffness parameter, the elastic constants, and the radio-frequency magnetic properties.With thin ~,40 mm!samples for which the static magnetization is along the magnetically soft ~111! direction the calculated transmission peak is augmented by a fringe pattern with more than 20secondarypeaks.Thisarisesfrominterferencebetweenspinandsoundwaves.Fortheconfigurationwith the static magnetization oriented along the magnetically hard ~100!direction the fringe pattern is absent. In this configuration the spin wave is a soft mode and the sample should develop a staticdistortion of the magnetization and lattice. The periodicity of this static distortion is tunable with anexternal magnetic field, although lattice imperfections may introduce hysteresis by pinning thedistortion. © 1997 American Institute of Physics. @S0021-8979 ~97!51308-9 # I. INTRODUCTION Terfenol-D ~Tb0.27Dy0.73Fe2!is a magnetic material hav- ing enormous magnetostriction.1,2Much work has been de- voted to characterizing the static and low-frequency proper-ties of Terfenol-D ~see, for example, Ref. 3 !. A review of the recent literature has not revealed any attempts to determinethe rf magnetic properties from a microwave experimentsuch as ferromagnetic resonance ~FMR !. This is not surpris- ing since Terfenol-D’s unique magnetostrictive behavior willnot be clearly exhibited in a reflection experiment unless thesample is extremely thin. 4,5The magnetoelastic properties, however, figure prominently in a microwave transmissionexperiment. It is the purpose of this article to present calcu-lations of the expected microwave transmission throughTerfenol-D and to draw attention to the qualitative featuresthat are due to the large magnetostriction. Experiments tomeasure this microwave transmission are currently underwayin our laboratory. II. CALCULATION A summary of the transmission calculation is presented here, the details are available elsewhere.6,7An incident mi- crowave beam is assumed to be normally incident on oneside of a slab of Terfenol-D which also has an static mag-netic field applied to it. Damped electromagnetic, sound, andspin waves are excited and transport energy across the slab;some of this energy is radiated in the form of microwavesinto the region on the opposite side of the slab. The response of the slab to the microwaves is described by Maxwell’s equations, Hooke’s law for the elastic con-tinuum, and the Landau–Lifshitz equation of motion for themagnetization as modified by Gilbert. The response of themedium at microwave frequencies is assumed small com-pared to static electromagnetic fields and deformations; thisallows the Landau–Lifshitz equation to be linearized. Therequirement that a wave of the form exp( ikx2i vt) satisfythese equations leads to a seventh-degree polynomial in k2 being zero. The seven roots of this polynomial nominally correspond to two electromagnetic waves, three soundwaves, and two spin waves. Magnetostriction causes somewaves to be strongly mixed for certain combinations of fre-quency and static magnetic field. Maxwell’s equations andthe equations of motion, once integrated across the materialdiscontinuities at the slab surfaces, yield boundary condi-tions which the waves must satisfy. It is the solution of theseboundary equations for the ratio of the transmitted amplitudeto the incident microwave field that is plotted in Figs. 1–3. The calculation requires material parameters describing Terfenol-D; the values used are listed in the caption of Fig.1. These are measured parameters with the exception of the g factor, magnetic damping, and exchange constant, which areestimates, and the elastic constants which are for a similarmaterial, 4Ho0.15Dy0.85Fe2. All of the measured parameters have been obtained at dc or kilohertz frequencies. Magnetoelastic effects are conventionally inserted into the equations of motion for the magnetization and the latticethrough the interaction energy 4 UME5B1( i513 eiiai21B2( i,j3,3 eijaiaj, ~1! where the eijare the strains defined by Kittel8and the aiare the direction cosines of the magnetization. The magnetostric-tion constants available in the literature, l 100andl111, are related to the Bby9 B1523 2~C112C12!l100 ~2! and B2523C44l111. ~3! TheCijare the conventional cubic elastic constants. In Terfenol-D it is10l11151.6431023, or equivalently B2, which is large. This leads to a large coupling between sound 5713 J. Appl. Phys. 81(8), 15 April 1997 0021-8979/97/81(8)/5713/3/$10.00 © 1997 American Institute of Physics [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.237.29.138 On: Thu, 27 Aug 2015 03:41:53waves involving11C44and the spin waves and the electro- magnetic wave influenced by the magnetization. This cou-pling is especially apparent if conditions are such that thesewaves have similar wavelengths.III. RESULTS The calculated transmission for various orientations of the magnetization relative to the crystal axes are shown inFigs. 1–3. Several features of the calculated transmission arenovel. Most obvious is the huge the transmission that is pos-sible at FMR ~see Fig. 1 !. In the units used in the calculation a transmission of 1 310 26is considered large. The calculated transmission amplitude through 40 mm of Terfenol-D is two orders of magnitude larger than this. The fringe pattern apparent in the transmission of Fig. 1 is primarily due to interference between two sound wavesthat propagate across the slab. The calculation for a 20- mm- thick slab indicates that, at applied fields less than requiredfor FMR, three waves, one of which is a spin wave, contrib-ute substantially to the transmission. This stands in sharpcontrast to nickel where, at FMR, only one wave has beenobserved to transport energy through the sample. 12–13Mul- tiple reflections are not important in Fig. 1 since the largemagnetostriction of Terfenol-D causes sound waves to havea substantial magnetic character which leads to moderate at-tenuation, due to ohmic losses and magnetic damping, overdistances of tens of microns. However, this attenuation issevere and the excitation of the wave is weak at FMR ifpropagation is along the ~111!direction, as shown in Fig. 2. In this case it is the transmission at fields removed fromFMR that is large. The fringe pattern detailed in the inset ofFig. 2 shows geometric resonances for multiple reflections ofa single propagating sound wave which occur as the mag-netic field alters the sound wavelength. Figure 3 shows the calculated transmission for the mag- netization oriented along the magnetically hard ~010!direc- tion. For this orientation the transmission at FMR is largebut, for the polarization of incident microwaves assumed, isdue to only one sound wave. The topology of the dispersion relations for the various waves excited in the Terfenol-D is unique. Essentially, thelarge magnetostriction strongly mixes all the sound waveswith the resonant spin wave and the extraordinary electro- FIG. 1. Transmission amplitude vs applied magnetic field. The calculation assumes a Terfenol-D sample thickness of ~a!20mm,~b!40mm, and ~c!50 mm. The waves propagate along the ~11¯0!direction and the static magnetic field is along the magnetically easy ~111!direction. Parameters used in the calculation are: gfactor 52.2;f516.95 GHz; 4 pM510 116 G; B1521.43108erg/cm3;B2522.33109erg/cm3;K1526.03105erg/cm3; K2522.03106erg/cm3; density: 9.036 g/cm3;C1151.4131012erg/cm3; C1256.4831011erg/cm3;C4454.8731011erg/cm3, Gilbert damping l53.753107s21; exchange stiffness A59.031027erg/cm; and resistivity: 60mVcm. The field at which ferromagnetic resonance occurs is noted with an arrow. FIG. 2. Transmission amplitude vs applied magnetic field. The calculationassumes a Terfenol-D sample 50 mm thick. The waves propagate along the ~111!direction and the static magnetic field is along the ~11¯0!direction. The field at which ferromagnetic resonance occurs is noted with an arrow. Theinset shows details of the high-field transmission on an expanded scale. FIG. 3. Transmission amplitude vs applied magnetic field. The calculationassumes a Terfenol-D sample 50 mm thick. The waves propagate along the ~100!direction and the static magnetic field is along the ~010!direction. The field at which ferromagnetic resonance occurs is noted with an arrow. 5714 J. Appl. Phys., Vol. 81, No. 8, 15 April 1997 G. Dewar [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.237.29.138 On: Thu, 27 Aug 2015 03:41:53magnetic wave ~see Fig. 4 !.~In high-symmetry directions, however, one or two of the sound waves can decouple fromthe spin-wave–electromagnetic-wave system. !This mixing results in the character of all these dispersion relationschanging, as outlined in the caption to Fig. 4. It is this strongmixing of the spin wave with the sound waves, caused by thelarge magnetostriction, that allows the spin wave to enterinto the transmission. In other materials the spin wave playsno appreciable role in transmission. 14 Another unique feature of the dispersion relations is ex- hibited by the curve labeled spin ~100!in Fig. 4. This shows how the curve labeled spin evolves if the applied magneticfield is reoriented and made large enough to overcome themagnetic anisotropy and align the magnetization in the mag-netically hard ~100!direction. This spin wave, which ordi- narily mixes with a sound wave, is driven to zero frequencyat a nonzero propagation constant, i.e., the spin wave modebecomes soft. It is the large magnetostriction that is respon-sible for the softening of the spin wave. This softening isinhibited if the magnetization lies along an easy directionand the anisotropy is sufficiently large.A proper minimization of the energies involved in deter- mining the static distortion of the lattice and magnetization,as described by James and Kinderlehrer, 15is beyond the scope of this article. However, a simple minimization of theelastic, magnetoelastic, exchange, anisotropy, and magneto-static energies assuming a sinusoidal variation of the magne-tization and lattice yields a 2 p/lfor the static distortion which is somewhat less than the kspinfor which v!0 ~kspin51.03106cm21for the conditions of Fig. 4 !. This static distortion’s wavelength can be tuned by adjusting theexternal magnetic field. The lower limit for the magneticfield is set by the anisotropy constants; the field must bestrong enough to align the net magnetization along the hard~010!axis. The upper limit is set by the field at which k spinis zero at v!0. Increasing field causes the curve labeled spin ~100!to rise vertically on Fig. 4 as the energy to create a spin wave increases. This range is 1.49 <H<4.87 kOe for the parameters used here. Any pinning centers would, of course,cause hysteresis and alter these limits somewhat. IV. CONCLUSION The calculated microwave transmission through Terfenol-D indicates that the transmission should be largeand show interference effects; this is due to Terfenol-D’slarge magnetostriction. Sound and spin waves are stronglymixed via magnetostriction and transmission experimentscould be used to measure the exchange stiffness ofTerfenol-D as well as other rf magnetic properties. Undercertain conditions, the resonant spin wave is a soft mode anda relatively simple static distortion of the lattice and magne-tization is predicted to exist. 1A. E. Clark and H. S. Belson, AIP Conf. Proc. 10, 749 ~1973!. 2A. E. Clark, J. E. Cullen, O. D. McMasters, and E. B. Callen, AIP Conf. Proc.29, 192 ~1976!. 3D. C. Jiles, J. Phys. D 27,1~1994!. 4C. Vittoria, J. N. Craig, and G. C. Bailey, Phys. Rev. B 10, 3945 ~1974!. 5T. Kobayashi, R. C. Barker, and A. Yelon, Phys. Rev. B 7, 3286 ~1973!. 6G. Dewar, Phys. Rev. B 36, 7805 ~1987!. 7G. Dewar, J. Appl. Phys. 64, 5873 ~1988!. 8C. Kittel, Introduction to Solid State Physics , 7th ed. ~Wiley, New York, 1996!, pp. 81–82. 9C. Kittel, Rev. Mod. Phys. 21, 541 ~1949!. 10R. Abbundi and A. E. Clark, IEEE Trans. Magn. MAG-13 , 1519 ~1977!. 11J. R. Cullen, S. Rinaldi, and G. V. Blessing, J. Appl. Phys. 49, 1960 ~1978!. 12K. Myrtle, B. Heinrich, and J. F. Cochran, J. Appl. Phys. 52, 2250 ~1981!. 13G. C. Alexandrakis, R. A. B. Devine, and J. H. Abeles, J. Appl. Phys. 53, 2095 ~1982!. 14J. F. Cochran, B. Heinrich, and G. Dewar, Can. J. Phys. 55, 787 ~1977!. 15R. D. James and D. Kinderlehrer, J. Appl. Phys. 76, 7012 ~1994!. FIG. 4. Dispersion relations for pertinent electromagnetic, sound, and spin waves. The propagation constants for all but one of the waves are along the~111!direction with an applied magnetic field of 500 Oe along the ~11¯0! direction; the line labelled spin ~100!describes a spin wave propagating in the~100!direction with an applied magnetic field of 2.5 kOe along the ~010! direction. The dispersion relation for the electromagnetic wave ~EM!at high frequency becomes a heavily damped spin wave at low frequency. Thewiggle at '17 GHz corresponds to FMR as well as to where the uncoupled spin wave dispersion relation would meet the frequency axis. The uncoupledspin wave dispersion relation intersects the three sound wave dispersionrelations and magnetoelastic coupling results in the dispersion relationsshown. At high frequency, the spin wave labelled spin, the slow transversesound ~ST!, the fast transverse sound ~FT!, and the longitudinal sound wave ~L!become at low frequency a slow transverse sound, a fast transverse sound, a longitudinal sound, and an electromagnetic wave, respectively. Thespin wave in the ~100!direction has zero frequency at nonzero k; it is a soft mode. 5715 J. Appl. Phys., Vol. 81, No. 8, 15 April 1997 G. Dewar [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.237.29.138 On: Thu, 27 Aug 2015 03:41:53

1.369930.pdf

Dynamic FE simulation of μMAG standard problem No. 2 (invited) B. Streibl, T. Schrefl, and J. Fidler Citation: Journal of Applied Physics 85, 5819 (1999); doi: 10.1063/1.369930 View online: http://dx.doi.org/10.1063/1.369930 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/85/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Enhanced M r and ( B H ) m a x in anisotropic R 2 Fe 14 B ∕ α Fe composite magnets via intergranular magnetostatic coupling J. Appl. Phys. 99, 08B506 (2006); 10.1063/1.2162818 Investigation of hard magnetic properties of nanocomposite Fe-Pt magnets by micromagnetic simulation J. Appl. Phys. 96, 3921 (2004); 10.1063/1.1792812 Nonuniform magnetic structure in Nd 2 Fe 14 B/Fe 3 B nanocomposite materials J. Appl. Phys. 93, 8119 (2003); 10.1063/1.1537702 Behavior of μMAG standard problem No. 2 in the small particle limit J. Appl. Phys. 87, 5520 (2000); 10.1063/1.373391 Dynamic micromagnetics of nanocomposite NdFeB magnets J. Appl. Phys. 81, 5567 (1997); 10.1063/1.364663 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 142.150.190.39 On: Mon, 22 Dec 2014 09:55:03Dynamic FE simulation of mMAG standard problem No. 2 invited B. Streibl, T. Schrefl,a)and J. Fidler Institute of Applied Physics, Technischen Universitat Vienna, Wiedner Hauptstrasse 8, A-1040 Vienna, Austria mMAG standard problem No. 2 was studied using a three-dimensional finite element simulation based on the solution of the Gilbert equation. Asymptotic boundary conditions were imposed inorder to compute the demagnetizing fields and a Gilbert-damping parameter ~ a51.0!was used to drive the system towards equilibrium. The coercivities observed for the thin elongated platelet onwhich standard problem No. 2 is based show a slight dependence on its size. The width of theparticle was varied from 1 to 30 times the exchange length while keeping the aspect ratio of 5:1:0.1unchanged. An external field is applied parallel to the @111#direction, giving values of the coercive field ranging from 20.056 to 20.04 in units of the saturation magnetization M s. With the external field applied parallel to the long axis of the particle a strong dependence of the coercivity on its sizeis found which can be attributed to different reversal mechanisms. © 1999 American Institute of Physics. @S0021-8979 ~99!28308-9 # I. INTRODUCTION The purpose of this work is to solve standard problem No. 2 which was proposed by the National Institute of Stan-dards and Technology. This problem consists of computingdemagnetization curves of a defect-free particle of well de-fined geometry ~see Fig. 1 !with the external field applied parallel to the @111#direction and of considering the ex- change as well as the magnetostatic interaction, withoutmagnetocrystalline anisotropy. No specific material param-eters are needed since the particle size is varied in units of the exchange length l ex5(2Am0/Js2)1/2and all fields are given in units of Ms. Yuan and co-workers1made calcula- tions that partly govern the specifications of standard prob-lem No. 2 which show similar switching mechanisms andcomparable switching fields to the results obtained in thiswork. In order to compute demagnetizing fields, asymptoticboundary conditions 2are used. II. MODEL AND SIMULATION METHOD To study the evolution of the system in time we use the Gilbert equation dJ dt52ugum0J3H1a JsJ3dJ dt, ~1! whereJdenotes the magnetic polarization with constant magnitude Jsandais the damping constant which was set to 1.0. The effective field His defined as the variational deriva- tive of the free energy and is given by H52A Js2¹2J1Hd1Hext2]wan ]J, ~2! whereHdis the demagnetizing field which can be deduced from a scalar potential fbyHd52¹fandHexdenotes the external field. In this work the magnetocrystalline anisotropyenergy density w anwas set to zero according to specifications ofmMAG standard problem No. 2. To discretize the Gilbert equation we used Galerkin’s method which consists of mul-tiplying the equation under consideration by test functions wi(r) and integrating it over the underlying volume. Each of the test functions is associated with a node iatrisuch that wi(rj)5dij. The extent of all the nodes spans the finite ele- ment mesh which in this work was divided into cubic ele-ments. Approximation of the unknown quantities ~polariza- tionJand scalar potential f!is made using linear combinations of the test functions. Those used for approxi-mating the scalar potential are of higher order than thoseused for the polarization Jin order to avoid a loss of accu- racy resulting from different orders of differentiation~D f5¹M!.3The Galerkin method finally leads to a system of nonlinear equations for each time step which is solvediteratively by means of the Newton–Rapshon method. Dueto the use of asymptotic boundary conditions ~ABCs !, 4in order to compute demagnetizing fields a fully implicitmethod for time integration can be used which is stable evenfor large time steps. 5The disadvantage of using ABCs is the fact that elements are needed outside the region of interest.This partly compensates the reduced computation time dueto the use of large time steps. In order to find out when thesystem has reached the equilibrium state the maximumchange of the polarization per unit time is checked everytime step. In this work the system is said to be in the equi-librium state when this quantity which is proportional to the FIG. 1. Particle configuration of mMAG standard problem No. 2. For the calculations in this article the external field is applied parallel to the @100#or parallel to the @111#direction.JOURNAL OF APPLIED PHYSICS VOLUME 85, NUMBER 8 15 APRIL 1999 5819 0021-8979/99/85(8)/5819/3/$15.00 © 1999 American Institute of Physics [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 142.150.190.39 On: Mon, 22 Dec 2014 09:55:03torque exerted on the polarization vectors falls below 1023Js2uguor when the energy starts increasing due to nu- merical errors. In the latter case the continuation of timeintegration makes no sense because the energy must neces-sarily fall when approaching the equilibrium state. For dis-cretization of the geometry of standard problem No. 2 ~see Fig. 1 !500 elements were used, 10 along width d, 50 along lengthL, and one along thickness tof the particle. It is found 6that for hard magnetic particles a discretization of two nodes per exchange length lexis enough to properly obtain the coercivities. However, numerical experiments showedthat the coercive field uHcuof well discretized particles ~morethan two nodes per exchange length lex!with zero magneto- crystalline anisotropy still increases with increasing meshdensity. This effect may be attributed to an underestimationof the exchange energy which dominates in particles withzero magnetocrystalline anisotropy. Therefore the coercivi-ties uHcuobtained from the simulations must be interpreted as lower limits, especially for the biggest particles consid-ered, those with widths d530l ex. III. RESULTS The above algorithm was applied to calculate the mag- netization reversal of the particle configuration given in Fig.1 with the applied field parallel to either the @100#or the @111#direction. The size of the particle was varied from d 5l extod530lexwhile keeping the length ratios unchanged. The demagnetization curves were calculated quasistaticallywith a fully saturated polarization configuration as the initialcondition. Starting from H ext5Msthe field was reduced in steps of 0.2 Ms, 0.05Ms, and 0.002 Msfor the external field in intervals @1.0–0.2 #,@0.2–0.0 #, and @0.0Hc#, respectively. The external field decreased after an equilibrium state, de-fined in Sec. II, was reached. FIG. 2. Remanent and transient magnetic states during irreversible switch- ing of a particle with d530lexfor an applied field parallel to the @100#axis. The arrows indicate the direction of the magnetic polarization. FIG. 3. Demagnetization curve of a particle with d530lexfor the external field applied parallel to the @111#direction. The circles denote the field values at which the equilibrium states where calculated using the Gilbertequation. The numbers given are the net magetizations in the remanent stateand the coercivity. TABLE I. Observed net magnetizations in the remanent state and coercivities for varying particle extensions. The external field is either parallel to the @100#or the @111#direction. Hexti111 100 d/lexMx/MsMy/MsMz/MsHc/MsMx/MsMy/MsMz/MsHc/Ms 1 0.999 0.029 20.004 20.056 0.999 0.001 0.000 20.380 5 0.999 0.006 0.000 20.056 0.999 0.001 0.000 20.148 10 0.998 0.020 0.000 20.054 0.999 0.001 0.000 20.078 20 0.973 0.081 0.000 20.050 0.996 0.000 0.000 20.056 30 0.963 0.078 0.000 20.046 0.987 0.000 0.000 20.0485820 J. Appl. Phys., Vol. 85, No. 8, 15 April 1999 Streibl, Schrefl, and Fidler [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 142.150.190.39 On: Mon, 22 Dec 2014 09:55:03Figure 2 gives the remanent and transient magnetic states during irreversible switching of a particle with d 530lexat an applied field, Hext520.048Ms, parallel to the @100#axis. For zero applied field a magnetic flower state is observed. At an applied field of Hext520.03Msthe flower state vanishes and end domains form. As a consequence, thenet magnetization is reduced from 0.98 to 0.8 M s. The end domains further develop into vortices that head towards eachother and lead to an increase of exchange energy and mag-netostatic energy. When the final ~closed !vortices are formed, flux closure causes a dramatic decrease in the de-magnetizing energy. In what follows the regions of the vor-tices with magnetization parallel to the field direction en-large. Thus the vortices expand in the direction of the longaxis and the centers of the vortices move to opposite sides oftheir original position as can be seen in Fig. 2. The vorticesmove further towards each other and thus cause an increasein the magnetic volume and surface charge density, leadingto an increase in the magnetostatic energy. When this strongly inhomogeneous magnetic state becomes uniform,the magnetostatic and exchange energies decrease rapidly. Figure 3 gives the calculated demagnetization curve of a particle with d530l exwhen the external field is applied par- allel to the @111#direction. The magnetization distributions given in Fig. 4 clearly show the transition from a uniformmagnetic state with the magnetization parallel to the externalfield to a nearly uniform state with the magnetization parallelto the long axis as the external field is reduced to zero. Acomponent of the external field parallel to the short axis ofthe particle supports the formation of end domains, whichare not observed in the remanent state for H extparallel to @100#. This difference in remanent states which has yet to be investigated may be attributable to either too loose a criterionfor equilibrium for the calculations with H extparallel to @100#or to the history of the applied field. When the external field becomes negative, the magnetization rotates uniformlywithin the center of the particle whereas the end domainsremain stable up to an external field of H ext520.038Ms.A t this critical external field value the demagnetization curveshows a steep decrease that is associated with the suddenreversal of the end domains towards the applied field direc-tion. This step in the demagnetization curve is only observedfor particles with d>20l exwhereas for smaller particles no end domains occur in the remanent state. Table I summarizes the numerical results obtained for the field applied in the @100#direction and the field applied in the@111#direction ~mMAG standard problem No. 2 !. Whereas a significant coercive field size dependence wasfound for H extparallel to @100#,Hcremains nearly constant as a function of particle size for Hextparallel to @111#. The numerical studies show that the magnetization reversalmechanism changes from a uniform rotation overhead to ahead domain wall motion 7to a vortex motion1when the field is applied parallel to the @100#direction. As described above, magnetization reversal is mainly governed by uniform rota-tion when the field is applied parallel to the @111#direction. The end domains formed in particles with d>20l excause a step in the demagnetization curve, but do not influence thecoercive field. 1S. Yuan, H. Bertram, J. Smyth, and S. Schultz, IEEE Trans. Magn. 28, 3171 ~1992!. 2B. Yang and D. R. Fredkin, J. Appl. Phys. 79, 5755 ~1996!. 3T. Schrefl, J. Fidler, K. J. Kirk, and J. N. Chapman, J. Magn. Magn. Mater.175, 193 ~1997!. 4A. Khebir, A. Kouki, and R. Mittra, IEEE Trans. Microwave Theory Tech.38, 1427 ~1990!. 5Y. Nakatani, Y. Uesaka, and N. Hayashi, Jpn. J. Appl. Phys., Part 1 28, 2485 ~1989!. 6W. Rave, K. Ramsto ¨ck, and A. Hubert, J. Magn. Magn. Mater. 183, 329 ~1998!. 7R. McMichael and M. J. Donahue, IEEE Trans. Magn. 32, 4167 ~1997!. FIG. 4. Magnetization distributions for different M/Msalong the demagne- tization curve.5821 J. Appl. Phys., Vol. 85, No. 8, 15 April 1999 Streibl, Schrefl, and Fidler [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 142.150.190.39 On: Mon, 22 Dec 2014 09:55:03

5.0029401.pdf

Appl. Phys. Lett. 117, 222406 (2020); https://doi.org/10.1063/5.0029401 117, 222406 © 2020 Author(s).Spin-wave focusing induced skyrmion generation Cite as: Appl. Phys. Lett. 117, 222406 (2020); https://doi.org/10.1063/5.0029401 Submitted: 12 September 2020 . Accepted: 15 November 2020 . Published Online: 01 December 2020 Zhenyu Wang , Z.-X. Li , Ruifang Wang , Bo Liu , Hao Meng , Yunshan Cao , and Peng Yan COLLECTIONS Paper published as part of the special topic on Mesoscopic Magnetic Systems: From Fundamental Properties to Devices MSFD2021 ARTICLES YOU MAY BE INTERESTED IN Magnon thermal Edelstein effect detected by inverse spin Hall effect Applied Physics Letters 117, 222402 (2020); https://doi.org/10.1063/5.0030368 Room temperature anomalous Hall effect in antiferromagnetic Mn 3SnN films Applied Physics Letters 117, 222404 (2020); https://doi.org/10.1063/5.0032106 Enhancement of spin–orbit torque and modulation of Dzyaloshinskii–Moriya interaction in Pt100-x Crx/Co/AlO x trilayer Applied Physics Letters 117, 222405 (2020); https://doi.org/10.1063/5.0030880Spin-wave focusing induced skyrmion generation Cite as: Appl. Phys. Lett. 117, 222406 (2020); doi: 10.1063/5.0029401 Submitted: 12 September 2020 .Accepted: 15 November 2020 . Published Online: 1 December 2020 Zhenyu Wang,1 Z.-X. Li,1Ruifang Wang,2 BoLiu,3Hao Meng,3Yunshan Cao,1and Peng Yan1,a) AFFILIATIONS 1School of Electronic Science and Engineering and State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science and Technology of China, Chengdu 610054, China 2Department of Physics and Institute of Theoretical Physics and Astrophysics, Xiamen University, Xiamen 361005, China 3Key Laboratory of Spintronics Materials, Devices and Systems of Zhejiang Province, Hangzhou 311305, China Note: This paper is part of the APL Special Collection on Mesoscopic Magnetic Systems: From Fundamental Properties to Devices. a)Author to whom correspondence should be addressed: yan@uestc.edu.cn ABSTRACT We propose a method to generate magnetic skyrmions through spin-wave focusing in chiral ferromagnets. A lens is constructed to focus spin waves by a curved interface between two ferromagnetic thin films with different perpendicular magnetic anisotropies. Based on the prin-ciple of identical magnonic path length, we derive the lens contour that can be either elliptical or hyperbolical depending on the magnonrefractive index. Micromagnetic simulations are performed to verify the theoretical design. It is found that under proper conditions, magnetic skyrmions emerge near the focus point of the lens where the spin-wave intensity has been significantly enhanced. A close investigation shows that a magnetic droplet first forms and then converts to the skyrmion accompanied by a change in topological charge. The phase diagramabout the amplitude and time duration of the exciting field for skyrmion generation is obtained. Our findings would be helpful for designingspintronic devices combining the advantages of skyrmionics and magnonics. Published under license by AIP Publishing. https://doi.org/10.1063/5.0029401 Skyrmionics 1–5and magnonics6–9are two emerging research fields in spintronics, which utilize skyrmions and spin waves (mag-nons when quantized) as carriers to encode, transmit, and processinformation, respectively. Magnetic skyrmions normally exist in chiralbulk magnets or magnetic thin films with Dzyaloshinskii–Moriya interaction (DMI). 10,11They are topologically protected spin textures and cannot be nucleated and annihilated under continuous magnetiza-tion deformation. In contrast, magnons are the low-energy excitationsin ordered magnets and can be created and destroyed due to theirbosonic nature. Realizing both skyrmion and magnon functionalities in a single spintronic device could significantly promote the development of mag-netic memory and logic elements as an alternative to the conventionalCMOS (complementary metal oxide semiconductor) computingtechnology. Indeed, the interaction between magnons and skyrmions has been extensively studied recently, such as magnon-skyrmion scattering, 12,13magnon-driven skyrmion motion,14,15and skyrmion- based magnonic crystals.16,17However, an important issue about the conversion between skyrmions and magnons has not been well inves-tigated although the spin-wave emission has been observed in theannihilation process or the core switching of magnetic skyrmions. 18,19 A common view is that it is rather difficult to convert spin waves tomagnetic skyrmions because the spin-wave energy is much lower than the barrier between the uniform ferromagnetic state and the skyrmion. Over the past few years, geometrical curvature effects in magne- tism20have attracted considerable attention due to their promising application potential and rich physics, including the enhanced stability of domain walls21and skyrmions22in magnetic nanotubes, chirality symmetry breaking in ferromagnetic M €obius rings,23and magnonic Cherenkov-like effect,24to name a few. Very recently, it has been dem- onstrated that the curved interface in two-dimensional ferromagneticfilms can be used to construct a lens for spin-wave focusing. 25,26The concept of spin-wave lens inspires us to accumulate energy to over- come the barrier for skyrmion formation. The idea is as follows: first, the spin-wave intensity near the focal point can be significantly enhanced so that the nonlinear effect becomes dominating; second, the focal-point magnetization oscillates strongly and might even be locally reversed, which is necessary for thenucleation of magnetic skyrmions. In this Letter, we demonstrate thatskyrmions can be created by spin-wave focusing in a magnetic film without introducing external defects as the nucleation sites. 27–29To achieve a perfect focusing without spherical aberration,26we design a spin-wave lens with a curved interface [see Eq. (6)below] between two ferromagnetic films with different perpendicular magnetic anisotropies Appl. Phys. Lett. 117, 222406 (2020); doi: 10.1063/5.0029401 117, 222406-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/apl(Ki;ði¼1;2Þ), as shown in Fig. 1 . The magnetic anisotropy change in this heterogeneous thin film can be realized by the recently discoveredeffect of the voltage-controlled magnetic anisotropy. 30Initially, the heterogenous films are uniformly magnetized along the þ^zdirection. The spin-wave dynamics is described by the Landau–Lifshitz–Gilbert(LLG) equation, @m @t¼/C0cl0m/C2Heffþam/C2@m @t; (1) where m¼M=Msis the unit magnetization vector with the saturated magnetization Ms,cis the gyromagnetic ratio, l0is the vacuum per- meability, and ais the Gilbert damping constant. The effective field Heffcomprises the exchange field, the DM field, the anisotropy field, and the dipolar field. The dipolar interaction is approximated by thedemagnetization field H d¼/C0Msmz^z. In the following, the interfacial DMI is considered. Neglecting the damping term ( a¼0), the spin- wave spectrum can be obtained by solving the linearized LLGequation, xðkÞ¼A /C3k2þxKi; (2) where A/C3¼2cA=Ms,w i t h Abeing the exchange constant, xKi¼2cKeff;i=Ms,Keff;i¼Ki/C0l0M2 s=2 is the effective anisotropy constant, and k¼ðkx;kyÞis the wave vector of the spin wave. From the dispersion relation (2), one can see that the DMI has no effect when the magnetization is perpendicular to the film plane.31,32 We approximate the refractive index of spin waves as follows:33 nsw¼ck xðkÞ; (3) where cis the speed of light in vacuum and k¼jkji st h ew a v en u m - ber of the spin wave. Thus, the relative refractive index of spin waves isdefined as the ratio of the magnonic wave number in the right domain(gray region in Fig. 1 ) to that in the left one (white region in Fig. 1 ), n¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x/C0xK2 x/C0xK1r : (4) In analogy to optics, we define the magnonic path length (MPL) as the distance spin waves propagate multiplied by the refractive index. Aperfect lens can be designed based on the identical MPL principle. We assume a parallel spin wave incident from the left which convergesinto a focal point ðx f;0Þin the right, as shown in Fig. 1 . Then, the MPL principle yields xþnffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxf/C0xÞ2þy2q ¼nxf; (5) and the lens contour is described by ðx/C0aÞ2 a2þy2 b2¼1; (6) where a¼n nþ1xfandb¼ffiffiffiffiffiffiffiffi n2/C01p nþ1xf. One can see that the lens shape is elliptical for n>1 and hyperbolical for n<1. Here, we focus on the n>1 case, as depicted in Fig. 1 . To verify our theoretical design, micromagnetic simulations are performed using MuMax3.34We consider a heterogeneous magnetic thin film with the length of 2000 nm, the width of 700 nm, and thethickness of 1 nm. The cell size of 2 /C22/C21n m 3is used in simula- tions. Magnetic parameters of Co are adopted in simulations:35 Ms¼5:8/C2105A=m,Aex¼15 pJ =m,D¼2.5 mJ =m2;K1¼8/C2105 J=m3,a n d K2¼6/C2105J=m3. In the dynamic simulations, a Gilbert damping constant of a¼0:001 is used to ensure a long-distance prop- agation of spin waves, and absorbing boundary conditions are adoptedin the dashed area in Figs. 2(a) and2(b) to avoid the spin-wave reflec- tion by the film edges. 36 We apply a sinusoidal monochromatic microwave field Hext¼h0sinðxtÞ^xin a narrow rectangular area [black bar in FIG. 1. Schematic of a spin-wave lens with an elliptical interface between two ferro- magnetic films with different perpendicular magnetic anisotropies K1;2. The static magnetization mis oriented along the þzdirection. The semi-major and semi- minor axes of the elliptical interface are aandb, respectively. A parallel incident spin wave (blue arrows) propagates through the interface and converges on thefocal point (red point). FIG. 2. (a) Snapshot of the spin wave with l0h0¼10 mT and x=2p¼80 GHz across the elliptical interface. The black bar denotes the exciting source of spinwaves. (b) Intensity of spin waves in (a). The black point represents the theoretical position of the focal point. Absorbing boundary conditions are shown as the dashed region in (a) and (b). (c) The profile of the spin-wave intensity along the xaxis at y¼0 in (b). The blue and red open dots indicate the spin-wave intensity at the exciting source and the focal point, respectively.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 222406 (2020); doi: 10.1063/5.0029401 117, 222406-2 Published under license by AIP PublishingFig. 2(a) ] to excite the incident spin waves. We set the amplitude and frequency of the oscillating field as l0h0¼10 mT and x=2p¼80 GHz. Based on Eq. (4), it is found that the relative refrac- tive index is frequency-dependent and n¼1.35 for 80 GHz. We set the semi-minor axis of the elliptical interface as b¼250 nm, and the corresponding aand xfcan be calculated to be 370 and 644 nm, respectively. Numerical results from magnetic simulations are shown inFig. 2(a) . We also calculate the spin-wave intensity using the equa- tion Iswðx;yÞ¼Ðt 0½dmxðx;y;tÞ/C1382dt,a sp l o t t e di n Fig. 2(b) .I n Figs. 2(a)and2(b), we can observe a significant focusing effect of spin waves in the right domain. However, it is noted that the focus point obtained from the numerical simulation is shifted along /C0xfrom the ideal posi- tion [black point shown in Fig. 2(b) ]. It is due to the ray optic approxi- mation for analyzing the spin-wave propagation, which requires the wavelength of spin waves much smaller than the lens size ( k/C28a;b). One can diminish such a deviation by increasing the size of the inter- face lens with respect to k.I nFig. 2(c) ,t h ep r o fi l eo ft h es p i n - w a v e intensity along the xaxis at y¼0 is plotted. It shows that the spin- wave intensity has been enhanced by one order of magnitude, which is comparable with that of the graded index lens.37,38It is worth men- tioning that the focus position xfcan be tuned by varying the excita- tion frequency or by applying a perpendicular static magnetic field.26 In such cases, additional spherical aberration may emerge, unless thelens shape is reconstructed based on (6). To create skyrmions, we increase the exciting field amplitude to l 0h0¼350 mT. A strong magnetization oscillation is observed around the focal point, as shown in Fig. 3(a) .W i t ht h ec o n t i n u o u s excitation of spin waves, more energies are harvested, leading to the formation of magnetic droplet, which can be easily driven by spinwaves [see Fig. 3(b) ]. The magnetic droplet is a strongly nonlinear and localized spin-wave soliton. 39,40In a chiral magnetic film, the trivial magnetic droplet is unstable due to the high DMI energy. Assisted by spin waves, the magnetic droplet is converted to a dynamical skyrmion att¼0.84 ns, as shown in Fig. 3(c) .W et h e nt u r no f ft h em i c r o w a v e field, and the system is relaxed toward an equilibrium state [see Fig. 3(d)]. After 1 ns, a stable skyrmion is formed, as shown in Fig. 3(e) . Figure 3(f) plots the time evolution of the topological charge Q ¼ð1=4pÞÐÐqðx;yÞdxdy with the topological charge density qðx;yÞ ¼m/C1ð@xm/C2@ymÞ. We observe an abrupt change in Qat 0.84 ns, which provides further evidence of the skyrmion creation. We emphasize that, in creating skyrmions (see supplementary material Video 1), the magnetic droplet acts as an indispensable inter- mediate between ferromagnetic and skyrmion states. However, it is hard to distinguish the energy variation during the transformation from the droplet to the skyrmion, which is due to spin waves in the fer- romagnetic film with a large size. To overcome this problem, we per-form additional simulations creating a skyrmion by the spin-transfer torque in a magnetic film with a smaller size (200 /C2200/C22n m 3). Figure 4(a) shows the time evolution of the free energy and topological charge. In Fig. 4(b) , the magnetization profile and topological charge density qare given. It is found that, although the energy of the sky- rmion is lower than the droplet, the skyrmion cannot be created directly from a ferromagnetic state. To find out the reason for such a transformation, we plot a schematic diagram of the magnetic droplet and skyrmion,41a ss h o w ni nt h el e f tp a n e lo f Fig. 4(b) . The main differ- ence between two solitons is the in-plane magnetization distribution in the circular domain wall separating the magnetized “down” state in thecenter region from the magnetized “up” state in the rest of the ferro- magnetic film. The magnetic droplet has no topology and can be trans- formed continuously from a ferromagnetic state, whereas the skyrmion has topology such that the continuous transformation from a ferro-magnetic state is impossible. Thus, the skyrmion creation requiresmore energy input from the external force compared to the droplet. InFig. 5(a) , the phase diagram of skyrmion creation induced by spin-wave focusing is shown. It was expected that the number N skof the generated skyrmions should increase with the amplitude h0and duration time Tdof the microwave field. However, the simulation results do not strictly follow this expectation. For example, spin waves excited by the microwave field with l0h0¼340 mT can produce one skyrmion, while no skyrmion is created under the exciting field withl 0h0¼360 mT. To figure out the reason, we plot the time evolution of the topological charge QinFig. 5(b) . The skyrmion generation and annihilation can be confirmed by the increase and decrease in jQj, FIG. 3. The creation process of magnetic skyrmions induced by the spin-wave focusing with the DMI ( D¼2.5 mJ =m2). The exciting field with l0h0¼350 mT is applied in (a)–(c) and is turned off in (d) and (e). The left column shows the x-com- ponent magnetization of the heterogeneous film. The z-component magnetization of the rectangular areas in the left column is enlarged in the right column. (f) Temporal evolution of the topological number Q. The microwave field starts at t¼0 and ends at t¼0.84 ns indicated by the gray dashed line.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 222406 (2020); doi: 10.1063/5.0029401 117, 222406-3 Published under license by AIP Publishingrespectively. One can see that the topological charge Qis always around 0 for 320 mT, and this indicates no skyrmion creation in the whole processes. For 340 mT, one abrupt change of Qfrom 0 to /C01a t t¼0.85 ns is observed, which represents the creation of a skyrmion. For 360 mT, the topological charge Qis around /C01o n l yi nas h o r t period from 0.6 to 0.72 ns and vanishes then. Under a higher excitingfield amplitude of 380 mT, we can observe multiple changes in Q, corresponding to more skyrmion creation and annihilation. A close observation of the skyrmion evolution shows that the skyrmion anni-hilation is induced by the interaction between the magnetic droplet and the skyrmion: after the skyrmion formation, magnetic droplets are still continuously generated by the lens and they crash into a skyrmion, leading to a droplet formation or a skyrmion annihilation accompanied by spin-wave emissions (see supplementary material Video 2). In addition, the fractional topological charge is found during the skyrmion creation process under 380 mT [see the green line in Fig. 5(b) ]. This is mainly due to spin waves in the ferromagnetic film. When we turn off the microwave field and set the damping as a¼1 to eliminate the influence of spin waves, a quasi-integer topological charge emerges (not shown). Here, the calculation of the topological charge is based on finite-difference derivatives, which would lead to a noninteger Q. This spurious deviation can be mitigated by using a lattice-based approach. 42The above results are obtained in magnetic metals, which usually have high perpendicular magnetic anisotropy and high damping. For the skyrmion creation, we apply a microwave field to excite spin waves with large amplitude and high frequency, which is difficult to be achieved in experiments. Fortunately, several methods have been pro- posed to solve this problem: for example, a large amplitude coherentspin wave can be excited by spin-polarized current 43and can be sus- tained by parametric pumping.44With the excitation of spin waves with high frequency (short wavelength), it is readily realized in experi- ments by using the magnetization precession in periodic ferromag- netic nanowires to drive very short spin waves in a neighboringmagnetic film 45or by a microwave voltage-controlled magnetic anisot- ropy pumping.46Moreover, it has been demonstrated that skyrmions can be hosted in magnetic insulators with perpendicular anisotropy and ultra-low damping,47–49which make our method more applicable from the materials point of view. The results reported here do not depend on the type of spin- wave lens. Other spin-wave lenses, such as graded index lenses37,38,50 and magnonic meta-lenses,51,52can also focus spin waves for the sky- rmion generation. Furthermore, our design can also be utilized to gen- erate the Bloch-type skyrmion, which is stabilized in the presence of the bulk DMI.53It is noted that a similar method of skyrmion genera- tion using spin waves was also reported in a crossbar geometrical heterostructure.54Different from spin-wave focusing, skyrmion crea- tion in Ref. 54results from the combination of the geometry change and the DMI-induced effective magnetic field. Besides spin waves,there have also been several important proposals about the skyrmion creation by magnetic microwave fields, 28,29,55,56local heating,57electric current,27,58–60and others.61–63However, creating skyrmions at a specific place using magnetic fields is not straightforward because FIG. 4. (a) Temporal evolution of the free energy Etotand topological charge Q for the skyrmion creation by the spin-transfer torque. (b) The left panel: schematic dia- gram of the magnetic droplet and skyrmion. The middle and right panels: magneti- zation profile and topological charge density distribution at 0.48 and 0.8 ns, whichcorrespond to the droplet and skyrmion, respectively. FIG. 5. (a) Phase diagram of skyrmion creation with respect to the amplitude h0 and duration time Tdof the exciting field. The black crosses denote no skyrmion creation, and the color dots represent the number of created skyrmions. (b)Temporal evolutions of the topological charge Qunder different field amplitudes.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 222406 (2020); doi: 10.1063/5.0029401 117, 222406-4 Published under license by AIP Publishinglocally applying a magnetic field over a nanoscopic region is challeng- ing. This problem can be conventionally resolved by fabricating ananoscopic defect on the ferromagnetic thin film, 28,29which, however, poses other issues for device performance and scalability. For sky- rmion creation via local heating with laser irradiation,57it needs to introduce a laser for each write head,64,65which is hard to utilize in an integrated circuit. As for the skyrmion creation by electric current, theJoule heating due to the excessive high critical current density is a bot- tleneck. Our proposal in this work well avoids these problems. In summary, we theoretically investigated the generation of mag- netic skyrmions induced by spin-wave focusing. The lens contour was derived based on the identical magnonic path length principle. Micromagnetic simulations were performed to demonstrate the effec- tive focusing of spin waves through the curved interface. By increasingthe amplitude of the exciting field, strong nonlinear effects emerge close to the focus of the spin-wave lens. We observed spin-wave focus- ing-induced magnetic skyrmion nucleation, mediated by unstablemagnetic droplets. Our findings provide a method to create skyrmions in the magnetic thin film without artificial defects and would promote the development of spintronic devices combining spin waves andskyrmions. See the supplementary material for animations showing the skyrmion generation. The amplitude of excitation field is h 0¼350 and 400 mT in Videos 1 and 2, respectively. We thank Z. Zhang for helpful discussions. This work was supported by the National Natural Science Foundation of China (NSFC) (Grant Nos. 12074057, 11604041, and 11704060) and theNational Key Research Development Program under Contract No. 2016YFA0300801. Z.W. acknowledges the financial support from the China Postdoctoral Science Foundation under Grant No. 2019M653063. Z.-X.L. acknowledges the financial support of the China Postdoctoral Science Foundation (Grant No. 2019M663461)and NSFC Grant No. 11904048. 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Barrier breakdown mechanism in nano-scale perpendicular magnetic tunnel junctions with ultrathin MgO barrier Hua Lv , Diana C. Leitao , Zhiwei Hou , Paulo P. Freitas , Susana Cardoso , Thomas Kämpfe , Johannes Müller , Juergen Langer , and Jerzy Wrona Citation: AIP Advances 8, 055908 (2018); View online: https://doi.org/10.1063/1.5007656 View Table of Contents: http://aip.scitation.org/toc/adv/8/5 Published by the American Institute of Physics Articles you may be interested in Perpendicular-anisotropy CoFeB-MgO magnetic tunnel junctions with a MgO/CoFeB/Ta/CoFeB/MgO recording structure Applied Physics Letters 101, 022414 (2012); 10.1063/1.4736727 Two breakdown mechanisms in ultrathin alumina barrier magnetic tunnel junctions Journal of Applied Physics 95, 1315 (2004); 10.1063/1.1636255 Zero-field spin transfer oscillators based on magnetic tunnel junction having perpendicular polarizer and planar free layer AIP Advances 6, 125305 (2016); 10.1063/1.4971229 Thermal stability analysis and modelling of advanced perpendicular magnetic tunnel junctions AIP Advances 8, 055909 (2017); 10.1063/1.5007690AIP ADV ANCES 8, 055908 (2018) Barrier breakdown mechanism in nano-scale perpendicular magnetic tunnel junctions with ultrathin MgO barrier Hua Lv,1,2,aDiana C. Leitao,1,2Zhiwei Hou,1,3Paulo P . Freitas,1 Susana Cardoso,1,2Thomas K ¨ampfe,4Johannes M ¨uller,4 Juergen Langer,5and Jerzy Wrona5 1INESC - Microsistemas e Nanotecnologias and IN - Institute of Nanoscience and Nanotechnology, Lisboa, Portugal 2Instituto Superior Tecnico (IST), Universidade de Lisboa, Lisboa, Portugal 3Henan University of Technology, Zhengzhou, Henan 450001, P .R. China 4Fraunhofer Institute for Photonic Microsystems IPMS, 01099 Dresden, Germany 5Singulus Technologies AG, 63796 Kahl am Main, Germany (Presented 9 November 2017; received 2 October 2017; accepted 30 October 2017; published online 18 December 2017) Recently, the perpendicular magnetic tunnel junctions (p-MTJs) arouse great interest because of its unique features in the application of spin-transfer-torque magnetoresis- tive random access memory (STT-MRAM), such as low switching current density, good thermal stability and high access speed. In this paper, we investigated cur- rent induced switching (CIS) in ultrathin MgO barrier p-MTJs with dimension down to 50 nm. We obtained a CIS perpendicular tunnel magnetoresistance (p-TMR) of 123.9% and 7.0
m2resistance area product ( RA) with a critical switching density of 1.41010A/m2in a 300 nm diameter junction. We observe that the extrinsic break- down mechanism dominates, since the resistance of our p-MTJs decreases gradually with the increasing current. From the statistical analysis of differently sized p-MTJs, we observe that the breakdown voltage ( Vb) of 1.4 V is 2 times the switching voltage (Vs) of 0.7 V and the breakdown process exhibits two different breakdown states, unsteady and steady state. Using Simmons’ model, we find that the steady state is related with the barrier height of the MgO layer. Furthermore, our study suggests a more efficient method to evaluate the MTJ stability under high bias rather than measuring Vb. In conclusion, we developed well performant p-MTJs for the use in STT-MRAM and demonstrate the mechanism and control of breakdown in nano-scale ultrathin MgO barrier p-MTJs. © 2017 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/1.5007656 I. INTRODUCTION The spin-transfer-torque magnetoresistive random access memory (STT-MRAM) is one of the most promising next generation memories due to its unique features, such as non-volatility, high density, high-speed operation, high endurance and low power consumption.1Hereby, STT-MRAM with perpendicular anisotropy has shown to have a better performance because of its lower switching current density, better thermal stability and higher access speed.2,3However, the dielectric breakdown of MTJ becomes a severe reliability issue upon shrinking the cell sizes and very thin barrier thick- nesses. There are two known types of breakdown mechanisms in MTJs, the intrinsic and the extrinsic mechanisms. The former is a physical deterioration process and leads to a sudden decrease of the device resistance, whereas the later is related to the pinhole growth inside the barrier and reveals a aEmail: hlv@inesc-mn.pt 2158-3226/2018/8(5)/055908/6 8, 055908-1 ©Author(s) 2017 055908-2 Lv et al. AIP Advances 8, 055908 (2018) FIG. 1. (a) SEM image showing the nano via over the nanopillar (two levels e-beam lithography). (b) Structure layout of p-MTJs, where the red arrow indicates the positive current direction. gradual decrease in it.4,5However, the study of breakdown mechanism in nano-scale perpendicular MTJs (p-MTJs) is still lacking.6,7 In this paper, we fabricated nano-scaled p-MTJs with ultrathin MgO barriers performing excellent perpendicular tunneling magnetoresistance (p-TMR) and very low switching current density. Our study demonstrates in detail how the breakdown happens step by step, applies a comprehensive understanding of the breakdown mechanism in such devices and suggests a more efficient method to improve the device stability under high bias stress to enhance the endurance. II. EXPERIMENTS AND METHODS Bottom pinned perpendicular TMR stacks had been deposited using Singulus TIMARIS ten cathode PVD system on a Cu/TaN electrode with CMP treatment. The layer stack was as follows (all thickness in nm): seed/ [Co(0.5)/ Pt(0.2)] 6/ Co(0.6)/ Ru(0.8)/ Co(0.6)/ [Pt(0.2)/ Co(0.5)] 3/ Pt(0.2)/ reference layer separation / FeCoB(1)/ MgO/ FeCoB(1.3)/ free layer enhancement / MgO/ cap. All layers had been deposited using dc magnetron sputter process except the MgO layers which were produced by rf-magnetron sputtering. In order to emulate BEOL thermal budget conditions required for embedded memory applications the wafers had been annealed for 80min in 400C after deposition. We structured pillars in a diameter range of 50 nm to 1000 nm. The pads were defined using direct write laser lithography, whereas the junctions were patterned by e-beam lithography (Raith 150).8The sidewalls were protected by SiO 2, deposited by PECVD (Electrotech system) after a two steps ion beam etching (IBE) junction definition, using a Nordiko3600 tool.9To reach the junctions, reactive ion etching (RIE) technology was adopted with the help of two level e-beam lithography,10 as shown in Fig. 1(a). Sputtered Cr(5nm)/Au(200nm) film was used as a top contact rather than Al metallization to improve the via filling and reach the top of the junctions. The electrical properties of p-MTJs were characterized by four-probe measurement technique. The switching current was measured by sweeping bias current between positive and negative direction, where the positive current is defined from bottom to top electrode as shown in Fig. 1(b). The breakdown behaviors were measured by increasing bias current gradually. III. RESULTS AND DISCUSS A. Current induced switching and breakdown In Fig. 2 (a), we show the example of a current induced switching (CIS) in a 300 nm diame- ter p-MTJ, where p-TMR of 123.9% and resistance area product ( RA) of 7.0
m2are obtained. The CIS curve shows a very low switching current density, where JAP!P= 1.41010A/m2and JP!AP= -2.81010A/m2. These values are consistent with switching current of around055908-3 Lv et al. AIP Advances 8, 055908 (2018) FIG. 2. (a) CIS in a 300nm diameter p-MTJ. (b) Breakdown process in p-MTJ: resistance cycle (black dots) and normalized pinhole area (blue squares) with bias current. (2-4)1010A/m2obtained in Ref. 2 and indicate that a relatively weak Gilbert damping is achieved in our devices.2,11In Fig. 2 (b), the breakdown process is characterized by increasing bias current. Hereby, the resistance starts to decrease gradually at 7 mA and to stabilize after 35 mA. This indicates the breakdown in our sample to be dominated by an extrinsic mechanism with the growth of a pinhole inside the barrier.4,5Comparing these results with the ones for thicker barrier MTJs, the breakdown in thinner barriers is dominated by extrinsic mechanisms rather than intrinsic mechanisms.4 If a pinhole exists in an MTJ, the applied bias current passes partially through a residue barrier and partially through pinhole, similar to a pair of resistors. Hence, the efficient resistance area product (RAeff) can be written as5 RAeff=A0 AApin RAjun + Apin RApin (1) where A0,ApinandAjunare the area of initial junction, pinhole and residue MgO barrier, respectively. The pinhole area can be obtained as5 Apin=A0RApin RAeff RAjunRAeff RAjunRApin! (2) In Fig. 2(b), we plot the normalized pinhole area as calculated from Eq. (2), where A0= 0.1 m2, RAjun= 8.96
m2,RApin= 1.43
m2. Hereby, the RAeffwas measured upon various bias current values. We can see that the pinhole area is zero if the current is below than 7 mA. The pinhole is created at 7 mA and by further increasing the current, the pinhole area increases nearly linearly. After 35 mA, the pinhole area is not changing with increasing current and we assume that the junction has been totally occupied by the pinhole. We can see that this breakdown is dominated by the pinhole growth. B. Mechanism and process of pinhole growth In Fig. 3(a), we plot the change in voltage with the bias current. We can distinguish three areas that exist during the breakdown process. In area (1), the linear increase of voltage with applying current indicates that the resistance is constant and the junction shows no degradation. In area (2), the breakdown occurs at a voltage of 0.58 V , hereafter referred as the breakdown voltage ( Vb). With increasing the bias current, the voltage stabilizes around 0.48 V and keeps constant in a large current range, we define this voltage as Vb,stable . In area (3), an ohmic behavior is observed since the device performance is entirely determined by the metallic pinhole after the complete breakdown of the junction. The current circuit of the MTJ with pinhole is shown in the inset in Fig. 3(a). According to Simmons’ model, the current density through the junction barrier ( Jjun) is determined by the bias voltage ( V), barrier height ( ') and thickness ( d):12 Jjun=e 42–hd8>><>>: 'eV 2! exp26666642ds 2m –h2 'eV 2!3777775 '+eV 2! exp26666642ds 2m –h2 '+eV 2!37777759>>=>>;(3)055908-4 Lv et al. AIP Advances 8, 055908 (2018) FIG. 3. (a) V oltage change with bias current. The inset shows a schematic drawing of the current circuit in p-MTJ with pinhole; (b) Change of current ( I) and current density ( J) flowing through residue MgO barrier and pinhole with increasing current. The gray area (1), cyan area (2) and yellow area (3) show the pMTJ before breakdown, during breakdown and after entire breakdown process, respectively. where e,~andmare the electron charge, reduced Planck’s constant and electron mass, respectively. We fitted the J-Vcurve of the parallel state in Fig. 2(b) and obtained '= 0.49 eV and d= 0.77 nm. The current flow through the junction ( Ijun) can be calculated from Eq. (2) and (3): Ijun=AjunJjun= A0Apin Jjun (4) Hence, the current passing through the pinhole Ipin=Ibias-Ijunand the associated current density inside the pinhole Jpin=Ipin/Apin. In Fig. 3(b), we plot the Ijun,Ipin,JjunandJpincalculated with previous equations for different bias current. In area (1), all of the applied current flows through the junction and Jjungrows linearly with increasing current, this is reasonable since the junction has not been broken down and the efficient junction area is not changing. In area (2), we find that JiunandJpinalmost remains constant during pinhole growth but Ijundecreases and Ipinincreases because of a pinhole area expansion and a junction area shrink. In area (3), the pinhole expands to the entire junction area and all of current passes through pinhole, resulting in an ohmic behavior of the device. This indicates that after the creation of the pinhole, an additionally applied current only flows through the pinhole without increasing the pinhole current density and, thus, the pinhole area only increases to pass the additional current. C. Unsteady state and steady state To identify the breakdown process dependence on junction size, we plot the distribution of Vb, Vb,stable andVsin Fig. 4(a), where Vsis resistive switching voltage. The Vbvalues measured range from 1.1-1.7 V , which are in line with literature measured under ns pulses13and constant current,14 respectively. The dispersion among the datapoints may be justified by excessive heating under high currents, which could be surely minimized if the measurements were done under short pulses. In our devices, Vbof around 1.4 V is two times higher than Vsof around 0.7 V , this allows us to have a large margin to set the switching voltage without reaching the breakdown bias. A similar relationship between VbandVsin real STT-MRAM devices was also mentioned by S. Amara- Dababi et al.7This indicates that our devices are capable for the requirements of real data storage technology. In Fig. 4(a), we find that Vb,stable is always smaller than Vb, which indicates that the initial breakdown is an unsteady state and ongoing breakdown process leads the system to a steady state. The steady state maintains in a large range of bias current and Vb,stable shows very small distribution with the junction dimensions from 50 nm to 1000 nm, which means that all the MTJs have the same voltage stress and current density. This implies that the steady state may be related with an intrinsic property of the materials and stack configuration. To understand this behavior, J-Vcurves in different MTJs were fitted by Eq. (3) and we find that Vb,stable are close to the values of barrier height ( '/e),055908-5 Lv et al. AIP Advances 8, 055908 (2018) FIG. 4. (a) Vb(red dots), Vs(pink stars) and Vb,stable (black squares) distributions on junction size, the dashed blue line indicates the average value of '/e, where 'is the barrier height obtained from J-Vcurve fitting with Eq. (3). (b) and (c) are schemes of unsteady and steady state during breakdown, respectively, where EFis the Fermi level of the ferromagnetic electrode, Jis the tunneling current density. the average barrier height of 0.49 eV , with a standard deviation of 0.18 eV , is indicated by dash line in Fig. 4(a). In Fig. 4(b) and (c), we show schematics of energy states and barrier shape of broken MTJs at unsteady and steady state, respectively. Fig. 4(c) indicates that the breakdown shall not hap- pen where the voltage is lower than the barrier height ( '/e). So to increase the breakdown voltage of MTJs, the higher barrier height should be preferred, which is consistent with the model dis- cussed by D.V . Dimitrov et al.14We can see that barrier height plays a very important role in barrier breakdown and could be used as a good reference value while improving the break- down performance. Thus, a more efficient method to optimize the stability of MTJs under high bias stress may just compare the barrier height rather than measuring the statistical breakdown voltage, since the former can be easily obtained and the later will permanently damage the samples. IV. CONCLUSIONS In this paper, we fabricated the high performance nano-scaled p-MTJs with a p-TMR of 123.9% and switching current density of 1.4 1010A/m2in 300 nm diameter devices. Based on the study of breakdown process, we find that the breakdown in our thin barrier p-MTJs is dominated by the extrinsic mechanism which is related with the growth of pinhole in the barrier. The variation of the current density simultaneously on the MgO barrier and pinhole during breakdown process suggests the pinhole area expansion after the initial breakdown. The statistical study shows that the initial breakdown is an unsteady state and the ongoing breakdown moves to a steady state and remains during a large current range. The steady state voltage shows a very small distribution with different MTJs dimension, which is related with the MgO layer barrier height. Our further study suggests an efficient method to evaluate the breakdown performance while improving the stability of MTJs under high bias. ACKNOWLEDGMENTS H. Lv acknowledge FCT grant: SFRH/BD/93597/2013. This project has received funding from the Electronic Component Systems for European Leadership Joint Undertaking under grant agree- ment No. 692519. This Joint Undertaking receives support from the European Union’s Horizon 2020 research and innovation program and Belgium, Germany, France, Netherlands, Poland, United Kingdom. D. C. Leitao acknowledges financial support through FSE/POPH. Z. Hou acknowledges the National Natural Science Foundation of China under Grants No. 51201059, Natural Science Foundation of Henan province No. 14A140027 and the Fund of HAUT No. 171208.055908-6 Lv et al. AIP Advances 8, 055908 (2018) 1M. Hosomi et al. , “A novel nonvolatile memory with spin torque transfer magnetization switching: Spin-ram,” IEDM. Tech. Dig. 56, 459–462 (2005). 2S. Ikeda et al. , “A perpendicular-anisotropy CoFeB–MgO magnetic tunnel junction,” Nat. Mater. 9, 721–724 (2010). 3Y . Wang et al. , “Compact model of magnetic tunnel junction with stochastic spin transfer torque switching for reliability analyses,” Microelectronics Reliability 54, 1774–1778 (2014). 4B. Oliver et al. , “Dielectric breakdown in magnetic tunnel junctions having an ultrathin barrier,” J. Appl. Phys. 91, 4348–4352 (2002). 5B. Oliver et al. , “Two breakdown mechanisms in ultrathin alumina barrier magnetic tunnel junctions,” J. Appl. Phys. 95, 1315–1322 (2004). 6W. Zhao et al. , “Failure analysis in magnetic tunnel junction nanopillar with interfacial perpendicular magnetic anisotropy,” Materials 9, 41–58 (2016). 7S. Amara-Dababi et al. , “Modelling of time-dependent dielectric barrier breakdown mechanisms in MgO-based magnetic tunnel junctions,” J. Phys. D: Appl. Phys. 45, 295002–295009 (2012). 8D. C. Leitao et al. , “Magnetoresistive nanosensors: Controlling magnetism at the nanoscale,” Nanotechnology 27, 045501-1–045501-11 (2015). 9S. Cardoso et al. , “Ion beam assisted deposition of MgO barriers for magnetic tunnel junctions,” J. Appl. Phys. 103, 07A905-1–07A905-9 (2008). 10B. Pires et al. , “Multilevel process on large area wafers for nanoscale devices,” (submitted 2017). 11B. Dieny and M. Chshiev, “Perpendicular magnetic anisotropy at transition metal/oxide interfaces and applications,” Rev. Mod. Phys. 89, 025008-1–025008-62 (2017). 12J. G. Simmons, “Generalized formula for the electric tunnel effect between similar electrodes separated by a thin insulating film,” J. Appl. Phys. 34, 1793–1803 (1963). 13B. Dieny, “Introductory course on magnetic random access memory,” Grenoble, France, (2011). 14D. V . Dimitrov et al. , “Dielectric breakdown of MgO magnetic tunnel junctions,” Appl. Phys. Lett. 94, 123110-1–123110-3 (2009).

1.3699414.pdf

STRUCTURE AND ENERGY OF MOVING DOMAIN WALLS E. Schlömann Citation: AIP Conference Proceedings 5, 160 (1972); doi: 10.1063/1.3699414 View online: http://dx.doi.org/10.1063/1.3699414 View Table of Contents: http://aip.scitation.org/toc/apc/5/1 Published by the American Institute of Physics160 STRUCTURE AND ENERGY OF MOVING DOMAIN WALLS E. Schlómann Raytheon Research Division, Waltham, Mass. 02154 ABSTRACT A rigorous solution of the micromagnetic equations for moving domain walls is presented assuming uniaxial anisotropy and uni- form wall velocity. This solution applies for wall velocities smaller than a critical velocity v1 H D1/2 )1/2 - 1 1 ^( a ) [(1 +a` ] Here 'y is the gyromagnetic ratio, H the anisotropy field, D the exchange constant, Q = 4irM %H a, and-M0 the saturation magnetiza- tion.tion. The wall energy E increases with increasing v according to E = E oa(v) , where E o is the energy of the wall at rest, a(v) is proportional to the reciprocal of the wall width, and is given by v2/72 H D = - (1 4_ + 2 + cs - a 2 (2)a At the critical velocity the derivative of the wall energy with res- pect to v becomes infinite, the wall energy itself remains finite. The "tails" of the domain wall (the regions where the magnetization approaches alignment with the anisotropy axis) can be considered as spin waves of imaginary wavenumber and frequency but real phase velocity. The parameter a in Eq. (2) is (apart from a con- stant factor) the imaginary part of the wavenumber. The domain velocity v is the same as the phase velocity of these spin waves. The wall mobility is velocity dependent, being inversely propor- tional to the wall energy in this velocity range. The structure of domain walls moving at speeds exceeding v 1 is discussed. It is well known that, for suitably small velocities v, the energy of a moving domain wall exceeds the energy of a static wall by a term proportional to v 2, which may be interpreted as due to an effective mass.' In a moving, plane wall the magnetization vec- tor has a component normal to this plane, and the wall contracts as the velocity increases. The analyses from which these results have been obtained are generally based on approximations that 3 become questionable when the velocity becomes large. Walker has obtained a rigorous solution of the micromagnetic equations of mo- tion for domain walls moving at a constant velocity in a material having uniaxial anisotropy. It is shown in the present paper that the solution applies when the velocity is smaller than the critical velocity v1 [ see Eq. (1) of abstract] and that at this critical velocity the de- rivative of the wall energy with respect to velocity becomes infinite, the energy itself remains finite.(1) 1MOVING DOMAIN WALLS 161 For the limiting case in which the saturation magnetization is much larger than the anisotropy field (cr a,) the existence of a critical velocity v crit has previously been pointed out by Bean and DeBlois 4 and by Enz.° Their expressions for v are equivalent y p critto the limit cr - o0 of Eq. (1). The z-axis of the coor- dinate system is chosen to coincide with the anisotropy axis, the x-axis parallel to the wall normal. Figure 1 shows in qualitative manner the distribution of magnetiza- tion in the stationary (broken lines) and moving wall (solid lines) . This figure also de- fines spherical coordinates 0 , 4) of the magnetization vector. The density of energy in the wall is cr= —1 f -2H cos 9 + (H + 47rM sin 2 4)) sin 2 0 + D(9' 2 + sin 2 a 4)' 2)] . (3)2 o a o Here M o is the saturation magnetization, H the magnetic field (applied along the anisotropy axis), H a the anisotropy field and D = 2A M0 the exchange constant. The prime denotes differentia- tion with respect to x. The four terms of Eq. (3) represent the Zeeman energy, the anisotropy energy, the dipolar energy, and the exchange energy. In the absence of damping the equations of motion for 0 and 4) are 9 = -1 6,E ^= —M s niel 60' 7 6,Esin9^=Mg 88(4) where 646060 and 641 6e are variational derivatives. If the wall moves at velocity v in the x-direction, the derivative with respect to time is equivalent to a derivative with respect to x. Thus the structure of the wall is determined by the following set of differ- ential equations v 9' = cr sin e sin 4) cos 4 - (sin 9) -1 (sin 2 e 4' )' -v sin 00' = isin 0 + (1+cysjn2 4)+4,12) sin 0 cos 0 -e". (5) Here a reduced notation has been introduced in which therime 1/ 2 denotes differentiation with respect to u = (x-vt) (H a/D)and = v D 1/ 2 H= H H a a= H^^(H a ) % a 47rM of a . (6) For H = 0 the structure equations (5) have the following simple solution = const. , sin 0 = [ cosh (au)] -1, cos 0 = - tanh (au) (7)Fig. 1. Structure of stationary (bro- ken lines) and moving (solid lines) domain walls. The walls lie in the yz plane. The arrows indicate the direction of magnetization. 2provided that a and 4) satisfy a2 = 1+0sin 24) 0.6 v 0.40.8 0.21.0 1.0 1.2 1.4 1.6 1.8 2.0 a Fig. 2. Domain wall velocity as a function of the parameter a, whichis proportional to the inverse of the wall width and proportional to the wall energy. Broken lines correspond to unstable wall structures.Consider now the energy (per unit area) of the moving wall. Inserting the solution (7) into Eq. (3) one finds after trivial calculations +co E = dx Eoa (9) -co162 E. SCHLÓHANN 0-sin 4 cos 4 va (8) The physical significance of the parameter a is that its inverse is proportional to the wall width. If the two consistency conditions (8) are solved for v as a func- tion of a (eliminating 0) one ob- tains Eq. (2) of the abstract. A graph showing the dependence of v on a is shown in Fig. 2. The maximum velocity is v 1 as given by Eq. (1) of the abstract. For v > v 1 our trial function does not yield a solution of the structure equations. where E = 2 M (H D) 1/ 2 is the of the wall. Thus the ovelocity a e ener ^ e stationary yu e ve ocity dependence of the energy is at once obvious from Fig. 2. For v « v 1 the energy at first increases proportional to v2. At v 1 the derivative of E with respect to v becomes infinite, the energy at this point is E = E (1 + 6 )1/ 4 (10)max o The second solution of the wall-structure equations, having ahigher energy (larger a) than the first, should be disregarded because it is unstable. The angle 4) (see Fig. 1) increases monotonically with v. Its maximum (reached at v 1) is given by sin2^ = [(1+0)1 2 - 1] 0 1 (11)max For materials with very large 0(such as permalloy, 0=10 3 to 10 4) Omax « 1, Emax >> E o , and the wall contracts substantially as the velocity increases to v1. For materials with 0 . 1 (materials for bubble devices) O rnax =45°, Emax =E 0, and the wall contracts only slightly as the velocity increases to v 1. (For a = 0. 5 4)max = 42°, 1.11 E ° .Emax ) In the regions far away from the center of the moving domain 3MOVING DOMAIN WALLS 163 wall the magnetization vector approaches alignment with the aniso- tropy axis. In these regions, the "tails" of the wall, a linearized theory, such as conventionally used in discussing spin waves, is applicable. The tails of the moving domain wall can in fact be con- sidered to be spin waves of imaginary wavenumber and imaginary frequency but real phase velocity. The parameter a in Eqs. (2) , (7), (8) d (9) is the imaginary part of the wavenumber divided by (Hal D) 1 2. For further details on this subject see Ref. 6. In the preceding discussion damping has been neglected for the sake of brevity. If a Gilbert damping term is taken into account the left hand sides in Eq. (5) become v(0' + k sin 0 0') and ?/(sin 0 4)' - X0') where X is a dimensionless damping parameter. In the subcritical region (v < v 1) the solution (7) is still applicable provided that v a (v) = H . (12) This implies that the mobility v/ H is velocity dependent, and that it is proportional to the wall width (i. e. , inversely proportional to the wall energy). A relation equivalent to (12), but expressed quite differently was first derived by Walker.? What happens to domain walls at velocities exceeding v 1? It is easy to see in one special case that the structure equations do not have a solution describing a wall that satisfies the following cri- teria 1.The magnetization approaches alignment with the aniso- tropy axis on both sides of the wall 2.The structure of the wall is stationary, as seen from the center of the wall. This is implied in the derivation of Eq. (5), in that time-derivatives are equated to space- derivatives. 3. The wall moves at v > v 1. ti(v cos 0 - sin 0 4)' )' = 0 (13) According to criterion 1 sin2 0 -÷ 0, cos 0 + 1 for x co. Thus Eq. (13) can be satisfied only for 'N) .". 0. Note that for a = 0 also v1 = 0. Although the proof given above is clearly limited to a very special case it suggests that in general (i. e. , for arbitrary (3) a solution of the wall-structure equations that satisfies all three criteria does not exist. It appears probable that a solution resem- bling our intuitive picture of a moving wall must depend upon x and t separately, not just through the combination x - vt. The difference between subcritical (v < v1) and supercritical (v > v 1) domain walls is analogous to the difference between lamer and turbulent flow of fluids. If an object is moved through a fluid at a low velocity the flow is laminar; it appears stationary as viewedThe non-existence of such a solution can easily be demonstrated for 0-= 0. For this special case the first line of Eq. (5) may be expressed as 4164 E. SCHLaMANN from the moving object. When the velocity is increased beyond a certain critical velocity the flow becomes turbulent; it no longer appears stationary as viewed from the moving object. Because of this analogy the term "turbulent wall structure" might be appro- priate for domain walls moving at v > v 1. The onset of turbulence may be expected to decrease the wall mobility because the turbulent wall would be followed by a region in which the magnetization has not returned completely to alignment with the anisotropy axis, a "wake". A similar physical picture has been proposed previously by Hagedorn and Gyorgy, 8 who argued that a wake should occur even at arbitrarily low velocities in mate- rials with low damping. Experimental evidence for a velocity dependent mobility in bubble materials has been described by Calhoun et al. 9 The observed critical velocity is of the order of 15 m/sec. Calculation according to Eq. (1) yields 46 m sec (for 47rM o = 40 Oe, A = 10 -7 erg/cm, D = 2A/M 0 = 6.3 X10 -6 Oe cm 2, Ha = 360 Oe, = 17.6 X 10 6 0e -1 sec -1). The difference may be attributable to the fact that the experiment was done on curved walls in thin plates, whereas the theory applies to plane walls in bulk material and to the influence of crystal defects. Stimulating discussions with C. E. Patton, J. P. Sage and H. J. Van Hook are gratefully acknowledged. REFERENCES 1. W. Dóring, Z. Naturforsch. 3a, 373 (1948). 2. L. Landau and E. Lifshitz, Physik. Z. Sovietunion 8, 153 (1935) . 3. L. R. Walker, unpublished work (1956) quoted by J. F. Dillon, Jr. in "Magnetism," Vol. III, edited by G. Rado and H. Suhl, Academic Press (1963). 4. C. P. Bean and R. W. DeBlois, Bull. Amer. Phys. Soc. { II], 4, 53 (1959) . 5.D. Enz, Helvetica Physica Acta 37, 245 (1964) . 6. E. Schlómann, Appl. Phys. Letters 19, 274 (1971). 7. Equation (4.8) of Ref. 3. 8. F. B. Hagedorn and E. M. Gyorgy, J. Appl. Phys. 32, 282S (1961). 9. B. A. Calhoun, E. A. Giess, and L. L. Rosier, Appl. Phys. Letters 18, 287 (1971) . 5

5.0022033.pdf

APL Mater. 8, 111112 (2020); https://doi.org/10.1063/5.0022033 8, 111112 © 2020 Author(s).Magnetization reversal signatures of hybrid and pure Néel skyrmions in thin film multilayers Cite as: APL Mater. 8, 111112 (2020); https://doi.org/10.1063/5.0022033 Submitted: 16 July 2020 . Accepted: 28 October 2020 . Published Online: 20 November 2020 Nghiep Khoan Duong , Riccardo Tomasello , M. Raju , Alexander P. Petrović , Stefano Chiappini , Giovanni Finocchio , and Christos Panagopoulos ARTICLES YOU MAY BE INTERESTED IN Room-temperature magnetic skyrmion in epitaxial thin films of Fe 2−xPdxMo3N with the filled β-Mn-type chiral structure Applied Physics Letters 117, 142401 (2020); https://doi.org/10.1063/5.0024071 Dynamic excitations of chiral magnetic textures APL Materials 8, 100903 (2020); https://doi.org/10.1063/5.0027042 Chip-scale nonlinear photonics for quantum light generation AVS Quantum Science 2, 041702 (2020); https://doi.org/10.1116/5.0020684APL Materials ARTICLE scitation.org/journal/apm Magnetization reversal signatures of hybrid and pure Néel skyrmions in thin film multilayers Cite as: APL Mater. 8, 111112 (2020); doi: 10.1063/5.0022033 Submitted: 16 July 2020 •Accepted: 28 October 2020 • Published Online: 20 November 2020 Nghiep Khoan Duong,1 Riccardo Tomasello,2 M. Raju,1 Alexander P. Petrovi ´c,1 Stefano Chiappini,3 Giovanni Finocchio,3,4,a) and Christos Panagopoulos1,a) AFFILIATIONS 1Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, S637371, Singapore 2Institute of Applied and Computational Mathematics, Foundation for Research and Technology – Hellas (FORTH), Heraklion, GR-70013 Crete, Greece 3Istituto Nazionale di Geofisica e Vulcanologia, Via di Vigna Murata 605, I-00143 Roma, Italy 4Department of Mathematical and Computer Sciences, Physical Sciences and Earth Sciences, University of Messina, I-98166 Messina, Italy a)Authors to whom correspondence should be addressed: gfinocchio@unime.it and christos@ntu.edu.sg ABSTRACT We report a study of magnetization reversals and skyrmion configurations in two systems, Pt/Co/MgO and Ir/Fe/Co/Pt multilayers, where magnetic skyrmions are stabilized by a combination of dipolar and Dzyaloshinskii–Moriya interactions (DMIs). The First Order Reversal Curve (FORC) diagrams of low-DMI Pt/Co/MgO and high-DMI Ir/Fe/Co/Pt exhibit stark differences, which are identified by micromag- netic simulations to be indicative of hybrid and pure Néel skyrmions, respectively. Tracking the evolution of FORC features in multilayers with dipolar interactions and DMI, we find that the negative FORC valley, typically accompanying the positive FORC peak near saturation, disappears under both reduced dipolar interactions and enhanced DMI. As these conditions favor the formation of pure Néel skyrmions, we propose that the resultant FORC feature—a single positive FORC peak near saturation—can act as a fingerprint for pure Néel skyrmions in multilayers. Our study thus expands on the utility of FORC analysis as a tool for characterizing spin topology in multilayer thin films. ©2020 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0022033 .,s I. INTRODUCTION Magnetic skyrmions have been realized in several material systems, most notably magnetic multilayer thin films that host nanoscale skyrmions at room temperature.1–3In such multilayers, the Dzyaloshinskii–Moriya interaction (DMI) arising at the ferro- magnet (FM)/heavy-metal (HM) interfaces is paramount in stabi- lizing the Néel spin textures that these skyrmions possess.4,5How- ever, the actual spin textures of these skyrmions have recently been proven to be more complex, owing to the competition between DMI and dipolar interactions between the thin film layers.6–8For most multilayer systems, skyrmions exhibit thickness-dependent magne- tization profiles, where a central-layer Bloch texture is sandwiched between Néel textures of opposite chiralities from the topmostand bottommost layers.7These are known as hybrid skyrmions, whereas uniform Néel-texture skyrmions throughout the multilayer, realizable in a high-DMI environment, are known as pure Néel skyrmions.7 Differences in the spin texture and chirality of skyrmions strongly influence their current-driven dynamics,7,9–11rendering the knowledge of their complete, three-dimensional spin textures cru- cial for spintronic material design. Distinguishing between hybrid and pure Néel skyrmions, however, requires sophisticated imag- ing methods, such as circular dichroism x-ray resonant magnetic scattering7,8or nitrogen-vacancy center magnetometry12in order to resolve the thickness-dependence of the magnetic textures. These techniques may not always be readily available in most research facilities. On the other hand, the interplay of dipolar interactions APL Mater. 8, 111112 (2020); doi: 10.1063/5.0022033 8, 111112-1 © Author(s) 2020APL Materials ARTICLE scitation.org/journal/apm and DMI, as well as other ubiquitous and tunable magnetic inter- actions in multilayers, directly affects the domain size, density, and level of disorder in the skyrmion configuration.3These parame- ters consequently influence the magnetization reversal processes13 and hysteretic behavior of spin textures,14which may provide indi- rect clues for inferring the inner complexity of three-dimensional skyrmions. The ability to identify these subtle processes has been demon- strated by the First Order Reversal Curve (FORC) technique, which provides a magnetic fingerprint of the interactions and reversal processes occurring in magnetic materials.15,16Recent studies have begun utilizing FORC in skyrmion-hosting multilayers to study field history control of zero-field skyrmion populations17,18while simul- taneously revealing magnetic reversal mechanisms influenced by the skyrmion configuration.17Indeed, the variety of magnetic interac- tions and skyrmion configurations realizable in different thin film heterostructures offers a rich resource for FORC studies. Making use of this approach, our work combines FORC and magnetic force microscopy (MFM) to study the mag- netization reversal and skyrmion properties in two systems, [Pt(3)/Co(0.9)/MgO(1.5)] ×15 and [Ir(1)/Fe(0.4)/Co(0.4)/Pt(1)] ×20 (thickness in nm in parentheses), hereafter referred to as Pt/Co/MgO and Fe(0.4)/Co(0.4), respectively. The former displaysskyrmions with a large average diameter of 105 ±21 nm, suggesting that their stability results primarily from magnetic dipolar interac- tions,19while the latter shows smaller 46 ±12 nm skyrmions, indi- cating the more dominant role played by the DMI ( D) in skyrmion formation.19FORC analysis and MFM imaging reveal distinct irre- versibility features in these two material systems. Using micro- magnetic simulations, we show that these two multilayers stabilize hybrid and pure Néel skyrmions, respectively, which may account for their distinct FORC features. To support this hypothesis, we apply our analysis to Pt/Co/MgO samples with different numbers of layer repetitions and also to Fe/Co multilayers with different fer- romagnetic compositions. Again, we observe a correlation between FORC features and relative strengths of dipolar interactions and DMI, which facilitate the transition from a hybrid to a pure Néel skyrmion texture.7This points toward a possible thermodynamic signature for high- Dmultilayers, which can stabilize pure Néel skyrmions. II. METHODS Multilayer thin films of Ta(3)/Pt(10)/[Ir(1)/Fe(0.2–0.5)/Co (0.4–0.8)/Pt(1)] 20/Pt(2) and Ta(3)/[Pt(3)/Co(0.9)/MgO(1.5)] 2-15/ Ta(4) were deposited on thermally oxidized silicon wafers at room FIG. 1 . Magnetic irreversibility and skyrmion configurations for low- and high-DMI systems. A set of FORCs for (a) low-DMI Pt/Co/MgO and (b) high-DMI Fe(0.4)/Co(0.4) multilayers. The dashed circles in (a) represent points where neighboring reversal curves diverge and converge, resulting in a sign change in the second derivative of the magnetization and, consequently, leading to the negative-valley/positive-peak pair. (c) FORC diagram of Pt/Co/MgO showing wide-spread irreversible regions terminated with a negative-valley/positive-peak pair near negative saturation. [(d)–(f)] Field-evolution of spin textures at different HRvalues in Pt/Co/MgO. A sparse array of ≈100 nm skyrmions is seen at μ0H=μ0HR=−110 mT. (g) FORC diagram of Fe(0.4)/Co(0.4) showing narrow irreversible regions constrained to the diagonal edge and terminated with a single positive-peak near negative saturation. [(h)–(j)] Field evolution of spin textures at different HRvalues in Fe(0.4)/Co(0.4). A dense array of ≈50 nm skyrmions is seen at −160 mT in Fe(0.4)/Co(0.4). All magnetic configurations were attained following saturation in a positive magnetic field at T = 300 K. Scale bars: 500 nm. APL Mater. 8, 111112 (2020); doi: 10.1063/5.0022033 8, 111112-2 © Author(s) 2020APL Materials ARTICLE scitation.org/journal/apm temperature (numbers in parentheses refer to the layer thickness in nm). Ir/Fe/Co/Pt samples were deposited using a Chiron ultra- high vacuum multi-source sputter tool, while Pt/Co/MgO samples were deposited using a Singulus Timaris ultra-high vacuum multi- target sputter tool. The base vacuum in each case is 1 ×10−8Torr, and sputtering is carried out in 1.5 ×10−3Torr of argon gas. Mag- netization measurements on these samples were performed using superconducting quantum interference device (SQUID) magnetom- etry, in a Quantum Design Magnetic Property Measurement System (MPMS), to obtain the saturation magnetization ( MS). Out-of-plane and in-plane hysteresis loops were also acquired to determine the uniaxial effective anisotropy values Keff. FORC measurements were then conducted on as-grown sam- ples using a Vibrating Sample Magnetometer (VSM) at room tem- perature. Each FORC measurement consists of a two-part sequence: (1) the sample is first saturated at a positive field and then brought to a reversal field HRwith (2) the magnetization of the sample is measured starting from HRand ending at 0 as the applied field is reversed. Repeating the sequence for multiple values of HR, we obtain a family of FORCs [Figs. 1(a) and 1(b)], used to compute the FORC distribution ρdefined as ρ=−1 2∂2M(HR,H) ∂HR∂H. (1) Plotting ρas a density plot against HRandHproduces a FORC dia- gram [Figs. 1(c) and 1(g)], which quantifies the degree of magnetic irreversibility for the magnetic field histories of the measured sam- ple. Each FORC diagram is studied by complementary MFM images, which capture the magnetic textures obtained by different field his- tories. The method used for acquiring and analyzing MFM images is similar to that described in Ref. 3. Micromagnetic computations were performed by means of a state-of-the-art micromagnetic solver, PETASPIN, which numer- ically integrates the Landau–Lifshitz–Gilbert (LLG) equation by applying the Adams–Bashforth time solver scheme,20 dm dτ=−(m×heff)+αG(m×dm dτ), (2) where αGis the Gilbert damping, m=M/Msis the normalized magnetization, and τ=γ0Mstis the dimensionless time, with γ0 being the gyromagnetic ratio and Msbeing the saturation magneti- zation. heffis the normalized effective magnetic field, which includes the exchange, interfacial DMI, uniaxial anisotropy, and Zeeman fields, as well as the magnetostatic field computed by solving the magnetostatic problem of the whole system.8,21 The [Pt(3 nm)/Co(0.9 nm)/MgO(1.5 nm)] 15and [Ir(1 nm)/Fe (0.4 nm)/Co(0.4 nm)/Pt(1 nm)] 20multilayers are simulated by 15 and 20 repetitions of a 1 nm FM, respectively. The ferromagnetic layers are coupled to one another by means of only the magne- tostatic field (exchange-decoupled layers) and are separated by a non-magnetic layer: 4 nm thick Pt/MgO for Pt/Co/MgO and 2 nm thick Ir/Pt for Fe(0.4)/Co(0.4). The ferromagnetic and spacer thick- nesses have been chosen to reduce the number of cells along the z-direction while matching the experimental layer thicknesses as closely as possible. For Pt/Co/MgO, the physical parameters used are as follows: saturation magnetization Ms= 1.586 MA/m, Keff= 49.56 kJ/m3(determined by SQUID measurements), exchange constantA= 12 pJ/m (based on Refs. 2 and 7), interfacial D= 0.5 mJ/m2(cal- culated from the periodicity of MFM-imaged domain walls, based on the method described in Ref. 22), and a discretization cell size of 2.5×2.5×1 nm3. An out-of-plane external field Hext= 110 mT is applied antiparallel to the skyrmion core as in the MFM measure- ments. For Fe(0.4)/Co(0.4), we used Ms= 841 kA/m, Keff= 109.38 kJ/m3,A= 11 pJ/m, D= 2.0 mJ/m2(according to Ref. 3), Hext= 160 mT (as in the MFM measurements), and a discretization cell size of 3×3×1 nm3. III. RESULTS AND DISCUSSION We first focus on the FORC diagram of Pt/Co/MgO, which has an estimated (Refs. 2 and 7) Dvalue of 0.5 mJ/m2. For this sample, we observe large regions of irreversibility extending all the way from HR= 0 mT to HR≈−180 mT. The first feature is a wide, positive- valued ridge from μ0HR= 0 mT to μ0HR≈−125 mT [Fig. 1(c)], coin- ciding with the transition from labyrinthine stripes to the skyrmion FIG. 2 . Micromagnetic simulations of skyrmion textures in low- and high-DMI sys- tems. Skyrmion diameter as a function of layer position for (a) Pt/Co/MgO at Hext = 110 mT and (b) Fe(0.4)/Co(0.4) at Hext= 160 mT. The insets depict the spa- tial distribution of the skyrmion magnetization at the layer positions indicated by the black arrows. The colors correspond to the normalized z-component of the magnetization, as shown in the color bar. APL Mater. 8, 111112 (2020); doi: 10.1063/5.0022033 8, 111112-3 © Author(s) 2020APL Materials ARTICLE scitation.org/journal/apm phase [Figs. 1(d)–1(f)], where ∼100 nm-diameter skyrmions emerge in a disordered configuration at μ0H=μ0HR=−110 mT [Fig. 1(f)]. Based on the interpretation of Ref. 17, we deduce that the large, positive-valued region of irreversibility for | H|≤|HR| in this HR range corresponds to skyrmion and stripe mergers taking place as the applied field decreases. As | μ0HR| increases from 125 mT to 180 mT, a pair of irreversible regions consisting of a negative-valley (blue) and a positive-peak (red) emerges [Fig. 1(c)]. This familiar pair feature arises from the sign change in the second derivative of the magnetization as neighboring reversal curves diverge and then converge in the high field regime [dashed circles in Fig. 1(a)]. The feature, which was observed near the out-of-plane saturation fields in FORC studies of other magnetic multilayers,15–17signifies the onset of skyrmion annihilation as the applied field increases along the diagonal edge, followed by skyrmion and stripe nucleation as the field is reduced along the Haxis. While the negative–positive pair feature frequently appears in magnetic multilayers, including the Ir/Fe(x)/Co(y)/Pt stacks, it does not appear for Fe(0.4)/Co(0.4), where D= 2.0 mJ/m2. No sign change in the second derivative of the magnetization is observed, and hence, only a single positive peak is seen as the sys- tem approaches saturation, i.e., from μ0HR≈−150 mT to μ0HR≈ −225 mT [Fig. 1(g)]. This feature is preceded by an elongated irre- versible ridge extending from μ0HR≈−50 mT to μ0HR≈−150 mT [Fig. 1(g)]. Unlike the sprawling irreversible feature in Pt/Co/MgO, the irreversible ridge for Fe(0.4)/Co(0.4) is narrower and localizedaround the diagonal edge of the FORC diagram. This indicates the presence of a large population of skyrmions, whose repulsive short-range interaction precludes skyrmion merger, thus resulting in less irreversible activity taking place as the applied field reverses from HR. Indeed, high-density skyrmions appear as early as μ0HR ≈−50 mT [Fig. 1(h)] and quickly transform into a dense array of small skyrmions ( ≈50 nm in diameter) as the applied field increases [Fig. 1(i)]. This configuration stands in sharp contrast to the sparse array of larger skyrmions ( ≈100 nm in diameter) observed in Pt/Co/MgO. Due to their larger size, the latter are likely to be strongly stabilized by dipolar interactions, thus exhibiting hybrid magnetization pro- files. The appearance of these hybrid spin textures may be linked to our observed differences in FORC features. To test this hypothesis, we performed micromagnetic simulations of the two systems and extracted their thickness-dependent spin textures. Figure 2 summarizes micromagnetic simulations for the two multilayers. In both cases, the skyrmion diameter is thickness- dependent, being larger in the middle layer and smaller in the exter- nal layers. This is attributed to the z-component of the magneto- static field.7The size of the skyrmion is larger in the Pt/Co/MgO sample than in Fe(0.4)/Co(0.4), in qualitative agreement with exper- imental measurements. A crucial difference between the two cases lies in the thickness-dependence of their respective spin textures. In Fe(0.4)/Co(0.4), the spin chirality is independent of the layer posi- tion and a pure Néel skyrmion is obtained in all the layers. This can FIG. 3 . Evolution of FORC distributions with the number of layer repetitions (N) in Pt/Co/MgO multilayers. [(a)–(d)] FORC diagrams for Pt/Co/MgO multilayers for N = 15, 10, 4, and 2, showing a gradual disappearance of the negative FORC valley as N decreases. Correspondingly, (e) the average skyrmion size decreases from ≈105 nm to ≈80 nm as N decreases from 15 to 4, reflecting a transition from a dipolar-dominant to a DMI-dominant regime of skyrmion stability. APL Mater. 8, 111112 (2020); doi: 10.1063/5.0022033 8, 111112-4 © Author(s) 2020APL Materials ARTICLE scitation.org/journal/apm be attributed to the strong DMI in Fe(0.4)/Co(0.4), which by over- coming the magnetostatic field dictates the skyrmion texture in all the layers, in agreement with previous theoretical results.7 On the other hand, a skyrmion in the Pt/Co/MgO exhibits a layer-dependent chirality (hybrid skyrmion), which gradually changes from Néel with an outward spin chirality at the bottomlayer, to an intermediate skyrmion mixing Néel-outward and Bloch- clockwise chiralities in the middle layer, and eventually to a Néel skyrmion with inward chirality at the top layer. This is ascribed to the small DMI value in Pt/Co/MgO, thus allowing the magnetostatic field to be dominant. The small DMI only affects the position of the Bloch skyrmion, which is not located in the middle layer, as expected FIG. 4 . Evolution of FORC distributions with Fe/Co compositions in Fe(x)/Co(y) multilayers. [(a)–(e)] FORC diagrams for various Fe(x)/Co(y) compositions: Fe(0.2)/Co(0.6), Fe(0.2)/Co(0.8), Fe(0.3)/Co(0.7), Fe(0.5)/Co(0.5), and Fe(0.4)/Co(0.4). Tuning the ferromagnetic compositions in this order brings the negative FORC valley closer to the diagonal edge: the negative valley eventually vanishes in Fe(0.4)/Co(0.4). (f) The skyrmion diameter correspondingly decreases from ≈97 nm to ≈46 nm across the samples, demonstrating an increasingly strong influence of the DMI that stabilizes pure Néel skyrmions in Fe(0.4)/Co(0.4). [(g)–(i)] Cross sections of the skyrmion textures, obtained by micromagnetic simulations, for samples Fe(0.2)/Co(0.8), Fe(0.5)/Co(0.5), and Fe(0.4)/Co(0.4). For Fe(0.2)/Co(0.8), we used the following parameters: Ms= 1238 kA/m, Keff= 287.01 kJ/m3,A= 10 pJ/m, D= 1.5 mJ/m2, and Hext= 190 mT. For Fe(0.5)/Co(0.5), we used the following parameters: Ms= 1010 kA/m, Keff= 59.05 kJ/m3,A= 10.3 pJ/m, D= 1.9 mJ/m2, andHext= 200 mT. APL Mater. 8, 111112 (2020); doi: 10.1063/5.0022033 8, 111112-5 © Author(s) 2020APL Materials ARTICLE scitation.org/journal/apm from the magnetostatic field, but is shifted upward to the tenth layer, consistent with previous findings.7,8 Comparing our micromagnetic simulations with the FORC fea- tures in Figs. 1(c) and 1(g), we found that the coexistence of a positive peak and a negative valley of the irreversibility coincides with the stabilization of hybrid skyrmions, stabilized by a combi- nation of DMI and dipolar interactions. On the other hand, the presence of a single positive peak coincides with the presence of pure Néel skyrmions, stabilized primarily by interfacial DMI interactions. The distinct FORC features observed in Fig. 1 and the hybrid and pure Néel skyrmion textures suggested by micromagnetic simula- tions thus suggest a potential correlation between FORC distribution features and the strengths of dipolar interactions and DMI, which influence the thickness-dependent skyrmion textures. To investigate this correspondence, we track the evolution of FORC distributions and skyrmion diameters in Pt/Co/MgO multi- layers with the dipolar interaction strength by reducing the num- ber of layer repetitions (N) progressively from 15 to 2. The results are encapsulated in Fig. 3, where the HRandHaxes of the FORC diagrams are normalized to the out-of-plane saturation field, deter- mined as the HRvalue at which irreversible features terminate. As the interlayer dipolar interaction weakens with reduced N, the FORC distributions transition from a negative–positive peak pair to a single positive-peak. Correspondingly, the observed skyrmion diameter decreases from ≈105 nm (for N = 15) to ≈80 nm (for N = 4), reflecting a transition from a dipolar-dominant regime to a DMI- dominant regime of skyrmion stability.19These observations suggest that the disappearance of the negative FORC valley correlates with a reduced dipolar interaction in the multilayer. Likewise, we also track the evolution of FORC features with the increase in DMI, achieved by varying the Fe/Co compositions of the [Ir(1)/Fe(x)/Co(y)/Pt(1)] 20heterostructure. Raising the Fe/Co com- position ratio while keeping their total thickness ≤1 nm effectively increases the DMI strength while also modifying other magnetic parameters. This results in a variation of the skyrmion size, density, and energetic stability, which can be correlated with key changes in the respective FORC diagrams. Figures 4(a)–4(e) compare the FORC features for Fe(0.2)/ Co(0.6), Fe(0.2)/Co(0.8), Fe(0.3)/Co(0.7), Fe(0.5)/Co(0.5), and Fe(0.4)/Co(0.4). Samples Fe(0.2)/Co(0.6), Fe(0.2)/Co(0.8), and Fe(0.3)/Co(0.7) display the familiar negative-valley/positive-peak feature similar to that exhibited by low- DPt/Co/MgO [Fig. 1(c)]. Meanwhile, Fe(0.5)/Co(0.5) shows a large positive peak together with a much smaller negative valley, and Fe(0.4)/Co(0.4) exhibits only a narrow positive-valued FORC ridge. The gradual disappear- ance of the negative-valley, the increase in the DMI strength, and the two-fold decrease in skyrmion diameter3,17[Fig. 4 (f)] again suggest a transition from a dipolar-dominant to a DMI-dominant regime of skyrmion stability,19thus hinting at a transition from hybrid to pure Néel skyrmions. To support this inference, we have performed additional micro- magnetic simulations for samples Fe(0.2)/Co(0.8) and Fe(0.5)/ Co(0.5) and compared them with the case of Fe(0.4)/Co(0.4). In Fe(0.2)/Co(0.8) with a DMI strength of 1.5 mJ/m2[Fig. 4(g)], we observe a hybrid skyrmion where the Bloch skyrmion is present in the 17th ferromagnetic layer. In contrast, the Bloch position for Pt/Co/MgO appears roughly at the center of the stack due to dipolar interaction dominance over DMI. In Fe(0.5)/Co(0.5)(D= 1.9 mJ/m2), no Bloch skyrmion is observed and the 3D skyrmion profile is purely Néel, with chirality inversion only in the topmost layer [Fig. 4(h)]. The skyrmion profile is a complete pure Néel texture in all the layers in the case of Fe(0.4)/Co(0.4), where D reaches 2.0 mJ/m2[Fig. 4(i)]. The results in Fe/Co are thus in full agreement with the analysis of the FORC diagrams. Together with the observations in samples with reduced layer repetitions, they sug- gest that the single-positive peak feature in the FORC diagram is indicative of pure Néel skyrmions in thin film multilayers. Lastly, we note that all magnetic interactions play a critical role in the minimization of domain wall energy, and their variations with layer additions result in the stabilization of skyrmion textures with varying spin configurations. In ultrathin films, accurately estimat- ing these interaction parameters remains a challenge, as these esti- mates are sensitive to the measurement techniques employed.23Our estimates for D- extracted from MFM-imaged zero-field domain textures, using the method described in Refs. 3 and 22- assume a maximum uncertainty of 25% in the absolute values of A,D.3Nev- ertheless, the trend of increasing Dwhile raising the Fe/Co com- position ratio remains robust, as indicated by the clear correspon- dence between FORC analysis, MFM imaging, and micromagnetic simulations. IV. CONCLUSION In summary, we investigated the magnetization reversals and skyrmion configurations for Pt/Co/MgO and Ir/Fe/Co/Pt multilay- ers using a combination of FORC measurements, MFM imaging, and micromagnetic simulations. Wide sprawling FORC regions with a characteristic negative-valley/positive-peak pair are indicative of large, hybrid skyrmions in low- DPt/Co/MgO. In contrast, a single positive FORC distribution peak is indicative of small, pure Néel skyrmions in high- DFe(0.4)/Co(0.4). By reducing the number of film layer repetitions in Pt/Co/MgO and tuning the thicknesses of Fe and Co in Fe(x)/Co(y) multilayers, we observe a transition of FORC features from a negative-valley/positive-peak pair to a single posi- tive peak in correspondence with a reduction in dipolar interactions and an increase in the DMI strength, respectively. Hence, we pro- pose that the single positive FORC feature can be a useful fingerprint for pure Néel skyrmions in multilayer systems. In addition to pro- viding an indicator for skyrmion spin chirality, the observed FORC features enable a robust assessment of the thermodynamic stabil- ity of skyrmions within a particular multilayer: the negative FORC valley vanishes as the stability rises. While additional spin imaging techniques are desirable for microscopically resolving the multitude of complex spin topologies,7,8,12FORC analysis can play an impor- tant role in the analysis of magnetic multilayers. Combining these techniques can efficiently address future challenges in designing and optimizing skyrmionic materials. AUTHORS’ CONTRIBUTIONS N.K.D. and R.T. contributed equally to this work. ACKNOWLEDGMENTS The work in Singapore was supported by the National Research Foundation (NRF) under NRF Investigatorship Programme (Refer- ence No. NRF-NRFI2015-04) and Academic Research Fund (AcRF) APL Mater. 8, 111112 (2020); doi: 10.1063/5.0022033 8, 111112-6 © Author(s) 2020APL Materials ARTICLE scitation.org/journal/apm Tier 3 from the Singapore Ministry of Education (MOE) (Reference No. MOE2018-T3-1-002). M.R. thanks the Data Storage Institute of Singapore for sample growth facilities. R.T. and G.F. thank the project “ThunderSKY” funded from the Hellenic Foundation for Research and Innovation and the General Secretariat for Research and Technology under Grant No. 871. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1C. Moreau-Luchaire et al. , “Additive interfacial chiral interaction in multilay- ers for stabilization of small individual skyrmions at room temperature,” Nat. Nanotechnol. 11(5), 444 (2016). 2S. Woo et al. , “Observation of room-temperature magnetic skyrmions and their current-driven dynamics in ultrathin metallic ferromagnets,” Nat. Mater. 15(5), 501–506 (2016). 3A. Soumyanarayanan et al. , “Tunable room-temperature magnetic skyrmions in Ir/Fe/Co/Pt multilayers,” Nat. Mater. 16(9), 898–904 (2017). 4A. Bogdanov and U. Rößler, “Chiral symmetry breaking in magnetic thin films and multilayers,” Phys. Rev. Lett. 87(3), 037203 (2001). 5A. 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1.3549704.pdf

Fast magnetization precession observed in L 1 0 -FePt epitaxial thin film S. Mizukami, S. Iihama, N. Inami, T. Hiratsuka, G. Kim, H. Naganuma, M. Oogane, and Y. Ando Citation: Applied Physics Letters 98, 052501 (2011); doi: 10.1063/1.3549704 View online: http://dx.doi.org/10.1063/1.3549704 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/98/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Magnetization reversal in perpendicularly magnetized L10 FePd/FePt heterostructures J. Appl. Phys. 116, 033922 (2014); 10.1063/1.4890936 Perpendicular magnetic anisotropy and magnetization of L10 FePt/FeCo bilayer films J. Appl. Phys. 115, 133908 (2014); 10.1063/1.4870463 Perpendicular magnetic anisotropy and spin reorientation transition in L10 FePt films J. Appl. Phys. 109, 07B760 (2011); 10.1063/1.3562506 Highly (001)-oriented Ni-doped L 1 0 FePt films and their magnetic properties J. Appl. Phys. 97, 10H309 (2005); 10.1063/1.1855271 Fabrication of L 1 0 ordered FePt alloy films by monatomic layer sputter deposition J. Appl. Phys. 93, 7238 (2003); 10.1063/1.1555363 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 132.241.216.43 On: Mon, 15 Dec 2014 05:18:37Fast magnetization precession observed in L10-FePt epitaxial thin film S. Mizukami,1,a/H20850S. Iihama,2N. Inami,2T . Hiratsuka,2G. Kim,2H. Naganuma,2 M. Oogane,2and Y . Ando2 1WPI-Advanced Institute for Materials Research, Tohoku University, Katahira 2-1-1, Sendai 980-8577, Japan 2Department of Applied Physics, Graduate School of Engineering, Tohoku University, Aoba 6-6-05, Sendai 980-8579, Japan /H20849Received 24 November 2010; accepted 10 January 2011; published online 31 January 2011 /H20850 Fast magnetization precession is observed in L10-FePt alloy epitaxial thin films excited and detected by all-optical means. The precession frequency varies from 45 to 65 GHz depending on the appliedmagnetic field strength and direction, which can be explained by a uniform precession model takingaccount of first- and second-order uniaxial magnetic anisotropy. The lowest effective Gilbertdamping constant has a minimum value of 0.055, which is about half that in Co/Pt multilayers andis comparable to Ni/Co multilayers with perpendicular magnetic anisotropy. © 2011 American Institute of Physics ./H20851doi:10.1063/1.3549704 /H20852 Nowadays, large amounts of digital information create ever increasing demands for much higher densities in mag-netic storage, in which magnetic materials with high-magnetic anisotropy play a key role. 1TheL10-ordered FePt alloy exhibits a very large uniaxial magnetic anisotropyenergy K u=66 Merg /cm3and a moderate saturation magnetization MS=1140 emu /cm3in bulk.2Thus, there have been lots of studies not only on fundamental propertiesbut also on advanced applications of magnetic storage 3and memory devices4using these alloys. In the recording pro- cess, memory can be written by various means, such as withmagnetic fields with or without assistance by laser heating 3,5 or microwaves6and spin-transfer-torque /H20849STT /H20850switching.7 In any process involving magnetization dynamics, the Gil- bert damping constant /H9251is one key indicator to optimize writing speeds and reduce power consumption. Recent stud-ies have shown that high- K umagnets including Pt possess large- /H9251,8–11so large- /H9251is also inferred in L10-FePt alloy, be- ing disadvantageous to the applications. Thus, it is quite ofimportance to investigate /H9251forL10-FePt alloys, especially for very thin films used in advanced applications. Here, wereport fast magnetization precession excited and detected byan all-optical means and an effective Gilbert damping con-stant in L1 0-FePt alloy epitaxial thin films. A 4 nm thick FePt epitaxial alloy film was prepared us- ing magnetron sputtering with a base pressure of less than2/H1100310 −7Pa. The FePt was deposited on the Pt /H2084910 nm /H20850/Cr/H2084940 nm/H20850buffered on a /H20849100/H20850-single crystal MgO substrate at a substrate temperature of 773 K and after cooling down wascapped by a Ta protecting layer. Structural analysis was per-formed by X-ray diffraction /H20849XRD /H20850and magnetization mea- surements were conducted by a superconducting quantuminterference device magnetometer. Details of procedures andstructural properties have been published elsewhere. 12Time- resolved magneto-optical Kerr effect /H20849MOKE /H20850was con- ducted by a conventional optical pump-probe set-up using aTi:sapphire laser combined with a regenerative amplifier inwhich the pulse laser wavelength, duration, and repetitionrate were /H11011800 nm, /H11011100 fs, and 1 kHz, respectively. 11Figure 1/H20849a/H20850shows an out-of-plane 2 /H9258-/H9275XRD pattern f o ra4n m thick FePt film. The /H20849001/H20850superlattice and /H20849002/H20850fundamental peaks for L10-FePt are observed clearly in the figure; while the /H20849200/H20850peaks for buffer and substrate are sharp, those for FePt are broadenedowing to interference fringes. The lattice constants a =3.90 Å and c=3.70 Å were evaluated from the /H20849001/H20850and /H20849111/H20850peaks observed in the 2 /H9258-/H9275XRD pattern with tilted substrate in the /H9273direction. The epitaxial relationship of MgO /H20849001/H20850/H20855100/H20856/H20648FePt /H20849001/H20850/H20855100/H20856was confirmed by the /H20849111/H20850 peak pole figure. Figure 1/H20849b/H20850shows magnetization curves for the film with the applied field parallel and perpendicular tothe film plane. The M S=1080 emu /cc is close to that in the bulk;2the squareness ratio is less than unity and coercivity is about 1 kOe. The hard axis hysteresis loop shows curvaturenear the zero applied field instead of a linear slope observedin perpendicularly-magnetized films and magnetization issaturated at /H1101130 kOe. Figure 2/H20849a/H20850shows Kerr signals as a function of pump- probe delay time for the film, measured with applied field H a/H20850Electronic mail: mizukami@wpi-aimr.tohoku.ac.jp. g. Intensity (a.u. ) (a) FePt (001)MgO (200)Pt (200)FePt (002)Cr (200) 20 30 40 50 60 70log 2θ/ω(o) (b) H// filmH /g1291film calc. -50 0 50-100001000M(emu/cc) H(kOe) FIG. 1. /H20849Color online /H20850/H20849a/H20850A2/H9258-/H9275X-ray diffraction pattern fo ra4n mt h i c k L10-FePt film. /H20849b/H20850Magnetization curves for a 4-nm- L10-FePt alloy epitaxial film with the applied field parallel and to perpendicular to the film plane.The solid curve is a theoretical calculation based on the uniform magneti-zation rotation model.APPLIED PHYSICS LETTERS 98, 052501 /H208492011 /H20850 0003-6951/2011/98 /H208495/H20850/052501/3/$30.00 © 2011 American Institute of Physics 98, 052501-1 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 132.241.216.43 On: Mon, 15 Dec 2014 05:18:37of 9.7 kOe directed at /H9258Hof 80° with different pump laser fluences Fp;/H9258His defined in Fig. 2/H20849b/H20850. The incident beams are almost parallel to the z-direction /H20851Fig.2/H20849b/H20850/H20852and the sig- nal is proportional to the z-component of magnetization, hence the precession signal intensity becomes large with alarge applied field near the film plane. The rapid decrease inKerr signals at zero delay time seen in Fig. 2/H20849a/H20850is typical of ultrafast demagnetization , at which M SandKuare reduced within hundreds of fs by a rising electron /H20849or spin /H20850tempera- ture resulting from pulse heating.13This leads to a sudden change in the effective magnetic anisotropy field and excitesa magnetization precession about the initial equilibriumangle /H9258/H20851Fig.2/H20849b/H20850/H20852. The oscillations in the Kerr signals cor- respond to this precession. The electron /H20849or spin /H20850temperature tends to recover its initial values through several transfer processes of heat that are manifested as exponential-like de-cays in the Kerr signals /H20851Fig.2/H20849a/H20850/H20852. This recovery rate de- pends not only on materials but also on the underlying nano-structure, pump-laser fluence, and other factors; thus, thisprocess is not easily modeled. 5,14–16Here, we assume that the oscillation signals can be fitted to a damped-harmonic func-tion added to an exponential-decaying background that ischaracterized by several time-independent parameters; spe-cifically, we have a+bexp/H20849− /H9263t/H20850+csin/H208492/H9266ft+/H9272/H20850exp/H20849−t//H9270/H20850, where aandbare the background magnitudes and /H9263is the background recovery rate. Parameters c,f,/H9272, and /H9270are the amplitude, frequency, phase, and relaxation time of magne-tization precession, respectively, as used previously.10,11In Fig.2/H20849a/H20850, the results of fitting are also shown as solid curves; the experimental data are well-fitted to the above equations.The validity of this fitting can be judged by whether theparameter values obtained are consistent with other measure-ments /H20849discussed later /H20850and can also be examined through the F p-dependence of fand 1 //H9270, as shown in Fig. 2/H20849c/H20850. Within the experimental errors, this shows negligible dependences atlow-F p, indicating that MSandKuare considered to be close to the original values during precession at low- Fp. Figure 3/H20849a/H20850shows the time-resolved Kerr signals for the thin films recorded with different /H9258HatH=9.7 kOe and Fp =0.39 mJ /cm2. Decayed precession signals are evident andthe data is well-fitted by the damped-harmonic functions /H20849solid curves /H20850, from which fvalues are extracted and plotted in Fig. 3/H20849b/H20850as a function of /H9258H. The fincreases from 57 to 65 GHz as /H9258Hdecreases from 80° to 40°, and this angular variation is weaker than those observed in films with perpen-dicular magnetic anisotropy /H20849PMA /H20850.11To understand this be- havior, we calculated fversus /H9258Husing the following equa- tions used in ferromagnetic resonance in PMA films:17,18 f=/H20849/H9253/2/H9266/H20850/H20881H1H2 with H1=Hcos/H20849/H9258H−/H9258/H20850+Hk1effcos 2/H9258 +Hk2/H208493 cos2/H9258sin2/H9258−cos4/H9258/H20850 and H2=Hcos/H20849/H9258H−/H9258/H20850 +Hk1effcos2/H9258−Hk2cos4/H9258. Here, /H9253is the gyromagnetic ratio defined as /H9253=g/H9262B//H6036with Lande’s g-factor g, while Hk1eff=2Ku1/MS+4Ku2/MS−4/H9266MSand Hk2=4Ku2/MS. The Ku1and Ku2are the first- and second-order uniaxial magnetic anisotropy constants, respectively. The /H9258is calculated from the equation: sin 2 /H9258=/H208492H/Hk1eff/H20850sin/H20849/H9258H−/H9258/H20850 +/H20849Hk2/Hk1eff/H20850cos2/H9258sin 2/H9258. The theoretical /H9258H-dependence of fis shown in Fig. 3/H20849b/H20850/H20849solid curve /H20850. The experimental and theoretical fversus Hdata are also shown in Fig. 3/H20849c/H20850.I n both these figures, the theoretical calculations have been fit-ted sufficiently to the experimental data. The best-fit param- eters are H k1eff=29.4 kOe, Hk2=18.8 kOe, and g=2.16. If Hk2is set to zero, the calculation does not yield weak angular variation, thus Ku2plays an important role in magnetization dynamics in this thin film. The respective Ku1andKu2take values 13 and 5.1 Merg /cm3, close to those for alloy films with L10-ordering parameter Sof around 0.5.19With these best-fit parameters, the hard-axis magnetization curve wasalso calculated using the equation for /H9258/H20851Fig.1/H20849b/H20850/H20852, which is in agreement with the experimental data, and this supportsthe validity of the analysis of time-resolved MOKE. While the 1 / /H9270versus /H9258HandHwere also obtained from the fitting, those deviated from the theoretical calculation of1/ /H9270=/H9251/H9253/H20849H1+H2/H20850/2 with the above best-fit parameters and any value of /H9251/H20849Fig.3/H20850. Theoretical 1 //H9270is roughly propor- tional to fand the deviation is possibly due to extrinsic mag- netic relaxation.11,18To gain quantitative information of Gil- bert damping, an effective damping constant, /H9251eff, defined as /H9251eff=1 /2/H9266f/H9270, was evaluated for the film /H20849Fig.4/H20850. From its-400-300-200-1000OKE signal (arb. unit)(a) (b) (c) 60 5060Hz) rad/s)0.24 mJ/cm2 0.39 mJ/cm2 0.78 mJ/cm2 1.6 mJ/cm2 2.0 mJ/cm2 2.4 mJ /cm2FP= HMθθθθ θθθθHz 020406080100-500400MO Delay time (ps)0 1 25055 3040f(GH Fp(mJ/cm2) 1/τ(Gr FIG. 2. /H20849Color online /H20850/H20849a/H20850Time-resolved Kerr signals for the epitaxial film ofL10-FePt with various pump laser fluences Fp, measured with applied field H=9.7 kOe at field angle /H9258H=80°, as defined in /H20849b/H20850. Solid curves plot the parameter-fitted damped-harmonic function of the data to obtain preces-sion frequency fand inverse of life-time 1 / /H9270for the film. /H20849b/H20850Schematic of time-resolved measurement geometry. A precession of magnetization Mis excited about /H9258which represents an initial equilibrium direction of M./H20849c/H20850Fp dependence of fand 1 //H9270. Solid curves are as visual guides.-1000(arb. unit)(b) (a) H=9.7 kOe θH=3 0o 40o 50o506070 204060f(GHz) 1/τ(Grad/s ) 020406080100-300-200MOKE signal Delay time (ps)(c)50 60o 70o 80o50oθH=8 0o0 30 60 90 θH(o) 02468100204060 20406080f(GHz) H(kOe) 1/τ(Grad/s ) FIG. 3. /H20849Color online /H20850/H20849a/H20850Time-resolved Kerr signals for the L10-FePt thin film with different field directions /H9258H, measured at applied field H =9.7 kOe and pump fluence Fp=0.39 mJ /cm2. A damped-harmonic func- tion has been parameter-fitted to the data /H20849solid curves /H20850. Precession fre- quency f/H20849/L50098/H20850and inverse of life-time 1 //H9270/H20849/H17034/H20850for the film as a function of /H9258H/H20849b/H20850andH/H20849c/H20850.I n /H20849b/H20850and /H20849c/H20850, theoretical calculations of fare fitted to the data /H20849solid curves /H20850and broken curves are typical theoretical calculations of 1//H9270with a Gilbert damping constant /H9251of 0.055.052501-2 Mizukami et al. Appl. Phys. Lett. 98, 052501 /H208492011 /H20850 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 132.241.216.43 On: Mon, 15 Dec 2014 05:18:37definition, /H9251does not exhibit any variation against /H9258HandH, but here, /H9251effvaries considerably. The /H9251effaccounts not only for/H9251but also for the extrinsic magnetic relaxation, so these trends are considered to reflect some dependence on extrinsicmagnetic relaxation and the true /H9251is then lower than /H9251eff values. The /H9251effhas a minimum of 0.055 at /H9258H=50° and H =9.7 kOe, so the true /H9251for this FePt thin film may be no greater than /H110110.06 although the proportion of /H9251to this mini- mum /H9251effis unclear, which depends on the H- and /H9258H-variation of extrinsic magnetic relaxation. Interestingly, this minimum /H9251effforL10-FePt alloy thin films is close to that assumed in the STT-switchingexperiments.7Also, this value is comparable to /H9251in/H1101115% Pt-doped permalloy films,20in the CoCrPt films,11and in the Co/Ni multilayer films18but is about half the /H9251values of Co/Pt multilayer films,8even though the Kuvalues for them are not larger than for this FePt film. These facts may indi-cate that /H9251is not scaled simply by Kueven though both /H9251 and Kuoriginate from spin-orbit interactions.21,22Further studies, e.g., L10-ordering or thickness dependence of /H9251, should be performed by measurements with a higher mag-netic field and/or by other techniques, although such explo-rations are outside the scope of this report. In summary, fast magnetization precessions in the L1 0-FePt alloy epitaxial thin film were investigated by all- optical time-resolved MOKE. Precession frequencies exhib-ited the variation with applied magnetic field strength anddirection, and these were well-explained by the uniform pre-cession model and were consistent with the hard-axis M-H curve. The lowest effective value for the Gilbert dampingconstant was determined in the FePt alloy film to be 0.055.This work was partially supported by the Funding Pro- gram for World-Leading Innovative R&D on Science andTechnology /H20849FIRST Program /H20850by JSPS and Grant-in-Aids for Scientific Research. 1D. Weller, A. Moser, L. Folks, M. E. Best, W. Lee, M. F. Toney, M. Schwickert, J.-U. Thiele, and M. F. Doerner, IEEE Trans. Magn. 36,1 0 /H208492000 /H20850. 2T. Klemmer, D. Hoydick, H. Okumura, B. Zhang, and W. A. Soffa, Scr. Metall. Mater. 33, 1793 /H208491995 /H20850. 3W. A. Challener, C. Peng, A. V. Itagi, D. Karns, W. Peng, Y. Peng, X. Yang, X. Zhu, N. J. Gokemeijer, Y.-T. Hsia, G. Ju, R. E. Rottmayer, M. A.Seigler, and E. C. Gage, Nat. Photonics 3, 220 /H208492009 /H20850. 4M. Yoshikawa, E. Kitagawa, T. Nagase, T. Daibou, M. Nagamine, K. Nishiyama, T. Kishi, and H. Yoda, IEEE Trans. Magn. 44, 2573 /H208492008 /H20850. 5C. Bunce, J. Wu, G. Ju, B. Lu, D. Hinzke, N. Kazantseva, U. Nowak, and R. W. Chantrell, Phys. Rev. B 81, 174428 /H208492010 /H20850. 6S. Okamoto, N. Kikuchi, and O. Kitakami, Appl. Phys. Lett. 93, 102506 /H208492008 /H20850. 7T. Seki, S. Mitani, K. Yakushiji, and K. Takanashi, Appl. Phys. Lett. 88, 172504 /H208492006 /H20850. 8A. Barman, S. Wang, O. Hellwig, A. Berger, and E. E. Fullerton, J. Appl. Phys. 101, 09D102 /H208492007 /H20850. 9G. Malinowski, K. C. Kuiper, R. Lavrijsen, H. J. M. Swagten, and B. Koopmans, Appl. Phys. Lett. 94, 102501 /H208492009 /H20850. 10S. Mizukami, E. P. Sajitha, F. Wu, D. Watanabe, M. Oogane, H. Na- ganuma, Y. Ando, and T. Miyazaki, Appl. Phys. Lett. 96, 152502 /H208492010 /H20850. 11S. Mizukami, D. Watanabe, T. Kubota, X. Zhang, H. Naganuma, M. Oo- gane, Y. Ando, and T. Miyazaki, Appl. Phys. Express 3, 123001 /H208492010 /H20850. 12N. Inami, G. Kim, T. Hiratsuka, H. Naganuma, M. Oogane, and Y. Ando, J. Phys.: Conf. Ser. 200, 052008 /H208492010 /H20850. 13M. van Kampen, C. Jozsa, J. T. Kohlhepp, P. LeClair, L. Lagae, W. J. M. de Jonge, and B. Koopmans, Phys. Rev. Lett. 88, 227201 /H208492002 /H20850. 14Y. Ren, H. Zhao, Z. Zhang, and Q. Y. Jin, Appl. Phys. Lett. 92, 162513 /H208492008 /H20850. 15Z. Xu, X. D. Liu, R. X. Gao, Z. F. Chen, T. S. Lai, H. N. Hu, S. M. Zhou, X. J. Bai, and J. Du, Appl. Phys. Lett. 93, 162509 /H208492008 /H20850. 16N. Kazantseva, D. Hinzke, U. Nowak, R. W. Chantrell, U. Atxitia, and O. Chubykalo-Fesenko, Phys. Rev. B 77, 184428 /H208492008 /H20850. 17C. Chappert, K. Le Dang, P. Beauvillain, and H. Hurdedquint, Phys. Rev. B34, 3192 /H208491986 /H20850. 18J.-M. Beaujour, D. Ravelosona, I. Tudosa, E. E. Fullerton, and A. D. Kent, Phys. Rev. B 80, 180415 /H208492009 /H20850. 19S. Okamoto, N. Kikuchi, O. Kitakami, T. Miyazaki, Y. Shimada, and K. Fukamichi, Phys. Rev. B 66, 024413 /H208492002 /H20850. 20J. O. Rantschler, R. D. McMichael, A. Castillo, A. J. Shapiro, W. F. Egelhoff, Jr., B. B. Maranville, D. Pulugurtha, A. P. Chen, and L. M.Connors, J. Appl. Phys. 101, 033911 /H208492007 /H20850. 21K. Gilmore, Y. U. Idzerda, and M. D. Stiles, Phys. Rev. Lett. 99, 027204 /H208492007 /H20850. 22P. Bruno, Phys. Rev. B 39, 865 /H208491989 /H20850.0.100.15eff 0.100.15eff(a) (b) θH=8 0oH= 9.7 kOe 02040608000.05αe θH(o)024681000.05 H(kOe)αe 50o FIG. 4. /H20849Color online /H20850Effective damping constant /H9251eff., defined as /H9251eff. =1 /2/H9266f/H9270, for the L10-FePt thin film as a function of /H20849a/H20850magnetic field direction /H9258Hand /H20849b/H20850strength H.052501-3 Mizukami et al. Appl. Phys. Lett. 98, 052501 /H208492011 /H20850 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 132.241.216.43 On: Mon, 15 Dec 2014 05:18:37

1.3317406.pdf

The ground state van der Waals potentials of the strontium dimer and strontium rare-gas complexes G. P. Yin, P. Li, and K. T. Tang Citation: J. Chem. Phys. 132, 074303 (2010); doi: 10.1063/1.3317406 View online: http://dx.doi.org/10.1063/1.3317406 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v132/i7 Published by the American Institute of Physics. Additional information on J. Chem. Phys. Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors Downloaded 15 Apr 2013 to 171.67.34.69. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissionsThe ground state van der Waals potentials of the strontium dimer and strontium rare-gas complexes G. P . Yin,1P . Li,1and K. T . T ang1,2, a/H20850 1Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, Sichuan, People’ s Republic of China 2Department of Physics, Pacific Lutheran University, Tacoma, Washington 98447, USA /H20849Received 27 November 2009; accepted 24 January 2010; published online 17 February 2010 /H20850 The entire ground state potential energy curve of the strontium dimer is accurately described by the Tang–Toennies potential model defined by the three dispersion coefficients and two well parameters.The predicted vibrational frequency, anharmonicity, and vibration-rotation coupling constant are inexcellent agreement with experiment. The Sr 2reduced potential is almost identical to that of Ca 2and Hg2, providing further evidence to the conjecture that the van der Waals dimer potentials of group IIA and group IIB elements have the same shape, which is different from that of rare-gas dimers.The potentials of Sr-RG complexes /H20849RG=He,Ne,Ar,Kr,Xe /H20850are generated by the same potential model with its parameters calculated with combining rules. These potentials are shown to have the same shape which is between that of the strontium and rare-gas dimers. © 2010 American Institute of Physics ./H20851doi:10.1063/1.3317406 /H20852 I. INTRODUCTION Diatomic alkaline earth molecules are excimer species. Interests in the van der Waals potentials of these dimers in-tensified in recent years, mainly because of cold atomphysics. 1In a recent paper,2we have shown that the entire ground state potential energy curve of calcium dimer can bedescribed by the Tang–Toennies /H20849TT/H20850potential model. 3Fur- thermore, the parameters of the potential obtained can becombined with the corresponding ones of the rare-gas dimersto give the potentials of the calcium rare-gas complexes. In the present article, the ground state strontium dimer potential is described and the potential curves of Sr-RG/H20849RG=He,Ne,Ar,Kr,Xe /H20850are determined in the same way. As far as we are aware, the present results are the first re- ported for the van der Waals potentials of SrNe, SrKr, andSrXe. The existence of the Sr 2molecule was first shown by laser spectroscopy in rare-gas matrices.4A Morse potential was constructed for the Sr 2ground state by Bergeman and Liao from the spectra of laser-excited photoluminescence ofthe dimer in gas phase. 5A more detailed analysis of the well resolved discrete fluorescence spectrum of the laser-inducedemission by Gerber et al. led to a set of Dunham coefficients and the Rydberg–Klein–Rees /H20849RKR /H20850potentials for the ground X 1/H20858g+state and the excited state, thought to be A1/H20858u+,o fS r 2.6The most recent and probably the most accu- rate experimental determination of Sr 2potentials was done by Stein et al.7with high resolution Fourier-transform spec- troscopy. A set of Dunham coefficients with 35 free param-eters for the ground state was constructed. The first few co-efficients /H20849such as Y 10,Y20,Y11/H20850, to which physical meaning can be attributed, are in very good agreement with the cor-responding ones found in Ref. 6. However, it was found thatthe rotational assignment of Gerber et al. has to be changed by four units for a consistent description. As a consequence,the equilibrium distance R eof the ground state potential of Stein et al. is larger by about 5% than that of the RKR potential in Ref. 6. The first theoretical calculations of Sr 2molecule were carried out with the density functional methods.8,9So far it is still a challenge for the density functional theory to describe the weakly bonded van der Waals molecules. These werefollowed by several sets of coupled cluster calculations withpseudopotential and single, double, and perturbative tripleexcitations /H20851CCSD /H20849T/H20850/H20852. 10–12Although this method gives a reasonable description of the interaction potentials, the re-sults depend on the chosen basis set and the core potential.With the same basis set and core polarization potential, someauthors also carried out multireference configuration interac-tion /H20849MRCI /H20850calculations. 11,12Compared to experiment, the binding is generally underestimated by CCSD /H20849T/H20850and over- estimated by MRCI. A nonrelativistic multiconfigurationself-consistent field /H20849MCSCF /H20850method with a two valence electron pseudopotential was used by Boutassetta et al. 13to calculate the low lying states of Sr 2molecules. The ground state potential is shifted to larger Rwith respect to the RKR potential of Gerber et al. but not as far as the potential of Stein et al. Recently an all electron relativistic valence bond configuration interaction calculation was carried out byKotochigova. 14The basis set was chosen to best reproduce the ground state RKR potential of Ref. 6. The equilibrium distance Reof the ground X1/H20858g+state is determined to be 8.35a0, which is only slightly smaller than the experimental value 8.40 a0of Gerber et al. but much smaller than the re- cent experimental value 8.83 a0of Stein et al. Far less is known about the interaction potentials be- tween strontium and rare-gas atoms. Many experiments onbroadening, shifting of resonance lines and redistribution of a/H20850Electronic mail: tangka@plu.edu.THE JOURNAL OF CHEMICAL PHYSICS 132, 074303 /H208492010 /H20850 0021-9606/2010/132 /H208497/H20850/074303/10/$30.00 © 2010 American Institute of Physics 132 , 074303-1 Downloaded 15 Apr 2013 to 171.67.34.69. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissionslight induced by collisions between strontium and rare-gas atoms have been carried out.15–20For quantitative interpreta - tions of these experiments, reliable potentials of Sr-RG arehighly desirable. There are several calculations of the SrHepotentials. 21–23Unfortunately, results from these computa - tions differ greatly from each other. Among all strontiumrare-gas complexes, SrAr is the only complex with its inter-action potential determined experimentally, and only in onesupersonic beam experiment. 24For this system a MCSCF calculation was carried out by Zhu et al.25The results are in reasonable agreement with experiment. An important development in theory is the ab initio cal- culations of the van der Waals dispersion coefficients. It iswell known that the interaction between two spherical sym-metric atoms in the asymptotic region is given by V/H20849R/H20850=−C 6 R6−C8 R8−C10 R10¯, /H208491/H20850 where Ris the internuclear separation, and Cnare the disper- sion coefficients. For Sr 2, there are a number of determina- tions for these coefficients in recent years.26–30They are gen - erally within 5% of each other. The values of C6,C8, and C10 of alkaline earth interactions calculated by Porsev and Derevianko30using relativistic many-body perturbation theory are believed to be accurate to 1%. It is important that the interaction potential energy prop- erly approach this asymptotic expression, especially for thecold atom collisions. In the accurate strontium dimer hybridpotential of Stein et al. , 7an elaborate nonlinear fitting pro - cedure is used to smoothly join Eq. /H208491/H20850to the potential in the intermediate potential well region at a judiciously chosenpoint R a. The intermediate potential is given in the form of /H20858i=0nai/H20851/H20849R−Rm/H20850//H20849R+bRm/H20850/H20852iwith n=20. This intermediate po- tential is then joined at another point Rito the short range repulsion in the form of A+B/R6. In this paper, we will show that with the given set of C6,C8,C10and the experimentally determined equilibrium distance Reand the well depth De, the TT potential3for Sr 2 can be constructed without any fitting. This potential is in excellent agreement with the multiparameter hybrid potentialof Stein et al. This enables us to easily compare the shapes of several types of van der Waals potentials. Furthermore, sincethe TT potential model can accurately describe both thehomonuclear strontium and rare-gas dimers, we investigatethe possibility of predicting the interaction potentials ofSr-RG complexes with combining rules. Atomic units will be used in all calculations. For com- parison with literature values, energy unit cm −1and length unit angstrom will also be used, /H20851For energy, 1 a.u. =2.1947 /H11003105cm−1; for distance, 1 a0=0.5292 Å. /H20852 II. METHOD A. The potential model The potential model proposed by TT in 1984 /H20849Ref. 3/H20850 will be used to generate the ground state interaction potentialof the strontium dimer. In this model, the short range repul-sive Born–Mayer potential Aexp /H20849−bR/H20850is added to the long range attractive potential which is given by the damped asymptotic dispersion series, V/H20849R/H20850=Ae −bR−/H20858 n=3nmax f2n/H20849bR/H20850C2n R2n. /H208492/H20850 Based on the form of the exchange correction to the disper- sion series, Tang and Toennies were led to the conclusionthat the damping function is an incomplete gamma function, f 2n/H20849bR/H20850=1− e−bR/H20858 k=02n/H20849bR/H20850k k!, /H208493/H20850 where bis the same as the range parameter of the Born– Mayer repulsion on the ground that both the repulsion anddispersion damping are consequences of the wave functionoverlap. Since its introduction, this damping function hasbeen widely used. For simple systems, such as H 2,H e 2, and HHe, damped dispersion can be numerically calculated;31–33 these accurately calculated damping functions are all in very good agreement with Eq. /H208493/H20850. The Born–Mayer repulsive po- tential is, of course, only an approximation, but experiencewith many systems indicates that it is usually a very goodapproximation in the van der Waals region. In principle therepulsive potential can be obtained a priori from theory, such as from a SCF calculation with an additional exchange dis-persion correction or from the asymptotic surface integralmethod. 34–36 For many systems, the first three coefficients C6,C8, and C10, which are usually available, are adequate, but for higher accuracy additional terms can be generated by the recurrencerelation 37 Cn+4=/H20873Cn+2 Cn/H208743 Cn−2. /H208494/H20850 Tested against the accurate values of H–H and He–He interactions,31,38this relation can predict C12,C14,C16to bet - ter than 4%. Generally, terms beyond C16do not make any appreciable contribution to the potential. Thus, the model potential is determined by five param- eters A,b,C6,C8, and C10. If the first three dispersion coef- ficients are available, only two parameters /H20849A,b/H20850need to be known in order to use the TT model. Furthermore, if the equilibrium distance Reand the well depth of the potential Deare known, then Aandbcan be determined in the fol- lowing way. It is useful to write the potential in its reduced form. Let x=R/Reand U/H20849x/H20850=V/H20849Rex/H20850/De, /H208495/H20850 so U/H20849x/H20850=A/H11569e−b/H11569x−/H20858 n=3nmax/H208751−e−b/H11569x/H20858 k=02n/H20849b/H11569x/H20850k k!/H20876C2n/H11569 x2n, /H208496/H20850 where074303-2 Yin, Li, and T ang J. Chem. Phys. 132 , 074303 /H208492010 /H20850 Downloaded 15 Apr 2013 to 171.67.34.69. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissionsA/H11569=A De,b/H11569=bRe,C2n/H11569=C2n DeRe2n. /H208497/H20850 It is convenient to define U/H20849n/H20850/H20849x/H20850=dnU/H20849x/H20850 dxn. /H208498/H20850 At the potential minimum R=Re/H20849x=1/H20850, the reduced potential provides us with two conditions, U/H208491/H20850=−1 , /H208499/H20850 U/H208491/H20850/H208491/H20850=0 . /H2084910/H20850 It follows that U/H208491/H20850=A/H11569e−b/H11569−/H20858 n=3nmax/H208751−e−b/H11569/H20858 k=02n/H20849b/H11569/H20850k k!/H20876C2n/H11569=−1 /H2084911/H20850 and U/H208491/H20850/H208491/H20850=−b/H11569A/H11569e−b/H11569−/H20858 n=3nmax b/H11569e−b/H11569/H20849b/H11569/H208502n /H208492n/H20850!C2n/H11569 +/H20858 n=3nmax/H208751−e−b/H11569/H20858 k=02n/H20849b/H11569/H20850k k!/H208762nC2n/H11569=0 . /H2084912/H20850 From the last equation we have A/H11569=/H20858 n=3nmax/H20875eb/H11569−/H20858 k=02n/H20849b/H11569/H20850k k!/H208762n b/H11569C2n/H11569−/H20858 n=3nmax/H20849b/H11569/H208502n /H208492n/H20850!C2n/H11569. /H2084913/H20850 Substituting A/H11569into Eq. /H2084911/H20850, /H20858 n=3nmax/H208751−e−b/H11569/H20858 k=02n/H20849b/H11569/H20850k k!/H20876/H208732n b/H11569−1/H20874C2n/H11569−/H20858 n=3nmax e−b/H11569/H20849b/H11569/H208502n /H208492n/H20850!C2n/H11569 =−1 . /H2084914/H20850 The only unknown in this equation is b/H11569. It can be expedi- ently solved by an iterative procedure on a computer. Thereis a simple program in the appendix of Ref. 39to do that. With a given set of dispersion coefficients, the program con-verts D eandReinto Aandb. B. Spectroscopic parameters Experimentally, the shape of the potential is best charac- terized by the spectroscopic parameters of vibration and ro-tation. The parameters are usually given by the coefficientsof a power series expansion of the energy respect to /H20849 v +1 /2/H20850kand /H20851J/H20849J+1/H20850/H20852l, where vandJare the quantum num- bers of vibration and rotation, respectively. That is, E/H20849v,J/H20850=/H20858 k=0/H20858 l=0Ykl/H20849v+1 2/H20850k/H20851J/H20849J+1/H20850/H20852l, /H2084915/H20850 where Yklare known as Dunham coefficients.40Physical meanings can be attached to the first few Dunham coeffi- cients: Y10/H11229/H9275e, /H2084916/H20850 Y20/H11229−/H9275e/H9273e, /H2084917/H20850Y11/H11229−/H9251e, /H2084918/H20850 where /H9275eis the classical frequency, /H9275e/H9273eis the anharmonic- ity, and /H9251eis the vibrotational coupling constant. The exact expressions for Y10,Y20, and Y11are given by Dunham.40 They differ from the expressions in Eq. /H2084916/H20850to Eq. /H2084918/H20850by terms in Be2//H9275e2./H20849For Sr 2,Be2//H9275e2=1.9/H1100310−7, see Sec. III. /H20850 By expanding the reduced potential function near the equilibrium into a power series with respect to the relativedistance x, these parameters can be expressed in terms of the expansion coefficients. Let U/H20849x/H20850be the reduced potential, U/H20849x/H20850=−1+ a 0/H20849x−1/H208502/H208491+a1/H20849x−1/H20850+a2/H20849x−1/H208502+¯/H20850, /H2084919/H20850 where a0=1 2U/H208492/H20850/H208491/H20850,a1=1 3U/H208493/H20850/H208491/H20850 U/H208492/H20850/H208491/H20850,a2=1 12U/H208494/H20850/H208491/H20850 U/H208492/H20850/H208491/H20850,¯ /H2084920/H20850 It can be shown that the following three dimensionless quan- tities can be expressed in terms of the expansion coefficientsa n,40,41 BeDe /H9275e2=1 4a0, /H2084921/H20850 /H9251e/H9275e Be2=−6 /H208491+a1/H20850, /H2084922/H20850 /H9273e/H9275e Be=−3 2/H20873a2−5 4a12/H20874, /H2084923/H20850 where Beis the rotational constant, and Be=h//H208498/H92662/H9262cRe2/H20850.I n terms of atomic units, Be/H20849a.u./H20850=1 2/H9262Re2, /H2084924/H20850 where both the equilibrium distance Reand the reduced mass /H9262should be in atomic units. If /H9262is given in amu, then it needs to be multiplied by a factor of 1822.8, since 1 amu=1822.8 m e, where meis the mass of an electron which is equal to one in atomic units. If Beis to be expressed in cm−1, as usually done in spectroscopy, then another conversion fac-tor has to be multiplied, B e/H20849cm−1/H20850=Be/H20849a.u. /H20850/H110032.194 74 /H11003105. /H2084925/H20850 Since U/H20849x/H20850of Eq. /H208496/H20850is an analytic expression, its de- rivatives can be calculated in a straight forward way, U/H208492/H20850/H208491/H20850=b/H115692A/H11569e−b/H11569+/H20858 n=3nmax e−b/H11569/H20849b/H115692+2nb/H11569/H20850/H20849b/H11569/H208502n /H208492n/H20850!C2n/H11569 −/H20858 n=3nmax/H208751−e−b/H11569/H20858 k=02n/H20849b/H11569/H20850k k!/H208762n/H208492n+1/H20850C2n/H11569,/H2084926/H20850074303-3 Strontium dimer and strontium rare gas J. Chem. Phys. 132 , 074303 /H208492010 /H20850 Downloaded 15 Apr 2013 to 171.67.34.69. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissionsU/H208493/H20850/H208491/H20850=−b/H115693A/H11569e−b/H11569−/H20858 n=3nmax e−b/H11569/H20851b/H115693+2nb/H115692+4n/H20849n+1/H20850b/H11569/H20852 /H11003/H20849b/H11569/H208502n /H208492n/H20850!C2n/H11569+/H20858 n=3nmax/H208751−e−b/H11569/H20858 k=02n/H20849b/H11569/H20850k k!/H208762n/H208492n+1/H20850 /H11003/H208492n+2/H20850C2n/H11569, /H2084927/H20850 U/H208494/H20850/H208491/H20850=b/H115694A/H11569e−b/H11569+/H20858 n=3nmax e−b/H11569/H20851b/H115694+2nb/H115693+2n/H208492n+3/H20850b/H115692 +4n/H20849n+1/H20850/H208492n+3/H20850b/H11569/H20852/H20849b/H11569/H208502n /H208492n/H20850!C2n/H11569 −/H20858 n=3nmax/H208751−e−b/H11569/H20858 k=02n/H20849b/H11569/H20850k k!/H208762n/H208492n+1/H20850/H208492n+2/H20850 /H11003/H208492n+3/H20850C2n/H11569. /H2084928/H20850 These derivatives can be used to find the expansion co- efficients an. With De,Re,a0,a1, and a2, the spectroscopic parameters /H9275e,/H9273e/H9275e, and/H9251ecan be easily determined from Eq. /H2084921/H20850to Eq. /H2084923/H20850. III. STRONTIUM DIMER POTENTIAL A. The ground state potential of Sr2 For Sr 2,De=4.929 /H1100310−3a.u. /H208491081.82 cm−1/H20850and Re =8.828 a0/H208494.672 Å /H20850are well established by Stein et al.7To- gether with the three dispersion coefficients, C6=31.03 /H11003102a.u., C8=37.92 /H11003104a.u., C10=42.15 /H11003106a.u., given in Porsev and Derevianko,30the Born–Mayer param - eters are determined with the simple program in the appendixof Ref. 39to be A= 44.73 a.u., b= 0.9699 a 0−1, for nmax=5 , /H2084929/H20850 A= 34.20 a.u., b= 0.9305 a0−1, for nmax=8 . /H2084930/H20850 The ground state potential energy curve of strontium dimer calculated from the TT model with these parameters isshown in Fig. 1. The potentials in the well region calculated forn max=5 and nmax=8 are indistinguishable in the scale of Fig. 1. This is because with Deand Refixed, the present model is to a large extent self-adjusting. The difference in thenumber of terms of the dispersion series is nearly completelycompensated by the changes in Aandbto produce the cor- rect shape near the bottom of the potential well. For R/H11271R e, the potential is dominated by the three leading dispersionterms. There are, of course, some differences in numericalvalues, but the differences are very small. For the sake ofclarity, from here on all numerical results are taken from thecalculations with n max=5 unless otherwise specified. In Fig. 1, the 22-parameter hybrid potential of Stein et al.,7which is valid in the region from 4 to 11 Å /H208497.6–20.8 a.u. /H20850for their spectroscopic data, is also shown. It is seen that the present potential is in excellent agreement with the ex-perimentally determined potential. The only discernible dif-ferences are in the region for R/H110214 Å, where the present potential is shifted slightly to the right of this experimentalhybrid potential. The functional forms of the hybrid potential for Sr 2used by Stein et al.7and for Ca 2used by Allard et al.42are exactly the same, except in the region of R/H11021Ri/H20849Ri=3.98 Å for Sr 2, Ri=3.66 Å for Ca 2/H20850. In this region, for Sr 2the potential is given by A+B/R6, whereas for Ca 2it is A+B/R12.I ti si n - teresting to note that in this region the TT potential for Sr 2is shifted slightly to the right of the experimental hybrid poten-tial, and, in the case of Ca 2, it is shifted slightly to the left.2 In view of the lack of data and the uncertainty of the func-tional form of the experimental hybrid potential in this re-gion, the present potential may not be less accurate than thehybrid potential. A more stringent test is to compare the spectroscopic parameters of the present potential with the experimentalDunham coefficients. With the present potential, the threedimensionless ratios /H20851Eqs. /H2084921/H20850–/H2084923/H20850/H20852for Sr 2are BeDe /H9275e2= 0.011 55;/H9251e/H9275e Be2= 22.155;/H9273e/H9275e Be= 22.162. /H2084931/H20850 The rotational constant Befor88Sr2calculated from Eq. /H2084924/H20850 is 0.017 55 cm−1. It follows that in the unit of cm−1 /H9275e= 40.54; /H9273e/H9275e= 0.3890; /H9251e= 0.000 168. /H2084932/H20850 These predictions can be directly compared with the three available sets of experiments. From Stein et al.7 /H9275e/H20849Y10/H20850= 40.32; /H9273e/H9275e/H20849−Y20/H20850= 0.3994; /H2084933/H20850 /H9251e/H20849−Y11/H20850= 0.000 168. From Gerber et al.6 FIG. 1. The ground state potential of strontium dimer. The solid red line is the present TT potential. The dashed black line is the experimental multipa-rameter hybrid potential of Stein et al. /H20849Ref. 7/H20850, which is valid from 4 to 11 Å/H208497.6–20.8 a.u. /H20850for their spectroscopic data. Note the only discernible difference between the two is in the region for Rless than 7.6 a.u. where the present potential is shifted slightly to the right.074303-4 Yin, Li, and T ang J. Chem. Phys. 132 , 074303 /H208492010 /H20850 Downloaded 15 Apr 2013 to 171.67.34.69. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions/H9275e/H20849Y10/H20850= 40.32 /H110060.02; /H9273e/H9275e/H20849−Y20/H20850= 0.405 /H110060.015; /H2084934/H20850 /H9251e/H20849−Y11/H20850=/H20849200/H110064/H20850/H1100310−6. From Bergeman and Liao5 /H9275e= 39.6 /H110061.0;/H9273e/H9275e= 0.45 /H110060.02; /H9251e= 0.0025. /H2084935/H20850 The Morse potential of Bergeman and Liao is not expected to yield reliable /H9273e/H9275eand/H9251e. Other than that, the predicted parameters are in good agreement with experiments. It is interesting to note that /H9275eand/H9273e/H9275egiven by Stein et al. are in close agreement with the corresponding ones given by Gerber et al. , but the well parameters /H20849Re,De/H20850de- termined in these two experiments are different from each other. To examine the effect of the well parameters, werepeated the calculation with R e/H20849=4.45 Å /H20850and De /H20849=1060 cm−1/H20850given by Gerber et al. The results are /H9275e= 37.29; /H9273e/H9275e= 0.3317; /H9251e= 0.000 188. /H2084936/H20850 They are all outside the experimental error bounds. The other input data to the TT potential are the disper- sion coefficients. We carried out another set of calculationswith the well parameters given by Stein et al. and the dis- persion coefficients C 6/H20849=3249 a.u. /H20850,C8/H20849=385 400 a.u. /H20850, andC10/H20849=42 500 000 a.u. /H20850given by Mitroy and Bromley.48 The results are /H9275e= 39.64; /H9273e/H9275e= 0.4179; /H9251e= 0.000 178. /H2084937/H20850 Compared with experiment, the agreement is not as good as those obtained with the dispersion coefficients of Porsev andDerevienko. For example, the value of /H9251ein Eq. /H2084932/H20850is the same as the one in Eq. /H2084933/H20850, but the /H9251ein Eq. /H2084937/H20850is 6% larger. These comparisons show that the validity of the potential model depends critically on the input data. With the disper-sion coefficients C 6,C8,C10of Porsev and Derevienko30and the well parameters Re,Deof Stein et al. ,7the TT model is able to describe the entire ground state potential energy curve of Sr 2in the van der Waals region with spectroscopic precision. B. Comparison of van der Waals potentials Homonuclear van der Waals diatomic molecules are formed from elements of three groups in the periodic table:group IIA, the alkaline earth metals; group IIB, Zn, Cd, andHg; and group VIII, rare gases. The strength of the bond is,however, very different for these groups. It is remarkable thatthey can all be described by the TT potential model. Thesituation is illustrated in Fig. 2/H20849a/H20850, where the present Sr 2 potential is compared with the potentials of Ca 2,H g 2,A r 2, and Kr 2. The parameters of these potentials are listed in Table I. The rare-gas dimer bond is purely dispersive, there- fore, the well depths Deof Ar 2and Kr 2are relatively shal- low. The mercury dimer bond is not purely dispersive but isstrengthened by a covalent exchange contribution. 45The po - tential well of the ground state Hg 2is more than three times as deep as the nearby rare-gas dimer. The alkaline earthdimer bond is also not purely dispersive because of the nears-pdegeneracy of the valence shell. Like the rare gases, thealkaline earth atoms have a closed-shell electron configura- tion. But unlike rare-gas atoms, the alkaline earths arechemically active due to low ionization potentials which al-low them to bind ionically with other atoms. This effect isreflected clearly in the well depths D eof the interatomic potentials. The well depth of Sr 2and of Ca 2are almost one order of magnitude larger than that of the nearby rare-gasdimer. The TT potential model was first derived for theweakly bonded systems such as the rare-gas dimers. Onlyrecently was this model shown to work just as well for themuch strongly bonded van der Waals systems. The uncannyability of the TT model to mimic the universal behavior ofthe van der Waals potential is once again demonstrated bythe present results. It is interesting to compare the shapes of these three types of van der Waals potentials, since they can all be de-scribed by the same TT model. In Fig. 2/H20849b/H20850, the reduced FIG. 2. /H20849a/H20850The ground state potentials of Ar2,K r2,H g2,C a2,S r2. They are representatives of group IIA /H20849Ca2,S r2/H20850, group IIB /H20849Hg2/H20850, and group VIII /H20849Ar2,K r2/H20850dimers. These potentials are all described by the TT potential model. /H20849b/H20850Comparison of the reduced potentials of Ar2,K r2,H g2,C a2,S r2. Note that the reduced potentials of group IIA /H20849Ca2,S r2/H20850and group IIB /H20849Hg2/H20850dimers are almost identical, while group VIII /H20849Ar2,K r2/H20850dimers have a different shape.074303-5 Strontium dimer and strontium rare gas J. Chem. Phys. 132 , 074303 /H208492010 /H20850 Downloaded 15 Apr 2013 to 171.67.34.69. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissionspotentials U/H20849x/H20850/H20851=V/H20849xRe/H20850/De/H20852of Sr 2,C a 2,H g 2,A r 2, and Kr 2 are plotted against x/H20851=R/Re/H20852. It is seen that these five sys- tems fall into two groups. The reduced potentials of Ar 2and Kr2are identical. This is well known that all rare-gas dimers are conformal. The fact that the reduced potentials of Sr 2, Ca2, and Hg 2fall almost on the same line is less familiar. Apparently this is due to the fact that they all have a closedouter electronic s 2shell, even though the well depths of Sr2/H208491082 cm−1/H20850and Ca 2/H208491102 cm−1/H20850are about three times as large as that of Hg 2/H20849380 cm−1/H20850. Thus, the present results provide further evidence to the conjecture that the ground state van der Waals dimer potentials of elements in groupIIA, with the possible exception of Be, and in group IIB havethe same shape, which is different from that of the rare-gasdimers. IV. THE GROUND STATE POTENTIAL OF STRONTIUM RARE-GAS COMPLEXES The fact that the interatomic potentials of both strontium and rare-gas dimers are expressible in terms of the TT po-tential model makes it likely that the interaction potentialbetween a strontium atom and a rare-gas atom can also bedescribed by the TT potential. We will first separately discussthe combining rules for short range Born–Mayer parametersand the long range dispersion coefficients. Then we will usethe results to construct the interaction potentials for stron-tium rare-gas complexes. A. Combining rules for Aandb In the TT model, the short range repulsive potential is expressed in terms of the Born–Mayer form Aexp /H20849−bR/H20850. There are several combining rules for Aand b.46–49 Böhm–Ahlrichs49tested four different sets of combining rules for Aandb,/H20851Eqs. /H2084930/H20850–/H2084933/H20850of Ref. 49/H20852. They found that all of them are useful but the results are not identical.Since these rules are based on heuristic arguments, their va-lidity can only be judged by the results they predict.As we have shown previously, 2the following set of com - bining rules /H20851Eq. /H2084931/H20850of Ref. 49/H20852are most appropriate for calcium rare-gas interactions: Aij=/H20851AiAj/H208521/2,bij=2bibj bi+bj, /H2084938/H20850 where a single index indicates the potential parameter for the like system. We assume that this set of rules is generallyapplicable to all alkaline earth rare-gas systems. With theBorn–Mayer parameters Aandbof the strontium dimer cal- culated in Eq. /H2084929/H20850and those of the rare-gas dimers given in Ref. 44, the corresponding Aandbfor the interaction of a strontium atom and a rare-gas atom calculated from Eq. /H2084938/H20850 are listed in Table II. B. Combining rules for C6,C8, and C10 For quite some time there exist a set of reliable combin- ing rules for dispersion coefficients which are valid for allsystems. The combining rule for C 6is extensively docu- mented and shown to be more accurate better than 1%. Thecombining rules for C 8andC10are not so widely known. Therefore we will present a parallel development for C6,C8, andC10to show that they have the same theoretical founda- tion and should be equally accurate. The combining rules for C6,C8, and C10follow from the Casimir–Polder theory of dispersion interaction.50,51In this theory, the individual dispersion coefficients are made up of different terms arising in the multipole expansion of the per-turbation operator, C 6=Cij/H208491,1 /H20850, /H2084939/H20850 C8=Cij/H208491,2 /H20850+Cij/H208492,1 /H20850, /H2084940/H20850 C10=Cij/H208491,3 /H20850+Cij/H208492,2 /H20850+Cij/H208493,1 /H20850, /H2084941/H20850 where Cij/H208491,1/H20850is the dipole-dipole, Cij/H208491,2/H20850is the dipole- quadrupole, Cij/H208492,2/H20850is the quadrupole-quadrupole, and Cij /H208491,3/H20850is the dipole-octupole interaction. They are given by the exact formula50,52TABLE I. Potential parameters for the dimers of some elements in groups IIA, IIB, and VIII, all in a.u. System Reference Ab C6 C8 C10 Sr2 Present 44.73 0.9699 31.03 /H1100310237.92/H1100310442.15/H11003106 Ca2 2 28.23 0.9987 21.21 /H1100310222.30/H1100310421.32/H11003106 Hg2 43 19.35 1.2727 392.0 12.92 /H1100310353.70/H11003104 Ar2 44 748.3 2.031 64.30 16.23 /H1100310249.06/H11003103 Kr2 44 832.4 1.865 129.6 41.87 /H1100310215.55/H11003104 TABLE II. Potential parameters for the strontium rare-gas systems, all in a.u. System Ab C6 C8 C10 Sr–He 43.32 1.4012 38.64 2670.0 198 500.0 Sr–Ne 94.47 1.3909 75.58 5413.0 417 100.0Sr–Ar 183.0 1.3129 297.2 22 650.0 1 804 000.0Sr–Kr 193.0 1.2761 442.6 35 100.0 2 886 000.0Sr–Xe 206.3 1.2301 697.6 59 350.0 5 172 000.0074303-6 Yin, Li, and T ang J. Chem. Phys. 132 , 074303 /H208492010 /H20850 Downloaded 15 Apr 2013 to 171.67.34.69. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissionsCij/H20849l1,l2/H20850=/H208492l1+2l2/H20850! 4/H208492l1/H20850!/H208492l2/H20850!/H208732 /H9266/H20874/H20885 0/H11009 /H9251l1i/H20849i/H9275/H20850/H9251l2j/H20849i/H9275/H20850d/H9275,/H2084942/H20850 where /H9251l1i/H20849i/H9275/H20850and/H9251l2j/H20849i/H9275/H20850are respective dynamic multipole polarizabilities at the frequency /H9275of atoms iandj, respec- tively. Note that Eq. /H2084942/H20850is an integral over imaginary fre- quency of the product of the dynamic polarizabilities of theinteracting atoms. This formulation reduces the original twocentered problem to a one centered problem of evaluating thefrequency dependent polarizabilities. To approximate the dy-namic dipole polarizability /H9251l/H20849i/H9275/H20850, a one term approximant /H20851/H9251l/H20849i/H9275/H20850/H20852is introduced,53 /H20851/H9251l/H20849i/H9275/H20850/H20852=/H9251l 1+ /H20849/H9275//H9024l/H208502,l= 1,2,3, /H2084943/H20850 where /H9251lis the static polarizability /H9251l/H208490/H20850and/H9024lis an effec- tive energy. With this approximation and the well known mathematical identity,54 2 /H9266/H20885 0/H11009ab /H20849a2+/H92752/H20850/H20849b2+/H92752/H20850d/H9275=1 a+b, one can easily show Cij/H208491,1 /H20850=3 /H9266/H20885 0/H11009 /H20851/H92511i/H20849i/H9275/H20850/H20852/H20851/H92511j/H20849i/H9275/H20850/H20852d/H9275=3 2/H20875/H92511i/H92511j/H90241i/H90241j /H90241i+/H90241j/H20876. /H2084944/H20850 Similarly, Cij/H208491,2 /H20850=15 4/H20875/H92511i/H92512j/H90241i/H90242j /H90241i+/H90242j/H20876, /H2084945/H20850 Cij/H208491,3 /H20850=7/H20875/H92511i/H92513j/H90241i/H90243j /H90241i+/H90243j/H20876, /H2084946/H20850 Cij/H208492,2 /H20850=35 2/H20875/H92512i/H92512j/H90242i/H90242j /H90242i+/H90242j/H20876. /H2084947/H20850 For homonuclear dipole-dipole interaction, Eq. /H2084944/H20850reduces to C6i=Cii/H208491,1 /H20850=3 4/H20849/H92511i/H208502/H90241i, /H2084948/H20850 where a single index indicates that the quantity is for the homonuclear system. Clearly, in this approximation, /H90241i=4 3C6i /H20849/H92511i/H208502. /H2084949/H20850 According to Tang’s theorem,53the approximant /H20851/H92511i/H20849i/H9275/H20850/H20852 with this /H90241imust intersect /H92511i/H20849i/H9275/H20850once and only once beside /H9275=0. Furthermore, in the integral for C6, the amount that /H92511i/H20849i/H9275/H20850is overestimated by /H20851/H92511i/H20849i/H9275/H20850/H20852before the point of in- tersection is completely compensated by the amount that /H92511i/H20849i/H9275/H20850is underestimated by /H20851/H92511i/H20849i/H9275/H20850/H20852after the point of inter- section. Therefore the combining rule obtained by substitut- ing Eq. /H2084949/H20850into Eq. /H2084944/H20850,C6ij=Cij/H208491,1 /H20850=2/H92511i/H92511jC6iC6j /H20849/H92511i/H208502C6j+/H20849/H92511j/H208502C6i, /H2084950/H20850 must be very accurate. A few representative interactions were used by Tang to show that is indeed the case.53This was confirmed with further testing by Kramer and Herschbach,55 by Zeiss and Meath,56and by Kutzelnigg and Maeder.57 Thakkar58carried out the most comprehensive testing with 210 interactions and found a rms error of only 0.52%. Based on the same principle, similarly accurate combin- ing rules for C8andC10can be derived.39For homonuclear interactions, Ci/H20849l1,l2/H20850=Ci/H20849l2,l1/H20850. With C8i=2Cii/H208491,2 /H20850, the ef- fective energy /H90242ican be solved from Eq. /H2084945/H20850, /H90242i=2C8i/H90241i 15/H92511i/H92512i/H90241i−2C8i. /H2084951/H20850 Solving for /H90243ifrom Eq. /H2084946/H20850, we have /H90243i=Ci/H208491,3 /H20850/H90241i 7/H92511i/H92513i/H90241i−Ci/H208491,3 /H20850, /H2084952/H20850 where Ci/H208491,3 /H20850can be expressed as Ci/H208491,3 /H20850=1 2/H20851C10i−Ci/H208492,2 /H20850/H20852=1 2/H20851C10i−35 4/H20849/H92512i/H208502/H90242i/H20852. /H2084953/H20850 Therefore with the polarizabilities /H92511i,/H92512i,/H92513iand the homonuclear dispersion coefficients C6i,C8i,C10i, the three ef- fective energies /H90241i,/H90242i,/H90243ican be determined. The hetero- nuclear coefficients C6ij,C8ij, and C10ijcan then be calculated from Eqs. /H2084939/H20850–/H2084941/H20850. In this paper, these expressions are used to calculate the dispersion coefficients of the unlike systems. For the homo-geneous rare-gas dimers, C 6i,C8i,C10i, and/H92511iare taken from Ref. 44and/H92512i,/H92513iare taken from Ref. 59. For strontium dimer, all polarizabilities and dispersion coefficients aretaken directly from Ref. 30. The dispersion coefficients of the interactions between a strontium atom and a rare-gasatom calculated from these combining rules are also shownin Table II. C. Results and comparison with previous determinations The van der Waals potentials of strontium rare-gas sys- tems are calculated from the TT model of Eq. /H208492/H20850with the parameters listed in Table II. Since the Born–Mayer param- eters and the dispersion coefficients are separately deter-mined, the potentials will depend on the number of termsincluded in the dispersion series. Therefore we carried outtwo sets of calculations: one with n max=5 and the other with nmax=8. The differences between these two sets provide an indication to the extent of the uncertainty of the present ap-proach. The full potential curves for all these five systems ob- tained by summing the dispersion series with n max=5 are shown in Fig. 3/H20849a/H20850. It is interesting to compare this figure with Fig. 3 of Ref. 2where the calcium rare-gas potentials are shown. These two sets of curves are very similar. Theyfollow the same trend in that the depth of the potential wellD eincreases regularly with the size of the rare-gas atoms going from He to Xe. The corresponding equilibrium posi-074303-7 Strontium dimer and strontium rare gas J. Chem. Phys. 132 , 074303 /H208492010 /H20850 Downloaded 15 Apr 2013 to 171.67.34.69. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissionstionReincreases very little from one system to another. The corresponding curves calculated with nmax=8 are very simi- lar with only a deeper potential well /H20849see Table III/H20850. It is interesting to note that the reduced potentials of these five systems have the same shape. In Fig. 3/H20849b/H20850, the present results U/H20849x/H20850are plotted against x. The five curves are essentially identical within a few percent. With the scale of the figure, the small differences are not noticeable. This in-variance with respect to the shape of the potential is not to beconstrued as the result of using the same TT potential func-tion for all cases. It is seen in Fig. 2/H20849b/H20850that the shape of the potential for the strontium dimer is different from that of therare-gas dimers, even though the same TT model is used todescribe all these potentials. The potential bowl of the stron- tium rare-gas complexes is wider than that of the rare-gasdimers but narrower than that of the strontium dimer. The predicted well depth D e, the equilibrium distance Re, the vibrational frequency /H9275e, and the anharmonicity /H9273e/H9275e of Sr-RG dimers with nmax=5 are listed in Table III. The numbers in the parentheses are obtained by summing thedispersion series with n max=8. There are only a limited num- ber of previous determinations of these potentials. The pa-rameters of all available Sr-RG potentials are listed in thetable for comparison. For SrHe, Lovallo and Klobukowski 23carried out CCSD /H20849T/H20850calculations for the ground state potential. They reported both the nonrelativistic and scalar relativistic poten-tials. In Table III, the relativistic corrected D eand Reare listed in the parentheses. It is seen that their well depths aremuch smaller than the present predictions. On the otherhand, the present results fall in between the MRCI results ofStienkemeier et al. 21and the surface integral results of Kleinekathöfer22and are in much better agreement with them. In addition to Deand Re, Lovallo and Klobukowski23 also reported the values of the dispersion coefficients C6 /H2084945.6, 45.0 a.u. /H20850andC8/H208494050, 3700 a.u. /H20850. The second num- ber in the parentheses is the relativistic corrected value. Asseen in Table II, they are quite different from our predictions C 6/H2084938.64 a.u. /H20850andC8/H208492670 a.u. /H20850. For these coefficients, it is almost certain that the present results are more accurate,since they are calculated from the well established combin-ing rules. FIG. 3. /H20849a/H20850The ground state van der Waals potentials of Sr-RG /H20849RG =He,Ne,Ar,Kr,Xe /H20850. These potentials are generated by the TT potential model with A,b,C6,C8,C10calculated from the corresponding parameters of the homonuclear strontium and rare-gas dimers with combining rules. /H20849b/H20850 The reduced potentials of SrHe, SrNe, SrAr, SrKr, and SrXe. They arealmost identical to each other. This reduced potential has a potential bowlwider than that of the rare-gas dimers but narrower than that of the strontiumdimer. TABLE III. Comparison of the well depth, the equilibrium distance, the vibrational frequency, and the anhar- monicity of the van der Waals ground state potentials of the strontium rare-gas systems. System ReferenceDe /H20849cm−1/H20850Re /H20849Å/H20850/H9275e /H20849cm−1/H20850/H9273e/H9275e /H20849cm−1/H20850 Sr–He Present 9.77 /H2084911.45 /H20850 5.19 /H208495.12 /H20850 16.4 /H2084917.9 /H20850 7.66 /H208497.61 /H20850 Surface integral, Ref. 22 7.11 5.54 MRCI, Ref. 21 12.71 5.71 CCSD /H20849T/H20850, Ref. 23 2.85 /H208492.96 /H20850 6.42 /H208496.34 /H20850 Sr–Ne Present 16.14 /H2084919.24 /H20850 5.35 /H208495.26 /H20850 10.0 /H2084911.0 /H20850 1.75 /H208491.73 /H20850 Sr–Ar Present 59.67 /H2084970.0 /H20850 5.38 /H208495.28 /H20850 14.3 /H2084915.3 /H20850 0.95 /H208490.93 /H20850 Expt., Ref. 24 68/H1100615 12.0 0.6 MCSCF, Ref. 25 63.45 6.21 8.7 Sr–Kr Present 88.12 /H20849102.5 /H20850 5.38 /H208495.27 /H20850 13.6 /H2084914.7 /H20850 0.59 /H208490.56 /H20850 Sr–Xe Present 133.8 /H20849154.9 /H20850 5.40 /H208495.30 /H20850 14.8 /H2084915.8 /H20850 0.45 /H208490.43 /H20850074303-8 Yin, Li, and T ang J. Chem. Phys. 132 , 074303 /H208492010 /H20850 Downloaded 15 Apr 2013 to 171.67.34.69. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissionsFor SrAr, there is one set of data from experiment. The spectroscopic constants were determined by Kowalski et al.24with excitation spectra in a supersonic jet. The experi - mental values of /H9275e,/H9275e/H9273e, and Deare listed in Table III. Only Dewas reported with an error bar. Theoretically, there is the MCSCF calculations of Zhu et al.25Their Deis in very good agreement with experiment, but their /H9275eis smaller than the experimental value. As seen in Table III, the present Dealso falls in the experimental error limits. The present /H9275eis larger than the experimental value by about the same amount. ForSrNe, SrKr, and SrXe, there is no previous potential that canbe directly compared with the present results. In the present Sr-RG potentials, the long range part should be very reliable because the dispersion coefficientsare calculated with the well established combining rules.However, the combining rules for the Born–Mayer param-eters are not nearly as precise. Therefore the accuracy of thepredicted well parameters D e,Reis not expected to be very high. Nevertheless, the predicted potentials may still be use-ful as the starting point for the interplay between theory andexperiment. In many collision experiments of ultracold at-oms, only the long range part of the potential is of criticalimportance. Even if higher accuracy is required near the bot-tom of the potential, the well parameters D e,Rein the TT potential function can be easily modified in such a way thatthe long range potential is maintained. V. DISCUSSION Interatomic potentials obtained either from spectroscopic data or from quantum chemical calculations are given at aseries of discrete interatomic separations R. Functions with 20 to 30 parameters have been proposed to fit these point-wise potential energies. Since these data points are usuallycentered on the potential minimum, they need to be joined tothe long range dispersion series at large Rin a physically appropriate manner. This is particularly important in coldatom collisions, since they are sensitive to this part of thepotential. Often the long range series is attached to the po-tential of the well region in an ad hoc manner at an arbitrary point near the outer end of the region of the data points. In contrast, the TT potential function used in this work is specified by only five parameters. Three dispersion coeffi-cients have well defined physical meaning and can be sepa-rately determined. The other two parameters can be deter-mined by the energy at the potential minimum. In this paper,we have shown that the entire Sr 2potential energy curve, from the long range to the well region and up into the repul-sive wall, can be accurately described by the TT potentialfunction without any fitting. The predicted spectroscopic pa-rameters /H9275e,/H9275e/H9273e, and/H9251efrom the TT potential, defined by C6,C8,C10of Porsev and Derevianko30andDe,Reof Stein el al,7are in excellent agreement with experiment. Further - more, we have shown that the three different types of van derWaals dimers of groups IIA, IIB, and VIII elements can allbe described by the same TT potential function. Thisstrongly suggests that the essential physics of van der Waalsinteractions is contained in the potential model. The TT po-tential of Eq. /H208492/H20850is not only a single continuous function,mathematically it is an entire function. Its derivatives exist everywhere and to all orders. 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1.2834399.pdf

Spin current studies in by ferromagnetic resonance and time- resolved magneto-optics B. Kardasz , O. Mosendz , and B. Heinrich Z. Liu and M. Freeman Citation: J. Appl. Phys. 103, 07C509 (2008); doi: 10.1063/1.2834399 View online: http://dx.doi.org/10.1063/1.2834399 View Table of Contents: http://aip.scitation.org/toc/jap/103/7 Published by the American Institute of Physics Spin current studies in Fe/Ag,Au/Fe by ferromagnetic resonance and time-resolved magneto-optics B. Kardasz,a/H20850O. Mosendz, and B. Heinrich Department of Physics, Simon Fraser University, 8888 University Dr., Burnaby, British Columbia V5A 1S6, Canada Z. Liu and M. Freeman Department of Physics, University of Alberta, Edmonton, Alberta T6G 2E1, Canadaand National Institute for Nanotechnology, Edmonton, Alberta T6G 2M9, Canada /H20849Presented on 9 November 2007; received 11 September 2007; accepted 30 October 2007; published online 21 February 2008 /H20850 A precessing magnetization within a magnetic double layer acts as a peristaltic spin pump which transports spin momentum but no net electric charge. Crystalline Fe single layersAu /12Fe /GaAs /H20849001 /H20850and double layers Fe12 //H20849Au,Ag /H20850/Fe16 /GaAs /H20849001 /H20850were prepared by molecular beam epitaxy, where the integers represent the number of Fe atomic layers, and /H20849Au,Ag /H20850 represents a set of gold and silver layers of different thicknesses. Ferromagnetic resonance /H20849FMR /H20850 was used to investigate spin diffusion in thick Au layers in Au /12Fe /GaAs /H20849001 /H20850samples. Time-resolved magneto-optical Kerr effect /H20849TRMOKE /H20850measurements are an ideal tool for investigating the propagation of spin currents in these structures. Spin currents generated by thebottom 16Fe layer propagated across the normal metal spacer and resulted in rf excitations in the top12Fe film. Landau-Lifshitz-Gilbert equations of motion modified by spin pump and spin sink effectswere used to interpret the FMR and TRMOKE measurements. The spin diffusion lengths in Au wereAg were found 34 and 170 nm, respectively. © 2008 American Institute of Physics . /H20851DOI: 10.1063/1.2834399 /H20852 The spin dynamics in the classical limit in a single ul- trathin ferromagnetic film can be described by the Landau-Lifshitz-Gilbert /H20849LLG /H20850equation of motion 1 /H9253/H11509M /H11509t=− /H20849M/H11003Heff/H20850+/H9251 /H9253/H20873M/H11003/H11509n /H11509t/H20874, /H208491/H20850 where nis the unit vector in the direction of the magnetiza- tionM,/H9253=g/H20841e/H20841/2mcis the absolute value of the gyromag- netic ratio, Heffis the effective field, and /H9251is the Gilbert damping parameter.1,2 In magnetic multilayers which include ferromagnetic /H20849F/H20850 and normal metal /H20849NM /H20850layers the magnetic dynamics gen- erates spin currents which propagate across the NM layersresulting in nonlocal dynamic coupling. Tserkovnyak et al. 3 and Heinrich et al.4have shown that the precessing magne- tization injects spin /H20849magnetic /H20850momentum into the NM layer. The resulting magnetic moment current per unit area isgiven by j m=−g/H9262B 8/H9266g˜↑↓n/H11003/H11509n /H11509t, /H208492/H20850 where g˜↑↓is proportional to the interface spin mixing con- ductance given to a very good approximation by double ofthe number of transverse channels in the NM. g ˜↑↓=1.7 n2/3, where nis the density of electrons per spin.2,5gis the spec- troscopic g-factor coefficient. The spin current creates an ac- cumulated spin /H20849magnetic /H20850momentum density in the adja- cent NM layer. In ballistic transport an accumulatedmagnetic momentum density propagates equally by the Fermi velocity vFin forward and backward directions per- pendicular to the F/NM interface. For small precessional mo-tion the rf component of nis nearly transversal to the static magnetization vector. In a magnetic double layer F1/NM/F2structure both spin currents in NM are fully absorbed in theFe1/NM and NM/F2 interfaces. When the resonance fre-quencies /H20849fields /H20850of the layers F1 and F2 are different, then spin pumping is mostly generated by the layer undergoingferromagnetic resonance /H20849FMR /H20850, the other layer is a passive spectator acting as a spin sink. The conservation of spin mo-mentum requires that the loss of spin momentum at theF1/NM interface is compensated by interface damping in F1satisfying the Gilbert damping phenomenology 2 /H9251=g/H9262Bg˜↑↓ 8/H9266Ms1 dF1, /H208493/H20850 where dF1is the thickness of the layer F1. For thick NM layers one has to include relaxation of the accumulated magnetic momentum density. The accumulatedmagnetic momentum density m NMin NM is described by the following diffusion equation: i/H9275mNM=D/H115092mNM /H11509x2−1 /H9270sfmNM, /H208494/H20850 where /H9275is the angular frequency, /H9270sfis the spin flip relax- ation time, xis the coordinate in the perpendicular direction to the interface, D=vF2/H9270el/3 is the diffusion coefficient, and /H9270elis electron momentum relaxation time constant.5The so- lutions of Eqs. /H208491/H20850and /H208494/H20850require boundary conditions. For aa/H20850Electronic mail: bkardasz@sfu.ca.JOURNAL OF APPLIED PHYSICS 103, 07C509 /H208492008 /H20850 0021-8979/2008/103 /H208497/H20850/07C509/3/$23.00 © 2008 American Institute of Physics 103 , 07C509-1 single magnetic layer structure F/NM the boundary condi- tions at the F/NM interface are5 jm− 0.5 vFmNM=−D/H11509mNM /H11509x. /H208495/H20850 For the outer interface we used a free magnetic moment con- dition /H11509mNM /H11509x=0 . /H208496/H20850 For a magnetic double layer structure F1/NM/F2 the bound- ary conditions at the F1/NM interface are equivalent to Eq./H208495/H20850. The boundary conditions at the NM/F2 interface are 5 −/H11509mNM /H11509x= 0.5 vFmNM. /H208497/H20850 The boundary conditions in Eq. /H208497/H20850are valid for the case when the layer F2 is off resonance and therefore contributesnegligibly to spin pumping. The coefficient 0.5 correspondsto the effective transmission coefficient from the NM to Flayers and is given by Eq. /H2084913/H20850in Ref. 5. The right hand side of Eq. /H208497/H20850represents the magnetic current from NM into F2 and acts as a driving field for the magnetic moment in F2. The purpose of the studies presented in this paper was to identify the spin diffusion coefficient and spin flip relaxationtime in Au. We carried out two experiments. Dand /H9270sfwere determined by FMR employing a single magnetic structureF/NM. Similar experiments were done by Mizukami et al. 6 on the Cu /permalloy /Cu /Pt films. In addition the propaga- tion of spin current in NM was investigated by time andspatial resolved Kerr effect technique. For this case we useda double magnetic layer F1/NM/F2 where F1 was used forspin pumping and the layer F2 was used as a detector of thespin current. The Fe films were deposited at room temperature on a commonly used 4 /H110036-GaAs /H20849001 /H20850reconstructed template. The 4 /H110036 surface reconstruction was obtained by annealing the GaAs wafer at /H11229600 °C following hydrogen cleaning and Ar +sputtering at 650 eV. The following structures were grown: /H20849a/H20850nAu /16Fe /GaAs /H20849001 /H20850, where n=20, 80, 150, 200, 250, 300 and the integers represent the number of atomic layers; and /H20849b/H2085020Au /12Fe /300Au /16Fe /GaAs /H20849001 /H20850 and 20Au /12Fe /300Ag /16Fe /GaAs /H20849001 /H20850. The FMR studies were carried out using standard micro- wave spectrometers using 10, 24, 36, and 73 GHz, see de-tails in Ref. 7. For both bulk and interface Gilbert damping /H9004His strictly linearly dependent on the microwave angular frequency /H9275,/H9004H=/H9251/H20849/H9275//H9253/H20850. The NM layer increases magnetic damping when its thickness becomes comparable to the spin diffusion length /H9254sd=vF/H20849/H9270sf/H9270el/3/H208500.5. For dNM/H11270/H9254sd/H9004His given only by the intrinsic Gilbert damping of the Fe layer. For dNM/H11271/H9254sdthe /H9004Hincreases by the loss of spin momentum in NM. The equations of motion /H208491/H20850and /H208494/H20850with the boundary conditions /H208495/H20850and /H208496/H20850were solved self-consistently, and were employed for fitting the measured spin pumping coefficient /H9251, see Fig. 1. The spin pumping Gilbert damping parameter, /H9251sp,a sa function of the Au thickness was fitted with the followingparameters: /H9251intr=3.5/H1100310−3,g˜↑↓=2.4/H110031015cm−2,/H9270el=1.2 /H1100310−14s,/H9270sf=15/H1100310−14s, and the Fermi velocity was as- sumed to be /H9271F=1.4/H11003108cm /s. The fitted parameters result in the spin diffusion length /H9254sdof 34 nm. It is interesting to note that Kurt et al.8studied the spin diffusion length by using current–perpendicular-to-plane /H20849CPP /H20850giant magnetore- sistance /H20849GMR /H20850measurements using polycrystalline Au /Cu spacers. They obtained /H9254sd=35 nm, which is very close to our result. The ratio r=12.5 indicates that in our samples /H9270sf is one order of magnitude larger than /H9270el. 20Au /12Fe /nAu /16Fe /GaAs /H20849001 /H20850structures were em- ployed in the study of propagation of spin currents across the NM film. Time-resolved magneto-optical Kerr effect/H20849TRMOKE /H20850measurements are an ideal tool for investigating the propagation of spin currents in these structures. Strobo-scopic measurements of magnetization precession in the10 GHz frequency range were carried out with picosecondtime resolution and submicrometer spatial resolution, using acoplanar transmission line carrying repetitive picosecondmagnetic excitation pulses. After excitation the 100 fs dura-tion laser pulses probed the top 12Fe layer via the perpen-dicular component of precessing magnetization /H20849polar MOKE /H20850at the delay time t D, see detailed description in Ref. 9. Spin currents generated by the bottom 16Fe layer propa- gated across the normal metal spacer and resulted in rf exci-tations of the top 12Fe film. The resonant frequencies of theFe layers are strongly affected by the interface anisotropies,see Ref. 10. Therefore the 12Fe and 16Fe films have their resonant frequencies 4.5 GHz apart and therefore the spincurrent induced magnetization precession in the 12Fe filmcan be in principle easily distinguished. However, the iden-tification of absorbed spin current is complicated by the pres-ence of a direct TRMOKE signal from the bottom 16Fe layerwhich becomes observable when the spacer thickness is lessthan 250 atomic layers. A Au spacer with the thickness of300 atomic layers was sufficient to suppress the signal fromthe bottom 16Fe film. No measurable MOKE signal was ob-served on the 300Au /16Fe /GaAs /H20849001 /H20850sample. Therefore, further studies with the Au spacer were carried out using the 20Au /12Fe /300Au /16Fe /GaAs /H20849001 /H20850structure. The time dependence of the picosecond resolved Kerr signal and its fast Fourier transform /H20849FFT /H20850are shown in Figs. 2/H20849a/H20850and FIG. 1. Dependence of the additional damping by spin pumping, /H9251sp, on the Au cap layer thickness dAuin the Au /16Fe /GaAs /H20849001 /H20850samples. The /H20849•/H20850 symbols represent the measured data from the /H9004Hdependence on micro- wave frequency f,/H9004H/H20849f/H20850./H9004H/H20849f/H20850followed well a linear dependence on f. The error bars were determined from small slope variations in the /H9004H/H20849f/H20850 measurements. The solid line shows fitting using the spin pumping theory with the following parameters.: g˜↑↓=2.4/H110031015cm−2,/H9270el=1.2/H1100310−14s, and /H9270sf=15/H1100310−14s.07C509-2 Kardasz et al. J. Appl. Phys. 103 , 07C509 /H208492008 /H20850 2/H20849b/H20850, respectively. In order to suppress low frequency noise in FFT spectra one had to subtract the background in thetime domain spectrum using a low-pass filtering method. Forthe Au spacer of thickness /H2084960 nm /H20850the spin current was sig- nificantly decreased by the loss of spin momentum in the Au spacer. FFT of the measured TRMOKE data showed mostlythe main resonance frequency corresponding to the 12Felayer. The spin current induced signal which occurred at f =10.6 GHz was buried in noise, see Fig. 2/H20849b/H20850and the inset. Computer simulations of the picosecond resolved transientmagnetization oscillations were carried out using Eqs. /H208491/H20850 and /H208494/H20850including the realistic shape of the magnetic pulse, and using the magnetic and spin diffusion parameters ob-tained from the FMR measurements, see above. The simula-tion resulted in a small spin current contribution, see Fig.2/H20849b/H20850. A detailed inspection of the measured and simulated FFT spectra suggests that the spin current contribution mightbe present in the measured data, see the inset to Fig. 2. How- ever, this signal is comparable to noise and can be consideredonly as a marginal evidence for spin current excitations. Forthis reason we decided to use a Ag spacer. Ag has a signifi-cantly smaller spin orbit interaction than Au and thereforeone expects a larger spin diffusion length, and with that, anoticeably increased signal. In addition, the optical proper-ties of Ag in the near infrared enhance the attenuation of thedirect signal from the bottom 16Fe layer compared to that ofthe Au spacer of equivalent thickness. Indeed Fig. 3/H20849b/H20850 clearly shows the spin pump signal for the 300Ag spacer. Forthe measured data the ratio rof the FFT areas corresponding to the spin pump and the main frequency components wasfound to be r=1.8/H110060.5%. In order to minimize the error weemployed well known Savitzky–Golay filtering method. The main advantage of this approach is that it tends to preservefeatures of the distribution such as relative maxima, minima,and width. The maximum spin current contribution /H20849negli- gible Ag spacer thickness /H20850results in r=3%. This clearly in- dicates that the 300Ag spacer already decreased the spin cur-rent at the Ag /12Fe interface. In order to obtain a comparable decrease in computer simulations to the mea-surements /H20849see inset of Fig. 3/H20850, one has to use the following parameters: /H9270el=5/H1100310−14s and/H9270sf=70/H1100310−14s resulting in r=1.6%. An estimated value of the spin diffusion length in Ag was found to be of 170 /H1100620 nm. A number of experiments using various techniques were reported in the literature on spin relaxation in metals. How-ever, we note that there is a lack of consensus as to the exactvalue of spin diffusion lengths in Ag and Au metals. 11 1B. Heinrich and J. F. Cochran, Adv. Phys. 42,5 2 3 /H208491993 /H20850. 2B. Heinrich, in Ultrathin Magnetic Structures III , edited by J. A. C. Bland and B. Heinrich /H20849Springer, Berlin, 2004 /H20850, Chap. 5, p. 143. 3Y. Tserkovnyak, A. Brataas, and G. Bauer, Phys. Rev. Lett. 88, 117601 /H208492002 /H20850. 4B. Heinrich, Y. Tserkovnyak, G. Woltersdorf, A. Brataas, R. Urban, and G. Bauer, Phys. Rev. Lett. 90, 187601 /H208492003 /H20850. 5Y. Tsekovnyak, A. Brrataas, and G. E. Bauer, Phys. Rev. B 66, 224403 /H208492002 /H20850. 6S. Mizukami, Y. Ando, and T. Miyazaki, Phys. Rev. B 66, 104413 /H208492002 /H20850. 7B. Heinrich, in Ultrathin Magnetic Structures II , edited by B. Heinrich and J. A. C. Bland /H20849Springer, Berlin, 1994 /H20850, Chap. 3.1, pp. 195–222. 8H. Kurt, W.-C. Chiang, C. Ritz, K. Eid, W. P. Pratt, Jr., and J. Bass, J. Appl. Phys. 93, 7918 /H208492003 /H20850. 9B. Choi and M. Freeman, in Ultrathin Magnetic Structures III , edited by J. A. C. Bland and B. Heinrich /H20849Springer, Berlin, 2004 /H20850, Chap. 6, p. 211. 10R. Urban, G. Woltersdorf, and B. Heinrich, Phys. Rev. Lett. 87, 217204 /H208492001 /H20850. 11J. Bass and J. W. P. Pratt, J. Phys.: Condens. Matter 19, 183201 /H208492007 /H20850. FIG. 2. /H20849a/H20850Time-resolved magnetization oscillations for the 20Au /12Fe / 300Au /16Fe /GaAs /H20849001 /H20850sample. The applied field H=500 Oe. /H20849b/H20850Fast Fourier transform /H20849FFT /H20850. The thick solid line corresponds to FFT of the data in/H20849a/H20850. The thin solid and dotted FFT lines show the simulations with a maximum spin current contribution /H20849infinite /H9254sd/H20850and that corresponding to /H9254sd=34 nm, respectively. The inset in /H20849b/H20850shows magnified FFT spectra al- lowing one to easily view the spin current contributions. /H20849c/H20850The phase of FFT obtained for the measured data. FIG. 3. /H20849a/H20850Time-resolved data for the 20Au /12Fe /300Ag /16Fe / GaAs /H20849001 /H20850sample. The applied field H=500 Oe. /H20849b/H20850FFT spectrum. The thick solid line represents FFT of the measured data. The thin solid linerepresents FFT of theoretical simulations using /H9254sd=151 nm. The inset pro- vides a magnified part of the FFT spectra for easy viewing. /H20849c/H20850The phase of FFT obtained for the measured data.07C509-3 Kardasz et al. J. Appl. Phys. 103 , 07C509 /H208492008 /H20850

1.5077025.pdf

Appl. Phys. Lett. 114, 112401 (2019); https://doi.org/10.1063/1.5077025 114, 112401 © 2019 Author(s).Thermally activated magnetization back- hopping based true random number generator in nano-ring magnetic tunnel junctions Cite as: Appl. Phys. Lett. 114, 112401 (2019); https://doi.org/10.1063/1.5077025 Submitted: 24 October 2018 . Accepted: 02 March 2019 . Published Online: 18 March 2019 Jianying Qin , Xiao Wang , Tao Qu , Caihua Wan , Li Huang , Chenyang Guo , Tian Yu , Hongxiang Wei , and Xiufeng Han COLLECTIONS This paper was selected as an Editor’s Pick Thermally activated magnetization back-hopping based true random number generator in nano-ring magnetic tunnel junctions Cite as: Appl. Phys. Lett. 114, 112401 (2019); doi: 10.1063/1.5077025 Submitted: 24 October 2018 .Accepted: 2 March 2019 . Published Online: 18 March 2019 Jianying Qin,1Xiao Wang,1TaoQu,2 Caihua Wan,1LiHuang,1Chenyang Guo,1Tian Yu,3Hongxiang Wei,1 and Xiufeng Han1,a) AFFILIATIONS 1Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, University of Chinese Academy of Sciences, Beijing 100190, China 2School of Physics and Astronomy, University of Minnesota-Twin Cities, Minneapolis, Minnesota 55455, USA 3College of Physical Science and Technology, Sichuan University, Chengdu 610065, China a)Email: xfhan@iphy.ac.cn ABSTRACT A true random number generator based on the magnetization backhopping process in nano-ring magnetic tunnel junctions is demonstrated in this work. The impact of environmental temperature ( T) and current pulse width ( s) on backhopping is investigated statistically by experi- ments, micromagnetic simulations, and theoretical analysis. The backhopping probability increases at high Tand wide s, as explained by the combined effect of thermal fluctuation and spin-transfer-torque noise. The magnetoresistance at backhopping is randomly distributed over alarge operational current range. This manifestation of backhopping in magnetic tunnel junctions can be used as the basic unit of a true ran- dom number generator. Published under license by AIP Publishing. https://doi.org/10.1063/1.5077025 Random numbers have been indispensably used in a wide variety of applications ranging from cryptography to statistics. Pseudo- random number generators (PRNGs)—which generate a sequence ofnumbers from a seed using a computer program—are often used because of their high operating rate of about 100 Gbps. 1However, there exist severe security limitations for the PRNGs because the gen-erated sequences can be decoded mathematically. Thus, true random number generators (TRNGs) based on physical processes with com- plete indeterminacy are desirable. Until now, several approaches, suchas quantum optics, jitter oscillators, and physical noise source amplifi- cation, 2–6have been developed to generate true random numbers (RNs). In general, enhanced technological applicability would greatlybenefit from CMOS compatibility, low power dissipation, and com- pactness. TRNGs based on the magnetization backhopping process in nano-ring shaped magnetic tunneling junctions (NR-MTJs) are prom-ising candidates with several advantages. First, according to previous reports, 7–13NR-MTJs can eliminate stray fields between each other, which will lead to high areal density, further resulting in a high RNgeneration rate. Second, the formation of the “Onion” state for the free layer magnetization in the NR-MTJs can reduce the energy barrier ofthe cell 14and eliminate the thermal turbulence on the sub-50 nm in- plane MTJs,15both of which favoring the functioning of the TRNGs as discussed below. Besides, backhopping occurs within nanoseconds,which allows the TRNG to work near GHz frequencies. Also, no ini- tializing operations are required before the generation of RNs. Finally, the simpler two-terminal device, high throughput, 5,16,17low power c o n s u m p t i o no fl e s st h a n1f J( R e f . 18) and theoretically infinite oper- ating times10of MTJs also make it a competitive candidate for TRNGs. In MTJ devices, the backhopping phenomenon occurs mainly in the current induced magnetization reorientation19–22process. At low current, spin transfer torque (STT) plays the role of a driving force that stably sets the two magnetic layers in either parallel (P) or anti- parallel (AP) states, which corresponds to low and high resistances,respectively. 7–11,23,24With further increase of current, the magnetiza- tion will be driven deviating from the stable state and fluctuatesbetween P and AP states. 15,25–32 In this article, the characteristics of backhopping in NR-MTJs are investigated. We find that the probability of the emergence of back-hopping depends on the measurement temperature Tand current Appl. Phys. Lett. 114, 112401 (2019); doi: 10.1063/1.5077025 114, 112401-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplpulse width s. The distribution of resistance in the backhopping cur- rent range is totally random, which is taken as the basis of the TRNG.All the experimental results are reproduced well by micromagnetic simulations, which also explains the mechanism of backhopping from the perspective of energy. The experiments are performed in two hundred NR-MTJ devi- ces. The ring shape is fabricated using electron beam lithography and has inner and outer radii of 40 nm and 60 nm, respectively, as shown inFig. 1(a) . The stack structure is depicted in Fig. 1(b) ,w i t ht h ef r e e layer and the reference layer being in-plane magnetized (more infor- mation is included in supplementary material Fig. S1). The measure- ment setup is shown in Fig. 1(b) . A square wave current pulse is injected into the MTJ device through a bias-tee to switch the magneti- zation of the free layer. Resistance measurements are performed after the pulse ends via the inductance terminal. All these measurementsare done in zero magnetic field. Figure 2(a) shows the current manipulated magnetization switch- ing process, which can be divided into three regions: (I) a normal STT switching hysteresis loop with the critical switching current I csand (II and III) two backhopping regions22,25,32with the starting current Ibh. Backhopping occurs when the current amplitude is above Ibh. Actually, 200 NR-MTJs are measured, but not all of them are observed with backhopping. We consider the probability of backhopping Pbh (the amount of backhopping MTJs out of the 200 ones) as functions of Tands. The details of the measurement method are discussed in the supplementary material ,P a r tV .F o re x a m p l e ,i n Fig. 2(b) , for a rela- tively short pulse, such as 10 ns, backhopping occurs in 22% of thedevices at room temperature. As sincreases to 1 ls,P bhis enhanced to 82% and then saturates at 92% when sreaches 1 ms. On the other hand, Talso impacts the behavior of backhopping, as revealed by Fig. 2(c). More than 75% of MTJs stay in the stable P state at low T(below150 K) and short s(shorter than 100 ns), in which case the backhop- ping does not happen. Then, a quick increase in Pbhoccurs when T andsincrease, until the percentage reaches 90% for slonger than 100 lsa n d Tabove 220 K. For devices operating at room temperature, a short pulse of s¼10 ns can drive more than 20% of the NR-MTJs into backhopping, which means that backhopping can be triggered at /C24100 MHz. It is also noticed that an asymmetry for P to AP and AP to P back- hopping exists in Fig. 2(b) , and the former circumstance shows a higher probability than the latter one. This is related to the current shift in pos- itive and negative directions in regions I and II shown in Fig. 2(a) , where /C0Ibh¼/C0580lA, while Ibh¼480lA. This may be a result of the asymmetric influence on bias behavior of spin torque,33and the deviation from an ideal ring shape inadvertently introduced during experimental fabrication. It can be compensated by reducing the cur-rent amplitude by /C070lA for P to AP transition (positive current direction) shown by the cyan triangles in Fig. 2(b) ,w h e r et h et w op r o b - abilities overlap well with each other. Besides, multiplied by the resis-tance of the AP state, which is around 1250 X, the backhopping voltage is about 750 mV. This value is quite smaller than the report of Min. et al. 22This can be attributed to the ring shape design, which can result in lower critical current due to the “onion” magnetization state.14 Previous research works have reported that backhopping occurs because of thermal activated magnetization perturbation,15,22,27,31 which originates similarly to the thermally assisted magnetization switching process. Min et al.22describe this mechanism as follows: Ics;bh¼Ics0;bh01/C0kBT Ecs;bhlns s0/C18/C19"# ; (1) where kBis the Boltzmann constant, Ecs;bhis the energy barrier of the critical switching or backhopping, Ics0;bh0characterizes the Ics;bhat 0 K, respectively, and 1/ s0is the attempt frequency. Ibhfollows the same rule of evolution as Icsthat larger critical current is needed to trigger backhopping at low Tand shorter s. This is consistent with the obser- vation shown in Figs. 3(a) and 3(b) that Ics;bhdepends on Tand ln(s/s0) linearly and monotonically. There always exists a gap between IbhandIcs, which means that the barrier of backhopping is larger than that of current induced magnetization switching. Accordingly, at room temperature, Ebhis estimated to be about 1.62 Ecs, while the ther- mal stability, depicted by K¼Ebh kBT, is derived to be about 56 and inversely proportional to Tshown in Fig. 3(a) . Actually, this value is relatively low compared to Kobtained in most of the MgO based MTJs in previous reports,34,35which directly leads to the instability of magnetization that can be excited into backhopping. The magnetization switching rate c1;2(for AP to P and P to AP, respectively) can be expressed as a function of Ibh31,32as follows: FIG. 1. (a) The SEM image of the NR-MTJs. (b) The multilayer structure and the setup of the measurement system. CFB stands for CoFeB. The current pulse comes from the arbitrary waveform generator (AWG) and is monitored with a digital storage oscilloscope (DSO). The resistance is measured with a Keithley 2400 volt-age meter. FIG. 2. (a) Current with sof 1lsi n d u c e d magnetization switching at RT in NR-MTJs. (b)Pbhfor different sat room temperature in both positive and negative current. (c)The contour map of P bhvsTands.H e r e , just the positive pulse current driven switch- ing is taken into consideration.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 114, 112401 (2019); doi: 10.1063/1.5077025 114, 112401-2 Published under license by AIP Publishingc1;2¼c0exp/C0C kBTðHK7HeffÞ217Ibh Ibh0/C18/C19 /C20/C21 ; (2) where Heffis the effective field which includes the contribution of mag- netocrystalline anisotropy and stray field, C is a constant in the energy scale, Hkis the free layer shape anisotropy (should be 0 in the NR- MTJs), and c0denotes the attempt frequency P1;2ðsÞ/C24c1;2 c1þc21/C0exp/C0ðc1þc2Þs ½/C138/C8/C9 : (3) Considering that the parameters in Eq. (2)vary much slower than the magnetization switching time in sub-nanoseconds, theswitching probability as functions of c 1;2andscan be derived into Eq. (3),w h e r e P1;2is the probability of magnetization in P and AP states, respectively.32P2stands for Pbht h ep o s i t i v ec u r r e n t ,t h em a g n e t i z a t i o n switching from the P state back to the AP state means the backhop-ping happens. By comparing P 1and P2as shown in Eq. (4),w h e r e A¼2CH2 eff kBand B¼2CH2 eff Ebh, it is clear that backhopping happens at higher temperatures and longer current pulses with a larger probabil- ity, which is consistent with the experimental results P2 P1/C24exp/C0A1 TþBlns s0/C18/C19 /C20/C21 : (4) In the following, the distribution of the resistance in the backhop- ping region is explored. In Fig. 2(a) ,Ibh¼480lAi nr e g i o nI I I .T h u s , we select the pulse current with the amplitude of 640 lA (larger than Ibh)a n d sof 10 ns at room temperature to inject into the NR-MTJs, which can ensure the occurrence of backhopping by larger probability.After the current pulse, a relaxation time of 1 s is adopted, which is long enough for the relaxation of the heating generated by current. Then, the resistance is read out by a low current with the amplitude of100 nA, which would not disturb the tested resistance state. The resultsof 20 000 testing cycles are summarized in Fig. 4(a) (only 200 results are plotted here for clear visibility), which shows no obvious regularity of the resistance distribution visually. It is found that the resistances are mainly distributed in the P or AP state, with a few intermediatevalues. By applying a reference sorter ðR maxþRminÞ/2, which classifies resistance values larger than the reference as 1 and otherwise as 0, a sequence of pure 0 and 1 is obtained as a mosaic pattern in Fig. 4(b) . A640lA deviation of trigger current does not affect the results, indi- cating the stability of the devices [shown in Fig. 4(b) andsupplemen- tary material Figs. S3(e) and S3(f)]. Then, the reliability of randomness o ft h en u m b e rs e q u e n c e( 2 0 0 0 0n u m b e r s )i se x a m i n e da c c o r d i n gt o the randomness testing suites of FIPS 140–1 and STS 2.1.1, issued bythe National Institute of Standards and Technology (NIST).5,36–39The results of both suites show that all the tests have been passed (see sup- plementary material , Table S1 and Fig. S2), indicating high quality of the randomness for the number sequence. In order to gain an insight into the backhopping state, micromag- netic simulations based on the Landau-Lifshitz-Gilbert (LLG) equa- tion,40,41as shown in the following equation, are performed: d^m dt¼/C0c^m/C2ð~Heffþ~gÞþa^m/C2d^m dt þaJ^m/C2ð^m/C2^mpÞþbJð^m/C2^mpÞ; (5) where c,a,and ^mare the gyromagnetic ratio, the damping constant, and the unit vector of local magnetization in the free layer, respec- tively. Assuming that the reference layer is solidly pinned by the syn- thetic anti-ferromagnetic layer of PtMn as shown in the M-H loop in supplementary material S1, the interlayer coupling from the reference layer acts as a constant bias field and cannot contribute much to back- hopping. Thus, we mainly focus on the impact of thermal fluctuation and the STT effect on the free layer magnetization operation.25,26,28,32 In LLG Eq. (5),~gis the thermal fluctuation field,42,43described by a white noise with an amplitude dependent on temperature. Inaddition, two STT terms: current induced in-plane Slonczewski torquea J19and field-like torque bJ44,45are implemented in the LLG equation. The factors aJis equal to/C22hcPJ 2eMsdandbJis/C15V2,w h e r e eis the magnitude of the electron charge, Pis the polarization constant, Jis the current density, Msis the free layer saturation magnetization and dis the thickness of the free layer. In the micromagnetic simulation, the fol-lowing parameters are used for the free layer: the thickness is 2 nm, M s ¼774 emu/cm3, the exchange constant A¼1.0/C210/C06erg/cm, the perpendicular uniaxial anisotropy K?¼2:79/C2106erg=cm3,t h ei n - plane anisotropy Kjj¼3:87/C2104erg=cm3;P¼0:2a n d a¼0.024. These parameters are derived from micromagnetic simulation fitting to the M-H loop and Ics.sis set to 10 ns. The current amplitude is FIG. 3. (a) and (b) Tandsdependence of Ics,Ibh, andK. FIG. 4. (a) Distribution of the resistance in the backhopping region under a bias current of 640 lA. The reference level is represented by the black dashed line. (b) Distribution of the logic values under different bias currents. The arrow representsthe direction of the zigzag arrangement of the 200 logic values. (c) Distribution of50 consecutive resistances obtained by micromagnetic simulation. (d) Schematic diagram of a TRNG device employing an array of NR-MTJs.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 114, 112401 (2019); doi: 10.1063/1.5077025 114, 112401-3 Published under license by AIP Publishingincreased by 30 lA steps to 900 lA, until backhopping occurs. The simulated Curie temperature from these parameters is 700 K, which is substantially above the device working temperature. Figure 5(a) shows the simulated current driven magnetization switching results. We can also see three regions, including backhop-ping, where I csandIbhare equal to 270 lAa n d4 8 0 lA, respectively, just as depicted by Fig. 2(a) . To demonstrate the mechanism of back- hopping, the magnetization motion related to the system energy evolu-tion is simulated as well, as shown in Figs. 5(b) and5(c). We can see that when the current increases to 270 lA, the increment of system energy resulting from the enlarging in-plane STT reaches the extreme value at 2 ns, which is sufficient to overcome the damping torque and achieve the magnetization switching from the /C0xdirection to the þx one. After the current is off, the injected energy from the STT quicklydissipates due to the damping effect, and the system relaxes to the low energy state. Then, with current enlarged to 360 lA, the energy also keeps increasing. The magnetization tends to switch away from theþxdirection at the beginning of 8 ns. But, this high energy state is not sufficient to overcome the energy barrier of backhopping. Thus, itreturns to the P state after the pulse is off. As a result, backhopping does not happen. However, when the current reaches 480 lA, the sys- tem energy reaches a maximum at 10 ns and the magnetization finallyswitches back to the /C0xdirection again. In previous research, 29it is reported that the quadratic dependence of field-like torque on the applied voltage leads to a steady precession of magnetization during the current pulse and backhopping is measured when the pulse ends.However, in our simulation, once the contribution of thermal fluctua-tion is removed, backhopping does not happen for any current up to900lA. In fact, the estimated field-like torque is about 10 Oe in the þxdirection and is negligible compared to the thermal fluctuationnoise, whose amplitude is around 38 000 Oe in this simulation. 42Thus, we can say that backhopping is caused by a combined effect of in- plane STT and thermal fluctuation. Both of these two factors canintroduce noise into the magnetic system. Thus, at this point, both STT and the thermal effect stimulate the system to a maximum energy state, as shown in Fig. 5(b) . Then, the system has equal possibility to f a l li n t ot h el o we n e r g ys t a t eo fPo rA P ,a ss h o w ni nt h ei n s e to f Fig. 5(b). The corresponding domain evolution under the processes men- tioned above is summarized in Fig. 5(d) :t h ec u r r e n to f2 7 0 lA pro- vides sufficient STT to switch the magnetization from the initial /C0x direction to the þxone; the current of 360 lAc a n n o ts w i t c ht h em a g - netization back to the /C0xdirection due to the high backhopping bar- rier; the current of 480 lA finally triggers the backhopping by switching the magnetization back to the /C0xdirection. The resistance distribution at 600 lA (in Region III) within 50 cycles is also simulated and depicted in Fig. 4(c) , in which the states of P and AP are purely randomly chosen with no preference and the self-correlation of the resistance state is 0, indicating a high quality random number sequence. Finally, the Tandsdependence of P bhis also simulated. At a low Tof 100 K, shown in Fig. 5(e) , even with sof 100 ns, less than 12% of all the devices show backhopping behavior. For sless than 1 ns, back- hopping is not present at all. Pbhincreases remarkably with increasing T,a sw e l la sf o rl o n g e r s.F o ra sof 100 ns at 300 K, approximately 52% of the scans show backhopping. Although other simulations with longer swere not conducted owing to excessive computational cost, the effects of Tand son backhopping can now semi-quantitatively reproduce the experimental results. It should be noticed that at room temperature, experimentally, 10 ns-pulse-width current can trigger a Pbhof 22%, while theoretically, this percentage is 20% for s¼1n s . Besides, analogous to MRAM, if a series of N NR-MTJs are arranged in parallel connection as shown in Fig. 4(d) , in general, a random number sequence containing 2Nnumbers can be obtained. Thus, the TRNG generating rate can be as high as /C24100 MHz, even /C241G H z with an infinite sequence length, which can cover most applications. Itis not easy to forwardly improve the functioning frequency because the magnetization switching duration is around the nanosecond scale. 46,47But, still some investigations reported that by optimizing the structure of the MTJs, the functioning time can be reduced to the fem- tosecond scale,48which sheds light on the application of the NR-MTJs based TRNGs at higher frequencies. In summary, backhopping is observed in nano-ring MTJ devices. We find that backhopping occurs more frequently at high Tand wide s. The resistance in the backhopping region appears to follow a truly random distribution. Micromagnetic simulations show that backhop- ping is stimulated by the combination of in-plane STT and thermal noise. The ability of backhopping to rapidly generate a sequence of random resistances suggests feasibility for a high quality TRNG that can work at up to gigahertz frequency generation rate. Seesupplementary material for a summary of the M-H loop of the film, the details of the randomness reliability of the numbersequences by the suites of NIST, the device lifetime exploration, the breakdown of the devices, and the method of determining P bh. This work was supported by the National Key Research and Development Program of China (MOST, Grant No. 2017YFA0206200) and FIG. 5. (a) The stimulated current switching magnetization results of the NR-MTJ at room temperature with a sof 10 ns. (b) and (c) The time evolution of the total energy and the component of magnetization in the xdirection for 270 lA, 360 lA, and 480 lA with sof 10 ns. (d) The magnetization and the domain motion under different currents in NR-MTJs. (e) Pbhcalculated from multiple scans versus Tand sby micromagnetic simulations.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 114, 112401 (2019); doi: 10.1063/1.5077025 114, 112401-4 Published under license by AIP Publishingthe National Natural Science Foundation of China (NSFC, Grant Nos. 11434014, 51620105004, and 11674373) and par tially supported by the Strategic Priority Research Program (B) (Grant No. XDB07030200), the International Partnership Program (Grant No. 112111KYSB20170090), and the Key Research Program of Frontier Sciences (Grant No. QYZDJ-SSWSLH016) of the Chinese Academy of Sciences (CAS). This work wasalso supported by the University of Minnesota Supercomputing Institute forcomputer time. REFERENCES 1Y. 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1.4957445.pdf

Perpendicular magnetic anisotropy in Co 2Fe0.4Mn0.6Si B. M. Ludbrook , B. J. Ruck , and S. Granville Citation: J. Appl. Phys. 120, 013905 (2016); doi: 10.1063/1.4957445 View online: http://dx.doi.org/10.1063/1.4957445 View Table of Contents: http://aip.scitation.org/toc/jap/120/1 Published by the American Institute of Physics Perpendicular magnetic anisotropy in Co 2Fe0.4Mn0.6Si B. M. Ludbrook,1B. J. Ruck,1and S. Granville2 1The MacDiarmid Institute for Advanced Materials and Nanotechnology, School of Chemical and Physical Sciences, Victoria University of Wellington, P.O. Box 600, Wellington 6140, New Zealand 2The MacDiarmid Institute for Advanced Materials and Nanotechnology, Robinson Research Institute, Victoria University of Wellington, P.O. Box 33436, Lower Hutt 5046, New Zealand (Received 10 March 2016; accepted 28 June 2016; published online 7 July 2016) We report perpendicular magnetic anisotropy (PMA) in the half-metallic ferromagnetic Heusler alloy Co 2Fe0.4Mn 0.6Si (CFMS) in a MgO/CFMS/Pd trilayer stack. PMA is found for CFMS thick- nesses between 1 and 2 nm, with a magnetic anisotropy energy density of KU¼1:5/C2106erg/cm3 fortCFMS¼1:5 nm. Both the MgO and Pd layer are necessary to induce the PMA. We measure a tunable anomalous Hall effect, where its sign and magnitude vary with both the CFMS and Pdthickness. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4957445 ] I. INTRODUCTION A magnetic thin film will generally prefer to have its magnetic moment lying in the plane of the sample owing to the large demagnetizing field. New magnetic deviceshave made it desirable to fabricate thin magnetic layerswhich instead have an easy axis of magnetization directed perpendicular to the film plane, a condition known as perpendicular magnetic anisotropy (PMA). 1In particular, spin-transfer-torque (STT) based devices require that a magnetic free layer can be easily flipped by a spin- polarized current while maintaining a high stability againstthermal fluctuations. 2Magnetic layers with PMA are well suited to optimize this trade-off for device applications. In order to realize a high efficiency STT device, a high degree of spin-polariz ation in the magnetic layers is also desirable. CoFeB, with approximately 65% spin polarization, has been the most widely studied material so far, because it can be grown with PMA and incorporatedinto device structures with ve ry large tunneling magneto- resistance (TMR). 3There is a strong motivation to incor- porate a half-metallic ferr omagnet (HMFM) with higher spin polarization in these devices, and Heusler alloysare promising in this regard. 4In particular, Co 2FeSi and Co2MnSi both have 100% spin polarization5–7and high Curie temperatures (ca. 1000/C14C). Recent studies have shown the intermediary compound Co 2FexMn (1–x)Si with x/C250:4 to be eminently promisin g for device applications with a low Gilbert damping parameter8and record 75% room temperature giant magnetoresistance ratio.9 Efforts to induce PMA in the Heuslers have focused on compounds containing Fe on MgO.10–15This was guided by earlier studies of CoFeB/MgO where the PMAis thought to have its origin in the Fe-O hybridization. 3 PMA has recently been reported in Co 2MnSi in CMS/Pd multilayer stacks on MgO,16and for Pd buffered Co2FexMn 1–xSi on MgO,17but the details of the PMA and the contribution of the various interfaces remain unclear.Here, we demonstrate PMA in MgO/Co 2Fe0.4Mn 0.6Si (CFMS)/Pd stacks, and show that both interfaces are im- portant for this effect.II. METHODS Samples were grown by DC magnetron sputtering in a Kurt J Lesker CMS-18 UHV system with a base pressure of2/C210 /C08Torr. Multilayer stacks were prepared on 10 /C210 mm Si/SiO 2substrates in the sequence Si/SiO 2/MgO(2)/CFMS (tCFMS)/ Pd(2.5), where the number in parentheses is the nomi- nal layer thickness in nm. Samples were grown at room tem-perature and post-growth annealed in-situ for 1 h at 300 /C14C with an in-plane magnetic field of 170 Oe. MgO was RF sput-tered with 100 W power in 3 mTorr Ar, giving a growth rate of0.05 A ˚/s. CFMS was DC sputtered at 100 W and 5 mTorr Ar, giving a growth rate of 0.43 A ˚/s, and Pd was DC sputtered at 175 W in 8 mTorr Ar, with a growth rate of 4.0 A ˚/s. Growth rates were calculated by growing a thick ( >50 nm) film and measuring the thickness with a Dektak profilometer. The com-position of the Heusler film was verified to be Co 2Fe0.4Mn0.6Si by energy dispersive X-ray ana lysis in a SEM. Magnetization and Hall resistance measurements were done in a QuantumDesign Superconducting Quan tum Interference Device (SQUID) and Physical Property Measurement System, respec- tively, at room temperature. The Hall measurements were made using sampleholders from Wimbush Science and Technology,with spring-loaded contacts in a van der Pauw geometry. III. RESULTS AND DISCUSSION A. Magnetization measurements In Figs. 1(a)–1(d) , magnetization measurements with the field applied parallel to the film plane and perpendicular tothe film plane demonstrate PMA for CFMS films with thick-nesses between 1 and 2 nm. Samples in this thickness rangeare easily magnetized out-of-plane, having a small saturationfield ( H s<100 Oe) and high remanence [Fig. 1(e)]. With the field applied in-plane, a larger applied field is required to satu-rate the magnetization, and the remanence is close to zero.Conversely, the magnetic behavior of a 3 nm thick CFMS film shown in Fig. 1(d) indicates an in-plane easy axis of magnet- ization. The saturation magnetization of all films is closeto the expected bulk value of /C251000 emu/cm 3, within the 10% uncertainty. While we observe no direct evidence of a 0021-8979/2016/120(1)/013905/4/$30.00 Published by AIP Publishing. 120, 013905-1JOURNAL OF APPLIED PHYSICS 120, 013905 (2016) magnetically dead layer, an analysis of the uncertainties allows us to place an upper bound of 0.5 nm on this quantity. The uniaxial magnetic anisotropy energy density KUis a quantitative measure of the PMA strength, and is determinedfrom the difference in area under the out-of-plane and in-plane magnetization curves, where positive values correspondto PMA. K Uis plotted as a function of tCFMS in Fig. 1(g), showing that the PMA is strongest for the tCFMS¼1:5n m film, having a value KU¼1:5/C2106erg/cm3. This is compa- rable with values found for other Heusler alloys, which aretypically 1 /C03/C210 6erg/cm3.10,16–18 The decreasing KUwith increasing film thickness and the transition to in-plane magne tic anisotropy (negative KU)i n Fig.1(g)is a feature of magnetic thin films due to the competi- tion between interface induced PMA and the volume andshape anisotropy, which tend to favor an in-plane easy axis. 19 The uniaxial anisotropy is given by KU¼KV/C02pM2 sþKS=t, where KVand KSare the bulk and interface anisotropy terms, respectively, and the term 2 pM2 sis due to the shape ani- sotropy. KU/C1tCFMS is plotted against tCFMS in Fig. 1(h),w h e r e the intercept of the linea r extrapolation indicates KS¼1:1 60:2 erg/cm2. The slope is equal to the effective volume contribution Kef f V¼KV/C02pM2 s¼/C05:3/C2106erg/cm3which favors an in-plane magnetization. The value of KSreported here is similar to that reported in Co 2FeAl10,20(KS¼0:8/C01:0 erg/cm2), but larger than what has been meas- ured in Co 2MnSi stacks17(KS¼0:5e r g / c m2) or multilayers16 (KS¼0:16 erg/cm2). This demonstrates the interfacial origin of the PMA in CFMS in these trilayers. B. Anomalous Hall effect in CFMS The Hall resistivity measured in a ferromagnetic mate- rial is empirically given by qxy¼RHHzþRSMz.21It is the sum of the normal Hall effect, linear in applied field ( Hz) with coefficient RH, and the anomalous Hall effect (AHE), which is proportional to the out-of-plane magnetization ( Mz) and a material dependent term RS. Therefore, measurements of the Hall effect can be used to probe PMA in thin films.The AHE in Figs. 2(a)–2(e) confirms the PMA for CFMS film thicknesses 1 nm /C20t CFMS/C202 nm, which shows 100% remanence and a coercive field of around 25 Oe. The thicker3 nm CFMS film in Fig. 2(e)shows an AHE characteristic of an in-plane easy magnetic axis, with a saturation field ofabout 3 kOe, in agreement with the magnetization measure-ments in Fig. 1(d). Data for the 0.75 nm CFMS film shown in Fig. 2(a) show superparamagnetic behavior, similar to observations inCoFeB thin films below the thickness threshold for PMA. 22 A large, hysteretic AHE becomes apparent only at low tem-perature, as shown in Fig. 2(f). This is probably an indication that the film is not continuous, with the discontinuousregions of the film acting as superparamagnetic particles at room temperature. C. The importance of the interface PMA in magnetic thin films generally results from a modification of the orbital angular momentum due toFIG. 1. (a)–(d) SQUID magnetization hysteresis loops measured with the field in-plane (cyan/open symbols) and perpendicular-to-plane (orange/closed symbols) for MgO/CFMS/Pd trilayers. For t CFMS¼1;1:5;2 nm the samples show PMA with an out-of-plane easy axis of magnetization, while thetCFMS¼3 nm sample in (d) has an in-plane easy axis. (e) The low field region shows the hysteresis and remanence for the out-of-plane measure- ment for tCFMS¼1:5 nm. (f) The low field region of the tCFMS¼3 nm sam- ple has a sharp change in magnetization with the field applied in-plane, although there is no hysteresis. (g) The uniaxial anisotropy is calculatedfrom the data in panels (a)–(d). (h) Plotting K U/C1tCFMS vstCFMS shows the vol- ume and surface contributions to the anisotropy. The solid line is a linear fit to the data excluding the tCFMS¼1 nm datapoint. FIG. 2. (a)–(e) Hall resistance of MgO/CFMS/Pd trilayers with 0 :75 /C20tCFMS/C203 nm. (f) Temperature dependent Hall measurements for the sam- ple with tCFMS¼0:75 nm showing evidence of superparamagnetism. The AHE coefficient shown in (g) is determined by the zero-field extrapolation of the positive high-field Hall data. R AHEshows an approximately linear dependence on tCFMS with a sign change between 1.5 and 2 nm.013905-2 Ludbrook, Ruck, and Granville J. Appl. Phys. 120, 013905 (2016) hybridization of orbitals at the interfaces. In MgO/CoFeB, it is thought to be the Fe-O hybridization at the interfacethat leads to the PMA 3,23although a thin metallic capping layer, often Ta, has been shown to contribute also.24,25PMA has also been observed and studied in Co/Pd and Co/Ptmultilayers, 26but in these cases, it is the Co 3 d-(Pd, Pt) 5 d hybridization that is understood to induce the PMA.27,28 While the data presented here indicate an interfacial origin of the PMA in CFMS thin films, samples with different inter- faces were prepared in order to understand which interface is important. (i) Pd(2.5)/CFMS(1.5)/Pd(2.5): SQUID measurements show an in-plane easy axis for this stack [Fig. 3(a)], demonstrating that the PMA is not from the CFMS- Pd interface alone, and that the MgO layer plays an important role. (ii) MgO(2)/CFMS(1.5)/MgO(2): Magnetization meas- urements of this stack in Fig. 3(b) show no evidence of PMA. Interestingly, Hall and resistivity measure- ments were not possible on this sample, suggesting that the metallic CFMS layer does not form a continu-ous film. Hall measurements were made on a similar stack with a thicker CFMS layer ( t CFMS¼3 nm) and are shown in Fig. 3(c). The AHE with a large satura- tion field is consistent with an in-plane easy axis. (iii) MgO(2)/CFMS(1.5)/Cu(3): Replacing the Pd capping layer with a Cu layer of similar thickness (3 nm) also destroyed the PMA. The Hall effect measurement in Fig.3(d) shows a large saturation field of 5 kOe and zero remanence. This demonstrates the importance of the CFMS/Pd interface for attaining PMA. (iv) Pd(2.5)/CFMS(2.0)/MgO(2): Reversing the order of layers (i.e., with MgO forming the top capping layer) also results in PMA, as shown for a 2 nm CFMS filmin Fig. 3(e). From this series of experiments, we conclude that both interfaces (MgO/CFMS and CFMS/Pd) are required toobtain PMA in CFMS in this thickness range, in agreement with previous work. 17We note that structural order in theCFMS induced by one of the layers could also play a role, as shown for Co 2FeAl in Ref. 15. Structural characterization of thicker films on the different underlayers is an interestingtopic for future work. The thickness of the Pd layer also plays a role in attaining PMA in the CFMS thin film. Figs. 3(f)–3(h) show Hall meas- urements for layers of the structure MgO(2)/CFMS(1.5)/Pd(t Pd)f o r tPd¼1:25;2:5;3:75 nm. For tPd/C212:5 nm, the CFMS has PMA, evidenced by the sharp, hysteretic AHE inFigs. 3(g)and3(h). For the thinnest Pd layer ( t Pd¼1:25 nm), the film reverts to an in-plane magnetic anisotropy with a sat-uration field of about 1 kOe and no remanence or coercivity.This is again similar to what is reported for CoFeB on MgO,where a Ta layer above a threshold thickness is required toattain PMA. 29 D. Tunable AHE in CFMS While the AHE is proportiona l to the out-of-plane mag- netization, its sign and magnitude also depend on a number ofother properties of the material, including the band structure,scattering mechanisms, and spin-polarization. 21The anoma- lous Hall resistance ( RAHE) is defined here as the zero-field extrapolation of the high-field Hall effect data for positive applied fields. For CFMS thin films in a MgO/CFMS/Pd stack, we observe a thickness dependent AHE, as shown in Fig. 2(g). There is a change in the sign of R AHEbetween tCFMS¼1:5 andtCFMS¼2:0 nm. This trend, and the sign change, appears independent of the magnetic anisotropy, which transformsfrom PMA to in-plane anisotropy between t CFMS¼2:0a n d tCFMS¼3:0 nm. It is worth noting that this AHE sign change also occurs with a change in the Pd capping layer thickness,s h o w ni nF i g s . 3(f)–3(h) . A composition dependent sign- change of the AHE has been reported in CFMS related tochanges in the bandstructure an d spin-dependent band-gap. 30 A similar interpretation might be applicable here, with a thick- ness dependent change in the electronic structure causing thechanges in the AHE. However, further work is required tounderstand the relative importance of the spin-polarization, FIG. 3. (a) SQUID magnetization of a Pd/CFMS(1.5)/Pd trilayer shows in-plane magnetic anisotropy. Similarly, a MgO/CFMS(1.5)/MgO trilayer in (b) s hows no evidence of PMA. (c) Hall measurements on a similar stack with a 3 nm CFMS layer show in-plane magnetic anisotropy, as expected. Replacing the Pd with Cu results in in-plane magnetic anisotropy, shown in the Hall measurement in (d). Reversing the order of the layers does not affect the PMA, shown f or a Pd/CFMS 2.0/MgO trilayer in (e). Hall measurements of MgO/CFMS/Pd trilayers with varying Pd thickness are shown in (f)–(h). A thin Pd layer (tPd¼1:25 nm) in (f) gives no PMA, while thicker Pd layers do, as shown by the sharp, hysteretic AHE observed for tPd/C212:5 nm in (g) and (h).013905-3 Ludbrook, Ruck, and Granville J. Appl. Phys. 120, 013905 (2016) electronic structure, and the various scattering channels that affect the AHE. IV. CONCLUSIONS In conclusion, we have demonstrated PMA in Co 2Fex Mn1–xSi with x¼0.4 for thin films in a MgO/CFMS/Pd tri- layer stack. Both the MgO and the Pd layers are necessary to generate the PMA. This stack shows promise for incorpora-tion into a spin-transfer-torque device with high thermal sta- bility, low switching current, and high output power. The observation of a thickness-tunable AHE is an interestingeffect, and further work is required to elucidate its origin. ACKNOWLEDGMENTS The authors are grateful to Andrew Best from Callaghan Innovation for technical support, and to Sarah Spencer fromthe Robinson Research Institute for the SEM measurements. B.M.L. gratefully acknowledges post-doctoral funding from the MacDiarmid Institute. This work was supported byproject funding from the MacDiarmid Institute and the New Zealand Ministry of Building, Innovation and Employment Magnetic Devices Contract No. RTVU1203. 1A. Hirohata, H. Sukegawa, H. Yanagihara, I. Zutic, T. Seki, S. Mizukami, and R. Swaminathan, IEEE Trans. Magn. 51, 1 (2015). 2S. Mangin, D. Ravelosona, J. A. Katine, M. J. Carey, B. D. Terris, and E. E. Fullerton, Nat. Mater. 5, 210 (2006). 3S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan, M. Endo, S. Kanai, J. Hayakawa, F. Matsukura, and H. Ohno, Nat. Mater. 9, 721 (2010). 4M. I. Katsnelson, V. Y. Irkhin, L. Chioncel, A. I. Lichtenstein, and R. A. de Groot, Rev. Mod. Phys. 80, 315 (2008). 5H. C. Kandpal, G. H. Fecher, C. Felser, and G. Sch €onhense, Phys. Rev. B 73, 094422 (2006). 6F. J. Yang, C. Wei, and X. Q. Chen, Appl. Phys. Lett. 102, 172403 (2013). 7M. Jourdan, J. Min /C19ar, J. Braun, A. Kronenberg, S. Chadov, B. Balke, A. Gloskovskii, M. Kolbe, H. Elmers, G. Sch €onhense, H. Ebert, C. Felser, and M. Kl €aui,Nat. Commun. 5, 3974 (2014).8T. Kubota, S. Tsunegi, M. Oogane, S. Mizukami, T. Miyazaki, H. Naganuma, and Y. Ando, Appl. Phys. Lett. 94, 122504 (2009). 9J. Sato, M. Oogane, H. Naganuma, and Y. Ando, Appl. Phys. Express 4, 113005 (2011). 10Z. Wen, H. Sukegawa, S. Mitani, and K. Inomata, Appl. Phys. Lett. 98, 192505 (2011). 11X. Li, S. Yin, Y. Liu, D. Zhang, X. Xu, J. Miao, and Y. Jiang, Appl. Phys. Express 4, 043006 (2011). 12M. Ding and S. Joseph Poon, Appl. Phys. Lett. 101, 122408 (2012). 13Y. Cui, B. Khodadadi, S. Schafer, T. Mewes, J. Lu, and S. A. Wolf, Appl. Phys. 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1.2162812.pdf

Multiscale micromagnetic simulation of giant magnetoresistance read heads O. Ertl, G. Hrkac, D. Suess, M. Kirschner, F. Dorfbauer, J. Fidler, and T. Schrefl Citation: Journal of Applied Physics 99, 08S303 (2006); doi: 10.1063/1.2162812 View online: http://dx.doi.org/10.1063/1.2162812 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/99/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Increases in effective head field gradients in exchange spring media Appl. Phys. Lett. 95, 172509 (2009); 10.1063/1.3257364 Analysis of thermal magnetic noise in spin-valve GMR heads by using micromagnetic simulation J. Appl. Phys. 97, 10N705 (2005); 10.1063/1.1851881 Detailed modeling of temperature rise in giant magnetoresistive sensor during an electrostatic discharge event J. Appl. Phys. 95, 6780 (2004); 10.1063/1.1652426 Modeling of enlarged back gaps in vertical giant magnetoresistance read heads J. Appl. Phys. 85, 5321 (1999); 10.1063/1.370239 Signal-to-noise optimization in narrow vertical and horizontal giant magnetoresistent head sensors (abstract) J. Appl. Phys. 81, 4840 (1997); 10.1063/1.364849 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 138.37.211.113 On: Tue, 07 Oct 2014 08:01:04Multiscale micromagnetic simulation of giant magnetoresistance read heads O. Ertl,a/H20850G. Hrkac, D. Suess, M. Kirschner, F. Dorfbauer, and J. Fidler Vienna University of Technology, Vienna 1040, Austria T. Schrefl Department of Engineering Materials, University of Sheffield, Mappin Street, Sir Robert Hadfield Building, Sheffield S1 3JD, United Kingdom /H20849Presented on 3 November 2005; published online 18 April 2006 /H20850 The Landau-Lifshitz-Gilbert equation and quasistatic Maxwell equations were solved simultaneously to calculate the read back signal of giant magnetoresistance read heads with a hybridfinite-element/boundary element method. The finite-element simulations show the influence of thesense current on the linearity of the reader, the effect of the exchange bias field on the sensorperformance, and the influence of the Gilbert damping constant on the decay time of the read backvoltage. All parts of the system, the layers of the giant magnetoresistance sensor, the hard biasmagnets, the shields, and the recording layer are treated micromagnetically. In addition, theinfluence of the sense current onto the magnetization is taken into account self-consistently. Thecurrent distribution in the giant magnetoresistance stack is calculated from local resistivity whichdepends on the magnetization of the free and of the pinned layer. © 2006 American Institute of Physics ./H20851DOI: 10.1063/1.2162812 /H20852 INTRODUCTION Read head simulations usually assume a homogenous sense current density through the giant magnetoresistance/H20849GMR /H20850element. In fact, the current distribution is inhomo- geneous due to different conductivities of the GMR elementlayers and furthermore due to the magnetoresistance itself. Inturn, the current distribution generates an additional mag-netic field and influences the magnetization. This was themain motivation to develop a dynamic micromagnetic modelincluding a current model, which allows solving the currentdistribution self-consistently by means of the finite-element/H20849FE/H20850method. Similar micromagnetic simulations were al- ready done by Gibbons et al. 1However, they used a finite- difference /H20849FD /H20850method, which is more difficult to apply on arbitrary shaped geometries than the FE method. Moreover,the FE method allows the simulation of multiscale geom-etries, because the mesh can be easily refined in the regionsof interest, especially the gap region and near the GMR ele-ment. Both, shields and hard bias magnets, can be taken intoaccount in the finite-element approach. Takano 2also used the FE method, in order to study the influence of shields on thereading performance in narrow track recording. Our model simultaneously solves the Landau-Lifshitz- Gilbert /H20849LLG /H20850equation and the quasistatic Maxwell equation using a hybrid /H20849FE/H20850/boundary element /H20849BE /H20850method. So the influence of the sense current is taken into account self-consistently. MODEL We define a region /H9024with a magnetic subset /H9023and a conductive subset /H9003with the conductivity /H9268. Further we im-pose the possibility that /H9023and/H9003can overlap, meaning that /H9268can also depend on the magnetization Mto take magne- toresistive effects into account. The magnetization dynamics of our system is described by the LLG equation of motion, /H11509M /H11509t=−/H20841/H9253/H20841 /H208491+/H92512/H20850M/H11003Heff−/H20841/H9253/H20841 Ms/H9251 /H208491+/H92512/H20850M/H11003/H20849M/H11003Heff/H20850, /H208491/H20850 where /H9251is the dimensionless Gilbert damping constant, Ms is the saturation magnetization, and /H9253is the electron gyro- magnetic ratio. The effective field Heffis composed of the anisotropy field Hani, the exchange contribution Hexch, the magnetic static field HM, and the current field Hcurrdue to the sensor current, Heff=Hani+Hexch+HM+Hcurr. /H208492/H20850 The magnetic static field is calculated using a hybrid FE/BE method as described by Fredkin and Koehler.3 Calculation of the current distribution The current field is determined by the current distribu- tion, which can be obtained using Ohm’s law, j=/H9268·E, /H208493/H20850 and Maxwell’s theory. Here the local conductivity /H9268depends on the material and is a function of space and of the magne-tization since magnetoresistive effects have to be considered, /H9268=/H9268/H20849r,M/H20850. /H208494/H20850 The conductivity /H9268of the GMR element is described by the macroscopic model4 a/H20850Electronic mail: ertl@magnet.atp.tuwien.ac.atJOURNAL OF APPLIED PHYSICS 99, 08S303 /H208492006 /H20850 0021-8979/2006/99 /H208498/H20850/08S303/3/$23.00 © 2006 American Institute of Physics 99, 08S303-1 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 138.37.211.113 On: Tue, 07 Oct 2014 08:01:04/H9268=/H92680/H208731+RGMR1 − cos /H9004/H9258 2/H20874−1 , /H208495/H20850 where RGMRis the GMR ratio, /H9004/H9258is the angle between the orientations of the magnetization of free and pinned layer,and /H92680is the conductivity for the parallel state. The electric field Ecan be split into two parts, an irrotational Efreeand a solenoidal part Eeddy, representing the electric field due to external applied voltage or current, and the electric field dueto eddy currents, E=E free+Eeddy. /H208496/H20850 Neglecting eddy currents, which can be partially taken into account by choosing higher Gilbert damping constants,5im- plies /H11612/H11003E=0 . /H208497/H20850 Thus the electrical field can be written as the gradient of a scalar electric potential /H9021, E=−/H11612/H9021. /H208498/H20850 In the quasistatic approach Kirchhoff’s current law is defined /H11612j=0 . /H208499/H20850 Combining Eqs. /H208493/H20850,/H208498/H20850, and /H208499/H20850yields /H11612/H20849/H9268/H11612/H9021/H20850=0 , /H2084910/H20850 which is solved with the FE method in the conducting region /H9003. Finally the current distribution jcan be calculated as fol- lowed: j=−/H9268/H11612/H9021. /H2084911/H20850 Calculation of the current field One way to calculate the current field is by using the Biot-Savart law, H/H20849r/H20850=1 4/H9266/H20885 /H9003j/H11003r−r/H11032 /H20841r−r/H11032/H208413dV/H11032. /H2084912/H20850 In the finite-element method the whole region /H9024is dis- cretized into tetrahedral elements. Since /H9024is the union set of /H9003and/H9023, all elements can be classified into magnetic and conductive elements. The integral in /H2084912/H20850is over the conduc- tive region /H9003. Within the framework of the finite-element method /H2084912/H20850splits into a sum of integrals over tetrahedrons H/H20849r/H20850=1 4/H9266/H20858 e/H33528/H9003/H20885 Vej/H11003r−r/H11032 /H20841r−r/H11032/H208413dV/H11032. /H2084913/H20850 Within each tetrahedron e, the current density is constant, when linear finite elements are used to solve /H2084910/H20850. For con- stant current density the volume integrals in /H2084913/H20850can be transformed into equivalent surface integrals.6Instead of e we can use the four vertices /H20849ijkl /H20850to denote a tetrahedron. Then the field generated by the current density can be written asH/H20849r/H20850=/H20858 /H20849ijkl /H20850jijkl/H110031 4/H9266/H20873/H20885 /H9004ijk1 /H20841r−r/H11032/H20841dA/H11032+/H20885 /H9004jlk1 /H20841r−r/H11032/H20841dA/H11032 +/H20885 /H9004ilk1 /H20841r−r/H11032/H20841dA/H11032+/H20885 /H9004ijl1 /H20841r−r/H11032/H20841dA/H11032/H20874. /H2084914/H20850 The integrals in /H2084914/H20850can be evaluated analytically.7With this discretization an interaction matrix Aijis derived, which gives the magnetic field Hfor the calculated current distri- bution j, H/H20849i/H20850=/H20858 j=1L Aijj/H20849j/H20850,1/H33355i/H33355K, /H2084915/H20850 with Lbeing the number of conductive elements, and Kthe number of magnetic elements. To reduce the memory re-quirement in the order of O/H20849L *K/H20850for storing the interaction matrix Aij, the concept of hierarchical matrices is used.8 RESULTS For our calculation we used a simple read head model, as shown in Fig. 1. Between two Permalloy shields, gapwidth 60 nm, there is the GMR element with a free layer/Cu/pinned layer/antiferromagnet stack of size 100 nm in widthand 80 nm in height. Laterally two hard bias magnets arepositioned to align the free layer magnetization orthogonal tothe pinned layer magnetization. The exchange bias betweenantiferromagnetic and pinned layer is taken into account by auniform magnetic field acting on the pinned layer only /H20849pin- ning field /H20850. The maximum pinning field for direct exchange bias coupling for realistic configurations does not exceed 50mT. Nevertheless we used for our simulations pinning fieldsof 50 and 100 mT to show that for such small GMR sensordimensions stronger pinning fields are needed for good sen-sor properties. In reality such strong pinning fields up to 0.5T can be achieved using GMR stacks with a syntheticantiferromagnet. 9 The total current through the GMR element, working in the current-in-plane /H20849CIP /H20850mode, was set to 3 mA. The resis- tivities of the materials in the spin valve was estimated fromRef. 10 and the GMR ratio was set to 10%. The calculated FIG. 1. The magnetic model used for the FEM calculations. The free layer /H20849white /H20850and the pinned layer /H20849dark gray /H20850are positioned between the two hard bias magnets /H20849light gray /H20850.08S303-2 Ertl et al. J. Appl. Phys. 99, 08S303 /H208492006 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 138.37.211.113 On: Tue, 07 Oct 2014 08:01:04total resistance of our GMR sensor in ground state was about 30/H9024. Ideally, the magnetization of the free layer and the pinned layer should be orthogonal to each other. Due to thedemagnetizing field of the pinned layer the free layer mag-netization is slightly rotated towards the antiparallel state.Depending on the current direction, the current field can ei-ther support or compensate this deflection. The influence ofthe sense current direction is described in more detail withFig. 2. The sense current direction also changes the sensorcharacteristics, as shown in Fig. 3. The sensitivity of thesensor element depends on the slope of the output voltage asa function of field. The slope at zero field is /H9004V//H9004H =38 mV/T and /H9004V//H9004H=17 mV/T for current directions /H20849a/H20850 and /H20849b/H20850, respectively. The magnetic field created by a sense current in direction /H20849a/H20850partly compensates the demagnetizing field of the pinned layer which in turn improves the sensitiv-ity. The resistance decreases when the external field over- comes the pinning field, so that the magnetization of thepinned layer is rotated. This can be avoided, if the pinning field is increased. Using a pinning field of 0.1 T enlarges therange of linearity. In addition to the response curve the decay of the read back voltage was calculated. We start from an external fieldof 0.03 T in pinned layer direction /H20849Fig. 4 /H20850. After switching off the field, we get different relaxation behaviors for differ-ent Gilbert damping constants. We have optimum dampingfor an intermediate damping value of 0.3 with a relaxationtime of 0.17 ns. The decay time increases to 0.4 and 0.6 nsfor a damping value of 0.1 /H20849undercritical damping /H20850and 1 /H20849overcritical damping /H20850, respectively. Considering future read heads with a reading frequency up to 2 GHz the decay timecannot be neglected any more. CONCLUSION A three-dimensional dynamic micromagnetic model which includes the magnetic field of arbitrary current distri-butions was developed and applied to the simulation of asimple GMR read head model. The GMR effect influencesthe conductivity and therefore the current distribution, whichfinally leads to a change of the output voltage. We are able tosimulate the dynamic behavior of GMR heads, such as relax-ation behavior or the output signal for a given data layer. Ourmodel can also be easily adapted for tunnel magnetoresis-tance /H20849TMR /H20850read heads. ACKNOWLEDGMENT This project was supported by the Austrian Science Fund /H20849Grant No. Y132-N02 /H20850. 1M. R. Gibbons, G. Parker, C. Cerjan, and D. W. Hewett, Physica B 275, 11 /H208492000 /H20850. 2K. Takano, IEEE Trans. Magn. 41, 696 /H208492005 /H20850. 3D. R. Fredkin and T. R. Koehler, IEEE Trans. Magn. 26,4 1 5 /H208491990 /H20850. 4S. X. Wang and A. M. Tarantorin, Magnetic Information Storage Technol- ogy /H20849Academic, San Diego, 1999 /H20850, p. 167. 5G. Hrkac, M. Kirschner, F. Dorfbauer, D. Suess, O. Ertl, J. Fidler, and T. Schrefl, J. Appl. Phys. 97, 10E311 /H208492005 /H20850. 6S. Pissanetzky, IEEE Trans. Magn. 29, 1282 /H208491993 /H20850. 7D. A. Lindholm, IEEE Trans. Magn. 20, 2025 /H208491984 /H20850. 8S. Kurz, O. Rain, and S. Rjasanow, Comput. Mech. 32,4 2 3 /H208492003 /H20850. 9Y. Huai, J. Zhang, G. W. Anderson, P. Rana, S. Funada, C.-Y. Hung, M. Zhao, and S. Tran, J. Appl. Phys. 85, 5528 /H208491999 /H20850. 10A. T. McCallum and S. E. Russek, Appl. Phys. Lett. 84,3 3 4 0 /H208492004 /H20850. FIG. 2. The effect of the sense current direction on the free layer /H20849FL/H20850.F o r current direction /H20849a/H20850on the left-hand side, the current field Hcurrpartially cancels the stray field HSof the pinned layer /H20849PL/H20850. For the current direction /H20849b/H20850on the right-hand side, the demagnetizing field of the pinned layer is supported by the current field. FIG. 3. The relative change in resistance over the external applied field. Thecurrent directions /H20849a/H20850and /H20849b/H20850generate a field at the free layer, which is antiparallel or parallel to the demagnetizing field of the pinned layer,respectively. FIG. 4. The output signal over time for a relaxation process from equilib-rium state at H ext=0.3 T to the ground state at Hext=0 T for different Gilbert damping constants.08S303-3 Ertl et al. J. Appl. Phys. 99, 08S303 /H208492006 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 138.37.211.113 On: Tue, 07 Oct 2014 08:01:04

1.5130452.pdf

AIP Advances 10, 025012 (2020); https://doi.org/10.1063/1.5130452 10, 025012 © 2020 Author(s).Highly (001) oriented MnAl thin film fabricated on CoGa buffer layer Cite as: AIP Advances 10, 025012 (2020); https://doi.org/10.1063/1.5130452 Submitted: 23 October 2019 . Accepted: 23 January 2020 . Published Online: 07 February 2020 Daiki Oshima , Takeshi Kato , and Satoshi Iwata COLLECTIONS Paper published as part of the special topic on 64th Annual Conference on Magnetism and Magnetic Materials Note: This paper was presented at the 64th Annual Conference on Magnetism and Magnetic Materials. ARTICLES YOU MAY BE INTERESTED IN Properties of magnetic tunnel junctions with a MgO/CoFeB/Ta/CoFeB/MgO recording structure down to junction diameter of 11 nm Applied Physics Letters 105, 062403 (2014); https://doi.org/10.1063/1.4892924 Spintronics with compensated ferrimagnets Applied Physics Letters 116, 110501 (2020); https://doi.org/10.1063/1.5144076 Ultrafast magnetization switching by spin-orbit torques Applied Physics Letters 105, 212402 (2014); https://doi.org/10.1063/1.4902443AIP Advances ARTICLE scitation.org/journal/adv Highly (001) oriented MnAl thin film fabricated on CoGa buffer layer Cite as: AIP Advances 10, 025012 (2020); doi: 10.1063/1.5130452 Presented: 6 November 2019 •Submitted: 23 October 2019 • Accepted: 23 January 2020 •Published Online: 7 February 2020 Daiki Oshima,1,a) Takeshi Kato,2 and Satoshi Iwata1 AFFILIATIONS 1Advanced Measurement Technology Center, Institute of Materials and Systems for Sustainability, Nagoya University, Nagoya, Aichi 464-8603, Japan 2Department of Electronics, School of Engineering, Nagoya University, Nagoya, Aichi 464-8603, Japan Note: This paper was presented at the 64th Annual Conference on Magnetism and Magnetic Materials. a)E-mail: oshima@nuee.nagoya-u.ac.jp ABSTRACT 5 nm- and 15 nm-thick (001) oriented MnAl films were fabricated on CoGa buffer layers with various thermal treatments. The insertion of the CoGa layer was effective to obtain the square out-of-plane hysteresis loop even in the MnAl thickness of 5 nm. Highly (001) oriented MnAl film was obtained by depositing Mn and Al on CoGa at a substrate temperature of 200○C followed by annealing at 500○C. The perpendicular magnetic anisotropy was estimated to be 7.4 ±0.2 and 8.5 ±0.4 Merg/cc for 5 nm- and 15 nm-thick MnAl, respectively. Lower anisotropy in 5 nm-thick MnAl may be due to the interdiffusion between the MnAl and CoGa layers. ©2020 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/1.5130452 .,s I. INTRODUCTION Mn-Ga1–3and Mn-Al4alloys are quite attractive for applica- tions to permanent magnet, magnetic recording media, and spin transfer torque magnetic random access memory (STT-MRAM), since they are made out of inexpensive and abundant elements as well as they have large uniaxial magnetic anisotropy, high spin polarization, low saturation magnetization and small Gilbert damp- ing.5–9Especially, Mn-Ga alloys such as L1 0-MnGa, are promising candidates as a ferromagnetic electrode for STT-MRAM because small Gilbert damping and large uniaxial anisotropy are required to achieve a sufficient thermal stability for data retention and a low writing current for low power operation simultaneously. L1 0- MnAl whose crystal structures similar to L1 0-MnGa, also has a large perpendicular anisotropy.10,11L10-MnAl films were grown on MgO(001) substrates with Cr-based alloy buffer layers to obtain large perpendicular anisotropy.12–15(001) oriented L1 0-MnAl films are known to grow epitaxially on the Cr-based alloys and reported to exhibit large perpendicular magnetic anisotropies of ∼107 erg/cc. However, magnetic properties of MnAl films on Cr-based alloy buffers with the thickness <10 nm have not been reported, which is necessary to consider this material for the application toSTT-MRAM. Reduction of the MnAl thickness will result in the degradation of its perpendicular magnetic anisotropy. Recently, it was reported that MnGa films grown on the CoGa buffer layers exhibited a square hysteresis loop with perpendicular easy axis even in the MnGa thickness of 1 nm, and surprisingly, the thin MnGa films were grown at room temperature.16–18Since both MnGa and MnAl have L1 0phase with similar lattice constants, the CoGa buffer will also be effective to realize highly oriented MnAl films with the thickness of several nm. In this study, we first report the growth of thin MnAl films on the CoGa buffer layers, where growth and post-annealing temperatures were varied to find the optimum growth condition. II. EXPERIMENTAL METHOD The samples were prepared on MgO(001) single crys- tal substrates by RF magnetron sputtering. The stack was Cr(2 nm)/MnAl(5 or 15 nm)/CoGa(0 or 30 nm)/Cr(20 nm)/MgO sub. The Ar gas pressure was 0.4 Pa for all the deposition. A Co40Ga60target was used for the deposition of the CoGa layer. The MnAl layer was fabricated by co-sputtering of Mn and Al targets and the nominal composition of the MnAl layer derived from the AIP Advances 10, 025012 (2020); doi: 10.1063/1.5130452 10, 025012-1 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv FIG. 1 . (a), (c), and (e) AFM images and (b), (d), and (f) M-Hcurves of 5 nm-thick MnAl films grown on CoGa layers with Ts= [(a) and (b)] 200, [(c) and (d)] 300, and [(e) and (f)] 400○C. Full scale of the height is 10 nm. sputtering rates of Mn and Al was 50:50 at.%. Both Cr layer and CoGa layer were deposited at 400○C and subsequently annealed at 600○C for 30 min. The MnAl layer was deposited at Ts, and some of the samples were annealed at Ta= 500○C for 30 min after the deposition. Finally, the Cr layer was deposited as a protective layer. Magnetic properties were measured by an alter- nating gradient field magnetometer. Crystal structures were charac- terized by X-ray diffractometer (XRD) with Cu K αradiation. Sur- face morphologies were observed by an atomic force microscope (AFM). A superconducting quantum interference device magne- tometer combining vibrating sample magnetometer (SQUID-VSM) was used for measuring magnetization curves under high magnetic fields. III. RESULTS AND DISCUSSIONS Figure 1 shows (a), (c), and (e) AFM images and (b), (d), and (f)M-H curves of 5 nm-thick MnAl films grown on the CoGa layer with Ts= [(a) and (b)] 200, [(c) and (d)] 300, and [(e) and (f)] 400○C. These samples were not annealed after the deposition of MnAl layer. Step signals observed in the M-Hcurves near zero magnetic field originate from the CoGa layer which exhibits a small magnetization depending on the annealing condition. The average roughness Ra of MnAl, which is obtained by calculating an average of the ver- tical deviation of each point from the mean height in the AFM image, increased from 0.52 to 2.2 nm with increasing Ts, and the island growth of MnAl was confirmed for Ts≥300○C. The hystere- sis loop of MnAl was also sensitive to Ts. The coercivity of MnAl increased with increasing Ts, and the abrupt change of magneti- zation near coercivity was not observed for Ts= 400○C. This sug- gests that the island structures act as pinning centers of the domain walls and prevents the smooth propagation of domain walls in the samples.Figure 2 shows out-of-plane XRD profiles of the 5 nm-thick MnAl films grown at Ts= 200, 300, 400○C on the CoGa buffer layer. Dotted lines indicate the 001 and 002 peak positions of bulk L1 0- MnAl. 001 and 002 peaks of CoGa and MnAl, where 002 peak of Cr overlaps to 002 CoGa, were observed for all the films, indicat- ing all layers are grown on the MgO substrate with (001) orien- tation. The appearance of the 001 peak of MnAl means the exis- tence of ordered L1 0-MnAl phase. The positions of 001 and 002 peaks largely deviate from those of bulk L1 0-MnAl. One may note that MnAl 002 peak shifts toward lower angle with increasing Ts, whereas 001 peak does not shift as much as the 002 peak. The rea- son is considered as follows. According to the previous report on the growth of MnGa on CoGa,18MnGa first grows pseudomorphically on the CoGa layer up to a certain thickness, and then the growth mode changes to Stranski–Krastanov (SK) mode.19In this case, FIG. 2 . Out-of-plane XRD profiles of 5 nm-thick MnAl films grown on CoGa layers atTs= 200, 300, and 400○C. The profiles are shown in logarithm scale. Dotted lines indicate the peak positions of bulk L1 0-MnAl. AIP Advances 10, 025012 (2020); doi: 10.1063/1.5130452 10, 025012-2 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv FIG. 3 . (a), (c), and (e) AFM images and (b), (d), and (f) M-Hcurves of 5 nm-thick MnAl films grown on [(a) and (b)] Cr and (c)-(f) CoGa layers at Ts= 200○C. The sample [(e) and (f)] was annealed at Ta = 500○C after the deposition. Full scale of the height is 10 nm. there exist two types of MnGa layers through the thickness, i.e., the pseudomorphically- and 3-dimensionally-grown MnGa layers. Sim- ilar growth mode will be observed in the present MnAl films. One may note that there are two peaks at 2 θ∼53 and 57○for the sam- ple with Ts= 200 and 400○C, which may correspond to 002 peaks of the two layers: pseudomorphically- and 3-dimentionally-grown MnAl. If we assume L1 0phase is existed only in the 3-dimensionally- grown layer, 001 peak is considered to come only from the 3- dimensionally-grown layer in which the strain is gradually relaxed, and the peak position will be close to that of bulk L1 0-MnAl. From the peak position of MnAl 001 of the 3-dimensionally-grown layer, it is considered that 002 peak from the 3-dimensionally-grown layer appears at lower angle side and that from the pseudomorphically- grown layer appears at higher angle side. Increase of Tschanged the ratio of the two peaks, indicating the change of the volume fraction between pseudomorphically- and 3-dimensionally-grown layers. Figure 3 shows (a), (c), and (e) AFM images and (b), (d), and (f)M-Hcurves of 5 nm-thick MnAl film grown (a) and (b) on the Cr layer and (c)-(f) on the CoGa layer with Ts= 200○C. No post-annealing was performed on the samples shown in Fig. 3 (a)– (d), whereas the sample shown in Fig. 3(e), (f) was annealed at Ta = 500○C after the MnAl deposition. All the films were confirmed to exhibit perpendicular magnetic anisotropy. For the MnAl film grown on the Cr layer, recesses on surface were observed in the AFM image, and small magnetization and large coercivity compared to the film grown on CoGa were confirmed. On the other hand, the post- annealed MnAl film grown on the CoGa exhibited square-shape hysteresis with large remanence Mr= 420 emu/cc ( Mrwas almost the same as the saturation magnetization of MnAl if we neglect the contribution from the CoGa layer near zero magnetic field) and smaller coercivity than that of the film without post-annealing. The surface flatness of the MnAl film was also improved from Ra= 0.52 to 0.27 nm by the post-annealing.Figure 4 shows XRD profiles of the 5 nm-thick MnAl films on Cr and CoGa layers grown at Ts= 200○C. One of the MnAl films grown on the CoGa layer was post-annealed at Ta= 500○C. MnAl 001 and 002 peaks were observed in all the samples, indicating L1 0 phase MnAl was grown with (001) orientation. This is consistent with the perpendicular anisotropy observed in hysteresis loops of all samples. The 001 and 002 peak intensities for the film grown on the CoGa layer were larger than those grown on the Cr layer, and further increase of the peak intensities was confirmed for the post-annealed sample. However, the peak position did not change significantly with the fabrication condition, which suggests similar growth mode in all samples. The MnAl layer on Cr may grow in SK mode similar to the case grown on the CoGa layer. However, the intensities of 001 and 002 peaks were smaller than those grown on the CoGa layer, which will be due to poor (001) orientation of MnAl on Cr. This FIG. 4 . Out-of-plane XRD profiles of 5 nm-thick MnAl films grown on Cr and CoGa layers at Ts= 200○C. One of the MnAl films grown on the CoGa layer was post-annealed at Ta= 500○C. The profiles are shown in linear scale. Dotted lines indicate the peak positions of bulk L1 0-MnAl. AIP Advances 10, 025012 (2020); doi: 10.1063/1.5130452 10, 025012-3 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv will be also related to low magnetization and squareness in the M-H curve of MnAl on Cr (see Fig. 3 (b)). The post-annealing of the MnAl film grown on the CoGa layer enhances the (001) orientation of the MnAl film, which is the reason of the improvement of the magnetic property shown in Fig. 3 (f). Figure 5 shows [(a) and (c)] AFM images and [(b) and (d)] M-H curves of 15 nm-thick MnAl films grown on [(a) and (b)] CoGa and [(c) and (d)] Cr layers at Ts= 200○C followed by annealing at Ta = 500○C. The insertion of the CoGa layer was also effective to improve the magnetic property of the thick MnAl film. How- ever, the magnetization reversal of 15 nm-thick MnAl on CoGa was not smooth compared to the case of 5 nm-thick MnAl. As mentioned above, we consider two layers: psuedomorphically- and 3-dimensionally-grown layers, exist in the MnAl film, and thicker film will have thicker 3-dimensionally-grown layer. The 3- dimensionally-grown layer will have defects, which will prevent the smooth propagation of domain walls. Figure 6 shows (a) out-of-plane and [(b) and (c)] in-plane XRD profiles of 5 nm- and 15 nm-thick MnAl films grown on the CoGa layer at Ts= 200○C followed by annealing at Ta= 500○C. Dotted lines indicate the peak positions of a bulk L1 0-MnAl. In out-of-plane XRD profiles, MnAl 001 and 002 peaks were clearly observed and the lattice parameters cwere estimated to be 0.328 and 0.346 nm for 5 nm- and 15 nm-thick MnAl, respectively. MnAl 200 and 220 peaks were confirmed as shoulders at high angle side of Cr, CoGa 110 and 200, respectively, in in-plane XRD profiles. For 5 nm-thick MnAl, these peaks were not visible as shown in Fig. 6 (b) and (c). One of the reasons is the overlap of these peaks with the Cr, CoGa 110 and 200 peaks. If we assume the overlap of the peaks, the lat- tice parameters aof 5 nm- and 15 nm-thick MnAl are estimated to bea= 0.408 ±0.04 and 0.400 ±0.03 nm, respectively. Using these val- ues, the unit cell volumes of MnAl for both cases are calculated to FIG. 5 . (a) and (c) AFM images and [(b) and (d)] M-Hcurves of 15 nm-thick MnAl films grown on [(a) and (b)] CoGa layers and [(c) and (d)] Cr layers at Ts= 200○C followed by post-annealing at Ta= 500○C. Full scale of the height is 10 nm. FIG. 6 . (a) out-of-plane and (b), (c) in-plane XRD profiles of 5 nm- and 15 nm- thick MnAl films grown on CoGa layers at Ts= 200○C followed by annealing at Ta= 500○C. In-plane profiles show the diffractions from the planes parallel to (b) MgO(100) and (c) MgO(110). bea2c∼0.055 nm3, which agrees well with the unit cell volume a2c = 0.0552 nm3of a bulk L1 0-MnAl. This suggests that the MnAl film grown on CoGa is significantly strained due to the lattice mismatch between MnAl and CoGa, while conserving the unit cell volume. Similar dependence of the lattice parameters and unit cell volume on the thickness are also reported in the previous paper for MnGa on CoGa.18The long-range order parameter S, which describes a degree of ordering of the atomic arrangement, was estimated from the integral intensity ratio of MnAl 001 and 002 peaks. The param- eters Swere∼0.85 and 1.0 for 5 nm- and 15 nm-thick MnAl, respectively. Finally, perpendicular magnetic anisotropy constants of the (001) oriented MnAl were estimated by using SQUID-VSM mea- surements. Figure 7 shows M-Hcurves of (a) 5 nm- and (b) 15 nm- thick MnAl films grown on the CoGa layers at Ts= 200○C followed by annealing at Ta= 500○C. The hysteresis seen in the in-plane M-Hcurve shown in Fig. 7(a) is due to the sample tilting dur- ing the measurement. The effective anisotropy field Hkeffestimated from the in-plane curve was Hkeff= 30.3 ±0.6 kOe for 5 nm-thick MnAl and Hkeff= 27.5 ±1.5 kOe for 15 nm-thick MnAl. The per- pendicular anisotropy constant Kuwas calculated as Ku=MsHkeff/2 + 2πMs2, and it was Ku= 7.4±0.2×106erg/cc for 5 nm-thick MnAl and Ku= 8.5±0.4×106erg/cc for 15 nm-thick MnAl. Those values are slightly lower than the bulk value of ∼107erg/cc. The- oretical calculations20,21are reported that the order parameter and AIP Advances 10, 025012 (2020); doi: 10.1063/1.5130452 10, 025012-4 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv FIG. 7 . SQUID-VSM measurements of (a) 5 nm- and (b) 15 nm-thick MnAl films grown on CoGa layers at Ts= 200○C followed by annealing at 500○C. The magnetization is normalized so that the saturation magnetization becomes a unity. c/aratio affects the magnetic moment of Mn atoms and Ku. When c/aratio decreases from 0.865 (15 nm) to 0.804 (5 nm), the Mn moment decreases form 2.38 μBto 2.25 μBwhile Kuincreases from 1.9 to 2.4 ×107erg/cc.20Based on the theoretical calculations, pseudomorphically-grown layer with small c/aratio will have large perpendicular anisotropy compared to 3-dimensionally-grown layer with large c/aratio. However, 15 nm-thick MnAl, which may have thicker 3-dimensionally grown layer, exhibited large Ku. The dif- ference in the order parameter between 5 nm- and 15 nm-thick MnAl should be taken into account, but it was 1.0 for 15 nm-thick MnAl and 0.85 for 5 nm-thick MnAl, which will reduces Kuby 10%.21Therefore, the difference in MsandKubetween the 5 nm- and 15 nm-thick MnAl films can not be explained only by the vari- ation of the order parameter and c/aratio (growth mode). One possible mechanism to explain the decrease in Kuand Msis the interdiffusion between the MnAl and CoGa layers which was not considered in theoretical calculations. In the previous study, the substitution of Al atoms by Ga or Co atoms was reported to mod- ifies Kuof MnAl films,22and the substitution by Ga increased Ku, while the substitution by Co decreased Ku. Therefore, the diffu- sion of Co atoms of the CoGa layer into the MnAl layer during the deposition or post-annealing will explain the reduction of Ku in 5 nm-thick MnAl. Further optimization of the thermal treatment may improve the magnetic property of thin MnAl grown on CoGa layers. IV. CONCLUSION 5 nm- and 15 nm-thick MnAl films were fabricated by using CoGa buffer layers and highly (001) oriented films were obtained by optimizing the condition of thermal treatments. 5 nm-thick MnAl film grown at a substrate temperature of 200○C on the CoGa lay- ers exhibited a square-shape hysteresis loop with perpendicular easy axis. Subsequent annealing of the film at 500○C improved (001) ori- entation and magnetic properties of the MnAl film. For 15 nm-thick MnAl films, the CoGa buffer layer was also effective to obtain highly (001) oriented MnAl with L1 0ordered structure. The perpendicular magnetic anisotropy constants were estimated to be 7.4 ±0.2×106 for 5 nm- and 8.5 ±0.4×106for 15 nm-thick MnAl. The decrease in the perpendicular anisotropy with decreasing the thickness of the MnAl film may be due to the interdiffusion between the MnAl and CoGa layers, and further improvement of the perpendicular mag- netic anisotropy in thin MnAl will be expected by suppressing the interdiffusion.ACKNOWLEDGMENTS The authors thank Mr. Kumazawa for his assistance with some experiments, and Prof. Tsukamoto and Dr. Yoshikawa of Nihon University for SQUID-VSM measurements. This work was sup- ported in part by JSPS KAKENHI (Grant Numbers 17H03249, 19K15044), Tatematsu Foundation, and the Project of Creation of Life Innovation Materials for Interdisciplinary and International Researcher Development of the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan. A part of this work was performed under the Research Program of “Dynamic Alliance for Open Innovation Bridging Human, Environment and Mate- rials” in the “Network Joint Research Center for Materials and Devices.” REFERENCES 1M. Hasegawa and I. Tsuboya, “Magnetic properties of ηphase in Mn-Ga system,” J. Phys. Soc. Jpn. 20, 464 (1965). 2I. Tsuboya and M. Sugihara, “Magnetic properties of Mn-Ga alloys with a high coercive force,” J. Phys. Soc. Jpn. 20, 170 (1965). 3T. A. Bither and W. H. Cloud, “Magnetic tetragonal phase in the Mn-Ga binary,” J. Appl. Phys. 36, 1501 (1965). 4H. Kono, “On the ferromagnetic phase in manganese-aluminum system,” J. Phys. Soc. Jpn. 13, 1444–1451 (1958). 5B. Balke, G. H. Fecher, J. Winterlik, and C. Felser, “Mn 3Ga, a compensated fer- rimagnet with high Curie temperature and low magnetic moment for spin torque transfer applications,” Appl. Phys. Lett. 90, 152504 (2007). 6H. Kurt, K. Rode, M. Venkatesan, P. Stamenov, and J. M. D. Coey, “High spin polarization in epitaxial films of ferrimagnetic Mn 3Ga,” Phys. Rev. B 83, 020405 (2011). 7L. J. Zhu, D. Pan, S. H. Nie, J. Lu, and J. H. Zhao, “Tailoring magnetism of mul- tifunctional Mn xGa films with giant perpendicular anisotropy,” Appl. Phys. Lett. 102, 132403 (2013). 8F. Wu, S. Mizukami, D. Watanabe, H. Naganuma, M. Oogane, Y. Ando, and T. Miyazaki, “Epitaxial Mn 2.5Ga thin films with giant perpendicular magnetic anisotropy for spintronic devices,” Appl. Phys. Lett. 94, 122503 (2009). 9S. Mizukami, F. Wu, A. Sakuma, J. Walowski, D. Watanabe, T. Kubota, X. Zhang, H. Naganuma, M. Oogane, Y. Ando, and T. Miyazaki, “Long-lived ultrafast spin precession in manganese alloys films with a large perpendicular magnetic anisotropy,” Phy. Rev. Lett. 16, 117201 (2011). 10S. H. Nie, L. J. Zhu, J. Lu, D. Pan, H. L. Wang, X. Z. Yu, J. X. Xiao, and J. H. Zhao, “Perpendicularly magnetized τ-MnAl (001) thin films epitaxied on GaAs,” Appl. Phys. Lett. 102, 152405 (2013). 11C. Navío, M. Villanueva, E. Céspedes, F. Mompeán, M. G. Hernández, J. Camarero, and A. Bollero, “Ultrathin films of L1 0-MnAl on GaAs (001): A hard magnetic MnAl layer onto a soft Mn-Ga-as-Al interface,” APL Mater. 6, 101109 (2018). 12M. Hosoda, M. Oogane, M. Kubota, T. Kubota, H. Saruyama, S. Iihama, H. Naganuma, and Y. Ando, “Fabrication of L1 0-MnAl perpendicularly magne- tized thin films for perpendicular magnetic tunnel junctions,” J. Appl. Phys. 111, 07A324 (2012). 13H. Saruyama, M. Oogane, Y. Kurimoto, H. Naganuma, and Y. Ando, “Fabri- cation of L1 0-ordered MnAl films for observation of tunnel magnetoresistance effect,” Jpn. J. Appl. Phys. 52, 063003 (2013). 14M. Oogane, K. Watanabe, H. Saruyama, M. Hosoda, P. Shahnaz, Y. Kurimoto, M. Kubota, and Y. Ando, “L1 0-ordered MnAl thin films with high perpendicular magnetic anisotropy,” Jpn. J. Appl. Phys. 56, 0802A2 (2017). 15M. S. Parvin, M. Oogane, M. Kubota, M. Tsunoda, and Y. Ando, “Epitaxial L1 0- MnAl thin films with high perpendicular magnetic anisotropy and small surface roughness,” IEEE Trans. Mag. 54, 3401704 (2018). 16K. Z. Suzuki, R. Ranjbar, A. Sugihara, T. Miyazaki, and S. Mizukami, “Room temperature growth of ultrathin ordered MnGa films on a CoGa buffer layer,” Jpn. J. Appl. Phys. 55, 010305 (2016). AIP Advances 10, 025012 (2020); doi: 10.1063/1.5130452 10, 025012-5 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv 17K. Z. Suzuki, R. Ranjbar, J. Okabayashi, Y. Miura, A. Sugihara, H. Tsuchiura, and S. Mizukami, “Perpendicular magnetic tunnel junction with a strained Mn- based nanolayer,” Sci. Rep. 6, 30249 (2016). 18K. Kunimatsu, K. Z. Suzuki, and S. Mizukami, “Pseudomorphic deposi- tion of L1 0MnGa nanolayers at room temperature,” J. Cryst. Growth 514, 8 (2019). 19I. N. Stranski and V. L. Krastanow, “Zur theorie der orientierten ausscheidung von ionenkristallen aufeinander,” Akad. Wiss. Lit. Mainz Math.-Natur. Kl. IIb 146, 797 (1939).20A. Sakuma, “Electronic structure and magnetocrystalline anisotropy energy of MnAl,” J. Phys. Soc. Jpn. 63, 1422 (1994). 21Y. Kota and A. Sakuma, “Relationship between magnetocrystalline anisotropy and orbital magnetic moment in L10-type ordered and disordered alloys,” J. Phys. Soc. Jpn. 81, 084705 (2012). 22S. Zhao, Y. Wu, Z. Jial, Y. Jia, Y. Xu, J. Wang, T. Zhang, and C. Jiang, “Evolution of intrinsic magnetic properties in L1 0Mn-Al alloys doped with substitutional atoms and correlated mechanism: Experimental and theoretical studies,” Phys. Rev. Appl. 11, 064008 (2019). AIP Advances 10, 025012 (2020); doi: 10.1063/1.5130452 10, 025012-6 © Author(s) 2020

1.373225.pdf

High frequency effects in perpendicular recording media S. J. Greaves, H. Muraoka, Y. Sugita, and Y. Nakamura Citation: Journal of Applied Physics 87, 4990 (2000); doi: 10.1063/1.373225 View online: http://dx.doi.org/10.1063/1.373225 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/87/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Changes in switching fields of CoCrPt – SiO 2 perpendicular recording media due to Ru intermediate layer under low and high gas pressures J. Appl. Phys. 105, 013926 (2009); 10.1063/1.3065525 The effects of oxygen on intergranular exchange and anisotropy dispersion in Co ∕ Pd multilayers for perpendicular magnetic recording media J. Appl. Phys. 99, 08E708 (2006); 10.1063/1.2162487 Reducing average grain and domain size in high-coercivity Co ∕ Pd perpendicular magnetic recording media through seedlayer engineering J. Appl. Phys. 97, 10N118 (2005); 10.1063/1.1855206 Recording performance and magnetization switching of CoTb/CoCrPt composite perpendicular media J. Appl. Phys. 91, 8058 (2002); 10.1063/1.1452273 CrPt 3 thin film media for perpendicular or magneto-optical recording J. Appl. Phys. 85, 4307 (1999); 10.1063/1.370351 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 137.149.200.5 On: Mon, 24 Nov 2014 16:26:48High frequency effects in perpendicular recording media S. J. Greaves,a)H. Muraoka, Y. Sugita, and Y. Nakamura RIEC, Tohoku University, Katahira 2-1-1, Aoba ku, Sendai 980-8577, Japan Simulations of perpendicular recording media were carried out to determine the effect of switching speed limitations upon recording performance. Simulations of hysteresis loops at various appliedfield sweep rates show that the onset of a switching speed limited increase in coercivity occurs forsweep rates in excess of 2 310 13Oe/s. Switching speeds of individual grains were found to be around 18 to 38 ps, depending on the medium thickness and the magnetization state of surroundinggrains. Recording simulations show that for media thicknesses of up to 180 Å, writing frequenciesof 5 Gbit/s are feasible before loss of output occurs. © 2000 American Institute of Physics. @S0021-8979 ~00!58508-9 # I. INTRODUCTION As recording data rates approach 1 Gb/s an understand- ing of high frequency phenomena in recording media is nec-essary. Since a finite time is required for the magnetizationof a grain to reverse, an upper limit to the recording fre-quency exists. Previous studies have mainly concentrated onlongitudinal media. 1–4In this paper various high frequency effects related to magnetization switching in perpendicularmedia will be discussed. II. SIMULATIONS A micromagnetic model based on the Landau–Lifshitz– Gilbert ~LLG!equation was used for the simulations. Calcu- lations of hysteresis loops, individual particle switching andrecording at various frequencies were carried out. In eachcase the same magnetic properties of saturation magnetiza-tion,M s, equal to 300 emu/cc and uniaxial anisotropy, Ku, of 1 3106erg/cc, oriented perpendicular to the film plane were used and the damping constant, a, was 0.05. Thermal fluctuations were introduced by means of random field termsadded to the effective field at each time step. The thermalfluctuations ensure that there is always a torque acting on themagnetic moments by preventing them from aligning withthe applied field. The parameters for each of the individualsimulations are shown in Table I. III. RESULTS A range of applied field sweep rates was used for the calculation of hysteresis loops. The sweep rate is given byR5dH/dtwheredHis the change in applied field at each cycle of the LLG equation and dtis the time step. Thus, the applied field variation with time formed a triangle-like wave-form. Figure 1 depicts the coercivity versus sweep rate formedia with various exchange coupling constants. To discussthe results we categorize granular media as those with ex-change coupling constants from zero to 5 310 28erg/cm and exchange coupled media as having Avalues from 5 31027erg/cm to 1 31026erg/cm. We find that at low sweep rates the coercivity of the exchange coupled media islower than that of the granular media, which is in agreement with experiments. The granular media has a coercivity closetoH k(2Ku/Ms), or6.66kOeatlowsweeprates.Inaddition, the variation of coercivity for sweep rates over the range 1011 to 1013Oe/s is much greater for the exchange coupled media than for the granular media. This is because the hysteresisloops of the exchange coupled media are square, whereasthose of the granular media are sheared with a slope of1/4 pMs. This makes the exchange coupled media more sus- ceptible to thermal fluctuations in the vicinity of Hc, result- ing in a larger variation of Hcwith sweep rate. Once the sweep rate exceeds 2 31013Oe/s the coercivities of all media show a rapid increase. This occurs because the switchingspeed limit of the media has been reached rather than as aresult of thermal fluctuations. The difference in coercivitiesamong the media in this regime is less pronounced than atlower sweep rates, with the exchange coupled media havingthe highest coercivity. We can calculate the switching fre-quency at each sweep rate, f s(R), using the relationship a!Electronic mail: simon@kiroku.riec.tohoku.ac.jpTABLE I. Parameters used in the simulation. Hysteresis loops at various sweep rates Time step 0.568 ps Temperature 300 K Exchange constant Zero to 1 31026erg/cm Cell size 100Å 3100Å 3300Å Field sweep rate 3.5 31010to 3.5 31014Oe/s Individual grain reversal time Time step 0.0114 ps Temperature 0.1 K Exchange constant Zero between grains 131026erg/cm in grains Cell size 14Å 314Å 314Å Grain size 98Å 398Å 3dÅ Recording of tracks at various frequencies Time step 0.284 ps Temperature 300 K Exchange constant Zero between grains 131026erg/cm in grains Cell size 100Å 3100Å 360Å Media thickness 60 to 300 ÅWrite head dimensions 2100Å 3700Å, single pole Head-disc spacing 200 ÅJOURNAL OF APPLIED PHYSICS VOLUME 87, NUMBER 9 1 MAY 2000 4990 0021-8979/2000/87(9)/4990/3/$17.00 © 2000 American Institute of Physics [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 137.149.200.5 On: Mon, 24 Nov 2014 16:26:48fs(R)5R/2Hc(R), which, for a field sweep rate of 2 31013Oe/s and a coercivity of about 7 kOe, corresponds to a switching frequency of about 1.4 Gbit/s. The value of fs increases with field sweep rate but at the expense of higher coercivity. For a sweep rate of about 5 31014Oe/s the coer- civity reaches 20 kOe, which is near the limit of availablewrite head fields. Using these numbers we obtain a value for f sof 12.5 Gbit/s. For the calculations of particle switching speeds an array of nine particles in three initial configurations was consid-ered as shown in Fig. 2. The particles were discretized intocubes and the length varied by adding extra cubes. A 9 kOefield was applied to the center particle only and the switchingtime of the particle was calculated, defined here as the timetaken for the magnetization to change from 190% ofM sto 290% ofMs. The results of the switching time simulations for the three configurations are shown in Fig. 3. We find that for thesaturated case of Fig. 2 the switching time is roughly propor-tional to the medium thickness. The switching times rangefrom 18 to 25 ps, depending upon the film thickness. Theswitching times increase significantly for the other two mag-netization configurations, with the slowest switching beingobserved for the reversed case of Fig. 2. In this configurationthe initial state is most stable and to switch the center particlerequires overcoming the demagnetizing field of the surround-ing particles. These results were calculated at a temperatureof 0.1 K, increasing the temperature results in a distributionof switching times with the same average time as shown inFig. 3. The switching times increase significantly if the defi-nition of switching time is changed to 199% to 299% ofM s. In this case the switching times for a 504 Å particle increase to 60, 92 and 144 ps for the saturated, chessboardand reversed cases, respectively. Plots of the hysteresis loopof the center particle show that the effect of the surroundingparticles, if they are uniformly magnetized in the same direc-tion, is to shift the hysteresis loop by about 1.5 kOe. Figure 4 shows the rate of magnetization switching, dM/dt, versus time for the 504 Å medium. The external field is applied at t511.4ps, after which there is a latent time before reversal begins. This latent time increases withmedium thickness and also depends upon the state of thesurrounding particles, but the switching always takes place inthe order shown in the figure. Raising the temperature of theparticles reduces the latent time and the order in which theparticles switch is randomized. The peak value of dM/dtis always greatest for the saturated case and lowest for the re-versed case, although the maximum switching rate valuesconverge as the medium thickness increases. Figure 5 shows the zcomponent of magnetization with depth in 504 Å long particles for the saturated and reversedcases at various times during the switching process. Themagnetization is switching from M z51MstoMz52Ms, but the time interval between each curve is not constant. Forthe saturated case ~Fig. 5, left !the reversal process is initi- ated by rotation of the magnetic moment at both ends of theparticle ~i.e., at 0 and 500 Å !. The magnetization at the ends FIG. 1. Effect of field sweep rate on coercivity for various exchange cou- pling constants. FIG. 2. Magnetization configuration of particles prior to application ofpulsed field. FIG. 3. Switching speeds for particles subject t oa9k O ep u lsed field. The surrounding particles are magnetized as shown in Fig. 2. FIG. 4.d/M/dt~divided by Ms!versus time for the 504 Å thick media. The field is applied at t511.4ps.4991 J. Appl. Phys., Vol. 87, No. 9, 1 May 2000 Greaveset al. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 137.149.200.5 On: Mon, 24 Nov 2014 16:26:48of the particle always leads the magnetization in the center during the reversal process. In the reversed case, magnetiza-tion reversal is again initiated at the ends of the particle,although the lag between the magnetization in the center ofthe particle and at the ends is much less. When M zbecomes less than 20.5Msthe magnetization in the center of the par- ticle takes the lead over that at the ends and continues to leaduntil saturation. Figure 6 shows two tracks written at a density of 254 kfci and a frequency of 1.76 Gbit/s: the media were initiallydemagnetized. Compared with the 60 Å medium, where thenoise is concentrated at the transitions between bits, the trackin the 300 Å medium is narrower and within the written bitsmany reversed grains can be observed. The number of re-versed grains increases with medium thickness, leading to a decrease in output. The reversed grains occur in the center ofthe bits because with a single pole write head the in-planecomponent of head field, which is responsible for initiatingreversal of the magnetization, is a minimum directly underthe head. Reversal begins in an annulus around the edge ofthe head and the last part of the bit to reverse is the centralarea. This situation is analogous to the reversed case of Fig.2 where reversal is opposed by the magnetostatic field of thesurrounding particles and takes the longest time. Thus, as thewriting frequency increases, deterioration in the recordedtracks first becomes apparent in the center of the recordedbits. Figure 7 shows the output, calculated from the average magnetization of the bits, versus writing frequency for vari-ous media thicknesses. At all frequencies the output de-creases as the medium thickness increases. However, for fre-quencies up to 5 Gbit/s the difference in output for mediathicknesses of 60 to 180 Å is small. A rolloff of outputoccurs above 5 Gbit/s, decreasing to zero at around 30Gbit/s. This corresponds to a switching speed of about 33 ps,in good agreement with the calculated switching times forthe single particles. IV. CONCLUSIONS The switching speed of individual particles was found to be as fast as 18 ps in a thin, 56 Å medium. Recording simu-lations show that a perpendicular medium of up to 180 Åthickness can be written to at frequencies of up to 5 Gbit/sbefore there is a significant loss of output. This assumes asquare wave head field. The output is likely to decrease fur-ther as the rise time of the head field increases. 1Q. Peng and H. N. Bertram, J. Appl. Phys. 81, 4384 ~1997!. 2H. Fang and J. G. Zhu, IEEE Trans. Magn. 32, 3584 ~1996!. 3H. N. Bertram and Q. Peng, IEEE Trans. Magn. 34, 1543 ~1998!. 4E. D. Boerner and H. N. Bertram, IEEE Trans. Magn. 33, 3052 ~1997!. FIG. 5. The Zcomponent of magnetization with depth in the center particle of Fig. 2 during the switching process. Left: saturated case, right: reversedcase. The time is increasing from the top to the bottom although the con-tours are not equally spaced in time. FIG. 7. Output vs write frequency for various media thicknesses. FIG. 6. Written tracks for 60 and 300 Å thick media. Depicted area 50.6 32mm2, write frequency 51.76Gbit/s, linear density 5254kfci.4992 J. Appl. Phys., Vol. 87, No. 9, 1 May 2000 Greaveset al. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 137.149.200.5 On: Mon, 24 Nov 2014 16:26:48

1.3067853.pdf

Anomalous stabilization in a spin-transfer system at high spin polarization Inti Sodemann and Ya. B. Bazaliy Citation: Journal of Applied Physics 105, 07D114 (2009); doi: 10.1063/1.3067853 View online: http://dx.doi.org/10.1063/1.3067853 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/105/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Dependence of spin-transfer switching characteristics in magnetic tunnel junctions with synthetic free layers on coupling strength J. Appl. Phys. 111, 07C905 (2012); 10.1063/1.3672240 Spin-transfer-induced magnetic domain formation J. Appl. Phys. 100, 073906 (2006); 10.1063/1.2357002 Spin-polarized current-driven switching in permalloy nanostructures J. Appl. Phys. 97, 10E302 (2005); 10.1063/1.1847292 Frequency modulation of spin-transfer oscillators Appl. Phys. Lett. 86, 082506 (2005); 10.1063/1.1875762 Spin-transfer effects in nanoscale magnetic tunnel junctions Appl. Phys. Lett. 85, 1205 (2004); 10.1063/1.1781769 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.170.6.51 On: Mon, 10 Aug 2015 23:53:01Anomalous stabilization in a spin-transfer system at high spin polarization Inti Sodemann1,a/H20850and Ya. B. Bazaliy1,2 1Department of Physics and Astronomy, University of South Carolina, Columbia, South Carolina 29208, USA 2Institute of Magnetism, National Academy of Science, Kyiv 03142, Ukraine /H20849Presented 13 November 2008; received 22 September 2008; accepted 3 November 2008; published online 12 February 2009 /H20850 Switching diagrams of nanoscale ferromagnets driven by a spin-transfer torque are studied in the macrospin approximation. We consider a disk-shaped free layer with in-plane easy axis and externalmagnetic field directed in plane at 90° to that axis. It is shown that this configuration is sensitive tothe angular dependence of the spin-transfer efficiency factor and can be used to experimentallydistinguish between different forms of g/H20849 /H9258/H20850, in particular, between the original Slonczewski form and the constant gapproximation. The difference in switching diagrams is especially pronounced at large spin polarizations, with the Slonczewski case exhibiting an anomalous region. © 2009 American Institute of Physics ./H20851DOI: 10.1063/1.3067853 /H20852 I. INTRODUCTION Spin polarized electric currents have been successfully used to switch the magnetization direction of nanoscale fer-romagnetic layers via the spin transfer effect. 1–6One of the questions of current-induced dynamics is the dependence ofspin-transfer efficiency, or Slonczewski factor g, on the angle between the polarization of incoming spin current and themagnetization direction. 7–10Such a dependence can be es- sential, and, for example, leads to the asymmetry betweenthe positive and negative switching currents. However, thereis still a lack of experimental tests for the precise functionalform of efficiency factor. It is expected that angular depen-dence of gwill become more important at high spin polar- izations where the constant efficiency approximation canfail, while constant gcan still be in good agreement with experimental results at low spin polarization. 11–14 Here we perform stability analysis for the equilibrium configurations of a bilayer spin-transfer device using theSlonczewski form for the efficiency factor and compare itwith a similar analysis that uses the constant efficiency ap-proximation. We observe that the switching diagram for theSlonczewski case displays a stability region and precessionalstates that are absent in the constant efficiency case. Theseanomalous regions become larger as the spin polarizationincreases. Our results may motivate further experimental ef-forts to directly measure the functional form of the efficiencyfactor at high spin polarizations. II. MACROSPIN DESCRIPTION OF THE DEVICE A typical device used to study the spin-transfer effect is a nanopillar, with two layers of ferromagnetic material sepa-rated by a normal paramagnetic metal /H20851see Fig. 1/H20849a/H20850/H20852. The magnetization of one layer /H20849polarizer /H20850is fixed and oriented along a unit vector s, while the magnetization of the other /H20849free layer /H20850,M=Mn, rotates and is described in the mac-rospin approximation by the Landau–Lifshitz–Gilbert equa- tion including the Slonczewski spin torque term, 1 n˙=/H9253 M/H20875−/H9254E /H9254n/H11003n/H20876+/H9253/H6036I 2eVMg/H20849/H9258,P/H20850/H20851n/H11003/H20849s/H11003n/H20850/H20852 +/H9251/H20851n/H11003n˙/H20852, /H208491/H20850 where /H9253is the gyromagnetic ratio, E/H20849n/H20850is the magnetic en- ergy of the free layer, and /H9251is the Gilbert damping constant. The strength of the spin torque is characterized by the effi-ciency factor g/H20849 /H9258,P/H20850, which depends on the angle /H9258between the magnetizations of the polarizer and the free layer and the degree of current spin polarization P/H33528/H208510,1 /H20852. In general the functional form of g/H20849/H9258,P/H20850is material and geometry dependent.7–10Here we will compare Slonczewski’s1form, g/H20849/H9258,P/H20850=1 fP/H20849/H9264P+ cos/H9258/H20850, /H208492/H20850 with/H9264P=3–4 /fP,fP=/H208491+P/H208503/4P3/2, and the g/H20849/H9258,P/H20850=const approximation. The magnetic energy of the free layer includes contribu- tions from the intrinsic anisotropy /H20849easy axis anisotropy with strength Haand direction aˆ/H20850, shape anisotropy /H20849easy plane anisotropy with normal vector pˆ/H20850, and the interaction energy with an external magnetic field H. Equation /H208491/H20850can be writ- ten as a/H20850Electronic mail: sodemann@physics.sc.edu.xzy SM H⊥a^n^ S NI(b) (a) θ ϕ FIG. 1. /H20849Color online /H20850/H20849a/H20850Typical nanopillar device with free layer on the top and polarizer at the bottom. /H20849b/H20850In-plane magnetic field configuration. Two stable directions of the free layer magnetization are labeled as N and S.JOURNAL OF APPLIED PHYSICS 105, 07D114 /H208492009 /H20850 0021-8979/2009/105 /H208497/H20850/07D114/3/$25.00 © 2009 American Institute of Physics 105 , 07D114-1 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.170.6.51 On: Mon, 10 Aug 2015 23:53:01n˙=/H9270/H20849n/H20850+/H9251n/H11003/H9270/H20849n/H20850, /H208493/H20850 where we have rescaled the time as T=t//H208491+/H92512/H20850and/H9270is defined as /H9270/H20849n/H20850=−/H11612/H9255/H20849n/H20850/H11003n+/H9275Ig/H20849/H9258,P/H20850n/H11003/H20849s/H11003n/H20850, /H208494/H20850 /H9255/H20849n/H20850=/H9275p 2/H20849pˆ·n/H208502−/H9275a 2/H20849aˆ·n/H208502−/H9275H/H20849hˆ·n/H20850. The newly defined constants are related to the already intro- duced parameters according to /H9275a=/H9253Ha,/H9275p=8/H92662/H9253M, /H208495/H20850 /H9275H=/H9253H,/H9275I=/H9253/H6036 2eVMI. All of them have dimensions of frequency making the com- parison between the terms of different origin straightforward.In accord with experimental situations, it is assumed that /H9275I/H11270/H9275p. We study a device with an in-plane easy axis and in- plane magnetic field perpendicular to it /H20851see Fig. 1/H20849b/H20850/H20852. Choosing the system of coordinates sˆ=aˆ=eˆz,hˆ=eˆx, and pˆ =eˆy, we obtain, from Eq. /H208494/H20850, the components of /H9270in spheri- cal coordinates /H9270/H9278=1 2sin 2/H9258/H20849/H9275psin2/H9278+/H9275a/H20850−/H9275Hcos/H9258cos/H9278, /H208496/H20850 /H9270/H9258=−/H9275p 2sin/H9258sin 2/H9278−/H9275Hsin/H9278−/H9275Ig/H20849cos/H9258/H20850sin/H9258. The equilibrium directions of the magnetization ncorre- spond to the solutions of the equation /H9270/H20849n/H20850=0. Here we con- sider the two in-plane equilibrium points. At /H9275I=0,/H9275H=0 these are the north /H20849N/H20850and the south /H20849S/H20850poles. For /H9275I=0, /H9275H/HS110050 they shift and approach the direction of magnetic field, finally merging at /H9275H=/H9275a. The shifted equilibrium points are still labeled by N and S /H20851Fig.1/H20849b/H20850/H20852. The stability of an equilibrium can be checked by ex- panding /H9270in angular deviations /H9254/H9258,/H9254/H9278, and writing Eq. /H208493/H20850 in an approximate form /H20873/H9278˙ /H9258˙/H20874=D/H20873/H9254/H9278 /H9254/H9258/H20874. /H208497/H20850 The equilibrium is stable when the real parts of both eigenvalues of Dare negative, or equivalently when matrix Dsatisfies Tr D/H110210 and det D/H110220 at the equilibrium. III. STABILITY REGIONS The modified positions of the N and S equilibria for /H9275H/HS110050,/H9275I/HS110050 are given bysin/H9258N,S=/H9275H /H9275a+O/H20873/H9275I /H9275P/H208742 , /H208498/H20850 sin/H9278N,S=−gN,S/H9275I /H9275p/H208731+/H9275a /H9275p/H20874+O/H20873/H9275I /H9275P/H208742 , for 0/H11021/H9275H/H11021/H9275a/H20849with gN,S=g/H20849/H9258N,S /H20850/H20850. The angle /H9258and the magnetic field strength /H9275Hhave a one-to-one correspondence and can be used interchangeably. The trace of D-matrix at these points can be found as TrD=−/H9275I/H20877g/H20849/H9258/H20850cos/H9258+d d/H9258/H20851g/H20849/H9258/H20850sin/H9258/H20852/H20878−/H9251/H20851/H9275pcos 2/H9278 +/H20849/H9275psin2/H9278+/H9275a/H20850/H208491 + cos2/H9258/H20850/H20852. /H208499/H20850 Approximations /H208498/H20850give the stability condition in the form /H9251/H208751+/H9275a /H9275p/H208491 + cos2/H9258/H20850/H20876/H11022−/H9275I /H9275p/H208512g/H20849/H9258/H20850cos/H9258+g/H11032/H20849/H9258/H20850sin/H9258/H20852 +O/H20873/H9275I /H9275P/H208742 . /H2084910/H20850 The determinant 1 1+/H92512detD=/H9275a/H20849/H9275p+/H9275a/H20850cos2/H9258+O/H20849/H9275I2/H20850/H20849 11/H20850 in the small current regime remains positive, so it does not play any role in the stability analysis in this case. In contrast,the trace Tr Dis more sensitive and can change sign as the current is varied. Moreover, the explicit appearance of g /H11032/H20849/H9258/H20850 in the formula leads to important differences in the switching diagrams for different forms of g/H20849/H9258/H20850. In the case of constant g-factor the stability condition for N- and S-equilibira can be written as /H9275I/H11125/H11007/H9251/H9275p+/H9275a/H208492− /H20849/H9275H//H9275a/H208502/H20850 2g/H208811− /H20849/H9275H//H9275a/H208502, /H2084912/H20850 where /H11022,/H11002 /H20849/H11021,+ /H20850corresponds to the N /H20849S/H20850stability region /H20849see Fig. 2/H20850. The switching current exhibits the 1 /cos/H9258di- vergence reported in the experiments for this regime.14 FIG. 2. Switching diagram for the g=const approximation. The value of gis chosen as the average value of Slonczewski’s g/H20849/H9258/H20850used in Fig. 3. Other parameters are set to /H9275a//H9275p=0.01, P=0.7, and /H9251=0.01. Stability regions for the N and S equilibria /H20849see text /H20850overlap forming the bistable region marked as B.07D114-2 I. Sodemann and Y . B. Bazaliy J. Appl. Phys. 105 , 07D114 /H208492009 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.170.6.51 On: Mon, 10 Aug 2015 23:53:01For the Slonczewski g-factor, the condition of stability for the N point is /H9275I/H11022−/H9251/H9275p+/H9275a/H208492− /H20849/H9275H//H9275a/H208502/H20850 2gN/H208811− /H20849/H9275H//H9275a/H208502+gN2fP/H20849/H9275H//H9275a/H208502, /H2084913/H20850 whereas for the S point the condition becomes /H9275I/H11124/H9251/H9275p+/H9275a/H208492− /H20849/H9275H//H9275a/H208502/H20850 2gS/H208811− /H20849/H9275H//H9275a/H208502−gS2fP/H20849/H9275H//H9275a/H208502, /H2084914/H20850 where /H11021 /H20849/H11022/H20850is the condition for /H9275/H11021/H9275H/H11569/H20849/H9275/H11022/H9275H/H11569/H20850, and/H9275H/H11569 designates the field for which the denominator of Eq. /H2084914/H20850 becomes zero and determines the onset of an stability behav-ior completely absent in the g-constant case /H20849Fig. 3/H20850. This field, or equivalently the angle characterizing the S point,depends only on polarization Pand can be found from cos /H9258c=− /H208811− /H20849/H9275H/H11569//H9275a/H208502=/H20881/H9264P2−1−/H9264P. /H2084915/H20850 In the “anomalous” regime /H9275H/H11022/H9275H/H11569a current of positive polarity stabilizes both N and S points, while the applicationof a sufficiently large negative current destabilizes bothpoints /H20849see Fig. 3/H20850. Moreover, there is a region in which none of the equilibria are stable, suggesting the existence of pre-cessional motion. The value of /H9275H/H11569becomes smaller as the polarization increases. In the limit P→1 it becomes zero, so that the anomalous region fills all the switching diagram. Inother words, as the polarization becomes larger, the differ-ences between the g-constant approximation and the Slonc- zewski form become quite dramatic. The position /H9258cof the S point at /H9275H/H11569is shown in Fig. 4. Substantial difference between the switching diagrams at large spin polarizations found in this study underscores thenecessity of developing new experiments capable of deter-mining the g/H20849 /H9258/H20850dependence. It also suggests that in the re- gime of large spin polarization the behavior of spin-transfer devices may experience qualitative changes. ACKNOWLEDGMENTS The authors are grateful to S. Garzon for many stimulat- ing discussions. 1J. C. Slonczewski, J. Magn. Magn. Mater. 159,L 1 /H208491996 /H20850. 2L. Berger, Phys. Rev. B 54, 9353 /H208491996 /H20850. 3M. Tsoi, A. G. M. Jansen, J. Bass, W.-C. Chiang, V. Tsoi, and P. Wyder, Phys. Rev. Lett. 80, 4281 /H208491998 /H20850. 4E. B. Myers, D. C. Ralph, J. A. Katine, R. N. Louie, and R. A. Buhrman, Science 285,8 6 7 /H208491999 /H20850. 5J. Z. Sun, J. Magn. Magn. Mater. 202,1 5 7 /H208491999 /H20850. 6J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers, and D. C. Ralph, Phys. Rev. Lett. 84, 3149 /H208492000 /H20850. 7J. C. Slonczewski, J. Magn. Magn. Mater. 247,3 2 4 /H208492002 /H20850. 8A. A. Kovalev, A. Brataas, and G. E. W. Bauer, Phys. Rev. B 66, 224424 /H208492002 /H20850. 9J. Xiao, A. Zangwill, and M. D. Stiles, Phys. Rev. B 70, 172405 /H208492004 /H20850. 10J. Xiao, A. Zangwill, and M. D. Stiles, Phys. Rev. B 72, 014446 /H208492005 /H20850. 11X. Wang, G. E. W. Bauer, and T. Ono, Jpn. J. Appl. Phys., Part 1 45, 3863 /H208492006 /H20850. 12H. Morise and S. Nakamura, Phys. Rev. B 71, 014439 /H208492005 /H20850. 13Y. B. Bazaliy, Phys. Rev. B 76, 140402 /H20849R/H20850/H208492007 /H20850. 14F. B. Mancoff, R. W. Dave, N. D. Rizzo, T. C. Eschrich, B. N. Engel, and S. Tehrani, Appl. Phys. Lett. 83, 1596 /H208492003 /H20850. FIG. 3. Switching diagrams for the Slonczewski’s form of the g-factor. Other parameters are the same as in Fig. 2./H20849b/H20850Regions of stability for north /H20849N/H20850and south /H20849S/H20850poles overlapping in the bistable region; region of preces- sional states /H20849P/H20850are shown. The onset of the anomalous stability behavior occurs at field /H9275H/H11569//H9275P=0.76.0.25 0.5 0.75 1PΠ/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt23Π/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt4ΠΘC FIG. 4. Critical angle /H9258cfor the onset of the anomalous stabilization as function of the polarization.07D114-3 I. Sodemann and Y . B. Bazaliy J. Appl. Phys. 105 , 07D114 /H208492009 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.170.6.51 On: Mon, 10 Aug 2015 23:53:01

1.4968813.pdf

Low-current, narrow-linewidth microwave signal generation in NiMnSb based single- layer nanocontact spin-torque oscillators P. Dürrenfeld , F. Gerhard , S. M. Mohseni , M. Ranjbar , S. R. Sani , S. Chung , C. Gould , L. W. Molenkamp , and J. Åkerman Citation: Appl. Phys. Lett. 109, 222403 (2016); doi: 10.1063/1.4968813 View online: http://dx.doi.org/10.1063/1.4968813 View Table of Contents: http://aip.scitation.org/toc/apl/109/22 Published by the American Institute of Physics Low-current, narrow-linewidth microwave signal generation in NiMnSb based single-layer nanocontact spin-torque oscillators P.D€urrenfeld,1,a)F.Gerhard,2S. M. Mohseni,3M.Ranjbar,1,b)S. R. Sani,4,c)S.Chung,1,4 C.Gould,2L. W. Molenkamp,2and J. A˚kerman1,4,5 1Department of Physics, University of Gothenburg, 412 96 Gothenburg, Sweden 2Physikalisches Institut (EP3), Universit €at W €urzburg, 97074 W €urzburg, Germany 3Department of Physics, Shahid Beheshti University, Tehran 1983969411, Iran 4Materials and Nanophysics, School of ICT, KTH Royal Institute of Technology, Electrum 229, 164 40 Kista, Sweden 5NanOsc AB, 164 40 Kista, Sweden (Received 9 September 2016; accepted 13 November 2016; published online 30 November 2016) We report on the fabrication of nano-contact spin-torque oscillators based on single layers of the epi- taxially grown half-metal NiMnSb with ultralow spin wave damping. We demonstrate magnetiza-tion auto-oscillations at microwave frequencies in the 1–3 GHz range in out-of-plane magnetic fields. Threshold current densities as low as 3 /C210 11Am/C02are observed as well as minimum oscil- lation linewidths of 200 kHz, both of which are much lower than the values achieved in conventionalmetallic spin-valve-based devices of comparable dimensions. These results enable the fabrication of spin transfer torque driven magnonic devices with low current density requirements, improved sig- nal linewidths, and in a simplified single-layer geometry. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4968813 ] Interest in spin wave generation on the nano-scale has increased tremendously in the last few years thanks to the emerging field of magnonics. 1,2In contrast to electronics, where only the electron charge is used, or spintronics, whereboth charge and spin are used, magnonics uses spin waves, or their quantized counterpart, magnons, to transmit and pro- cess information. An important figure of merit in magnonics is a materi- al’s spin wave damping. Of all materials known to date, yttrium iron garnet (YIG) has the lowest Gilbert dampingconstant of less than 10 /C04and is hence the material of choice in insulator based magnonics.3In metal based magnonics, permalloy (Py, Ni 80Fe20) and CoFeB are typically used,4,5 thanks to their relatively limited damping of typically 7/C210/C03and 4/C210/C03, respectively.6,7More recently, metal- lic Heusler and half-Heusler alloys have been intenselyresearched. 8,9since they combine near-100% spin polariza- tion10with a lower damping. One of the most important and fundamental magnonic functions is the generation of well defined spin waves. Whilethis has traditionally been achieved using antenna structures, the recent demonstration of current tunable propagating spin waves 11generated by spin transfer torque12,13(STT) in nano-contact spin torque nano-oscillators14(NC-STNOs) provides a much more efficient, and controllable, method; it also lends itself directly to continued miniaturization. While almost all STNOs use either a giant magnetoresis- tance (GMR) spin valve or a magnetic tunnel junction stack,it was recently shown that nano-contacts (NC) on single NiFe layers with asymmetric interfaces can generate magne-tization auto-oscillations and microwave signals without theneed for a separate fixed magnetic layer. 15Although the two interfaces generate STT of opposite sign, if the two interfa-ces to the ferromagnetic layer are asymmetric, a net STT canresult from the different degrees of spin filtering at the twointerfaces. 16Indirect evidence of such phenomena has also been detected through static resistance measurements.17–20 The single layer design has a number of important advan- tages. Not only does it avoid any extrinsic spin wave damp-ing from interlayer coupling, it also greatly simplifies boththe material growth and the device fabrication as fewer dis-parate thin film materials are required and hence allows for amore straightforward development and design of completemagnonic circuits and systems. For optimal magnonic per-formance, it would hence be beneficial to demonstrate singlelayer spin wave generation in ultra-low damping materials. In this work, we demonstrate the fabrication of NC- STNOs made from a single layer of the epitaxially grown half-Heusler alloy NiMnSb, 21which has been shown to exhibit low magnetic damping.22–24In recent studies of spin valves with the half-metallic Heusler compound Co 2(Fe,Mn)Si, the large spin polarization led to high output powers in nanopillars25,26 as well as in the point-contact geometry.27In our NC-STNOs, the oscillations have low threshold currents I thand, at the same time, narrow linewidths. 30 nm and 40 nm thick half-Heusler alloy NiMnSb films were grown by Molecular Beam Epitaxy on InP (001) sub-strates after deposition of a 200 nm thick buffer layer of(In,Ga)As 21,28and were capped in situ by 1.5 nm sputter- deposited copper. 8 /C216lm2mesas were defined by optical lithography and Ar ion milling. Subsequently, the samplewas covered by 50 nm of reactively sputtered SiO 2. The nanocontact (NC) and two ground contacts were defined ona)Present address: School of Electronic Science and Engineering, Nanjing University, 210093 Nanjing, China. b)Present address: School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, Arizona 85287, USA. c)Present address: Department of Materials Science and Engineering,Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA. 0003-6951/2016/109(22)/222403/4/$30.00 Published by AIP Publishing. 109, 222403-1APPLIED PHYSICS LETTERS 109, 222403 (2016) top of the mesa by electron beam lithography and etched into the SiO 2by CHF 3/CF 4/Ar-based reactive ion etching (see the inset of Fig. 1). In a final step, the devices were elec- trically connected with 1.3 lm thick copper waveguides made using photolithography and lift-off. The magnetic properties of the 40 nm thick NiMnSb film were measured using alternating gradient magnetometry(AGM) and ferromagnetic reso nance (FMR) on film pieces that have undergone the same processing steps as the final NC-STNO devices. In-plane AGM measurements are shown inFig.1for three different crystallographic directions of the sub- strate. Along the [110] directio n, the magnetization shows a switching behavior with a coerc ivity of less than 1 Oe, while in the [/C22110] direction, a gradual rota tion of the magnetization is seen. Along the [010] direction, partial rotation is seen before the magnetization reversal, representing an intermediate behavior compared to the previous two. Thus, the uniaxial in-plane anisotropy with an easy axis along [110] is likely to bedominant in our layer, which is consistent with previousreports on the magnetic properties of NiMnSb films of similarthickness. 21–23,28Fig. 2shows results from broadband field- swept FMR measurements along the in-plane easy axis of thefilm, taken with a NanOsc PhaseFMR system, where themicrowave field is generated from a coplanar waveguide. The absorption data (upper inset of Fig. 2) are fit with the derivative of an asymmetric Lorentzian lineshape to extract the linewidthand resonance fields. 29The effective magnetization is then determined from a fit to the Kittel e quation, taking into account an additional in-plane uniaxial anisotropy field. The effectivemagnetization and anisotropy field are l 0Meff¼0.826 T and 2KU M0¼26 Oe, respectively. The Gilbert damping constant of a¼(1.060.2)/C210/C03is calculated from a linear fit to the FMR linewidth and is thus less than one fifth of the values typ- ically achieved in Py. To characterize the microwave properties, the devices were connected through a bias-tee to a þ32 dB low-noise amplifier and their output was measured in the frequencydomain with a spectrum analyzer. The sample was mounted on a rotatable stage, allowing the angle between the external magnetic field and the film plane to be changed. All meas-urements were conducted at room temperature. A measurement of an NC-STNO from the 30 nm thick Heusler layer is shown in Fig. 3. The device consists of a NC with a diameter of 90 nm and has a dc resistance of 22.4 Xat 1 mA of current. The magnetic field of 3500 Oe is oriented atan angle of 86 /C14out-of-plane. Under these conditions, the auto-oscillations are visible starting from a current ofI DC¼/C01.8 mA, which results in an estimated threshold cur- rent density of j C/C253/C21011Am/C02in the NC. This value is remarkable, as it is much lower than the values being reportedfor NC-STNOs of comparable contact size, whereas lowthresholds are typically observed for vortex excitations. 30The integrated output power, see Fig. 3(b), amounts to tens of fW FIG. 1. In-plane magnetization measurements of the single-layer Heusler stack showing an easy axis along the [110] direction of the substrate. The insets show the layer structure of the NC-STNO and a scanning electron microscopy top-view picture of our devices. FIG. 2. Ferromagnetic resonance study of the 40 nm NiMnSb layer. The solid line is a fit to the Kittel equation yielding l0Meff¼0.826 T. Lower inset: FMR linewidth as a function of frequency. The linear fit results in aGilbert damping parameter a¼1.0/C210 /C03. Upper inset: FMR absorption measurement data for f0¼14, 15, and 16 GHz, from left to right. FIG. 3. (a) Power spectral density (PSD) map of a single-layer NiMnSb NC- STNO with an NC diameter of 90 nm as a function of bias current in an external field of 3500 Oe, tilted 86/C14with respect to the sample plane. Inset: PSD at I DC¼/C03.8 mA, where the solid line is a Lorentzian fit to the data; (b) and (c) integrated power and linewidth of the signal, respectively.222403-2 D €urrenfeld et al. Appl. Phys. Lett. 109, 222403 (2016) and is thus much lower than in spin-valve-based devices. In our single layer layout, the electrically measured signals are caused by the anisotropic magnetoresistance in the NiMnSb, which is /C240.1% in our stack ( supplementary material ), i.e., more than an order of magnitude lower than the typical GMR in spin valves. The linewidth of the oscillations is plotted in Fig.3(c) as it was determined from fitting the power spectral density with a Lorentzian function [see the inset of Fig. 3(a)]. To investigate the underlying nature of the observed auto-oscillations, we additionally implement spin torque- driven ferromagnetic resonance (ST-FMR) spectroscopy on another device, based on a 90 nm NC on top of the 40 nmthick NiMnSb layer. A pulsed microwave current of fre- quency f rfand power P rfis hereby applied through the bias- tee, leading to a dc mixing voltage, V mix, in the case of reso- nant excitations in the system,31which can then be extracted through a lock-in amplifier. The auto-oscillation data of the device at a current of /C09 mA in a field oriented at 86/C14are shown in Fig. 4(a), where the frequency varies between 1.8 GHz and 3.1 GHz over a field range of 3500 Oe. The line- width repeatedly takes up values as low as 200 kHz, whileincreasing above 10 MHz at the numerous mode changes. ST-FMR measurements are done at f rf¼2.8 GHz with Prf¼0.1 mW, as this frequency lies well within the auto- oscillation range of the device. Selected ST-FMR curves for various bias currents, see Fig. 4(b), show the evolution of the resonance peak. The shape of the resonance peaks is charac-terized by a sharp increase below and a more gradual decrease above the resonance field, independent of theapplied bias and it is also observed for a lower f rf(not shown). We here define the resonance field Hresas the point of maximum V mixandHresalways shifts to lower values with the application of larger microwave currents, as shownin the inset of Fig. 4(c) for zero applied bias. Both the shape of the resonance and the shifting are signs of a nonlinear res- onance 32with a fold-over, which are commonly observed for large precessional motions. The resonance field as a function of bias current is plotted as black squares in Fig. 4(c) along- side the conditions of field and current at which auto- oscillations at f¼2.8 GHz have been observed (red circles). While a negative bias current increases the amplitude of theST-FMR signal and reduces the resonance field, a positive bias current does not change the resonance field significantly but decreases the signal amplitude until it is vanished formore than þ2 mA. Despite the apparent step at I ¼/C05m A between ST-FMR and auto-oscillation data, which can be attributed to the inherent shifting from measuring ST-FMRwith a finite microwave current, we can conclude that the auto-oscillation mode and the ST-FMR resonances at small bias are the identical mode. The saturation magnetization and the anisotropies of our extended NiMnSb film are known, and we can thus exclude a uniform FMR-like precession, which would relate to fre-quencies of /C245.1 GHz at a field of 7000 Oe, to be the reason for the measured ST-FMR resonances. Any spin wave modes with a finite wave vector in the uniformly magnetized layer,e.g., due to the confined geometry, would then just be to appear at even higher frequencies. This leads us to the assumption that the microscopic nature of the observed sig-nals, auto-oscillations and ST-FMR, relates to the movement of a magnetostatic object in a non-uniformly magnetized film. Such objects can experience a substantial STT only dueto the in-plane current getting polarized in the inhomogene- ously magnetized layer 33without the need for a polarizing layer. They are also known to couple efficiently to micro-wave currents. 34,35Epitaxially grown NiMnSb was shown to be able to host a magnetic vortex in 1 lm diameter disks, even for large out-of-plane fields.36However, due to the size of our 8 lm/C216lm large mesas and the therefore expected sub-GHz gyration frequency,36a single vortex in our devices is unlikely. The formation of multiple smaller vortices ordomains, as were observed in even larger structures of the epitaxially grown Heusler alloy Co 2(Fe,Mn)Si,37and their excitation by STT remains as one possible cause for ourmeasured electrical data. NiMnSb is known to exhibit a small perpendicular anisotropy, 23which might play a role in the magnetization formation in our mesas. The exact config-uration of the magnetization underneath the NC is, however, beyond the scope of this letter. It is most likely governed by pinning sites, which could be either due to intrinsic latticedefects from the epitaxial growth 37or have been introduced during the nanocontact fabrication, both leading to random- ness in the electrical signal from our devices. The device-to-device repeatability of auto-oscillations between the nominally identical devices is relatively poor. Of more than 30 tested NC-STOs with 80–100 nm diameterNCs, approximately 60% show microwave oscillations at conditions similar to those discussed here, i.e., with a strong out-of-plane field component. Less than 10% show anFIG. 4. (a) Auto-osc illation frequency and linewidth from a further device as a function of the applied field with an angle of 86/C14and I¼/C09 mA. The dashed line marks the applied field corresponding to f¼2.8 GHz. (b) ST- FMR data for various bias currents at frf¼2.8 GHz and P rf¼0.1 mW. The curves are vertically shifted for clarity. (c) ST-FMR resonance fields (black squares) and auto-oscillation conditions (red circles) for f¼2.8 GHz as a function of bias current. The inset shows the shift of the resonance field with applied microwave power for I ¼0m A . T h e d a s h e d line marks P rf¼0.1 mW.222403-3 D €urrenfeld et al. Appl. Phys. Lett. 109, 222403 (2016) excitation of two auto-oscillation modes, despite the pres- ence of just one NC, a further hint towards a non-uniform magnetization in our mesas ( supplementary material ). The single-layer NiMnSb NC-STNOs are more likely to producemicrowave signals when the field is rotated a few degrees away from the perpendicular direction, as the small in-plane component might play a role in the excitations underneaththe nanocontact. A field-dependent magnetoresistance and auto-oscillation measurement for a 90 /C14perpendicular direc- tion can be found in the supplementary material . Neither field nor current hysteresis of auto-oscillations has been observed in our devices. We can therefore consider the mag-netization configuration underneath the NC as an intrinsic magnetostatic object rather than being nucleated by STT. 38 Besides, no significant differences in oscillation linewidth and auto-oscillations’ conditions could be seen between devices from the 30 nm and 40 nm thick NiMnSb films. In summary, we have shown STT-induced auto-oscilla- tions in single-layer NiMnSb films. The NC-STNOs have significantly decreased threshold currents and very low line- widths, compared to metallic NC-STNOs. The auto-oscillations are generally observed in magnetic fields with a strong out-of-plane component and show signals similar to magnetic vortex gyrations in nanopillars, yet with unexpect-edly large frequencies. ST-FMR measurements verify the presence of intrinsic non-uniformities, which occur with a certain probability underneath the NC. While this random-ness depicts a drawback in our devices, we think that the realization of NiMnSb-based single-layer NC-STNOs presents an important step forward towards the implementa-tion of this materials class into spintronic devices. See supplementary material for details on anisotropic magnetoresistance and multi-mode excitations. We acknowledge R. K. Dumas for valuable discussions. Support from the Swedish Foundation for Strategic Research (SSF), the Swedish Research Council (VR), and the Knut andAlice Wallenberg Foundation is gratefully acknowledged. This work was supported by the European Commission FP7 Contract No. ICT-257159 “MACALO.” J.A ˚. is a Royal Swedish Academy of Sciences Research Fellow supported by a grant from the Knut and Alice Wallenberg Foundation. 1S. Neusser and D. Grundler, Adv. Mater. 21, 2927 (2009). 2V. V. Kruglyak, S. O. Demokritov, and D. Grundler, J. Phys. D: Appl. Phys. 43, 264001 (2010). 3H. Chang, P. Li, W. Zhang, T. Liu, A. Hoffmann, L. 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1.4799248.pdf

Spin rectification enabled by anomalous Hall effect Hang Chen, Xiaolong Fan, Hengan Zhou, Wenxi Wang, Y. S. Gui et al. Citation: J. Appl. Phys. 113, 17C732 (2013); doi: 10.1063/1.4799248 View online: http://dx.doi.org/10.1063/1.4799248 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v113/i17 Published by the American Institute of Physics. Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 12 May 2013 to 128.118.88.48. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsSpin rectification enabled by anomalous Hall effect Hang Chen,1Xiaolong Fan,1,a)Hengan Zhou,1Wenxi Wang,1Y . S. Gui,2C.-M. Hu,2 and Desheng Xue1 1The Key Lab for Magnetism and Magnetic Materials of Ministry of Education, Lanzhou University, Lanzhou 730000, People’s Republic of China 2Department of Physics and Astronomy, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada (Presented 17 January 2013; received 4 November 2012; accepted 9 January 2013; published online 15 April 2013) We report the observation of a transverse dc voltage which appears when a radio frequency (rf) current flows along the longitudinal direction of a ferromagnetic Hall device. This effect is fullyexplained through the spin rectification enabled by the anomalous Hall effect, which is nonlinear coupling between the dynamic magnetization and the rf current. The observed resonant feature and angular dependent line shape are related to the magnetization precession driven by a rf magneticfield. This suggests a method for detection of spin dynamic and rf magnetic field vector. VC2013 American Institute of Physics .[http://dx.doi.org/10.1063/1.4799248 ] The Anomalous Hall Effect (AHE) arises from an asym- metric scattering of the conduction electrons induced byintrinsic or extrinsic processes that depend on the spin-orbit interaction. 1The resultant large Hall voltage is suitable for sensors, memory, and magnetic logic applications.2,3In this paper, the AHE has been explored for dynamic applications especially within the radio frequency (rf) region. Due to a nonlinear coupling between the conduction electron and localspin, the out-of-plane component of the dynamic magnetiza- tion can directly rectify rf currents into a time-independent Hall signal in ferromagnetic materials. Based on such a rectifi-cation principle, the complex spin dynamics in the gigahertz (GHz) range as well as the propagation of electromagnetic waves can be measured by a static electrical approach, whichwill be a high sensitivity measurement technique. A Hall geometry is made of a single Co 90Zr10ferromag- netic layer by using laser exposure and lift-off method, asshown in Fig. 1(a). The Co 90Zr10layer with thickness of 20 nm were prepared by rf sputtering onto glass substrates attached to a water-cooling system with background pressure less than5/C210 /C05Pa.4The longitudinal resistance and Hall resistance were measured using a lock-in amplifier (Stanford SR830), with a modulation frequency of 1.31 kHz and a current of100lA. The microstructure was characterized by a high reso- lution Transmission Electron Microscope (TEM, F30, FEI). All the measurements were performed at room temperature. Figure 1(b) shows the TEM image of a 20 nm Co 90Zr10 layer, with an amorphous structure and a few Co nano-grains embedded within such a matrix. This amorphous and nano-crystalline combined structure made Co 90Zr10layer as a soft magnetic material with excellent high frequency respond.4,5 On the other hand, such a structure also has significantly suppressed the anisotropic magnetoresistance (AMR).6As shown in Fig. 1(c), the AMR ratio, which is defined as Dq=qk, is only /C00.01%, where qk¼144lXcm is the longi- tudinal resistivity when magnetization Mis parallel tocurrent, Dq¼/C00:014lXcm is the resistivity decrement when Mperpendicular to current. Figure 1(d)shows the Hall resistivity as a function of a perpendicular applied magnetic field, from which a saturated anomalous Hall resistivity qH¼1:15lXcm is obtained. We begin our theory from the generalized Ohm’s law E¼qkjþDqmðj/C1mÞ/C0qHj/C2m,w h e r e jis current density vector, m¼M=M0the unit vector of the magnetization, and M0the saturation magnetization. Considering that Dq(/C00.014 lXcm) is two orders smaller than qH(1.15 lXcm) for the Co90Zr10layer, the second term in the law can be ignored. If we send a rf current ~j¼jxe/C0ixtalong the longitudinal direction into the cross, it will simultaneously induce a rf magnetic field he/C0iðxt/C0UÞ.H e r e , x¼2pfis the frequency and Uis the phase of the rf field with respect to ~j. Due to the torque of the rf field, the magnetization will precess ( mte/C0iðxt/C0wÞ) around its equilib- rium direction ( m0), i.e., m¼m0þmte/C0iðxt/C0wÞ,w h e r e mtis FIG. 1. (a) Top view micrograph of device. (b) TEM image of a 20 nm Co90Zr10layer, a number of Co nanograins are indicted by the white circles. (c) Symbols represent longitudinal resistivity qxxas a function of h, the angle between magnetic field and xaxis. The solid line is a qk/C0Dqsin2hfit based on the principle of AMR. (d) Hall resistivity qxyas a function of per- pendicular applied magnetic field.a)Electronic mail: fanxiaolong@lzu.edu.cn 0021-8979/2013/113(17)/17C732/3/$30.00 VC2013 American Institute of Physics 113, 17C732-1JOURNAL OF APPLIED PHYSICS 113, 17C732 (2013) Downloaded 12 May 2013 to 128.118.88.48. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsthe amplitude of the dynamic magnetization unit vector, wis the phase lag between ~jandm. Consequently, a Hall voltage appears VyðtÞ¼Ðw 0Eydy¼qHwJx½m0zcosðxtÞþmtzcosðxtÞ cosðxtþwÞ/C138,w h e r e w¼300lm is the distance between Hall contact leads. After a time averaging of VyðtÞ, a dc Hall voltage is generated, Vy¼1 TðT 0VyðtÞdt¼qHwJx 2mtzcosw; (1) where T¼2p=xis the period of the rf current. Equation (1) is the expression for the AHE spin rectification, wherein thedc voltage V yis proportional to the amplitude of the out-of- plane component of the dynamic magnetization. As a corol- lary, peaks associated with magnetization resonance shouldappear when the V yis measured as a function of applied magnetic field or frequency. We begin our experiment by applying a dc magnetic field Halong the xdirection( h¼0/C14is the in-plane angle between Hand the xaxis), and inputting a rf signal (14.5 GHz, 20 dBm) with a modulation frequency of 3.52kHz into the device. Then V ywas measured as a function of Husing a lock-in amplifier. As shown in Fig. 2(a), two reso- nant signals appear at H¼61.79 kOe. The resonant signal in negative or positive field is found to be a Lorentz (disper- sive) line shape, which is symmetrical (anti-symmetrical) about its resonance position H0. A systematical measurement ofVyðHÞwithin a frequency range of 2 GHz to 18 GHz was performed. Figure 2(b) presents the relationship between H0 and frequency, which can be well fitted by the Kittel equa- tionx¼cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M0ðHkþjHjÞp , as the solid lines show. Here, c is the gyromagnetic ratio, Hkis the in-plane anisotropy field, and values of M0¼10.7 kOe, Hk¼48 Oe, and c¼18.8 GHz/ kOe were determined from the fitting. The dispersion curve is a striking proof that the resonant signals presented in Fig.2(a)are due to ferromagnetic resonance (FMR).A further analysis of the resonant line shape of VyðHÞ depends on a detailed calculation of mtzandwin Eq. (1).I ti s known that it is v, the susceptibility tensor, that links the dynamic response of magnetization to the rf field via mt¼vh. In our case, the rf field is induced by the rf current ~j, mtz¼½vzzhzþivzy0ðhycosh/C0hxsinhÞ/C138=M0; (2) where hi(i¼x, y, z ) is the icomponent of h.vzzandivzy0are susceptibility tensor elements for the x0y0zcoordinate system as shown in Fig. 1(a), which are given by solving the Landau-Lifshitz-Gilbert equation,7 vzz;zy0¼Azz;zy0DHðH/C0H0ÞþiDH2 ðH/C0H0Þ2þDH2;with Azz¼cH0M0 aGxð2H0þM0Þ;Azy0¼M0 aGð2H0þM0Þ; (3) where DHis the linewidth and aGthe Gilbert damping.8In order to solve the final expression of VyðHÞ, we have to deal with win Eq. (1).wis the phase of dynamic magnetization with respect to ~j, which can be separated into two compo- nents, one of which is Uthe phase of rf field with respect to ~j, the other is Hthe phase of dynamic magnetization with respect to the rf field, i.e., w¼HþU.9Thus, the production ofmtzcoswin Eq. (1)is given by mtzcosw¼Re½mtz/C138cosUþIm½mtz/C138sinU; (4) wherein Re ½mtz/C138¼mtzcosHand Im ½mtz/C138¼mtzsinH.B y substituting Eqs. (2)–(4)into Eq. (1),aHdependent expres- sion of Vyis given by Vy¼VDDHðH/C0H0Þ ðH/C0H0Þ2þDH2þVLDH2 ðH/C0H0Þ2þDH2;(5a) with VD¼qHwjx 2M0½AzzhzcosUþAzy0ðhycosh/C0hxsinhÞsinU/C138; (5b) VL¼qHwjx 2M0½AzzhzsinU/C0Azy0ðhycosh/C0hxsinhÞcosU/C138:(5c) B a s e do nE q . (5), the resonant signal due to the AHE spin rectification shows a linear combination of a dispersive line shape which is proportional to DHðH/C0H0Þ=½ðH/C0H0Þ2 þDH2/C138and a Lorentz one which is proportional to DH2= ½ðH/C0H0Þ2þDH2/C138. Then the data shown in Fig. 2(a)was fitted by using Eq. (5a). The fit curves (solid lines) show good consis- tencies with the experiments results. The VyðHÞcurves meas- ured at different frequencies were treated in the same way. Figure 2(c) presents the linewidth DHas a function of fre- quency. By a linear fit of DH¼DH0þax=c, a Gilbert damp- ing constant aG¼0:011 and DH0¼4:8 Oe, which result from structural imperfections for the Co 90Zr10layer to be obtained. Now we will have a closer look on the line shape of the curve shown in Fig. 2(a). Both signals with different line shapes have been proved to be the consequences of FMR. FIG. 2. (a) The dc Hall voltage due to AHE spin rectification. Symbols are raw data measured at h¼0/C14and curves are line shape fits by using Eq. (5a).( b ) Dispersion curves of the resonant signals presenting in VyðHÞ.( c )T h el i n e - width as a function of frequency, which indicates a Gilbert damping of FMR.17C732-2 Chen et al. J. Appl. Phys. 113, 17C732 (2013) Downloaded 12 May 2013 to 128.118.88.48. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsThe right one happens at h¼0/C14, the left one happens at h¼6180/C14. It is reasonable to predict that the line shape will be changed gradually from dispersive to Lorentz if the mag-netization were dragged from h¼0 /C14toh¼6180/C14by an applied magnetic field. This prediction is confirmed by the data shown in Fig. 3(a), wherein the VyðHÞcurves were measured at different hwith a fixed rf signal (14.5 GHz 20 dBm). Offsets of these curves have been applied, so the gradually changed line shape can be observed. In order to reveal the physics that dominates the change of line shape, curves have been fitted by using Eq. (5a). The in- plane angular dependence of H0as well as the line shape amplitudes ( VLandVD) are shown in Figs. 3(b)and3(c).T h e resonance field H0represents an oscillation with a period of 180/C14, which can be fitted using H0¼x2=ðc2M0Þ/C0Hkcos2h that is an expression of the in-plane uniaxial anisotropy partici- pated FMR. The fit gives Hk¼49 Oe which is consistent with early result. The line shape amplitudes VLandVDshow a com- bination of cos h-like oscillations and angular-independent backgrounds, as shown in Fig. 3(c). This result can be well e x p l a i n e db yE q s . (5b)and(5c), which can be further refined to VD¼VDzþVDycosh/C0VDxsinh; (6a) VL¼VLz/C0VLycoshþVLxsinh; (6b) where VDiðVLiÞði¼x;y;zÞis the contribution of hito the am- plitude of dispersive (Lorentz) line shape. It is clear that the angular-independent background contribution VDz¼69 nV andVLz¼67 nV of the data shown in Fig. 3(c)arises from hz. The cos h-like parts are attributed to hy,VDy¼/C075 nV and VLy¼/C060 nV. The hxdominated parts which were expected to be a sin htrend did not appear in the data, which suggeststhat there was no hxpresented in our device. Based on the above discussion, it is clear that the rf field induced by ~jhasz and ycomponents, and the ratio between them is jhy=hzj ¼cH0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V2 DyþV2 Lyq =ðxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V2 DzþV2 Lzp Þ¼0:35. Moreover, the phase lag between the rf field and ~jisU¼arctan ðVLz=VDzÞ ¼44/C14.10Therefore, the AHE spin rectification effect is capa- ble of detecting rf field vectors at the subwavelength scale. It is worth noting the difference between spin rectifica- tion enabled by AHE and that enabled by AMR. In Egan and Juretshke’s work, a mixing voltage signal from both effectshas been studied. 11,12In the theory of AMR rectification,13it is the square of the in-plane longitudinal component of dynamic magnetization that rectifies the rf current into a dcvoltage. It has been found that the AMR rectification signal vanishes at high symmetry points ( h¼0 /C14;90/C14;180/C14, and 270/C14; see Ref. 14Fig. 3for example). This symmetry is attributed to the differential effect of AMR, which gives an overall sin2 hterm. However, for AHE rectification, it is the out-of-plane component of dynamic magnetization that is re-sponsible for the rectification, and there is no overall in- plane angular dependent term [see Fig. 3(a)]. For an arbitrary single magnetic material, both AMR and AHE would con-tribute the rectification signal. The in-plane angular depend- ent amplitude of the signal would be the straightforward criteria to distinguish them. In a word, spin rectification enabled by AHE has been studied by using a selected Co 90Zr10layer. It has been dem- onstrated that it is the out-of-plane component of the dynamic magnetization that rectifies rf current into a dc volt- age along the Hall direction via AHE. This effect has beenproved to be a significant tool to study spin dynamic and to detect the rf field vector including its phase. The project was supported by the National Basic Research Program of China (Grant No. 2012CB933101),NSFC (Grant Nos. 11034004, 50925103, 61102001, and 11128408), and the Fundamental Research Funds for the Central Universities (No. lzujbky-2011-49). 1N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Rev. Mod. Phys. 82, 1539 (2010). 2J. Moritz, B. Rodmacq, S. Auffret, and B. Dieny, J. Phys. D: Appl. Phys. 41, 135001 (2008). 3A. Gerber, A. Milner, M. Karpovsky, B. Lemke, H.-U. Habermeier, J. Tuaillon-Combes, M. Ngrier, O. Boisron, P. Mlinon, and A. Perez,J. Magn. Magn. Mater. 242–245 , 90 (2002). 4Z. M. Zhang, X. L. Fan, M. Lin, D. W. Guo, G. Z. Chai, and D. S. Xue, J. Phys. D: Appl. Phys. 43, 085002 (2010). 5X. L. Fan, D. S. Xue, M. Lin, Z. M. Zhang, D. W. Guo, C. J. Jiang, and J. Q. Wei, Appl. Phys. Lett. 92, 222505 (2008). 6T. R. McGuire and R. I. Potter, IEEE Trans. Magn. 11, 1018 (1975). 7N. Mecking, Y. S. Gui, and C.-M. Hu, Phys. Rev. B 76, 224430 (2007). 8T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004) 9A. Wirthmann, X. L. Fan, Y. S. Gui, K. Martens, G. Williams, J. Dietrich, G. E. Bridges, and C.-M. Hu, Phys. Rev. Lett. 105, 017202 (2010). 10For a general case, one has to use three phases Ux;Uy, andUzfor each component of h. See M. Harder, Z. X. Cao, Y. S. Gui, X. L. Fan, and C.-M. Hu, Phys. Rev. B 84, 054423 (2011). 11W. G. Egan and H. J. Juretshke, J. Appl. Phys. 34, 1477 (1963). 12H. J. Juretshke, J. Appl. Phys. 31, 1401 (1960). 13Y. S. Gui, N. Mecking, X. Zhou, G. Williams, and C.-M. Hu, Phys. Rev. Lett.98, 107602 (2007). 14L. H. Bai, Y. S. Gui, A. Wirthmann, E. Recksiedler, N. Mecking, C.-M. Hu, Z. H. Chen, and S. C. Shen, Appl. Phys. Lett. 92, 032504 (2008).FIG. 3. (a) Gradually changed line shape from Dispersive (D) at h¼0/C14to Lorentz (L) at h¼6180/C14. (b) Resonant position ( H0) as a function of h. (c) Symbols are the angular dependence of line shape amplitude, accompanied by theoretical fits based on Eqs. (5b)and(5c).17C732-3 Chen et al. J. Appl. Phys. 113, 17C732 (2013) Downloaded 12 May 2013 to 128.118.88.48. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions

1.4895480.pdf

Tunable magnetization dynamics in disordered FePdPt ternary alloys: Effects of spin orbit coupling L. Ma, S. F. Li, P. He, W. J. Fan, X. G. Xu, Y. Jiang, T. S. Lai, F. L. Chen, and S. M. Zhou Citation: Journal of Applied Physics 116, 113908 (2014); doi: 10.1063/1.4895480 View online: http://dx.doi.org/10.1063/1.4895480 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/116/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Magneto-optical Kerr effect in L10 FePdPt ternary alloys: Experiments and first-principles calculations J. Appl. Phys. 115, 183903 (2014); 10.1063/1.4872463 Spin-orbit interaction tuning of perpendicular magnetic anisotropy in L10 FePdPt films Appl. Phys. Lett. 104, 192402 (2014); 10.1063/1.4876128 Laser induced spin precession in highly anisotropic granular L10 FePt Appl. Phys. Lett. 104, 152412 (2014); 10.1063/1.4871869 Low spin-wave damping in amorphous Co40Fe40B20 thin films J. Appl. Phys. 113, 213909 (2013); 10.1063/1.4808462 Ultrafast optical study of spin wave resonance and relaxation in a CoFe/PtMn/CoFe trilayer film J. Appl. Phys. 105, 07D304 (2009); 10.1063/1.3063675 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.217.58.222 On: Tue, 25 Nov 2014 23:36:01Tunable magnetization dynamics in disordered FePdPt ternary alloys: Effects of spin orbit coupling L. Ma,1S. F . Li,2P . He,3W. J. Fan,1,a)X. G. Xu,4Y. Jiang,4T. S. Lai,2,b)F . L. Chen,1 and S. M. Zhou1 1Shanghai Key Laboratory of Special Artificial Microstructure and Pohl Institute of Solid State Physics and School of Physics Science and Engineering, Tongji University, Shanghai 200092, China 2State Key Laboratory of Optoelectronic Materials and Technologies, School of Physics and Engineering, Sun Yat-Sen University, Guangzhou 510275, China 3Department of Physics, Fudan University, Shanghai 200433, China 4State Key Laboratory for Advanced Metals and Materials, School of Materials Science and Engineering, University of Science and Technology Beijing, Beijing 100083, China (Received 22 April 2014; accepted 1 September 2014; published online 18 September 2014) The magnetization dynamics of disordered Fe 0.5(Pd1/C0xPtx)0.5alloy films was studied by time-resolved magneto-optical Kerr effect and ferromagnetic resonance. The intrinsic Gilbert damping parameter a0 and the resonance linewidth change linearly with the Pt atomic concentration. In particular, the induced in-plane uniaxial anisotropy constant KUalso increases for xincreasing from 0 to 1. All these results can be attributed to the tuning effect of the spin orbit coupling. For the disordered ternary alloys, an approach is proposed to control the induced in-plane uniaxial anisotropy, different fromconventional thermal treat methods, which is helpful to design and fabrications of spintronic devices. VC2014 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4895480 ] INTRODUCTION Fundamental research on the magnetization dynamics has become an interest because of its crucial importance inthe storage industry. The magnetization dynamics can be studied by both ferromagnetic resonance (FMR) 1,2and time- resolved magneto-optical Kerr effect (TRMOKE)3,4through incorporating Landau-Lifshitz-Gilbert equation (LLG).5,6 The energy dissipation is described by a phenomenological Gilbert damping parameter a,which governs the time needed for a nonequilibrium magnetic state to return to equilibrium. This parameter manifests through the frequency dependent resonance linewidth in FMR technique and the decaying rateof magneto-optical Kerr signal with the time delay in TRMOKE technique. Manipulation of the magnetic damping parameter in magnetic disordered alloys has attracted much attention in the past decade. 7–10This is because magnetic disordered alloys are widely used in spintronic devices and their mag-netic damping parameter is of determinant role in device functionalities such as current driven magnetization switch- ing process. The mechanism of the damping enhancement isextremely complicated. For example, the damping parameter in permalloy is enhanced by doping of either rare-earth (RE) metals or transition metals, 11–14which is attributed to the interaction of the orbital moments of the RE metals and the conduction electrons and the combined effect of the density of states near Fermi level and the spin polarization of 5d element,15respectively. The enhancement of the magnetic damping parameter may also be related to the spin orbit cou- pling (SOC).16Therefore, the mechanism of the magneticdamping parameter in disordered alloys needs further investigation. In this work, we study the magnetization dynamics of disordered Fe0.5(Pd 1/C0xPtx)0.5alloy films by TRMOKE and FMR. Since their in-plane uniaxial anisotropy is muchsmaller than the demagnetization energy, it cannot be deter- mined well by the TRMOKE. Instead, it can be identified accurately by the in-plane angular FMR spectra. The intrin-sic damping parameter a 0is measured by the TRMOKE. The uniaxial anisotropy and the a0are both found to increase with increasing x. The increase of the two physical parame- ters is ascribed to the SOC enhancement when the SOC of heavier Pt atoms is larger than that of Pd atoms when Pd atoms are replaced by heavier Pt.17In particular, the a0in the disordered alloys changes as a linear function of the Pt/ Pd atomic concentration, in a way different from that of L1 0 FePtPd.17Remarkably, the present work provides a clue to tune the induced magnetic anisotropy in disordered alloys, which is different from the approach to control the prepara- tion procedures.18 EXPERIMENTS A series of disordered Fe 0.5(Pd 1/C0xPtx)0.5(¼FePdPt) ter- nary alloy films with 0 /C20x/C201 was deposited on polished single crystal MgO (001) substrates at room temperature by DC magnetron sputtering. The FePdPt alloy target was formed by placing small pieces of Pt and Pd onto an iron tar-get. The base pressure of the deposition system was 1 :0/C2 10 /C05Pa and the Ar pressure was 0.35 Pa. The rate of deposi- tion was about 0.1 nm/s. The film thickness was determinedto be 22 61 nm by x-ray reflectivity at small angles with Cu K aradiation. The microstructure and alloy composition were measured by transmission electronic microscopy (TEM) anda)Electronic address: stslts@mail.sysu.edu.cn b)Electronic address: eleanorfan@163.com 0021-8979/2014/116(11)/113908/5/$30.00 VC2014 AIP Publishing LLC 116, 113908-1JOURNAL OF APPLIED PHYSICS 116, 113908 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.217.58.222 On: Tue, 25 Nov 2014 23:36:01Energy Dispersive X-ray Spectroscopy (EDX). The analyti- cal accuracy in the EDX is about 62%. In-plane hysteresis loops and the saturation magnetization M Swere measured by the vibrating sample magnetometer (VSM). The saturation magnetization MSalmost does not change with xand is in the region from 886 to 930 emu/cm3for all x. The in-plane uniaxial anisotropy and magnetization dynamics are studied by FMR and TRMOKE. The FMR spectra at 9.78 GHz were carried out at room temperature. An in-plane magnetic fieldwas applied at different orientations. The TRMOKE was measured by a pump-probe technique. The measurements were set up with a pulsed Ti: sapphire laser combined with aregenerative amplifier at the wavelength of 800 nm. The du- ration time and repetition rate of the linearly polarized output pulse were 150 fs and 1 kHz, respectively. The pump beamis focused to a spot of 150 lmin diameter, whereas the probe spot is located at the center of the pump spot with much smaller diameter. The external magnetic field was applied atan angle of 13 /C14with respect to the film normal direction. The oscillation of Kerr signal becomes weak with the time delay because the energy transfers from the spins to the lat-tice. All measurements were performed at room temperature. According to the LLG equation, one has the following dispersion equation: 5,6,19,20 x c/C18/C192 ¼1 M2 Ssin2h@2E @h2@2E @u2/C0@2E @h@u !22 43 5; (1) where the gyromagnetic ratio c¼ge=2mcwith the mass ( m) and charge ( e) of electrons, the speed ( c) of light in vacuum, and the factor g.MSis the saturation magnetization. The h andurefer to the orientation of the magnetization vector with respect to the zaxis and the in-plane easy axis, respec- tively. For magnetic films with in-plane uniaxial anisotropy,the free energy density Econsists of the Zeeman, demagnet- ization and uniaxial anisotropy energies. In spherical coordi- nates, it is described as follows: 1 E¼/C0 ~M/C1~Hþ2pM2 Scos2h/C0KUsin2hsin2u; (2) where KUis the in-plane uniaxial anisotropy constant. The equilibrium angular position of the magnetization vector canbe obtained by the following equation, @E @h¼0;@E @u¼0. Moreover, the relaxation time in TRMOKE measurements obeys the following equation:3 1 s¼ac 2Hcosh/C0hH ðÞ /C04pMScos2hþHsinhH sinh/C02KU MS/C20/C21 ;(3) where hHanduHdescribe the orientation of the Hin spheri- cal coordinate system. RESULTS AND DISCUSSION The microstructural properties of the ternary FePdPt films are shown in Fig. 1. For all samples, no diffraction peak exists in the XRD spectra except for MgO (200) peak, asshown in Fig. 1(a). This is because all samples studied here were fabricated at ambient temperature, whereas thechemically ordered FePdPt alloys must be deposited at elevated temperatures or post-annealed after deposition. 17 The ring patterns of selected area TEM diffraction inFig. 1(c) further show the sample is polycrystalline. 21 Therefore, the present FePdPt films are polycrystalline disor- dered alloys. The TEM image of cross-sectional FePd sample in Fig. 1(b) shows that the microstructure changes with the film depth. Similar phenomena are often observed.22 The in-plane FMR spectra of the FePt film were shown in Fig. 2(a), where uHchanges from zero to 180/C14. Only one FIG. 1. XRD h–2hspectra for typical samples (a) and TEM cross sectional image of the FePd sample (b), where the (1) region refers to the MgO sub-strate, and the (2) and (3) regions are located at different depths in the FePd layer. In the (2) region, which is close to the MgO/FePd interface, the film favors to be amorphous. In the (3) region, which is far away from the inter- face, the film starts to become polycrystalline. 21In (c), the selected area electron diffraction pattern of FePd is given for the (3) region in (b). FIG. 2. The measured FMR spectra for FePt at different angles uH(a), the angular dependent resonance field (b), and remanent ratio (c) for typical samples. Solid lines in (b) and (c) are fitted results and serve a guide to the eye, respectively.113908-2 Ma et al. J. Appl. Phys. 116, 113908 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.217.58.222 On: Tue, 25 Nov 2014 23:36:01uniform resonance peak exists in the spectra, indicating the uniform magnetization distribution in the film, and it shifts with the uH. The angular dependences of the resonance field are shown in Fig. 2(b). Along the easy axis, i.e., uH¼0, the resonance field achieves a minimal value and it has a maximal value at uH¼90/C14. Angular dependent in-plane hysteresis loops were measured and the remanent ratio acquires a maximal value near uH¼0 and a minimal value atuH¼90/C14, as shown in Fig. 2(c). Static and dynamic measurements both prove that an in-plane uniaxial anisot- ropy is induced in FePdPt films during deposition.18The measured angular dependence in Fig. 2(b) can be fitted by Eqs. (1)and(2). For example, both candKUare deduced to be 1 :83/C2107Oe/C01s/C01and 1 :84/C2104erg/cm3for FePt, respectively. Figure 3(a) shows the Kerr signal as a function of the time delay for FePtPd films with x¼0.4 under various mag- netic fields. The polar Kerr rotation oscillates with the timedelay. The precessional frequency and the oscillation ampli- tude both increase with increasing H. This is because the component of the magnetization normal to the film planebecomes large as the Hincreases. The measured oscillation of the Kerr signal can be fitted by the following equation 23,24 hK¼a0þb0/C3expð/C0t=t0ÞþA/C3expð/C0t=sÞsinð2pftþu0Þ, where the parameters A,s,f,andu0are the amplitude, relax- ation time, frequency, and phase of the magnetization pre- cession, respectively. Here, a0,b0, and t0correspond to the background signal owing to the slow recovery process. The precession frequency fand relaxation time sare fitted. For example, the sis found to decrease from 809.7 to 211.3 (ps) with xincreasing from 0 to 1 under H¼5.6 kOe. Figures 3(b) and3(c) show the dependence of the fandsonHfor typical samples, respectively. The Hdependence of the fis almost identical for all samples since the in-plane uniaxial anisotropy is much weaker than the shape anisotropy 2 pM2 S and the saturation magnetization almost does not changewith x. For a specific sample, the Hdependence of the fcan be well fitted by Eqs. (1)and(2), where the effect of the in- plane uniaxial anisotropy is neglected. The gfactor is found to fall in the range of 1.99–2.14 for all samples and is close to the values deduced in the FMR technique. Figure 3(b) shows that the Hdependence of scan be fitted by Eq. (3) assuming that the aisindependent of the H. It is indicated that the extrinsic inhomogeneous contribution to the mag- netic damping can be neglected.24–28Accordingly, the intrin- sica0can be obtained. Moreover, fitted results show that the sdecreases at higher Halthough it changes little within a small Hregion. Figures 4(a)–4(d) show the xdependence of the KU, the coercivity HC, the intrinsic a0, and the resonance peak-to- peak linewidth DHppat the easy axis. The KUincreases by a factor of two for xincreasing from 0 to 1. Figure 4(b) shows that the coercivity also increases from 6 Oe to 24 Oe when the atomic concentration of Pt changes from 0 to 1.0, inagreement with the K U. The intrinsic a0is enhanced by a fac- tor of five for xbetween 0 and 1.0, as shown in Fig. 4(c). Thea0of the present disordered binary FePt alloys is equal to 0.044, close to the reported results of 0.04–0.06,29,30but much larger than that of 0.006 for pure Fe films.31,32As shown in Fig. 4(d), the resonance linewidth DHppshows a variation trend similar to that of the a0. The measured line- width is expected to consist of extrinsic and intrinsic terms DH0andDHin.33Since the latter DHinis proportional to both thefand the intrinsic a0according to the following equa- tion34DHin¼2a0x=ðffiffiffi 3p cÞ. With the value of the a0deter- mined by the TRMOKE, the DHincan be calculated, as shown in Fig. 4(d). It is shown that the measured resonance linewidth is close to the intrinsic one and the inhomogeneity contribution DH0can be neglected. These phenomena agree with the weak dependence of the relaxation time on the Hin Fig.3(c)and the existence of one unique uniform resonance in the FMR spectra in Fig. 2(a)albeit the apparent change of the microstructure with the film depth in Fig. 1(b). It is there- fore suggested that either the magnetization or the induced magnetic anisotropy is identical for amorphous and polycrys-talline FePd. FIG. 3. Measured TRMOKE results for x¼0.4 with H¼1.6, 4.8, and 8.0 (kOe) (a), the f(b) and s(c) as a function of Hfor typical samples. Solid lines in (a), (b), and (c) are fit results. In (b), all samples have close depend- encies of the fonH. In (c), the fitted results (red line) agree well with the measured data assuming that the salmost does not change with the reso- nance frequency and the H.FIG. 4. Ku(a),HC(b),a0(c), and the measured DHpp(solid box) and the intrinsic DHin(circle) (d) as a function of x. The solid lines in (a), (b), (c), and (d) serve a guide to the eye. The error bars in (c) are added.113908-3 Ma et al. J. Appl. Phys. 116, 113908 (2014) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.217.58.222 On: Tue, 25 Nov 2014 23:36:01For the disordered FePdPt alloys, the KUanda0both increase with increasing Pt atomic concentration. For L1 0 FePtPd alloys, the perpendicu lar magnetic anisotropy and the damping parameter are also found to increase with x.24 As leading parameters of the a0, the lattice constant, the density of states near Fermi level, Curie temperature,electron-phonon scattering rate , and spin and orbital mag- netic moments are all expected to change little with x. Therefore, the changes of a 0in the ordered and disordered ternary alloys are attributed to the tuning effect of the SOC strength nwhen Pd atoms are replaced by heavier Pt atoms.14,17,24,35–38With the precessional magnetization, the shape of the Fermi surface is in the non-equilibrium state and the electron-hole pairs are produced due to the SOC. Meanwhile, the relaxation of the pairs is alsostrongly related to the SOC strength, where the energy and angular momentum are transfe rred from the spin system to the lattice. 39Within the torque-correlation model,36,37for example, the intrinsic a0is proportional to n3andn2for either intraband or interband electron-hole transitions, respectively. Therefore, the a0should increase with increasing xalthough the effective SOC strength is not known for the disordered FePdPt alloys. It is noted that the present disordered FePdPt alloys exhibit different magneticdynamics, compared with the L1 0FePtPd. The a0changes linearly with xfor disordered FePtPd alloys, whereas it changes slowly and sharply at low and high xfor L1 0 FePtPd alloys, respectively. For high atomic concentra- tions, the intrinsic a0and the KUin disordered FePtPd alloys are much smaller than those of L1 0FePtPd.24The large discrepancy in both the magnetic anisotropy and the damping parameter between ordered and disordered FePdPt alloys is caused by their different crystalline struc-tures. The Fe, Pt, and Pd atoms in disordered alloys are randomly distributed and the al loys are of FCC structure, whereas the Fe and Pt/Pd atoms in ordered alloys are alter-natively stacked along the c-axis and the lattice constant along the caxis is smaller than those along the aand b axes, leading to lower structural symmetry. Ordered anddisordered alloys may have different orbital moments for F e ,P t ,a n dP da t o m s . 40In order to further reveal the mech- anism of the magnetic damping in the disordered magneticalloys, 11–15,24,36–38detailed theoretical investigations are required. Moreover, the present approach to tune the in-plane uniaxial anisotropy i n disordered alloys is differ- ent from the varying pre paration procedures.18For the Ni-Fe alloy films, the induced in-plane anisotropy during deposition depends on the fabrication condition such as thesubstrate temperatu re during deposition. CONCLUSION In summary, we have studied the magnetization dynam- ics in disordered FePdPt ternary alloy films by FMR andTRMOKE. The intrinsic a 0and, in particular, the induced KUincrease with increasing atomic concentration of Pt, which is explained as a result of the larger SOC energy ofheavier Pt atoms, compared with that of Pd ones. The varying Pt/Pd atomic concentration allows us to tune themagnetization dynamics properties and the induced uniaxial anisotropy in disordered alloys, which is of practical applica- tions in spintronic devices. 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5.0021093.pdf

J. Appl. Phys. 128, 100902 (2020); https://doi.org/10.1063/5.0021093 128, 100902 © 2020 Author(s).Point defects in two-dimensional hexagonal boron nitride: A perspective Cite as: J. Appl. Phys. 128, 100902 (2020); https://doi.org/10.1063/5.0021093 Submitted: 07 July 2020 . Accepted: 26 August 2020 . Published Online: 09 September 2020 Jijun Zhang , Rong Sun , Dongliang Ruan , Min Zhang , Yanxi Li , Kai Zhang , Faliang Cheng , Zhongchang Wang , and Zhi-Ming Wang COLLECTIONS This paper was selected as Featured ARTICLES YOU MAY BE INTERESTED IN Progress on and challenges of p-type formation for GaN power devices Journal of Applied Physics 128, 090901 (2020); https://doi.org/10.1063/5.0022198 Magnetic droplet solitons Journal of Applied Physics 128, 100901 (2020); https://doi.org/10.1063/5.0018251 Anisotropic properties of monolayer 2D materials: An overview from the C2DB database Journal of Applied Physics 128, 105101 (2020); https://doi.org/10.1063/5.0021237Point defects in two-dimensional hexagonal boron nitride: A perspective Cite as: J. Appl. Phys. 128, 100902 (2020); doi: 10.1063/5.0021093 View Online Export Citation CrossMar k Submitted: 7 July 2020 · Accepted: 26 August 2020 · Published Online: 9 September 2020 Jijun Zhang,1,2,3 Rong Sun,2Dongliang Ruan,1Min Zhang,1Yanxi Li,4Kai Zhang,5 Faliang Cheng,1,a) Zhongchang Wang,2,a) and Zhi-Ming Wang3,a) AFFILIATIONS 1Guangdong Engineering and Technology Research Centre for Advanced Nanomaterials, School of Environment and Civil Engineering, Dongguan University of Technology, Dongguan 523808, China 2International Iberian Nanotechnology Laboratory (INL), Braga 4715-330, Portugal 3Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu 610054, China 4Department of Materials Science and Engineering, Stanford University, Stanford, California 94305, USA 5Key Laboratory of Nanodevices and Applications, i-Lab, Suzhou Institute of Nano-Tech and Nano-Bionics (SINANO), Chinese Academy of Sciences, Suzhou 215123, China a)Authors to whom correspondence should be addressed: zhongchang.wang@inl.int ,chengfl@dgut.edu.cn , and zhmwang@uestc.edu.cn ABSTRACT Two-dimensional (2D) hexagonal boron nitride (h-BN) is one of the most promising materials for many technological applications ranging from optics to electronics. In past years, a property-tunable strategy that involves the construction of electronic structures of h-BN throughan atomic-level design of point defects has been in vogue. The point defects imported during material synthesis or functionalization bydefect engineering can endow h-BN with new physical characteristics and applications. In this Perspective, we survey the current state of theart in multifunction variations induced by point defects for 2D h-BN. We begin with an introduction of the band structure and electronic property of the pristine h-BN. Subsequently, the formation and characterization of the most obvious point defects and their modulation in electronic structures of h-BN nanomaterials are envisaged in theory. The experimental results obtained by atom-resolved transmission elec-tron microscopy, magnetic measurement, and optical measurements have provided insights into the point defect engineered structures andtheir corresponding emerging properties. Finally, we highlight the perspectives of h-BN nanomaterials for heterostructures and devices. ThisPerspective provides a landscape of the point defect physics involved to demonstrate the modulation of the structure and functionalities in h-BN and identify the roadmap for heterostructure and device applications, which will make advances in electronics, spintronics, and nanophotonics. Published under license by AIP Publishing. https://doi.org/10.1063/5.0021093 I. INTRODUCTION Two-dimensional (2D) hexagonal boron nitride (h-BN), known as “white graphene, ”is an important member of the 2D material family. 1–8Defect-free h-BN with boron (B) and nitrogen (N) atoms strongly bonded in a 2D honeycomb lattice is thermally and chemically stable and has a high mechanical strength.9,10These properties render h-BN widely used as thermally conductive fillers, anti-oxidation lubricants, and protective coatings, etc.11Recently,the realization of 2D heterostructures has aroused widespread utili- zation of 2D h-BN, servicing as a substrate for graphene,12–15an encapsulant for metal halides,16and a gate dielectric for transition metal dichalcogenides.17–19Moreover, due to the higher electroneg- ativity of N than B atoms, the electrons in sp2-hybridized B –N bonds cannot be delocalized and are more confined to the Natoms. This phenomenon leads to the polarity of B –N bonds and an ionic character of h-BN. 20Thus, h-BN behaves as an insulator with a wide bandgap of ∼6e V .21,22The wide gap in the defect-freeJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 100902 (2020); doi: 10.1063/5.0021093 128, 100902-1 Published under license by AIP Publishing.h-BN has aroused considerable interest in regulating its electronic structure and functionalization of its physical properties. Band engineering in bulk crystals has already been investi- gated for many years. In solid materials, like semiconductors andmetals, lattice imperfections often impose strong influence on theirelectrical, optical, and mechanical performance. 23–25The approach to tailor electronic structures of solids can be traced to the modula- tion of atomic structures by point defects, line defects, interfacialdefects, and volume defects. 24For 2D systems, it sparks extensive explorations in engineering electronic and atomic structures sincethe discovery of graphene and other layered materials. 26–352D materials are often abundant in structural defects, which can be induced in synthesis or subsequent treatment.36–40In 2D systems, structural defects are more likely to appear in the form of pointdefects and edges. 41–44The edge atoms in a 2D layer may exhibit distinct unoccupied states and electronic states, which are variable with the edge configurations.45–48The point defects, including vacancies, adatoms, anti-sites, substitutions, and dopant atoms, arethe most common defects in 2D systems. 49–51It has already been proven that point defects can alter or even create many novel physi-cal or chemical properties in 2D materials. 52–56Graphene, the most promising material in electronics, is nonmagnetic and inert in most chemical reactions. The hydrogen atoms adsorbed on a carbonlattice, however, can induce local magnetic ordering in graphene. 53 By doping nitrogen or boron atoms into graphene, the inert gra- phene becomes active in catalytic reactions.57,58 Compared to the zero-bandgap graphene, the wide bandgap (∼6 eV) of h-BN allows the band engineering in a wide range.59–64 Point defects can create energy levels in the bandgap and inject charge-scattering centers in h-BN layers, which can alter local elec- tronic structures and endow magnetic and photonic functionalities.65–69In theory, when adsorbed on an h-BN lattice, hydrogen or fluorine atoms often cause an asymmetric charge dis-tribution, making the B or N atoms unsaturated, which results inlocal ferromagnetic or antiferromagnetic orders. 70Furthermore, point defects hosted in h-BN are particularly important in quantum technologies, as they act as color centers for quantumemissions and can be tailored to obtain emitters with narrow linewidth and in ultra-brightness. 71However, the new energy states in h-BN created within the bandgap by intrinsic defects or impurities make it difficult to control and access the regulated properties effectively.72–75Nevertheless, much effort has been devoted to producing 2D h-BN with structural perfection and to introducingpoint defects in a precisely controllable way. The deposition of high-quality h-BN is currently being achieved via the use of high- purity reactants, the choice of substrates, and the control of deposi-tion conditions. 76–78To realize the precise control of point defects, a lot of advanced techniques have been applied, including heattreatment, etching, plasma processing, and electron radiation. 79–85 Studies show that these steps can sharply improve the purity of the single photon by reducing the general autofluorescence of pristineh-BN samples. In addition, the electronic states arising from thepoint defects can be further engineered by using external fields,including electrical field, 86–89magnetic field,90,91strain field,92–97 and pressure.29The engineering of defect states allows the control of functional properties in h-BN at the atomic scale. As a result,the unique electronic properties accompanying the emergingnew properties have made h-BN attractive for fundamental physics and potential applications ranging from electronics and spintronics to nanophotonics. 98–101 In this Perspective, we present the recent progress of local microstructure characteristics and functionality variation of h-BNthrough tuning point defects. We start by classifying the formation of point defects, including vacancies, adatoms, anti-sites, dopant atoms, and substitutional impurities, and how band structure reactsto the formation of defects. We will mainly focus on the pointdefects in the few-layered h-BN and their corresponding electronic,magnetic, and optical properties. We discuss the modulation of electronic and optical properties through defect engineering and the construction of heterostructures and devices with new function-alities. Finally, we present an overview of the challenges of this areaand the promising directions toward future applications of pointdefect engineered h-BN are outlined. II. ATOMIC AND ELECTRONIC STRUCTURE OF PRISTINE HEXAGONAL BN The bulk h-BN has a layered structure, and the crystal struc- ture and parameters of the monolayer h-BN is the same as those of graphene. However, as a two-atom system, B and N atoms contrib-ute a completely different electronic structure to the monolayerh-BN from graphene. Graphene is a zero-bandgap metal, while themonolayer h-BN is an insulator or a wide bandgap semiconductor. A. Atomic structure of pristine h-BN Bulk h-BN has a layered structure similar to graphite. Between the atomically flat layers, B and N atoms are alternatively stackedvia van der Waals (vdW) interaction, exhibiting an AB stacking pattern [ Fig. 1(a) ]. Within a 2D layer, B and N atoms are linked in asp 2-hybridized honeycomb lattice by strong covalent bonds [Fig. 1(b) ].11The bulk h-BN belongs to the space group of P63mmc, showing a layer spacing of 3.30 −3.33 Å and lattice con- stants of 2.504 Å.20The difference in electronegativity between B and N atoms generates polarity of B –N bonds, endows the covalent B–N bonds with a remarkable ionic character, and makes h-BN a wide-gap insulator.102 B. Electronic structure of pristine h-BN To unveil the electronic characteristics, different theoretical modes have been employed to calculate electronic structures ofh-BN in both bulk and monolayer forms. Currently, the theoreticalsimulations can be categorized into three aspects: the tight binding (TB), density functional theory (DFT), and Green ’s function quasi- particle (GW) methods. 99The differences in theoretical models and parameters based on simulation methods make the calculatedbandgaps of h-BN scatter over a broad range. 103Previous theoreti- cal analysis based on the TB approximation has indicated that the bandgap at the K corner point of the monolayer h-BN is 5.4 eV104 or 5.5 eV.105Under this approximation, Robertson106studied the binding energy and wave function of B 1 s→2pcore exciton, which is small and bound by the medium-range part of the core-hole potential, thereby impacting h-BN ’s electronic state. With the struc- tural and electronic optimization, a direct bandgap of 4.90 eV atJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 100902 (2020); doi: 10.1063/5.0021093 128, 100902-2 Published under license by AIP Publishing.the K point for the monolayer h-BN is predicted by the DFT approach, as shown in Fig. 1(d) .107Different from the results described above, the local density approximation (LDA) and gener- alized gradient approximation (GGA) based on the DFT schemes generally predict indirect bandgaps. The calculations using theLDA functional show that the monolayer h-BN possesses an indi-rect bandgap ranging from 4.27 eV to 4.58 eV between the valence band maximum (VBM) near the K-point and the conduction band minimum (CBM) at various points (K, Γ, or M-point). 108,109In thecase of the GGA functionals, the predicted bandgaps are about 0.2∼0.4 eV larger than that determined by the LDA showing VBM near the K-point and CBM mostly at the Γ-point.69,107,110 Furthermore, the corresponding bandgap of the monolayer h-BN is increased to 6.0 ∼8.43 eV calculated by the GW method, which is closer to the experimental data.103,111This is because, as one of the most accurate theoretical approaches for band structure simulation, the GW method considers the electron –electron interactions between atoms.112Yet, the agreement on the direct/indirect band FIG. 1. Atomic and band structure of h-BN. Atomic structure of (a) bulk and (b) monolayer h-BN. (c) Electronic band structure of the bulk h-BN calculated by the LDA (black lines) and GW (purple lines) calculations. (d) Electronic band structure of the monolayer h-BN calculated through the DFT approach. Panel (a) is reproduced with permission from Alem et al. , Phys. Rev. B 80, 155425 (2009). Copyright 2009 American Physical Society. Panel (b) is reproduced with permission from Tran et al. ,N a t . Nanotechnol. 11,3 7–41 (2016). Copyright 2016 Springer Nature. Panel (c) is reproduced with permission from Arnaud et al. , Phys. Rev. Lett. 96, 026402 (2006). Copyright 2006 American Physical Society. Panel (d) is reproduced with permission from Ekuma et al. , Phys. Rev. Lett. 118, 106404 (2017). Copyright 2017 American Physical Society.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 100902 (2020); doi: 10.1063/5.0021093 128, 100902-3 Published under license by AIP Publishing.structures of the monolayer h-BN has not been reached in theoreti- cal calculations. This contradiction may be caused by the very flat conduction and valence bands along the M –K–H and K –H direc- tions, respectively. Recently, Elias et al.113have confirmed in exper- iments that the monolayered h-BN presents a direct bandgap of6.1 eV via the combination of photoluminescence and reflectance spectroscopy in the deep-ultraviolet. It should be pointed out that the bandgap characteristics can be significantly affected by the stacked-structure modes of h-BNlayers. 114,115A slight shift in the settings of interlayer interactions could generate pronounced differences in the calculated electronic structures.99The bulk h-BN with various stacking structures is cal- culated to show a bandgap ranging from 2.9 to 4.9 eV, which isreduced by 0.2 ∼1.5 eV compared with that of the monolayer h- BN. 103,110,111For an AB stacked h-BN, both the LDA and GW sim- ulations consistently show an indirect (M –H) bandgap structure [Fig. 1(c) ].116The AA stacking has an indirect (K –M) bandgap structure as predicted by the LDA, while a direct bandgap at the Kpoint is predicted by the GGA. 109Similarly, both direct and indi- rect bandgaps of the bulk h-BN have been reported experimen-tally. 21,114,117Thus, the investigation on band structures of the stacked h-BN remains challenging. III. FORMATION AND ELECTRONIC STRUCTURE OF POINT DEFECTS IN HEXAGONAL BN With the adoption of electronic structure modulation strategy through point defects, many theoretical and experimental effortshave been devoted to clarifying the underlying physical mechanism.Clarifying the fundamental understanding of the interplay betweendefects and electronic states can tailor local band configurations and increase material ’s tuning flexibility, thereby facilitating h-BN toward many potential applications. In this section, the category ofpoint defects and their formation mechanisms as well as the theo-retical electronic structures will be discussed. A. Point defects in h-BN 2D h-BN often contains various types of defects, such as vacancies, interstitial atoms, dislocations, grain boundaries, andedges. 48,59,118 –120All these defects in h-BN will not only change its local atomic structure but also engineer its electronic structures. In the following, we will discuss point defects in h-BN nanosheetsfrom both the theoretical and experimental aspects. 1. Point defects identified by molecular dynamics calculations Figure 2 summarizes the atomic structures of the Stone –Wales (SW), boron –boron (BB), anti-site, tetrahedron defects, and inter- stitials in h-BN. The SW, BB, anti-site, and tetrahedron defects arecalculated by classical molecular dynamics with temperatures ranging from 0 K to 4000 K. 121The SW defect is where four hexa- gons are transformed into two pentagons and two heptagons[Fig. 2(a) ]. This transformation is appeared at 0 ∼10 K and derived from the 90° rotation of a pair of B –N bonds without removing or adding any atoms. We can imagine that the pronounced crystallo- graphic distortions caused by the lattice variation should play asignificant role in dominating the local energy in the 2D planes. Generally, the SW defect is known to appear in graphene 122but has not been experimentally observed in h-BN yet. When the tem-perature rises to the melting point ( ∼4000 K), a BB defect is detected. The BB defect originates from a broken BN bond, asshown in Fig. 2(b) . The B atom transforms to a metastable state with the B –N bond elongated from 1.47 Å to 1.53 Å and the forma- tion of two B –B bonds in a much larger bond length (1.9 Å). The NB 3tetrahedron [ Fig. 2(c) ] is another in-plane defect and forms where one N atom pops up above three in-plane B atoms. The NB 3 tetrahedron defect shows a formation energy of 4.4 eV and is unsta- ble. To better describe the lattice parameters and formation ener- gies of the h-BN systems, Los et al. have implemented the extended Tersoff potential in molecular dynamics simulations at zero tem-perature. 123The formation energy of SW is calculated to be 7.20 and 7.34 eV by the optimized methods, smaller than the values (8.6 and 8.8 eV) obtained by classical molecular dynamics.121The anti- site [ Fig. 2(d) ], which is mostly found in semiconductors, occurs when the locations of B and N atoms in a hexagonal lattice switch.The formation energy of anti-site is calculated to be 8.1 eV. Thelength of B –B and N –N bonds is 1.60 Å and 1.44 Å, respectively, and the B –N bond decreases from 1.47 Å to 1.36 Å. The interstitial defects are formed by inserting an extra atom between the bilayerh-BN. The intercalated atoms are chemically bonded with eachlayer and vertically or diagonally bonded between the bilayer h-BN [Figs. 2(e) and 2(f)]. The formation energy of interstitials ranges from 6.6 to 8.0 eV, depending on species of the intercalated atomsand bonding structures. Note that it often takes much effort invisualizing these defects experimentally. 2. Vacancies in h-BN investigated by transmission elec- tron microscopy The missing lattice atom is the simplest point defect in h-BN. Vacancies, including boron-vacancy (V B), nitrogen-vacancy (V N), and boron-nitrogen-vacancy (V BN), have already been experimen- tally observed by the state-of-the-art scanning transmission electron microscopy (STEM). As shown in Figs. 3(a) –3(d), a high density of atomic defects in the mechanical exfoliated monolayer h-BN are observed bySTEM. 124In order to avoid the damage of atomic-layer samples125 and increase the resolution for characterizing B and N atoms, theexperiment was operated in an ultrahigh-resolution aberration-corrected JEOL ARM-200F at an accelerating voltage of 60 kV.From the results of annular dark-field (ADF) STEM images andtheoretical calculations, vacancies with different atomic structures have been distinguished as V B,VN,VBN, and V B3N.I n Fig. 3(a) , lattice atoms in the perfect and defective regions (circled regions)are discriminated in the monolayer h-BN. From the enlarged image[Fig. 3(b) ] and simulated ADF STEM image [ Fig. 3(c) ], it is easy to distinguish the individual B from N atoms by the image contrast of the spots, which is approximately proportional to the square of the atomic number (Z 2).126Thus, the heavier N atoms appear in a brighter image contrast and B atoms are present in a darker imagecontrast. Figures 3(b) and3(c)show images of a single V B, where a less bright B atom disappeared from a hexagonal network. With the formation of V B, substantial deformation appears in the localJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 100902 (2020); doi: 10.1063/5.0021093 128, 100902-4 Published under license by AIP Publishing.structure with the V Bedge terminated by three doubly coordinated N atoms. Figure 3(d) presents a statistical analysis of the density for the four types of point defects. Obviously, V Bpresents the highest density of 1.1 × 105μm−2, about two orders of magnitude higher than that of V Nand V BNand about five times higher than that of V B3N. These statistical data confirm that V Bis the most fre- quently appeared point defect in as-prepared h-BN samples. However, the point defects in reduced types are more generally created by ion/electron irradiation of h-BN nanosheets.62,85,98Alem et al.20prepared h-BN nanosheets with triangle-shaped boron vacancies in different sizes through the reactive ion etchingmethod. Figure 3(e) shows the models of the observed monova- cancy (V B) and multivacancy (V B3N). As one of the multi- vacancies, V B3Ncan be created by removing one N atom and three surrounded B atoms, which presents the same orientation as V B. They have found that the B-typed vacancies with an N-terminated zigzag configuration are formed predominantly throughout the atomic-layer h-BN.It becomes increasingly interesting when the monovacancy is produced in a suspended bilayer h-BN. Alem et al.61have uncovered the structural distortion of V Bin the bilayer h-BN by combining advanced STEM with DFT calculations, as shown in Figs. 3(f) –3(k). At the V Bsite, the bilayer h-BN experiences a strong interlayer bonding, which is noticeably different from the bilayer graphenewhere the isolated defects are independent between the layers. 48The atomic structure models [ Figs. 3(f) and 3(i)] illustrate the recon- struction of one and two B-N interlayer covalent bonds across thelayer with V Band the intact layer. The interlayer bonds are caused by the Coulomb interaction of the highly negative charged vacancysites. As a result of the interlayer bonding reconstruction, both the in-plane and out-of-plane atomic displacements at the vacancy sites are visualized, as seen in the reconstructed aberration correctedhigh-resolution (HRTEM) images in Figs. 3(h) and3(k). The local structural distortions at the V Bsite are stable and a lattice symmetry breaking from the symmetric p3configuration to the asymmetric pmconfiguration is detected. The structural distortions and the FIG. 2. Point defects predicted theoretically in the monolayer h-BN. (a) Stone –Wales, (b) boron –boron, (c) tetrahedron defect, (d) anti-site, (e) B interstitial vertically bound between the bilayer h-BN, and (f) B interstitial diagonally bound between the bilayer h-BN. Panels (a) –(c) are reproduced with permission from Slotman et al. , J Phys. Condens. Matter 25, 045009 (2013). Copyright 2013 IOP Publishing Ltd. Panels (d) –(f) are reproduced with permission from Los et al. , Phys. Rev. B 96, 184108 (2017). Copyright 2017 American Physical Society.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 100902 (2020); doi: 10.1063/5.0021093 128, 100902-5 Published under license by AIP Publishing.FIG. 3. Observation of N-terminated vacancies in h-BN nanosheets. (a) STEM-ADF image of the monolayer h-BN containing boron vacancies in the monolayer h-BN , reproduced from Los et al ., Phys. Rev. B 96, 184108 (2017). Copyright 2017 American Chemical Society. (b) Enlarged image of a single V Bin (a). (c) Simulated STEM-ADF image of a single V B. (d) Statistical densities of V B,VN,VBN, and V B3N. Reproduced with permission from Wang et al. , Nano Lett. 18, 6898 (2018). Copyright 2018 American Chemical Society. (e) Atomic models of V Band V B3N. Reproduced with permission from Alem et al. , Phys. Rev. B 80, 155425 (2009). Copyright 2009 American Physical Society. (f) –(k) Probing structural distortions at boron monovacancies in the bilayer h-BN. Models [(f) and (g)], simulated [(g) and ( j)], and recon- structed [(h) and (k)] AC-HRTEM images of the (upper) single bonded bilayer h-BN and (bottom) double bonded bilayer h-BN created upon a boron vacancy. Reproduced with permission from Alem et al. , Phys. Rev. Lett. 106, 126102 (2011). Copyright 2011 American Physical Society.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 100902 (2020); doi: 10.1063/5.0021093 128, 100902-6 Published under license by AIP Publishing.reduction of lattice symmetry observed in experiments exactly coin- cide with the simulated results, as shown in Figs. 3(g) and3( j). It is worth mentioning that the single nitrogen vacancy (V N), which forms due to the absence of one nitrogen atom, is ultralow indensity ( Fig. 3 ) .S i m i l a r l y ,t h em u l t i p l ev a c a n c i e sa r eg e n e r a l l yr e s o l v e d as V B3Nand V B6N3 rather than V B1N3. The favored generation of boron vacancies is reasonable, since B is more easily to be removed by electron beam bombardment or knock-on damage at a smaller thresh-old beam energy of 74 ∼84 keV as compared with that of N atoms (84∼140 keV), resulting in vacancies terminated with N atoms. 127,128 However, at elevated temperatures, the species of vacancy becomes different. Boron-terminated vacancies in h-BN are discov- ered above 500 °C.129,130Cretu et al .129visualized the boron- terminated tetravacancy (V B1N3) in aberration-corrected STEM using an accelerating voltage of 60 kV at 500 °C [ Fig. 4(a) ]. TheVB1N3 defect is created where one B atom and three surrounding N atoms are removed from the perfect hexagonal lattice [ Fig. 4(b) ]. In VB1N3, the distance between two neighboring B atoms is 2.02 ± 0.10 Å, much smaller than that of the perfect h-BN(2.504 Å). The reduced distance between the coordinated B atomsindicates a structural relaxation around V B1N3. Interestingly, they observed an abnormal spectrum for B atoms at the V B1N3 site in electron energy-loss spectroscopy (EELS) mapping. The spectrumexhibits a shift in the π* peak and a new peak within the π* and σ* peaks. The calculated electron density map and boron K edge ofthe EELS curves in Fig. 4(c) demonstrate that the introduction of vacancies breaks the symmetry of charge distribution, making the π* peak shift to a lower energy experimentally at 191.1 eV and theo- retically at 191.6 eV. The asymmetry also reduces the energy of theσ* peak, and the localized orbitals of the edge boron atoms induce FIG. 4. Formation and dynamics of B-terminated vacancies in h-BN at elevated temperatures. (a) ADF image and (b) atomic model of a V B1N3 tetravacancy in the mono- layer h-BN probed at 500 °C. (c) Electron density map (top) and the boron K edge EELS signature (bottom) of the V B1N3 tetravacancy simulated by first-principles calcula- tions. Panels (a) –(c) are reproduced with permission from Cretu et al. , Phys. Rev. Lett. 114, 075502 (2015). Copyright 2015 American Physical Society. (d) –(f) HRTEM images (left) and corresponding atomic models (right) of (d) triangle defects at 500 °C and (e) hexagon- and (f) parallelogram-shaped defects at 900 ° C. Reproduced with permission from Pham et al. , Nano Lett. 16, 7142 (2016). Copyright 2016 American Chemical Society.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 100902 (2020); doi: 10.1063/5.0021093 128, 100902-7 Published under license by AIP Publishing.a shoulder peak at 196.2 eV (theoretically at 195.2 eV). Furthermore, Pham et al.130have investigated the stability and dynamics of B-terminated vacancies in h-BN at 80 keV from roomtemperature to 1000 °C. As depicted in Figs. 4(d) –4(f), the parallelogram- and hexagon-shaped defects are preferentially gen-erated above 700 °C, while the triangular defects are exclusively observed below 500 °C. The long-range and stable N- and B-terminated zigzag edges can coexist in a same defect with acorner of 120°. These studies on the relationship between thevacancy structures and temperatures provide a feasible way to regu-late the shapes and styles of h-BN defects. 3. Dopant atoms in h-BN Dopant atoms, e.g., hydrogen,131 –133carbon,134 –136 oxygen,135 –137fluorine,46,68,138and metal atoms88,139,140areanother typical kind of defect s in h-BN. These impurity atoms often occur in 2D crystal samples because of the special process of the chemical vapor deposition or the subsequent treatments.The B and/or N atoms in h-BN ca n be substituted during these processes. Krivanek et al . 135have employed an atom-by-atom analysis to resolve and identify the substitutions in the monolayer h-BN. Figure 5(a) presents the ADF STEM image of the substitu- tional C and O atoms in the monolayer h-BN. The hexagonalring marked by the green circle contains three darker B atomsand three brighter N atoms. The hexagonal ring marked by theyellow circle consists of six atoms of similar brightness with their intensities in the middle of B and N atoms. To interpret the devi- ations observed in the yellow circle, the as-recorded image hasbeen deconvolved and smoothed [ Fig. 5(b) ].Figure 5(c) displays the line profiles of ADF intensity taken across the arrows markedinFig. 5(b) . Profile X –X 0starts from two atoms surrounded in FIG. 5. Identification of atomic substitutions in the monolayer h-BN by an aberration-corrected STEM. (a) As recorded and (b) deconvolved ADF STEM image of a mono- layer h-BN. (c) Line profiles of the ADF intensity recorded along the lines X –X0and Y –Y0in panel (b). (d) The simulated atomic structure of h-BN with substitutional impu- rity atoms (red: B, yellow: C, green: N, and blue: O) overlaid on the deconvolved ADF STEM image. Reproduced with permission from Krivanek et al. , Nature 464, 571 (2010). Copyright 2010 Springer Nature.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 100902 (2020); doi: 10.1063/5.0021093 128, 100902-8 Published under license by AIP Publishing.the yellow circle, and profile Y –Y0starts with an especially bright atom. The ADF intensity of the beginning atoms of X –X0and Y – Y0lines is obviously different from that of the B and N atoms recorded in the profiles. Upon the atomic number (Z) of B(Z = 5), N (Z = 7), C (Z = 6), and O (Z = 8), the first two atoms inthe profile X –X 0are considered to be C atoms, and the first atom in the profile Y –Y0is an O atom. Figure 5(d) shows the calculated atomic structure by the DFT, which is overlaid on the filteredADF STEM image. Interestingly, C atoms only choose to substi-tute B –N pairs rather than individual B or N atoms, while an O atom tends to substitute a single N atom. It should be noted that the above substitutions do not occur until the B and/or N atoms are ejected from the perfect h-BN layer by electron beam,forming the vacancies. Such a substitutional process could beregarded as electron beam-induc ed lattice damage and defect healing process. Their work provides a promising method to con- struct custom-designed 2D structures at the atomic level.B. Electronic structure of point defects in h-BN The introduction of point defects into the few-layered h-BN will impose remarkable deviations in coordination environment and chemical bonding from the original lattice, thus contributing to electronic structure engineering. Figure 6 summarizes the point defect-related electronic structures in the monolayer h-BN system investigated by the DFT calculations. Figures 6(a) and6(b) sketch the relaxed geometries (top) and the corresponding simplified band structure diagrams (bottom) of V Band V N.88The V Bdefect under- goes a Jahn –Teller distortion together with its symmetry reduced from the D3 h to C2v. Due to the crystal field splitting and spin exchange splitting, an occupied state (bσ 2) in the spin-up channel shifts down to the valence band, while the hole levels located above the valence band make the charge-neutral V Ba triple acceptor [Fig. 6(a) ]. In contrast to V B,VNretains the D3 h symmetry. The VN-related states largely originated from conduction-band orbitals. FIG. 6. Point defect-related electronic states in the monolayer h-BN. (a) and (b) Atomic models (top) and simplified band-structure diagrams (bottom) of V Band V N. Reproduced with permission from Huang et al. , Phys. Rev. Lett. 108, 206802 (2012). Copyright 2012 American Physical Society. (c) and (d) Geometrical structure and spin density of V NNBand V NCB. Reproduced with permission from Sajid et al. , Phys. Rev. B 97, 064101 (2018). Copyright 2018 American Physical Society. (e) Summary of point defect-related charge-state transition levels for h-BN. Reproduced with permission from Weston et al. , Phys. Rev. B 97, 214104 (2018). Copyright 2018 American Physical Society. (f) Dependence of the bandgap on the O Ncontents. Reproduced with permission from Weng et al. , Adv. Mater. 29, 1700695 (2017). Copyright 2017 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 100902 (2020); doi: 10.1063/5.0021093 128, 100902-9 Published under license by AIP Publishing.The energy levels of the V Ndefect occur in the gap with the eσ state close to the CBM and the occupied aπstate in the spin-up channel [ Fig. 6(b) ]. The V Ndefect can behave as both a donor and an acceptor. Figures 6(c) and6(d) show the relaxed geometries and calculated spin density of V NNBand V NCB, respectively.118The VNNBis a typical anti-site defect with a nitrogen vacancy neighbor- ing an N substitution for B atoms [ Fig. 6(c) ]. The V NNBdefect has a C2v symmetry. The unpaired electron density caused by V NNB defects is localized largely on pzorbitals of the substitutional N atom (N 1). The V NCBis a carbon-related defect constructed by an N vacancy with one of the surrounding B atoms substituted by a C atom [ Fig. 6(d) ]. The geometrical structure of V NCBis calculated to be C2v symmetry. For V NCB, the spin density mainly arises from both carbon sp2and two boron (B 1and B 2)s p2orbitals. Figure 6(e) summarizes the charge-state transition levels introduced by pointdefects, including vacancies, anti-sites, interstitials, and substitu- tions. 72When a lattice defect is present in a monolayer h-BN, the gap states associated with the atomic levels of the defect appear. Thetunable bandgaps make it reliable to control charge states of theh-BN system depending on the intensity of electron interactions.Furthermore, the concentration of point defects also plays an important role in manipulating the bandgaps of h-BN. As shown in Fig. 6(f) , the N atoms in the honeycomb lattice are randomly sub- stituted by O atoms in the O-doping process. With the increase inthe O concentration from 0 to 16.7 at. %, the calculated bandgaps decrease monotonically from 4.56 to 4.34 eV. 69 IV. PROPERTIES OF HEXAGONAL BN The performance of h-BN is strongly interlinked with its elec- tronic structure. According to the unique electronic states inducedby defects, it is expected that novel physical properties, such asmagnetic and optical properties, can be achieved and controlled inh-BN systems. A. Magnetism in h-BN A h-BN nanosheet with a perfect structure is a typical non- magnetic material. However, the structural defects like vacancies,adatoms, dopants, or edges might produce local magneticorders. 141 –144Based on the first-principles calculations, the removal of B or/and N atoms can break the threefold symmetry of h-BN and induce spin splitting and large magnetic moment.145 –147Du et al.145characterized the magnetism of the monolayer h-BN with nanodots, nanoholes, and antidots by the DFT calculation andfound that the B-terminated nanodot (V N) lacks magnetism due to the edge reconstruction. In contrast, the B-terminated nanoholes show a magnetic nature owing to the enhanced structural stability. More interestingly, the N-terminated vacancies are calculated to bestable above room temperature and endow the monolayer h-BNwith spin transport anisotropy and strong magnetism. Experimentally, the ammoniated defective h-BN nanosheets are found to exhibit robust room-temperature ferromagnetism. 148 Using an anthracene vapor-assisted transport method, h-BN nano- sheets with ∼1.46 vol. % V Ndefect are prepared. The as-prepared defective h-BN nanosheets show a room-temperature ferromagnetic character with saturation magnetization ( Ms)o f∼206 memu/g and coercivity ( Hc) of 205 Oe. To enhance the vacancy-related magneticproperty of h-BN, N-containing atmosphere (N 2and NH 3) heat treatments were applied. After annealing in N 2and NH 3,t h e defective h-BN nanosheets show an enhancement in ferromagne-tism. M sand Hcof the NH 3-treated h-BN nanosheets are 456 memu/g and 378 Oe at 5 K and 336 memu/g and 157 Oe at300 K, respectively. As shown in Figs. 7(b) and7(c),t h em i s s i n g N atoms in the hexagonal lattice break the crystal symmetry and cause an imbalance in spin-pol arization. When annealed in the NH 3atmosphere, the N atoms (electronegative) in NH 3can be easily adsorbed on B atoms (electropositive) at the V Nsites, forming new π–πattractive interactions. The new π–πinteractions will induce extra magnetic moments and should be responsible for the enhanced magnetic properties. In the case of adatoms, Huang et al.88and Li et al.139pre- dicted that doping transition-metal (TM) atoms adsorbed on thevacancy sites can easily tailor their local magnetism. Various chem- ical doping of h-BN by Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, and Zn atoms 88has been calculated as a potential strategy to endow the monolayer h-BN with ferromagnetism. In addition, the doping oflight atoms (H, C, O, and F) into the honeycomb BN lattice hasalso been reported to yield spontaneous spin polarization and room temperature ferromagnetism both in theory and in experi- ments. 69,70,143,149Zhao et al.141have synthesized C-doped h-BN (B-C-N) nanosheets. In the honeycomb lattice, B and/or N atomsare randomly substituted by C atoms, as shown schematically in Fig. 7(d) . The substitution prefers to occur at the outermost layer of h-BN nanosheets, resulting in an atomic ratio (C/B + C + N) of 8%.The C substitutions have endowed the nonmagnetic h-BN withroom-temperature ferromagnetism [ Fig. 7(e) ]. However, the mag- netic moment of B –C–N nanosheets is much lower than that depicted in Fig. 7(a) . To investigate whether C is the origin of the ferromagnetism, the B –C–N nanosheets are oxidized in O 2at 923 K for various time to remove C dopant. After burning for30 min, the ferromagnetic character of B –C–N nanosheets disap- pears [ Fig. 7(f) ]. The vanishing of ferromagnetic upon the removal of C dopants from the honeycomb lattice confirms that the obtained ferromagnetism of B –C–N nanosheets is attributed to the substitutional C-doping. Additionally, F-doping has also beenverified as a feasible way to acquire room-temperature ferromag-netism in h-BN. 68The XPS spectra in Fig. 7(g) evidently show the B –Na n dB –F bonding, which confirms the formation of F-BN nanosheets. The 8.1% F-doped h-BN exhibits Msof 18 memu/g [ Fig. 7(h) ], where the F atoms attached on B atoms yield partial charge and unpaired electrons in the neighboring three N atoms. With the increase in the F concentration, Ms becomes larger. The ferromagnet ism of the F-doped h-BN is very weak but robust at room temperature, in good agreement withthe previous results. 46,143In addition, the ferromagnetic character is found to coexist with the antiferromagnetic feature in the fluo- rinated h-BN [ Fig. 7(i) ]. The competition between the two mag- netic behaviors may cause frustrated magnetization, which hasalso been reported in graphene and MoS 2systems.150,151 Furthermore, it has calculated that the spin configuration of the TM/h-BN system can be converted from the high-spin state to the low-spin state by exerting an external electric field.88The mag- netic states of vacancy-engineered h-BN systems can be tuned bybiaxial strain, 146and the magnetic coupling stability of F-dopedJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 100902 (2020); doi: 10.1063/5.0021093 128, 100902-10 Published under license by AIP Publishing.h-BN can also be modulated by the external strain.70Thus, future research on the magnetism in h-BN systems could concentrate on the large-scale defect engineering and external-field tunable micro- scopic magnetic state.B. Optical properties and si ngle photon emitters in h-BN Bulk and multilayer h-BN are wide-bandgap semiconductors and exhibit no optical absorption of electromagnetic spectrum at the visible wavelengths (390 ∼700 nm). Therefore, they appear FIG. 7. Ferromagnetism in h-BN. (a) –(c) N monovacancies induced robust room-temperature ferromagnetism in h-BN nanosheets, reproduced from Machado-Charry et al., Appl. Phys. Lett. 101, 132405 (2012). Copyright 2012 American Chemical Society. (a) Hysteresis loops of as-prepared, N 2-annealed, and NH 3-annealed h-BN nano- sheets. (b) Atomic models of the defective h-BN with a V Ndefect. (c) Sketch showing the mechanism of the ammonia functionalization enhanced ferromagnetism in the defective h-BN. Reproduced with permission from Li, ACS Appl. Mater. Interfaces 9, 39626 (2017). Copyright 2017 American Chemical Society. (d) –(f) Carbon-doped boron nitride (B-C-N) nanosheets with high temperature ferromagnetism. (d) Schematic of the B –C–N. (e) Hysteresis loops of h-BN and B –C–N nanosheets. (f) Hysteresis loops of B –C–N nanosheets suffered high temperature burning in O 2. Reproduced with permission from Zhao et al. , Adv. Funct. Mater. 24, 5985 (2014). Copyright 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. (g) –(i) Room temperature ferromagnetism in fluorinated h-BN nanosheets. (g) XPS spectrum of B 1 s peak. (h) Hysteresis loop of h-BN nanosheets doped with 8.1% fluorine. (i) T emperature dependence of zero field cooling (ZFC) and field cooling (FC) susceptib ility under a mag- netic field of 500 Oe. Reproduced with permission from Radhakrishnan et al. , Sci. Adv. 3, e1700842 (2017). Copyright 2017 American Association for the Advancement of Science.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 100902 (2020); doi: 10.1063/5.0021093 128, 100902-11 Published under license by AIP Publishing.white in color and are generally referred to as “white graphene. ”152 –155However, pure h-BN nanosheets exhibit a strong absorption of the ultraviolet light between 200 and 220 nm.156 –158 A sharp luminescence peak and a couple of s-like free exciton absorption bands are observed at ∼215 nm in h-BN single crys- tals.117Pure h-BN nanosheets also exhibit an obvious cathodolumi- nescence emission in the deep ultraviolet range at room temperature.159,160However, due to the large bandgap, pure h-BN nanosheets only tend to emit ultraviolet light and present limitedoptical activity. 11Recently, defect engineering has been developed with the purpose of regulating bandgaps of h-BN for light activity enhancement. Atomic defects are highly desirable in ultraviolet and visible light luminescence, where they serve as color centers.161 –164 For instance, atomic defects such as B vacancy, N vacancy, and adatoms have been found to generate enhanced cathodolumines-cence and photoluminescence. 165 –169These atomic defects offer a potential way in designing high-efficient quantum fluorescent materials, rendering h-BN applicable to optoelectronics andnanophononics. Comtet et al. 82have compared the quantum emission property of h-BN with and without plasma treatment. As shown in Fig. 8(a) , the CVD-grown h-BN nanosheets without plasma treatment present blinking spots in two colors (red and green). Two emission spectracenter have been found at λ 1≈585 nm and λ2≈640 nm, signifying two types of defects within the CVD-grown h-BN [ Fig. 8(b) ]. However, for the exfoliated h-BN nanosheet treated by oxygen plasma [ Figs. 8(c) and8(d)], most of the samples depict only one single emission wavelength ( λ≈585 nm). The preferential creation of one type emitter indicates that there is only one type of active defectexisting throughout the h-BN sample after plasma treatment. This optical behavior confirms that thermal annealing represents a useful way in purifying quantum emitters, thereby enabling precise controlof defect type and concentration. In order to acquire quantum emis-sion with high bright and narrow emission lines, Schell et al. 163have studied the dependence of the quantum emission on the excitation wavelength. As shown in Fig. 8(e) , the quantum emitters marked in the white and yellow circles exhibit strikingly different strengths whenscanned using 530 nm, 550 nm, and 600 nm light. This phenomenonindicates that the emitters have a complex level structure and differentexcitation spectra. Two emission lines are detected at 656 and 676 nm in PL spectra [ Fig. 8(f) ]. And, the brightness of these two emission lines changes with the excitation wavelength ranging from 530 to620 nm. The brightest emission for the 656 nm line is excited at590 nm light. While for the 676 nm line, the brightest emission is recorded at 540 nm. Furthermore, the saturation curves, which have a positive correlation with the quantum efficiency, also present a strongexcitation wavelength dependence [ Fig. 8(g) ]. Thus, it is important to match the excitation wavelength to the emitter so as to acquire effi-cient quantum emission. Nevertheless, the quantum emissi on efficiency is still highly needed to be enhanced. 165A plenty of methods such as laser irradia- tion and ion implantation have been applied to regulate the species,distribution, and concentration of the point defects, which are key toenhance the emission efficiency. 89,163,167,170Kianinia et al.165have pro- posed a two-laser repumping strategy and investigated the photody- namic property and nonlinear behavior of the emitter in h-BN, asshown in Figs. 8(h) –8(k). In their work, two pairs of lasers areadopted: 675 nm (basic laser) plus 532 nm (repumping laser) and 708 nm (basic laser) plus 532 nm (repumping laser). Figures 8(h) and 8(i)d i s p l a yt h ee m i t t e rb r i g h t n e s sm a p sa n dP Ls p e c t r ao fa single defect excited by two lasers (675 nm plus 532 nm). A highlynonlinear increase in the emission intensity is yielded with the linearincrease in the excitation power of the 532 nm repumping laser. The repumping effect also causes a fast saturation of the emitters [Fig. 8( j) ]. When excited by a sole 708 nm laser, the saturation power of quantum emitter is ∼14 mW. While changed to the pair lasers, the saturation power is reduced by nearly one order of magnitude. Thephotokinetics of the nonlinear enhanced quantum emission can be elaborated by a four-level system, which contains the ground state, excited state, a fast-decaying intermediate state, and a long-lived meta-stable state. Imposing the 532 nm laser on the 675 nm laser willrepump the system from the intermediate state to the excited state andinhibit their fast decay to the ground state, thus suppressing the quan- tity of the intermediate state. As a co nsequence, the sharply decreased quantity of the intermediate state le a d st oar e d u c t i o ni nt h es a t u r a t i o n power. Furthermore, higher image resolution can even be achievedwith the co-excitation laser [ Fig. 8(k) ]. The sub-diffraction resolution of 63 ± 4 nm is obtained when the excitation power of 708 nm laser increases to 60 mW. This work offers potential ways in endowing h-BN with a high quantum efficiency and imaging the emitters with asuper-high resolution. The quantum functionality of the 2D nanosheets can be mod- ulated over a wide range by external fields (e.g., strain, magnetic field, electrical field, and temperature) via lowering their structuralsymmetry and energy levels. 171 –176Exarhos et al.177have demon- strated a strong magnetic-field-dependent photoluminescence phe-nomenon in h-BN at room temperature and found that, when an in-plane magnetic field is exerted along the h-BN nanosheet, the emitters show changes in color and brightness. To systematicallyinvestigate the magnetic-dependence of quantum emission in theh-BN system, researchers have employed a measurement system asillustrated in Fig. 9(a) . By rotating the sample and adjusting mag- netic goniometer, arbitrary field orientations ( α,ϵ, and β) can be obtained. For in-plane magnetic fields ( β= 0°), the PL variation exhibits both increased and decreased quantum emissions withrespect to the sample orientations [ Fig. 9(b) ]. The aligned 90° modulation period suggests a strong anisotropic PL pattern, which persists when changing the magnetic field strength. With the increase in the in-plane magnetic fields, the PL increases monoton-ically at α= 90° and reaches a saturation state at the field of ∼600 G [Fig. 9(c) ]. However, at α= 45°, the PL variation becomes non- monotonic by increasing first with the magnetic field and then decreasing at 70 G, which eventually falls far below the zero-fieldemission rate, as denoted by the dashed line. A strong dependenceof the PL variation on the out-of-plane magnetic fields ( β= 90° and 45°) has also been detected, which indicates an underlying 180° symmetry other than the 90° periodicity. The magnetic-field depen- dent PL indicates the presence of optically addressable spin defectsin h-BN. The field-dependent PL variation is likely to be producedby a spin-dependent inter-system crossing a transition betweentriplet and singlet manifolds. This work paves the way for the development of h-BN in 2D quantum spintronics. On the other hand, Grosso et al. 83reported that strain field can provide a dynamic way to modify single quantum emission inJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 100902 (2020); doi: 10.1063/5.0021093 128, 100902-12 Published under license by AIP Publishing.FIG. 8. Efficient quantum emission in h-BN nanosheets. (a) –(d) Plasma effect on single quantum emission properties. Reconstructed map of individual emitters, distribu- tion of emission wavelength, and center wavelengths of different samples for [(a) and (b)] CVD-grown h-BN nanosheets and [(c) and (d)] exfoliated h-B N nanosheets irradi- ated by an O plasma for 30 s. Reproduced with permission from Comtet et al. , Nano Lett. 19, 2516 (2019). Copyright 2019 American Chemical Society. (e) –(g) The dependence of single quantum emission efficiency on the excitation wavelength. (e) Confocal maps of h-BN nanosheets using 530 nm, 550 nm, and 600 nm ex citation. (f) PL map, the excitation wavelengths range from 530 to 620 nm. (g) Saturation curves for the emission lines of 656 nm and 676 nm. Reproduced with permissio nf r o m Schell et al. , Adv. Mater. 30, 1704237 (2018). Copyright 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. (h) –(k) Enhanced nonlinear photoemission in h-BN by a two-laser repumping scheme. (h) PL maps revealing a nonlinear enhanced emitter brightness upon two lasers. (i) PL spectra of single defect excited by one laser and two lasers. ( j) Saturation curves showing a fast saturation of the quantum emitters by two-laser repumping. (k) Negative ground-state depletion (GSD) i mages of the single defect under excitation with 708 nm laser (top) and 708 nm + 532 nm laser (bottom). Reproduced with permission from Kianinia et al. , Nat. Commun. 9, 874 (2018). Copyright 2018 Springer Nature.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 100902 (2020); doi: 10.1063/5.0021093 128, 100902-13 Published under license by AIP Publishing.h-BN. Both the tensile and compression strain have been applied to h-BN nanosheets via bending the polycarbonate substrate under a Poisson ’s ratio of 0.37 [ Fig. 9(d) ]. The applied strains lead to spec- tral shifts from −3.1 to 6 meV/% for single photon emitters in h-BN nanosheets [ Fig. 9(e) ]. The simulated strain-dependent spec- tral distribution [ Fig. 9(f) ] coincides with the experimental results,confirming the non-linear and non-monotonic nature of the quantum emitters. The energy shift has been theoretically ascribed to the strain-induced lattice deformation at the V NNBdefect site [Fig. 9(e) , inset]. The strain produces a change in the N –B bond length, which deforms molecular orbitals and perturbs their energylevels, thereby allowing to tune the energy of quantum emissions. FIG. 9. Modulation of single quantum emission from point defects in h-BN at room temperature. (a) –(c) Magnetic control of photoluminescence. (a) The setup for applying a magnetic field to h-BN nanosheets. β: in the x –z plane, the angle of the magnetic field and the sample plane (x –y plane). α,ϵ: in the sample plane, the orientation of the absorptive and emissive dipole, respectively. (b) Sample orientation dependence of PL variation under an in-plane magnetic field of 890 G. (c) In -plane magnetic dependence of PL variation. Reproduced with permission from Exarhos et al. , Nat. Commun. 10, 222 (2019). Copyright 2019 Springer Nature. (d) –(f) Strain modulated single photon emission. (d) Schematic of the experimental setup for applying strain to h-BN nanosheets. (e) Energy shift of single photon emission tu ned by the strain field. (f) Calculated optical response as a function of strain field. The inset shows the lattice structure of V NNBunder the strain field in different directions. Reproduced with permission from Grosso et al. , Nat. Commun. 8, 705 (2017). Copyright 2017 Springer Nature. (g) and (h) Pressure modulated defect emission. (g) Pressure-dependent PL spectra. (h) Pressure-dependent PL peak energies. The inset shows the lattice structure of V NNBthat leads to a single-photon emission peak shift under pressure. Reproduced with permission from Xue et al. , ACS Nano 12, 7127 (2018). Copyright 2018 American Chemical Society.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 100902 (2020); doi: 10.1063/5.0021093 128, 100902-14 Published under license by AIP Publishing.In addition, the quantum emission can also be manipulated by anomalous pressure. Xue et al .174have explored the pressure- dependent PL spectra of h-BN nanosheets at low temperaturesand demonstrated that the PL emission lines respond to the pres-sure in three different forms: a redshift, a blueshift, and even atransition from redshift to blueshift under high pressures. As shown in Figs. 9(g) and9(h), the PL peak exhibits a redshift with a pressure coefficient of 1.31 meV/GPa (peak A) with theincrease in the pressure from 0.69 to 2.03 GPa. Subsequently, ablueshift at a rate of 0.72 meV/GPa (peak B) takes place in the PLpeak as the pressure continues to increase. Based on theoretical calculations, the pressure characteristics of the PL spectra origi- nate from the V NNBdefect and relate to the competition between the strain-induced intralayer interaction and interlayer interac-tion. With successive experimental progress in tuning the band-structure and optical properties of h-BN nanosheets, one may expect that external physical fields would play a vital role in future quantum science and technology. V. CONCLUSION AND FUTURE PERSPECTIVES Recent advances in point defect engineering for 2D h-BN toward electronic, magnetic, and optical properties have been sum- marized in this Perspective. Experimental progress on point defectsformation and the corresponding band structures is presentedtogether with theoretical calculations for better comprehending theelectronic characteristics of 2D h-BN with various defects. We stress the exciting magnetic and optical properties (photolumines- cence and single quantum emission) of h-BN, which are directlycorrelated with the defect engineered electronic structures.Attention has also been paid to the external field modulation ofsingle quantum emission from point defects in h-BN in order to obtain a controllable and scalable quantum emission to satisfy the optical, photoelectrical sensing, and quantum spintronics applica-tions. Some principal challenges and perspectives in this field areaddressed for future investigation as follows. Although extensive research studies have been conducted in point defect engineering of the h-BN system, the studies in thisfield are still in their infancy, leaving ample room for future explo-ration. For instance, numerous methods have been employed toinduce point defects into h-BN, such as plasma etching, high- energy electron irradiation, and chemical-vapor deposition. However, the defects are usually in low density and appear in arandom way. Deterministic construction of precisely located defectsin a large quantity is crucial for fulfilling aspirations of scalabilityand nanodevice applications. Thus, more diverse and multiple methods for creating atomic defects should be exploited to prepare a defect-confined h-BN system. What is remarkable is that non-magnetic h-BN can be induced to either ferromagnetic or antiferro-magnetic nature via defect engineering, yet the nano-magnetismappears very weak and at a specific area. Hence, the macroscopic-scale magnetism in the h-BN system remains to be studied. The optical properties of h-BN can also be effectivelytuned by vacancy construction and external fields. The formerstrategy results in exciton emission by inducing localized states within the original wide gap, while the latter leads to the manipula- tion of vacancies and new emission behaviors. Further defectengineering encourages more efforts on the efficient and scalable quantum photon emission in h-BN. Resolving the correlation between quantum emission and defect structure and drawing outthe optical diversity of the defect-emission are another two chal-lenges in quantum applications. As a consequence, the explorationin large-scale defect engineering and structure –property correlation is still essential to boosting the functionalities and applications of the h-BN system. Being analogous to graphene in structure but exhibiting its own fascinating properties, the h-BN nanomaterial has emerged asan ideal platform as a support-, barrier-, or protection-layer for other 2D materials ( Fig. 10 ). 178 –182In Fig. 10(a) , h-BN-encapsulated graphene or/and MoX 2(X = S, Se) layers are created into electrochemical devices, opening possibilities for 2Delectrochemical energy conversion and storage. 183Additionally, h-BN provides the possibility in modulating physico-chemical characteristics when in association with other 2D materials. Graphene/h-BN heterostructures have currently aroused greatinterest due to their unique electronic band structure, which is sus-ceptive to the crystallographic alignment between graphene andh-BN. In Figs. 10(b) –10(d) , h-BN acts as an extraordinary environ- ment for graphene plasmons. 12The h-BN/graphene/h-BN hetero- structure yields simultaneously low plasmon damping and strongfield confinement. As for the plasmon damping, graphene dissi-pates plasmon energy electronically from intrinsic thermal phonons and h-BN dissipates the energy of plasmons via dielectric losses. This enables the blossoming of nanoscale optoelectronicdevices such as single-plasmon nonlinearities, plasmon lenses, andlight absorbers. Beyond plasmons, h-BN can also fulfill heterostruc-tures and devices with high mobility, reduced local charge fluctua- tions, and degenerated energy levels, which can create new physical phenomena of graphene, such as the moiré superlattice, quantumHall effects, Hofstadter spectrum, and long-lived phonon polari-tons. 13,29,30,184In terms of the semiconducting transitional metal dichalcogenides (TMDCs) system, the inert h-BN behaves as a tun- neling buffer layer to improve electrical contact, as a dielectric mate- rial to increase the carrier mobility and as a tunnel barrier for therecombination of excitons. 19As 2D topological insulators, semicon- ducting TMDCs systems face challenges in the observation of thequantum spin Hall effect due to structural or chemical instabilities. The h-BN-supported WTe 2device exhibits a high-temperature quantum spin Hall effect up to 100 K [ Figs. 10(e) and 10(f) ].185 Figures 10(g) and10(h) demonstrate a light-emitting diode, which is constructed by stacking graphene (transparent conductive layer), h-BN (tunnel barrier), and MoS 2(materials for quantum wells).18 The device exhibits an extrinsic quantum efficiency of ∼8.4%, which is comparable to the value of modern-day organic light-emitting diodes. For the magnetic 2D materials, a long-range mag-netic order was observed in pristine 2D crystals since 2017, includ- ing semiconducting CrI 3,C r 2Ge2Te6, and metallic Fe 3GeTe 2.186 –192 The currently discovered 2D magnets provide opportunities for atomically thin magneto-optic and magnetoelectric devices,although they are unstable in air atmosphere. In view of this point,the air-stable h-BN has been generally utilized to encapsulate these oxidizable and deliquescent magnetic layers, thereby creating multi- functional magnetoelectric devices. In Figs. 10(i) –10(k) , a magnetic tunnel junction based on graphite –CrI 3–graphite has beenJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 100902 (2020); doi: 10.1063/5.0021093 128, 100902-15 Published under license by AIP Publishing.FIG. 10. Perspectives on future applications of h-BN nanosheets. (a) Atomic models of various vertical heterostructures composed of h-BN and other 2D materi als. Reproduced with permission from Bediako et al. , Nature 558, 425 (2018). Copyright 2018 Springer Nature. (b) –(d) Highly confined low-loss plasmons in graphene –boron nitride heterostructures. Reproduced with permission from Woessner et al. , Nat. Mater. 14, 421 (2015). Copyright 2015 Springer Nature. (e) and (f) Quantum spin Hall effect up to 100 K in a graphite-monolayer WT e 2-h-BN heterostructure. Reproduced with permission from Wu et al. , Science 359, 76 (2018). Copyright 2018 American Association for the Advancement of Science. (g) and (h) Light-emitting diode heterostructures including several graphene, TMDCs, and h-BN. Reprod uced with permission from Withers et al. , Nat. Mater. 14, 301 (2015). Copyright 2015 Springer Nature. (i) –(k) A tetralayer CrI 3magnetic tunnel junction encapsulated in h-BN layers. Reproduced with permission from Klein et al. , Science 360, 1218 (2018). Copyright 2018 American Association for the Advancement of Science.FIG. 10. Perspectives on future appli- cations of h-BN nanosheets. (a) Atomic models of various vertical heterostructures composed of h-BN and other 2D materials. Reproduced with permiss ion from Bediako et al. , Nature 558, 425 (2018). Copyright 2018 Springer Nature. (b) –(d) Highly confined low-loss plasmons in graphene –boron nitride heterostructures. Reproduced with permission from Woessner et al. , Nat. Mater. 14, 421 (2015). Copyright 2015 Springer Nature. (e) and (f) Quantum spin Hall effect up to 100 K in a graphite-monolayer WT e 2-h-BN heterostructure. Reproduced with permission from Wu et al. , Science 359, 76 (2018). Copyright 2018 American Association for the Advancement of Science. (g) and (h) Light-emitting diode heterostructures including several graphene, TMDCs, and h-BN. Reproduced with permission from Withers et al. , Nat. Mater. 14, 301 (2015). Copyright 2015 Springer Nature. (i) –(k) A tetralayer CrI 3magnetic tunnel junction encapsulated in h-BN layers. Reproduced with permission from Klein et al. , Science 360, 1218 (2018). Copyright 2018 American Association for the Advancement of Science.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 100902 (2020); doi: 10.1063/5.0021093 128, 100902-16 Published under license by AIP Publishing.sandwiched between two h-BN flakes, which shows a total magneto- resistance of 550% and a characteristic of inelastic electron tunnel- ing.193With the protection of h-BN layers, the assembling and external field control of magnetic, magnetoelectric, and magneto-optical 2D devices composed of degradable components canbecome a reality. This fascinating combination of physics and many other properties induced by h-BN offers exciting opportunities for 2D heterostructures in a wide range of applications, such as highlydurable field emitters, compact UV laser devices, atomic tunnelingdevices, flexible and transparent electronics, switches, 194 –196and spintronic devices.197To further construct highly engineered hetero- structures, a diverse range of material and device components could be mixed, matched, and combined. Simultaneously, a new class ofelectronic, magnetic, and optoelectronic devices with extraordinaryperformances or new functionalities could be created. ACKNOWLEDGMENTS This work was financially supported by the National Natural Science Foundation of China (NSFC) (Grant Nos. 51728202 and 21775022), the Natural Science Foundations of Guangdong Province (No. 2017A030310603), China Postdoctoral ScienceFoundation (No. 2020M673174), and the Guangdong ProvincialKey Platform and Major Scientific Research Projects for Collegesand Universities (Nos. 2015KCXTD029 and 2016KCXTD023). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1Y. Tian, B. Xu, D. Yu, Y. Ma, Y. Wang, Y. Jiang, W. Hu, C. Tang, Y. Gao, K. Luo, Z. Zhao, L. M. Wang, B. Wen, J. He, and Z. Liu, Nature 493, 385 (2013). 2A. Gupta, T. Sakthivel, and S. Seal, Prog. Mater. Sci. 73, 44 (2015). 3G. Lu, T. Wu, Q. Yuan, H. Wang, H. Wang, F. 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1.4939709.pdf

Switching probabilities of magnetic vortex core reversal studied by table top magneto optic Kerr microscopy G. Dieterle , A. Gangwar , J. Gräfe , M. Noske , J. Förster , G. Woltersdorf , H. Stoll , C. H. Back , and G. Schütz Citation: Appl. Phys. Lett. 108, 022401 (2016); doi: 10.1063/1.4939709 View online: http://dx.doi.org/10.1063/1.4939709 View Table of Contents: http://aip.scitation.org/toc/apl/108/2 Published by the American Institute of PhysicsSwitching probabilities of magnetic vortex core reversal studied by table top magneto optic Kerr microscopy G.Dieterle,1A.Gangwar,1,2J.Gr€afe,1M.Noske,1J.F€orster,1G.Woltersdorf,3H.Stoll,1 C. H. Back,2and G. Sch€utz1 1Max Planck Institute for Intelligent Systems, Stuttgart 70569, Germany 2Department of Physics, University of Regensburg, Regensburg 93053, Germany 3Department of Physics, University of Halle, Halle 06120, Germany (Received 16 November 2015; accepted 27 December 2015; published online 11 January 2016) We have studied vortex core reversal in a single submicron Permalloy disk by polar Kerr microscopy. A sophisticated lock-in-technique based on repetitive switching of the magnetic vortexcore and a continuous calibration allows for a reliable determination of the switching probability. This highly sensitive method facilitates the detection of a change in the magnetic moment of the tiny magnetic vortex core which is about 1.5 /C210 /C017Am2. We have investigated vortex core switching caused by excitation of the vortex core gyromode with varying frequencies and amplitudes. The frequency range in which switching occurs was found to broaden with increasing excitation amplitude, whereby the highest frequency in this range shifts stronger to higher frequencies than the lowest frequency to lower frequencies. The experimental results are in good agreement with micromagnetic simulations. VC2016 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License . [http://dx.doi.org/10.1063/1.4939709 ] Magnetic vortices are found in small suitably shaped ferromagnetic platelets. In a magnetic vortex structure, themagnetization curls in-plane. However, at the center of thedisk, one finds a region, in which the magnetization points out-of-plane. Typically, this vortex core has a diameter of 10–20 nm. 1This magnetic texture can be excited to a gyra- tion around its equilibrium position (gyromode) with charac- teristic frequencies in the 100 MHz to some GHz range depending on the sample dimensions.2Using synchrotron based X-ray microscopy reversal of the vortex core by low external ac magnetic fields, exciting the vortex gyromode3 has been demonstrated triggering a new chapter in vortex dynamics. Later, unidirectional vortex core reversal by exci-tation with in-plane rotating magnetic fields has been achieved. 4,5In this letter, we present how to measure vortex core reversal by varying the excitation frequency and ampli-tude with a commercial table-top Magneto-Optical Kerr Effect (MOKE) microscope with a lateral resolution of 2 lm combined with a highly sensitive lock-in-technique. Withthis technique, vortex core reversal can be detected and, in particular, the switching probability can be determined highly precise compared to other methods. The experiments were performed on a single Permalloy disk (500 nm in diameter, 50 nm thick, protected from oxida- tion by a 4 nm thick aluminum layer) deposited on top of a crossed copper strip line which is used to generate in-plane rotating fields 5(cf. Fig. 1(b)). These structures are prepared on top of silicon nitride membranes by e-beam lithography and lift-off-processing. No additiona l dielectric layers have been added to enhance the Kerr signal.6The membranes are not necessary for the MOKE measurements but allow for addi- tional time resolved scanning transmission X-ray microscopyexperiments on the same sample. The laser-spot of a scanning Kerr microscope (NanoMOKE3, Durham Magneto Optics) is focused on thePermalloy disk (Fig. 1). The measurements are performed in the polar Kerr geometry which is sensitive to the out-of-plane component of the magnetization. The diameter of the laser spot is about 2 lm in diameter, whereas the vortex core has a much smaller diameter of about 10–20 nm. In order toreduce drift effects and to improve the signal to noise ratio, a lock-in-technique is applied. This lock-in technique is explained in detail below. The data acquisition is synchron-ized with the RF-setup used for vortex core switching and the Kerr signal is read out continuously 100 000 times per second during the whole measurement time of typically2 minutes. For probing vortex core reversal and to determine the switching probability, the measurement consists of a fre-quent repetition of a sequence of six different steps A, B, C, D, E, and F during which either an excitation is applied or the Kerr signal is recorded (cf. Fig. 2): (A) In step A, the vortex core polarity is prepared into a known state, e.g., vortex core up. This state is the initial state for the switching experiments. For this purpose, an FIG. 1. (a) Schematic of the measurement setup. The laser is placed on a Permalloy disk located on a copper strip line. (b) SEM-image of a typical Permalloy sample for vortex core switching experiments. The red circle illustrates the area illuminated by the laser spot. 0003-6951/2016/108(2)/022401/4 VCAuthor(s) 2016. 108, 022401-1APPLIED PHYSICS LETTERS 108, 022401 (2016) excitation which selectively (i.e., only one polarity will be switched, whereas the other polarity will not be changed) switches the vortex core up is used. This isrealized by gyromode excitation using in-plane rotatingmagnetic fields as investigated in Refs. 4and5. (B) During step B, the vortex core is pointing up and the Kerr signal is recorded in order to obtain a referencevalue of the Kerr signal for vortex core up. (C) In step C, the excitation of interest is applied. This is the excitation whose switching behavior is to bedetermined. (D) During step D, the vortex core is pointing either up or down depending on whether it has been switched in stepC, and the Kerr signal is recorded in order to obtain a Kerr signal for this vortex core state. The following steps E and F are implemented to contin- uously calibrate the switching probability 1. (E) In step E, the vortex core polarity is selectively switched down. In order to achieve this, the same excitation as instep A is used but with a reversed sense of rotation ofthe field. (F) During step F, the vortex core is pointing down and the Kerr signal is recorded to obtain a reference value of the Kerr signal for vortex core down. The difference DKerr(B,D) of the Kerr signal recorded during steps B and D is proportional to the switching proba-bility between steps B and D, and the difference DKerr(B,F) in the Kerr signal recorded during steps B and F acts as a cal- ibration of the switching probability 1. By comparingDKerr(B,D)t oDKerr(B,F), the switching probability P between steps B and D can be determined P¼ DKerr B ;DðÞ DKerr B ;FðÞ: (1) This measurement sequence is repeated about 200 000 times with a repetition rate of about 2 kHz and the Kerr signalrecorded in the steps B, D, and F is averaged separately over all repetitions. The lock-in technique described above was applied to investigate the switching behavior of the vortex core under excitations with bursts of in-plane rotating magnetic fieldswith varying amplitudes (0.68 mT, 1.91 mT, and 3.81 mT) and frequencies in the range from 300 MHz to 1000 MHz, i.e., around the resonance frequency of the vortex core gyro-mode. The duration of the excitation was kept constant at 100 ns. Switching the vortex core from the “up” into the “down” state requires a counterclockwise sense of rotationof the excitation field. In steps A and E, bursts of in-plane rotating fields with the rotation sense in step A being clock- wise and in step E being counterclockwise are used. Therotation frequency of the bursts is 530 MHz, which is close to the resonance frequency of the gyrotropic mode, the am- plitude is 2.7 mT, and the burst length is 10 ls. This reliably ensures that the vortex core polarity is “up” in step B and “down” in step F. At this point, it is important to note that the excitation whose switching behavior is to be investigateddoes not need to be similar to the one used in steps A and E. Using this method, applying vortex core reversal by gyro- mode excitation to reliably switch the vortex core can, forexample, be utilized to study vortex core reversal by excita- tion with pulses or multi GHz in plane rotating fields. The measured switching probabilities are shown in Fig. 3(a). For a low excitation amplitude of 0.68 mT (marked by red squares), switching only occurs in a very narrow region near the resonance frequency of 480 MHz. At higher excita-tion amplitudes of 1.91 mT and 3.81 mT (marked by green circles and blue triangles, respectively), the frequency region, in which switching occurs, increases. Moreover, thisincrease in the frequency range is more pronounced for higher frequencies than for lower frequencies. Another feature of the switching behavior we observe is that in between the frequency ranges, in which switching occurs and those in which no switching occurs, there is a transition region, in which the switching probability assumes FIG. 2. Schematic of the lock-in-tech- nique. In step A, the vortex core polar- ity is prepared in a known state, e.g., vortex core up. During step B, the Kerr signal is recorded to get a reference value for vortex core up. In step C, the excitation of which the switching behavior is to be investigated isapplied. During step D, the Kerr signal is recorded to detect switching. In step E, the vortex core polarity is selec- tively switched to the other state. During step F, the Kerr signal is recorded to get a reference value for vortex core down.022401-2 Dieterle et al. Appl. Phys. Lett. 108, 022401 (2016)values between 0 and 1. With increasing frequency, the switching probability monotonously increases or decreases,respectively, in these regions. From parameters for which the switching probability should be 1, the error in the determination of the switchingprobability of 1 can be estimated. The standard deviation of 19 measured probabilities was calculated to be 0.06. The typ- ical measured difference in the Kerr signal was in the rangeof 30 ldeg to 40 ldeg. Moreover, the difference in the Kerr rotation for vortex core up and down can be estimated to be about 15 ldeg, 7which is in the same order of magnitude as the values measured in the experiments. A discussion, in which we rule out contributions to the measured probabilities other than vortex core switching, can be found in thesupplementary material. To estimate the sensitivity of the measurement technique related to volume based techniques, the magnetic moment of the vortex structure was calculated by using micromagnetic simulations to be 1.5 /C210/C017Am2. Additionally, micromagnetic simulations for a corre- sponding Permalloy disk have been carried out with the MuMax3 micromagnetic simulation program8applying in plane rotating fields with the same rotation sense (counter- clockwise), burst length (100 ns), and with the same initial vortex core polarity (“up”) as in the experiment. The simula- tion cell size was (4.17 nm)3and for Permalloy, a saturation magnetization of M sat¼6.0/C2105A/m, an exchange constant of A ex¼1.3/C210/C011J/m, and a Gilbert damping parameter of a¼0.007 (comparable to the Gilbert damping parameter measured in Refs. 9and10) were used. Micromagnetic simu- lations with a higher Gilbert damping parameter of a¼0.015 have also been carried out and the results are shown in supple- mentary Figure S1, but no significant differences were found.In the simulations, fluctuations resulting from finite tempera- tures are not included. The simulations were performed for different frequencies and amplitudes, and by determining thevortex core polarity after the end of the excitation the switch- ing behavior was extracted. The results are shown in Fig. 3(b) and are in good agree- ment with the experimental findings. The micromagnetic simulations reproduce the switching behavior concerning the center frequency and the width of the frequency ranges, inwhich switching occurs, as well as the observed difference in the shift of the lowest and highest frequency in this fre- quency range. This difference and the broadening of the frequency range in which switching occurs can be explained as follows. Fig. 3(c) shows the maximum values of the vortex core velocity dur-ing the simulations for different excitation amplitudes and frequencies. As the vortex core velocity for a fixed frequency is increasing with increasing excitation amplitude, the fre-quency range where a distinct velocity is exceeded is expanding. And since it is a crucial for vortex core reversal by gyromode excitation whether a certain velocity of about350 m/s is reached, 11this leads to the observed broadening of the frequency range in which switching occurs. Furthermore, the curves in Fig. 3(c)are asymmetric: When the frequency is detuned away from the resonance frequency, the velocity is stronger decreasing when the frequency is decreased than when it is increased. This results in the observed difference inthe frequency shifts. Compared to the experiment, the amplitudes required for switching the vortex core are higher in the simulations. Such adifference has already been reported 12and was attributed to the influence of finite temper atures and the formation of a Bloch point during the vortex core reversal which is problem-atic in micromagnetic simulatio ns. Furthermore, sample rough- ness could influence the switching threshold. 13Moreover, in contrast to the experimental results, the micromagnetic simula-tions do not show a continuous transition in the switching probability between the frequency ranges in which switching occurs and those in which it does not occur. This differencecan be explained by fluctuations, f or example, originating from finite temperature effects or from noise generated by the RF- setup. Furthermore, defects c ould influence the dynamics and FIG. 3. (a) Experimental results for the switching probability for different amplitudes and frequencies of the in-plane rotating magnetic field used for the excitation. The solid lines are a guide to the eye. (b) Results of micro- magnetic simulations for vortex core switching determined from the final state after the end of the excitation. The solid lines are a guide to the eye (and do not indicate switching probabilities between 0 and 1). (c) Maximumvortex core velocity extracted from micromagnetic simulations. The solid lines are a guide to the eye. The dashed lines in parts (b) and (c) show that the transition to the range in which switching occurs coincides with the fre- quencies at which the critical velocity is reached.022401-3 Dieterle et al. Appl. Phys. Lett. 108, 022401 (2016)in conjunction with fluctuations further contribute to nondeter- ministic switching, as, for example, vortex core trapping could influence the initial vortex core position,14or small deviations in the vortex core trajectory may be crucial for an interactionof the vortex core with a defect. The presented measurement technique allows, due to the large number of repetitions of the switching experiment, foranalyzing stochastic events with a high accuracy compared to other methods and without the need of synchrotron radiation. Using this measurement technique, vortex core reversal by gyromode excitation with different excitation frequencies and amplitudes has been investigated. A broadening of the fre-quency range in which switching occurs has been observed being more pronounced for higher frequencies. The findings could be explained by evaluating micromagnetic simulationswhich are in good agreement with the experiments. 1T. Shinjo, T. Okuno, R. Hassdorf, K. Shigeto, and T. Ono, Science 289(5481), 930 (2000). 2A. A. Thiele, Phys. Rev. Lett. 30(6), 230 (1973). 3B. Van Waeyenberge, A. Puzic, H. Stoll, K. W. Chou, T. Tyliszczak, R. Hertel, M. Fahnle, H. Bruckl, K. Rott, G. Reiss, I. Neudecker, D. Weiss, C. H. Back, and G. Schutz, Nature 444(7118), 461 (2006). 4M. Curcic, B. Van Waeyenberge, A. Vansteenkiste, M. Weigand, V. Sackmann, H. Stoll, M. F €ahnle, T. Tyliszczak, G. Woltersdorf, C. H. Back, and G. Sch €utz,Phys. Rev. Lett. 101(19), 197204 (2008).5M. Curcic, H. Stoll, M. Weigand, V. Sackmann, P. Juellig, M. Kammerer, M. Noske, M. Sproll, B. Van Waeyenberge, A. Vansteenkiste, G.Woltersdorf, T. Tyliszczak, and G. Sch €utz,Phys. Status Solidi B 248(10), 2317 (2011). 6N. Qureshi, H. Schmidt, and A. R. Hawkins, Appl. Phys. Lett. 85(3), 431 (2004); U. J. Gibson, L. F. Holiday, D. A. Allwood, S. Basu, and P. W.Fry,IEEE Trans. Magn. 43(6), 2740 (2007). 7See supplementary material at http://dx.doi.org/10.1063/1.4939709 for an estimation of the expected change in the polar Kerr signal, a discussion inwhich effects other than vortex core switching are ruled out and results ofadditional micromagnetic simulations with a Gilbert damping parameterofa¼0.015. 8A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez, and B. Van Waeyenberge, AIP Adv. 4(10), 107133 (2014). 9T. Weindler, H. G. Bauer, R. Islinger, B. Boehm, J. Y. Chauleau, and C. H. Back, Phys. Rev. Lett. 113(23), 237204 (2014). 10A Gilbert damping parameter of a¼0.0072 has been measured in Ref. 9 in Permalloy films prepared in the same deposition chamber. 11K. Yu Guslienko, K.-S. Lee, and S.-K. Kim, Phys. Rev. Lett. 100(2), 027203 (2008). 12M. Noske, A. Gangwar, H. Stoll, M. Kammerer, M. Sproll, G. Dieterle, M.Weigand, M. F €ahnle, G. Woltersdorf, C. H. Back, and G. Sch €utz,Phys. Rev. B 90(10), 104415 (2014); A. Thiaville, J. M. Garc /C19ıa, R. Dittrich, J. Miltat, and T. Schrefl, ibid. 67(9), 094410 (2003). 13A. Vansteenkiste, M. Weigand, M. Curcic, H. Stoll, G. Sch €utz, and B. Van Waeyenberge, New J. Phys. 11(6), 063006 (2009). 14K. Kuepper, L. Bischoff, Ch. Akhmadaliev, J. Fassbender, H. Stoll, K. W. Chou, A. Puzic, K. Fauth, D. Dolgos, G. Sch €utz, B. Van Waeyenberge, T. Tyliszczak, I. Neudecker, G. Woltersdorf, and C. H. Back, Appl. Phys. Lett. 90(6), 062506 (2007).022401-4 Dieterle et al. Appl. Phys. Lett. 108, 022401 (2016)

1.392839.pdf

4,517,664 43.40.Ph SEISMIC APPARATUS Carl O. Berglund, assignor to Teledyne Exploration Company 14 May 1985 (Class 367/163); filed 12 November 1981 This patent is concerned with a particular hydrophone designed for use in marine seismic streamers or towed hydrophone arrays of the type used in geophysical exploration. Somewhat more than the usual amount of design 4,485,678 43.40.Rj ROTOR DIAGNOSTIC AND BALANCING SYSTEM Frank Fanuele, assignor to Mechanical Technology, Incorporated 4 December 1984 (Class 73/660); filed 27 September 1982 This system is intended to provide means for rapid diagnosis and trim balancing of vibrations of rotors, such as are used in aircraft engines. The outputs of vibration and speed sensors mounted on an engine are processed as the machine is spun up, allowed to coast down, and run at several steady speeds. The processed data are compared to four predetermined criteria to determine the location of the greatest vibration, the presence of a rotor shift, the sources of various vibration components, and the possible presence of misalignment. The results are analyzed to determine whether the observed unbalance problem can be corrected by trim balancing, and to evaluate the required trim balance weights and locations.--EEU information is given. The hydrophone is of the acceleration canceling type, and is provided with a means to safeguard against permanent damage when it is inadvertently carried to greater than the normal operating depth.-- SAC 4,480,959 43.40.Tm DEVICE FOR DAMPING VIBRATIONS OF MOBILE TURBINE BLADES 4,500,978 43.40.Ph SEISMOGRAPHIC METHOD AND APPARATUS USING SCALED SOUND SOURCES Antoni M. Ziolkowski and William E. Lerwill, assignors to Seismo- graph Service Corporation 19 February 1985 (Class 367/142); filed 16 July 1982 Serge P. L. Bourgnignon, Raymond R. Choque, and Lucien P. Pham, assignors to S.N.E.C.M.A. 6 November 1984 (Class 426/220 R); filed in France 12 March 1982 A series of wedge-shaped shims is placed between the turbine blades and a retaining ring, so that centrifugal forces forcing the shim outward result in pushing the blades against a second retaining ring, thereby increas- ing friction between the blades and this second ring.--EEU A seismic method and apparatus are presented utilizing at the same location two point sources which produce impulsive signals having a known difference. This is said to simplify obtaining the subsurface properties in the earth. An example is given in which a 10-cubic-in. and an 80-cubic-in. gun are fired at a depth of 30 ft and at firing pressures of 2000 psi.--SAC 4,482,125 43.40.Tm LAMP SUPPORTING UNIT FOR ABSORBING SHOCKS AND VIBRATIONS 4,516,227 43.40.Ph SUBOCEAN BOTTOM EXPLOSIVE SEISMIC SYSTEM Kenneth R. Wener and Anthony R. Tinkle, assignors to Marathon Oil Company 7 May 1985 (Class 367/150}; filed 4 December 1981 A means is provided for implantation of a seismic source or detector in the subocean bottom, with a signal cable extending to an anchored com- munication buoy. A means of retrieving the implanted unit is also pro- vided.--SAC Richard Ziernicki, assignor to Over-Lower Company 13 November 1984 (Class 248/604); filed 22 September 1982 This unit is intended to protect lamps mounted atop mobile light towers from vibrations acting in any direction. Lamp fixtures are fastened to a supporting arm 32 by means of bolts 34 and 36. This arm is resiliently $4 4,519,053 43.40.Ph FORCE OR PRESSURE FEEDBACK CONTROL FOR SEISMIC VIBRATORS John W. Bedenbender and Gilbert H. Kelly, assignors to Texas Instru- ments, Incorporated 21 May 1985 (Class 367/190); filed 1 October 1984 "A seismic vibrator source having a hydraulic vibrator coupled to vi- brate a ground pad is provided with one or more force transducers such as, for example, load cells,. strain gauges, or piezoelectric elements for measur- ing the force applied to the earth. Signals indicative of the pressure force are applied to a controller for the hydraulic vibrator to adjust control signals to prevent decoupling of the ground pad from the earth during ground seismic operations."---SAC suspended by means of an octahedral arrangement of springs 42 from a square frame 30, which is attached at the top of a telescoping pole 24.-- EEU 4,482,592 43.40.Tm VIBRATION ISOLATION PAD James H. Kramer, assignor to the B. F. Goodrich Company 13 November 1984 (Class 428/67); filed 2 September 1983 To an upper, relatively rigid, layer of ultra-high molecular weight poly- ethylene, there is bonded first a layer of vulcanized elastometric material and then a layer of resilient vulcanized elastometric projections. This pad, which can readily be cut to size, is intended to be installed directly under vibration-producing machines.EEU 1451 J. Acoust. Soc. Am. 78(4), Oct. 1985; 0001-4966/85/101451-01500.80; ¸ 1985 Acoust. Soc. Am.; Patent Reviews 1451 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 130.88.90.110 On: Fri, 19 Dec 2014 12:42:59

1.5049470.pdf

Reuse of AIP Publishing content is subject to the terms at: <a href="https://publishing.aip.org/authors/rights-and-permissions">https://publishing.aip.org/authors/rights- and-permissions</a>. Downloaded to: 37.9.46.146 on 21 November 2018, At: 22:12A Luneburg lens for spin waves N. J. Whitehead , S. A. R. Horsley , T. G. Philbin , and V. V. Kruglyak Citation: Appl. Phys. Lett. 113, 212404 (2018); doi: 10.1063/1.5049470 View online: https://doi.org/10.1063/1.5049470 View Table of Contents: http://aip.scitation.org/toc/apl/113/21 Published by the American Institute of PhysicsA Luneburg lens for spin waves N. J. Whitehead, S. A. R. Horsley, T. G. Philbin, and V. V. Kruglyaka) Department of Physics and Astronomy, University of Exeter, Stocker Road, Exeter EX4 4QL, United Kingdom (Received 22 July 2018; accepted 3 November 2018; published online 20 November 2018) We report on the theory of a Luneburg lens for forward-volume magnetostatic spin waves and verify its operation via micromagnetic modelling. The lens converts a plane wave to a point source,and vice versa, by a designed graded refractive index, realized by modulating either the thickness or the saturation magnetization in a circular region. We find that the lens enhances the wave ampli- tude by about 5 times at the lens focus, and 47% of the incident energy arrives in the focal region.A lens with small deviations from the optimal profile can still result in good focusing if the index is graded smoothly. Published by AIP Publishing. https://doi.org/10.1063/1.5049470 It is often useful to manipulate a wave as it travels through a material, and this can be achieved by designing asuitable graded refractive index. This is a well-established field in optics, 1and similar ideas have also been applied to other areas of wave physics. In magnonics,2,3the study of spin waves,4the theme of “graded index magnonics”5has been gaining interest recently as the parameter space of magnetic materials is fur-ther exploited to confine, 6,7direct,8–10generate11–13or focus14spin waves. In Ref. 14, a graded decrease in the mag- netization was induced by locally heating an in-plane magne-tized yttrium-iron-garnet (YIG) film. This heating profile acted as a focusing or defocusing lens for backward volume and Damon-Eshbach magnetostatic spin waves, respectively.Magnonic lens designs with abrupt boundaries were demon- strated in Refs. 15and16. However, such abrupt changes in magnetic parameters induce unnecessary reflection of spinwaves, which is partly avoided in the case of smoothly changing a material parameter. Alternatively, focusing can be achieved using curved magnonic sources, 17in which case no modulation of the magnonic index is needed, and via reflection from a curved magnetic boundary.18However, non-rotationally symmetric designs15,16and those based on in-plane magnetized films14only work for a particular direc- tion of spin wave incidence. In graded index optics, there are a several rotationally symmetric lens designs. One example is the Luneburg lens,19 designed to focus a plane wave to a point, or conversely, toconvert a point source to a plane wave. This profile has beenstudied in many other areas of wave physics. 20–23As such, the Luneburg lens may have an important role in future wave- based computing circuitry, to launch plane waves from anantenna, or to increase the amplitude of incoming plane waves to be read by the same antenna. To read/launch a plane wave from/to a different direction, one only needs to move theantenna to the corresponding point on the lens edge, without having to reconfigure the lens. In this work, we demonstrate theoretically how a Luneburg lens for spin waves can be realized in a thin mag- netic film.The refractive index profile n(r) for a Luneburg lens is given by n LðrÞ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C0ðr=RÞ2q ;r/C20R; 1; r>R;8 < :(1) where ris the radial coordinate and Ris the radius of the lens. This profile, along with the ideal operation of the lens,is shown in Fig. 1. For light propagating in an isotropic non-dispersive medium, the graded refractive index is given by nðrÞ/C17kðrÞ k0¼c vðrÞ; (2) where k0(k) and c(v) are the wave number and the speed of light in vacuum (the graded medium), respectively. In thiscase, the dispersion relation x(k), where xis the angular fre- quency, is linear, isotropic and there is no band gap in thespectrum. As a result, the graded index has the same spatial profile for different frequencies. For spin waves, the medium is always dispersive. Indeed, the spin wave dispersion relation x(k) has a gap at k¼0, is non-linear, and may depend upon the mutual orientation ofthe wave vector and magnetization. Hence, the magnonicrefractive index can usually be defined only for a fixed fre- quency and perhaps its vicinity. So, any profile of the mag- netic field or material parameters required to make a specialgraded profile of the index is generally frequency-dependent. To make a Luneburg lens for spin waves at a particular frequency, we thus need to ensure that k(r)/k 0obeys Eq. (1), where k0is now the reference wave number of the spin waves outside the lens. To avoid anisotropy, we choose FIG. 1. (a) The Luneburg lens, outlined by the dashed line, focuses rays (red lines) to a diffraction-limited spot on the opposite edge of the lens. (b) Refractive index profile described by Eq. (1).a)Electronic mail: V.V.Kruglyak@exeter.ac.uk 0003-6951/2018/113(21)/212404/5/$30.00 Published by AIP Publishing. 113, 212404-1APPLIED PHYSICS LETTERS 113, 212404 (2018) to work with forward-volume magnetostatic spin waves (FVMSWs), propagating in the plane of a perpendicularly magnetized thin ferromagnetic film. These waves have the following isotropic dispersion relation (given in Gaussian units)24 k¼1 s2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /C0ð1þjÞp arctan1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /C0ð1þjÞp ! ; (3) where j¼XH X2 H/C0X2;X¼x 4pcM;XH¼Hi 4pM: (4) cis the gyromagnetic ratio, Mis the magnetization length, Hi¼Hex/C04pMis the static internal magnetic field, Hex is the applied (external) magnetic field, and sis the film thickness. There are three parameters in Eqs. (3)and(4)that we can manipulate to vary the wave number and the magnonic index: s,MandHex. Interestingly, if we vary the film thick- ness and fix all other parameters, there is a simple relation between the index and the thickness outside, sR, and inside, s(r), the lens nðrÞ¼sR=sðrÞ: (5) Equation (5)shows that graded magnonic index profiles obtained via the film thickness modulation do not depend on the FVMSW frequency. The profile of s(r) required to make a Luneburg lens is given in Fig. 2(a). The film in the center of the profile needs to beffiffiffi 2p times thinner than outside of the lens, which suggests that the lens will not be too sensitive to small thickness variations. Note that Eqs. (3)and (4) neglect changes in the static demagnetizing field due to thenon-uniform thickness profile. This approximation is justi- fied if s/C28R, since the deviation of the demagnetizing field from its thin film value of 4 pMscales quadratically with Ds/ R, where Dsis the maximum change in thickness. The profile of the magnetization or applied magnetic field required to make the Luneburg lens for spin waves can- not be written in an explicit form. However, in contrast to the thickness, it is significantly easier to model changes in these quantities in finite-difference micromagnetic simula-tions. So, to demonstrate the operation of a spin wave Luneburg lens, we vary the saturation magnetization in space. We determine the magnetization profile M(r) [shown in Fig. 2(b)] required to produce the lens from Eqs. (3)and (4). There are two features to notice. First, M(r) needs to increase in the center of the lens. This cannot be achieved bya local heating of the sample 14but requires a local cooling instead. Second, the required maximum change in the mag- netization is just 1.7%, which is rather small. For a fixed thickness, this magnetization change depends on the fre-quency and external field, as detailed in the supplementary material . We now describe how the lens is designed and tested in micromagnetic simulations using MuMax3 software. 25We define a 1 /C20.5 mm2YIG-like film in the x/C0yplane, with a fixed thickness of s¼2lm. Periodic boundary conditions are applied in both in-plane directions. The Gilbert damping constant is 10/C04. The saturation magnetization is MR¼140 kA/m outside the lens and varies inside the lens as shown in Fig. 2(b). The cell size is 0.5 /C20.5lm2in the film plane, and is equal to the thickness in the zdirection. The wavelength kof the studied spin waves is much greater than both the cell size and the exchange length. So, any effects of the exchange interaction are irrelevant in our model. First,we magnetize the sample by an out-of-plane magnetic field of 200 mT. Then, we apply a burst of microwave magnetic field in the region to the left of the lens profile. The micro-wave field is parallel to the x-axis and has central frequency of 1 GHz (corresponding to k¼33.9lm), bandwidth of 0.1 GHz and amplitude 0.1 mT. The spatiotemporal profile ofthe burst is designed to launch a Gaussian spin wave packet propagating towards the lens. The use of the wave packets in combination with a suitably long sample (in the direction ofthe wave packet travel) allows us to avoid using absorbing boundary conditions, 26which can still cause spurious reflec- tions from the gradients in the damping constant. The Luneburg lens profile is designed in the geometrical optics approximation,27i.e., for k/C282R. From comparisons with other studies,28,29and to keep the simulation size rea- sonable, we use R/C256k. We use Eqs. (1)–(4)to define 255 concentric circular regions in MuMax3, between which the saturation magnetization changes in equal steps to form theM(r) profile required for the Luneburg lens. The snapshots of the x-component of the reduced dynamic magnetization, m x¼Mx/MR, are shown in Fig. 3for different moments of time. The wavefronts behave as expected: the wavelength decreases in the region of increased refractive index, curving the wavefronts towardsthe lens’s focus. In addition, the wavefronts are slowed within the lens. We see the effect of this after the wave has left the lens, when the focused energy is re-emitted from thefocal spot. To evaluate the degree of focusing, Fig. 4(a)shows the maximum amplitude of m xattained in each cell of the model over the entire duration of the simulation. The largest ampli- tude is indeed attained in the focus of the lens. The corre- sponding video shows the amplitude of the wave movingthrough the lens in time. Figure 4(b) shows the spin wave energy density near the focus at the time when the maximum amplitude over the simulation is achieved. The energy ismostly concentrated around 6k/2 of the ideal focus. However, the peak is shifted along xfrom the ideal position, similar to Ref. 30. Increasing the size of the lens with respect tokshould bring the focal spot closer to the lens edge. To evaluate the beam waist, Fig. 4(c)shows the energy density cross-sections along yfor the line i shown in panels (a) and FIG. 2. The thickness (a) and magnetization (b) profiles required to make a Luneburg lens.212404-2 Whitehead et al. Appl. Phys. Lett. 113, 212404 (2018)(b) and for the xposition of the actual focus peak. The waist, measured as the full width at half maximum (FWHM) of the peak at the actual focus, is around 23 lm or 0.67 k,w h i c hi s reasonable for a diffraction-limited lens. At the actual andideal focal points, the peak amplitudes of m xare 5 and 4.7 times greater than the unfocused amplitude, respectively. This enhancement of the wave amplitude may be useful when read- ing an incoming plane wave using an antenna. There is little data on the amplitude at the focus of similar lenses outside ofmagnonics. However, Refs. 23and31reported an amplitude increase in 3–4 times for lens radii 2–3 times the wavelength. Now, we evaluate the fraction of the wave packet energy that reaches the focal region. First, we sum the energy over the rectangular region before the lens, as shown in Fig. 4(a), at 14 ns, i.e., once the packet starts moving and before it encounters the lens. The region has an xextent of 300 lm, which completely encompasses the wave packet length, and yextent of 2 R, so that only the portion of the wave packet that enters the lens is counted. Then, we sum the energy inthek/C2kregion centered around the actual focus peak, as shown in Fig. 4(b), at the time when the peak amplitude of the duration of the simulation is reached. As a result, we findthat 46% of the incident energy arrives in the focal region. Note that this is a pessimistic way to count the energy reach- ing the focus, since we include the effect of damping, 9,32 responsible for the energy loss of around 7% in our case. In thesupplementary material , we consider yet another method to quantify the lens efficiency. We compare the Fourier amplitudes for the positive and negative wave numbers that result from the wave’s interaction with the lens. Using this method, we find that no more than 13% of the wave is reflected overall. Next, we examine how sensitive the results are to devia- tions from the ideal magnetization profile. We run simulationsfor three parabolic profiles [Fig. 5(a)]o b t a i n e db yfi t t i n gt h e Luneburg profile to MðrÞ=M R¼a/C0br2,w h e r e aandbare fitting parameters.33The best fit profile has an M(0) error of 5% relative to the ideal value and still acts almost identically to the Luneburg profile, yielding a 5-times increase in the spin wave amplitude at the actual focal spot. We do not show this result here—rather, we present the results for the 630% error profiles in Figs. 5(b)–5(d) .I fM(0) is increased by 30% above the ideal value, the lensing effect is strengthened, creating a narrower focus and increasing the peak amplitude of mxby5.7 times. If M(0) is decreased by 30% below the ideal value, the peak enhancement of the mxamplitude decreases to 3.8 times of the incident wave’s amplitude. Figure 5(b)compares the corresponding energy density cross-sections along the direction of incidence. We find that the þ30% error, Luneburg, and /C030% error profiles have FWHMs of 0.8 k, 1.1k, and 1.3 k, respectively. Comparing the energy of the wave packet before entering the lens with the energy in the k /C2kfocus regions [Figs. 5(c) and5(d)34], we find that 49% and 37% of the incident energy arrives in the focus region fortheþ30% and /C030% error profiles, respectively. Recall that the Luneburg profile received 46% of the incident energy atthe focal region. So, if the magnetization in the center is increased, the lens may produce a somewhat tighter focus than the Luneburg profile. This may be beneficial in a systemwhere only the focusing power matters. Our analysis so far shows that the lens profile can devi- ate from the ideal Luneburg profile and still produce a rea-sonable lensing effect. However, only the actual Luneburg profile can both reliably focus a plane wave to a spot and convert a point source to a plane wave, as we show in Fig. 6. To create these images (and their associated videos), wehave used the same parameters as previously, except wehave extended the size of the model (to suppress spurious interference) and introduced a source with a continuous- wave (CW) temporal and Gaussian spatial profile. Thesource is either located at the ideal focus on the edge of thelens, or at the actual focus determined from the focusing study. As before, we compare the ideal Luneburg lens to the 630% error profiles. Figure 6shows that the Luneburg profile creates a plane wave, albeit with some interference due to reflections within the lens. The þ30% and /C030% profiles focus the outgoing FIG. 3. Snapshots of mxare shown as the wave packet moves through the Luneburg lens (black circle) at times of (a) 16 ns, (b) 45 ns, (c) 80 ns, and (d) 106 ns. Multimedia view: https://doi.org/10.1063/1.5049470.1 FIG. 4. (a) Maximum amplitude of mxattained across the model over the duration of the simulation. The rectangular box indicates the region used to calculate the incident spin wave energy. (b) Energy density, W, near the focus region at the time of peak amplitude. The white square has a side of kand is centered on peak of the actual focus spot (black cross). (c) Energy density cross-sections for the line i from panels (a) and (b) (blue line) and at the x position of the actual focus (black line), at the times when the maximumamplitude occurs. Multimedia view: https://doi.org/10.1063/1.5049470.2212404-3 Whitehead et al. Appl. Phys. Lett. 113, 212404 (2018)wavefronts too much and too little for both source positions, respectively. Positioning the source at the actual focus of the /C030% profile is a good improvement. However, the wave- fronts are still not completely parallel to each other, and thewave amplitude is still lower than in the other two cases. All the results presented here are valid for the specific choice of the frequency, applied magnetic field and film thickness. In the supplementary material , we show how the required magnetization profile depends on the field and fre- quency values used in the design. In addition, we test how the lens behaves when the incident wave frequency is differ-ent from the frequency it was designed for. We find that the lens can still focus waves with frequencies different from the optimal value, although with reduced amplitude and/or largerFWHM. To conclude, we have demonstrated how to form a Luneburg lens in a magnetic material with perpendicular mag-netization. The simple relation between the refractive indexand the magnetic film thickness for purely magneto-dipole waves suggests the thickness as a parameter of choice for designing broadband graded magnonic index profiles, validfor a range of field and magnetic parameter values. In contrast, the Luneburg profile of the saturation magnetization is fre- quency and field dependent. Notably, the magnetizationchange required to achieve Luneburg focusing may be rather small. Furthermore, magnetization profiles different from the Luneburg could still focus spin waves. Yet, the Luneburg pro-file is still optimal for focusing a plane wave to a point and converting a point source to a plane wave, with the detector/ source positioned at the lens edge. Finally, we note that thetheory can easily be extended to spin waves of shorter wave-length/higher frequency. The only requirement is that the dis- persion of the relevant spin waves must be isotropic. The dispersion itself could be defined analytically, or even com-puted from results of micromagnetic simulations. 35Seesupplementary material for analyses of the depen- dence of the M(0) value for the Luneburg profile on the spin wave frequency and the applied magnetic field; for the dependence of the magnonic index on magnetization and field for different frequencies; for the behavior of the lens forwaves with sub-optimal frequencies; and for the reciprocal space/time domain analysis of the lens efficiency. We thank F. B. Mushenok for helping to create the wave packet excitation in the micromagnetic simulations and C. A. Vincent for technical support throughout. The research leading to these results has received funding from the Engineering and Physical Sciences Research Council of the United Kingdom, via the EPSRC Centre for Doctoral Training in Metamaterials (Grant No. EP/L015331/1), andfrom the European Union’s Horizon 2020 research and innovation program under Marie Skłodowska-Curie Grant Agreement No. 644348 (MagIC). SARH would like to thank the Royal Society and TATA (RPG-2016-186). 1D. T. Moore, “Gradient-index optics: A review,” Appl. Opt. 19, 1035–1038 (1980). 2V. V. Kruglyak, S. O. Demokritov, and D. Grundler, “Magnonics,”J. Phys. Appl. Phys. 43, 264001 (2010). 3A. A. Serga, A. V. Chumak, and B. Hillebrands, “YIG magnonics,” J. Phys. D: Appl. Phys. 43, 264002 (2010). 4A. I. Ahiezer, V. G. Bar’Yakhtar, and S. V. Peletminski K%,Spin Waves (Wiley, NY, 1968). FIG. 5. (a) The Luneburg profile (black) is shown with parabolas with either 5% (red, dashed) or 630% (orange and blue, respectively) errors in M(0). (b) Energy density cross-sections along xat the yposition of the actual focus are shown for the lens profiles from (a). Spatial maps of the spin wave energy density are shown for (c) þ30% and (d) /C030% error profiles. Line i in (b) and intersections of lines i and ii in (c) and (d) show the ideal positions of the focus. FIG. 6. Snapshots of mxat 82 ns when a CW-Gaussian source is positioned near (a) and (b) the Luneburg lens, (c) and (d) the þ30% error profile, and (e) and (f) the /C030% error profile. The source is centered on the lens edge for (a), (c), and (e) and on the actual focus for (b), (d), and (f). Guide lines (black, dashed) are provided in the plane wave regions. Multimedia views:https://doi.org/10.1063/1.5049470.3 ;https://doi.org/10.1063/1.5049470.4 ;https:// doi.org/10.1063/1.5049470.5 ;https://doi.org/10.1063/1.5049470.6 ;https:// doi.org/10.1063/1.5049470.7 ;https://doi.org/10.1063/1.5049470.8212404-4 Whitehead et al. Appl. Phys. Lett. 113, 212404 (2018)5C. S. Davies and V. V. Kruglyak, “Graded-index magnonics,” Low Temp. Phys. 41, 760–766 (2015). 6C. Bayer, S. O. Demokritov, B. Hillebrands, and A. N. Slavin, “Spin-wave wells with multiple states created in small magnetic elements,” Appl. Phys. Lett. 82, 607–609 (2003). 7E. V. Tartakovskaya, M. Pardavi-Horvath, and R. D. McMichael, “Spin wave localization in tangentially magnetized films,” Phys. Rev. B 93, 214436 (2016). 8C. S. Davies, A. Francis, A. V. Sadovnikov, S. V. Chertopalov, M. T.Bryan, S. V. Grishin, D. A. Allwood, Y. P. Sharaevskii, S. A. Nikitov, and V. V. Kruglyak, “Towards graded-index magnonics: Steering spin waves in magnonic networks,” Phys. Rev. B 92, 020408 (2015). 9P. Gruszecki and M. Krawczyk, “Spin-wave beam propagation in ferro- magnetic thin films with graded refractive index: Mirage effect and pro- spective applications,” Phys. Rev. B 97, 094424 (2018). 10M. Vogel, R. Aßmann, P. Pirro, A. V. Chumak, B. Hillebrands, and G. von Freymann, “Control of spin-wave propagation using magnetisation gradients,” Sci. Rep. 8, 11099 (2018). 11C. S. Davies, A. V. Sadovnikov, S. V. Grishin, Y. P. Sharaevskii, S. A. Nikitov, and V. V. Kruglyak, “Generation of propagating spin waves from regions of increased dynamic demagnetising field near magnetic antidots,” Appl. Phys. Lett. 107, 162401 (2015). 12C. S. Davies and V. V. Kruglyak, “Generation of propagating spin waves from edges of magnetic nanostructures pumped by uniform microwave magnetic field,” IEEE Trans. Magn. 52, 2300504 (2016). 13N. J. Whitehead, S. A. R. Horsley, T. G. Philbin, A. N. Kuchko, and V. V. Kruglyak, “Theory of linear spin wave emission from a Bloch domain wall,” Phys. Rev. B 96, 064415 (2017). 14O. Dzyapko, I. V. Borisenko, V. E. Demidov, W. Pernice, and S. O. Demokritov, “Reconfigurable heat-induced spin wave lenses,” Appl. Phys. Lett. 109, 232407 (2016). 15G. Csaba, A. Papp, and W. Porod, “Spin-wave based realization of optical computing primitives,” J. Appl. Phys. 115, 17C741 (2014). 16J.-N. Toedt, M. Mundkowski, D. Heitmann, S. Mendach, and W. Hansen, “Design and construction of a spin-wave lens,” Sci. Rep. 6,3 3 1 6 9 (2016). 17A. V. Kozhevnikov, Y. V. Khivintsev, G. M. Dudko, V. K. Sakharov, A. S. Dzhumaliev, S. L. Vysotskii, A. V. Stal’makhov, and Y. A. Filimonov, “Filtration of surface magnetostatic waves in yttrium iron garnet films of variable width excited by focusing transducers,” Tech. Phys. Lett. 44, 705–708 (2018).18S. Choi, S.-K. Kim, V. E. Demidov, and S. O. Demokritov, “Double-con-tact spin-torque nano-oscillator with optimized spin-wave coupling: Micromagnetic modeling,” Appl. Phys. Lett. 90, 083114 (2007). 19R. K. Luneburg and M. Herzberger, Mathematical Theory of Optics (University of California Press, Berkeley, Los Angeles, 1964). 20A. D. Falco, S. C. Kehr, and U. Leonhardt, “Luneburg lens in silicon pho- tonics,” Opt. Express 19, 5156–5162 (2011). 21T .Z e n t g r a f ,Y .L i u ,M .H .M i k k e l s e n ,J .V a l e n t i n e ,a n dX .Z h a n g ,“ P l a s m o n i c Luneburg and Eaton lenses,” Nat. Nanotechnol. 6, 151–155 (2011). 22J. A. Dockrey, M. J. Lockyear, S. J. Berry, S. A. R. Horsley, J. R. Sambles, and A. P. Hibbins, “Thin metamaterial Luneburg lens for surface waves,” Phys. Rev. B 87, 125137 (2013). 23S.-H. Kim, “Sound focusing by acoustic Luneburg lens,” preprint arXiv:1409.5489 [cond-mat.mtrl-sci] (2014). 24D. D. Stancil and A. Prabhakar, Spin Waves (Springer, Boston, 2009). 25A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez, and B. Van Waeyenberge, “The design and verification of MuMax3,” AIP Adv. 4, 107133 (2014). 26G. Venkat, H. Fangohr, and A. Prabhakar, “Absorbing boundary layers for spin wave micromagnetics,” J. Magn. Magn. Mater. 450, 34–39 (2018). 27U. Leonhardt and T. G. Philbin, “Chapter 2 transformation optics and the geometry of light,” in Progress in Optics , edited by E. Wolf (Elsevier, 2009), Vol. 53, pp. 69–152. 28M. M. Mattheakis, G. P. Tsironis, and V. I. Kovanis, “Luneburg lenswaveguide networks,” J. Opt. 14, 114006 (2012). 29J. D. de Pineda, R. C. Mitchell-Thomas, A. P. Hibbins, and J. R. Sambles, “A broadband metasurface Luneburg lens for microwave surface waves,” Appl. Phys. Lett. 111, 211603 (2017). 30P. Rozenfeld, “The electromagnetic theory of three-dimensional inho- mogeneous lenses,” IEEE Trans. Antennas Propag. 24, 365–370 (1976). 31F. Gaufillet and E. Akmansoy, “Graded photonic crystals for Luneburglens,” IEEE Photonics J. 8, 2400211 (2016). 32M. Dvornik, A. N. Kuchko, and V. V. Kruglyak, “Micromagnetic method of s-parameter characterization of magnonic devices,” J. Appl. Phys. 109, 07D350 (2011). 33In this case, the lens radius Ris 2.5% greater than before. 34Note that, in Figs. 5(c)and5(d), and 4(b), the color scale is set to the max- imum value attained in Fig. 5(c), for ease of comparison. 35M. Dvornik, Y. Au, and V. V. Kruglyak, “Micromagnetic simulations in magnonics,” Top. Appl. Phys. 125, 101 (2013).212404-5 Whitehead et al. Appl. Phys. Lett. 113, 212404 (2018)

1.2165136.pdf

Magnetization dynamics driven by the combined action of ac magnetic field and dc spin-polarized current G. Finocchio, I. Krivorotov, M. Carpentieri, G. Consolo, B. Azzerboni, L. Torres, E. Martinez, and L. Lopez-Diaz Citation: Journal of Applied Physics 99, 08G507 (2006); doi: 10.1063/1.2165136 View online: http://dx.doi.org/10.1063/1.2165136 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/99/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Ultrafast switching of a nanomagnet by a combined out-of-plane and in-plane polarized spin current pulse Appl. Phys. Lett. 95, 012506 (2009); 10.1063/1.3176938 Spin-current-induced dynamics in ferromagnetic nanopillars of lateral spin-valve structures J. Appl. Phys. 105, 07D110 (2009); 10.1063/1.3058621 Nanoring magnetic tunnel junction and its application in magnetic random access memory demo devices with spin-polarized current switching (invited) J. Appl. Phys. 103, 07E933 (2008); 10.1063/1.2839774 Effect of the classical ampere field in micromagnetic computations of spin polarized current-driven magnetization processes J. Appl. Phys. 97, 10C713 (2005); 10.1063/1.1853291 Spin-polarized current-driven switching in permalloy nanostructures J. Appl. Phys. 97, 10E302 (2005); 10.1063/1.1847292 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.186.1.81 On: Tue, 26 Aug 2014 14:31:31Magnetization dynamics driven by the combined action of ac magnetic field and dc spin-polarized current G. Finocchioa/H20850 Dipartimento di Fisica della Materia e Tecnologie Fisiche Avanzate, University of Messina, Salita Sperone 31, 98166 Messina, Italy I. Krivorotov Department of Physics and Astronomy, University of California at Irvine, Irvine, California 92697-4575 M. Carpentieri, G. Consolo, and B. Azzerboni Dipartimento di Fisica della Materia e Tecnologie Fisiche Avanzate, University of Messina,Salita Sperone 31, 98166 Messina, Italy L. T orres, E. Martinez, and L. Lopez-Diaz Departamento de Fisica Aplicada, Universidad de Salamanca, Plaza de la Merced s/n,37008 Salamanca, Spain /H20849Presented on 1 November 2005; published online 18 April 2006 /H20850 The spin-polarized current flowing through a ferromagnet can apply a torque able to drive magnetization reversal or to excite persistent dynamical states of magnetization. In the present workwe simulate the dynamic behavior of nanomagnets in nanopillar spin valve structures due to thecombined action of ac magnetic fields and dc spin-polarized current. The simulations are performedusing a micromagnetic model which includes the effect of the spin-polarized current. We alsodiscuss possible experimental realizations of the ac-field-assisted current-induced switching. Twoparticular cases of the ac-field polarization are studied: circular and linear. We find that in both casesac field accelerates the switching process. © 2006 American Institute of Physics . /H20851DOI: 10.1063/1.2165136 /H20852 A spin-polarized current /H20849SPC /H20850flowing through a nano- magnet applies torque to its magnetization that can eitherinduce magnetization reversal or excite persistent dynamicalstates of magnetization. The existence of this spin transfertorque was predicted theoretically 1,2and confirmed by a number of experiments.3–5Theory predicts that this torque differs fundamentally from the one exerted by magneticfields, 1,2,6,7and it is better understood as an effective nega- tive damping torque.8The potential applications of this effect are nonvolatile magnetic random access memory and nanos-cale microwave devices such as microwave sources. 3,4 In this paper, we will focus our attention on nanopillars with a ferromagnet/normal metal/ferromagnet /H20849FNF /H20850geom- etry, with one of the ferromagnetic layers thicker /H20851pinned layer /H20849PL/H20850/H20852than the other /H20851free layer /H20849FL/H20850/H20852. Since spin torque is an interfacial effect, it does not excite dynamics in the PL.When PL and FL are parallel /H20851parallel state /H20849PS/H20850/H20852, the struc- ture presents low electrical resistance, while for PL and FLantiparallel /H20851antiparallel state /H20849APS /H20850/H20852, high-resistance state is observed. A three-dimensional /H208493D/H20850finite difference micro- magnetic solver has been used to model nanopillars /H20851Permal- loy /H20849Py/H2085010 nm ”Cu 5 nm ”Py 2.5 nm /H20852with rectangular sec- tions /H20849S1:L x/H11003Ly/H11003Lz=60 nm /H1100320 nm /H110032.5 nm and S2: Lx /H11003Ly/H11003Lz=120 nm /H1100340 nm /H110032.5 nm /H20850.9,10 In order to take into account the effect of the SPC, an additional first-principles term, as deduced by Slonczewski,1 has been added in the Gilbert equation. This term gives riseto two new terms in the equivalent Landau-Lifshitz-Gilbert equation which in dimensionless form is /H208491+/H92512/H20850dm d/H9270=− /H20849m/H11003heff/H20850−/H9251m/H11003/H20849m/H11003heff/H20850 −g/H20841/H9262B/H20841j /H20841e/H20841/H92530Ms2Lz/H9273/H20849m·p,/H9261/H20850/H20851m/H11003/H20849m/H11003p/H20850 −/H9251/H20849m/H11003p/H20850/H20852, /H208491/H20850 where gis the gyromagnetic splitting factor, /H92530is the gyro- magnetic ratio, /H9262Bis the Bohr magneton, jis the applied current density, Lzis the thickness of the FL, eis the electron charge, m=M/MSis the magnetization of the FL, p =P/MSis the magnetization of the PL, MSis the saturation magnetization, and d/H9270=/H92530MSdtrepresents the dimensionless time step. The scalar function /H9273/H20849m·p,/H9261/H20850was deduced by Slonczewski,1the polarizing factor /H9261for Py is 0.3.1The perpendicular applied current density j=jzˆwas assumed to be spatially uniform. In our computations the effective fieldincludes the following contributions: h eff=hexch+hani+hext+hM+hAmp+hAF, /H208492/H20850 where hexch,hani,hext, and hMare the standard micromag- netic contributions from exchange, anisotropy, external, anddemagnetizing fields. h AmpandhAFare the Ampere field and magnetostatic coupling between the FL and the PL,respectively. 9,10The induced Ampere field was computed by means of appropriate numerical techniques developed toevaluate solenoidal fields. 10 a/H20850Electronic mail: gfinocchio@ingegneria.unime.itJOURNAL OF APPLIED PHYSICS 99, 08G507 /H208492006 /H20850 0021-8979/2006/99 /H208498/H20850/08G507/3/$23.00 © 2006 American Institute of Physics 99, 08G507-1 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.186.1.81 On: Tue, 26 Aug 2014 14:31:31Our study points to a strategy to decrease the current pulse duration needed for the switching process in nanopillarspin valve structures. This strategy employs the combinedaction of magnetic field /H20849ac along the yand zaxes /H20850and dc SPC. Two particular cases of the ac-field polarization havebeen studied: circular, h ext=h0cos/H20849/H9275t/H20850yˆ+h0sin/H20849/H9275t/H20850zˆ, and linear /H2084945° degrees in the y-zplane /H20850,hext=h0cos/H20849/H9275t/H20850yˆ +h0cos/H20849/H9275t/H20850zˆ, where h0and/H9275are the amplitude of the com- ponents and the frequency of the ac field, respectively. The results obtained for both types of ac field are similar, so wewill primarily refer to the case of linear polarization. In anexperiment, linearly polarized microwave magnetic fieldmay be realized by placing a spin valve nanopillar device ontop of a microstrip. A microwave drive voltage of frequency /H9275,Vd/H20849/H9275/H20850, applied to the microstrip will result in a linearly polarized ac magnetic field applied to the nanopillar. Micromagnetic simulations show that the combined ac- tion of ac magnetic field and dc SPC changes the switchingbehavior from PS to APS and vice versa compared to thatdue to the dc SPC only. Figure 1 shows the switching time 11 /H20849tS/H20850versus the value of the applied current /H20851top S1 /H20849left PS →APS, right APS →PS/H20850and bottom S2 /H20849left PS→APS, right APS →PS/H20850/H20852. For all sets of parameters used in our simulations, the presence of an applied ac magnetic fieldinvariably decreases the switching time. We used the following Py material parameters: M S =8.6/H11003105A/m, hani=0, and /H9251=0.02. The nanopillar has been discretized into an array of cubic cells of side of2.5 nm. Simulations performed using cells of 5 /H110035 /H110032.5 nm 3gave the same results qualitatively and quantita- tively that the switching times vary by less than 5%. A con-stant time step of 60 fs has been used: simulations performedusing smaller time step values yielded the same results. The obtained results are qualitatively the same for both structures, so we concentrate on those for S1. In particular,Fig. 2 shows the temporal evolution of the three componentsof the magnetization /H20849/H20855m x/H20856,/H20855my/H20856,/H20855mz/H20856/H20850 /H20851mx/H20849initial /H20850/H112290.99xˆ/H20852for PS to APS switching due to dc SPC /H20849J=−0.8 /H11003108A/cm2/H20850. Figure 3 shows the same simulation due to combined action of dc SPC+ac magnetic field: 1 mT, 10 GHz. The switching event due to the combined action ofthe dc SPC and the ac field takes place 0.5 ns before that dueto dc SPC alone. An experimental strategy to decrease the switching time was presented in Ref. 12, where a dc precharging currentexcites the magnetization to a precession trajectory therebyaccelerating the switching induced by a subsequent currentpulse. The role of the ac field in our simulations is similar tothat of the precharging current. It causes deviation of the y andzcomponents of the magnetization from their initial val- ues faster than when the dc SPC alone is applied /H20849Fig. 2 compared to Fig. 3 /H20850; consequently the torque due to the cur- rent is larger during the initial steps of the switching process.The switching proceeds via a nucleation process as illus-trated by Fig. 4 that shows the domain configuration at pointsA/H20849top/H20850/H20849Fig. 3, 1.3 ns /H20850and B /H20849bottom /H20850/H20849Fig. 2, 1.3 ns /H20850. The difference between these two configurations is due to the ac FIG. 1. A comparison between the switching time vs the switching current with and without ac magnetic field /H20851top S1 structure /H20849left PS→APS, right APS→PS/H20850and bottom S2 structure /H20849left PS→APS, right APS →PS/H20850/H20852. FIG. 2. Temporal evolution of /H20855m/H20856from the PS to the APS due to dc SPC, J=−0.8 /H11003108A/cm2. FIG. 3. Temporal evolution of /H20855m/H20856from the PS to the APS due to dc SPC+ac field /H20849J=−0.8 /H11003108A/cm2+ac magnetic field linear: 1 mT, 10 GHz /H20850.08G507-2 Finocchio et al. J. Appl. Phys. 99, 08G507 /H208492006 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.186.1.81 On: Tue, 26 Aug 2014 14:31:31field assisting the magnetization nucleation process for the cells in the left side of the structure. Reversal of the polar-ization vector of the ac field reverses the side of the structurein which the nucleation process takes place thus confirmingour conjecture of the field-assisted nature of the nucleationprocess. In order to explain the detailed mechanism of the ac- field-accelerated switching, spatial configuration of the ex-cited spatial modes has been studied. This is achieved bymeans of a micromagnetic spectral mapping technique simi-lar to the one used in Refs. 13–15. The procedure consists ofperforming Fourier transform of the magnetization temporalevolution for each computational cell and then plotting thetwo-dimensional /H208492D/H20850spatial distribution of the spectral power at the specific frequency of the mode to be analyzed.A gray scale is used to describe the power of the mode foreach cell /H20849white small, black large /H20850; it is proportional to the sum of the square of the amplitude of Fourier transform ofthex,y, and zcomponents of magnetization of that cell and that frequency. Figure 5 shows the spatial configuration ofthe main mode /H208497.4 GHz /H20850excited during the first 1.5 ns of the simulations of Figs. 2 and 3, without /H20849top/H20850and with /H20849bot- tom /H20850ac magnetic field. These show that the presence of ac magnetic field stimulates the excitation of the main modethat drives the switching process in the left side of the struc- ture helping in this way the nucleation process. In contrast to the strategy presented in Ref. 12, our ap- proach also accelerates the switching from APS to PS. Forexample, Fig. 6 shows the temporal evolution of /H20855m x/H20856from APS to PS due to dc SPC alone /H20849solid line /H20850and dc SPC +ac magnetic field /H20849dotted line /H20850,J=0.6/H11003108A/cm2. To confirm the ubiquity of the ac-field acceleration of the SPC-driven switching, we have performed simulations for awide range of the system parameters: frequency of the acmagnetic field ranging from 8 to 12 GHz and h 0ranging from 0.5 to 2 mT. All the simulations performed showed thatthe combined action of an ac magnetic field and dc SPCaccelerates the switching in both directions. In summary, employing a combined action of an ac mag- netic field and the dc SPC we have predicted a strategy todecrease the switching time for both APS →PS and PS →APS processes. 1J. Slonczewski, J. Magn. Magn. Mater. 159,L 1 /H208491996 /H20850;195, L261 /H208491999 /H20850;247, 324 /H208492002 /H20850. 2L. Berger, Phys. Rev. B 54, 9353 /H208491996 /H20850. 3J. A. Katine et al. , Phys. Rev. Lett. 84, 3149 /H208492000 /H20850. 4S. I. Kieselev et al. , Nature /H20849London /H20850425,3 8 0 /H208492003 /H20850. 5M. AlHajDarwish et al. , Phys. Rev. Lett. 93, 157203 /H208492004 /H20850. 6S. Urazhdin et al. , Phys. Rev. Lett. 91, 146803 /H208492003 /H20850. 7A. Fabian et al. , Phys. Rev. Lett. 91, 257209 /H208492003 /H20850. 8M. D. Stiles and A. Zangwill, Phys. Rev. B 66, 014407 /H208492002 /H20850; I. Krivo- rotov et al. , Science 307, 228 /H208492005 /H20850. 9L. Torres, L. Lopez-Diaz, E. Martinez, M. Carpentieri, and G. Finocchio, J. Magn. Magn. Mater. 286, 381 /H208492005 /H20850. 10M. Carpentieri et al. , J. Appl. Phys. 97, 10C713 /H208492005 /H20850. 11In this paper we define the switching time from the PS to the APS and vice versa, as the time elapsed from the instant the current pulse reaches itsmaximum value until the moment the normalized xcomponent of the magnetization achieves the value of −0.9 or 0.9, respectively. 12T. Devolder et al. , Appl. Phys. Lett. 86, 062505 /H208492005 /H20850. 13M. Grimsditch et al. , Physica B 354,2 6 6 /H208492004 /H20850. 14R. D. McMichael and M. D. Stiles, J. Appl. Phys. 97, 10J901 /H208492005 /H20850. 15D. V. Berkov, and N. L. Gorn, Phys. Rev. B 71, 052403 /H208492005 /H20850. FIG. 4. Snapshots of the magnetization configuration marked in Figs. 2 and 3 points A and B, respectively. FIG. 5. Spatial distribution of power in the main mode /H208497.4 GHz /H20850excited for the simulation of Fig. 2 /H20849black corresponds to large power /H20850. Top: SPC only; bottom: SPC+ac linear magnetic field. FIG. 6. Temporal evolution of /H20855mx/H20856from the APS to the PS, J=0.6 /H11003108A/cm2. solid line: dc SPC; dotted line: dc SPC+ac field /H20849current +ac magnetic field linear: 1 mT, 10 GHz /H20850.08G507-3 Finocchio et al. J. Appl. Phys. 99, 08G507 /H208492006 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.186.1.81 On: Tue, 26 Aug 2014 14:31:31

1.3068429.pdf

Microwave generation of tilted-polarizer spin torque oscillator Yan Zhou, C. L. Zha, S. Bonetti, J. Persson, and Johan Åkerman Citation: J. Appl. Phys. 105, 07D116 (2009); doi: 10.1063/1.3068429 View online: http://dx.doi.org/10.1063/1.3068429 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v105/i7 Published by the American Institute of Physics. Related Articles Nonlinear phase shifters based on forward volume spin waves J. Appl. Phys. 113, 113904 (2013) Continuous-wave coherent imaging with terahertz quantum cascade lasers using electro-optic harmonic sampling Appl. Phys. Lett. 102, 091107 (2013) Double-corrugated metamaterial surfaces for broadband microwave absorption J. Appl. Phys. 113, 084907 (2013) Preface to Special Topic: Intense terahertz sources for time-resolved studies of matter Rev. Sci. Instrum. 84, 022501 (2013) Characterization of the THz radiation source at the Frascati linear accelerator Rev. Sci. Instrum. 84, 022703 (2013) Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 19 Mar 2013 to 132.203.235.189. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsMicrowave generation of tilted-polarizer spin torque oscillator Yan Zhou,1,a/H20850C. L. Zha,1S. Bonetti,1J. Persson,1and Johan Åkerman1,2,b/H20850 1Department of Microelectronics and Applied Physics, Royal Institute of Technology, Electrum 229, 164 40 Kista, Sweden 2Department of Physics, Göteborg University, 412 96 Göteborg, Sweden /H20849Presented 14 November 2008; received 16 September 2008; accepted 5 November 2008; published online 20 February 2009 /H20850 Microwave frequency generation in a spin torque oscillator /H20849STO /H20850with a tilted fixed layer magnetization is studied using numerical simulation of the Landau–Lifshitz–Gilbert–Slonczewskiequation. The dependence of the STO free layer precession frequency on drive current is determinedas a function of fixed layer tilt angle. We find that zero-field STO operation is possible for almostall tilt angles, which allow for great freedom in choosing the detailed layer structure of the STO.©2009 American Institute of Physics ./H20851DOI: 10.1063/1.3068429 /H20852 I. INTRODUCTION Spin momentum transfer from a spin-polarized current to localized magnetic moment was first introduced theoreti-cally by Slonczewski 1and Berger2in 1996, and demon- strated experimentally in 1999.3One of the most appealing application of spin transfer devices is the possibility of real-izing compact, current tunable microwave oscillators, so-called spin torque oscillators /H20849STOs /H20850. 4In addition to its com- pact form factor with typical lateral dimensions in the 100nm range, the STO also offers high spectral purity, an ex-tremely wide frequency tuning range and the possibility ofintegration with most Si and III-V semiconductor technolo-gies. As a consequence the STO has great potential to replacetraditional ultrabroadband oscillator technologies, such as theyttrium iron garnet /H20849YIG /H20850oscillator. STOs typically operate under the application of large external magnetic fields, and as in the case of YIG oscilla-tors, this both complicates the final device and may lead toincreased power consumption. Recently, it has been shownthat if a fixed layer with a magnetization perpendicular to thefree layer is employed in a spin-valve structure, it is possibleto observe microwave generation in zero applied magneticfield. 5One drawback of this design is the rotational symme- try of the magnetizations in this device, which by definitioncancels any output signal. To generate an output one employsa readout layer /H20849spin-valve based /H20850to sense the oscillation. Adding additional layers with associated pinning layers fur-ther complicates the stacks and renders it less appealing forproduction. Besides the perpendicular STO, two other typesof novel STOs, the “wavy” torque STO and the vortex STO,have been experimentally demonstrated to yield microwavegeneration under essentially zero magnetic field. 6,7While the former lacks in output signal strength, the latter is limited tomaximum frequencies around 1 GHz and hence less attrac-tive for microwave generation. More recently we have shown, using numerical simula- tions, that a novel STO design, where the magnetization ofthe fixed layer is tilted with respect to the free layer, is ca- pable of generating microwave signal in zero applied mag-netic field and without the need of an extra readout layer. 8As a realistic choice of fixed layer material, we suggested L10 /H20849111/H20850FePt and L10/H20849101 /H20850FePt with tilt angles of /H9252=36° and 45°, respectively. In this work we extend the angular range toall angles between 0° and 90° and find somewhat unexpect-edly that zero field operation should be possible at almost allfixed layer angles. We study the detailed magnetization dy-namics as a function of drive current and fixed layer angle,and also the associated microwave frequency and effectivemagnetoresistance /H20849MR /H20850of our device. II. MODEL AND DESCRIPTION The schematic of the tilted-polarizer STO /H20849TP-STO /H20850is shown in Fig. 1. The fixed layer magnetization is tilted out of the film plane and forms an angle /H9252with the free layer easy x-zplane. While Min principle may take on any angle with respect to the x-axis, we here limit the analysis to Mlying in thex-zplane. As a consequence of the tilt angle /H9252, the spin polarized current is now composed of two spin polarizationcomponents: the in-plane component /H20849p x/H20850and the perpendicular-to-the-plane component /H20849pz/H20850. As shown in our previous study,8the perpendicular component pzof the spin polarized current is able to generate precession in the free a/H20850Electronic mail: zhouyan@kth.se. b/H20850Electronic mail: akerman1@kth.se. Free layerm/CID1/CID1/CID1 Free layerz/CID1/CID1 Spacer Fixed layer xy z /CID1M x /CID1 FIG. 1. /H20849Color online /H20850Schematic of a TP-STO. Mis the tilted fixed layer magnetization. NM is nonmagnetic layer and mstands for the magnetization of the free layer. xis the easy axis and x-yis the easy plane of the free layer magnetization. Mlies in the x-zplane with angle /H9252with respect to the x-axis.JOURNAL OF APPLIED PHYSICS 105, 07D116 /H208492009 /H20850 0021-8979/2009/105 /H208497/H20850/07D116/3/$25.00 © 2009 American Institute of Physics 105, 07D116-1 Downloaded 19 Mar 2013 to 132.203.235.189. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionslayer m, while the in-plane component of the fixed layer provides a means to detect the oscillations through giant MR/H20849GMR /H20850or tunneling MR. In our simulations we assume an asymmetric Sloncze- wski type torque incorporated into the standard Landau–Lifshitz–Gilbert equation: dmˆ dt=−/H9253mˆ/H11003Heff+/H9251mˆ/H11003dmˆ dt+/H9253/H6036/H9257/H20849/H9272/H20850J 2/H92620MS,freeedmˆ/H11003/H20849mˆ /H11003Mˆ/H20850, /H208491/H20850 where mˆis the unit vector of the free layer magnetization, MS,freeis its saturation magnetization, /H9253is the gyromagnetic ratio, /H9251is the Gilbert damping parameter, and /H92620is the mag- netic vacuum permeability. The last term represents theasymmetric spin torque, where /H6036is the reduced Planck con- stant, dis the free layer thickness, eis the electron charge, and /H9272is the angle between mˆand Mˆ. The applied field is held at zero throughout this work. Thus the effective field iscomposed of an easy-axis anisotropy field H kalong the x axis and an out-of-plane demagnetization field Hdalong the z axis: Heff=/H20849Hkeˆxmx−Hdeˆzmz/H20850//H20841m/H20841. The current density Jis defined as positive when it flows from the fixed layer to the free layer. We use the general asymmetric angular dependence of the spin torque amplitude,9–15 /H9257/H20849/H9272/H20850=2/H9264/H208491+/H9273/H20850 2+/H9273/H208491 + cos /H9272/H20850, /H208492/H20850 where /H9264is the spin polarization efficiency constant and /H9273is the GMR asymmetry parameter describing the deviationfrom sinusoidal angular dependence. The following general-ized expression is used to describe the angular dependence ofMR, 9–15 r=R/H20849/H9272/H20850−RP RAP−RP=1 − cos2/H20849/H9272/2/H20850 1+/H9273cos2/H20849/H9272/2/H20850, /H208493/H20850 where ris the reduced MR, and RPandRAPdenote the re- sistance in the parallel and antiparallel configurations, re-spectively. As material parameters we use Cu as spacer, Permalloy /H20849Py/H20850as the free layer, and FePt as the fixed layer. The lateral dimension of the Py thin film free layer is assumed to be anelliptical shape of 130 /H1100370 nm 2, with a thickness of 3 nm. The thickness of the fixed layer FePt is 20 nm. The values ofsome parameters used in the calculation are listed asfollows: 8,12,16/H9251=0.01, /H20841/H9253/H20841=1.76 /H110031011Hz /T, Ms =860 kA /m,Hk=0.01 T, Hd=1 T, /H9264=0.35, and /H9273=4. III. RESULTS AND DISCUSSION The dynamic phase diagram of frequency fas a function of the driving current density Jand tilt angle /H9252is shown in Fig.2/H20849a/H20850. The critical current for the onset of the STO pre- cession is plotted in the inset of Fig. 2/H20849a/H20850for both current polarities. While the critical current is very high at small tiltangles, it gradually decreases with increasing /H9252and ap- proaches values that have been attained in perpendicularlypolarized STOs. It can also be seen that the working currentwindow has a nonmonotonic dependence on the tilt angle and is very asymmetric with respect to the current polarity.For positive Jthe current window has a maximum around /H9252/H1101122°. Above this angle, the full frequency range is acces- sible by varying the current. However, below /H9252=22°, the frequency range is essentially cutoff at the lower end. Fornegative Jthe /H9252dependence is much less dramatic and the entire frequency range is accessible at all tilt angles. In Fig. 2/H20849b/H20850, we plot the effective MR as a function of the driving current density Jand tilt angle /H9252. The effective MR is calculated by the difference between the maximumand minimum values of resistance for each oscillation andnormalized by the maximum resistance variation betweenparallel and antiparallel configurations of the spin-valvestructure. Thus it is correlated with the output power fromthe device. The effective MR is generally larger at lowerangles than at higher angles for both current polarities, asshown in Fig. 2/H20849b/H20850. In the inset, we show the development of the precession orbits with increasing /H9252at fixed current den- sityJ=1/H11003108A/cm2. With increasing tilt angle and hence a larger perpendicular component of the spin polarizationcurrent, the orbit will be pushed further away from the filmplane /H20849x-yplane /H20850. The farther away from the film plane /H20849i.e., the smaller the cone angle /H20850the smaller the effective MR. Figure 2/H20849b/H20850also suggests that positive drive current is favor- able for large output TP-STO operation; only in a very lim-ited beta range does negative current polarity produce any sizeable effective MR. To optimize the effective MR it ap-pears that a choice of /H9252=10° –40° should be reasonable. The oscillation frequency dependence on driving current at selected angles is plotted in Fig. 3. It is noteworthy that the maximum operating frequency /H2084929 GHz /H20850is independent of tilt angle at negative drive current and only starts depend- 7090(a) 0.45 J<0 J>030f(GHz) egree)5070 00 00.150.30J>0Jc(108A/cm2) 1020 /CID1(de 300 1 53 04 56 07 59 00.00 /CID1(degree) 0 10 90(b)r 0 ee)70r 100% 50% -1-0.50zA B C /CID1(degre 3050 0% -1 0 1 -101 x y C 10 ABC J(108A/cm2)- 101234567 8 J(108A/cm2) FIG. 2. /H20849Color online /H20850/H20849a/H20850Frequency and /H20849b/H20850effective MR vs drive current density at tilt angles from 0° to 90°, respectively. Inset in /H20849a/H20850: critical current density vs tilt angle for both current polarities. Inset in /H20849b/H20850: precession orbits for different points A /H20849/H9252=10° /H20850,B /H20849/H9252=20° /H20850, and C /H20849/H9252=30° /H20850atJ=1 /H11003108A/cm2.07D116-2 Zhou et al. J. Appl. Phys. 105, 07D116 /H208492009 /H20850 Downloaded 19 Mar 2013 to 132.203.235.189. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsing on /H9252for values below 35° at positive current polarity. For yet smaller angles in the positive current range, themaximum operating frequency drops dramatically and alsoexhibits a nonmonotonic current dependence /H20849the complete plots for 15° and 25° are not shown /H20850. The strong curvature and final decrease in operating frequency is due to the asym-metric spin torque employed in the calculation. 12,15 Experimentally, a tilted easy-axis fixed layer can be re- alized by growing L0/H20849111/H20850FePt thin films with high magne- tocrystalline anisotropy /H20849Ku/H110117/H11003107erg /cm3/H20850, and a mag- netization with an orientation /H9252=36° and 45°, with respect to the film plane.8,17To achieve a continuously variable tilt angle, amorphous alloys such as TbFeCo /H20849Ref. 18/H20850may be employed or CoCr films, which show a thickness dependenttilt angle. 19–21 In summary, we investigated a new type of tilted polar- izer STO, which combines both high frequency tuning range,zero field operation, and relatively high effective MR. Wehope our study can stimulate further experimental effort andcan serve as a guideline in search of the parameter space,which combines wide operating frequency and optimizedpower output for TP-STO-based devices.ACKNOWLEDGMENTS Support from The Swedish Foundation for strategic Re- search /H20849SSF /H20850, The Swedish Research Council /H20849VR/H20850, and the Göran Gustafsson Foundation is gratefully acknowledged.Johan Åkerman is a Royal Swedish Academy of SciencesResearch Fellow supported by a grant from the Knut andAlice Wallenberg Foundation. 1J. C. Slonczewski, J. Magn. Magn. Mater. 159,L 1 /H208491996 /H20850. 2L. Berger, Phys. Rev. B 54, 9353 /H208491996 /H20850. 3E. B. Myers, D. C. Ralph, J. A. Katine, R. N. Louie, and R. A. Buhrman, Science 285,8 6 7 /H208491999 /H20850. 4J. Z. Sun, IBM J. Res. Dev. 50,8 1 /H208492006 /H20850. 5D. Houssameddine, U. Ebels, B. Delaet, B. Rodmacq, I. Firastrau, F. Ponthenier, M. Brunet, C. Thirion, J.-P. Michel, L. Prejbeanu-Buda,M.-C.Cyrille, O.Redon,and B. Dieny, Nature Mater. 6, 447 /H208492007 /H20850. 6O. Boulle, V. Cros, J. Grollier, L. G. Pereira, C. Deranlot, F. Petroff, G. Faini, J. Barnas, and A. Fert, Nat. Phys. 3,4 9 2 /H208492007 /H20850. 7V. S. Pribiag, I. N. Krivorotov, G. D. Fuchs, P. M. Braganca, O. Ozatay, J. C. Sankey, D. C. Ralph, and R. A. Buhrman, Nat. Phys. 3,4 9 8 /H208492007 /H20850. 8Y. Zhou, C. L. Zha, S. Bonetti, J. Persson, and J. Akerman, Appl. Phys. Lett. 92, 262508 /H208492008 /H20850. 9I. N. Krivorotov, D. V. Berkov, N. L. Gorn, N. C. Emley, C. Sankey, D. C. Ralph, and R. A. Buhrman, Phys. Rev. B 76, 024418 /H208492007 /H20850. 10J. Slonczewski, J. Magn. Magn. Mater. 247, 324 /H208492002 /H20850. 11S. Urazhdin, R. Loloee, and W. P. Pratt, Phys. Rev. B 71, 100401 /H208492005 /H20850. 12J. Xiao, A. Zangwill, and M. D. Stiles, Phys. Rev. B 70, 172405 /H208492004 /H20850. 13J. Barnas, A. Fert, M. Gmitra, I. Weymann, and V. K. Dugaev, Phys. Rev. B72, 024426 /H208492005 /H20850. 14J. Barnas, A. Fert, M. Gmitra, I. Weymann, and V. K. Dugaev, Mater. Sci. Eng., B 126, 271 /H208492006 /H20850. 15J. Xiao, A. Zangwill, and M. D. Stiles, Phys. Rev. B 72, 014446 /H208492005 /H20850. 16M. Gmitra and J. Barnas, Phys. Rev. Lett. 99, 097205 /H208492007 /H20850. 17C. L. Zha, B. Ma, Z. Z. Zhang, T. R. Gao, F. X. Gan, and Q. Y. Jin, Appl. Phys. Lett. 89, 022506 /H208492006 /H20850. 18S. Shiomi, J. Kato, S. Saito, T. Kobayashi, and M. Masuda, Jpn. J. Appl. Phys., Part 2 33, L1159 /H208491994 /H20850. 19D. P. Ravipati, W. G. Haines, J. M. Sivertsen, and J. H. Judy, J. Appl. Phys. 61, 3149 /H208491987 /H20850. 20M. R. Khan, D. J. Seagle, N. C. Fernelius, and J. I. Lee, J. Appl. Phys. 61, 3161 /H208491987 /H20850. 21M. R. Khan, J. I. Lee, D. J. Seagle, and N. C. Fernelius, J. Appl. Phys. 63, 833 /H208491988 /H20850.FIG. 3. /H20849Color online /H20850TP-STO operating frequency vs current density at different tilt angles.07D116-3 Zhou et al. J. Appl. Phys. 105, 07D116 /H208492009 /H20850 Downloaded 19 Mar 2013 to 132.203.235.189. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

1.4875618.pdf

Tunable eigenmodes of coupled magnetic vortex oscillators Max Hänze, Christian F. Adolff, Markus Weigand, and Guido Meier Citation: Applied Physics Letters 104, 182405 (2014); doi: 10.1063/1.4875618 View online: http://dx.doi.org/10.1063/1.4875618 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/104/18?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Direct imaging of phase relation in a pair of coupled vortex oscillators AIP Advances 2, 042180 (2012); 10.1063/1.4771683 Broadband probing magnetization dynamics of the coupled vortex state permalloy layers in nanopillars Appl. Phys. Lett. 100, 262406 (2012); 10.1063/1.4729825 Magnetic vortex dynamics on a picoseconds timescale in a hexagonal Permalloy pattern J. Appl. Phys. 107, 09D302 (2010); 10.1063/1.3358223 Direct observation of the vortex core magnetization and its dynamics Appl. Phys. Lett. 90, 202505 (2007); 10.1063/1.2738186 Vortex dynamics in coupled ferromagnetic multilayer structures J. Appl. Phys. 99, 08F305 (2006); 10.1063/1.2173630 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 160.45.67.132 On: Wed, 21 May 2014 09:36:29Tunable eigenmodes of coupled magnetic vortex oscillators Max H €anze,1,a)Christian F . Adolff,1Markus Weigand,2and Guido Meier1,3 1Institut f €ur Angewandte Physik und Zentrum f €ur Mikrostrukturforschung, Universit €at Hamburg, 20355 Hamburg, Germany 2Max-Planck-Institut f €ur Intelligente Systeme, Heisenbergstr. 3, 70569 Stuttgart, Germany 3The Hamburg Centre for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg, Germany (Received 7 March 2014; accepted 27 April 2014; published online 7 May 2014) We study the magnetization dynamics of coupled vortices in arrays of Permalloy disks via analytical calculations and scanning transmission x-ray microscopy. The Thiele approach is used to derive linear equations of motion of the vortices. Thereby, vortex motions following a nanosecond field pulse are described by a superposition of eigenmodes that depend on the vortex polarizations.Eigenmodes are calculated for a specific polarization pattern of a 3 /C23 vortex array. With magnetic field pulses distinct oscillations are excited and imaged in space and time. The calculated eigenmodes precisely describe the measured oscillations. VC2014 AIP Publishing LLC . [http://dx.doi.org/10.1063/1.4875618 ] Coupled modes play a fundamental role in solid state physics. For example, the propagation of sound waves in crys- tals can be described by coupled harmonic oscillators.Although the interactions, in such models, may be limited to the next neighbors of a lattice site, the vast number and the order of all participating oscillators determines the actualbehavior. 1–3Here, we show that the concept of coupled har- monic oscillators is applicable to a system of coupled mag- netic vortex oscillators. Vortices form in magnetic nanodisksof suitable geometry. 4They have a magnetization configura- tion curling in the plane either clockwise or counter-clockwise with an out-of-plane component at the center position pointingeither up or down. This entails two state parameters, chirality cand polarization p. 5The gyrotropic mode representing the fundamental excitation of the vortex ground state can be com-pared to the oscillation of a harmonic oscillator. 6Coupling occurs when the center-to-center distance of a pair of vortices is less than twice the diameter of the disks.7,8Vortex oscilla- tions have been studied in pair-coupled vortices,9–12in one-dimensional vortex chains,13and larger two dimensional vortex arrays.14–16In contrast to systems of coupled harmonic oscillators, the interaction strength can be tuned dynamically in dependence on the polarizations.17–20Using self-organized state formation, we are able to investigate distinct polarizationconfigurations experimentally. 21,22Eigenmodes of these con- figurations are calculated in the following as they lead to a deeper understanding of the experimentally observed oscilla-tions. Analogies with photonic crystals 23are drawn in order to interpret the results obtained by scanning transmission x-ray microscopy at the MAXYMUS microscope of the BESSY IIsynchrotron in Berlin, Germany. The Thiele equation describes the motion of a single magnetic vortex in a particle model. 24Here, we calculate the eigenmodes of a vortex array with coupled Thiele equations. The eigenmodes that means the eigenvectors and eigenfre- quencies fully characterize all undriven motions of coupledvortices. In principle, in an array of ferromagnetic disks Gilbert damping has to be considered. However, it turns outthat in the underdamped case considered here the damping term can be neglected when calculating the eigenmodes. For a single vortex without damping, the Thiele equationreads 24,25 _~x¼1 G2 0~G/C2~Fð~x1; :::;~xNÞ; (1) where ~x2~x1; :::;~xN fg is the deflection of a vortex core, G0 is the absolute value of the gyrovector ~G,25~Fis the force pointing perpendicular to the direction of motion, and Nis the number of vortices in the system. In the absence of external fields, the vortex cores oscillate with their eigenmo-des. In this undriven case a vortex is subjected to the force ~F¼~F harmþ~Fint, with the components ~Fharm ¼/C0jðx;y;0Þt25and ~Fint¼/C0 ð Ix;Iy;0Þt.~Fharmdescribes the force due to a two-dimensional harmonic potential with the stiffness coefficient jthat confines a core in a disk. ~Fint determines the interaction with the other vortices. The gyro- vector of the vortex points in z-direction ~G¼pG 0~ez, with the polarization p2f /C0 1;1g. Equation (1)transforms using ~x¼ðx;yÞt;~I¼ðIx;IyÞtand the 90/C14rotation matrix ~r90into _~x¼/C0p G0~r90ðj~xþ~IÞ;with ~r90¼0/C01 10/C18/C19 :(2) The out-of-plane component is zero and has been left out. The velocity _~xdepends on the interaction term ~Iand the deflection ~x. There are N coupling terms ~Iiand deflection terms ~xi, with i2½1;N/C138forNvortices. By summarizing the vortex deflections to a single vector ~u¼ð~x1;~x2;/C1/C1/C1;~xNÞt and the interaction terms to a vector ~J¼ð~I1;~I2;/C1/C1/C1;~INÞt, the equation of motion for the vortices can be written in the compact form _~u¼/C0~p G0~R90ðj~uþ~JÞ;with (3) ~p¼diagNðp1;p2; :::;pNÞ;~R90¼diagNð~r90; :::; ~r90Þ:(4) The interaction terms ~Iiare given by18,26 a)Electronic mail: mhaenze@physnet.uni-hamburg.de 0003-6951/2014/104(18)/182405/4/$30.00 VC2014 AIP Publishing LLC 104, 182405-1APPLIED PHYSICS LETTERS 104, 182405 (2014) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 160.45.67.132 On: Wed, 21 May 2014 09:36:29~Ii¼XN j6¼icicjgxxijxjþgxyijyj gyxijxjþgyyijyj ! ; (5) where ciandcjare the chiralities and gare the coupling coef- ficients of the vortices. Vortex pairs ijwith a big interdis- tance have coupling coefficients that are comparably smallerthan vortex pairs with a reduced interdistance. The coupling coefficients gare computed by numerical integration as described in Ref. 21taking into account the overestimation of the coupling strength by the rigid vortex model. 7,18,27The calculation is done for all vortex pair interactions separately. Thus, vortices at the boundaries of the array have differentinteraction terms than vortices inside the vortex array. ~I ið~x1; :::;~xNÞlinearly depends on all deflections ~xi. Thus, the interaction vector ~Jcan be expressed by an interaction ma- trix ~Zthat fulfills the equation ~J¼~Z/C1~uand has the matrix elements ~Z¼ðzijÞwith zij¼ð1/C0dijÞcicjgxxijgxyij gyxijgyyij ! ;(6) where dijdenotes the Kronecker delta. Equation (3)is trans- formed into a system of first order linear differential equa- tions with solutions of the form ~u¼~/C23ekt _~u¼/C0~p G0~R90ðj1þ~ZÞ~u¼:~M~u: (7) Thus, the system of Thiele equations has been reduced to an eigenvalue problem ~M~/C23¼k~/C23,w h e r e kare eigenvalues and ~/C23 are eigenvectors of the system. For our system, eigenvalues represent the frequencies and eigenvectors the motions forany alignment of vortices. It builds the basis to understand the experimental system as coupled harmonic oscillators. In the following, the eigenfrequencies and motions are presented for the specific case of a 3 /C23 vortex array. The polarization of every single vortex influences the eigenfre- quencies of the array. Nine coupled vortices lead to a numberof 2 9¼512 possible polarization patterns. These patterns have different eigenmodes and cannot be analyzed individu- ally. Thus, for reasons of simplicity, we focus on one polar-ization pattern accessible by self-organized state formation as it has been described in detail in Ref. 21. This pattern has columns of alternating polarizations and will be addressed asstripe pattern. We obtain nine eigenmodes that are depicted in Fig. 1with its frequencies and motions. The motions are characterized by the relative phases and the relative ampli-tudes between the vortices during one oscillation period. The phases indicate the positions of the vortices for a snapshot in time. Eigenfrequencies are scattered around the resonancefrequency of 238 MHz for an isolated vortex (see Fig. 1(a)). Obviously, the eigenfrequencies are not degenerate. For some eigenmotions, the amplitude of a vortex is completelysuppressed, for example, for the central vortex of mode 2 (see Fig. 1(b)). Other modes have comparable amplitudes but different phase relations. This can be seen by comparingmode 4 and mode 6. These spatially non-uniform trajectory amplitudes and phases are characteristic for standing waves resulting from the fixed boundary condition in arrays of afinite number of disks. 13We have to keep in mind that the sense of rotation depends on the polarization. Consequently, the phase relations between vortices with negative and posi-tive polarity change along a period of rotation. Although the chiralities do not alter the eigenfrequencies they result in a phase shift of 180 /C14. Accordingly, Fig. 1(b) shows snapshots of the phase relations of an array with equal chiralities. The vortex parameters chosen here are equivalent to those of the experimental samples. In the following measurements on 3 /C23 vortex arrays are presented. Arrays of Permalloy (Ni 80Fe20) disks depicted in Fig. 2are prepared with electron-beam lithography, thermal evaporation, and lift-off-processing on 100 nm thick silicon nitride membranes transparent for soft x-rays. The disks have ad i a m e t e ro f2 lm, a thickness of 60 nm, and a center-to-cen- ter distance of 2.25 lm. The vortices are excited by a mag- netic field generated by a current applied to a stripline in coplanar waveguide geometry above the array. Self-organizedstate formation establishes stripe patterns consisting of col- umns of alternating polarizations. The alignment is either par- allel or perpendicular to the exciting magnetic field using stategeneration frequencies of 225 MHz or 260 MHz, respectively. A stripe pattern is exemplarily depicted in Fig. 2(c).I nan e x tFIG. 1. (a) Calculated eigenfrequencies of the polarization pattern with col- umns of alternating polarization. The frequency 238 MHz of a single non- interacting vortex is indicated by a black line. (b) Trajectories of the eigenm- odes. Phase relations of the depicted snapshot are indicated by black dots. The polarization is color-coded (red and blue for up and down) to indicate the sense of rotation. FIG. 2. (a) Scanning electron micrograph of the coplanar waveguide placed above the vortex arrays. The inset shows the time structure of the pulse witha width of 1.8 ns. (b) Scanning transmission x-ray micrograph of a vortex array. (c) Magnetic XMCD contrast within the region of the vortex cores.182405-2 H €anze et al. Appl. Phys. Lett. 104, 182405 (2014) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 160.45.67.132 On: Wed, 21 May 2014 09:36:29step, we apply a field pulse to deflect the vortex cores from their equilibrium positions. Depending on the polarity and the chirality of a vortex, the deflection points in or opposite to the direction of the pulsed magnetic field.28When the pulse is over, the system freely gyrates back into its equilibrium posi- tion. If the linearity of the presented model applies to the experiment the emerging oscillations should be a superposi-tion of eigenmotions. Vortex dynamics is measured using scanning transmission x-ray microscopy and compared to the calculated eigenmodes. 29When the stripline is placed above all the vortices of an array, a field pulse acts on all vortices equally. The chiralities only influence the phase relations ofthe vortex motions while the polarizations additionally influ- ence the eigenfrequencies. In the experiment, the chirality cannot be controlled. The vortex chiralities are randomly dis-tributed. However, the dependency of the coupling dynamics can be eliminated by considering the effective in-plane mag- netizations instead of the core position vectors. 13Thus, with- out loss of generality, we illustrate the dynamics of a vortex array after a pulse deflection using an array with equal chiral- ities. In Fig. 3(b)), a snapshot of motions immediately after a short pulse is shown for the stripe pattern pointing either per- pendicular or parallel to the lines of constant polarization. It also depicts the eigenmotions 1 and 7 that correspond to theseinitial deflections. The deflections are approximately described by a single eigenmode. Due to the assumption that the initial deflection is generated with only one eigenmode,the vortices should gyrate with the corresponding eigenfre- quency after the field pulse ends. To evaluate the frequency of gyration, the motion of each vortex core is measured for a pe-riod of approximately 60 ns for both orientations of the stripe pattern. Its trajectory is determined and Fourier transformed. The sum of the squared magnitudes of all Fourier transforms,which is equivalent to the absorption of the system, 30is depicted in Fig. 3. For parallel and perpendicular deflections clearly distinguishable peaks are observed at different fre- quencies. Due to the finite sampling and the finite damping inthe sample the peaks are broadened. Using the material pa- rameters of Permalloy and the geometry of the disks, the pre- dicted frequencies of both eigenmodes are plotted. They are inexcellent agreement with the experimental results. Additional measurements have been performed, where the stripline is placed only over the first three elements of an array. A short pulse leads to an initial deflection of the first row of elements. This initial state is explained by a superposition of multipleeigenmodes. The initial deflection is decently approximated by the three eigenmodes 1, 4, and 7 depicted in Fig. 1.A l l three modes show a deflection of the top line of constantpolarization that concurs with the deflection direction of the vortices. When all three modes are combined the amplitudes of the two lower lines of constant polarization are compen-sated and the initial pulse deflection is achieved. In Fig. 4,t h e absorption of the array is depicted along with the theoretical eigenmodes. The frequencies of the three observed peaks arein good agreement with the theoretical eigenfrequencies depicted by black bars. The ratio of the peak amplitudes is in reasonable agreement with the theoretical expectation. In allexperiments, the trajectories of excited vortices show slightly different amplitudes and frequencies due to variations in the prepared disks and noise in the measurements. The vorticesare periodically excited with a period of about 60 ns. This delay is sufficient for the vortices to freely gyrate back into their equilibrium positions. Resulting remaining deflections atthe time of the field pulse may suppress the amplitudes of the depicted eigenmotions and explain small differences between theory and experiment. We have demonstrated that a coupled system of equa- tions of motion based on the Thiele model is a powerful tool to explain and understand dynamics in vortex arrays. Timeresolved and spatially resolved scanning transmission x-ray microscopy proves that the predictions of the model are reli- able. This experimental demonstration of vortex coupling FIG. 4. Fourier transform of vortex gyrations in the excited stripe pattern. The initial excitation is intentionally limited to only the first row of vorticesin this experiment. Calculated eigenfrequencies are indicated by black lines. The inset depicts the stripline configuration along with the initial deflection.FIG. 3. (a) Fourier transforms of vortex gyrations in the excited stripe pat- tern. The theoretically expected frequencies of the excited eigenmodes are indicated by black lines. (b) Experimental setup of the excitation for perpen- dicular (eigenmode 1) and parallel (eigenmode 7) alignment of the stripe pattern with respect to the exciting magnetic field. The polarizations are indicated by black and white dots. Resulting pulse deflection of the stripepattern and excited eigenmotion (lower right). All chiralities are set to 1. The deflection direction depends on the polarization.182405-3 H €anze et al. Appl. Phys. Lett. 104, 182405 (2014) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 160.45.67.132 On: Wed, 21 May 2014 09:36:29paves the way for experiments with larger artificial vortex crystals. For a raised number of eigenmodes such a system can be described by concepts like a density of states and a band structure.14These tunable band structures of vortex crystals suggest analogies to band structure engineering of photonic crystals. We thank Ulrich Merkt and Andreas Vogel for fruitful discussions and Michael Volkmann for superb technical as- sistance. We acknowledge the support of the Max-Planck-Institute for Intelligent Systems (formerly MPI for Metals Research), Department Sch €utz and the MAXYMUS team, particularly Michael Bechtel and Eberhard Goering.Financial support of the Deutsche Forschungsgemeinschaft via the Sonderforschungsbereich 668 and the Graduiertenkolleg 1286 is gratefully acknowledged. Thiswork has been supported by the excellence cluster “The Hamburg Centre for Ultrafast Imaging—Structure, Dynamics, and the Centre of Matter at the Atomic Scale” ofthe Deutsche Forschungsgemeinschaft. 1V. V. Kruglyak, S. O. Demokritov, and D. Grundler, J. Phys. D: Appl. Phys. 43, 264001 (2010). 2B. Lenk, H. Ulrichs, F. Garbs, and M. M €unzenberg, Phys. Rep. 507, 107 (2011). 3S.-K. Kim, J. Phys. D: Appl. Phys. 43, 264004 (2010). 4T. Shinjo, T. Okuno, R. Hassdorf, K. Shigeto, and T. Ono, Science 289, 930 (2000). 5A. Wachowiak, J. Wiebe, M. Bode, O. Pietzsch, M. Morgenstern, and R.Wiesendanger, Science 298, 577 (2002). 6B. Van Waeyenberge, A. Puzic, H. Stoll, K. W. Chou, T. Tyliszczak, R. Hertel, M. F €ahnle, H. Br €uck, K. Rott, G. Reiss, I. Neudecker, D. Weiss, C. H. Back, and G. Sch €utz,Nature 444, 461 (2006). 7A. Vogel, A. Drews, T. Kamionka, M. Bolte, and G. Meier, Phys. Rev. Lett. 105, 037201 (2010). 8J. Mej /C19ıa-L /C19opez, D. Altbir, A. H. Romero, X. Batlle, I. V. Roshchin, C.-P. Li, and I. K. Schuller, J. Appl. Phys. 100, 104319 (2006).9S. Sugimoto, Y. Fukuma, S. Kasai, T. Kimura, A. Barman, and Y. Otani, Phys. Rev. Lett. 106, 197203 (2011). 10A. Vogel, T. Kamionka, M. Martens, A. Drews, K. W. Chou, T. Tyliszczak, H. Stoll, B. Van Waeyenberge, and G. Meier, Phys. Rev. Lett. 106, 137201 (2011). 11A. Vogel, A. Drews, M. Weigand, and G. Meier, AIP Adv. 2, 042180 (2012). 12H. Jung, K.-S. Lee, D.-E. Jeong, Y.-S. Choi, Y.-S. Yu, D.-S. Han, A.Vogel, L. Bocklage, G. Meier, M.-Y. Im, P. Fischer, and S.-K. Kim, Sci. Rep. 1, 59 (2011). 13D.-S. Han, A. Vogel, H. Jung, K.-S. Lee, M. Weigand, H. Stoll, G. Sch €utz, P. Fischer, G. Meier, and S.-K. Kim, Sci. Rep. 3, 2262 (2013). 14J. Shibata and Y. Otani, Phys. Rev. B 70, 012404 (2004). 15A. Y. Galkin, B. A. Ivanov, and C. E. Zaspel, Phys. Rev. B 74, 144419 (2006). 16A. Vogel, M. H €anze, A. Drews, and G. Meier, Phys. Rev. B 89, 104403 (2014). 17Y. Liu, Z. Hou, S. Gliga, and R. Hertel, Phys. Rev. B 79, 104435 (2009). 18J. Shibata, K. Shigeto, and Y. Otani, Phys. Rev. B 67, 224404 (2003). 19A. Barman, S. Barman, T. Kimura, Y. Fukuma, and Y. Otani, J. Phys. D: Appl. Phys. 43, 422001 (2010). 20A. Vogel, M. Martens, M. Weigand, and G. Meier, Appl. Phys. Lett. 99, 042506 (2011). 21C. F. Adolff, M. H €anze, A. Vogel, M. Weigand, M. Martens, and G. Meier, Phys. Rev. B 88, 224425 (2013). 22S. Jain, V. Novosad, F. Y. Fradin, J. E. Pearson, V. Tiberkevich, A. N. Slavin, and S. D. Bader, Nat. Commun. 3, 1330 (2012). 23S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, S. John, K. Busch, A. Birner, U. G €osele, and V. Lehmann, Phys. Rev. B 61, R2389 (2000). 24A. A. Thiele, Phys. Rev. Lett. 30, 230 (1973). 25B. Kr €uger, A. Drews, M. Bolte, U. Merkt, D. Pfannkuche, and G. Meier, Phys. Rev. B 76, 224426 (2007). 26K. Yu. Guslienko, B. A. Ivanov, V. Novosad, Y. Otani, H. Shima, and K. Fukamichi, J. Appl. Phys. 91, 8037 (2002). 27O. V. Sukhostavets, J. Gonz /C19alez, and K. Y. Guslienko, Phys. Rev. B 87, 094402 (2013). 28S.-B. Choe, Y. Acremann, A. Scholl, A. Bauer, A. Doran, J. St €ohr, and H. A. Padmore, Science 304, 420 (2004). 29The spatial resolution with the zone plate used in the present experiment is 25 nm. The maximum temporal resolution is 40 ps. 30A. Drews, B. Kr €uger, G. Selke, T. Kamionka, A. Vogel, M. Martens, U. Merkt, D. M €oller, and G. Meier, Phys. Rev. B 85, 144417 (2012).182405-4 H €anze et al. Appl. Phys. Lett. 104, 182405 (2014) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 160.45.67.132 On: Wed, 21 May 2014 09:36:29

1.367914.pdf

Dependence of anti-Stokes/Stokes intensity ratios on substrate optical properties for Brillouin light scattering from ultrathin iron films J. F. Cochran, M. From, and B. Heinrich Citation: Journal of Applied Physics 83, 6296 (1998); doi: 10.1063/1.367914 View online: http://dx.doi.org/10.1063/1.367914 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/83/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Brillouin light scattering study of spin waves in NiFe/Co exchange spring bilayer films J. Appl. Phys. 115, 133901 (2014); 10.1063/1.4870053 Brillouin light scattering observation of the transition from the superparamagnetic to the superferromagnetic state in nanogranular ( SiO 2 ) Co films J. Appl. Phys. 104, 093912 (2008); 10.1063/1.3009339 High-intensity Brillouin light scattering by spin waves in a permalloy film under microwave resonance pumping J. Appl. Phys. 102, 103905 (2007); 10.1063/1.2815673 Thickness dependence of magnetic anisotropy in uncovered and Cu-covered Fe ∕ Ga As ( 110 ) ultrathin films studied by in situ Brillouin light scattering J. Appl. Phys. 99, 08J701 (2006); 10.1063/1.2165927 Brillouin light scattering investigations of structured permalloy films J. Appl. Phys. 81, 4993 (1997); 10.1063/1.364881 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.59.222.12 On: Mon, 01 Dec 2014 21:13:08Dependence of anti-Stokes/Stokes intensity ratios on substrate optical properties for Brillouin light scattering from ultrathin iron films J. F. Cochran,a)M. From, and B. Heinrich Department of Physics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada Brillouin light scattering experiments have been used to investigate the intensity of 5145 Å laser light backscattered from spin waves in 20 monolayer thick Fe ~001!films. The experiments have shown that the ratio of frequency upshifted light intensity to frequency downshifted light intensitydepends upon the material of the substrate used to support the iron films. For a fixed magnetic fieldand for a fixed angle of incidence of the laser light this intensity ratio is much larger for an iron filmdeposited on a sulphur passivated GaAs ~001!substrate than for an iron film deposited on a Ag ~001! substrate.Thedatahavebeencomparedwithacalculationthattakesintoaccountmultiplescatteringof the optical waves in the iron film and in a protective gold overlayer. The observations are inqualitative agreement with the theory, except for angles of incidence greater than 60°. © 1998 American Institute of Physics. @S0021-8979 ~98!37011-5 # The ratio of upshifted scattered light intensity ~anti- Stokes line !to downshifted scattered light intensity ~Stokes line!in Brillouin light scattering experiments ~BLS!per- formed on ultrathin magnetic films depends on the opticalproperties of the substrate that support the film. 1This is il- lustrated in Fig. 1 for BLS experiments performed on ironfilms 20 monolayers ~ML!thick. Figure 1 ~a!shows scattered light intensity versus frequency shift for a 20 ML iron filmgrown on a Ag ~001!single crystal by means of molecular beam epitaxy. 2The iron film was covered by 18 ML of gold. The BLS measurements were carried out ex situ. A magnetic field of 1.00 kOe was applied in the plane of the film andperpendicular to the plane of incidence of the incident 5145Å laser light. The p-polarized laser light was incident at 45°, and the scattered light was collected in the backscatteringconfiguration. 3The data shown in Fig. 1 ~b!were collected in the backscattering configuration for p-polarized light4,5inci- dent at 45°, and using an applied magnetic field of 1.08 kOe.The iron film was 20 ML thick, and was grown by means ofmolecular beam epitaxy on a sulphur passivated GaAs ~001! single crystal. 6The iron film was covered by 20 ML of gold and the BLS data were measured ex situ. It is clear from a comparison of Figs. 1 ~a!and 1 ~b!that~i!the ratio of up- shifted to downshifted scattered light intensity is quite dif-ferent for these two specimens; and ~ii!the observed fre- quency shift is different for the two specimens. Both of thesedifferences can be attributed to the effect of different sub-strates. The change in frequency between the two specimenscan be primarily attributed to a difference in perpendicularuniaxial surface anisotropy, see Table I. The dependence ofthe intensity ratios on substrate material is a consequence ofthe dependence of the magnitude and phase of the opticalelectric field in the iron film on the optical properties of thesubstrate material. A 20 ML thick iron film is thin comparedwith the penetration depth of 5145 Å light in iron, conse-quently the optical electric field amplitudes in the iron filmdepend very strongly on the optical properties of thesubstrate. 1Data for thin iron films grown on Ag and GaAs sub- strates have been compared with calculations carried out us-ing the formalism described by Cochran and Dutcher. 7Rel- evant parameters used in the calculations are listed in TableI. The magnetic parameters listed in Table I give a good a!Electronic mail: jcochran@sfu.ca FIG. 1. Light intensity vs frequency shift for 5145 Å light scattered from a 20 ML thick Fe ~001!film; the laser light was incident on the specimen at an angle of 45°. The scattered light was collected in the backscattering con-figuration, and intensities were normalized to the maximum intensity of thefrequency upshifted peak. ~a!Ag~001!/20 ML Fe/18 ML Au with a field of 1.0 kOe applied in the film plane and directed along ~100!.~b! GaAs ~001!/20 ML Fe/20 ML Au with a field of 1.08 kOe applied in the specimen plane and directed along ~110!. In both cases the magnetic field was applied perpendicular to the plane of incidence of the laser light.JOURNAL OF APPLIED PHYSICS VOLUME 83, NUMBER 11 1 JUNE 1998 6296 0021-8979/98/83(11)/6296/3/$15.00 © 1998 American Institute of Physics [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.59.222.12 On: Mon, 01 Dec 2014 21:13:08description of the dependence of spin-wave frequencies on applied magnetic field, see Fig. 2. The optical dielectric con-stants listed for 5145 Å light ~2.41 eV !were obtained from Johnson and Christy ~Iron, 8Silver9!, Joenson et al.10~gold!, and Aspnes and Studna11~GaAs !. The observed upshifted to downshifted intensity ratios are plotted as a function of applied magnetic field in Fig. 3for light incident at 45° on the specimens. The solid lines areintensity ratios calculated using the parameters listed inTable I. The data exhibit a decreasing intensity ratio withincreasing magnetic field; this decrease with field is repro-duced by the calculations. However, the calculated intensity ratios tend to be smaller than the observed ratios, especiallyin the case of the GaAs substrate. The calculated intensityratios are very sensitive to the amplitudes and phases of theoptical electric field components in the iron film, and thesefield components in turn, especially their phases, are sensi-tive to the film thicknesses and the dielectric parameters usedto describe them. The ratio of upshifted to downshifted light intensity has been investigated as a function of the angle of incidence ofthe light for fixed magnetic field. The results of angular mea-TABLE I. Parameters used to calculate the frequencies and ratios of upshifted to downshifted scattered light intensities for the Ag ~001!/20Fe/18Au and GaAs ~001!/20Fe/20Au used for this work. The perpendicular uniaxial anisotropy energy was taken to have the form Eu52(Ku/d)(mz/Ms)2ergs/cc, where dis the iron film thickness and mzis the magnetization component perpendicular to the plane of the Fe film. Fe film: Thickness d520 monolayers 528.6 Å Saturation magnetization, 4 pMs521.4 kOe Exchange stiffness parameter, A52.0331026 Gilbert damping parameter, G5108Hz Optical dielectric constant, e5(20.40, 16.44) Resistivity, r51.031025Vcm ~a!Ag~001!/20Fe/18Au Cubic anisotropy parameter, K154.03105ergs/cc Uniaxial anisotropy parameter, Ku51.04 ergs/cm2 ~b!GaAs ~001!/20Fe/20Au Effective cubic anisotropy parameter for Halong ~110!, K153.73104ergs/cc Uniaxial anisotropy parameter, Ku51.42 ergs/cm2 Au film: 2.04 Å per monolayerOptical dielectric constant, e5(23.75, 2.75) Resistivity, r52.3531026Vcm Ag substrate: Optical dielectric constant, e5(210.70, 0.33) Resistivity, r51.5931026Vcm GaAs substrate: Optical dielectric constant, e5(17.65, 3.19) Resistivity, r51Vcm FIG. 2. Spin-wave frequency vs applied magnetic field. The data were ob- tained from Brillouin light scattering experiments on 20 ML thick films withthe field applied in the specimen plane. The angle of incidence of the laserlight was 45°. 3—Ag ~001!/20 ML Fe ~001!/18 ML Au. The field was ap- plied along the ~100!direction; ~a!calculated using the parameters listed in Table I. 1—GaAs ~001!/20 ML Fe ~001!/20 ML Au. The field was applied along the ~110!direction; ~b!calculated using the parameters listed in Table I. FIG. 3. The ratio of frequency upshifted to frequency downshifted Brillouinbackscattered light intensity vs magnetic field applied in the specimen plane.The 5145 Å laser light was incident at 45°. 3—Ag ~001!/20 ML Fe ~001!/18 ML Au, the field was applied along ~100!;~a!calculated using the param- eters listed in Table I. 1—GaAs ~001!/20 ML Fe ~001!/20 ML Au, the field was applied along ~110!;~b!calculated using the parameters listed in Table I. The vertical error bars correspond to an estimated 20% uncertainty in thedata.6297 J. Appl. Phys., Vol. 83, No. 11, 1 June 1998 Cochran, From, and Heinrich [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.59.222.12 On: Mon, 01 Dec 2014 21:13:08surements on a specimen grown on a silver substrate are shown in Fig. 4, and results for an iron film grown on aGaAs substrate are shown in Fig. 5. The angular variation ofthe incident light is expressed in terms of the spin-wavewave vector component parallel with the film plane, Q 5(4 p/l)sinu, where l55145 Å is the wavelength of the incident light and uis the angle of incidence. The data ex- hibit an increasing intensity ratio with increasing Q, i.e., with increasing angle of incidence. The calculations, indi-cated by the solid lines in Figs. 4 and 5, also display anincreasing intensity ratio with increasing angle of incidence.According to theory, the intensity ratio varies with angle ofincidence because of an interference between the scatteredlight originating from the optical electric field components inthe iron film that are parallel and perpendicular to the filmplane. In the limit of normal incidence the optical electricfield in the iron film has only a component parallel to theplane, there is no interference effect, and therefore the up-shifted and downshifted intensity ratio must approach unityin the limit of Q50. This expectation is borne out by the experimental observations. For reasons which we do not un-derstand, the intensity ratio data shown in Fig. 5, and ob-tained using an applied field of 2.2 kOe, are in better agree-ment with the ratios calculated for an applied field of 1 kOe@curve ~i!#than with the curve calculated using a field of 2.2 kOe@curve ~ii!#. This appears to be a coincidence. However, the drop off in intensity ratio observed for Qlying between 2.0 and 2.2 310 5cm21~ubetween 55° and 65° !, and ob- served for both the silver substrates and the GaAs substrates,appears to be real. It may be caused by some unknown in-strumental effect. For the GaAs substrate the absolute inten-sity at u565° was also much reduced over that measured at u555° contrary to theoretical expectations. The origin of this rather sudden drop off in scattered light intensity is un-known: no such effect was observed for specimens grown onsilver substrates. It may be associated with the observedrough iron growth obtained using a GaAs substrate. 6A de- crease in intensity ratio at large angles of incidence runscounter to the rather sharp increase in the upshifted to down-shifted intensity ratio for Qapproximately equal to 2.25 310 5cm21reported by Moosmu ¨ller, Truedson, and Patton for thin permalloy films sputtered on silicon.12A monotonic dependence of the intensity ratio on angle of incidence of thelaser light was reported by Camley et al. 5for 100 Å thick polycrystalline iron films. The authors would like to thank S. Watkins for the sul- phur passivated GaAs substrates used in this work, and T.Monchesky for communicating to us the results of his 36GHz microwave measurements on 20 ML Fe films grown onthese GaAs substrates. We would also like to thank the Natu-ral Sciences and Engineering Research Council of Canadafor grants that supported this work. 1M. G. Cottam, J. Phys. C 16, 1573 ~1983!. 2Specimens were prepared in a layer-by-layer growth mode using molecu- lar beam epitaxy as described by B. Heinrich, Z. Celinski, J. F. Cochran,A. S. Arrott, and K. Myrtle, J. Appl. Phys. 70, 5769 ~1991!. 3J. R. Sandercock, in Topics in Applied Physics Vol. 51, Light Scattering in Solids III , edited by M. Cardona and G. Gu ¨ntherodt ~Springer, Berlin, 1982!, p. 173. 4In the backscattering configuration the intensity of the scattered light is the same for both p- ands-polarized incident light; see R. E. Camley and M. Grimsditch, Phys. Rev. B 22, 5420 ~1980!; also Ref. 5. 5R. E. Camley, P. Gru ¨nberg, and C. M. Mayr, Phys. Rev. B 26, 2609 ~1982!. 6A buffer layer of GaAs was grown on a GaAs ~001!wafer by means of metalorganic chemical vapor deposition and the resulting surfaces werepassivated using a H 2S treatment at 400 °C. The iron film was deposited on the sulphur passivated GaAs surface at room temperature by means of molecular beam epitaxy. The iron growth was rough and exhibited noRHEED oscillations. 7J. F. Cochran and J. R. Dutcher, J. Magn. Magn. Mater. 73, 299 ~1988!. There is an error in the last term of R3on p. 309; the bracketed term should read ( Hy1Hz18pMs) not (Hy1Hz14pMs). 8P. B. Johnson and R. W. Christy, Phys. Rev. B 9, 5056 ~1974!. 9P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 ~1972!. 10P. Joenson, J. C. Irwin, J. F. Cochran, and A. E. Curzon, J. Opt. Soc. Am. 63, 1556 ~1973!. 11D. E. Aspnes and A. A. Studna, Phys. Rev. B 27, 985 ~1983!. 12H. Moosmu ¨ller, J. R. Truedson, and C. E. Patton, J. Appl. Phys. 69, 5721 ~1991!. FIG. 5. The ratio of frequency upshifted to frequency downshifted Brillouin backscattered light intensity vs the in-plane component of the spin-wavewave vector for the specimen. GaAs ~001!/20 ML Fe ~001!/20 ML Au. A magnetic field of 2.2 kOe was applied in the specimen plane along the ~110! direction, and 5145 Å laser light was used for the measurements. ~i!Calcu- lated for a field of 1.0 kOe and ~ii!calculated for a field of 2.2 kOe using the parameters listed in Table I. The vertical error bars correspond to an esti-mated 20% uncertainty in the data. FIG. 4. The ratio of frequency upshifted to frequency downshifted Brillouinbackscattered light intensity vs the in-plane component of the spin-wavewave vector for the specimen Ag ~001!/20 ML Fe ~001!/18 ML Au. A mag- netic field of 1.0 kOe was applied in the specimen plane along the ~100! direction, and 5145 Å laser light was used for the measurements. The solidline was calculated using the parameters listed in Table I. The vertical errorbars correspond to an estimated 20% uncertainty in the data.6298 J. Appl. Phys., Vol. 83, No. 11, 1 June 1998 Cochran, From, and Heinrich [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.59.222.12 On: Mon, 01 Dec 2014 21:13:08

1.4903474.pdf

Enhanced gyration-signal propagation speed in one-dimensional vortex-antivortex lattices and its control by perpendicular bias field Han-Byeol Jeong and Sang-Koog Kima) National Creative Research Initiative Center for Spin Dynamics and Spin-Wave Devices, Nanospinics Laboratory, Research Institute of Advanced Materials, and Department of Materials Science and Engineering, Seoul National University, Seoul 151-744, South Korea (Received 17 September 2014; accepted 22 November 2014; published online 4 December 2014) We report on a micromagnetic simulation study of coupled core gyrations in one-dimensional (1D) alternating vortex-antivortex (V-AV) lattices formed in connected soft-magnetic-disk arrays. Insuch V-AV lattices, we found fundamental standing-wave gyration modes as well as significantly enhanced gyration-signal speed, as originating from combined strong exchange and dipole interactions between the neighboring vortices and antivortices. Collective core oscillations in theV-AV networks are characterized as unique two-branch bands, band gap, and width of which are remarkably variable and controllable by externally applied perpendicular fields. The gyration propagation speed for the parallel polarization ordering is much faster ( >1 km/s) than that for 1D vortex-state arrays, and variable remarkably by application of perpendicular static fields. This work provides a fundamental understanding of the coupled dynamics of topological solitons as well as an additional mechanism for fast gyration-signal propagation; moreover, it offers an efficientmeans of significant propagation-speed enhancement that is suitable for information carrier applications in continuous thin-film nanostrips. VC2014 AIP Publishing LLC . [http://dx.doi.org/10.1063/1.4903474 ] Nontrivial spin textures, especially the magnetic vortex in confined potential wells such as ferromagnetic nanodots, have been widely studied owing not only to their intriguing gyration dynamics1,2and dynamic core-switching behaviors3,4 but also their potential applications for information-storage5 and -processing6–9as well as microwave devices.10,11Studies on the dynamics of single vortices have been extended to theircoupled systems. Shibata et al. 12reported on combined ana- lytical and numerical calculations of coupled vortex-gyration modes in two vortex-state disks. Recently, further two-vortexpairs 7,12–14and 1D or 2D periodic arrays of coupled vorti- ces6,8,9,15–18were intensively studied to understand the fundamental modes of coupled vortex gyrations and theircontrollability. In the meantime, the magnetic antivortex, the topologi- cal counterpart of the vortex, has been found to exhibit coregyration and related switching behaviors similar to vortex dynamics. 19–21Periodic vortex and antivortex arrangements have often been found as parts of cross-tie walls.22,23 Isolated antivortices by contrast, due to their unstable state, possibly can be formed in specially designed geometric con- finements.19–21,23Although the dynamics of single antivorti- ces19–21and their dynamic interactions with vortices11,24,25 have been reported in earlier studies, the dynamics of coupled vortices and antivortices and their coupled gyrationpropagation have not been well understood in the magnonic band aspect or in terms of gyration-signal propagation through 1D alternating vortex-antivortex (V-AV) lattices. In this letter, we report on the fundamentals of coupled vortices and antivortices in 1D V-AV lattices, specifically interms of their coupled excitation modes, unique two-branch band structures, and gyration-signal propagation speed. We found that gyration-signal propagations are much faster (>1 km/s) than those for 1D disk arrays composed only of vortex states, as originating from combined exchange and dipolar interactions between neighboring vortices and anti- vortices. Also, we discovered that the unique band structureand gyration-propagation speed are controllable according to the strength and direction of perpendicular static fields. In the present study, we used, for a model system, alter- nating V-AV lattices in a connected triple-disk structure wherein the diameter of each disk is 2 R¼303 nm, the thick- ness T¼20 nm, and the center-to-center interdistance D int¼243 nm, for cases of R<Dint<2R(see Fig. 1).26 Unlike physically separated disks (i.e., Dint>2R), this con- nected triple-disk structure has two antivortices betweenneighboring vortices of the same counter-clockwise (CCW) chirality in a sufficiently stable metastable state. In the initial state, the core magnetizations of the three vortices are FIG. 1. Model geometry of connected triple-disk structure. The individual disks have an equal diameter 2 R¼303 nm, a thickness T¼20 nm, and a disk center-to-center distance Dint¼243 nm. The color and height display the in-plane magnetization and out-of-plane magnetization components, respectively. The chirality of the three vortices is CCW, as indicated by the white arrows, and the polarizations of the three vortices and two antivortices are upward and downward, respectively.a)Author to whom correspondence should be addressed. Electronic mail:sangkoog@snu.ac.kr 0003-6951/2014/105(22)/222410/5/$30.00 VC2014 AIP Publishing LLC 105, 222410-1APPLIED PHYSICS LETTERS 105, 222410 (2014) upward (called polarization p¼þ1), and those of the two antivortices are downward ( p¼/C01), resulting in the antipar- allel polarization ordering. We numerically calculated the motions of local magnetizations (cell size: 3 /C23/C2Tnm3) using the OOMMF code,27which employs the Landau- Lifshitz-Gilbert equation.28We used the typical material parameters of Permalloy (Ni 80Fe20, Py): saturation magnetization Ms¼8.6/C2105A/m, exchange stiffness Aex¼1.30/C210/C011J/m, and zero magnetocrystalline anisotropy. To excite all of the coupled modes in such a model sys- tem, we displaced the core only of the left-end disk up to/C2454 nm in the /C0xdirection by applying a static local field of 200 Oe in the þydirection. Upon turning off the local field, the five individual cores of the vortices and antivortices weremonitored to trace the trajectories of the coupled core motions under free relaxation. All of the simulation results noted hereafter were obtained up to 200 ns after the field wasturned off. Figure 2(a)shows the trajectories of the individ- ual core motions with their position vectors X¼(X, Y). Owing to the direct excitation of the first core (noted as V 1), a large amplitude of the core gyration, starting at X¼/C054 nm toward the center position, is observed. The vor- tex gyration of the first disk is then propagated to the nextantivortex (AV 2), and then further propagates through the whole chain, as evidenced by the large gyration amplitudes of the remaining vortices, indicated as V 3and V 5. These large gyration amplitudes imply that the vortex gyration is well propagated to the next vortices through the neighboring antivortices, without significant energy loss. To elucidate theobserved modes, we plotted frequency spectra in the left col- umn of Fig. 2(b), as obtained from fast Fourier transforma- tions (FFTs) of the xcomponents of the individual core oscillations. From the first vortex to the last one, contrasting FFT powers between the major five peaks are observed. The distinct peaks are denoted as x i, with i¼1, 2, 3, 4, and 5. The individual peaks in the frequency domains correspond tox/2p¼0.23, 0.45, 0.55, 0.98, and 1.11 GHz, respectively. Each peak in each frequency spectrum is located at the same corresponding frequency from V 1through V 5, indicating that there are five distinct modes. Because of the intrinsic damp-ing of each core gyration, those peaks are somewhat broad- ened and overlapped with the neighboring peaks. There are noticeable differences between the vortex- and antivortex-core motions: in the AV 2and AV 4spectra, the x3peak dis- appears, while in the V 3spectrum, the x2peak disappears and the x4peak becomes very weak. In order to obtain those data with a better spectral resolution, we carried out further simulations with the same model but with an extremely lowdamping constant, a¼0.0001. The resultant FFT spectra (right column of Fig. 2(b)) show that the five major peaks become sharper. 29Moreover, for cases of antivortex gyration motions, the higher-frequency x4andx5modes, which were very weak for a¼0.01, are much stronger for a¼0.0001, due not only to the negligible energy loss but also to thestrong exchange interaction between the antivortices and the neighboring vortices. We note that such a kind of frequency spectra could be obtained experimentally as demonstrated oncoupled vortices by Sugimoto et al. 14 Fig. 3shows the spatial correlation between the five cores’ motion for each mode, obtained by making inverseFFTs of all of those peaks of each mode with a¼0.0001. The trajectories of the orbiting cores are compared for all of the modes, along with the corresponding profiles of the Y components. The Ycomponent profiles are different mark- edly between the modes. The most noteworthy feature is the fact that the collective core motions show a standing-waveform with a different overall wavelength for a given mode. The orbiting radii of the five cores for a given mode are sym- metric with respect to the center of the whole system (i.e., atV 3) and are also completely pinned at the imaginary points at both ends, denoted AV 0and AV 6, as reported in Refs. 9 and18. The lower x1,x2, and x3modes and the higher x4 FIG. 2. (a) Trajectories of gyration motions of individual vortex cores (red lines) and antivortex cores (blue lines). (b) FFTs of xcomponents of individ- ual core-position vectors from their own center positions. The left (right) column is the result with a¼0.01 (0.0001). The five major peaks are denoted xiwhere i¼1, 2, 3, 4, and 5, and are marked by the gray vertical lines. FIG. 3. Spatial distributions of individual core positions in one cycle period (2p/x) of gyration starting from about 100 ns for five different collective motions (modes). The core motions’ trajectories are magnified for clear comparison (though the magnifications are different for the different modes). The wide arrows represent the directions of the effective magnetizations hMiinduced by their own core shifts. The shapes of all of the trajectories are eccentric along either the xoryaxis, because the strong exchange inter- action between the antivortices and the neighboring vortices is additionallyemployed in such a connected-disk model, with its high asymmetry between thexandyaxes.222410-2 H.-B. Jeong and S.-K. Kim Appl. Phys. Lett. 105, 222410 (2014)andx5modes are quite different in terms of their standing- wave forms. For the x1mode, all of the cores move in phase, while for the x2mode, the core of V 3acts as a node, and for thex3mode, the two antivortex cores act as standing-wave nodes. For the x4andx5modes, the two antivortices are highly excited relative to the three vortices. Also, the anti- vortices’ core motions are permeated into the bonding axis,thus resulting in higher energy states due to their strong exchange interaction with the neighboring vortices. The dif- ference between the individual modes can be understood by the relative phases between the vortex’s and antivortex’s effective magnetizations, hMi, induced by their own core shifts and consequently by their dynamic dipolar interaction, as discussed in Ref. 9. The relative phases of the hMi between the neighboring core gyrations determine themode’s average dynamic dipolar energy. However, the strong exchange interaction in such a connected thin-film strip must also be taken into account. Due not only to theunknown potential wells of the antivortices but also the com- plex asymmetric exchange interaction terms between the vortices and antivortices, it is difficult—in such V-AV latti-ces—to separately extract the individual contributions of the exchange and dipolar interactions to the individual core motions in each mode. Next, on the basis of the above findings, we extended our study to longer 1D chains, as shown in Fig. 4(a). Here, we consider two different polarization orderings withthe same CCW chirality for all of the individual vortices: parallel and antiparallel orderings represented by ( p V,pAV) ¼(þ1,þ1) and ( þ1,/C01), respectively. For such V-AV fi- nite lattices, the thickness was set to 40 nm to make them more stable than in a thinner strip, and we applied a static local field of 200 Oe in the /C0ydirection only in the left-end disk. The other simulation conditions were the same as those for the earlier model shown in Fig. 1. From the FFTs of the x components of all of the 25 core positions, we obtaineddispersion curves in the reduced zone scheme for the parallel and antiparallel polarization orderings, as shown in the fourth column of Fig. 4(b). The dispersions were asymmetric with respect to the wavenumber k¼0, because the initial core motion was excited only in the left vortex and was then propagated toward the þxdirection. Therefore, the modes with positive group velocities were relatively strong as com- pared with those with the negative group velocities. In suchband structures, there are two distinct higher- and lower- frequency branches, as expected from the two different types of standing-wave forms found in the earlier V-AV model shown in Fig. 1. It is noteworthy that such dispersions are quite analogous to collective gyrations in the acoustic and optical branch , respectively, as observed in only vortex-state lattices consisting of alternating different materials 18or in diatomic lattice vibrations. We also note that the lower band consists of quantized flat-shaped local modes. In the FFTs of the coupled V-AV gyrations, we applied a periodic boundary condition: as a bi-object array, the wave number is set to k¼pm=N/C22dint, where Nis the number of the unit basis, /C22dintis its lattice constant, and mis an arbitrary integer in the range of/C0kBZ/C20k<kBZwith the first Brillion zone boundary kBZ¼p=/C22dint.30At each of the N-discrete kvalues, there are two corresponding frequencies, thus leading to 2 Nnormal modes,18,30,31although such quantized modes are not clearly shown in the much narrow higher band. In both polarization orderings, the bandgap between the two branches is almost the same, /C240.25 GHz, and the lowest frequency of the higher branches is /C241.23 GHz. The band width of the lower branch is as wide as /C240.9 GHz, due to the strong exchange-dipole interaction between vortices and antivortices in such connected-disk arrays. The overall shapeof the lower branch is concave up; that is, the frequency is lowest at k¼0 and highest at k¼k BZ, for both polarization orderings.18However, the shape of the higher branch varies according to the polarization ordering: concave down for the antiparallel polarization ordering, and almost flat for the par- allel one. In the lower branch, as kapproaches the kBZ, thex value reaches the angular eigenfrequency x0of isolated Py disks of the given dimensions (here x/2p¼0.97 GHz). This result reveals that all of the antivortices act as nodes in thestanding-wave form and, thus, do not contribute to the lower band at k¼k BZbut dominantly contribute to the higher band atk¼kBZ. The flat higher band for the parallel polarization ordering is owed to the fact that the dynamic interaction energy averaged over one cycle of gyration is almost equal for the entire krange. The xvalue in the higher flat band would provide the angular eigenfrequency of a virtual sys- tem composed of isolated antivortices (here /C241.23 GHz). Next, in order to control the observed two-branch band structure and the gyration-signal propagation speed in such V-AV lattices, we applied perpendicular bias fields Hzof FIG. 4. (a) Model geometry of 1D chains comprising 13 vortices and 12 antivortices between the neighboring vortices. Odd (even) index numbers from 1 to 25 represent vortices (antivortices). The in-plane curling magnetization of all of the vortices is CCW. The red (blue) dots display upward (downward) c ore orien- tation. (b) Two-branch band-structure variation with Hzfor both parallel and antiparallel polarization orderings in V-AV lattice shown in Fig. 4(a).222410-3 H.-B. Jeong and S.-K. Kim Appl. Phys. Lett. 105, 222410 (2014)different strength and direction. It is known that the eigenfre- quency of a gyrotropic mode varies with Hz, as expressed by x¼x0ð1þpHz=HsÞ, with x0the angular eigenfrequency at Hz¼0, and Hsthe perpendicular field for the saturation of a given system’s magnetization.32,33In our numerical calcula- tions, x0was estimated to be linearly proportional to Hzas well (not shown here). Thus, it is interesting to examine howcoupled gyrations in such V-AV lattices, and consequently how the resultant band structure, vary with H z. In our further simulations, we excited vortex and antivortex gyrations by the same method as described earlier, but under the applica- tion of different values of Hzat intervals of 1 kOe in the Hz¼/C03t oþ3 kOe range. Figure 4(b) compares the con- trasting band structures for the indicated Hzvalues. The band widths of the lower and higher bands and their bandgapsmarkedly vary with H zand differ from the parallel to anti- parallel polarization ordering. For example, at Hz¼/C03.0 kOe, in the case of ( pV,pAV)¼(þ1,þ1), the higher flat band becomes stronger and more flat over a wide range of k, and the lower band becomes weak, while in the case of ( pV, pAV)¼(þ1,/C01), the higher flat band becomes very weak and the lower band becomes relatively strong. Also in the case of ( pV,pAV)¼(þ1,þ1), the lower band width increases with Hz, while in the case of ( pV,pAV)¼(þ1,/C01), it increases with Hzuntil Hz¼1 kOe and then decreases again. We also plotted the angular frequencies xBZatk¼kBZ for the lower and higher bands (see left of Fig. 5(a)). As for (pV,pAV)¼(þ1,þ1),xBZof the lower and the higher bands are linearly proportional to Hzin the given Hzrange, with slopes of 0.12 GHz/kOe and 0.21 GHz/kOe, respectively. Infact,x BZfor the higher and lower bands correspond to x0’s of the isolated antivortex and vortex, respectively. Thus, the linear dependences of xBZonHzfor both bands are associ- ated with the variation of the x0of isolated vortices and anti- vortices with Hz, as expressed by x¼x0ð1þpHz=HsÞ. Using this equation, linear fits to the data yield the two fittingparameters: x 0/2p¼0.9560.002 (GHz), Hs¼8.260.08 (kOe) for the lower band and x0/2p¼1.260.004 (GHz), Hs¼5.960.01 (kOe) for the higher band. These two values ofx0/2p¼0.95 and 1.2 GHz are close to the x0of isolated vortices and antivortices. The larger slope for the higherband is the result of the higher x0and lower Hs. In the case of (pV,pAV)¼(þ1,/C01), on the other hand, the xBZof the higher band is linearly decreased and then, after crossing about Hz¼1 kOe, increased, as shown in the right panel of Fig. 5(a). The xBZof the lower band, meanwhile, shows exactly the reverse effect. The higher (lower) band in the Hz<1 kOe range seems to follow the relation x¼ x0ð1/C0Hz=HsÞ(x¼x0ð1þHz=HsÞ), whereas in the Hz>1 kOe range, x¼x0ð1þHz=HsÞ(x¼x0ð1/C0Hz=HsÞ). These results reveal that in the Hz<1 kOe ( Hz>1kOe) range, antivortices play a dominant role in the higher (lower) band and that vortices do the same in the lower (higher)band. That is why the band-structure variation with H zis rather more complicated in the case of ( pV,pAV)¼(þ1,/C01) than in the case of ( pV,pAV)¼(þ1,þ1). From the technological perspective, as with domain- wall motions in a given nanostrip, gyration-signal propaga- tion through alternating V-AV lattices can be used as aninformation carrier. From displacements of the individual cores from their own center positions in the whole system with time, we estimated the gyration-signal propagationspeeds versus H zfor both polarization ordering cases. Figure 5(b) shows that the resultant propagation speed is linearly proportional to Hzin the given Hzrange for the parallel polarization ordering (open squares), but that for the antipar- allel ordering (open circles), the speed decreases with the increase in jHzjand, thereby, is at the maximum at Hz¼0. The speed difference ratio between the parallel and antiparal- lel orderings increases markedly with Hz, as shown in Fig. 5(c). For example, the difference ratio increases to /C24135% at Hz¼3 kOe, compared with 28% at Hz¼0 and 2% atHz¼/C03 kOe. This remarkable variation of the speed dif- ference ratio with Hzis the result of increases in the intrinsic x0’s of both the vortex and antivortex with Hzfor ( pV, pAV)¼(þ1,þ1) and the result of decreases in the antivor- tex’s x0for ( pV,pAV)¼(þ1,/C01). These results are very promising from the technological point of view, owing to the following several advantages of this signal-propagation mechanism and its applicability to any potential spin-basedsignal-processing devices: (1) Such V-AV lattices can be made of simple, single-material, round-shaped modulated nanostrips; (2) the parallel polarization ordering betweenneighboring vortices and antivortices is readily available by application of H z; (3) such faster gyration-propagation speed is caused by the combined exchange and dipolar couplingbetween the neighboring vortices and antivortex, and accord- ingly the propagation speed is higher than 1 km/s and controllable by the application of H z. In summary, we studied the dynamics of coupled vorti- ces and antivortices in their alternating 1D periodic arrays. Standing-wave discrete modes and their dispersion relationswere found to have lower and higher branches. We found faster gyration-signal propagations in continuous nanostripes composed of alternating vortex and antivortex lattices for theparallel polarization ordering than for the antiparallel one; both are much faster than only-vortex-state arrays. Also, the band structures and gyration-propagation speed are markedlyvariable by means of the perpendicular bias field. Such core- gyration signals can be detectable due to relatively large variations in the in-plane magnetizations of vortex andFIG. 5. (a) Plots of angular frequency at k¼kBZfor lower and higher bands in both polarization ordering cases. The solid and dashed lines represent theresults of linear fits to the data. (b) Gyration-signal propagation speed in both polarization ordering cases and (c) their difference ratio.222410-4 H.-B. Jeong and S.-K. Kim Appl. Phys. Lett. 105, 222410 (2014)antivortex motions. This work provides not only fundamen- tal insights into the dynamic interactions between different types of topological solitons but a robust means for signifi- cant enhancements of gyration-signal propagation speed insoft magnetic thin-film nanostrips. This research was supported by the Basic Science Research Program through the National ResearchFoundation of Korea funded by the Ministry of Science, ICT & Future Planning (Grant No. 2014001928). 1K. Y. Guslienko, B. A. Ivanov, V. Novosad, Y. Otani, H. Shima, and K. Fukamichi, J. Appl. Phys. 91, 8037 (2002). 2S. Kasai, Y. Nakatani, K. Kobayashi, H. Khono, and T. Ono, Phys. Rev. Lett. 97, 107204 (2006). 3B. Van Waeyenberge, A. Puzic, H. Stoll, K. W. Chou, T. Tyliszczak, R. Hertel, M. F €ahnle, H. Br €uckl, K. Rott, G. Reiss, I. Neudecker, D. 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Tebble, Feromagnetism and Ferromagnetic Domains (North Holland, Amsterdam, 1965). 23K. Shigeto, T. Okuno, K. Mibu, T. Shinjo, and T. Ono, Appl. Phys. Lett. 80, 4190 (2002). 24K. Kuepper, M. Buess, J. Raabe, C. Quitmann, and J. Fassbender, Phys. Rev. Lett. 99, 167202 (2007). 25J. Miguel, J. S /C19anchez-Barriga, D. Bayer, J. Kurde, B. Heitkamp, M. Piantek, F. Kronast, M. Aeschlimann, H. A. D €urr, and W. Kuch, J. Phys.: Condens. Matter 21, 496001 (2009). 26The geometrical parameters used here were optimized to achieve vortex and antivortex arrays in at least metastable states, thereby allowing coupled core gyrations to propagate well along the chains. When Dintbecomes longer, it is difficult to form antivortices and their gyrations; whereas when Dintbecomes shorter, the neighboring vortices and antivortices are in too-close proximity, so that they might be annihilated during their coupled gyrations. 27Seehttp://math.nist.gov/oommf for more information about the code. 28L. D. 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Lett. 105, 222410 (2014)Applied Physics Letters is copyrighted by the American Institute of Physics (AIP). Redistribution of journal material is subject to the AIP online journal license and/or AIP copyright. For more information, see http://ojps.aip.org/aplo/aplcr.jsp

1.3143036.pdf

Arbitrary amplitude ion-acoustic solitary excitations in the presence of excess superthermal electrons N. S. Saini, I. Kourakis, and M. A. Hellberg Citation: Phys. Plasmas 16, 062903 (2009); doi: 10.1063/1.3143036 View online: http://dx.doi.org/10.1063/1.3143036 View Table of Contents: http://pop.aip.org/resource/1/PHPAEN/v16/i6 Published by the American Institute of Physics. Related Articles Effect of nonthermality of electrons on the speed and shape of ion-acoustic solitary waves in a warm plasma Phys. Plasmas 19, 072301 (2012) Dynamic characteristics of gas-water interfacial plasma under water Phys. Plasmas 19, 063507 (2012) Nonlinear electrostatic excitations of charged dust in degenerate ultra-dense quantum dusty plasmas Phys. Plasmas 19, 062107 (2012) The interaction between two planar and nonplanar quantum electron acoustic solitary waves in dense electron- ion plasmas Phys. Plasmas 19, 062105 (2012) Arc-based smoothing of ion beam intensity on targets Phys. Plasmas 19, 063111 (2012) Additional information on Phys. Plasmas Journal Homepage: http://pop.aip.org/ Journal Information: http://pop.aip.org/about/about_the_journal Top downloads: http://pop.aip.org/features/most_downloaded Information for Authors: http://pop.aip.org/authors Downloaded 12 Jul 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://pop.aip.org/about/rights_and_permissionsArbitrary amplitude ion-acoustic solitary excitations in the presence of excess superthermal electrons N. S. Saini,1,a/H20850I. Kourakis,1,b/H20850and M. A. Hellberg2,c/H20850 1Centre for Plasma Physics, Department of Physics and Astronomy, Queen’s University Belfast, Belfast BT7 1NN, Northern Ireland, United Kingdom 2School of Physics, University of KwaZulu-Natal, Private Bag X54001, Durban 4000, South Africa /H20849Received 13 March 2009; accepted 4 May 2009; published online 9 June 2009 /H20850 Velocity distribution functions with an excess of superthermal particles are commonly observed in space plasmas, and are effectively modeled by a kappa distribution. They are also found in somelaboratory experiments. In this paper we obtain existence conditions for and some characteristics ofion-acoustic solitary waves in a plasma composed of cold ions and /H9260-distributed electrons, where /H9260/H110223/2 represents the spectral index. As is the case for the usual Maxwell–Boltzmann electrons, only positive potential solitons are found, and, as expected, in the limit of large /H9260one recovers the usual range of possible soliton Mach numbers, viz., 1 /H11021M/H110211.58. For lower values of /H9260, modeling the presence of a greater superthermal component, the range of accessible Mach numbers is reduced.It is found that the amplitude of the largest possible solitons that may be generated in a given plasma/H20849corresponding to the highest allowed Mach number for the given plasma composition /H20850falls off with decreasing /H9260, i.e., an increasing superthermal component. On the other hand, at fixed Mach number, both soliton amplitude and profile steepness increase as /H9260is decreased. These changes are seen to be important particularly for /H9260/H110214, i.e., when the electrons have a “hard” spectrum. © 2009 American Institute of Physics ./H20851DOI: 10.1063/1.3143036 /H20852 I. INTRODUCTION The scope of the article at hand embraces the nonlinear dynamics of ion-acoustic /H20849IA/H20850waves under the effect of a non-Maxwellian electron velocity distribution with excesssuperthermal particles, represented by a /H9260distribution. The basic prerequisites of our study are outlined in the followingparagraphs. Plasmas are often characterized by a particle distribution function with a high energy tail and they may thus deviatesignificantly from a Maxwellian. 1–3Both space and labora- tory plasma environments may have such an excess super-thermal electron population due to velocity space diffusion,which may lead to an inverse power-law distribution at avelocity much higher than the electron thermal speed. 4–6 Such behavior is effectively modeled by a kappa /H20849or gener- alized Lorentzian /H20850distribution function,1,6–8which appears to be more appropriate than a thermal /H20849Maxwellian /H20850distribu- tion in a wide range of plasma situations. The commonly used three-dimensional, isotropic kappa /H20849/H9260/H20850distribution is given by6,8 f/H9260/H20849v/H20850=n0 /H20849/H9266/H9260/H92582/H208503/2/H9003/H20849/H9260+1/H20850 /H9003/H20849/H9260−1 2/H20850/H208731+v2 /H9260/H92582/H20874−/H20849/H9260+1/H20850 , /H208491/H20850 where n0is the species equilibrium number density, /H92582=/H20851/H20849/H9260 −3 /2/H20850//H9260/H20852/H208492kBT/m/H20850is the effective thermal speed, modified by the spectral index /H9260, with Tthe kinetic temperature and m the species mass, and /H9003/H20849x/H20850is the gamma function. Here v2 =vx2+vy2+vz2obviously denotes the square norm of the veloc-ityv. Clearly, for a physically realistic thermal speed, one requires /H9260/H110223/2. At very large values of the spectral index /H9260, the velocity distribution function approaches a Maxwelliandistribution. Low values of /H9260represent distributions with a relatively large component of particles with speed greaterthan the thermal speed /H20849“superthermal particles” /H20850and an as- sociated reduction in “thermal” particles, as one observes ina “hard” spectrum. First applied by Vasyliunas 1to model observations of particle energy distributions in space-based experiments, the /H9260distribution is widely used to fit velocity distributions ob- served in space plasmas, often with 2 /H11021/H9260/H110216. Examples in- clude measurements of plasma sheet electron and ion distri-butions /H20849 /H9260i=4.7 and /H9260e=5.5 /H20850,9and observations in the earth’s foreshock /H208493/H11021/H9260e/H110216/H20850.10Modelers have also used /H9260distribu- tions with low values of /H9260, e.g., Pierrard and co-workers11,12 developed a Lorentzian ion exosphere model and associated solar wind model with coronal electrons satisfying 2 /H11021/H9260e /H110216. Although there is no completely satisfactory theory for the persistence and apparent ubiquity of /H9260distributions in space, works by Treumann and co-workers,13,14Leubner,4 and Collier,15provided heuristic explanations or pointers to- ward a full explanation. It has been argued that a combination of kappa distribu- tions models multicomponent plasmas more effectively thana superposition of Maxwellians. 16,17Indeed, recent observa- tions of the electron velocity distribution function in Saturn’smagnetosphere appear to confirm this view. 5 By integrating the kappa distribution function over ve- locity space, one can obtain the number density of the cor-responding plasma constituent /H20849s/H20850, which affects the charge balance via Poisson’s equation. An important characteristica/H20850Electronic mail: nssaini@yahoo.com and ns.saini@qub.ac.uk. b/H20850Electronic mail: i.kourakis@qub.ac.uk. c/H20850Electronic mail: hellberg@ukzn.ac.za.PHYSICS OF PLASMAS 16, 062903 /H208492009 /H20850 1070-664X/2009/16 /H208496/H20850/062903/9/$25.00 © 2009 American Institute of Physics 16, 062903-1 Downloaded 12 Jul 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://pop.aip.org/about/rights_and_permissionsof the kappa distribution function is that the dependence of the density on the electrostatic potential differs from the fa-miliar exponential form obtained from the Maxwell–Boltzmann distribution. The consequences will be obvious inour analysis below, both from an analytical and a numericalpoint of view. The linear properties of plasmas in the presence of a kappa distribution with excess superthermal particles havebeen investigated rather extensively. A modified plasma dis-persion function for such a “superthermal” plasma was intro-duced in Ref. 16for integer /H9260, and was extended to a gen- eralized plasma dispersion function for arbitrary real /H9260.18,19 The usual plasma dispersion function /H20849derived for the Max- wellian case /H20850is obtained by both these approaches in the limit of infinite /H9260, as expected. It has been shown20that the generalized plasma dispersion function, Z/H9260could provide a plasma diagnostic in space, in that wave data recorded in the magnetosphere could be used to find the appropriate /H9260value characterizing the electron distribution function, clearly dif-ferentiating it from results calculated using a Maxwellianassumption. Similarly, it was found that wave experimentscan act as a diagnostic for the distribution function in a labo-ratory plasma when use is made of the Z /H9260plasma dispersion function.6 The effect of superthermal electrons on linear IA waves propagating in a magnetized plasma was studied in Ref. 19 Interestingly, the presence of a high energy tail leads to asignificant variation in the damping rate of electrostaticplasma waves, as compared to Maxwellian plasmas, 21so ex- cess superthermality was found in that case to enhance Lan-dau damping. The founding blocks of a nonlinear theory for IA plasma excitations were provided four decades ago, with a study ofsmall-amplitude nonlinear excitations, 22and an arbitrary am- plitude theory for IA solitary waves.23,24A model was pro- posed to study the dynamics of solitary waves in an electron-ion plasma, 23and a domain for the Mach number /H20849M /H33528/H208511,1.58 /H20852/H20850was found for the existence of solitary waves. The pseudopotential method developed by Sagdeev23for nonlinear IA excitations /H20849later extended to describe magne- tized plasmas25/H20850predicted that only positive potential distur- bances may occur in simple electron-ion plasmas. Neverthe-less, negative potential solitary structures have later beenshown to exist in the presence of two electronpopulations, 26,27and/or in multi-ion plasma compositions or dusty plasmas.28–31 It may be added for completeness that another approach to velocity space nonthermality is provided by the so calledTsallis distribution. 32Like the kappa distribution, the Tsallis distribution represents a family of distribution functions,governed by a single parameter /H20849q, in this case /H20850, and possess- ing a power-law structure /H20849the power is given by 1 //H208511−q/H20852/H20850, with the Maxwellian as a limiting case, when q→1. Unfor- tunately, although there are similarities, there is no simpletransformation between the Tsallis and kappa distribution/H208491/H20850, as the forms of the argument and the power do not both fit the same transformation. However, one may wish to usean approximate relationship given by /H9260→1//H20849q−1/H20850, in that, forq/H110221, an increase in qincreases the fraction of superther-mal electrons relative to that of the Maxwellian, which is equivalent to a decrease in /H9260. Recently, existence conditions have been found for IA solitons in a plasma composed ofcold ions and electrons modeled by a Tsallis distribution. 33 The main results were that /H20849i/H20850as for the conventional IA solitons based on a Maxwellian distribution, only positivesolitons were found, and /H20849ii/H20850the accessible range in Mach number found for a Maxwellian, /H208511,1.58 /H20852, is reduced as qis increased beyond q=1, i.e., increasing the superthermal ex- cess reduces the range of propagation speeds available to thesolitary structure. These results are qualitatively recoveredby our analysis here. The aim of our investigation is to elucidate the effect of electron superthermality, as manifested through the com-monly observed kappa distribution, on the propagation char-acteristics of nonlinear IA excitations in a simple electron-ion plasma. We rely on a pseudopotential method toinvestigate the occurrence and characteristics of arbitraryamplitude IA waves. We shall determine the range of permit-ted Mach number values for the existence of solitary IAwaves in a plasma with excess superthermal electrons, andwill, in particular, demonstrate their dependence on “super-thermality” /H20849via /H9260/H20850. Recall that the limit /H9260→/H11009leads to the Maxwellian case, so that the known Mach number domain/H208511,1.58 /H20852/H20849Ref. 23/H20850is recovered in this case /H20849see Fig. 1/H20850. The layout of the paper is as follows. The analytical model equations are presented in Sec. II. In Sec. III, wedevelop a pseudopotential theory and determine the range ofpermitted velocity values for the existence of solitary struc-tures. We proceed by numerically evaluating and discussingthe propagation velocity range and the effects of superther-mality in Secs. IV and V , respectively. Our results are thensummarized in Sec. VI. kM FIG. 1. /H20849Color online /H20850IA soliton existence domain in the parameter space of /H9260and Mach number, M. Solitons may be supported in the region between the two curves. The lower, dashed curve represents the minimum /H20849soliton /H20850 condition, M1, and the upper, solid curve the infinite compression limit, M2.062903-2 Saini, Kourakis, and Hellberg Phys. Plasmas 16, 062903 /H208492009 /H20850 Downloaded 12 Jul 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://pop.aip.org/about/rights_and_permissionsII. GOVERNING MODEL EQUATIONS We consider a two-component plasma consisting of cold ions /H20849charge qi=+Ze, mass mi/H20850, described by the fluid-moment equations, and electrons /H20849qe=−e, mass me/H20850, assumed to obey a kappa ve- locity distribution. The fluid equations for the ions /H20849in the absence of pressure effects /H20850read /H11509ni /H11509t+/H11509/H20849niui/H20850 /H11509x=0 , /H208492/H20850 /H11509ui /H11509t+ui/H11509ui /H11509x=−qi mi/H11509/H9021 /H11509x, /H208493/H20850 and the two fluids are coupled through Poisson’s equation, /H115092/H9021 /H11509x2=−4/H9266e/H20849niZ−ne/H20850, /H208494/H20850 where ni,ui, and/H9021are the ion number density, the ion mean velocity, and the electrostatic potential, respectively. The as-sumption of charge neutrality at equilibrium yields n i0Z−ne0=0 , /H208495/H20850 where the index “0” denotes the unperturbed /H20849equilibrium /H20850 number density values. We adopt a kappa distribution for the electrons, and by integrating over velocity space obtain the electron numberdensity, 8 ne=ne0/H208751−e/H9021 /H20849/H9260−3 2/H20850kBTe/H20876−/H9260+1 /2 , /H208496/H20850 where the real parameter /H9260measures the deviation from Maxwellian equilibrium. We stress that the latter is recoveredin the limit of infinite /H9260at every step. Normalizing by appropriate scaling quantities, the num- ber density for the electrons may be written in dimensionlessform as n e=/H208731−/H9278 /H9260−3 /2/H20874−/H9260+1 /2 . /H208497/H20850 The normalized ion continuity and momentum equations, and Poisson’s equation are /H11509n /H11509t+/H11509/H20849nu/H20850 /H11509x=0 , /H208498/H20850 /H11509u /H11509t+u/H11509u /H11509x=−/H11509/H9278 /H11509x, /H208499/H20850 /H115092/H9278 /H11509x2=−n+/H208731−/H9278 /H9260−3 /2/H20874−/H9260+1 /2 , /H2084910/H20850 where the fluid velocity ui, the particle density ni, and the electrostatic potential /H9021are scaled as u=ui/cs,n=ni/ni0, and /H9278=/H9021//H90210, respectively. Here, ni0is the equilibrium ion den- sity. We have made use of the quasineutrality relation /H208495/H20850 above. Space and time variables are scaled by the Debyelength /H9261D,e=/H20849kBTe/4/H9266ne0e2/H208501/2=/H20849kBTe/4/H9266Zni0e2/H208501/2, and the inverse ion plasma frequency /H9275p,i−1=/H208494/H9266ni0Z2e2/mi/H20850−1 /2. Fi- nally, the potential scale reads /H90210=kBTe/e. The characteristic IA sound speed used for velocity normalization is then cs /H11013/H20849ZkBTe/mi/H208501/2. However, we should note that this expression for the sound speed is applicable to an electron-ion plasma in whichthe electron density satisfies a Boltzmann distribution. Debyeshielding is altered in a plasma with a /H9260distribution, and thus an effective /H9260-dependent Debye length is found.34–37 Hence, the true sound speed in the plasma model under con- sideration, with electron density as given by Eq. /H208496/H20850, is kappa dependent and differs from cs, as will be seen later. We should like to emphasize that the normalization used does not contain /H9260at all, and thus the full dependence on /H9260 of all variables is exhibited in the normalized expressions,and will be reflected in the numerical work that follows, as isthe case for the true sound speed. We note that we havesubstituted explicitly for /H9258as given in the clarification fol- lowing Eq. /H208491/H20850, and hence the potential is written in terms of the kinetic temperature Tebased on a Maxwellian of equal number density and average kinetic energy.1,16,18,34–36 III. ARBITRARY AMPLITUDE SOLITARY WAVE THEORY Anticipating the existence of arbitrary amplitude travel- ing solitary waves, we assume that all fluid variables in theevolution equations depend on a single variable /H9264=x−Mt /H20849where Mis the Mach number, i.e., the pulse propagation velocity normalized by the sound speed, here taken to be thenormalization value, c s/H20850. This is the well-known pseudopo- tential /H20849so-called “Sagdeev” /H20850method, leading to a number of ordinary differential equations in a variable of /H9264; details can be found, e.g., in Refs. 38and39. Using the above transfor- mation in Eqs. /H208498/H20850–/H2084910/H20850, the fluid equations become −M/H11509n /H11509/H9264+/H11509/H20849nu/H20850 /H11509/H9264=0 , /H2084911/H20850 −M/H11509u /H11509/H9264+u/H11509u /H11509/H9264+/H11509/H9278 /H11509/H9264=0 , /H2084912/H20850 /H115092/H9278 /H11509/H92642=−n+/H208731−/H9278 /H9260−3 /2/H20874−/H9260+1 /2 . /H2084913/H20850 After integrating Eqs. /H2084911/H20850and /H2084912/H20850and applying appro- priate boundary conditions for localized perturbations, viz.,n→1,u→0, and /H9278→0a t/H9264→/H11006/H11009, we write −Mn+nu=−M, /H2084914/H20850 i.e., u=M/H208731−1 n/H20874, /H2084915/H20850 and062903-3 Arbitrary amplitude ion-acoustic solitary excitations … Phys. Plasmas 16, 062903 /H208492009 /H20850 Downloaded 12 Jul 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://pop.aip.org/about/rights_and_permissions−Mu+u2 2=−/H9278. /H2084916/H20850 From Eqs. /H2084915/H20850and /H2084916/H20850, we obtain n=1 /H208811−2/H9278 M2. /H2084917/H20850 The reality condition M2/H113502/H9278is then imposed; note that this requirement of a physically realistic density limits positivepotential values only. Substituting Eq. /H2084917/H20850into Eq. /H2084913/H20850, multiplying the re- sulting equation by d /H9278/d/H9264, integrating, and applying the boundary conditions, d/H9278/d/H9264→0a t/H9264→/H11006/H11009, we find that Poisson’s equation takes the form 1 2/H20873d/H9278 d/H9264/H208742 +V/H20849/H9278/H20850=0 , /H2084918/H20850 where the /H20849Sagdeev-type /H20850pseudopotential V/H20849/H9278/H20850is given by V/H20849/H9278/H20850=M2/H208731−/H208811−2/H9278 M2/H20874+1−/H208731−/H9278 /H9260−3 /2/H20874−/H9260+3 /2 . /H2084919/H20850 Equation /H2084918/H20850can be regarded as the “pseudo-energy- balance equation” for an oscillating particle of unit mass,with position /H9278, time /H9264, velocity d/H9278/d/H9264, and potential V/H20849/H9278/H20850. We recall, in view of the forthcoming analysis that the Max- wellian limit23is recovered for /H9260→/H11009. In order for solitary solutions to exist, the following re- quirements must be fulfilled: /H20849i/H20850 V/H20849/H9278=0/H20850=dV/H20849/H9278/H20850/d/H9278/H20841/H9278=0=0/H20849at the origin /H20850, which rep- resents the requirement that both the electric field and the charge density be zero far from the localized IAsolitary structures, and /H20849ii/H20850 d 2V/H20849/H9278/H20850/d/H92782/H20841/H9278=0/H110210/H20849i.e.,V/H20849/H9278/H20850has a maximum at the origin /H20850so that the sign of the derivative of the charge density is compatible with the sign of the electric fieldat large distances. Finally, as imposed by the reality of /H9278, from Eq. /H2084918/H20850, /H20849iii/H20850V/H20849/H9278/H20850/H110210 in the region 0 /H11021/H20841/H9278/H20841/H11021/H20841/H9278m/H20841; here /H9278mdenotes the positive root /H20849/H9278max/H20850, for positive potential excita- tions /H20849or conversely the negative root /H20849/H9278min/H20850, for nega- tive potential excitations /H20850. A. Soliton existence conditions The origin at /H9278=0 defines the equilibrium state, which should represent a local maximum of V/H20849/H9278/H20850. From Eq. /H2084919/H20850,i t is clear that both V/H20849/H9278=0/H20850=0 and dV/H20849/H9278=0/H20850/d/H9278=0 are satis- fied at equilibrium. The requirement /H20879d2V d/H92782/H20879 /H9278=0=1 M2−1−1 /H9260−3 /2/H110210 /H2084920/H20850 constitutes the soliton /H20849existence /H20850condition to be fulfilled. The root of d2V/d/H92782/H20841/H9278=0in terms of the Mach number M defines a critical value as a lower limit for M, i.e.,M1/H11013/H20873/H9260−3 /2 /H9260−1 /2/H208741/2 /H113491. /H2084921/H20850 For a fixed value of /H9260, soliton solutions may exist only for values of the Mach number satisfying M/H11022M1. It may easily be shown36that this /H9260-dependent expression for M1is actu- ally the IA speed in a two-component plasma with kappa-distributed electrons, normalized with respect to the conven-tional IA speed, c s/H11013/H20849ZkBTe/mi/H208501/2. Thus the existence condition, M/H11022M1implies, as expected, that for solitary waves to exist, they must be traveling at a speed exceedingthe true sound speed. Note that the simple value M 1=1 is recovered for the limit /H9260→/H11009/H20849simply implying supersonic excitations for IA waves in e-iplasmas with Maxwellian electrons /H20850. It is straightforward to see the influence of excess superthermal electrons /H20849via/H9260/H20850on this soliton velocity thresh- old. In particular, M1decreases monotonically with decreas- ing/H9260from the “conventional” value of M1=1 found for large /H9260, and as /H9260→3/2,M1→0. A second /H20849upper /H20850velocity limit for the existence of posi- tive potential solitons arises from the physical requirement ofa real ion number density, as expressed by Eq. /H2084917/H20850. For /H9278 →M2/2, the density nbecomes infinite /H20849and so would the pressure /H11011n/H9253in a warm ion model with polytropic index /H9253/H20850. Accessible values of the Mach number are those for whichthe Sagdeev well yields a root /H9278mbefore this infinite com- pression limit is reached, and hence we find the largest pos-sible value of Mby imposing the requirement V/H20849 /H9278=M2/2/H20850 /H113500. The upper limit on the speed of the solitary waves /H20849say, M2/H20850, expressed in terms of the Mach number, is thus ob- tained by solving the associated equation, M22+1−/H208731−M22 2/H9260−3/H20874−/H9260+3 /2 =0 , /H2084922/H20850 forM2.A s/H9260→/H11009, the last term tends to an exponential form, and hence the upper Mach number limit will then take on theconventional value of 1.58. At the opposite extreme, it caneasily be shown that as /H9260→3/2,M2→0. Summarizing, assuming the kappa-dependent electron density function given by Eq. /H208496/H20850, positive potential solitary wave solutions of the ion fluid system of equations exist forvalues of the Mach number Min the range M 1/H11021M/H11021M2. Clearly, both of these limits vary with kappa, and we need toinvestigate their dependence on physical parameters. Relying on the analytical toolbox outlined above, we have performed a parametric investigation, in order to studythe properties of arbitrary amplitude solitary waves, as de-duced from the pseudopotential V/H20849 /H9278/H20850given by Eq. /H2084919/H20850. Our findings are presented and discussed in the following. IV. PROPAGATION VELOCITY OF LOCALIZED EXCITATIONS Let us first consider the dependence of the critical Mach number values M1andM2on the presence of excess super- thermal electrons /H20849superthermality /H20850via/H9260, and hence explore the range of accessible Mach numbers as a function of /H9260. For the lower velocity threshold, M1, this can be inferred analyti- cally upon simple inspection of Eq. /H2084921/H20850, as commented on062903-4 Saini, Kourakis, and Hellberg Phys. Plasmas 16, 062903 /H208492009 /H20850 Downloaded 12 Jul 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://pop.aip.org/about/rights_and_permissionsin Sec. III A. Recalling the fact that this soliton existence condition represents the requirement of superacoustic propa-gation speed /H20849i.e.,M 1is the true IA speed /H20850, we see that the true sound speed in this kappa distribution plasma has alower value than in a Maxwellian plasma, i.e., an increase insuperthermal /H20849and the associated reduction in thermal /H20850elec- trons causes the linear IA wave to propagate at lower speed.This has been shown in a rigorous manner in Ref. 36. Unlike the lower limit, the variation of the upper veloc- ity limit imposed by infinite compression of the ions, viz., M 2, the root of Eq. /H2084922/H20850, can only be studied numerically. It is found that M2decreases monotonically as /H9260decreases. In Fig. 1we have plotted the lower and upper limits, M1and M2, respectively, over the range 3 /2/H11349/H9260/H1134920, and hence show the permitted range of Mach numbers, which satisfy M1/H11021M/H11021M2and thus support IA solitons in such kappa plasmas. We see that as /H9260is decreased, the available range of Mach numbers over which positive potential IA solitons mayexist is reduced. A few comments are in order, regarding the physical interpretation of Fig. 1. First we note that both curves show an asymptotic behavior as /H9260is increased. As expected, the two limiting Mach numbers tend to 1 and 1.58, respectively,as is well known for the Maxwell–Boltzmann case. 23,24That these values are already closely approached from /H9260/H1122910 agrees with earlier studies, where linear wave behavior inplasmas with values of /H9260above /H1101510/H20849roughly /H20850was found to be practically equivalent to that in a Maxwellian plasma.19 In this figure, one sees that as expected, both the lower and the upper Mach number limits tend to zero as /H9260→3/2, the limiting value of /H9260. Recalling that a decrease in /H9260mea- sures the deviation from the Maxwellian behavior through anincrease in the superthermal electron component and a con-comitant decrease in the thermal part of the electron velocitydistribution function, we note that higher superthermality re-sults in the shrinking of the permitted region for soliton ve-locities, compared to what is found for a Maxwellian plasma. V. ROLE OF SUPERTHERMALITY We wish to study the effect of superthermality on the solitary wave characteristics. First, we consider two values of /H9260, viz., /H9260=16, which is pseudo-Maxwellian /H20849Fig. 2/H20850, and/H9260 =4, which is strongly non-Maxwellian, with a large super- thermal component, and has been found to occur in spaceplasmas /H20849Fig. 3/H20850. In each case we present Sagdeev potential plots which represent positive potential solitary wave struc-tures, calculated for a range of values of the Mach number, M, lying in the range, M 1/H11021M/H11021M2. The amplitude of the solitary electrostatic potential structures /H20849measured by the magnitude of the root, /H9278m/H20850is seen in both figures to increase monotonically as the Mach number is increased, thus show-ing that behavior of this kind, known for the Maxwell–Boltzmann case, applies to low kappa also. Specifically, as Mis increased from its lowest value to the largest soliton propagation speed plotted, the normalized electrostatic po-tential amplitude increases from effectively zero to /H112291.13 for the pseudo-Maxwellian case /H20849Fig. 2/H20850, but to a somewhat lower value, viz., /H112290.77 for the low- /H9260case /H20849Fig. 3/H20850. Thesefigures thus indicate that the presence of additional super- thermal particles appears to reduce the maximum solitonamplitude. We also see that in both figures the well depth of the Sagdeev potential curve increases monotonically and dra-matically, as the Mach number is increased from close to thelower limit to just below the upper limit. Whereas for thepseudo-Maxwellian case the maximum well depth reaches anormalized value of 0.3, in the presence of stronger super-thermality it is reduced to /H112290.2. The actual numbers in- volved in this well depth have less physical significance thanthe changes in well depth. It follows from Eq. /H2084918/H20850that the well depth is proportional to the square of the maximumelectric field, i.e., it is related to the maximum slope of theelectrostatic potential profile representing the solitary wavestructure. From these two figures we thus deduce that theaddition of superthermal particles associated with a lowervalue of /H9260gives rise to a reduction in the steepness of theV(f) f/CID10.5 0.5 1.0 /CID10.3/CID10.2/CID10.1 FIG. 2. /H20849Color online /H20850Variation of V/H20849/H9278/H20850for/H9260=16 and different values of Mach number, M. From top to bottom: Dotted curve: M=0.97; dashed curve: M=1.10; dotted-dashed curve: M=1.23; long-dashed curve: M =1.36; and solid curve: M=1.50. V(f) f/CID10.4 /CID10.2 0.2 0.4 0.6 0.8 /CID10.20/CID10.15/CID10.10/CID10.050.05 FIG. 3. /H20849Color online /H20850Variation of V/H20849/H9278/H20850for/H9260=4 and different values of Mach number, M. From top to bottom: Dotted curve: M=0.85; dashed curve: M=0.95; dotted-dashed curve: M=1.05; long-dashed curve: M =1.15; and solid curve: M=1.24.062903-5 Arbitrary amplitude ion-acoustic solitary excitations … Phys. Plasmas 16, 062903 /H208492009 /H20850 Downloaded 12 Jul 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://pop.aip.org/about/rights_and_permissionssoliton profile. These aspects are explored further in Figs. 4 and5. In Fig. 4we present a set of curves that show the solitary wave amplitude as a function of the increment in Mach num-ber over the soliton existence condition, M 1/H20849i.e., M−M1/H20850, for a wide range of values of the parameter /H9260, running from a true Maxwellian /H20849/H9260=50 /H20850to a strongly non-Maxwellian form /H20849/H9260=3/H20850. A widely cited qualitative aspect of the weakly supersonic, small-amplitude, Korteweg–de Vries soliton theory is that larger excitations propagate at higher speedsand are narrower /H20849i.e., “taller is faster and thinner” /H20850. Consid- ering first the Maxwellian curve, we note that the potentialincreases monotonically with M−M 1from zero up to the ion compression cutoff at M=1.58 /H20849M−M1=0.58 /H20850that the rate of increase is effectively linear for smaller amplitude soli- tons, relatively close to the lower Mach number limit, andthat the slope of the curve gradually decreases as the Mach number is increased. Not only do we see in Fig. 4that the monotonic behavior is exhibited well beyond the small-amplitude range, but alsothat it applies whether one has the usual Maxwell–Boltzmann plasma or a kappa distribution that is highly non-Maxwellian. For smaller /H9260one finds that at fixed values of M−M1the associated solitary wave potential is lower, i.e., superthermality reduces the amplitude of the solitons. In ad-dition, as we have already seen, the upper Mach numbercutoff, M 2, decreases with increasing superthermality /H20849de- creasing /H9260/H20850and the accessible range for solitons, M2−M1, also decreases. These effects are most dramatic as /H9260is re- duced from 5 to 3. It should be noted that this monotonic behavior is not found universally for arbitrary amplitude acoustic solitons,but appears to relate specifically to solitary waves whoseexistence domain is restricted by the linear wave speed andan infinite compression or rarefaction, as is the case here. It has, for instance, been observed that in some cases where theupper cutoff in the existence domain arises from the exis-tence of a double layer, the amplitude does not increasemonotonically over the full range of accessible Machnumbers. 29,40 In Fig. 5we have plotted Sagdeev potential curves for a set of values of /H9260in the range /H2084916,3 /H20850, i.e., scanning the range from effectively Maxwellian to strongly non-Maxwellian,but this time choosing Mach numbers that are very close to/H20849in fact, within 0.0015 of /H20850the relevant upper limit, M 2, for the value of /H9260under consideration. As the pseudopotential curve breaks down /H20849ends /H20850at the upper cutoff, the curves in this figure only just cross the axis /H20849yield a root /H20850, and as a result graphical representation clearly showing the root isdifficult. It will be noted that some of the curves obviouslycross the axis, while the others “touch” the axis. Carefulnumerical evaluation confirms that they too do yield roots.Bearing in mind the fact that we have seen that the amplitudeincreases monotonically with Mat fixed /H9260, it follows that we are effectively exploring the largest soliton amplitudes thatcan be supported by a plasma with a given value of /H9260.A s found in Fig. 4, we observe that as the superthermal compo- nent increases with decreasing /H9260from the pseudo- Maxwellian case /H20849/H9260=16 /H20850, the largest soliton amplitudes that may be achieved decrease monotonically, the normalized po- tentials dropping from about 1.13 to 0.62, the value found for /H9260=3. The well depth, and thus the steepness of the profile of these “largest” solitons, is also found to fall off monotoni-cally as the superthermal component of the distribution func-tion increases with falling /H9260. Having established what occurs when kappa is kept constant, we turn next to a set of calcu-lations for which the Mach number Mis kept constant, and /H9260 varied. Figure 6depicts the variation of the pseudopotential V/H20849/H9278/H20850with/H9278for fixed Mach number, M=1.1, and different values of /H9260ranging from 10 down to 3. It will be recalled that for /H9260=10 we observed an accessible range of Mach numbers that was approaching that for a Maxwellian distri-bution. We now see that as we introduce a higher proportionof superthermal electrons /H20849i.e., for decreasing /H9260/H20850, the ampli-0.0 0.2 0.4 0.60.00.20.40.60.81.01.2 φm M-M1κ=50 κ=16 κ=10 κ=7 κ=5 κ=3 FIG. 4. /H20849Color online /H20850Variation of /H9278mwith M−M1for different values of /H9260. The dotted curve corresponds to /H9260=3, the dashed curve to /H9260=5, the dotted- dashed curve to /H9260=7, the dotted-dotted dashed curve to /H9260=10, the short- dashed curve to /H9260=16, and the solid curve to /H9260=50. /CID11.0 /CID10.5 0.5 1.0 /CID10.3/CID10.2/CID10.10.1V(/CID1) /CID1 FIG. 5. /H20849Color online /H20850Variation of V/H20849/H9278/H20850for different values of /H9260, and values of Mach number, M, given by M=M2−0.0015. The dotted curve corre- sponds to /H9260=3, the dashed curve to /H9260=5, the dotted-dashed curve to /H9260=7, the long-dashed curve to /H9260=10, and the solid curve to /H9260=16.062903-6 Saini, Kourakis, and Hellberg Phys. Plasmas 16, 062903 /H208492009 /H20850 Downloaded 12 Jul 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://pop.aip.org/about/rights_and_permissionstude of the solitary electrostatic potential structures increases from a normalized value of 0.4 for /H9260=10, to 0.6 for /H9260=3. This increase in amplitude is indirectly associated with theincrease of the superthermal electron component. We recall that the Mach number is measured relative to a fixed “sound speed,” which does not take account of the factthat the true sound speed decreases with decreasing /H9260.I ti s not unusual to normalize with respect to such a fixed char-acteristic speed when different parameters are being varied.However in this case it follows that as we decrease /H9260at fixed M, we are actually increasing the value of /H20849M−M1/H20850. More- over we have, of course, earlier shown that /H9278mincreases as /H20849M−M1/H20850increases. Thus, for fixed M, a decrease in /H9260causes an increase in the amount by which the chosen value of Mach number exceeds the local threshold /H20849the true sound speed /H20850. Hence it follows that decreasing /H9260yields larger soli- tons, and as found in this figure, it is thus associated withincreasing soliton amplitude over the range 3 /H11349 /H9260/H1134910. We also see that the depth of the Sagdeev pseudopoten- tial well increases dramatically from 0.01 to 0.14 over thisrange. Thus the maximum slope of the soliton profile in-creases with decreasing /H9260over this range, i.e., as the super- thermal component is increased, the amplitude increases andthe soliton profile also becomes steeper. It is of interest to see whether this effect applies even for very strongly non-Maxwellian plasmas. In Fig. 7, we present some examples of the pseudopotential V/H20849 /H9278/H20850for a couple of cases with a very strong superthermal component, with /H9260 ranging from 1.8 to 2.0, and M=0.62. Such a very hard spectrum, with an extreme accelerated superthermal compo-nent, may be found near very strong shocks associated withFermi acceleration. 18We see that both the amplitude of the soliton and the maximum slope of the soliton profile arelarger for the lower values /H20849e.g., /H9260=1.8 /H20850than for the higher value, /H9260=2. Thus over this narrow range, too, the increase in superthermality gives rise to an increase in soliton amplitudeand steepness, at fixed M. We have found analogous results even for values of /H9260a little above 1.6. Finally, we take this discussion a step further with Fig. 8,showing plots of amplitude against /H9260at four fixed values of Mach number, M, ranging from 1.0 to 1.3. In each case we observe that as /H9260is decreased, the potential at fixed Mrises. There are cutoffs at lower values of /H9260for the higher values of M,a sM2falls below the Mach number under consideration. It is noticeable that the lowest curve, for M=1.0, rises much more steeply than the others. This is presumably associatedwith the rapid fall-off of M 1with decreasing /H9260below about 5, and the resulting rapid rise in /H20849M−M1/H20850in that range /H20849see Fig. 1/H20850, which is associated with a rise in amplitude, /H9278m/H20849see Fig. 4/H20850. In summary, the results of our calculations show that the answer to the question of how soliton amplitude varies with /H9260depends significantly on how the question is asked. From Figs. 6–8one may wish to argue that increased superther- mality causes larger amplitude solitons. That is indeed thecase at fixed M, and thus at increasing values of /H20849M−M 1/H20850,V(/CID1) /CID1 /CID10.6 /CID10.4 /CID10.2 0.2 0.4 0.6 /CID10.15/CID10.10/CID10.050.05 FIG. 6. /H20849Color online /H20850Variation of V/H20849/H9278/H20850for fixed M=1.1 and different values of /H9260. Dotted curve: /H9260=3; dashed curve: /H9260=4; dotted-dashed curve: /H9260=6; and solid curve: /H9260=10.V(/CID1) /CID1 /CID10.2 /CID10.1 0.1 0.2 /CID10.03/CID10.02/CID10.010.01 FIG. 7. /H20849Color online /H20850Variation of V/H20849/H9278/H20850for fixed M=0.62 and different values of /H9260. Dotted curve: /H9260=1.8; dashed curve: /H9260=1.85; dotted-dashed curve: /H9260=1.9; and solid curve: /H9260=2.0. 2 4 6 8 10 12 14 1 60.00.20.40.60.81.0 φm κM=1.3 M=1.2 M=1.1 M=1.0 FIG. 8. /H20849Color online /H20850Variation of /H9278mwith/H9260for different values of the Mach number, M. The dotted curve corresponds to M=1.0; the dashed curve toM=1.1; the dotted-dashed curve to M=1.2; and the solid curve to M=1.3.062903-7 Arbitrary amplitude ion-acoustic solitary excitations … Phys. Plasmas 16, 062903 /H208492009 /H20850 Downloaded 12 Jul 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://pop.aip.org/about/rights_and_permissionsand its association with increasing /H9278m. On the other hand, Figs. 2–5clearly show that the largest possible values of soliton amplitude for a given value of /H9260, attainable for M chosen so that /H20849M2−M/H20850is small, actually decrease with de- creasing values of /H9260. Overall, our results show that superthermality /H20849as mea- sured by the value of the parameter /H9260/H20850plays a significant role in the modification of solitary electrostatic IA structures andtheir behavior, but that the resultant behavior depends on theexperiment that one is carrying out. Finally we note that as is the case both for electrons with a Maxwellian distribution and for a Tsallis distribution, onlypositive solitons have been observed. A wide-ranging nu-merical search did not reveal any negative potential solitons. VI. CONCLUSIONS In this paper we have studied the existence conditions and the characteristics of IA solitary waves propagating in aplasma composed of cold fluid ions and electrons whose ve-locity distribution is modeled by a kappa distribution. First, itis noted that only positive potential IA solitary structures areobserved in such a plasma, as is the case for a conventionalelectron-ion plasma in which the electrons are Maxwellian. Itthus appears that the presence of additional superthermal par-ticles does not make qualitative changes to this importantaspect of soliton behavior, unlike the addition of an addi-tional species, and that the changes are essentially quantita-tive only, as outlined above. This also agrees with the resultsobtained for a Tsallis distribution. 33 Second, the limiting case for the Maxwellian distribution23is recovered for /H9260→/H11009, as expected. Third, we have shown that the lower Mach number limit for the existence of IA solitons decreases with the presenceof a greater superthermal component, i.e., with decreasing /H9260. This lower threshold, which tends to zero as /H9260→3/2, repre- sents the true IA speed in the plasma model under discussion.The upper limit, associated with the ion infinite compressionlimit, cannot be expressed in a simple closed form, but has tobe found numerically. It decreases more rapidly with de-creasing /H9260than the lower limit, and hence distributions that may be modeled by lower values of /H9260can support solitons only over a narrower range of accessible Mach numbers. Thereduction in accessible solitary wave propagation speedsagrees qualitatively with that found for the case in which theelectrons have a Tsallis distribution. 33 At fixed kappa, that is, for a given velocity distribution function, soliton amplitude and soliton profile steepness bothincrease monotonically as the Mach number is increasedfrom the threshold value. An interesting result is that thelargest possible soliton that can be supported at a fixed valueof /H9260is found to decrease as /H9260decreases. This observation is in line with the facts that as /H9260is decreased, the range of available Mach numbers /H20849M1toM2/H20850decreases, and that soli- ton amplitude and profile steepness increase monotonicallywith Mach number /H20851through /H20849M−M 1/H20850/H20852at fixed /H9260. On the other hand, for a fixed soliton propagation speed /H20849M/H20850within the accessible range, greater superthermality yields an increase in soliton amplitude, and more pronouncedsteepness of the soliton profile. This behavior follows be- cause as the threshold Mach number /H20849where the amplitude vanishes /H20850decreases with decreasing /H9260, fixed Mis increas- ingly greater than the lower limit, and hence larger solitonamplitudes are generated. These quantitative changes areseen to be particularly important for very low values of /H9260, such as /H9260/H110214, i.e., in the presence of a hard spectrum. Thus, in a plasma in which the electrons have a kappa distribution with lower values of /H9260, IA solitons of fixed Mach number have a larger amplitude, and are steeper in their pro-file, than is the case for conventional solitons occurring in aplasma whose electrons satisfy a Maxwell–Boltzmann veloc-ity distribution. This is because of the increased excess su-perthermal /H20849“tail” /H20850electrons and associated decrease in the thermal component, associated with lower /H9260, which give rise to a lower soliton threshold, the true IA speed for the plasmamodel under discussion. However, the largest possible soli-tons that may be generated in such a kappa plasma with aspecific velocity distribution are found to be smaller thanthose found in a Maxwell–Boltzmann plasma. The results reported in this paper may be of importance in the interpretation of localized electrostatic disturbancesobserved in space plasmas, where /H9260distributions are very common, as well as in laboratory plasmas, in which the pres-ence of an acceleration mechanism may lead to electron ve-locity distributions that are well modeled by a /H9260distribution. ACKNOWLEDGMENTS Useful discussions with Thomas Baluku, Richard Mace, and Frank Verheest are gratefully acknowledged. The workof N.S.S. and I.K. was supported by a UK EPSRC Scienceand Innovation award in Plasma Physics /H20849CPP grant EP/ D06337X/1 /H20850. N.S.S. would like to thank Guru Nanak Dev University, Amritsar, India for providing leave. Part of thework was carried out by I.K. during a research visit to theUniversity of Sydney. I.K. is grateful to the UK Royal Soci-ety for the award of a travel grant, and to the University ofSydney for its hospitality and local support provided duringthat visit. The research is also supported in part by the Na-tional Research Foundation of South Africa /H20849NRF /H20850. Any opinion, findings, and conclusions or recommendations ex-pressed in this material are those of the authors and thereforethe NRF does not accept any liability in regard thereto. 1V . M. Vasyliunas, J. Geophys. Res. 73, 2839, DOI:10.1029/ JA073i009p02839 /H208491968 /H20850. 2M. P. Leubner, J. Geophys. Res. 87, 6335, DOI:10.1029/ JA087iA08p06335 /H208491982 /H20850. 3T. P. Armstrong, M. 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1.2838596.pdf

Neutral modes of a two-dimensional vortex and their link to persistent cat’s eyes M. R. Turner,1,a/H20850Andrew D. Gilbert,1and Andrew P. Bassom2 1Mathematics Research Institute, School of Engineering, Computing and Mathematics, University of Exeter, Exeter EX4 4QF , United Kingdom 2School of Mathematics and Statistics, University of Western Australia, Crawley 6009, Australia /H20849Received 1 August 2007; accepted 14 December 2007; published online 14 February 2008 /H20850 This paper considers the relaxation of a smooth two-dimensional vortex to axisymmetry after the application of an instantaneous, weak external strain field. In this limit the disturbance decaysexponentially in time at a rate that is linked to a pole of the associated linear inviscid problem/H20849known as a Landau pole /H20850. As a model of a typical vortex distribution that can give rise to cat’s eyes, here distributions are examined that have a basic Gaussian shape but whose profiles have beenartificially flattened about some radius r c. A numerical study of the Landau poles for this family of vortices shows that as rcis varied so the decay rate of the disturbance moves smoothly between poles as the decay rates of two Landau poles cross. Cat’s eyes that occur in the nonlinear evolutionof a vortex lead to an axisymmetric azimuthally averaged profile with an annulus of approximatelyuniform vorticity, rather like the artificially flattened profiles investigated. Based on the stability ofsuch profiles it is found that finite thickness cat’s eyes can persist /H20849i.e., the mean profile has a neutral mode /H20850at two distinct radii, and in the limit of a thin flattened region the result that vanishingly thin cat’s eyes only persist at a single radius is recovered. The decay of nonaxisymmetric perturbationsto these flattened profiles for larger times is investigated and a comparison made with the result fora Gaussian profile. © 2008 American Institute of Physics ./H20851DOI: 10.1063/1.2838596 /H20852 I. INTRODUCTION Fluid flows at high Reynolds numbers can be dominated by the dynamics and interactions of long-lived vorticalstructures. 1–4The dynamics and stability of these vortices are particularly important in areas of fluid dynamics such asmeteorology 5,6and magnetohydrodynamics.7Many studies have considered the behavior of an axisymmetric vortex sub-jected to a transient, nonaxisymmetric strain. If this strain isweak then in the early stages the vortex may evolve dynami-cally into an axisymmetric state; any nonaxisymmetric com-ponents of the vorticity become finely scaled through spiralwind-up in the underlying flow field. 8During this process, the far-field form of the stream function decays exponentiallyin time and rotates with a fixed angular velocity which is fastcompared to the decay rate. This behavior of the vortex istermed a quasimode 9–11and has been observed both in plasma experiments10/H20849where the governing equations are isomorphic to the Euler equations /H20850and in numerical studies.12 Put simply, a quasimode is essentially a wave located in the core of a vortex and, while it appears to be a single mode,it can more accurately be characterized as a wave-packet ofcontinuum modes. The quasimode is a solution of the linear-ized Euler equations that is not separable in time, and decayson a much longer time-scale than the turn-over time-scale ofthe underlying vortex. Quasimodes of an axisymmetric cir-cular vortex have been studied analytically via Laplace trans-forms of the linearized Euler equation; 9,10,13an attraction of this method is that the angular velocity and the decay rate ofthe disturbance stream function can be deduced from the realand imaginary parts of a simple pole of the governing sys-tem. The location of this pole, known as a Landau pole, canbe calculated by analytic continuation techniques 9and it fol- lows that the decay rate and angular velocity of the quasi-mode depend only on the form of the axisymmetric baseprofile of the vortex. Moreover, the decay rate of the quasi-mode is very sensitive to the gradient of the axisymmetricbase profile at the critical radius where the angular velocityof the quasimode equals that of the vortex. 11 An illustration of an azimuthal wave number n=2 dis- turbance to a Gaussian vortex is shown in Fig. 1. At the earlier time /H20851Fig. 1/H20849a/H20850/H20852there is a linear combination of a spiral structure in which vorticity depends on rand/H9258, and a mode in which the sign of the vorticity depends just on /H9258: this is the quasimode, and corresponds to an elliptical distor-tion of the whole vortex that rotates /H20849much like a Kelvin mode on a top-hat vortex /H20850and decays because of interaction with spiral wind-up of fluctuations at the critical radiuswhere the fluid corotates with the mode. At the later time/H20851Fig.1/H20849b/H20850/H20852the quasimode has decayed to low levels, leaving behind the spiral structure, in which vorticity is of fine scaleand essentially passive. When an axisymmetric vortex is placed within a suffi- ciently strong irrotational strain field, it is well known thatthe vorticity can evolve into a tripole in which a pair of cat’seyes rotate around a coherent vortex core. 10,11,14,15The vor- ticity within the cat’s eyes becomes wound up and homog-enized; moreover, if cat’s eyes persist /H20849i.e., the far-fielda/H20850Author to whom correspondence should be addressed. Electronic mail: M.R.Turner@ex.ac.uk. Telephone: /H11001441392 725280. Fax: /H11001441392 217965.PHYSICS OF FLUIDS 20, 027101 /H208492008 /H20850 1070-6631/2008/20 /H208492/H20850/027101/10/$23.00 © 2008 American Institute of Physics 20, 027101-1stream function neither grows nor decays /H20850then they can be associated with a neutral mode of the mean vortex profile.On the other hand, if they are not maintained, decay occursat a rate that corresponds to a Landau pole. 15Numerical simulations of the Navier–Stokes equations by Turner andGilbert 15show how finite thickness cat’s eyes can be gener- ated by forcing an axisymmetric vortex with a rotating strainfield. When the strain field is switched on, cat’s eyes formaround the radius where the angular velocity of the vortexequals that of the strain field. If the strain field is of highamplitude or left on for a long duration, then when it isturned off, cat’s eyes may persist at a radius whose positiondepends upon how the perturbation vorticity feeds back tochange the axisymmetric basic profile. 15In contrast, some earlier studies16have indicated that when cat’s eyes gener- ated on a Gaussian vortex are vanishingly thin, they are lo-calized at a single specific radius. The full simulations 15 could not access this limit of infinitesimal thickness, as un-feasibly long runs at very small amplitudes and very highReynolds numbers would have been required. The purpose of the current work is to ascertain whether finite thickness cat’s eyes on a Gaussian vortex can be main-tained at many radii or whether they are restricted to a singlespecific radius. Can these cat’s eye structures superimposedon the vortex be long lived and, if so, what are the keyproperties of the underlying axisymmetric profile that enableit to support a neutral mode? We shall examine these issuesby calculating the Landau poles of a model axisymmetricprofile with a small flattened region or “defect.” Our chosenprofile /H20851detailed in Eq. /H2084918/H20850below /H20852comprises a standard Gaussian form supplemented by a three-parameter family ofdistributions; in essence the three parameters control the am-plitude, thickness, and location of the defect. This form ofvorticity distribution is quite typical of the azimuthal averageof a vortex that can sustain a cat’s eyes structure throughvorticity homogenization in the eyes. The present paper is laid out in the following way. In Sec. II we formulate the Landau pole problem and then inSec. III describe how the poles move in the complex plane asthe three parameters of the defect are varied. The Landaupoles are tracked by solving a suitable eigenvalue problemformulated in Laplace transform space. The results obtainedsuggest that a profile with a thin homogenized layer of vor-ticity can support a neutral mode and, in the limit of a van-ishingly thin layer, the position of the neutral mode is aspredicted by the asymptotic work of Le Dizès. 16We also examine the large-time behavior of the far-field stream func-tion and achieve this by solving the linearized Euler equationusing a Keller box method. 17Some concluding remarks are made in Sec. IV. II. FORMULATION OF THE LANDAU POLE PROBLEM We consider a vortex governed by the two-dimensional, incompressible Euler equations in standard plane polar coor-dinates /H20849r, /H9258/H20850, /H11509t/H9024−r−1/H20849/H11509r/H9023/H11509/H9258/H9024−/H11509/H9258/H9023/H11509r/H9024/H20850=0 , /H208491/H20850 /H116122/H9023=−/H9024,/H116122/H11013/H11509r2+r−1/H11509r+r−2/H11509/H92582. /H208492/H20850 In this system, /H9024/H20849r,/H9258,t/H20850denotes the vorticity and /H9023/H20849r,/H9258,t/H20850 is the corresponding stream function, which can found by inverting Eq. /H208492/H20850and is permitted to grow no faster than ln r for large r. We decompose the vorticity and stream function distri- butions into an axisymmetric part and a weak nonaxisym-metric perturbation, /H9024=/H9024 0/H20849r/H20850+/H9280ˆ/H9275n/H20849r,t/H20850ein/H9258+ c.c. + O/H20849/H9280ˆ2/H20850, /H208493/H20850 /H9023=/H90230/H20849r/H20850+/H9280ˆ/H9274n/H20849r,t/H20850ein/H9258+ c.c. + O/H20849/H9280ˆ2/H20850; /H208494/H20850 here the azimuthal mode number n/H333561,/H9280ˆ/H112701, and “c.c.” de- notes the complex conjugate. In this limit, Eqs. /H208491/H20850and /H208492/H20850 reduce to the linearized Euler equations for the perturbationquantities, /H11509t/H9275n+in/H9251/H20849r/H20850/H9275n+in/H9252/H20849r/H20850/H9274n=0 , /H208495/H20850 /H9004/H9274n=−/H9275n,/H9004/H11013/H11509r2+r−1/H11509r−r−2n2. /H208496/H20850 The angular velocity /H9251/H20849r/H20850and the quantity /H9252/H20849r/H20850are given by /H9251/H20849r/H20850=−r−1/H11509r/H90230,/H9252/H20849r/H20850=r−1/H11509r/H90240, /H208497/H20850 where the axisymmetric quantities are coupled via /H90240=−r−1/H11509r/H20849r/H11509r/H90230/H20850. The properties of Laplace transforms of Eq. /H208495/H20850have been studied in depth by Briggs et al.9and Schecter et al. ,10 so we only highlight the main points here. /H20849Further details can be obtained from those papers and from Ref. 15./H20850The quantity nthat appears in the above equations identifies the particular Fourier mode under investigation; notice that xy −6 −6 66 xy −6 6−66 (b) (a)FIG. 1. An n=2 disturbance to a Gaussian vortex at /H20849a/H20850 t=200 and /H20849b/H20850t=1000. The scale is white/black, where the vorticity is greater/less than 0.5 times themaximum/minimum vorticity.027101-2 Turner, Gilbert, and Bassom Phys. Fluids 20, 027101 /H208492008 /H20850n=1 is a special case of a pure translation, which gives no dynamical effects; hence, it is of no interest here.18,19We shall take n=2 in all our subsequent calculations for two main reasons. First, and conveniently, this mode is next se-quentially in a general multipole expansion after the n=1 mode. On more physically significant grounds, the multipoleexpansion of a single vortex in a large scale flow will bedominated by the n=2 component in the presence of long distance interactions with other vortices; for general n, the effect of one vortex on another falls off /H11008R −n, where Ris their separation, so that the largest dynamical effect corre-sponds to n=2. We would also expect qualitatively similar results to those described here for the minority of flows forwhich the n=2 mode is completely absent so that the multi- pole moment with some other n/H110222 dominates the dynamical effects internal to the vortex. Notice that although our calcu-lations will be restricted to the n=2 case, we shall retain a general nin our formulation to illustrate the role of the mode number in the structure of the final equations. We define the Laplace transform pair with respect to tby f ¯/H20849p/H20850=/H20885 0/H11009 eiptf/H20849t/H20850dt, /H208498/H20850 f/H20849t/H20850=−1 2/H9266/H20885 /H11009+i/H9268−/H11009+i/H9268 e−iptf¯/H20849p/H20850dp, /H208499/H20850 with any real /H9268/H110220. Combining Eqs. /H208495/H20850and /H208496/H20850and taking the Laplace transform with respect to tleads to an equation for/H9274¯n/H20849r,p/H20850of the form /H20875/H115092 /H11509r2+1 r/H11509 /H11509r−n2 r2+n/H9252/H20849r/H20850 p−n/H9251/H20849r/H20850/H20876/H9274¯n/H20849r,p/H20850=−i/H9275n/H20849r,0/H20850 p−n/H9251/H20849r/H20850. /H2084910/H20850 If the nth multipole moment Qn/H20849t/H20850is defined by Qn=/H20885 0/H11009 rn+1/H9275n/H20849r,t/H20850dr /H20849n/H333561/H20850, /H2084911/H20850 then the evolution of ln /H20841Re/H20849Q2/H20850/H20841for a Gaussian vortex acted upon by a weak instantaneous strain12is shown in Fig. 2. Three distinct regimes of decay are apparent. Regime A isthe quasimode period of exponential decay with decay rate /H9253G/H11015−0.0063 and angular velocity /H9251G/H110150.0089 calculated via Landau pole methods,15while regimes B and C are re- lated to the large- rform of the vorticity and the remnant of the vorticity in the core, respectively.12Schecter et al.10give the quasimode decay rate as /H208490.226−0.079 i/H20850/H90240/H208490/H20850, where we have taken /H90240/H208490/H20850=1 /4/H9266. The minor difference between the angular velocity values of Turner and Gilbert15and Schecter et al.10can be traced to the fact that the former was calcu- lated using an infinite domain and the latter a large finitedomain. The multipole moment can be written in terms of the far-field form of the stream function byQ n/H20849t/H20850= lim r→/H11009/H20849nrn/H9274n−rn+1/H11509r/H9274n/H20850 /H20849see Ref. 12/H20850. We impose the boundary condition /H9274n=0 at r=r0where we will take the limit r0→/H11009, so as to simulate the evolution of a disturbance in an infinite domain. Equation /H2084910/H20850is solved by means of the appropriate Green’s function9,13that leads to the Laplace transform of the quantity Qn/H20849t/H20850expressed as Q¯n/H20849p/H20850=/H20879−irn+1 /H9023L/H20849r,p/H20850/H20885 0rs/H9023L/H20849s,p/H20850/H9275n/H20849s,0/H20850 r/H20849n/H9251/H20849s/H20850−p/H20850ds/H20879 r=r0, /H2084912/H20850 where the function /H9023L/H20849r,p/H20850satisfies the homogeneous form of Eq. /H2084910/H20850, /H20875/H115092 /H11509r2+1 r/H11509 /H11509r−n2 r2+n/H9252/H20849r/H20850 p−n/H9251/H20849r/H20850/H20876/H9023L/H20849r,p/H20850=0 , /H2084913/H20850 subject to the condition /H9023L/H208490,p/H20850=0. It is clear that Eq. /H2084913/H20850potentially has a singularity at the point rs, where /H9251/H20849rs/H20850=p/n. If the axisymmetric vorticity pro- file/H90240/H20849r/H20850is a smooth decreasing function and if /H9252/H20849rs/H20850van- ishes, then it can be shown that there is a neutral mode of the vortex.9Conversely, if /H9252/H20849rs/H20850/HS110050 and /H90240/H20849r/H20850is monotonically decreasing on the whole domain r/H33528/H208510,r0/H20852, then there are no discrete eigenmodes of the vortex.9In the first case the func- tion Q¯n/H20849p/H20850possesses a simple pole, which on inverting the Laplace transform gives rise to an exponential term in Qn/H20849t/H20850. In the second case, however, we have no simple poles of Q¯n/H20849p/H20850and hence no exponential terms in Qn/H20849t/H20850. All the be- havior of Qn/H20849t/H20850is contained in the branch cut along the real p-axis in the range n/H9251/H20849r0/H20850/H33355p/H33355n/H9251/H208490/H20850. Analytical continua- tion techniques enable this branch cut to be deformed below the real p-axis by moving the radial contour of integration in Eq. /H2084912/H20850above the real r-axis. If the branch cut is bent sufficiently, then one or more so-called Landau poles pj /H20849j=1,2,3, …/H20850appear in the analytical continuation of Q¯n/H20849p/H20850 /H20851the number of poles that are seen depends both on how far the contour is deformed and the exact form of /H90240/H20849r/H20850/H20852. When the inversion contour is deformed around a pole and the-30-20-100 12000 9000 6000 3000 0A CBln|Re(Q )|2 t FIG. 2. Plot of ln /H20841Re/H20849Q2/H20850/H20841for a Gaussian vortex as a function of t.T h e dashed line gives the regime A decay rate /H9253G=−0.0063 from a Landau pole calculation.027101-3 Neutral modes of a 2-D vortex and cat’s eyes Phys. Fluids 20, 027101 /H208492008 /H20850branch cut, this leads to a term in Qn/H20849t/H20850of the form e−ipjt; although such a term can dominate the early evolution of Qn/H20849t/H20850, it does not correspond to a normal mode. This Landau pole identifies a complex decay rate − ipj=/H9253Q−in/H9251Q, where /H9253Q/H110210 and /H9251Qare the decay rate and angular velocity, re- spectively. When there are NLandau poles in the analytical continuation of Q¯n/H20849p/H20850, then Qn/H20849t/H20850takes the form Qn/H20849t/H20850=/H20858 j=1N fje−ipjt+ a branch cut contribution, /H2084914/H20850 where the constants fjdepend on the initial conditions. The angular velocity /H9251Qof each quasimode identifies a corre- sponding critical radius rQat which the fluid particles coro- tate. This radius is found by solving /H9251/H20849rQ/H20850=/H9251Q, /H2084915/H20850 where we have used Eq. /H208497/H20850to write /H9251/H20849r/H20850as /H9251/H20849r/H20850=1 r2/H20885 0r s/H90240/H20849s/H20850ds. For a pure Gaussian vortex /H9251G/H110150.0089, which gives rG/H110154.22. With the vortex profiles we consider in this study, small/H9251Qcorresponds to a large critical radius rQ, while if /H9251Q/H11015/H9251/H208490/H20850, then rQ/H110150. With a linear combination of contri- butions such as in Eq. /H2084914/H20850it is not clear just how Qn/H20849t/H20850 might behave during its early evolution, as the relative val- ues of the quantities fjare important. Nevertheless, after a sufficiently long initial transient, it will be the Landau polewith smallest decay rate that will eventually dominate. To calculate the positions of the Landau poles we used contours in the complex r-plane, parametrized by s,o ft h e general form r/H20849s/H20850= tan/H20877/H9266 2/H20875s+i/H92671sexp/H208731 /H92672/H20849s−1/H20850/H20874/H20876/H20878, /H2084916/H20850 where /H92671and/H92672are real constants. As smoves from 0 to 1 so the corresponding r-path joins zero to /H20849real /H20850infinity and a typical form of the contour is sketched in Fig. 3/H20849a/H20850.Equation /H2084913/H20850is integrated from s=s˜/H112701t o s=1 along Eq. /H2084916/H20850using a Runge–Kutta method. As r→0, it is known that/H9023L/H20849r,p/H20850/H11011rn, and thus we impose the initial conditions /H9023L/H20849r/H20849s˜/H20850,p/H20850=rn/H20849s˜/H20850,/H11509/H9023L /H11509s=nr/H11032/H20849s˜/H20850rn−1/H20849s˜/H20850, /H2084917/H20850 ats=s˜/H112701. Of course, in general /H9023L/HS110050a t s=1, and thus a Newton iteration on the eigenvalue pis used to ensure that /H9023Lvanishes at this endpoint. An alternative strategy for solving Eq. /H2084913/H20850subject to homogeneous boundary condi- tions relies on a global eigenvalue method20that generates a discrete approximation to the continuous spectrum as well asfinding any poles around which the inversion contour has tobe deformed. The global method is used to give an initialstarting value to the eigenvalue pfor the iterative local method, which is generally the more accurate. An example ofthe global solution to Eq. /H2084913/H20850is shown in Fig. 3/H20849b/H20850, which illustrates the continuous spectrum together with four Lan-dau poles. Bearing in mind our objective to investigate when a cat’s eyes structure might be relatively long lived, we consider thethree-parameter family of axisymmetric profiles given by /H9024 0/H20849r;/H9254,/H9280,rc/H20850=1 4/H9266e−r2/4+/H9254rc/H20849r−rc/H20850 8/H9266 /H11003exp/H20873r 4rc/H208492−rc2/H20850−1 2−/H20849r−rc/H208502 /H92802/H20874./H2084918/H20850 Here,/H9254,/H9280, and rcdenote the amplitude, thickness, and radius of the defect, respectively. Some sample profiles are shownin Fig. 4, which illustrates the different forms of /H9024 0/H20849r/H20850as the three parameters are varied. The form of Eq. /H2084918/H20850is that of a Gaussian vortex that has been flattened off in the vicinity ofthe critical r c; as mentioned earlier, this type of profile is characteristic of those for which cat’s eyes are observed. Theparameter /H9254is a measure of the degree of flattening imposed; clearly, when /H9254=0, the Gaussian is unaltered and at the other extreme /H20849/H9254=1/H20850, there is a turning point and an inflection point at r=rc/H20851/H90240/H11032/H20849rc/H20850=/H90240/H11033/H20849rc/H20850=0/H20852, which we shall refer to as00.30.60.91.21.5 0 3 6 9 12 15 Re(r)Im(r) -0.03-0.02-0.010 0 0.02 0.04 0.06 0.08Im(p) Re(p) (a) (b) FIG. 3. Plot of /H20849a/H20850the contour in the complex r-plane given by Eq. /H2084916/H20850with/H92671=1.7 and /H92672=2 and /H20849b/H20850a discrete approximation to the continuous spectrum and four Landau poles in the complex p-plane for an axisymmetric vorticity profile /H2084918/H20850, with /H9254=0.1,/H9280=0.5, and rc=4.027101-4 Turner, Gilbert, and Bassom Phys. Fluids 20, 027101 /H208492008 /H20850a “flat region” for brevity. This zone can be thought of as the azimuthal average of a vortex that has cat’s eyes at a radiusr cof thickness /H9280, which have been formed through some nonlinear mechanism. However, vortices containing cat’seyes, which have been generated in fully nonlinear simula-tions, do not have a smooth azimuthal profile 15but, instead, typically contain a flattened region with fine scale structuredue to the nonlinear interactions. They also generally havesharper vorticity gradients around the flattened region thanare present in the idealized profile /H20851Eq. /H2084918/H20850/H20852. Thus, our model used here might be helpful for understanding the be-haviors of thin cat’s eyes, but nonlinear interactions need tobe considered for our results to be confirmed. III. RESULTS In this section we describe results of the Landau pole calculations and the linear code simulations for the profilesgiven by Eq. /H2084918/H20850. We shall have frequent cause to refer to the real and imaginary parts of Landau poles p jand thus it is to be remembered that these quantities correspond to nmul- tiples of the angular velocity /H9251Qand the decay rate /H9253Qof the vortex quasimode, respectively. A. Landau pole calculations The Landau pole calculations rely on the parameters /H92671 and/H92672in Eq. /H2084916/H20850being chosen so that the corresponding contour of the continuous spectrum in the p-plane is bent sufficiently to reveal the poles we wish to follow. All theresults here used /H92671/H33528/H208511.5,5 /H20852and/H92672/H33528/H208510.4,2.2 /H20852with the particular values governed by the nature of the underlying vorticity /H90240/H20849r/H20850. In the limits of small /H9280and extreme rc/H20849either small or large /H20850, we found that it was not always possible to track all the poles of interest as the contour in the p-plane often becomes very knotted and intricate. Moreover, thepoles then become easily confused with the continuous spec-trum. That said, however, we were able to capture the behav-ior of the poles sufficiently well to make some useful obser-vations. We have a three-dimensional parameter space /H20849 /H9254,/H9280,rc/H20850 to investigate and for the parameter ranges considered, wegenerally found that the values of /H92671and/H92672in Eq. /H2084916/H20850led to three Landau poles of interest in the p-plane; the only excep- tion occurs when /H9254=1 when an extra pole appears. We ex- plored our parameter space systematically by fixing the pa-rameters two at a time and then allowing the third to vary.Figure 5plots the imaginary parts Im /H20849p/H20850= /H9253Qof three Landau poles as the basic vorticity changes. Figure 5/H20849a/H20850relates to a defect of moderate thickness /H9280=0.5 positioned at rc=4, whose amplitude 0 /H33355/H9254/H333551. For /H9254=0 we have a Gaussian profile, which has a Landau pole at pG=0.0177−0.0063 i, which corresponds to pole 1. As /H9254is increased from zero, so poles 2 and 3 move in quickly fromthe large negative imaginary region of the p-plane. One of the poles /H20849pole 3 /H20850has a larger decay rate than the other two for all the values of /H9254considered and hence is expected to be insignificant for the initial decay of the perturbation. Poles 1and 2 have very similar decay rates for 0 /H11021 /H9254/H110210.2, so it is unclear at which rate the vortex quasimode will decay in this region. As we approach the case of a flat region at r=rcin /H90240/H20849r/H20850/H20849i.e., the amplitude of the defect /H9254=1/H20850, then we find that the original Gaussian quasimode Landau pole /H20849pole 1 /H20850 again has the smallest decay rate, and hence should dominatethe initial decay of the perturbation. In Fig. 5/H20849b/H20850the defect has a small fixed amplitude /H9254=0.1 and is positioned at the point rc=4, but has a variable thickness /H9280. The two Landau poles that had similar decay rates in Fig. 5/H20849a/H20850continue to do so over the whole range of /H9280 investigated. Pole 3 is again insignificant over most of the /H9280-domain as it has a much larger decay rate than the other two, although, as /H9280becomes smaller /H20849i.e., as the defect thins /H20850, the presence of this third pole becomes more important. Lastly, the defect in Fig. 5/H20849c/H20850has a fixed small amplitude /H20849/H9254=0.1 /H20850and is of moderate thickness /H20849/H9280=0.5 /H20850, but is moved around by varying rc. Close to rc=4, the two poles, which were almost coincident in Fig. 5/H20849a/H20850, remain near to each other, and actually swap places. We would therefore expectthe decay rate of the perturbation to change between thesetwo values with a corresponding jump in frequency /H20849cf. Fig. 10later /H20850. Pole 3 again remains insignificant for the values of r cconsidered here; however, for large rc/H20849i.e., when the de- fect is positioned well away from the vortex core /H20850this pole tends to the Landau pole value for a Gaussian vortex notedearlier. As r cis moved towards the origin, with /H9280still mod- erately large, the core of the vortex becomes increasinglydeformed and sharp gradients of vorticity form around thedefect. Consequently, results with r c/H110211 and moderate /H9280need to be treated with care although if the thickness of the defect /H9280is reduced as rcmoves towards the origin then results for smaller values of rccan be attained. Figure 5/H20849a/H20850suggests that when there is a flat region in the profile /H20849/H9254=1/H20850, there appears to be a single Landau pole which dominates the decay of the perturbation. This is ex- amined further in Fig. 6, which shows the real and imaginary parts of the significant Landau poles as functions of the de-fect position r cfor amplitude /H9254=1 and thickness /H9280=0.5. We see that there are now four Landau poles which appear in theregion of the p-plane we consider. Figure 6/H20849b/H20850shows that poles 3 and 4 are not important in the evolution of the per-turbation, as their decay rates are larger than the other two.0.06 0.04 0.02 5 4 3 2 1 0 rΩ0 FIG. 4. Figure showing /H90240/H20849r/H20850for/H9254=0 /H20849solid line /H20850;/H9254=1,/H9280=0.5, rc=3 /H20849dashed line /H20850,a n d/H9254=1,/H9280=0.3, rc=2 /H20849dotted line /H20850.027101-5 Neutral modes of a 2-D vortex and cat’s eyes Phys. Fluids 20, 027101 /H208492008 /H20850When rc/H110224.5, there are two poles close together, and thus we expect these to combine together to affect the decay rateof perturbations to the vortex. For 1 /H11021r c/H110214.5, there is just one pole that dominates the decay of the vortex perturbationand it is this pole on which we concentrate now. This pole is significant because at rc/H110152.6, the vortex has a decay rate of only/H9253Q/H11015−6.4/H1100310−5, and thus at this location the vortex perturbation will decay extremely slowly. Indeed, this decay-0.01-0.0050 0 0.2 0.4 0.6 0.8 1 δIm(p)1 2 3 -0.03-0.02-0.010 0.2 0.4 0.6 0.8 1 εIm(p)1 2 3 -0.015-0.01-0.005 2 3 4 5 62Im(p)1 3 rc(a) (b) (c) FIG. 5. Plot of the imaginary parts of three Landau poles as a function of /H20849a/H20850/H9254for /H20849/H9280,rc/H20850=/H208490.5,4 /H20850,/H20849b/H20850/H9280for /H20849/H9254,rc/H20850=/H208490.1,4 /H20850, and /H20849c/H20850rcfor /H20849/H9254,/H9280/H20850=/H208490.1,0.5 /H20850. 00.020.040.060.08 1 2 3 4 5 6 71 Re(p) 23 4 rc-0.012-0.008-0.0040 1 2 3 4 5 6 723 4Im(p) rc1 (a) (b) FIG. 6. Plot of /H20849a/H20850the frequency or real part Re /H20849pj/H20850=n/H9251Qand /H20849b/H20850the decay rate or imaginary part Im /H20849pj/H20850=/H9253Qof four Landau poles as a function of rcfor /H9254=1.0 and /H9280=0.5.027101-6 Turner, Gilbert, and Bassom Phys. Fluids 20, 027101 /H208492008 /H20850is so slow that it suggests that only tiny changes to the details of the defect might be sufficient to ensure that /H9253Q=0, thereby forming a neutral mode. Figure 7illustrates Im /H20849p/H20850=/H9253Qfor the Landau pole with the smallest decay rate, as a function of defect position rc, for /H9254=1 and various values of /H9280. We see that as the defect thick- ness/H9280decreases from /H9280=0.5 /H20849solid line /H20850the maximum growth rate value increases. However, when /H9280=0.3 /H20849dotted line /H20850we see that there are now two maxima in the growth rate curve and these both have /H9253Q=0; i.e., they are both neutral modes of the vortex. As /H9280is reduced further, both the neutral modes remain, but their positions separate. This sug-gests that cat’s eyes of a given finite thickness /H9280, which have an azimuthal average profile that contains an annulus of ho-mogenized vorticity, can persist at two distinct radii. As /H9280→0 we expect to recover the result of Le Dizès,16who showed that vanishingly thin cat’s eyes can only possess aneutral mode at the one radius r LD=3.44 /H20849using our scalings /H20850. Figure 8/H20849a/H20850plots the value of rcat which the maximum value of Im /H20849p/H20850=/H9253Qoccurs as a function of the defect thick-ness/H9280/H20849with/H9254=1/H20850. When /H9280/H113510.35, there are two positions at which the maximum /H9253Qoccurs; at these points /H9253Qis zero and thus these correspond to neutral modes. Figure 8/H20849b/H20850shows the dependence of 2 /H9251Q=Re /H20849p/H20850, where the minimum value of /H9253Qoccurs as a function of the thickness /H9280. The neutral mode that moves towards the origin /H20849branch 2 /H20850has an angular ve- locity that increases as /H9280is reduced, and in the limit /H9280→0, so /H9251Q→1/4/H9266. Le Dizés16has shown that a vanishingly thin critical layer cannot exist at the origin, and we leave the issueof the limit /H9280→0 along branch 2 as a topic for future study. Linear extrapolation suggests that the position and angularvelocity of the branch 1 neutral mode tend to r c=3.448 and /H9251Q=0.0127 as /H9280→0. Both of these results agree with the results rLD=3.44 and /H9251LD=0.0127 derived by Le Dizès16 with errors of less than 1%. An explanation for the occurrence of the two neutral modes seen in Figs. 7and8is offered by Fig. 9. This shows the form of rQ−rc, found by solving Eq. /H2084915/H20850, as a function ofrcfor/H9254=1 and the five chosen defect thicknesses used in Fig.7. We see that when /H9280=0.4 or 0.5, the critical radius rc always exceeds rQ. This implies that the critical radius of the quasimode never occurs at the flat center of the defect in the-0.0002-0.00010 1 1.5 2 2.5 3 3.5 4Im(p) rc FIG. 7. Figure showing the decay rate /H9253Q=Im /H20849p/H20850for the Landau pole with the smallest decay rate as a function of rcwith/H9254=1 and /H9280=0.5 /H20849solid line /H20850, 0.4 /H20849large dashed line /H20850,0 . 3 /H20849small dashed line /H20850, 0.2 /H20849dotted line /H20850, and 0.1 /H20849dot-dashed line /H20850, from the middle to top right. 00.511.522.533.5 0 0.1 0.2 0.3 0.4 0.51 2 εrc 0.020.040.060.08 0 0.1 0.2 0.3 0.4 0.51Re(p) ε2 (a) (b) FIG. 8. Plot of /H20849a/H20850rcand /H20849b/H20850Re/H20849p/H20850=2/H9251Q, where the maximum value of /H9253Qoccurs as a function of /H9280. The numbers indicate the two branches of solutions.-0.16-0.12-0.08-0.040 1 1.5 2 2.5 3 3.5 4r− r c Q rc FIG. 9. Plot of rQ−rcagainst rcfor/H9254=1 and the values of /H9280and the line styles from Fig. 7with/H9280=0.5 at the bottom to /H9280=0.1 at the top. The solid horizontal line represents the line rQ=rc.027101-7 Neutral modes of a 2-D vortex and cat’s eyes Phys. Fluids 20, 027101 /H208492008 /H20850profile; in other words, the singular point rQof Eq. /H2084913/H20850 never occurs where /H9252=0. On the other hand, the curves for other thicknesses have rQ=rctwice at the two values of rc given in Fig. 8/H20849a/H20850. Thus, the positions of the neutral modes correspond to points where the quasimode critical layer oc-curs exactly at the center of the flat region of the profile, i.e.,the singular point of Eq. /H2084913/H20850,r=r Q, is where /H9252=0. As the thickness /H9280of the flat region is reduced from 0.5 /H20849reading up the curves /H20850, we find that the peak of the growth rate curve moves towards the line rQ=rcand at /H9280/H110150.35, the curve and line become tangent to one another and there is a singleneutral mode of the vortex. For values of /H9280/H113510.35, there are two neutral modes; one of these moves towards rc=3.448 and the other approaches rc=0 as the defect thickness /H9280→0. B. Comparison with linear code Next we compare our Landau pole calculations with re- sults determined by numerically solving Eq. /H208495/H20850using a Keller box17routine from the NAG suite; see Ref. 12for more details. The scheme imposes an external flow whichinstantaneously distorts the vortex and then allows it toevolve freely over time. This problem is of initial value typeand thus the decay rate and angular velocity measured fromthe solution of Eq. /H208495/H20850are time dependent. To estimate the values /H9253Qand/H9251Q, we took the best linear fit of the data between two time values that occur just before the vortexleaves the linear decay part of its evolution /H20849i.e., just before the vortex leaves regime A in a case such as Fig. 2/H20850. These times were decided by a preliminary run of the code, fromwhich we estimated the instant t 2when the vortex changes regime. The earlier time was then taken to be t1=t2−1000, and then a re-run of the code gave decay rates and angularvelocities that agree very well with the Landau pole theory. The results outlined in Sec. III A show that in certain regions of our three-dimensional parameter space, one Lan-dau pole dominates over the others, and hence we expect themultipole moment Q 2/H20849t/H20850to decay exponentially with a decayrate and angular velocity given by this Landau pole. How- ever, this is not completely obvious because this quasimodedecay is just a transient /H20849see regime A in Fig. 2/H20850and thus there is only a finite time over which a decay rate can bemeasured before the contribution from the branch cut takesover. Hence, to assess at what rate the vortex perturbationsactually decay in the early stages, we used the numericalscheme above to compare with our Landau pole results. Figure 10shows results from the solution to the linear equations /H208495/H20850and /H208496/H20850together with the Landau poles corre- sponding to the smallest decay rates for amplitude /H9254=1 and thickness /H9280=0.5 as in Fig. 6. Figure 10/H20849a/H20850demonstrates very good agreement between the predictions of the linear code and the real part of the Landau pole with the larger of thetwo angular velocities, at least for 2.5 /H11021r c/H110215. However, when the flat defect is centered at rc/H110155, a sudden change in the angular velocity of the perturbation occurs and the linearcode results correlate with the Landau pole with the smallerof the two angular velocities once r c/H110225. The reason for this frequency jump is suggested in Fig. 10/H20849b/H20850, which shows the imaginary part of the Landau pole /H20849=/H9253Q/H20850. The linear code gives a growth rate that agrees with the Landau pole with smallest decay rate for 2.5 /H11021rc/H110215; however, at rc/H110155 an- other Landau pole acquires a smaller decay rate, and thus thevortex perturbation switches onto this new rate, with the cor-responding angular velocity given by this pole /H20851see Fig. 10/H20849a/H20850/H20852. The reason the two methods are not in complete agreement for r c/H110225 is because when the decay rates of these two Landau poles become close, the vortex decays with alinear combination of the two decay rates and frequenciesand distinguishing between them is not particularly easy. Figure 11considers the far-field amplitude ln /H20841Q 2/H20849t/H20850/H20841for a vorticity profile with amplitude /H9254=1, thickness /H9280=0.5, and defect position rc=5.2 /H20849solid line /H20850. This value of rclies close to the point where the two growth rate curves meet in Fig.6/H20849b/H20850. The two straight lines represent two decay rates: the first given by the best fit analysis of the linear code /H20849dashed line /H20850and the second by the dominant Landau pole /H20849dotted0.010.020.030.04 2.5 3 3.5 4 4.5 5 5.5 6Re(p) rc-0.003-0.002-0.0010 2.5 3 3.5 4 4.5 5 5.5 6Im(p) rc (a) (b) FIG. 10. Plot of /H20849a/H20850the real part and /H20849b/H20850the imaginary part of the growth rate calculated via the linear code /H20849dotted line /H20850as a function of rcfor/H9254=1 and /H9280=0.5. The solid line and the large dashed line show the two Landau poles with largest growth rates in this region of interest. For rc/H113515, the solid and dotted lines are indistinguishable.027101-8 Turner, Gilbert, and Bassom Phys. Fluids 20, 027101 /H208492008 /H20850line /H20850. We can see that due to the oscillations in ln /H20841Q2/H20841a best fit line would depend on the time interval chosen in which totake the approximation and because this initial exponentialdecay is only a transient we cannot integrate until one Lan-dau pole clearly dominates the solution; this is the reason forthe small discrepancy between the results in Fig. 10once r c/H110225. The decay rate given by the Landau pole in Fig. 11 /H20849dotted line /H20850is in good agreement with the decay rate of ln/H20841Q2/H20841, and hence the vortex perturbation really does dimin- ish at the rate corresponding to that Landau pole with theslowest decay rate. The Landau pole calculation in Fig. 7of Sec. III A shows that for defect thicknesses /H9280/H110210.35, there are two po- sitions rc, where Im /H20849p/H20850=/H9253Q=0, and hence two radii that can support a neutral mode. We can confirm that a vortex does admit these two neutral modes by using the linear code forthe basic profile with /H9254=1 and /H9280=0.3. The outcome is illus- trated in Fig. 12; it is concluded that the linear code /H20849dashed line /H20850and the Landau pole result /H20849solid line /H20850are in excellentagreement, and the frequencies n/H9251Q=Re /H20849p/H20850from the two calculations /H20849not shown /H20850turn out to be indistinguishable over this range of rc. Figure 13plots ln /H20841Re/H20849Q2/H20849t/H20850/H20850/H20841for the Gaussian vortex and four vorticity profiles, all with defect amplitude /H9254=1 and thickness /H9280=0.5, but with various positions in the range 1/H33355rc/H3335510. The Gaussian vortex /H20849curve 1 /H20850, discussed in Sec. II, has an initial period of exponential decay with /H9253G/H11015−0.0063 followed by an interval of decay related to the large- rform of the vorticity and finally a region of algebraic decay linked to the remnant of vorticity in the core of thevortex /H20849see regimes B and C of Fig. 2/H20850. What makes Fig. 13 so interesting is that all the results with /H9254=1 /H20849curves 2–5 /H20850 appear to decay at exactly the same rate and with the angularvelocity of the Gaussian vortex /H20849curve 1 /H20850, at least up to some time. Thereafter, they start to decay exponentially at a ratedetermined by the dominant Landau pole. Finally, when /H20841Q 2/H20841 drops below some threshold, the vortex perturbation beginsto decay algebraically /H20849see curve 4 /H20850as for the /H9254=0 case. Curve 4 with rc=6 exhibits a small region of nonexponential decay at t/H110152500 before decaying at the Landau pole value. The decay rates for rc=10 and 9 /H20849curves 2 and 3, respec- tively /H20850both tend to the exponential decay rate given by the dominant Landau pole. At larger values of rc, the amplitude threshold for the algebraic decay may be reached before thisperiod of exponential decay occurs; in this case the exponen-tial decay may not be observed. The features in real space of a quasimode for a profile /H2084918/H20850with a defect are illustrated by Fig. 14, which can be compared with the purely Gaussian case in Fig. 1. This figure shows a n=2 disturbance to a profile with a defect at r c=2.85. Again we see a combination of spiral wind-up coupled with the modal behavior, in which the elliptical dis-tortion gives a sign of vorticity that depends on angle /H9258 rather than on radius. At the early time, the picture in Fig.14/H20849a/H20850is similar to the Gaussian case in Fig. 1/H20849a/H20850with the flat defect at r c=2.85 clearly visible. However, at late times in Fig.14/H20849b/H20850we note that the elliptical distortion is still present /H20849unlike in the Gaussian case /H20850, showing that the quasimode-30-20-100 0 4000 8000 12000 16000 tln|Q | 2 FIG. 11. Plot of ln /H20841Q2/H20841against time for the case /H9254=1,/H9280=0.5, and rc=5.2. The dashed line is the growth rate given by the linear code and the dottedline is the growth rate give by the dominant Landau pole. -6e-05-4e-05-2e-050 1 1.5 2 2.5 3 3.5Im(p) rc FIG. 12. Plot of Im /H20849p/H20850=/H9253Qcalculated by the linear code /H20849dashed line /H20850and the Landau pole code /H20849solid line /H20850as a function of rcfor/H9254=1, and /H9280=0.3.-30-20-100 0 5000 10000 15000 200005 4 2 3ln|Re(Q )|2 t1 FIG. 13. Curve 1 shows a plot of ln /H20841Re/H20851Q2/H20849t/H20850/H20852/H20841when/H9254=0. Curves 2–5 all relate to the case when /H9254=1 and /H9280=0.5, but with rc=10, 9, 6, and 1, respectively.027101-9 Neutral modes of a 2-D vortex and cat’s eyes Phys. Fluids 20, 027101 /H208492008 /H20850persists in this case, and we have a neutral mode. We can also see how the defect interacts with the vorticity in theneighborhood of r c=2.85, where the spiral arms are broader. It is in this region where cat’s eyes /H20849of a width of order /H9280=0.3 /H20850could be maintained in the full nonlinear problem. IV. CONCLUSIONS In this paper we have studied how an axisymmetric vor- tex relaxes after being subjected to a weak, instantaneous,nonaxisymmetric forcing. This has been accomplished usinga two-pronged numerical approach; one of these relies on thetracking of Landau poles in the complex plane while theother is a Keller-box strategy applied to the linear problem.Our underlying vortex has been taken to be a Gaussian typewhich has been modified by inclusion of zone where theprofile is artificially flattened. After a transient, the nonaxi-symmetric perturbation decays with a decay rate and fre-quency fixed by the Landau pole that has the least negativeimaginary part. The time taken for the perturbation to start todecay at this Landau pole rate depends upon the structure ofthe axisymmetric profile. For a profile with a flat region of thickness /H9280/H113510.35 po- sitioned at r=rc, we have shown that two neutral modes exist at two distinct values of r=rc. Moreover, in the limit as the defect thickness /H9280→0, one of these neutral modes moves to the point rc=3.448, in accord with the asymptotic results of Le Dizès.16Thus, this study suggests that when cat’s eyes below a critical thickness are generated in a vortex via anonlinear mechanism, they can persist at two distinct radii.However, nonlinear mechanisms rarely give axisymmetricprofiles as clean as the ones studied in this paper and the finescale structure within the eyes often becomes significant. It isclear that further detailed numerical work is required to as-certain whether our conclusions can be extended to nonlinearsituations. ACKNOWLEDGMENTS This work was undertaken on the EPSRC Grant No. EP/ D032202/1.1B. Fornberg, “A numerical study of 2-D turbulence,” J. Comput. Phys. 25, 1/H208491977 /H20850. 2J. C. McWilliams, “The emergence of isolated coherent vortices in turbu- lent flow,” J. Fluid Mech. 146,2 1 /H208491984 /H20850. 3R. Benzi, G. Paladin, S. Patarnello, P. Santangelo, and A. Vulpiani, “In- termittency and coherent structures in two-dimensional turbulence,” J.Phys. A 19, 3771 /H208491986 /H20850. 4M. E. Brachet, M. Meneguzzi, H. Politano, and P. L. Sulem, “The dynam- ics of freely decaying two-dimensional turbulence,” J. Fluid Mech. 194, 333 /H208491988 /H20850. 5T. A. Guinn and W. H. Schubert, “Hurricane spiral bands,” J. Atmos. Sci. 50, 3380 /H208491993 /H20850. 6G. B. Smith and M. T. Montgomery, “Vortex axisymmetrization: Depen- dence on azimuthal wave-number or asymmetric radial structure changes,”Q. J. R. Meteorol. Soc. 121, 1615 /H208491995 /H20850. 7K. Bajer and H. K. Moffatt, “On the effect of a central vortex on a stretched magnetic flux tube,” J. Fluid Mech. 339,1 2 1 /H208491997 /H20850. 8M. V. Melander, J. C. McWilliams, and N. J. Zabusky, “Axisymmetriza- tion and vorticity-gradient intensification of an isolated two-dimensionalvortex through filamentation,” J. Fluid Mech. 178,1 3 7 /H208491987 /H20850. 9R. J. Briggs, J. D. Daugherty, and R. H. Levy, “Role of Landau damping in crossed-field electron beams and inviscid shear flow,” Phys. Fluids 13, 421 /H208491970 /H20850. 10D. A. Schecter, D. H. E. Dubin, A. C. Cass, C. F. Driscoll, I. M. Lansky, and T. M. O’Neil, “Inviscid damping of asymmetries on a two-dimensional vortex,” Phys. Fluids 12, 2397 /H208492000 /H20850. 11N. J. Balmforth, S. G. Llewellyn Smith, and W. R. Young, “Disturbing vortices,” J. Fluid Mech. 426,9 5 /H208492001 /H20850. 12A. P. Bassom and A. D. Gilbert, “The spiral wind-up of vorticity in an inviscid planar vortex,” J. Fluid Mech. 371,1 0 9 /H208491998 /H20850. 13N. R. Corngold, “Linear response of the two-dimensional pure electron plasma: Quasimodes for some model profiles,” Phys. Plasmas 2,6 2 0 /H208491995 /H20850. 14C. Macaskill, A. P. Bassom, and A. D. Gilbert, “Nonlinear wind-up in a strained planar vortex,” Eur. J. Mech. B/Fluids 21,2 9 3 /H208492002 /H20850. 15M. R. Turner and A. D. Gilbert, “Linear and nonlinear decay of cat’s eyes in two-dimensional vortices, and the link to Landau poles,” J. Fluid Mech. 593, 255 /H208492007 /H20850. 16S. Le Dizès, “Non-axisymmetric vortices in two-dimensional flows,” J. Fluid Mech. 406, 175 /H208492000 /H20850. 17H. B. Keller, “A new difference scheme for parabolic problems,” in Nu- merical Solution of Partial Differential Equations, II (SYNSP ADE 1970)(Proceedings of the Symposium) , University of Maryland, College Park, MD, 1970 /H20849Academic, New York, 1971 /H20850. 18A. J. Bernoff and J. F. Lingevitch, “Rapid relaxation of an axisymmetric vortex,” Phys. Fluids 6, 3717 /H208491994 /H20850. 19J. F. Lingevitch and A. J. Bernoff, “Distortion and evolution of a localized vortex in an irrotational flow,” Phys. Fluids 7, 1015 /H208491995 /H20850. 20P. J. Schmid and D. S. Henningson, Stability and Transition in Shear Flows /H20849Springer, New York, 2001 /H20850. xy −6−6 66 xy −6 6 −66 (a) (b)FIG. 14. An n=2 quasimode within a profile /H2084918/H20850with /H9254=1,/H9280=0.3, and rc=2.85 at /H20849a/H20850t=200 and /H20849b/H20850 t=1000. The gray scale is as in Fig. 1.027101-10 Turner, Gilbert, and Bassom Phys. Fluids 20, 027101 /H208492008 /H20850

1.4802266.pdf

Thickness dependence of current-induced domain wall motion in a Co/Ni multi-layer with out-of-plane anisotropy Hironobu Tanigawa, Tetsuhiro Suzuki, Shunsuke Fukami, Katsumi Suemitsu, Norikazu Ohshima et al. Citation: Appl. Phys. Lett. 102, 152410 (2013); doi: 10.1063/1.4802266 View online: http://dx.doi.org/10.1063/1.4802266 View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v102/i15 Published by the American Institute of Physics. Additional information on Appl. Phys. Lett. Journal Homepage: http://apl.aip.org/ Journal Information: http://apl.aip.org/about/about_the_journal Top downloads: http://apl.aip.org/features/most_downloaded Information for Authors: http://apl.aip.org/authors Downloaded 22 Apr 2013 to 129.10.107.106. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsThickness dependence of current-induced domain wall motion in a Co/Ni multi-layer with out-of-plane anisotropy Hironobu Tanigawa,1,a)Tetsuhiro Suzuki,1Shunsuke Fukami,2,3Katsumi Suemitsu,1 Norikazu Ohshima,1and Eiji Kariyada1 1Production and Technology Unit, Renesas Electronics Corporation, Sagamihara, Kanagawa 252-5298, Japan 2Center for Spintronics Integrated Systems, Tohoku University, Sendai, Miyagi 980-8577, Japan 3Green Innovation Research Laboratories, NEC Corporation, Tsukuba, Ibaraki 305-8508, Japan (Received 7 January 2013; accepted 5 April 2013; published online 18 April 2013) Thickness dependence of current-induced domain wall (DW) motion in a perpendicularly magnetized [Co/Ni] Nmultilayered wire containing Ta/Pt capping and Pt/Ta seed layers has been studied. The thickness of the magnetic layer was controlled by the stacking number, N.T h e threshold current density for driving DW had a local minimum at N¼3 and the velocity of DW motion decreased with N. Estimation of carrier spin polarization from measurements of DW velocity revealed that a thinner Co/Ni stack adjacent to the Pt layers reduced the carrier spin polarization and the strength of adiabatic spin transfer torque. VC2013 AIP Publishing LLC [http://dx.doi.org/10.1063/1.4802266 ] Magnetization switching through electric-current-induced spin transfer torque has been intensively investigated due to its potential to realize magnetic devices with low power con-sumption and high reliability. The electrical manipulation of domain walls (DWs) 1is regarded as a possible technique for writing in magnetic devices such as magnetic random accessmemories (MRAMs). 2Recently, many researchers have stud- ied DW motion in ferromagnetic wires with perpendicular magnetic anisotropy (PMA), which consists of multilayeredfilm with total thickness in the range from sub-nanometer to several nanometers. 3–17A typical material for the DW motion is multilayered Co/Ni.5,7,9–12,14,15,17In a 3.4-nm-thick Co/Ni film with symmetric capping and seed layers, it has been reported that an adiabatic spin transfer torque drives the DW in the direction of electron flow.11Theoretical prediction based on the adiabatic torque indicates that the threshold cur- rent density decreases with the thickness of the magnetic layer.18,19In fact, a smaller threshold current density is seen in a system of ultrathin (approximately 1-nm thick) Co/Ni/Co with TaN capping and Pt/TaN seed layers15than in a 3.4-nm- thick Co/Ni system.11However, in Ref. 15, the DW drives against the electron flow and the DW motion reaches a veloc- ity of a few hundred m/s at room temperature; these are in contradiction to the adiabatic model. Similar characteristicswere observed for a 0.7-nm-thick Co wire with AlO xcapping and Pt seed layers.4,13A spin Hall current20,21injected from the Pt layer or Rashba spin-orbit torque22,23at the Co/Pt inter- face due to the structural inversion asymmetry16has been sug- gested as candidates for the origin of these results. To exclude these interfacial effects and investigate the dependence of DWmotion characteristics on thickness, here we investigated the threshold current density and velocity of DW motion in multi- layered Co/Ni wires with thicknesses ranging from 1.2 to5.7 nm and symmetric stack structure. Multilayered Co/Ni films were deposited by dc magne- tron sputtering onto a Si substrate covered with SiO 2. The film was composed of Ta (3 nm)/Pt (1.6 nm)/Co (0.3 nm)/[Ni(0.6 nm)/Co (0.3 nm)] N/Pt (1.6 nm)/Ta (3 nm). The stacking number Nwas varied from 1 to 6 in order to control the thickness of the magnetic layer. The Ta (3)/Pt (1.6) and Pt(1.6)/Ta (3) layers are the capping and seed layers. Figure 1(a) shows magnetization curves of the multilayered films measured with a vibrating sample magnetometer at roomtemperature along the out-of-plane direction. The magnetiza- tion easy axis of each film was perpendicular. The inset of Fig. 1(a) shows the magnetic moment as a function of the thickness t Co/Ni of the magnetic layer. Linear-fitting to the plot points (red line) revealed no magnetic dead layer at the Co/Pt interface. The temperature dependence of saturationmagnetization M Swas also evaluated (Fig. 1(b)). All samples followed the same general trend on this point, with MS decreasing as Tincreased and approaching zero at about 700 K (Curie temperature TC), except for the sample with N¼1. As for N¼1, the TCwas approximately 650 K. Figure 2(a) is a schematic view of the experimental setup for the DW motion measurement along with an image of the sample taken with a scanning electron microscope. FIG. 1. Magnetization characteristics for [Co/Ni] Nfilms with Nfrom 1 to 6: (a) magnetization curves at room temperature along the out-of-plane direc- tion and (b) saturation magnetization MSas a function of temperature. The inset of Fig. 1(a)shows magnetic moment as a function of thickness tCo/Ni of the magnetic layer. The plane dimensions Sof the samples for magnetization measurements are 10 mm /C210 mm. MSwas estimated by dividing the mag- netic moment by S/C2tCo/Ni.a)Electronic mail: hironobu.tanigawa.yn@renesas.com 0003-6951/2013/102(15)/152410/4/$30.00 VC2013 AIP Publishing LLC 102, 152410-1APPLIED PHYSICS LETTERS 102, 152410 (2013) Downloaded 22 Apr 2013 to 129.10.107.106. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsThe [Co/Ni] Nfilm was deposited onto a Si substrate covered with SiO 2, in which Cu electrodes were embedded and pat- terned into the wires forming a cross shape. The widths wof the wires in which the DWs were propagated were in therange from 70 to 200 nm and the width of the wire for Hall resistance measurement was approximately 100 nm. The dis- tance lbetween electrode A and the Hall cross was 2 lm. The upper panel of Fig. 2(b) shows the evolution of t totr (¼l/Rw) as a function of tCo/Ni obtained from measuring the resistances of devices having l¼2lm and w¼200 nm, where ttotis the total thickness of the stack and ris the conduc- tivity. Interface scattering is known to produce a positive cur-vature in the relation between t totrandttot; this trend is seen when the ttotbecomes small compared with the electron mean free path.6,24Accordingly, the positive curvature appearing in the thin tCo/Ni region in our experiment s hould indicate scatter- ing at the interface between Co and Pt. The lower panel of Fig.2(b)shows the estimated shunt current ratio in part of the ferromagnetic layer as a function of tCo/Ni;t h ev a l u e sw e r e estimated from the sheet resistances measured with various stack structures. The shunt curr ent ratio decreased with thinner tCo/Ni a n dd r o p p e dt o8 %a t tCo/Ni¼1.2 nm ( N¼1) due to the increased interface scattering. We evaluated the threshold current density for DW motion Jthfor each [Co/Ni] Ndevice. The measurement sequence was basically the same as was used for our previous reports.9,10The duration of the pulsed current injection into the Co/Ni wires was 200 ns. The current amplitude through the Co/Ni wires was measured from the voltage detected by an oscilloscope with a 50- Xtermination. The measurement was performed at room temperature. We used the shunt cur- rent ratio shown in Fig. 2(b) to convert the current to current density Jin the ferromagnetic part. Figure 2(c)shows typical Hall resistance measurements for [Co/Ni] Ndevices with wof approximately 130 nm. In all devices other than that with N¼1, an abrupt change in the Hall resistance was seen in the region J<0, whereas no change was observed for J>0; this corresponds to the DW motion in the direction of electron flow. In the device with N¼1(tCo/Ni¼1.2 nm), no such change was observed anywhere in the range of current used in measurement ( J<0.81/C21012A/m2), in contrast to the result reported by Ryu et al.15Figure 2(d)shows the required external field strength at which the DW was depinned through the cen- t e ro ft h eH a l lc r o s s Hdep, and (e) shows the average Jth, both as functions of tCo/Ni. We repeated the measurements five times for fifty elements at each tCo/Ni.Hdepwas in the range from 500 to 600 Oe and slightly increased with tCo/Ni. Meanwhile, Jthhad a local minimum value of 0.46 /C21012A/m2atN¼3 (tCo/Ni¼3.0 nm). In a similar way, the dependences of Hdep andJthon wire width in the range from 70 to 200 nm were also measured for stacks with each N. While the Hdepof the device with 70-nm-wide wires was appr oximately 2.5 times greater than that for the 200-nm width ( Hdepffi1000 and 400 Oe, respectively) probably due to the difference in the pinningeffect of the Hall cross on the DW, 3,17no dependence of Jthon wwas observed for any of the stacks (these results are not shown). This indicates that the driving force of the DW motionwe observed is adiabatic spin transfer torque. 11 Next, DW velocity vs. current density was measured and the carrier spin polarization Pwas estimated. Tomeasure the velocity of DW motion, samples with w¼130 nm and l¼6lm were used. The details of the mea- surement sequence and analysis are described in Ref. 17. The duration of one pulsed current used in measuring DWvelocity was 30 ns. Figure 3(a)shows the average velocity v of DW motion as a function of Jin wires with N¼2, 3, 4, 5, and 6. We used one element at each Nand repeated the mea- surement 10 times at each Jto estimate v. The vincreased with Jand reached saturation above a certain Jbecause Joule heating led to increased device temperature. 12The sat- urated value of vfell markedly with N. This is likely due to the differences between the device temperatures at each J. Temperatures Tdof devices with thinner tCo/Ni films tended to be higher during the application of current due to their higher resistivity, which can be seen in the upper panel of Fig.2(b).25,26A second possible cause is the dependence of the carrier spin polarization PonN. A third is interfacial effects such as the Rashba torque and/or spin Hall current. However, since our Co/Ni system has a symmetric stack,these interfacial effects should be so small compared with those in the systems described in Refs. 20and22that we can exclude them as significant factors. Consequently, we esti-mated the dependence of PonT dfor each device on the basis of the adiabatic spin transfer model.27,28When Jis suffi- ciently larger than Jth(the so-called precessional regime29), FIG. 2. (a) Experimental setup for the DW motion measurement. (b) The ttotr(upper panel) and shunt current ratio in the ferromagnetic layer (lower panel) as functions of tCo/Ni. The blue dashed line indicates a linear approxi- mation with zero intercept to the plot points for the region of thicker tCo/Ni. (c) Typical results of DW motion measurement. (d) External field strengths required for the DW to pass through the center of the Hall cross Hdepand (e) threshold current density Jthas a function of tCo/Ni.152410-2 Tanigawa et al. Appl. Phys. Lett. 102, 152410 (2013) Downloaded 22 Apr 2013 to 129.10.107.106. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsthe DW velocity is expressed by v¼glBPJ/2eMS, where gis theg-factor, lBis the Bohr magneton, and eis the charge of an electron. Hence, the temperature dependence of Pcan be estimated from the MS–Tcurve shown in Fig. 1(b)and the v/Jin Fig. 3(a). Figure 3(b) shows Pas a function of Td. These results were derived through the method for obtaining P2explained in Ref. 14. Here, we adopted measured data at theJof larger than 0.7 /C21012A/m2to derive the v/Jfor esti- mating P. The Tdduring the 30-ns pulsed current injection was calculated from the device resistance at the given Jand the temperature coefficients of device resistance (inset of Fig.3(b)).10,30For all stacks, Pdecreased with increasing Td. More importantly, the Pvalues for N¼2 and 3 were signifi- cantly lower than those for N¼4, 5, and 6. While the behav- ior of Pvs.TdforN¼4, 5, and 6 is similar to that previously reported at Td<500 K,14the decrease in Pbeing insignifi- cant for N¼2 and 3 in comparison with the decreases for N¼4, 5, and 6 was found. This difference could be due to the temperature dependence of spin-dependent conductivitiesr upandrdown, because the resistance in layers with thinner tCo/Ni is more insensitive to temperature, as can be seen in the inset of Fig. 3(b).31The values of rup¼(1þP)/2qand rdown¼(1/C0P)/2q, where qis resistivity, are related to pho- non or magnon scattering.32The end result is that a thinner Co/Ni layer adjacent to a Pt layer will have lower carrierspin polarization. The mechanism presumably responsible for this is spin-scattering events in the Pt layer, and the fre- quency of these events is related to the magnitude of theGilbert damping constant a. Thinner Co/Ni multilayered films having higher values of ahave been demonstrated. 33 The decrease of Pin a thinner tCo/Ni might be caused by the dissipation of angular momentum into the Pt layer.34 Finally, we discuss the dependence of Jthon thickness (seen in Fig. 2(e)). The theoretical value of Jthis given by Jth¼eMScDHK/glBP, where cis the gyromagnetic ratio, D is the DW width parameter, and HKis the hard-axis anisot- ropy field of the DW.27,28The physical constants that maybe dominant in terms of dependence on thickness are D,HK, and P. As described above, theory predicts that Jthwill decrease with the thickness of the ferromagnetic layer, and this tendency is due to the decrease in HKif we assume a sta- ble Bloch-type DW and constant carrier spin polariza- tion.18,19The results of 2-D simulation confirmed that Bloch-type DWs would preferentially be formed and thatDwas insensitive to thickness in our Co/Ni system. Hence, the dependence of J thontCo/Ni shown in Fig. 2(e) could account for the dependence of both HKandPontCo/Ni. That is, the increase in Jthseen for Ngreater than 3 was caused by the increase in HK, while the increase in Jthseen for Nless than 3 was caused by the decrease in P. Similarly, the reason that no DW motion was seen at N¼1 would be due to P being lower than that for N¼2, resulting in a Jthabove the range of Jwe used. This implies that, in ultrathin systems such as Co/Ni/Co15and a single layer of Co,4,13along with asymmetric capping and seed layers containing Pt, interfacialeffects can readily appear due to dilution of the spin transfer torque which drives the DW in the direction of the electron flow. In summary, we have investigated the dependence on thickness of the threshold current density and the DW veloc- ity in a perpendicularly magnetized [Co/Ni] Nmultilayered wire having a symmetric structure. A local minimum of the threshold current density for driving the DW with respect to the film thickness and a decrease in the velocity of the DWmotion with decreasing thickness of the ferromagnetic layer were seen. Estimation of the carrier spin polarization from the measurement of the DW velocity suggested that thinnerCo/Ni layers adjacent to the Pt layers had lower carrier spin polarization. We believe that this conclusion will provide a useful clue in clarifying the mechanism behind the DWmotion in systems with an ultrathin ferromagnetic layer and asymmetric capping and seed layers. We thank Professor D. Chiba and Professor T. Ono for fruitful discussions. A portion of this research was supported by the Japan Society for the Promotion of Science (JSPS) through its “Funding Program for World-Leading InnovationR&D on Science and Technology (FIRST Program).” 1L. Berger, J. Appl. Phys. 55, 1954 (1984). 2S. Fukami, T. Suzuki, K. Nagahara, N. Ohshima, Y. Ozaki, S. Saito, R. Nebashi, N. Sakimura, H. Honjo, K. Mori, C. Igarashi, S. Miura, N. Ishiwata, and T. Sugibayashi, Dig. Tech. Pap. – Symp. VLSI Technol. 2009 , 230. 3D. Ravelosona, S. Mangin, J. A. Katine, E. E. Fullerton, and B. D. Terris, Appl. Phys. Lett. 90, 072508 (2007). 4T. A. Moore, I. M. Miron, G. Gaudin, G. Serret, S. Auffret, B. Rodmacq, A. Schuhl, S. Pizzini, J. Vogel, and M. Bonfim, Appl. Phys. Lett. 93, 262504 (2008). 5H. Tanigawa, T. Koyama, G. Yamada, D. Chiba, S. Kasai, S. Fukami,T. Suzuki, N. Ohshima, N. Ishiwata, Y. Nakatani, and T. Ono, Appl. Phys. Express 2, 053002 (2009). 6M. Cormier, A. Mougin, J. Ferr /C19e, A. Thiaville, N. Charpentier, F. Pi /C19echon, R. Weil, V. Baltz, and B. Rodmacq, Phys. Rev. B 81, 024407 (2010). 7C. Burrowes, A. P. Mihai, D. Ravelosona, J.-V. Kim, C. Chappert, L. Vila, A. Marty, Y. Samson, F. Garcia-Sanchez, L. D. Buda-Prejbeanu, I. Tudosa, E. E. Fullerton, and J. P. Attan /C19e,Nat. Phys. 6, 17 (2010). 8L. San Emeterio Alvarez, K.-Y. Wang, S. Lepadatu, S. Landi, S. 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1.5121157.pdf

Appl. Phys. Lett. 115, 142406 (2019); https://doi.org/10.1063/1.5121157 115, 142406 © 2019 Author(s).Write-error rate of nanoscale magnetic tunnel junctions in the precessional regime Cite as: Appl. Phys. Lett. 115, 142406 (2019); https://doi.org/10.1063/1.5121157 Submitted: 22 July 2019 . Accepted: 19 September 2019 . Published Online: 01 October 2019 Takaharu Saino , Shun Kanai , Motoya Shinozaki , Butsurin Jinnai , Hideo Sato , Shunsuke Fukami , and Hideo Ohno ARTICLES YOU MAY BE INTERESTED IN Spin-orbit torque induced magnetization switching in ferrimagnetic Heusler alloy D0 22- Mn3Ga with large perpendicular magnetic anisotropy Applied Physics Letters 115, 142405 (2019); https://doi.org/10.1063/1.5125675 Direct observation of terahertz emission from ultrafast spin dynamics in thick ferromagnetic films Applied Physics Letters 115, 142404 (2019); https://doi.org/10.1063/1.5087236 Interface engineering towards enhanced exchange interaction between Fe and FeO in Fe/ MgO/FeO epitaxial heterostructures Applied Physics Letters 115, 141603 (2019); https://doi.org/10.1063/1.5112093Write-error rate of nanoscale magnetic tunnel junctions in the precessional regime Cite as: Appl. Phys. Lett. 115, 142406 (2019); doi: 10.1063/1.5121157 Submitted: 22 July 2019 .Accepted: 19 September 2019 . Published Online: 1 October 2019 Takaharu Saino,1Shun Kanai,1,2,a) Motoya Shinozaki,1 Butsurin Jinnai,3 Hideo Sato,1,2,4,5 Shunsuke Fukami,1,2,3,4,5 and Hideo Ohno1,2,3,4,5 AFFILIATIONS 1Laboratory for Nanoelectronics and Spintronics, Research Institute of Electrical Communication, Tohoku University, Sendai 980-8577, Japan 2Center for Spintronics Research Network, Tohoku University, Sendai 980-8577, Japan 3WPI-Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 4Center for Innovative Integrated Electronic Systems, Tohoku University, Sendai 980-0845, Japan 5Center for Science and Innovation in Spintronics, Tohoku University, Sendai 980-8577, Japan a)Author to whom correspondence should be addressed: sct273@riec.tohoku.ac.jp .Tel.:þ81-22-217-5555. Fax:þ81-22-217-5555. ABSTRACT We investigate the write-error rate (WER) of spin-transfer torque (STT)-induced switching in nanoscale magnetic tunnel junctions (MTJs) for various pulse durations down to 3 ns. While the pulse duration dependence of switching current density shows a typicalbehavior of the precessional regime, WER vs current density is not described by an analytical solution known for the precessionalregime. The measurement of WER as a function of magnetic field suggests that the WER is characterized by an effective damping con- stant, which is significantly larger than the value determined by ferromagnetic resonance. The current density dependence of WER is well reproduced by a macrospin model with thermal fluctuation using the effective damping constant. The obtained finding implies alarger relaxation rate and/or thermal agitation during STT switching, offering a previously unknown insight toward high-reliabilitymemory applications. Published under license by AIP Publishing. https://doi.org/10.1063/1.5121157 Spin-transfer torque (STT)-induced magnetization switching in nanoscale magnetic tunnel junctions (MTJs) 1–4is a key ingredient for magnetoresistive random access memories (MRAMs), which havestarted to be commercialized recently. While the first generation STT-MRAMs will replace a part of embedded flash memories, much effortis now devoted to the application of STT-MRAMs as cache memories,where the write-error rate (WER) in the high-speed ( /C24nanoseconds) regime is one of the most critical indices. The WER is also an interest- ing subject in fundamental research since it relates to the dynamics of collective spin systems under an interplay of energy relaxation, ther-mal fluctuation, and STT. To date, a number of studies on WER havebeen reported; analytical formulation and numerical calculation werecarried out, 5–8while WER was measured as a performance index of integrated MTJs in test chips.9–12 It is known that there are two regimes of STT switching depend- ing on the time scale: thermally activated regime and precessionalregime. 13For current pulses with duration slonger than several tensof nanoseconds, the switching probability, or WER, obeys the Arrhenius law against an effective energy barrier modified by the external magnetic field and/or STT, falling into the thermally activated regime, where critical current linearly increases with decreasing log s.7 On the other hand, in the precessional regime for sshorter than several nanoseconds, switching/nonswitching is determined by the total amount of transferred angular momentum, leading to a linearrelationship of the switching current with 1/ s. The analytical solution of WER in the precessional regime is derived by considering the dis- persion of the initial magnetization angle due to the thermal fluctua- tion (described later for more details). However, it has not been clarified how accurately this solution describes the experimental results and what factors fill the gap between the model and experiment if any. In this paper, we measure WER of nanoscale MTJs by nanosecondpulses under various magnetic fields and discuss the mechanism of WER through a comparison of experimental results with the analytical solution and numerical simulation. Appl. Phys. Lett. 115, 142406 (2019); doi: 10.1063/1.5121157 115, 142406-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplA standard stack structure for high performance MTJs, Ta (5 nm)/ Pt (5 nm)/[Co (0.4 nm)/Pt (0.4 nm)] /C26/Co (0.4 nm)/Ru (0.4 nm)/[Co (0.4 nm)/Pt (0.4 nm)] /C22/ C o( 0 . 4 n m ) / T a( 0 . 2 n m ) / C o 0.19Fe0.56B0.25 (1 nm)/MgO/Co 0.19Fe0.56B0.25 (1.6 nm)/Ta (0.2 nm)/Co 0.19Fe0.56B0.25 (1 nm)/MgO/Ru (5 nm)/Ta (50 nm), is fabricated on a sapphire sub- strate by dc/rf magnetron sputtering. The bottom CoFeB coupled with a synthetic ferrimagnetic structure through Ta (0.2 nm) corresponds to the reference layer, whereas CoFeB (1.6 nm)/Ta (0.2 nm)/CoFeB (1 nm) corresponds to the free layers with a double MgO interface.14Both the reference and free layers possess a perpendicular easy axis. The stack is processed into MTJs with electron-beam lithography, reactive-ion etch- ing, and Ar-ion milling. We fabricate MTJ devices with a coplanarw a v e g u i d em a d eo fC r( 5n m ) / A u( 1 0 0n m )o nas a p p h i r es u b s t r a t ef o r the high-frequency measurement. After processing, MTJs are annealed at 300 /C14C for 1 h in vacuum under a perpendicular magnetic field of 0.4 T. The resistance area product RAis determined to be 2.9 Xlm2 from a linear relationship between R/C01andAf o rM T J sw i t hd i a m e t e r D>37 nm, where Ais measured using a scanning electron microscope. The effective diameter of each MTJ is electrically determined from the above RAand measured R. In this paper, we focus on an MTJ with D¼24 nm, where WER for 50-ns pulses is confirmed to be well described by an analytical solution of a thermally activated regime witha macrospin approximation (see the supplementary material ). The WER is measured by using an electrical circuit shown in Fig. 1(a) . Positive voltage is defined as a direction where electrons flow from the free layer to the reference layer. We apply a 100- ls-long waveform 1.6 /C210 4times using an arbitrary waveform generator (AWG), where the unit waveform consists of initialization, write, and read pulses as shown in Fig. 1(b) . The amplitude and duration of the write pulses ( Vwrite,s) are varied, whereas those for initialization and reading are fixed at (420 mV, 6 ls) and ( /C090 mV, 60 ns), respectively. A high-speed oscilloscope records the transmitted voltage for each read pulse of 100- ls-long waveform, i.e., generating a histogram of 1.6/C2104results. Examples of histograms for a series of current densi- tiesJare shown in Figs. 1(c)–1(f) , indicating an increase in switching probability from conductive parallel (P) to less conductive antiparallel(AP) states with increasing J. The magnitude of Jis determined from the transmitted voltage V T,write for write voltage: J¼VT,write /AZ0, where Z0i st h ec h a r a c t e r i s t i ci m p e d a n c e5 0 X. WER is calculated by the number of unswitched events divided by total trials. Standard devi- ations of the transmitted voltage VT,read for read voltage for P and APstates are almost identical, and the ratio of the peak-to-peak distance to the deviations is /C249 ,w h i c hi sl a r g ee n o u g ht od i s c u s sW E Rd o w nt o /C243/C210/C06. The entire measurement including the creation of the his- togram for one set of conditions takes three minutes, allowing highthroughput evaluation of WER. WER with different Jfors¼3–20 ns is shown in Fig. 2(a) .T h e increase in sorJmonotonically reduces WER as expected. In the pre- cessional regime, the total amount of transferred angular momentum governs the switching as described earlier and the analytical solutionof WER is derived by considering thermal fluctuation at the initialstate as 7,12 WER ¼1/C0exp/C04Dexp/C02sacl0Heff KJ JC0/C0Hz Heff K/C01/C18/C19 /C20/C21 /C26/C27 ;(1) where Dis the thermal stability factor, ais the Gilbert damping con- stant, cis the gyromagnetic ratio, l0is the permeability of vacuum, HKeffis the effective perpendicular anisotropy field, JC0is the intrinsic critical current density, and Hzis the external perpendicular magnetic field. For a constant WER condition, the switching current density Jth under Hz¼0 is expressed as Jth¼/C01 sJC0 2acl0Heff Klog/C01 4Dlog WERðÞ/C18/C19 þJC0: (2)FIG. 1. Schematic diagram of (a) measurement setup and (b) applied pulse voltage waveform. (c)–(f) Histogram of transmitted voltage VT,read for 1.6 /C2104trials with different current densities J. FIG. 2. (a) Write-error rate (WER) as a function of write-current density Jfor various pulse durations s. (b) 1/ sdependence of switching current density Jth. (c) WER as a func- tion of Jwiths¼3 ns. The plot is obtained by experiment. The blue and gray curves are fittings with Eqs. (1)and (3), respectively. The inset shows the same result shown by the logarithmic scale.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 115, 142406 (2019); doi: 10.1063/1.5121157 115, 142406-2 Published under license by AIP PublishingEquation (2)indicates a linear relationship between Jthand 1/ s,w i t h an intercept of JC0.Figure 2(b) shows Jthfor WER ¼0.5 as a function of 1/s.F o r s<15 ns (1/ s>0.07 ns/C01),Jthlinearly increases with 1/ s, in accordance with the analytical model. Next, we compare the model of WER with experiment. The squares in Fig. 2(c) show measured WER as a function of Jfors¼3 ns, whereas the solid (blue) line denotes the fitted curve using Eq. (1), where parameters were adjusted to minimize error on the logarithmic scale (inset). As can be seen in the figure, the fit does not reproduce the experimental results well. We also note that while Eq. (1)indicates a linear relationship between log(WER) and Jat WER /C281, the experimental results show a nonlinear behavior even at WER /C2410/C03. To quantitatively evaluate the degree of nonlinearity, we introduce the following phenomenologi-cal equation to approximate the experimental result: log WERðÞ ¼0 J<J 1 ðÞ /C0CJ/C0J1ðÞmJ>J1 ðÞ;( (3) where exponent mdenotes a nonlinearity index in WER vs J(m¼1 corresponds to a linear relationship), Cis a constant, and J1is the threshold current density. The result is shown with the dashed (gray) curve in Fig. 2(c) ,w h e r e m¼1.76 and J1¼20.9 MA/cm2are obtained by minimizing error on the logarithmic scale for WER >10/C04. Figure 3 shows the sdependence of m(black circle), where mincreases with increasing s. We also plot mobtained by fitting Eqs. (3)to(1)at WER >10/C04by a dashed line, where m¼1.12. The gap between the black circles and dashed line indicates that the analytical model is not sufficient for describing WER in this regime. To probe further, we conduct a numerical simulation based on a macrospin approximation, including the effect of thermal agitation during switching, which is not taken into account in the analyticalmodel. We use the Landau-Lifshitz-Gilbert equation for the polar magnetization angle h, dh dt¼/C0 acl0Heff KcoshþsSTT/C0acl0Hzno sinhþcl0hT;h;(4) where tis the time, hT,h(t) is the thermal field with the Gaussian distribu- tion satisfying hhT,h(t)i¼0a n d h(hT,h(t))2i¼2akBT/cl0MStfreeADt,15 sSTTis the STT amplitude given by c/C22hl0Jg(h)/MStfree,/C22his the Dirac con- stant, g(h) is the spin transfer efficiency determined from the tunnel magnetoresistance (TMR) ratio,1MStfreeis the magnetic moment perunit area (3.3 Tnm, determined by vibrating sample magnetometry), kB is the Boltzmann constant, Dtis the time step of simulation, and Tis the temperature 300 K. l0HKeffis determined to be 350 mT from a switching measurement for MTJs with similar D.hatt¼0 is set to follow the Gaussian distribution with hhi¼0a n d hh2i¼1/{2D(1þHz/HKeff)} (see thesupplementary material ). The first term in Eq. (4)represents the competition of STT with the torque induced by anisotropy and externalfields. The second term represents the thermal effect, which increases withaaccording to fluctuation-dissipation theorem. 15WER is calculated by computing time evolution of the magnetization vector 2 /C2104times for each sSTT. Each of the initial magnetization directions is obtained by calculating the magnetization trajectory for 20 ns under zero Jstarting from the energy equilibrium direction. In order to compare the results with experiment, we determine musing Eq. (3)while varying sSTT.T h e red squares in Fig. 3 are the results using a¼0.007, which are obtained from a homodyne-detected ferromagnetic-resonance (FMR) measure-ment for an MTJ with the same free-layer structure. 16It is known that the linewidth of FMR for nanoscale MTJ gives a larger value than that of the blanket film due to several extrinsic effects such as the two-magnonscattering and inhomogeneity of anisotropy. 17–19The value a¼0.007 is obtained after excluding these factors by using the perpendicular magne- tization configuration and thus represents an intrinsic a.20,21The results fora¼0.02, 0.04, and 0.06 shown by the circle, triangle, and reversed tri- angle, respectively, are also included. As can be seen, a¼0.04 and 0.06 reproduce the experimental trend well, suggesting that the effective a larger than that of the intrinsic one ( ¼0.007) governs the WER. Note that there are two effects of the damping onto the WER under STT.First, damping changes WER by changing torque against the STT duringswitching irrespective of thermal agitation, as is expressed in Eqs. (1) and(4). Second, with STT, the magnetization distribution can be differ- ent from Boltzmann distribution because the system is not in an equilib- rium state. The thermal field is still proportional to the square root ofthe damping constant; 15thus, the WER depends on the damping constant. From Eq. (1), the derivative of log(WER) with respect to Hzin the precessional regime is expressed as dflog WERðÞ g dHz¼2sacl0 log 10x expxðÞ/C01; (5) where x¼4Dexp{–2 sacl 0HKeff(J/JC0–Hz/HKeff/C01)}. In our experi- mental condition at WER <10/C02,x/C281 is satisfied and the right-hand side of Eq. (5)converges to –2 sacl 0/log10, allowing deter- mination of effective acharacterizing WER. Figure 4(a) shows WER vsJat/C030 mT <l0Hz<30 mT and s¼5 ns. The Hzdependence of WER with various Jis shown in Fig. 4(b) . We calculate the same dependence using macrospin simulation including thermal agitation.The simulation results with a¼0.007 are shown in Fig. 4(c) .T h e slopes that represent the effective damping are different between the experiment and simulation. Figure 4(d) shows the experimental result (closed circles) of the slope d{log(WER)}/d H zatHz¼0a saf u n c t i o n of log(WER) together with the simulation results (open symbols) forvarious a.The experimental result is again not reproduced by a¼0.007 but well by a¼0.06 using Eq. (5), consistent with what we found in Fig. 3 . The experiment being reproduced by almost an order of magnitude greater aappears to indicate additional relaxation path- ways of magnetization dynamics, because the thermal torque increaseswith aaccording to the fluctuation-dissipation theorem, 15which mayFIG. 3. The nonlinearity index mas a function of pulse duration s. Experimentally obtained data are plotted by the closed circle. The dashed line and open symbolsare obtained by Eq. (1)and macrospin simulation with different sand damping con- stants afitted by Eq. (3), respectively.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 115, 142406 (2019); doi: 10.1063/1.5121157 115, 142406-3 Published under license by AIP Publishinglead to thermally activated regimelike nonlinear behavior. Note that the derived “effective damping” is a specific damping that character-izes WER. In this regard, the effective damping discussed here couldbe different from another effective damping that characterizes the fig-ure of merit D/I c0and intrinsic damping determined from FMR.16 One of the possible factors for enhanced relaxation is phase dissipa- tion, which is enhanced with inhomogeneity in the device; e.g., atcrystal defect through two-magnon scattering, 19a tt h ed e v i c ee d g e through change of magnetic properties during the process,22and at interface of the stack.23Enhanced dissipation could result in larger apparent damping through the thermal effect and effective energyrelaxation. 24,25We also note that several studies revealed that the domain wall propagation is a dominant switching mechanism inMTJs with a diameter of down to 40 nm. 26–28The domain wall width in our device is derived to be 30 nm, which is larger than the diameterof MTJ, suggesting that the macrospin picture should reasonablydescribe our experiment. WER in the thermally activated regime isconfirmed to be well described by the macrospin model as shown inthesupplementary material , while, interestingly, WER in the preces- sional regime is not. This incoherency can also be a factor for the sig-nificantly large effective damping as the special incoherency shouldlead to an energy dissipation. From the application viewpoint, the increase in min principle improves WER. Our results suggest that even with the fast pulse duration of /C24ns, the thermal effect reduces switching error than what is expected from the analytical solution. Simultaneously, however, increasing mleads to read disturbance, an unintentional switching by a read current pulse. Thus, the comprehen-sive understanding of physics, especially the effective damping, charac- terizing the STT switching probability in nanoscale MTJs is important to understand and design highly reliable high-speed STT-MRAMs. In summary, we investigate the write-error rate (WER) in nano- scale perpendicular magnetic tunnel junctions (MTJs) with a diameterof 24 nm for various pulse durations sdown to 3 ns. We measure 1.6/C210 4switching events using an arbitrary waveform generator and high-speed oscilloscope. WER as a function of current density J,s,a n d magnetic field Hzis compared with the analytical solution and macro- spin simulation. While the switching current density Jthshows a linear dependence on 1/ sfors<15 ns, which is a typical characteristic of the precessional regime, log(WER) vs Jis not reproduced by the corre- sponding analytical solution. The measurement of WER under Hz reveals that WER is characterized by an apparent damping constant a ’0.06, with which the experimental results are well described. The larger value of apparent damping than the intrinsic one suggests anenhanced relaxation rate and/or thermal effect. See the supplementary material for the WER in the thermally activated regime and the thermal field in the LLG equation. The authors thank S. Kubota, N. Ohshima, T. Funatsu, J. Igarashi, Y. Takeuchi, T. Hirata, I. Morita, H. Iwanuma, and K.Goto for discussion and technical support. This work was supported in part by the ImPACT Program of CSTI, JST-OPERA Program Grant No. JPMJOP1611, JSPS Kakenhi No. 19H05622,Core-to-Core Program of JSPS, and Cooperative Research Projectsof RIEC. REFERENCES 1J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). 2L. Berger, Phys. Rev. B 54, 9353 (1996). 3S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan, M. Endo, S. Kanai, J. Hayakawa, F. Matsukura, and H. Ohno, Nat. 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1.3592298.pdf

Effect of parallel and antiparallel configuration on magnetic damping in Co/Ag/Co/Gd S. Demirtas, M. B. Salamon, and A. R. Koymen Citation: Journal of Applied Physics 109, 113919 (2011); doi: 10.1063/1.3592298 View online: http://dx.doi.org/10.1063/1.3592298 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/109/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Spin pumping effects for Co/Ag films J. Appl. Phys. 112, 053906 (2012); 10.1063/1.4748165 Magnetic properties of antidots in conventional and spin-reoriented antiferromagnetically coupled layers J. Appl. Phys. 111, 07B921 (2012); 10.1063/1.3679602 Reactive sputtering synthesis of Co – Co O ∕ Ag nanogranular and multilayer films containing core-shell particles J. Appl. Phys. 101, 09E504 (2007); 10.1063/1.2671688 Combination of ultimate magnetization and ultrahigh uniaxial anisotropy in CoFe exchange-coupled multilayers J. Appl. Phys. 97, 10F910 (2005); 10.1063/1.1855171 Correlation between perpendicular exchange bias and magnetic anisotropy in IrMn ∕ [ Co ∕ Pt ] n and [ Pt ∕ Co ] n ∕ IrMn multilayers J. Appl. Phys. 97, 063907 (2005); 10.1063/1.1861964 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.252.67.66 On: Sun, 21 Dec 2014 12:05:28Effect of parallel and antiparallel configuration on magnetic damping in Co/Ag/Co/Gd S. Demirtas,1,2M. B. Salamon,1,a)and A. R. Koymen3 1School of Natural Sciences and Mathematics, University of Texas at Dallas, Richardson, Texas 75080, USA 2Physics Department, Middle East Technical University, Ankara, 06531 Turkey 3Physics Department, University of Texas at Arlington, Arlington, Texas 76019, USA (Received 17 February 2011; accepted 14 April 2011; published online 10 June 2011) When two ferromagnetic layers have a common resonance frequency, the usual spin-pumping broadening may be reduced by dynamic exchange coupling. Utilizing the antiferromagnetic coupling between Co and Gd, we explore the change in ferromagnetic resonance accompanying a spontaneous transition from parallel to antiparallel magnetic alignment of two Co films below the compensationtemperature T comp. Above T comp, the data are consistent with conventional models. However, a rapid doubling of magnetic damping accompanies the realignment, which is reminiscent of resistivity changes in giant magnetoresistance. VC2011 American Institute of Physics . [doi: 10.1063/1.3592298 ] I. INTRODUCTION The discovery of giant magnetoresistance (GMR) by Baibich et al.1has led to important applications in magnetic recording and data storage. Nonetheless, a fundamental understanding of the microscopic mechanism remains a sub-ject of continuing research. 2–11More recently, attention has turned to the possibility of pure spin transport in magnetic multilayers.12Driven by ferromagnetic resonance, pure spin currents in a trilayer structure can flow from films in reso- nance through spacer layers and into adjacent ferromagnetic layers. Such structures have been proposed as single-termi-nal spin batteries. 13 There is, by now, an extensive amount of literature on ferromagnetic resonance (FMR) in ferromagnet-normalmetal-ferromagnet trilayers. 12–18When the two layers reso- nate at sufficiently different frequencies and are not coupled by static Ruderman–Kittel–Kasuya–Yosida exchange, eachlayer exhibits FMR independently. 17When, however, the FMR frequencies (or fields for resonance) are sufficiently close, the two layers resonate collectively in symmetric andantisymmetric modes that have different linewidths and reso- nant frequencies. 19This so-called dynamical coupling15can only occur when the magnetizations of the two layers arenearly collinear; dynamical coupling is absent or reduced when the anisotropy and pinning fields induce nonparallel equilibrium alignments. So far, there has been no exploration of the nature of dynamical coupling on GMR-type changes in alignment, mainly because the fields for resonance tend tofavor collinearity. Exploiting the strong antiferromagnetic coupling between Co and Gd, 20we have fabricated a GMR-like struc- ture that, in an applied field, spontaneously reverses the rela- tive magnetic orientation of the two Co layers as the temperature is reduced. We follow the FMR signal on cool-ing first through the Curie temperature of Gd and then upon reversal from parallel (P) to antiparallel (AP) alignment ofthe two Co layers below the compensation temperature T comp. We consider these results in the context of both spin- pumping12and the so-called breathing-Fermi-surface models of ferromagnetic damping.21–23 II. EXPERIMENT Both bilayer (Co/Gd) and multilayer samples (Co/Ag/ Co/Gd) were prepared at room temperature using a dc mag- netron sputtering system. The base pressure of the depositionchamber was 10 /C09Torr. Ultrahigh purity argon gas was used and the deposition pressure was 3 mTorr. An in situ quartz thickness monitor, calibrated by a stylus profilometer, meas-ures the deposition thicknesses. Samples were sputtered from pure Gd, Co, and Ag targets on Si (100) substrates. The Ag layers, 20 nm in thickness, were used as buffer layers inall samples. The Co(1)/Ag/Co(2)/Gd multilayer was created with a 10 nm nonmagnetic Ag spacer between the two 4 nm thin Co layers, which is thick enough to suppress long rangeexchange interactions. An Ag spacer-layer thickness of 10 nm is less than the spin diffusion length of Ag which is approximately 170 nm. 24A 10 nm Ag cap layer completed the deposition. The Curie temperature TCof the 10 nm Gd thin film is 240 K, somewhat below the bulk value. The dy- namical resonance of the thin films was measured in a stand-ard Bruker 10 GHz electron paramagnetic resonance spectrometer between 77 and 300 K. The operating fre- quency during the experiments was 9.39 GHz and the exter-nal magnetic field was in the plane of the sample surface. The peak- to-peak linewidth was measured from the deriva- tive FMR curve. III. RESULTS AND DISCUSSION We produce the reversal of the Co magnetization through a judicious choice of layer thicknesses such that the magnetic moment of the Gd will exceed that of Co at lowtemperatures. In Fig. 1, we demonstrate this effect via the low-field magnetization of a Ag(10 nm)/Co(4 nm) bilayer on Gd(10 nm). The sample was zero-field cooled to 50 K aftera)Author to whom correspondence should be addressed. Electronic mail: salamon@utdallas.edu. 0021-8979/2011/109(11)/113919/4/$30.00 VC2011 American Institute of Physics 109, 113919-1JOURNAL OF APPLIED PHYSICS 109, 113919 (2011) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.252.67.66 On: Sun, 21 Dec 2014 12:05:28which a field of 1 T was applied to saturate the Gd layer, aligning the antiferromagnetically-coupled Co layer opposite to the field direction. The field was then reduced to 100 Oe and the magnetization tracked upon warming. Since the Gdmagnetization decreases with increasing temperature, the bilayer moment decreases, reaching a minimum at the com- pensation temperature T comp¼170 K, above which point the Co moment aligns with the field, allowing the overall mag- netization to increase as the Gd magnetization decreases to- ward zero at its Curie temperature. The absorption spectrum as a function of the applied magnetic field for the Co(1)/Ag/Co(2)/Gd multilayer at room temperature is shown in Fig. 2. Several field sweeps to 1500 Oe are made to ensure the saturation of the layers that align with the field. Two Lorentzian derivative fits are also shownin Fig. 2to identify two different resonances at two different external fields at the same frequency. The existence of two overlapping, but distinguishable, lines enables us to identifythe narrower line with the symmetric mode and the broader line with the antisymmetric mode in the regime where there is modest dynamical coupling. 19In this case, the width of the symmetric line is approximately that of the intrinsic width of each layer separately, while the antisymmetric line is further broadened by spin damping. Figure 3shows the temperature dependence of the line- width. With decreasing temperature, the two lines broaden slightly down to the Curie temperature of the 10 nm Gd layer(T C¼240 K). Below TConly a single, broader line [from the Co(1) layer, with Co(2) now a spin sink] is observed, whose width continues to gradually increase with decreasing tem-perature to the compensation point, T comp¼170 K. Below Tcomp, the linewidth increases much more strongly with decreasing temperature, exceeding the resonant field below100 K. The resonance properties of dynamically (and statically) coupled ferromagnetic bilayers have been treated in detail byTserkovnyak et al. 19The linearized and generalized Landau- Ginzburg-Gilbert equation is then dm! i dt¼xiz!/C2m! iþxxm! i/C2m! jþaim! i/C2dm! i dt þa0m! i/C2dm! i dt/C0m! j/C2dm! j dt/C19 :/C18 (1) The frequencies, xiandxx, are the resonance frequencies of each layer and the static exchange coupling between them. Each layer has an intrinsic Gilbert damping26parameter, ai, while a0is the effective damping parameter due to spin- pumping. When the resonant frequencies of the two layers are sufficiently close, they tend to lock together into symmet- ric and antisymmetric modes, with the symmetric modeclose to the intrinsic linewidth and the antisymmetric mode FIG. 1. Magnetic moment as a function of temperature for the [Co 4 nm/Gd 10 nm] bilayer. The minimum corresponds to Tcomp. Here, the external mag- netic field is 100 Oe, denoted by an arrow at the top. For T>Tcomp, the larger Co moment aligns with the external field and the smaller Gd moment points in the opposite direction. However, for T<Tcomp, the relative orienta- tions switch, as sketched. At T¼Tcomp, Co and Gd moments tend to cancel, but may align perpendicular to the field with a small canted moment parallel to the field. (See Ref. 20.) FIG. 2. FMR absorption spectra at 9.39 GHz for the [Co 4 nm/Ag 10 nm/Co 4 nm/Gd 10 nm] film at room temperature (black curve). Linewidths were found by making two Lorentzian derivative fits (dotted lines) to the overall absorption spectrum. The resonance fields are at 630 Oe and 466 Oe for the higher and lower field fits, respectively We identify the broader lower field line with the antisymmetric resonance, and the narrower line with the sym- metric mode. FIG. 3. FMR linewidth as a function of temperature for parallel and antipar- allel alignment of Co layers in the [Co 4 nm/Ag10 nm/Co 4 nm/Gd 10 nm]film. Where two signals are observed, circles track the temperature depend- ence of the symmetric line and the full squares track that of the antisymmet- ric mode. Below 240 K, relaxation of the Co(2) magnetization by ordered Gd suppresses dynamic coupling, leaving only the Co(1) resonance signal. The Co(1) linewidth increases sharply below 170 K, consistent with T comp from Fig. 1.113919-2 Demirtas, Salamon, and Koymen J. Appl. Phys. 109, 113919 (2011) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.252.67.66 On: Sun, 21 Dec 2014 12:05:28strongly broadened. Tserkovnyak et al.19explored the effect of static exchange, xx=0, on the two lines. The tendency of the two modes to lock at the same frequency is suppressed,but the dynamic exchange still strongly affects the effective Gilbert parameters when the resonances of the two layers are sufficiently close. In order to consider whether spin-pumping and dynamic exchange can explain our results, we assume that the two Co layers have nearly the same resonant frequencies and that thetwo Co/Ag interfaces are the same. We allow the possibility of static exchange coupling. Above the ordering temperature of the Gd layer, we consider the two resonances in Fig. 2to be the symmetric (narrower) and antisymmetric (broader) resonances, shifted from each other by 150 Oe. Assuming the demagnetization of the two resonances are the same, weestimate x 1/x2/C01/C250.1; the dynamically coupled resonan- ces are, however, shifted from their decoupled values. For the symmetric resonance, the Gilbert parameter is approxi-mately a 1/C25100 Oe/1.16 H r/C250.03 for H r¼3.3 kOe at 9.39 GHz. In the absence of static exchange coupling, there should be no mutual spin pumping because | x1/x2/C01|>a1. With static exchange, however, the spin-pumping regime becomes extended and, with the parameters used by Tser- kovnyak et al.,19the linewidth of the antisymmetric mode is larger by approximately 0.5 a0Hr/C2550 Oe from our results. This gives us an estimated value of a0¼0.03. When the Gd layer orders below TC¼240 K, the intrin- sic damping parameter a2of the Co(2) layer increases25due to relaxation to the nonresonant Gd magnetic moment. This effectively suppresses the dynamic spin coupling, leavingthe width of the symmetric mode as ( a 1þa0)Hr/C25200 Oe, as observed at TC. We assume that the antisymmetric resonance becomes too broad to observe. The continuedincrease in linewidth with decreasing temperature reflects the so-called “conductivitylike” increase of a 1that is intrin- sic to Co.28 Clearly, below the compensation point, when the Co(1) and Co(2) layers are antiparallel, all dynamic coupling would disappear even if a2were to remain small. If there were strong static exchange coupling, the magnetization of the Co(1) layer could be rotated away from the applied field or be broken into domains. However, we took great care tosaturate the magnetization before making each sweep through the resonance, which should have mitigated such sources of inhomogeneous broadening. Furthermore, the do-main formation would tend to move the resonant field to higher values due to the noncollinearity or a reduced demag- netization factor. No such increase in the field for resonanceis observed. Band calculations of Co-normal metal multilayers reveal strong localization of the Co majority band electrons 7when successive layers are aligned antiparallel. Inasmuch as Co is close to being a half-metal, the majority band electrons con- tribute most of the density of states at the Fermi surface ofCo. One consequence, calculated by Binder et al ., 3is that normal-metal impurities at the interface are seven times more effective at scattering in the AP configuration com-pared with P alignment. While the thickness of our Ag spacer may prevent strong localization, we suggest that thechange from P to AP alignment results in a strong increase ofa 0at both interfaces,27resulting in the observed increasing linewidth below the compensation temperature. As one other possibility, we note that the conductivity- like increase in the linewidth from TCtoTcomp suggests that the intrinsic relaxation parameter a1is dominated by the so- called torque-correlation (or breathing-Fermi-surface) proc- esses.29Due to the spin-orbit coupling, the time-dependent magnetization induces a redistribution of occupation num-bers around the Fermi surface. These changes generally lag the instantaneous magnetization, creating currents that exert a damping torque on the magnetization. Gilmore et al . 23 have shown that aBFSðTÞ¼csðTÞ 2l0mX nkCnðkÞjj2/C0@f @e/C18/C19 ; (2) where s(T) is the orbital relaxation time of the conduction electrons, CnðkÞis the torque matrix element from the breathing-Fermi-surface effect, and ð/C0@f=@eÞis the negative derivative of the Fermi function. The sum is over band indi- ces. By artificially changing the Fermi energy in their bandcalculations, Gilmore et al. 23demonstrated that the summa- tion in Eq. (2)follows the density of states at the Fermi sur- face for Co and other ferromagnetic metals. Hence, thestrong tendency, found by Shep et al., 7for the majority elec- trons in Co to change from delocalized to localized, as the relative orientation of Co layers in the Co/Cu multilayerschanges from parallel to antiparallel alignment may increase the local density of states in the AP configuration, thereby increasing the intrinsic Gilbert parameter, a 1. IV. SUMMARY In conclusion, we have fabricated trilayer structures in which the relative orientation of two Co layers spontaneously changes from parallel to antiparallel as the temperature isreduced. Above the ordering temperature of the Gd under- layer, the dynamically coupled moments exhibit two eigenmo- des at slightly different applied fields. In the symmetric mode,the precessing moments are in phase, so that the back-currents cancel interface damping. For the antisymmetric mode, there is no cancellation and the Gilbert parameter is the sum of theintrinsic component of each layer and twice the interfacial contribution. This is described in considerable detail in the review article by Tserkovnyak et al. 19While the behavior is dominated by intrinsic damping, spin-pumping, and changes in dynamic exchange above the compensation temperature, strong broadening of the resonance line is found as the mag-netization of an underlying Gd layer reverses the relative alignment of the two Co films. We argue that the increased width is a consequence of quantum confinement of the major-ity-spin electrons of Co, leading to an enhancement of inter- face scattering (spin-pumping relaxation) and/or an increase in intrinsic relaxation via the torque-correlation process. ACKNOWLEDGMENTS We have benefited from helpful comments from Y. Tserkovnyak. One of us (A.R.K.) wishes to acknowledge the113919-3 Demirtas, Salamon, and Koymen J. Appl. Phys. 109, 113919 (2011) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.252.67.66 On: Sun, 21 Dec 2014 12:05:28support of the Welch Foundation through Grant No. Y-1215. S.D. acknowledges the partial support from Tubitak-Bideb (Turkish Science and Technical Research Council-Scientist Support Program) during the preparation of the manuscript’sfinal form. 1M. N. Baibich, J. M. Broto, A. Fert, N. V. Dau, F. Petroff, P. Etienne, G. Creuzet, A. Friederich, and J. Chazelas, Phys. Rev. Lett. 61, 2472 (1988). 2P. Zahn, J. Binder, I. Mertig, R. Zeller, and P. H. Dederichs, Phys. Rev. Lett. 80, 4309 (1998). 3J. Binder, P. Zahn, and I. Mertig, J. Appl. Phys. 89, 7107 (2001). 4R. E. Camley and J. Barna ´s,Phys. Rev. Lett. 63, 664 (1989). 5P. M. Levy, S. Zhang, and A. Fert, Phys. Rev. Lett. 65, 1643 (1990). 6P. Zahn, I. Mertig, M. Richter, and H. Eschrig, Phys. Rev. Lett. 75, 2996 (1995). 7K. M. Schep, P. J. Kelly, and G. E. W. Bauer, Phys. Rev. Lett. 74, 586 (1995). 8S. F. Lee, Q. Yang, P. Holody, R. Loloee, J. H. Hetherington, S. Mah- mood, B. Ikegami, K. Vigen, L. L. Henry, P. A. Schroeder, W. P. Pratt, Jr., and J. Bass, Phys. Rev. B 52, 15426 (1995). 9K. Xia, P. J. Kelly, Ge. E. W. Bauer, I. Turek, J. Kudrnovsky, and V. Drchal, Phys. Rev. B 63, 064407 (2001). 10K. M. Schep, J. B. A. N. van Hoof, P. J. Kelly, G. E. W. Bauer, and J. E. Inglesfield, Phys. Rev. B 56, 10805 (1997). 11M. D. Stiles and D. R. Penn, Phys. Rev. B 61, 3200 (2000). 12B. Heinrich, G. Woltersdorf, R. Logan, and E. Simanek, J. Appl. Phys. 93, 7545 (2003).13A. Brataas, Y. Tserkovnyak, G. E. W. Bauer, and B. I. Halperin, Phys. Rev. B 66, 060404(R) (2002). 14Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. B 66, 224403 (2002). 15B. Heinrich, G. Woltersdorf, R. Urban, and E. Simanek, J. Appl. Phys. 93, 7545 (2003). 16Y. Tserkovnyak, G. A. Fiete, and B. I. Halperin, Appl. Phys. Lett. 54, 5234 (2004). 17B. Heinrich, Y. Tserkovnyak, G. Woltersdorf, A. Brataas, R. Urban, and G. E. W. Bauer, Phys. Rev. Lett. 90, 187601 (2003). 18R. Urban, G. Woltersdorf, and B. Heinrich, Phys. Rev. Lett. 87, 217204 (2001). 19Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin, Rev. Mod. Phys. 77, 1375 (2005). 20S. Demirtas, M. R. Hossu, R. E. Camley, H. C. Mireles, and A. R. Koy- men, Phys. Rev. B 72, 184433 (2005). 21J. Kunes and V. Kambersky ´,Phys. Rev. B 65, 212411 (2002). 22K. Gilmore, Y. U. Idzerda, and M. D. Stiles, Phys. Rev. Lett. 99, 027204 (2007). 23K. Gilmore, Y. U. Idzerda, and M. D. Stiles, J. Appl. Phys. 103, 07D303 (2008). 24B. Kardasz, O. Mosendz, B. Heinrich, Z. Liu, and M. Freeman, J. Appl. Phys. 103, 07C509 (2008). 25S. Demirtas, R. E. Camley, Z. Celinsky, M. R. Hossu, A. R. Koymen, C. Yu and M. J. Pechan e-print arXiv:1002.4889v1 [cond-mat.mtrl-sci]. 26T. L. Gilbert, Phys. Rev. 100, 1243 (1955); IEEE Trans. Mag. 40, 3443 (2004). 27D. Steiauf and M. Fa ¨hnle, Phys. Rev. B 72, 064450 (2005). 28S. M. Bhagat and P. Lubitz, Phys. Rev. B 10, 179 (1974). 29V. Kambersky, Can. J. Phys. 48, 2906 (1970).113919-4 Demirtas, Salamon, and Koymen J. Appl. Phys. 109, 113919 (2011) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.252.67.66 On: Sun, 21 Dec 2014 12:05:28

1.3204306.pdf

A biomechanical model of cardinal vowel production: Muscle activations and the impact of gravity on tongue positioning Stéphanie Buchaillarda/H20850and Pascal Perrier ICP/ GIPSA-Lab, UMR CNRS 5216, Grenoble INP , 38402 Saint Martin d’Hères, France Yohan Payan TIMC-IMAG, UMR CNRS 5525, Université Joseph Fourier, 38706 La Tronche, France /H20849Received 18 March 2008; revised 21 July 2009; accepted 23 July 2009 /H20850 A three-dimensional /H208493D/H20850biomechanical model of the tongue and the oral cavity, controlled by a functional model of muscle force generation /H20849/H9261-model of the equilibrium point hypothesis /H20850and coupled with an acoustic model, was exploited to study the activation of the tongue and mouth floormuscles during the production of French cardinal vowels. The selection of the motor commands tocontrol the tongue and the mouth floor muscles was based on literature data, such aselectromyographic, electropalatographic, and cineradiographic data. The tongue shapes were alsocompared to data obtained from the speaker used to build the model. 3D modeling offered theopportunity to investigate the role of the transversalis, in particular, its involvement in theproduction of high front vowels. It was found, with this model, to be indirect via reflex mechanismsdue to the activation of surrounding muscles, not voluntary. For vowel /i/, local motor commandvariations for the main tongue muscles revealed a non-negligible modification of the alveolar groovein contradiction to the saturation effect hypothesis, due to the role of the anterior genioglossus.Finally, the impact of subject position /H20849supine or upright /H20850on the production of French cardinal vowels was explored and found to be negligible.©2009 Acoustical Society of America. /H20851DOI: 10.1121/1.3204306 /H20852 PACS number /H20849s/H20850: 43.70.Bk, 43.70.Aj /H20851AL/H20852 Pages: 2033–2051 I. INTRODUCTION Speech movements and acoustic speech signals are the results of the combined influences of communicative linguis-tic goals, perceptual constraints, and physical properties ofthe speech production apparatus. To understand how thesedifferent factors combine and interact with each other re- quires an efficient approach that develops realistic physicalmodels of the speech production and/or speech perceptionsystems. The predictions of these models can then be com-pared with experimental data, and used to infer informationabout parameters or control signals that are not directly mea-surable or the measurement of which is difficult and notcompletely reliable. Such a methodological approach under-lies the present work, in which a biomechanical model of thevocal tract has been used to study muscle control in vowelproduction, its impact on token-to-token variability, and itsconsequences for tongue shape sensitivity to changes in head/H20849supine versus upright /H20850orientation. The findings are inter- preted in the light of our own experimental data and datapublished in the literature. Biomechanical models of the tongue and vocal tract have been in use since the 1960s, and their complexity hasincreased with the acquisition of new knowledge about ana-tomical, neurophysiological, and physical characteristics ofthe tongue, as well as with the vast growth in the computa-tional capacities of computers. All these models have signifi-cantly contributed to the increase in knowledge about tongue behavior and tongue control during speech production, andmore specifically about the relations between muscle recruit-ments and tongue shape or acoustic signal /H20849see, in particular, Perkell, 1996 , using his model presented in Perkell, 1974 ; Kakita et al. , 1985 ;Hashimoto and Suga, 1986 ;Wilhelms- Tricarico, 1995 ;Payan and Perrier, 1997 ;Sanguineti et al. , 1998 ;Dang and Honda, 2004 /H20850. With a more sophisticated three-dimensional /H208493D/H20850vocal tract model, based on non- linear continuum mechanics modeling, and taking into con-sideration a number of recent experimental findings, thisstudy aims at deepening and extending these former worksfor vowel production. The model consists of a 3D biomechanical model of the tongue and the oral cavity, controlled by a functional modelof muscle force generation /H20851/H9261-model of the equilibrium point hypothesis /H20849EPH /H20850/H20852and coupled with an acoustic model. It is a significantly improved version of the model originally de-veloped in GIPSA-Lab by Gérard and colleagues /H20849Gérard et al. , 2003 ,2006 /H20850. The oral cavity model was developed so as to give as realistic a representation as possible of theanatomy and of the mechanical properties of the oral cavity.The original modeling was based on the data of the Visible Human Project, and further adapted to the anatomy of a spe-cific subject. For this subject, different kinds of data /H20851x-ray, computed tomography /H20849CT/H20850images, and acoustic data /H20852were available. The parameters used in this model were either ex-tracted from the literature, derived from experimental data,or adapted from the literature. This modeling study is insepa-rable from a thorough experimental approach. In addition to a/H20850Author to whom correspondence should be addressed. Electronic mail: stephanie.buchaillard@gmail.com J. Acoust. Soc. Am. 126 /H208494/H20850, October 2009 © 2009 Acoustical Society of America 2033 0001-4966/2009/126 /H208494/H20850/2033/19/$25.00 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 138.251.14.35 On: Tue, 23 Dec 2014 02:45:13a careful and accurate account of anatomical, mechanical, and motor control facts, the model implements a number ofhypotheses about the hidden parts of the speech productionsystem. Simulation results, their interpretation, and the cor-responding conclusions aim at opening new paths for furtherexperimental research that could validate or contest theseconclusions. The main characteristics of the model /H20849geometry, me- chanical properties, and model of control /H20850are presented in Sec. II. The model includes improvements in the anatomicaland morphological descriptions and in the strain/stress func-tion, as well as a control model of muscle activation /H20849Sec. II/H20850. The model is first used /H20849Sec. III /H20850in order to characterize the muscle activation patterns associated with the productionof the French cardinal vowels. Starting from these patterns,the relation between internal muscle strain and muscle acti-vations is systematically studied. In Sec. IV, the sensitivity ofthe postural control of the tongue /H20849and hence of the formant frequencies /H20850to changes in motor commands is precisely studied for /i/, which is often described in the literature as avery stable vowel due to specific combinations of muscleactivations. Finally, the impact on tongue positioning ofchanges in gravity orientation is assessed /H20849Sec. V /H20850. Perspec- tives and further developments are discussed in Sec. VI. II. MODELING THE ORAL CAVITY Modeling the oral cavity by a finite element approach requires meshing the structure of interest, specifying its me-chanical properties and defining a motor control scheme.Then, the simulation of movements in response to motorcommands requires solving the body motion equations.These different aspects will be described in this section. The primary goal of our work is the development of a model which allows a better understanding of how motorcontrol and physical aspects combine and interact to deter-mine the characteristics of speech production signals. Hence,a high degree of realism is essential in the design of themodel, not only concerning the geometrical properties butalso the mechanical and control aspects. The model described below is an improved version of the model developed by Gérard and colleagues /H20849Gérard et al. , 2003 ,2006 /H20850. The original model was based on the Visible Human Project® data for a female subject and thework of Wilhelms-Tricarico /H208492000 /H20850. It was then adapted to a specific male subject, PB henceforth. Major differences be-tween the current version and those of Gérard and colleagues/H20849Gérard et al. , 2003 ,2006 /H20850lie in /H208491/H20850the motor control scheme /H20849muscle forces are now computed via the /H9261-model of the EPH /H20850,/H208492/H20850the constitutive law for the tongue tissues /H20851the law inferred by Gérard et al. /H208492005 /H20850from indentation mea- surements of fresh cadaver tissues was modified to match theproperties of living tissues; in addition, the law now dependson the level of muscle activation /H20852,/H208493/H20850the modeling of the hyoid bone /H20849a new scheme was also developed to deal with hyoid bone mobility and to model the infrahyoid and digas-tric muscles /H20850. Modifications were also made to the tongue mesh, the muscle fibers, the bony insertions, and the areas ofcontact between the tongue and the surrounding surfaces,namely, the mandible, the hard palate, and the soft palate. The 3D vocal tract model was also coupled with an acousticmodel. A. Geometrical and anatomical structures A precise description of the tongue anatomy will not be given here. A thorough description, which lies at the root ofthis work, can be found in Takemoto /H208492001 /H20850. The tongue model represents the 3D structure of the tongue of a malesubject /H20849PB/H20850, for whom several sets of data have been col- lected in the laboratory in the past 15 years. This model ismade of a mesh composed of hexahedral elements. The ana-tomical location of the major tongue muscles is specified viasubsets of elements in the mesh. Figure 1shows the imple- mentation of the 11 groups of muscles represented in themodel and known to contribute to speech production. Nineof them exert force on the tongue body itself, while the othertwo, depicted in the last two panels /H20851Figs. 1/H20849j/H20850and1/H20849k/H20850/H20852, are considered to be the major mouth floor muscles. Of course,due to the elastic properties of tongue tissues, each muscle islikely to induce strain in all the parts of the tongue andmouth floor. On rare occasion the muscle shape is somewhatunnatural because the tongue muscles were defined as a sub-set of elements of the global mesh. This is, for example, thecase with the inferior longitudinalis /H20849IL/H20850. However, when activated, the force generated by the IL appeared correct inamplitude and direction. The insertion of the different partsof the geniglossus on the mandible can also appear odd: inhuman beings, GGp emanates from the lower surface of theshort tendon that reduces crowding of the fibers at the man-dibular symphysis by allowing GGp to arise from below andthe radial fibers to arise from above. In the model, the tendonis not represented and the origins of GGp, GGm, and GGa FIG. 1. Mesh representation /H20849gray elements /H20850of lingual and mouth floor muscles as subsets of tongue elements /H20849global mesh /H20850/H20849anterior oblique view /H20850./H20851/H20849a/H20850–/H20849c/H20850/H20852anterior, medium, and posterior part of the genioglossus, /H20849d/H20850 styloglossus, /H20849e/H20850hyoglossus, /H20849f/H20850verticalis, /H20849g/H20850transversalis, /H20849h/H20850inferior lon- gitudinalis, /H20849i/H20850superior longitudinalis, /H20849j/H20850geniohyoid, and /H20849k/H20850mylohyoid. The muscle fibers are represented in red. The yellow squares and the bluedots represent the muscle insertions on the mandible and the hyoid bone,respectively. 2034 J. Acoust. Soc. Am., Vol. 126, No. 4, October 2009 Buchaillard et al. : Modeling study of cardinal vowel production Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 138.251.14.35 On: Tue, 23 Dec 2014 02:45:13are all on the mandibular symphysis. This results in a some- what too large region of insertion on the mandible. Only arefined mesh structure would allow a better muscle definitionin this area. It is generally accepted that a muscle can possibly be divided into a number of functionally independent parts. Fortongue muscles this possibility exists, but little work hasbeen done in the past concerning this issue. Some proposalswere the results of ad hoc choices made in order to explain measured two-dimensional /H208492D/H20850or 3D tongue shapes /H20849e.g., the most recent proposal for the styloglossus in Fang et al. , 2008 /H20850. Some more physiologically based studies used elec- tromyographic /H20849EMG /H20850signals, generally assuming that these signals reflect the underlying motor control. Among thesestudies, the one carried out by Miyawaki et al. /H208491975 /H20850 showed evidence for different activities in different parts ofthe genioglossus. However, EMG activity is the result of acombination of efferent and afferent influences and it cannotbe seen as a direct image of the underlying control. In addi-tion, as emphasized by Miyawaki et al. /H208491975 /H20850, if subdivi- sions exist in a muscle, we do not know in what manner theyare voluntarily controlled /H20849p. 101 /H20850. We believe that the only reliable way to address this issue would be to look at themotor unit distribution within tongue muscles. To our knowl-edge, we lack information on the localization of motor unitterritories in human tongue muscles. One way to know moreabout it could be to study the architecture of the muscles,with the underlying hypothesis that structurally separatedmuscle parts could be innervated by independent motorunits. Slaughter et al. /H208492005 /H20850carried out such a study for the human superior longitudinalis /H20849SL/H20850, and they found that this muscle consists of a number of in-series muscle bundles thatare distributed along the front-back direction. However, theycould not provide clear evidence for the fact that thesemuscle bundles are innervated by independent motor units.In the absence of convincing physiological evidence, and inorder to limit the complexity of the model, only the genio-glossus, for which a consensus seems to exist, was subdi-vided: three independent parts called the GGa /H20849anterior ge- nioglossus /H20850, the GGm /H20849medium genioglossus /H20850, and the GGp /H20849posterior genioglossus /H20850were thus defined. To mesh the hard and soft structures forming the oral cavity, data of different kinds such as CT scans, MRI data,and x-ray data, all collected for PB, were exploited. In addi-tion to the tongue and mouth floor meshes, the model /H20849Fig.2/H20850 includes a surface representation of the mandible, the softpalate, the hard palate, and the pharyngeal and laryngealwalls as well as a volumetric mesh /H20849tetrahedral elements /H20850of the hyoid bone. A set of six pairs of springs /H20849right and left sides /H20850, emerging from the hyoid bone, are used to represent the elastic links between this mobile bone and fixed bonystructures associated with the anterior and posterior belly ofthe digastric, infrahyoid muscles /H20849sternohyoid, omohyoid, and thyrohyoid muscles /H20850, as well as the hyo-epiglottic liga- ments. The relative positions of the different articulators were carefully adjusted so as to represent well PB’s morphology ina seated position and at rest, just as they are described bylateral x-ray views of PB’s oral cavity. The final tongueshape in the midsagittal plane at rest was also adapted so as to match the corresponding x-ray view. This induced somegeometrical changes to the original shape proposed in Gérard et al. /H208492006 /H20850, because the MRI data used for the original design of that model corresponded to the subject in the su-pine position; gravity was then shown in that case to influ-ence tongue shape. B. Mechanical properties The lingual tissues were modeled with a non-linear hyper-elastic constitutive law, more precisely a second orderYeoh constitutive law /H20849Gérard et al. , 2005 ,2006 /H20850. Two dif- ferent constitutive equations were introduced: one describesthe passive behavior of tongue tissues and the other onemodels the strain/stress relation for active muscle tissues asan increasing function of muscle activation. For a particularmesh element, the passive or the active constitutive law isused according to whether this element belongs to a passiveor to an active region /H20851i.e., a region made of activated muscle /H20849s/H20850/H20852. The passive constitutive law was directly derived from the non-linear law proposed by Gérard et al. /H208492005 /H20850, which was derived from measurements on a fresh cadaver.However, since the stiffness of tissues measured shortly afterdeath is known to be lower than that measured in in vivo tissues, the constitutive law originally proposed by Gérard et al. /H208492005 /H20850was modified. To our knowledge, one of the most relevant in vivo mea- surements of human muscle stiffness is the one carried outbyDuck /H208491990 /H20850, who proposed a value of 6.2 kPa for the Young’s modulus for a human muscle at rest and a maximumvalue of 110 kPa for the same muscle once contracted. TheYoung’s modulus measured by Gérard et al. /H208492005 /H20850on a cadaver tongue at low strain is 1.15 kPa, which is signifi-cantly smaller than Duck’s /H208491990 /H20850in vivo measurements. FIG. 2. Oblique anterior view of the 3D tongue mesh in the whole oral cavity for a rest position /H20849tongue mesh in magenta, mandible in cyan, hyoid bone in yellow, translucent soft palate, pharyngeal and laryngeal walls ingray, infra- and supra-hyoid muscles represented as magenta lines /H20850. J. Acoust. Soc. Am., Vol. 126, No. 4, October 2009 Buchaillard et al. : Modeling study of cardinal vowel production 2035 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 138.251.14.35 On: Tue, 23 Dec 2014 02:45:13This difference is not surprising, since in living subjects a basic muscle tonus exists, even at rest. Hence, it was decidedto multiply both second order Yeoh law coefficients origi-nally proposed by Gérard et al. /H208492005 /H20850by a factor of 5.4, in order to account properly for the Young’s modulus at restmeasured by Duck /H208491990 /H20850. Multiplying both coefficients by the same factor allows preservation of the overall non-linearshape of the Yeoh constitutive law /H20849Fig. 3/H20850. This new law specifies the properties of passive tongue tissues. In order toaccount for the stiffening associated with muscle activationas measured by Duck /H208491990 /H20850, it was decided for the elements belonging to an activated muscle to multiply the coefficientsof the Yeoh constitutive law for passive tissues by a factorthat is a function of muscle activation. Thus, an activation-related constitutive law was defined for the active muscles.The multiplying factors were chosen by taking into accountthe fact that the contributions of the different muscles to theYoung’s modulus of an element combine in an additive man-ner. The basic idea is that an activation of a muscle leads toan increase in its Young’s modulus. Given c 10,randc20,rthe Yeoh parameters for tongue tissues at rest, the parametersc i0/H20849e,t/H20850/H20849i/H33528/H208531,2 /H20854/H20850at time tfor an element ebelonging to the tongue or mouth floor, are given by ci0/H20849e,t/H20850=ci0,r/H208731+ /H20858 muscles mp1/H20849m/H20850/H20875/H20858 fibers f/H33528mA/H20849f,t/H20850p2/H20849f,e/H20850/H20876/H20874, /H208491/H20850 where p1is a positive muscle-dependent factor, A/H20849f,t/H20850is the activation level for the macrofiber fat time t/H20851see Eq. /H208492/H20850 below /H20852, and p2/H20849f,e/H20850is a factor equal to 1 if ebelongs to m and if the fiber fruns along the edges of e, 0 otherwise. The multiplying factor p1was chosen in order to main- tain the stiffness value below 110 kPa, when maximalmuscle activation is reached. Since tongue tissues are considered to be quasi- incompressible, a Poisson coefficient equal to 0.499 wasused. Furthermore, tongue tissue density was set to1040 kg m −3, close to water density. Currently, only the tongue and the hyoid bone /H20849with the springs connecting it to fixed bony structure /H20850are modeled as movable structures and need to be mechanically character-ized. The hyoid bone was considered as a rigid body and itsdensity /H208492000 kg m −3/H20850was estimated based on values pub- lished in the literature /H20849Dang and Honda, 2004 /H20850. The same stiffness coefficient /H20849220 N m−1/H20850was chosen for all the springs connecting the hyoid bone to solid structures; this value enabled us to reproduce displacements of the hyoidbone that were consistent with data published in Boë et al. /H208492006 /H20850. C. Motor control: Implementation of postural control with short latency feedback The motor control scheme implemented is based on the /H9261version of the EPH /H20849Feldman, 1986 /H20850. This theory is known to be controversial in the motor control domain. The maincriticisms are about the fact that this theory claims that thetime variation of motor control variables does not result fromany inverse kinematics or inverse dynamics processes /H20849see, for example, Gomi and Kawato, 1996 orHinder and Milner, 2003 /H20850. However, the defenders of the EPH theory have sys- tematically provided refutations of these criticisms that sup-port the value of the model in research /H20849e.g., in Gribble and Ostry, 1999 orFeldman and Latash, 2005 /H20850. Our own work has also shown that speech motor control based on the EPHgives a good account of complex kinematic patterns with a2D biomechanical model of the vocal tract /H20849Payan and Per- rier, 1997 ;Perrier et al. , 2003 /H20850. From our point of view, this motor control theory seems particularly interesting forspeech production because it provides the framework for adiscrete characterization of continuous physical signals at amotor control level, thanks to the link that can be made be-tween successive equilibrium points and targets; it thus al-lows a connection to be made between the discrete phono-logical units and the physical targets that underlie continuousarticulatory and acoustic signals /H20849Perrier et al. , 1996 /H20850.I na d - dition, the EPH integrates short latency feedback to contrib-ute to the accuracy of speech gesture, which is for us a cru-cial feature for speech production control /H20849Perrier, 2006 /H20850. Hence, the approach used in our previous modeling workwith the 2D biomechanical model of the vocal tract was ex-tended to the 3D model. 1. Adjustment of feedback delay The implementation chosen for the EPH follows the ap- proach proposed by Laboissière et al. /H208491996 /H20850and further de- veloped by Payan and Perrier /H208491997 /H20850. In the model, bundles of fibers are represented by way of macrofibers /H20849specified as ordered lists of mesh nodes along the edges of elements /H20850that represent the main directions of muscle fibers in the differentparts of the tongue. In the current version of the model, aunique activation threshold was defined for each muscle/H20849three for the genioglossus, which was divided into three parts that are assumed to be separately controlled: the ante-rior, posterior, and medium parts /H20850. Every muscle was as- sumed to be controlled independently. Obviously, synergiesand antagonisms exist in tongue muscles. However, there isno evidence in the literature supporting the hypothesis thatthese muscle coordinations are implemented in humans frombirth. It is much more likely that coordinated muscle activa-tions are the result of learning and that they could be task−0.5 −0.3 −0.1 0.1 0.3 0.5−150−120−90−60−30030 En gineerin gstrainEngineering stress (kPa) FIG. 3. /H20849Color online /H20850Stress/strain hyperelastic constitutive law /H20849Yeoh sec- ond order material /H20850for lingual tissues. The dotted curve represents the origi- nal law obtained from fresh cadaver tissues /H20849c10=192 Pa and c20=90 Pa /H20850, the dashed curve represents the law used in the current model for passivetissues /H20849c 10=1037 Pa and c20=486 Pa /H20850, and the solid line represents the law used for the maximal activation /H20849c10=10.37 kPa and c20=4.86 kPa /H20850. 2036 J. Acoust. Soc. Am., Vol. 126, No. 4, October 2009 Buchaillard et al. : Modeling study of cardinal vowel production Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 138.251.14.35 On: Tue, 23 Dec 2014 02:45:13specific. Our modeling approach is in line with this state- ment. The design of our biomechanical model gives the larg-est possible number of degrees of freedom to the system tobe controlled and does not impose a priori hypotheses thatcould bias our study. It allows future work on the emergenceof muscle coordinations through task specific learning. Foreach muscle, the motor command /H9261 muscle was determined for the longest macrofibers lmax; the/H9261value for each macrofiber of the same muscle was then determined by simply multiply-ing the /H9261 muscle value by the ratio of the macrofiber length at rest over lmax. For a given macrofiber, the muscle activation Atakes into account the difference between the macrofiber lengthand the motor command /H9261, as well as the lengthening/ shortening rate. A stretch reflex delay d, which corresponds to the propa- gation delay for the electrical signals to travel along the re-flex arc plus the synaptic time and the integration time ofthese signals at the interneurons, is taken into account forfiber length and velocity intervening in the computation of A. In their model of the mandible, Laboissière et al. /H208491996 /H20850 proposed a delay of 10 ms, and in their tongue/jaw modelSanguineti et al. /H208491998 /H20850suggested a delay of 15 ms. In the present model, dwas set to 17 ms, based on the data of Ito et al. /H208492004 /H20850. Simulations conducted for dranging from 5 to 20 ms showed that this value had a limited impact ontongue motion; the trajectory, peak velocity, acceleration, orforce levels were altered, but in a limited range so that thechoice of this value did not seem to be critical within thisrange of variation. The sensitivity of the activation to the lengthening/shortening rate l˙is modulated by a damping co- efficient /H9262, considered for the sake of simplicity as constant and identical for all the muscles. /H9262was chosen to be equal to 0.01 s to ensure the stability of the system, following numer-ous simulations A/H20849t/H20850=/H20851l/H20849t−d/H20850−/H9261/H20849t/H20850+ /H9262l˙/H20849t−d/H20850/H20852+. /H208492/H20850 Muscle activation is associated with the firing of the moto- neurons /H20849henceforth MNs /H20850. Hence Ais either positive or zero /H20849ifAis mathematically negative, it is set to zero /H20850. A zero value corresponds to the MN fire threshold; beyond thisthreshold, the MN depolarization becomes possible: thehigher the activation A, the higher the firing frequency of MNs. As long as the activation Ais zero, no force is gener- ated. Force varies as an exponential function of the activa-tion /H20849see below /H20850. 2. Feedback gain: A key value for postural control stability Active muscle force M˜is given as a function of the activation A/H20849t/H20850by the following equation:M˜/H20849t/H20850= max /H20851/H9267/H20849expcA/H20849t/H20850−1/H20850,/H9267/H20852, /H208493/H20850 with/H9267a factor related to the muscle capacity of force gen- eration and ca form parameter symbolizing the MN firing gradient. The determination of the parameter /H9267, which modulates the force generation capacity, is based on the assumptionthat, for a fusiform muscle, /H9267is linked in a first approxima- tion to the cross-sectional area of the muscle. The values arebased on the work of Payan and Perrier /H208491997 /H20850for the tongue muscles, except for the transversalis, which was non-existentin a 2D tongue model, and with some adaptations for theverticalis, the implementation of which was slightly differ-ent. For the mouth floor muscles, /H9267values were estimated from the data of van Eijden et al. , 1997 , and were measured on the model for the transversalis. This muscle force capacity/H20849Table I/H20850was distributed among the different macrofibers proportionally to the volume of the surrounding elements.Given a fiber fbelonging to a muscle m, its capacity of force generation /H9267fibis such that /H9267fib/H20849f/H20850=/H9267/H20849m/H20850/H20858eV/H20849e/H20850/H11003p/H20849e,f/H20850 S, /H208494/H20850 where eis an element belonging to m,V/H20849e/H20850is the volume of e,p/H20849e,f/H20850is a parameter equal to 1 if fis located inside the muscle, 0.5 on a muscle face /H20849exterior surface of a muscle excluding muscle corners /H20850, and 0.25 on a muscle edge /H20849ex- terior surface of a mesh, corners only /H20850.Sis a normalization term, such that the /H9267fibvalues for the different fibers of m sum up to /H9267/H20849m/H20850. Parameter cis an important factor for stability issues since it determines how feedback information included in theactivation influences the level of force. Original values for c found in the literature /H20849c=112 m −1,Laboissière et al. , 1996 /H20850 brought about dramatic changes in the muscular activationlevel for a small variation in the muscle length. This gener-ated mechanical instabilities. Therefore, parameter cwas de- creased. After several trials, cwas fixed to 40 m −1. This value is not the only one that ensured a stable mechanicalbehavior of the model. A large range of values was possible.The value 40 m −1was chosen because it provides a fair com- promise between the level of reflex activation and stability/H20849Buchaillard et al. , 2006 /H20850. The influence of muscle lengthening/shortening velocity on the force developed is also included. The model accountsfor the sliding filaments theory /H20849Huxley, 1957 /H20850by calculating the total muscle force Fwith the following equation /H20849Labois- sière et al. , 1996 /H20850:TABLE I. Cross-sectional areas and corresponding force generation capacities /H20849/H9267/H20850. GGa GGm GGp Sty HG Vert Trans IL SL GH MH Area /H20849mm2/H20850 82 55 168 109 295 91 227 41 86 80 177 /H20849/H9267/H20850/H20849N/H20850 18 12 37 24 65 20 50 9 19 17.5 39 J. Acoust. Soc. Am., Vol. 126, No. 4, October 2009 Buchaillard et al. : Modeling study of cardinal vowel production 2037 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 138.251.14.35 On: Tue, 23 Dec 2014 02:45:13F/H20849t/H20850=M˜/H20849t/H20850/H20873f1+f2arctan/H20873f3+f4l˙/H20849t/H20850 r/H20874+f5l˙/H20849t/H20850 r/H20874, /H208495/H20850 where l˙is the lengthening/shortening velocity and ris the muscle length at rest. The parameters used are based on thework of Payan and Perrier /H208491997 /H20850for rapid muscles, but are slightly different: f 1=0.7109, f2=0.712, f3=0.43, f4 =0.4444 s, and f5=0.0329 s. D. Lagrangian equation of motion and boundary conditions The Lagrangian equation of motion that governs the dy- namic response of the finite element system is given by Mq¨/H6023+Cq˙/H6023+Kq/H6023=F/H6023, /H208496/H20850 where q/H6023is the nodal displacements vector, q˙/H6023andq¨/H6023are its first and second derivatives, Mis the mass matrix, Cis the damping matrix, Kis the stiffness matrix, and F/H6023is the load vector /H20849the reader can refer to Bathe, 1995 for a detailed description of the finite element method /H20850. A Rayleigh damping model was chosen for the defini- tion of the damping matrix: C=/H9251M+/H9252K./H9251and/H9252were set to 40 s−1and 0.03 s, respectively, in order to have a damping close to the critical one in the range of modal frequency from3t o1 0H z /H20849Fig.4/H20850. The load vector F /H6023includes the muscle forces computed for every macrofiber /H20851Eq. /H208495/H20850/H20852, the gravity and contact forces between tongue and vocal tract walls. Two kinds of boundary conditions were introduced through the definition of no-displacement constraints tomodel muscular insertions and the management of contacts.Muscle insertions on the bony structures /H20849inner anterior and lateral surface of the mandible and hyoid bone /H20850were imple- mented and they match as well as possible the informationabout PB’s anatomy that was extracted from x-ray scans.During speech production, the tongue comes into contactwith the hard and soft tissues that compose the vocal tractwalls. Consequently, the contacts were modeled between thetongue and the set hard palate/upper dental arch, the softpalate, and the set inner surface of the mandible/lower dentalarch. The modeling of contacts is non-linear. A face-to-facedetection was used to avoid the interpenetration of the sur-faces in contact, which are potentially in contact with thetongue. A relatively low Coulomb friction was used, since friction is assumed to be limited due to the saliva. The con-tacts are managed through an augmented Lagrangianmethod, which corresponds to an iterative series of penaltymethods. The partial differential equation /H208496/H20850was solved by the ANSYS ™ finite element software package, based on a com- bination of Newton–Raphson and Newmark methods. E. Acoustic modeling A model of sound synthesis, including the determination of the 3D area function of the vocal tract, was coupled withthe mechanical model. The computation of the area function from the mesh node coordinates was achieved by using MATLAB ® software. Before computing the area function, the surface of thetongue was interpolated using 35 periodic cubic splines inorder to get a more accurate detection of the constrictionlocations in the vocal tract. This processing and its use forthe computation of the area function make the implicit as-sumption that the spatial sampling of the tongue surface pro-vided by the finite element mesh is sufficient to allow acorrect interpolation of the tongue surface from the positionsof the nodes. A set of planes, which will be referred to ascutting planes below, was computed for the vocal tract in itsrest position. These cutting planes, orthogonal to the sagittalplane, are approximately perpendicular to the vocal tractmidline at rest. For a given vocal tract configuration, theintersections between the cutting planes and the surface ofthe tongue /H20849approximated by a set of periodic cubic splines /H20850, of the mandible, of the hyoid bone, of the hard and softpalates, and of the pharyngeal and laryngeal walls were com-puted. On every cutting plane, a closed contour based onthese intersections and representing the shape of the vocaltract was computed and approximated by periodic cubicsplines. The inner surface of each of the thus-determinedclosed contours was calculated. The lips, which are not partof the biomechanical model, were represented by a singlecylinder, whose length and section represented lip protrusionand aperture, respectively. To determine the distance betweentwo consecutive cutting planes, and thus to compute thelength of the path from the glottis to the lips, it was decidedto compute the distance between the centers of gravity oftwo successive surfaces. This distance approximates the av-erage distance traveled by the acoustic wave between twoconsecutive cutting planes. An acoustic model /H20849analog har- monic of the vocal tract /H20850was used to generate the spectrum of the signal produced from the area function. 1 III. MUSCLE ACTIVATIONS DURING FRENCH ORAL VOWEL PRODUCTION A. Muscle activations To study the postural control of speech sounds, the best approach would consist of roaming the motor commandspace of the biomechanical model in a systematic and com-prehensive way, using, for example, a Monte Carlo method,in order to characterize the links between motor commands,tongue shapes, and acoustics, following the approach of Per-0 5 10 15 2011.261.522.533.5 Frequency (Hz)Criticaldamping rat io FIG. 4. Ratio damping over critical damping versus frequency for /H9251 =40 s−1and/H9252=0.03 s /H20849Rayleigh damping model /H20850. For a modal frequency from 3 to 10 Hz, the damping ratio is below 1.26, i.e., close to the criticaldamping /H20849ratio equal to 1 /H20850. 2038 J. Acoust. Soc. Am., Vol. 126, No. 4, October 2009 Buchaillard et al. : Modeling study of cardinal vowel production Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 138.251.14.35 On: Tue, 23 Dec 2014 02:45:13rieret al. /H208492005 /H20850for their 2D model. However, such an ap- proach is currently impossible with this 3D model, becauseof the running time /H20849around 40 min for a 100 ms simulation, with ANSYS ™ 11.0 and Windows XP SP2 running on a Pen- tium IV CPU at 3 GHz and 1 GB of RAM /H20850. Consequently, it was necessary to work with a more limited number of simu-lations to study muscle activations in vowel production andthe sensitivity of the vowel configurations to changes in mo-tor commands. The results presented in this section wereobtained based on the 300 simulations, all carried out with afixed mandible. These simulations resulted from a specificchoice of motor commands guided primarily by studies withour model of the individual impact of each muscle on tongueshape /H20849see below /H20850. Our objectives were to generate a very good match of the tongue shapes classically observed in themidsagittal plane for French oral vowels by means of cinera-diographic data /H20849Bothorel et al. , 1986 /H20850. EMG studies by Miyawaki et al. /H208491975 /H20850andBaer et al. /H208491988 /H20850were also used as sources of complementary information on the main tongueand mouth floor muscles activated during vowel production.Acoustic signals were synthesized from the final vocal tractshape, and the formants were calculated. The selection of the optimal vowels has involved a mostly qualitative evaluation of the similarity between thecomputed tongue shapes and the 3D tongue shapes measuredfor the speaker PB /H20849CT data /H20850. A quantitative comparison of the simulated tongue shapes with the measured 3D shapeswas not possible and would not have been very informative,mainly for two reasons: /H208491/H20850In our vocal tract model, the jaw is fixed. It is known that a variety of jaw positions is possible for the samesound without endangering the quality of its perception,and, in particular, producing speech with a fixed jawdoes not prevent the speakers from producing satisfac-tory vowels with fair formants, as shown by bite blockexperiments /H20849Mooshammer et al. , 2001 /H20850; however, this articulatory perturbation has an impact on the tongueshape considered in its entirety. /H208492/H20850The model is a symmetrical one while human subjects are never symmetrical. Hence a detailed comparison ofthe constriction shape was not possible. This is why oursimulations were essentially assessed in terms of globaltongue elevation, proximity to the palate, and front/backposition of the constriction in the vocal tract. However, aquantitative evaluation of the simulated and measuredformant patterns was carried out. Only the simulations obtained for the extreme vowels /i, a, u/ will be presented in this paper. The results correspond tothe shape and position of the tongue at the end of the simu-lated movement. For single muscle activations, movementlasted 400 ms while it lasted only 200 ms for the vowels /H20849for the three vowels, steady-state equilibrium positions werereached /H20850. In all cases, the movement started from rest posi- tion. 1. Impact of individual muscles on tongue shape Figures 5and 6show the individual impact of the tongue and mouth floor muscles on the tongue shape in themidsagittal plane and in the 3D space from a front view perspective. Target motor commands were defined such thata single muscle was activated during each simulation. Forthe only activated muscle, the command /H20849i.e., the threshold muscle length above which active muscle force is generated /H20850 was set either to 75% or 85% of the muscle length at rest /H20849the smaller the percentage, the larger the activation; hence, alarger percentage was chosen for larger muscles to avoid toostrong deformations /H20850. For the other muscles, the motor com- mands were set to a large enough value so as to prevent thesemuscles from generating forces; for example, a commandtwice as large as the muscle length at rest ensures that thismuscle will remain inactive throughout a simulation. Thesesimulations show that the role of the individual muscles inour model matches well with classic knowledge inferredfrom experimental data and clarify their impact on thetongue shape. The anterior genioglossus moves the tonguedownward in its front part, essentially in the region close tothe midsagittal plane /H20849tongue grooving in the palatal region /H110156m m /H20850. This downward movement is associated with a slight backward movement in the pharyngeal region /H20851Fig. 5/H20849a/H20850,/H110151.6 mm /H20852. Note that the backward movement is much smaller than the one predicted by 2D /H20849Payan and Perrier, 1997 /H20850or 2.5D /H20849Dang and Honda, 2004 /H20850models. This can be explained by the fact that in these models the volume con-servation is, in fact, implemented as a surface conservationproperty in the midsagittal plane. In our model, volume con-servation causes the changes that are generated in one part ofthe tongue to be compensated not only in the other parts ofthe midsagittal plane but also in the whole tongue volume.Indeed, a slight enlargement of the tongue is observed in thetransverse direction /H20849up to 2.2 mm /H20850. It can also be noticed that the limited backward motion is consistent with datashowing that a larger expansion may occur in the transverseplane, local to the compression, while a small expansion oc-cur in the same plane /H20849Stone et al. , 2004 /H20850. The medium part of the genioglossus lowers the tongue in its dorsal region/H20849/H110155m m /H20850and moves the apical part forward /H20849/H110153m m /H20850and upward, while an enlargement of the tongue is observed in the transverse direction /H20849up to 1.4 mm /H20850. The GGp enables the tongue to be pushed forward /H20849/H110155.3 mm /H20850; this forward movement is associated with an elevation of the tongue /H20849/H110152.4 mm /H20850due to the apex sliding on the anterior part of the mandible /H20851Fig.5/H20849c/H20850/H20852. However, the elevation of the tongue is less strong than what was predicted by the 2D and the 2.5Dmodels. As for the GGa, it leads to the enlargement of thetongue in the transverse direction /H20851/H110151.3 mm in the apical area and 1.6 mm in the pharyngeal area, Fig. 6/H20849c/H20850/H20852 . The sty- loglossus /H20849Sty/H20850causes downward /H20849/H110159.6 mm /H20850and backward /H20849/H110157m m /H20850displacements of the tongue tip, producing an el- evation of the dorsal part of the tongue and a lowering of the apical region /H20851Figs. 5/H20849d/H20850and6/H20849d/H20850/H20852. No change is observed in the transverse direction. Changes in the midsagittal plane aresimilar to the predictions of 2D or 2.5D models. The hyoglo-ssus generates a backward movement in the pharyngeal part/H20849/H110155m m /H20850, an apex elevation /H20849upward displacement of /H110154.7 mm /H20850, and a lowering of the tongue in its dorsal part /H20851Fig.5/H20849e/H20850/H20852. An enlargement is observed in the transverse di- rection in the pharyngeal part /H20851Fig.6/H20849e/H20850,/H110155m m /H20852. The ver- J. Acoust. Soc. Am., Vol. 126, No. 4, October 2009 Buchaillard et al. : Modeling study of cardinal vowel production 2039 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 138.251.14.35 On: Tue, 23 Dec 2014 02:45:13ticalis provokes only a very small lowering in the palatal region /H20849below /H110150.5 mm /H20850associated with a very slight back- ward movement in the pharyngeal part /H20849below /H110150.5 mm /H20850 /H20851Fig.5/H20849f/H20850/H20852. Its contraction also widens the tongue /H20849/H110151.4 mm in the apical area /H20850. Its impact will then be essentially indi- rect: by stiffening its elements in the palatal part of thetongue, it will modify the action of other muscles. The trans-verse muscle induces essentially a reduction in the tonguewidth in the transverse direction /H20851up to 2.1 mm in the supe- rior part of the tongue, Fig. 6/H20849g/H20850/H20852. Due to the volume conser- vation property, this change spreads over the whole tongue inthe midsagittal plane, generating at the same time a smallforward movement of the apex and a small backward move-ment in the pharyngeal part. The inferior longitudinalis low-ers the tongue tip /H20849/H110155m m /H20850and moves it backward /H20849/H110156m m /H20850. A small backward movement of the tongue is also observed in the pharyngeal region /H20849/H110151.3 mm /H20850/H20851Fig.5/H20849h/H20850/H20852.I n the transverse direction, a slight enlargement is observed in the dorsal region /H20849/H110151m m /H20850. The activation of the superior longitudinalis mainly induces an elevation /H20849/H1101512 mm /H20850and a backward movement /H20849/H1101511 mm /H20850of the tongue tip with a slight backward movement in the pharyngeal part /H20849/H110151.7 mm /H20850/H20851Fig. 5/H20849i/H20850/H20852. The geniohyoid essentially movesthe hyoid bone forward and downward, which induces a slight lowering in the dorsal region /H20849/H110150.7 mm /H20850/H20851Fig.5/H20849j/H20850/H20852. Finally, the mylohyoid elevates the mouth floor in its mid- sagittal part /H20849up to 4 mm /H20850 and moves the dorsal part of the tongue slightly upward /H20851Fig.5/H20849k/H20850/H20852. The analysis of the influ- ences of individual tongue muscles revealed possible syner-gies and antagonisms between muscles: GGp, Sty, and GHcan act in synergy to produce an elevation of the tongue inthe palatal region; in this part of the tongue they act antago-nistically with the GGa and the GGm. The Sty and GGm areantagonists for the control of the vertical position of the dor-sal part of the tongue. The GGp, SL, and GGm contribute tothe tongue tip elevation and their action can be counteractedby that of the IL, the Sty, and the GGa. As for the control ofthe width of the tongue in the transverse direction, Transtends to reduce it in the whole tongue body; GGa and GGmare the main muscles enlarging it in the palatal part, whileHG contributes to its enlargement in the pharyngeal part. 2. Simulations of French vowels In order to generate the 300 simulations used to deter- mine the muscle activation patterns for the French vowels,6 8 10 12 144681012 antero−posterior axis (cm)longitudinal axis (cm) (a) GGa6 8 10 12 144681012 antero−posterior axis (cm)longitudinal axis (cm) (b) GGm6 8 10 12 144681012 antero−posterior axis (cm)longitudinal axis (cm) (c) GGp6 8 10 12 144681012 antero−posterior axis (cm)longitudinal axis (cm) (d) Sty 6 8 10 12 144681012 antero−posterior axis (cm)longitudinal axis (cm) (e) HG6 8 10 12 144681012 antero−posterior axis (cm)longitudinal axis (cm) (f) V ert6 8 10 12 144681012 antero−posterior axis (cm)longitudinal axis (cm) (g) Trans 6 8 10 12 144681012 antero−posterior axis (cm)longitudinal axis (cm) (h) IL6 8 10 12 144681012 antero−posterior axis (cm)longitudinal axis (cm) (i) SL6 8 10 12 144681012 antero−posterior axis (cm)longitudinal axis (cm) (j)G H6 8 10 12 144681012 antero−posterior axis (cm)longitudinal axis (cm) (k) MH FIG. 5. /H20849Color online /H20850Impact of the activation of individual lingual and mouth floor muscles on tongue shape /H20849400 ms command duration, sufficient to reach mechanical equilibrium /H20850. The contours of the articulators /H20849tongue and hyoid bone, mandible, hard and soft palates, and pharyngeal and laryngeal walls /H20850are given in the midsagittal plane /H20849tongue tip on the left /H20850. For every simulation, the target motor command of the only activated muscle equals 75% of the muscle length at rest, except for the activation of the long muscles Sty, IL, and SL /H2084985% of the muscle length at rest /H20850. The dotted contours correspond to the tongue shape in its rest position. 2040 J. Acoust. Soc. Am., Vol. 126, No. 4, October 2009 Buchaillard et al. : Modeling study of cardinal vowel production Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 138.251.14.35 On: Tue, 23 Dec 2014 02:45:13the timing of the motor commands was as follows: at time t=0, the central commands were equal to the muscle length at rest; then they varied linearly for a transition time of30 ms up to the target values. Coarse sets of motor com-mands were first determined for each vowel, guided by priorknowledge of the tongue shapes and by literature data. Thevalues of the commands for the main muscles involved in theproduction of the vowels were then made to vary within amore or less wide range around their primary value. Therange was determined according to the tongue shape sensi-tivity to their modification. Within the set of 300 simulations, the best motor com- mands for French extreme vowels /H20849Table II/H20850were selected based on the obtained tongue shapes and the formant pat-terns. Optimal vowels were chosen in order to get the bestmatch between the tongue shape in the midsagittal planewith PB’s MRI data, and between the formants computed with the formants measured from PB’s acoustic data. Thetongue shapes and the formant patterns obtained for the 300simulations were compared to the MRI data and the formantpatterns collected from subject PB. For each French extremevowel, the motor commands providing the best match of thetongue shape experimentally measured on PB in the midsag-ittal plane and of the corresponding formant patterns havebeen selected as reference motor commands /H20849Table II/H20850. The corresponding tongue shapes are represented in Fig. 7/H20849ob- lique anterior and posterior views /H20850and the formants are given in Table III /H20849the lip aperture and protrusion are also indicated in this table /H20850. Table IVsummarizes the force levels computed at the end of the selected simulations for everytongue and mouth floor muscle. The values indicated corre-spond to the algebraic sum of the force levels computed for−3 −2 −10 1 2 367891011 transverse axis (cm)longitudinal axis (cm) (a) GGa−3 −2 −10 1 2 367891011 transverse axis (cm)longitudinal axis (cm) (b) GGm−3 −2 −10 1 2 367891011 transverse axis (cm)longitudinal axis (cm) (c) GGp−3 −2 −10 1 2 367891011 transverse axis (cm)longitudinal axis (cm) (d) Sty −3 −2 −10 1 2 367891011 transverse axis (cm)longitudinal axis (cm) (e) HG−3 −2 −10 1 2 367891011 transverse axis (cm)longitudinal axis (cm) (f) V ert−3 −2 −10 1 2 367891011 transverse axis (cm)longitudinal axis (cm) (g) Trans−3 −2 −10 1 2 367891011 transverse axis (cm)longitudinal axis (cm) (h) IL −3 −2 −10 1 2 367891011 transverse axis (cm)long itudinalaxis (cm) (i) SL−3 −2 −10 1 2 367891011 transverse axis (cm)longitudinal axis (cm) (j)G H−3 −2 −10 1 2 367891011 transverse axis (cm)longitudinal axis (cm) (k) MH−3 −2 −10 1 2 367891011 transverse axis (cm)longitudinal axis (cm) (l) Rest position FIG. 6. /H20849Color online /H20850Impact of the activation of individual lingual and mouth floor muscles on tongue shape /H20849400 ms command duration, sufficient to reach mechanical equilibrium /H20850/H20849frontal view /H20850. For every simulation, the target motor command of the only activated muscle equals 75% of the muscle length at rest, except for the activation of the long muscles Sty, IL, and SL /H2084985% of the muscle length at rest /H20850. The shape of the tongue at rest is given on the bottom right. TABLE II. Motor commands used for the production of French cardinal vowels and for / ./. These values are given as a percentage of the muscle length at rest. Values below 1 therefore correspond to a voluntary activation. Vowel GGa GGm GGp Sty HG Vert Trans IL SL GH MH /i/ 1.03 1.05 0.60 0.90 1.23 1.13 1.05 1.02 1.09 0.76 0.75 /./ 0.98 0.98 0.98 1.00 1.00 1.00 1.00 1.00 1.00 0.98 0.94 /u/ 1.20 1.20 0.91 0.84 1.25 1.35 0.95 0.98 1.20 0.95 0.80 /a/ 0.75 1.10 1.00 1.10 0.70 0.85 1.30 1.00 1.20 1.05 1.05 J. Acoust. Soc. Am., Vol. 126, No. 4, October 2009 Buchaillard et al. : Modeling study of cardinal vowel production 2041 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 138.251.14.35 On: Tue, 23 Dec 2014 02:45:13each macrofiber. It is not a true value of the force exerted on the tongue, but it provides a fair idea of its order of magni-tude. Figure 7reflects the traditional relationships between the French extreme vowels /H20849anterior versus posterior, low versus high /H20850while the formants are consistent with the clas- sic published values and with acoustic data obtained for thespeaker PB /H20849Table V/H20850. We note a good correspondence be- tween the formants of the acoustic data measured for PB andthose obtained with the simulations. The average differencebetween the formants that were measured and those thatwere simulated is below 3.3% for the first four formants. Thedifference does not exceed 4.3% for the first formant /H20849vowel /u//H20850and 10.2% for the second formant /H20849vowel /i/ /H20850. Due to the redundancy of the system /H20849some pairs of muscles interact as agonist-antagonists /H20850, the commands were also chosen such that the amount of force generated by thedifferent muscles remains reasonable. Only the extreme car-dinal vowels will be presented in detail. a. Muscle activation pattern in vowel /i/ . Figure 8shows the tongue shapes obtained by simulation and those obtainedexperimentally for the speaker PB /H20849CT data /H20850. Some discrep-ancies can be seen in the tongue posterior part, but the de- limitation of the tongue contours in this area is less precise/H20849the delimitation of the tongue body on CT images is a te- dious and less obvious task in this part of the body, due to thepresence of the hyoid bone, epiglottis, and other soft tissues /H20850 and acoustically less relevant than in the anterior part. Thisfigure shows a good correspondence between the experimen-tal results and the computed data, in particular, in the anteriorpart of the tongue, which plays an important role in the pro-duction of vowel /i/. As expected from Fig. 5, since vowel /i/ is an anterior and high vowel /H20851Fig.7/H20849a/H20850/H20852, the model predicted the GGp, GH, and MH muscles to play a fundamental role inits production. In addition to their slight impact on thetongue geometry /H20849see Sec. III A 1 /H20850, the GH and MH muscles can help stiffen the mouth floor, thanks to a significantpropagation of the stress into the lingual tissues. Activatedalone, the styloglossus pulls the tongue backward /H20851Fig.5/H20849d/H20850/H20852; this movement is here counterbalanced by the strong GGpactivation, while both muscles elevate the tongue in the pala-tal region. For the transversalis and the anterior genioglos-sus, the motor commands /H20849/H9261commands of the EPH /H20850areTABLE III. Lip aperture laand protrusion lpchosen for the determination of the vocal tract area function /H20849based onAbry et al. ,1 9 8 0 /H20850and the values of the first four formants for the simulation of French oral vowels /H20849extreme cardinal vowels and / .//H20850. These values were computed with WINSNOORI software. Vowella /H20849cm2/H20850lp /H20849cm/H20850F1 /H20849Hz/H20850F2 /H20849Hz/H20850F3 /H20849Hz/H20850F4 /H20849Hz/H20850 /i/ 3 0.5 321 2095 2988 4028 /./ 1.5 0.8 502 1235 2407 3612 /u/ 0.3 1.5 298 723 2547 3450 /a/ 4.5 0.8 667 1296 2875 3948 681012 −202681012 antero−posterior axis (cm) tranverse axis (cm)vertical axis (cm) (a) vowel /i/681012 −202681012 antero−posterior axis (cm)transverse axis (cm)vertical axis (cm) (b) vowel /u/681012 −202681012 antero−posterio r axis (cm)transverse axis (cm)vertical axis (cm) (c) vowel /a/ 681012−202681012vert icalaxis(cm ) antero−posterior axis (cm)transverse axis (cm) (d)vowel /i/681012−202681012vertical axis (cm) antero−posterior axis (cm)transverse axis (cm) (e)vowel /u/681012−202681012vertical axis (cm) antero−posterior axis (cm)transverse axis (cm) (f)vowel /a/ FIG. 7. /H20849Color online /H20850Final tongue shape for the simulation of the French cardinal vowels /H20849first row: anterior oblique view; second row: posterior oblique view /H20850. 2042 J. Acoust. Soc. Am., Vol. 126, No. 4, October 2009 Buchaillard et al. : Modeling study of cardinal vowel production Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 138.251.14.35 On: Tue, 23 Dec 2014 02:45:13larger than the muscle lengths at rest /H20849Table II/H20850. From a mo- tor control perspective, these two muscles can consequentlybe seen as being in their rest state, and their activation is theresult of reflex loops /H20849Table IV/H20850. The transversalis reflex ac- tivation avoids an overwidening of the tongue that wouldotherwise result from the combination of the GGp activation/H20849see Sec. III A 1 /H20850, while ensuring a contact between the pala- tal arch and the lateral borders of the tongue in the alveolarregion. The GGa reflex activation limits tongue elevation inthe median alveolar region, thus creating the slight groovecharacteristic of an /i/. The voluntary activations are consis-tent with the EMG data of Baer et al. /H208491988 /H20850, except for the Sty, for which no activity was measured by these authors forvowel /i/. With the model, the combined activation of theGGp and Sty is essential to precisely control the location ofthe constriction for high vowels. This co-activation is con-sistent with our previous findings with a 2D tongue model/H20849Payan and Perrier, 1997 /H20850. Qualitatively the tongue shape proposed for /i/ is in good agreement with different kinds of data published in the lit-erature. This is true for the 2D shape in the midsagittal plane,which is consistent with Bothorel et al. /H208491986 /H20850data for French speech sounds. It is also true for the 3D distributionof the contacts between the hard palate and the upper dentalarch on the one side and the tongue lateral borders on theother side. These contacts are represented in Fig. 10/H20849a/H20850. The surface of contact stretches over the whole hard palate and isalso extended to the inner aspects of the molars. In addition,we note the presence of contacts between the apex and themandible inner surface, behind the lower incisors /H20849not shown /H20850. These observations are consistent with the EPG data ofStone and Lundberg /H208491996 /H20850/H20849Fig.9/H20850andYuen et al. /H208492007 /H20850 for English vowels. b. Muscle activation pattern in vowel /u/ . The model produces vowel /u/, a posterior and high vowel /H20851Fig.7/H20849b/H20850/H20852, essentially with the activation of the styloglossus, the mylo-hyoid, and the transversalis /H20849Table II/H20850. As with vowel /i/, the model requires the activation of the MH to stiffen the mouthfloor and thus contribute to the tongue elevation, due to thecomplementary action of other muscles. The styloglossus al- lows the tongue to be pulled both backward and upward. TheGGp is also active. It increases the size of the vocal tractback cavity by propelling the tongue forward and contributesto the upward movement of the tongue. The transversaliscontributes to the limitation of the tongue widening, but thisis not its only role. Indeed, for this vowel the model uses anactive recruitment of the transversalis in order to facilitatethe tongue elevation, due to the incompressibility of the lin-gual tissues /H20849note, however, that the amount of force gener- ated by the transversalis is close to that used in the produc-tion of /i/ /H20850. The motor commands proposed in our model are consistent with the EMG data of Baer et al. /H208491988 /H20850. Here again, the 2D tongue shape in the midsagittal plane is ingood agreement with data of Bothorel et al. /H208491986 /H20850 . In our simulation, the tongue tip is located in the midheight of thetongue. Figure 10/H20849b/H20850shows the distribution of the contacts of the tongue dorsum and the tongue tip with the surroundingstructure, namely, the hard and soft palates, the superior den-tal arch, and a part of the pharyngeal walls. The figure showsthat the tongue post-dorsal surface is laterally in contact withthe inner surface of the molars and, further back, with thelateral sides of the pharyngeal walls. The contacts betweenthe tongue and hard palate observed in the simulations areconsistent with the EPG data of Stone and Lundberg /H208491996 /H20850 /H20849Fig.9/H20850andYuen et al. /H208492007 /H20850. However, EPG data do not provide information on possible contacts between the tongueand velum. c. Muscle activation pattern in vowel /a/ . Vowel /a/, a posterior and low vowel, was essentially produced in themodel with the activations of the HG and GGa muscles /H20851Fig. 7/H20849c/H20850and Table II/H20852. The HG pulls the tongue backward and downward but also rotates the tongue tip toward the palateTABLE IV. Final force levels /H20849in newtons /H20850observed for every tongue and mouth floor muscle during the production of the French cardinal vowels and / ./. The levels of force indicated correspond to the algebraic sum of the forces computed for every macrofiber. Bold cells represent voluntarily activated muscles. Vowel GGa GGm GGp Sty HG Vert Trans IL SL GH MH /i/ 0.51 0 25.82 6.90 0 0 1.61 0 0 3.37 13.89 /./ 0.11 0.13 0.95 0.04 0.04 0.00 0.25 0 0.10 0.10 1.62 /u/ 0 0 6.73 7.42 00 1.83 0.47 0 1.02 6.79 /a/ 3.34 0 1.91 0 8.21 2.31 0 0.16 0 0 0.78 TABLE V. Values of the first four formants based on acoustic data obtained for the speaker PB. The values were averaged over ten repetitions of everyone of the extreme cardinal vowels in different contexts. VowelF1 /H20849Hz/H20850F2 /H20849Hz/H20850F3 /H20849Hz/H20850F4 /H20849Hz/H20850 /i/ 311 2308 3369 4126 /u/ 285 792 2783 4055 /a/ 661 1291 2657 3717 FIG. 8. Superimposition of the shape of the tongue for the speaker PB /H20849CT data /H20850/H20849dense mesh /H20850and the shape of the tongue obtained by simulation /H20849coarse mesh /H20850for vowel /i/. J. Acoust. Soc. Am., Vol. 126, No. 4, October 2009 Buchaillard et al. : Modeling study of cardinal vowel production 2043 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 138.251.14.35 On: Tue, 23 Dec 2014 02:45:13/H20851Fig.5/H20849e/H20850/H20852. The GGa limits the apex rotation by flattening the tongue tip and maintaining it in contact with the inner sur-face of the mandible, thus preventing the creation of a sub-lingual cavity and increasing the size of the anterior cavity.The GGp activation is a reflex activation, since from themotor command point of view it is in its rest state /H20849see Table II/H20850; the GGp limits the backward movement of the tongue and thus avoids the occlusion of the vocal tract in the laryn-gopharyngeal region. The motor commands are in agreementwith the EMG data of Baer et al. /H208491988 /H20850. The tongue shape in the midsagittal plane is in agreement with data of Bothorel et al. /H208491986 /H20850. The lateral borders of the tongue are in contact with the lower dental arch over its entire length, but not withthe palate. The lower surface of the tongue anterior part ispartially in contact with the inner surface of the mandible.For vowel /a/, the EPG data of Stone and Lundberg /H208491996 /H20850 andYuen et al. /H208492007 /H20850reported an either extremely limited or non-existent contact between tongue and palate; the re-sults obtained are consistent with their data. B. Highlighting the role of the transverse muscle in midsagittal tongue shaping A 3D biomechanical tongue model allows the study of the transverse muscle action during speech production. Sincespeech has experimentally mainly been studied in the sagittaldomain, the potential role of this muscle has essentially beenignored. However, it could be of great importance in speechproduction, since it is the only muscle able to directly act ontongue deformations in the transverse dimension orthogonalto the sagittal plane.The role of the transverse muscle in the midsagittal de- formation of the tongue was recently observed by Gilbert et al. /H208492007 /H20850for swallowing through the analysis of diffusion-weighted MRI measurements. They found, in par-ticular, that the recruitment of the transversalis is used togenerate depressions in the tongue to facilitate the movementof the food toward the pharynx. Unfortunately, similar ex-perimental observations do not yet exist for speech, and it isa strength of our 3D model that it offers the possibility toquantitatively assess the role of the transversalis in speechproduction. As a matter of fact, the simulations of vowelproduction reported in Sec. III A 2 highlighted the funda-mental role of this muscle in the maintenance of the tonguedimension along the transverse direction and its influence onmidsagittal shaping. These results have been obtained in thecontext of our motor control model, based on the EPHtheory, which gives an account of the postural control in aparticularly effective way, thanks to the integration of reflexactivation in the muscle force generation mechanisms. In-deed, the model predicts that for vowel /i/ /H20849and also for the high anterior vowels /y/ and /e/ not presented here /H20850, the transverse muscle is active, despite the fact that the motorcommands for this muscle were those of the rest position orhigher /H20851see Eq. /H208492/H20850/H20852. This is the result of a reflex activation /H20849or limited active contraction /H20850due to the lengthening of the transverse fibers induced by the centrally activated musclesthat mainly act on the tongue shape in the sagittal plane. Thisreflex activation limits the amplitude of the deformations inthe transverse dimension and, in turn, due to the incompress-ibility of tongue tissues, it increases the deformations in thesagittal plane. According to the simulations, a voluntary ac-tivation of the transversalis would lead to a decrease in thetongue width that does not seem compatible with the produc-tion of high anterior vowels, unless this decrease can becompensated by the action of other muscles. Hence, the com-bination of a voluntary co-activation of the transversalis andof other tongue muscles could also be considered as an al-ternative to the proposed reflex activation of the transversa-lis. Such a strategy is realistic, but it would imply the acti-vation of a larger number of muscles acting antagonistically,inducing an increase the amount of force necessary to pro-duce high anterior vowels. Our simulations do not rule out FIG. 9. EPG data for American English vowels /i/ and /u/. Reprinted from Stone, M., and Lundberg, A., J. Acoust. Soc. Am., “Three-dimensionaltongue surface shapes of English consonants and vowels,” 99/H208496/H20850, 3728– 3737, 1996. Copyright© 1996, Acoustical Society of America. FIG. 10. Illustration of the contacts between the tonguedorsum and tongue tip, and surrounding structures ofthe vocal tract for vowels /i/ /H20849a/H20850and /u/ /H20849b/H20850/H20849superior view, apex at the top /H20850. The surrounding structures, rep- resented by translucent gray meshes, include the hardpalate, the upper dental arch, the velum, and the pha-ryngeal walls close to the velum. The entire tonguemesh is represented, but only the tongue surface ele-ments used for the detection of potential contacts be-tween the tongue and the surfaces listed above are col-ored. Red elements represent tongue surfaces in contactwith the surrounding structures, yellow elements repre-sent tongue surfaces close to the surrounding structures,and blue elements represent tongue surfaces far fromthe surrounding structures. 2044 J. Acoust. Soc. Am., Vol. 126, No. 4, October 2009 Buchaillard et al. : Modeling study of cardinal vowel production Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 138.251.14.35 On: Tue, 23 Dec 2014 02:45:13the possibility of a voluntary activation of the transversalis. However, such a strategy does not sound like an economicalway to control tongue shapes for high vowels. As alreadymentioned above, 2D or 2.5D models, such as those of Payan and Perrier /H208491997 /H20850orDang and Honda /H208492004 /H20850, could only account for tongue incompressibility in the sagittal plane dueto a simplifying assumption assimilating volume conserva-tion and area preservation in this very plane. In a way, thissimplifying approach implicitly included the role of thetransverse muscle, without formalizing it in explicit terms.We have seen in Sec. III A 1 that this hypothesis led to par-tially inaccurate conclusions concerning the role of musclestaken individually. Our 3D modeling approach allows theseformer conclusions to be corrected and emphasizes the indi-rect role of the transverse muscle in the shaping of thetongue midsagittally /H20851Fig.7/H20849a/H20850/H20852. Based on simulations made with their 2D model, Perrier et al. /H208492000 /H20850concluded that the main directions of deforma- tion for the tongue during speech production as observed fordifferent languages /H20851namely, the factors front and back rais- ing of the parallel factor analysis /H20849PARAFAC /H20850ofHarshman et al. /H208491977 /H20850, see Jackson, 1988 ;Maeda, 1990 ;Nix et al. , 1996 ;Hoole, 1998 , or more recently Mokhtari et al. , 2007 /H20852 did not result from a specific speech control, but emergednaturally from the actions of the major tongue muscles/H20849GGp, GGa, HG, and Sty /H20850. Similar conclusions could be drawn from Honda’s /H208491996 /H20850EMG data. The results concern- ing the role of individual muscles in our 3D model can beused to reformulate these conclusions more accurately. Themain directions of deformation could indeed emerge natu-rally, provided that the tongue widening along the transversedirection is strictly controlled by the reflex transversalis ac-tivation. This reflex activation, based on the use of the motorcommands at rest, is not likely to be speech specific, since itallows the tongue to remain within the space determined bythe dental arches, possibly in order to avoid biting problems/H20851several observations indeed show a widening of the tongue for edentulous people /H20849Kapur and Soman, 1964 /H20850/H20852. Taking into account this reflex limitation of tongue width seems tobe essential to understanding the precise control process ofthe place of articulation in the vocal tract. IV. VARIABILITY OF MOTOR COMMANDS AND TONGUE POSITIONING ACCURACY FOR VOWEL /i/ A. Methodology The accuracy of speech motor control is an important and still unsolved issue. Indeed, speech movements can be asshort as a few tens of milliseconds, so it is traditionally sug-gested that cortical feedback, involving long latency loops,can only be used to monitor speech after its production andnot during on-going production /H20849see, for example, Perkell et al. , 2000 for details /H20850. Tongue positioning has to be very accurate though for the production of some sounds, such asfricatives and high vowels. This apparent contradiction /H20849the absence of cortical feedback versus the accuracy require-ment /H20850suggests that speech motor control has developed into a very efficient process to ensure, in a simple way, accuracyand stability of tongue positioning. This efficient treatmentand accuracy can be seen as the result of the high amount of training and experience in speaking that speakers have. For the high vowel /i/ more specifically, it has been ar- gued that control accuracy would come from a combinationof biomechanical effects, namely, the co-contraction of theGGp and the GGa associated with tongue/palate contacts/H20849Fujimura and Kakita, 1979 /H20850. This effect is called the “satu- ration effect.” Using a rudimentary 3D tongue model,Fujimura and Kakita /H208491979 /H20850showed that the tongue was sta- bilized during the production of /i/ when laterally pressedagainst the palate, due to the combined action of the GGaand GGp, which stiffened the tongue. Our 3D model, whichintegrates numerous improvements as compared to Fujimura and Kakita’s /H208491979 /H20850original model /H20849smaller mesh elements, non-linear tissue elasticity, gravity, stiffening due to activa-tion, and accurate model of contacts /H20850, offers a powerful con- text to revisit this hypothesis and to better understand howthe different biomechanical factors interact. With the currentmodel, a number of simulations were realized around thereference tongue shape for /i/ to evaluate the articulatory andacoustic sensitivity of the vowel to changes provided to themotor commands. The tongue shape variations as well as theformant variations resulting from small changes in the cen-tral commands were studied for this vowel, so as to betterunderstand the patterns of variability observed during its pro-duction. The motor commands defined previously /H20849see Sec. III and Table II/H20850formed the basis of this study. The motor commands of the main tongue muscles /H20849i.e., the anterior, medial and posterior genioglossus, the styloglossus, the hyo-glossus, the transversalis, the lingual inferior and superiormuscles, and the mylohyoid /H20850were independently modified. For the GGa, GGp, Sty, MH and Trans, the motor commandswere modified by /H110062%,/H110065%,/H110068%, and /H1100610% around their values at target. For the GGm, HG, IL, and SL, whichwere not active during the production of vowel /i/ in ourmodeling, the motor commands were only modified by −2%,−5%, −8%, and −10%, since an increase in their valueswould leave them inactive. The same lip protrusion and ap-erture parameters as previously applied were used to gener-ate the acoustic signals and to determine the formants asso-ciated with the different area functions. Table VIindicates the first three formants for each of the 56 simulations. B. Results Figure 11shows the scatter plots in the midsagittal plane for six nodes on the tongue surface obtained from the simu-lations. Results are presented in the upper left panel for thevariations in all muscle commands together, and in the otherpanels, more specifically, for the variations in the commandsto three muscles that play a major role in the production ofvowel /i/: the styloglossus and the anterior and posterior ge-nioglossus. For the global results /H20849upper left panel /H20850,3 /H9268el- lipses characterizing the node position dispersion with aGaussian statistical model are superimposed on the data.Considering first the influence of all muscles taken together,the following observations can be made. In the pharyngealand velopharyngeal regions /H20849three most posterior nodes /H20850, the major axes of the dispersion ellipses essentially correspond J. Acoust. Soc. Am., Vol. 126, No. 4, October 2009 Buchaillard et al. : Modeling study of cardinal vowel production 2045 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 138.251.14.35 On: Tue, 23 Dec 2014 02:45:13to a displacement along the front-to-back direction /H20849from the pharyngeal to the velopharyngeal position, lengths of the ma-jor axes 3.0, 3.9, and 4.4 mm, respectively, lengths of thesmall axes 1.5, 1.4, and 0.8 mm, angles of the major axeswith the antero-posterior axis 133°, 147°, 173° /H20850. In the pala- tal and alveopalatal parts of the tongue /H20849second and third nodes from the front /H20850, the ellipses have no clear direction and they tend to be more circular. In addition, the maximal vari-ability is smaller than in the back part of the tongue /H20849lengths of the major axes 2.9 and 3.5 mm, respectively, lengths ofthe small axes 2.4 and 2.1 mm /H20850. Finally, in the apical part /H20849most anterior node /H20850, a very strong correlation is observed between elevation and forward movement. This leads to aglobal ellipse orientation similar to the one observed in thetongue blade region, but much stronger and clearer and withmuch more variation along the principal axis /H20849length of the major axis 8.3 mm, length of the small axis 3.4 mm, andangle of the major axis 147° /H20850. These observations are in quite good agreement with experimental data published in the literature about vowelvariability. See, in particular, Perkell and Nelson, 1985 ; Beckman et al. , 1995 ;o r Mooshammer et al. , 2004 : the front-back orientation of the variability in the velar regionand the reduced variability in the palatal and alveopalatalregions /H20849the region of constriction for /i/ /H20850were already ob- served by these authors. In addition, the absence of clearorientation of the ellipses in the region of constriction wasalso observed in two of the three subjects studied byMooshammer et al. /H208492004 /H20850while Perkell and Nelson /H208491985 /H20850 andBeckman et al. /H208491995 /H20850rather observed ellipses parallel to the palatal contour in this region. The large variability inthe apical part was observed by Mooshammer et al. /H208492004 /H20850, but not by Perkell and Nelson /H208491985 /H20850andBeckman et al. /H208491995 /H20850. Note, however, that this specific aspect of the dis- placement of the apex relative to that of the tongue body hasalready been observed many times by different authors, inparticular, Perkell /H208491969 /H20850. Our model allows one to look more specifically at the biomechanical factors influencing these articulatory patterns.Looking at the variability associated with the variation in theGGa, GGp, and Sty activations separately, it can be observedthat the angle of the main ellipses in the three posterior nodesis similar to the orientation of the scatter plots generated bythe Sty and GGa. However, for the GGp, the largest variabil-ity is also observed in the front-back direction. The reductionin the variability in the region of constriction is observedboth for the Sty and the GGp, while the GGa, in contrast,TABLE VI. First, second and third formants computed after a local modi- fication of the motor commands for vowel /i/. The formants for vowel /i/ aregiven in Table III. The motor commands of the vowel /i/ were modified for 9 muscles independently by /H110062%,/H110065%,/H110068% or /H1100610%. The formants obtained following these modifications are given. In the /H9261-model, an in- crease in the motor commands corresponds to a decrease in the muscleactivation; therefore a modification by +10% corresponds to the lowest levelof activation for a given muscle, whereas a modification by −10% corre-sponds to the highest level of activation. F1 /H20849Hz/H20850 F2 /H20849Hz/H20850 F3 /H20849Hz/H20850 GGp +10% 350 2084 2942 +8% 340 2073 2931+5% 322 2080 2950+2% 326 2090 2976−2% 317 2105 2992−5% 309 2102 3014−8% 305 2102 3024 −10% 302 2111 3026 GGa +10% 273 2078 3035 +8% 273 2078 3035+5% 283 2082 3012+2% 307 2099 3008−2% 333 2082 2967−5% 351 2071 2928−8% 368 2062 2903 −10% 375 2049 2895 MH +10% 329 2101 2977 +8% 325 2096 2976+5% 322 2095 2986+2% 321 2095 2982−2% 319 2091 2983−5% 318 2100 2986−8% 317 2096 2985 −10% 316 2096 2980 IL −2% 321 2097 2980 −5% 322 2094 2969−8% 322 2098 2964 −10% 321 2103 2958 GGM −2% 321 2085 2979 −5% 322 2057 2960−8% 321 2019 2938 −10% 322 2001 2924 HG −2% 320 2094 2986 −5% 323 2087 2973−8% 326 2079 2963 −10% 331 2076 2957 Sty +10% 272 2009 3017 +8% 278 2018 3000+5% 296 2050 2994+2% 311 2076 2981−2% 334 2135 3003−5% 348 2167 2924−8% 360 2214 2814 −10% 364 2251 2766 Trans +10% 316 2118 3049 +8% 317 2117 3043+5% 318 2112 3027+2% 320 2108 3004−2% 321 2083 2966TABLE VI. /H20849Continued. /H20850 F1 /H20849Hz/H20850 F2 /H20849Hz/H20850 F3 /H20849Hz/H20850 −5% 321 2069 2944 −8% 321 2056 2920 −10% 319 2048 2913 SL −2% 322 2094 2989 −5% 327 2104 2977−8% 342 2103 2958 −10% 346 2106 2912 2046 J. Acoust. Soc. Am., Vol. 126, No. 4, October 2009 Buchaillard et al. : Modeling study of cardinal vowel production Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 138.251.14.35 On: Tue, 23 Dec 2014 02:45:13shows the largest variability in this region. This can be inter- preted in the light of the palatal contacts for vowel /i/ /H20849see Fig.10/H20850. The GGp and Sty act on the position of the whole tongue body, whose variability in the constriction region islimited by the palatal contacts. The GGa influences only thecenter of the front part of the tongue, which is not in contactwith the palate. It can be noted that in our simulations thestyloglossus generates the largest variability, as compared tothe other muscles, except as explained above in the constric-tion region. This phenomenon is intrinsically linked to theapproach that was used in our simulations. In the context ofthe/H9261-model, since the styloglossus macrofibers are longer than in the other muscles, a given percentage of variationgenerated a larger change in the commands for the styloglo-ssus and then, in turn, larger changes in force level. Thisapproach could have influenced the global amount of vari-ability depicted in the upper left panel of Fig. 11, but not its relation with the node position on the tongue, neither interms of orientation nor of amplitude, except for the tonguetip variation, which is largely dependent on the styloglossus. As concerns the other muscles /H20849not depicted in Fig. 11/H20850, their impacts are smaller, but some interesting observationscan be mentioned. The transversalis shows a notable contri-bution in the variability in the velopharyngeal region /H20849third node from the back /H20850, where an increase in its activation in- duces backward displacements of the tongue dorsum. Themylohyoid participates in the up-down displacements, withan amplitude of approximately 1.5 mm in the pharyngealregion. C. The saturation effect for vowel /i/ revisited From the simulations of the consequences of motor command variability for tongue shape for vowel /i/, interest-ing conclusions can be drawn for this vowel concerning theinfluence of the GGa and of its variability on the vocal tract shape and formants. Contrary to what could be inferred fromthe statistical processing of articulatory speech data /H20849see, for example, Badin et al. , 2002 /H20850, the central tongue groove ob- served for vowel /i/ in many languages does not seem to bea consequence of the combined activations of the GGp andSty muscles. It is, in fact, obtained in our model very spe-cifically by activating the GGa. As mentioned above, thisstatement is consistent with Fujimura and Kakita’s /H208491979 /H20850 hypothesis of a co-activation of the GGp and GGa in theproduction of /i/. However, the variability patterns generated with our model, together with their interpretation in terms of the re-spective influence of each muscle, strongly suggest that thereis no saturation effect, which would facilitate the accuratecontrol of the constriction area for /i/. This observation ques-tions Fujimura and Kakita’s /H208491979 /H20850original hypothesis as well as the numerous follow-up contributions that have usedthis hypothesis to explain the control of high front vowels, inparticular, those of Perkell et al. /H208492000 /H20850andBadin et al. /H208491990 /H20850. In agreement with the work of the previous authors, our model tends to confirm that the tongue is indeed stabilized inits entirety by these palatal contacts, and that this shouldcontribute to simplifying its motor control. However, in con-trast to Fujimura and Kakita’s /H208491979 /H20850tongue model, which was quite rudimentary because of the computational limita-tions existing at that time, our model shows that the variabil-ity of the GGa activation leads to a variation in the alveolar groove with noticeable consequences for its formant pattern/H20849see below /H20850. This variation is highly localized in the globally well-stabilized tongue, but it is fundamental to acoustics, be-cause it plays on the constriction size. The amplitudes of variation for the first three formants6 8 10 129101112 antero−posterior axis (cm)longitudinal axis (cm) (a) Global results6 8 10 129101112 antero−posterior axis (cm)longitudinal axis (cm) (b) Results for GGp 6 8 10 129101112 antero−posterior axis (cm)longitudinal axis (cm) (c) Results for GGa6 8 10 129101112 antero−posterior axis (cm)longitudinal axis (cm) (d) Results for St y FIG. 11. /H20849Color online /H20850Displacement scatter plots /H20849circles /H20850for vowel /i/ in the midsagittal plane. Only the surface of the tongue is represented. Panel /H20849a/H20850 summarizes the results obtained for the nine muscles whose motor commands were modified. The 3 /H9268ellipses of dispersion are also represented, and their major axes are drawn. Panels /H20849b/H20850–/H20849d/H20850represent the dispersion obtained when modifying the motor commands of the posterior genioglossus /H20849b/H20850, the anterior genioglossus /H20849c/H20850, and the styloglossus /H20849d/H20850only. J. Acoust. Soc. Am., Vol. 126, No. 4, October 2009 Buchaillard et al. : Modeling study of cardinal vowel production 2047 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 138.251.14.35 On: Tue, 23 Dec 2014 02:45:13were as follows: /H9004F1/H11015103 Hz, /H9004F2/H11015250 Hz, /H9004F3 /H11015283 Hz /H20849Table VI/H20850.2An important part of the variability is due to the styloglossus /H20849impact on F1, F2, and F3, but see our remark above about the force level variation for thismuscle /H20850, but other muscles also have a noticeable influence, either on the first, second, or third formant. The F1 variabil-ity is due in great part to the modifications in the level ofactivation of the GGa and Sty, and secondarily of the GGpand SL. According to the model, the variability of F2 resultsmainly from the modification of the Sty and GGm motorcommands, while that of F3 is due to the Sty and Trans. The variability of F1 for /i/ has important consequences; indeed the perception of vowel /i/ is sensitive to F1 varia-tions in French /H20849one can easily move from /i/ to /e/ /H20850. Like- wise for F3, too low an F3 value moves the perception from/i/ to /y/ /H20849Schwartz and Escudier, 1987 /H20850. It can therefore be concluded that the articulatory variability generated in thesimulations is too important to ensure proper perception ofvowel /i/. It is necessary to reduce it. This need for an activereduction in the articulatory variability is consistent with theobservations made by Mooshammer et al. /H208492004 /H20850with Ger- man speakers: they concluded from their study that the po-tential saturation effect related to the interaction betweentongue and palate did not seem to be sufficient for theirspeakers to meet the perceptive requirements of the Germanvowel system, and that a specific control adapted to the in-dividual palate shape of each speaker was necessary to limitthe articulatory variability and its consequences for percep-tion. This is then quite an important result as it throws back into question a widely made assumption to explain the pre-cise control of the vowel /i/, namely, the saturation effect. V. IMPACT OF GRAVITY ON LINGUAL MOVEMENTS With the increasing use of MRI systems, numerous speech data are acquired while the subject is lying on his orher back. Due to the change in the orientation of gravita-tional forces in relation to the head, this position is likely toalter the vocal tract shape and its control. This is why manystudies have tried to compare the production of speechsounds and speech articulations /H20849either vowels or conso- nants /H20850for subjects when they are sitting, standing, or lying /H20849Weir et al. , 1993 ;Tiede et al. , 2000 ;Shiller et al. , 2001 ; Stone et al. , 2007 /H20850. Our model allows the impact of gravity to be tested and quantitatively assessed. With this aim inview, the pattern of activation needed to keep the tongue inits neutral position was first studied in the presence of agravitational field in an upright and in a supine position.Then, the influence of gravity on the tongue shape during theproduction of vowels was evaluated together with its impacton the acoustic signal. A. Impact of gravity in the absence of active and reflex muscle activation First, the impact of gravity alone on the tongue shape and position was studied: the force generator was deactivated/H20849no internal force could be generated, whether active or re-flex forces /H20850. The final tongue shape is given in Fig. 12for upright and supine positions starting from the rest positionand after a 1 s movement. For a standing subject, there is aclear lowering of the tongue body, which is particularlymarked in the posterior radical part of the tongue but is alsovisible in its apical region /H20849approximately 1.5 mm /H20850. For a lying down subject, the gravity alone produces a strongbackward displacement of the tongue body, with a displace-ment of the tongue tip equal to 9 mm. These results showthat tongue muscle activations are required to maintain thetongue in its rest position, whether the subject is lying on hisback or standing. Reflex activation obtained with motor commands equal to muscle lengths at rest is not sufficient to maintain thetongue in the rest position, as shown by Fig. 13f o ra1s simulation. A small backward displacement of the apex/H20849/H110151m m /H20850and of the rear part of the tongue is visible in the upright position, as well as a more limited rotation of the apex in the supine position than in the absence of muscleactivations /H20849displacement of the tongue tip /H110153m m /H20850. A lim- ited voluntary activation of the GGp and GGa combined witha stronger activation of the MH associated with the reflexactivation of the other tongue and mouth floor muscles cancompensate for the gravity effect /H20849commands indicated in Table IIfor vowel / .//H20850. Based on the model, the MH activa- tion strengthens the mouth floor and limits the lowering ofthe tongue inferior region /H20851Fig.5/H20849k/H20850/H20852. The GGp action pre- vents the backward displacement of the tongue /H20851Fig. 5/H20849c/H20850/H20852 and the GGa counteracts the GGp action in the apical anddorsal areas, limiting the tongue elevation /H20851Fig.5/H20849a/H20850/H20852. A good equilibrium between the activation of these three muscles,based on numerous simulations, leads to the stabilization ofthe tongue in a “neutral” upright position. Correspondingforce levels computed at the end of the simulation for everytongue and mouth floor muscle are given in Table IV/H20849vowel /.//H20850.6 8 10 124681012 antero−posterior axis (cm)long itudinalaxis (cm) FIG. 12. /H20849Color online /H20850Final tongue position in the midsagittal plane for 1 s simulation under the influence of gravity alone. The neutral position of thetongue and the hyoid bone /H20849rest position for a subject in upright position /H20850is represented by a dotted line, the final shape for the tongue and hyoid bonefor a subject in upright position by a dashed line, and for a subject in supineposition by a solid line. Other solid lines correspond to the contours of themandible, hard and soft palates, and pharyngeal and laryngeal walls. 2048 J. Acoust. Soc. Am., Vol. 126, No. 4, October 2009 Buchaillard et al. : Modeling study of cardinal vowel production Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 138.251.14.35 On: Tue, 23 Dec 2014 02:45:13B. Impact of gravity on French oral vowels The impact of the subject position /H20849upright or supine /H20850on vowel production was studied by modifying the orientationof the gravitational field. Simulations were realized for thesupine position for the ten French oral vowels with the samecommands and the same timing as in the upright position.The tongue shapes and positions for the supine and uprightpositions were compared, as well as the force levels fortongue and mouth floor muscles. The differences in tongue shape and formant values be- tween upright and supine positions were negligible for allvowels. However, differences were noticed in the level offorces developed by the GGp, with an increase in supineposition that is variable across vowels: the peak and the finalforces increased on the orders of 8% for /a/, 7% for /u/, and1% for /i/. On the whole, this modification in the force levelaffects in percentage more posterior than anterior vowels.This is consistent with the fact that the production of frontvowels necessitates a strong force from the GGp anyway, incomparison to which the gravitational force becomesquasinegligible. Our results are also in agreement with ex-perimental observations, in which an increase in the GGpactivity in supine position is commonly observed /H20851see, for example, the EMG data of Niimi et al. /H208491994 /H20850andOtsuka et al. /H208492000 /H20850/H20852. In the model, the tongue weight, which is on the order of 1 N, is small as compared to the muscular forces. Hence,feedback activation efficiently counteracts the effects ofgravity orientation changes and limits tongue shape varia-tion. This small shape variation is in contradiction to experi-mental values typically found in the literature. For instance,Badin et al. /H208492002 /H20850reported a more important backward dis- placement of the tongue for both vowels and consonants insupine position /H20849MR images /H20850compared to upright position /H20849cineradiofilm images /H20850, which they attributed to the tongueweight. Shiller et al. /H208491999 /H20850found differences in the formant values between the upright and supine positions for vowels/a/ and / /H9255/. They found that when the head was in the supine orientation, the jaw was rotated away from occlusion, whichled them to conclude that the nervous system did not com-pletely compensate for changes in head orientation relativeto gravity. In the current model, the fact that the model has afixed jaw position /H20849the same for the supine and upright ori- entation /H20850could in part explain the absence of notable differ- ences. However, it should be mentioned that recent experi-mental findings provide good support for our simulationresults. Indeed, Stone et al. /H208492007 /H20850showed that the impact of gravity was low /H20849or even negligible /H20850for some speakers when vowels were pronounced in context and not in an isolatedmanner as has thus far been the case. VI. CONCLUSION A 3D finite element model of the tongue has been pre- sented which was used to study biomechanical aspects andtongue control during vowel production. The model providesa high level of realism both in terms of compliance withanatomical and morphological characteristics of the tongueand in terms of soft tissue modeling hypotheses /H20849geometrical and mechanical non-linearity /H20850. The tongue and mouth floor muscles were controlled using a force generator based on theEPH theory. Simulations with the model coupled with anacoustic analog of the vocal tract allowed muscle activationpatterns to be proposed for the French oral vowels whichwere consistent with the EMG data published in the literatureand which generated realistic tongue shapes, tongue/palatecontact patterns, and formant values. The simultaneousanalysis of these activation patterns and of the actual muscleforces generated for each vowel revealed, among otherthings, a systematic feedback activation of the transversalis.This suggests that this muscle is used to maintain the dimen-sion of the tongue quasi-constant along the transverse direc-tion orthogonal to the sagittal plane. This role is very impor-tant for the control of tongue shape in the midsagittal plane,since, due to tongue tissue incompressibility, it allows moredeformation in this plane. This is consistent with the recentexperimental observations made by Gilbert et al. /H208492007 /H20850for swallowing. The results obtained from the simulations haveled us to conclude that the main directions of tongue defor-mation in the midsagittal plane /H20851as described by the classic front and back raising factors of Harshman et al. /H208491977 /H20850/H20852 could naturally emerge from the combined action of the ma-jor tongue muscles and of the transversalis playing the roleof a “size maintainer” in the transverse direction. This con-clusion is in line with Perrier et al. /H208492000 /H20850, who suggested that these main directions of deformation are not speech spe-cific, but are intrinsically linked to tongue muscle arrange-ments. The muscle activation patterns proposed for each French vowel served as a basis for further studies. The patterns ofarticulatory variability, and their associated acoustic variabil-ity, were analyzed for local changes in the central commandsfor vowel /i/. These results cast doubt over the idea, gener-ally accepted since the work of Fujimura and Kakita /H208491979 /H20850,6 8 10 124681012 antero−posterior axis (cm)longitudinal axis (cm) FIG. 13. /H20849Color online /H20850Final tongue position in the midsagittal plane for 1 s simulation under the influence of the reflex activation alone. The neutralposition of the tongue and the hyoid bone /H20849rest position for a subject in upright position /H20850is represented by a dotted line, the final shape for the tongue and hyoid bone for a subject in upright position by a dashed line, andfor a subject in supine position by a solid line. Other solid lines correspondto the contours of the mandible, hard and soft palates, and pharyngeal andlaryngeal walls. J. Acoust. Soc. Am., Vol. 126, No. 4, October 2009 Buchaillard et al. : Modeling study of cardinal vowel production 2049 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 138.251.14.35 On: Tue, 23 Dec 2014 02:45:13that a muscular saturation due to a simultaneous co- activation of the GGa and GGp muscles would facilitate theaccurate control of /i/. Indeed, the tongue grooving in theconstriction region was shown to be sensitive to change inthe GGa activation with a significant impact on the F1 for-mant value. The impact of gravity was also considered.Simulations showed the importance of low-level feedback inthe postural control for the rest position, as well as the im-pact of the head orientation on the tongue shape and position.These results are at odds with data published in the literaturefor isolated sound production, but they find support in therecent work of Stone et al. /H208492007 /H20850on the production of vow- els in context. Further work will be required to significantly reduce the computation time, and thus increase the number of simula-tions and refine the results. Studies have also been under-taken to assess the contribution of this model to medicalapplications, in particular, the surgical planning of tongueexeresis, with lingual tissue resection and reconstruction pro-cesses. First results have proved to be promising and showthe potential of such a model /H20849Buchaillard et al. , 2007 /H20850. The results obtained for the planning of tongue surgeries and thecomparison with patients’ data should also provide particu-larly interesting information about the compensation pro-cesses and the motor control mechanisms. ACKNOWLEDGMENTS The authors wish to thank Ian Stavness for helpful com- ments and suggestions. This project was supported in part bythe EMERGENCE Program of the Région Rhône-Alpes andby the P2R Program funded by the CNRS and the FrenchForeign Office /H20849POPAART Project /H20850. 1Program written by Pierre Badin /H20849ICP/GIPSA-Lab /H20850. 2It should be noted that these values were obtained with tongue motions starting from a resting state. Modifying this starting state would have animpact on the formant values, but due to the simulation durations, suffi-cient to reach an equilibrium position, and to the model of motor control,the variations of the formant values should remain limited. Abry, C., Boë, L.-J., Corsi, P., Descout, R., Gentil, M., and Graillot, P. /H208491980 /H20850.Labialité et phonétique : Données fondamentales et études expéri- mentales sur la géométrie et la motricité labiales (Labiality and phonetics:Fundamental data and experimental studies on lip geometry and mobility)/H20849Publications de l’Université des langues et lettres de Grenoble, France /H20850. 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1.1465511.pdf

Biased switching in an interacting pair of magnetic particles C. Xu, P. M. Hui, J. H. Zhou, and Z. Y. Li Citation: Journal of Applied Physics 91, 5957 (2002); doi: 10.1063/1.1465511 View online: http://dx.doi.org/10.1063/1.1465511 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/91/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Effect of dipole interaction on microwave assisted magnetization switching J. Appl. Phys. 107, 033904 (2010); 10.1063/1.3298929 Computer simulations of magnetization switching of elongated magnetic particles with shape defects J. Appl. Phys. 93, 9865 (2003); 10.1063/1.1575494 Defect related switching field reduction in small magnetic particle arrays J. Appl. Phys. 93, 7038 (2003); 10.1063/1.1557399 Biased switching of small magnetic particles Appl. Phys. Lett. 75, 1143 (1999); 10.1063/1.124623 Magnetization switching in small ferromagnetic particles: Nucleation and coherent rotation J. Appl. Phys. 85, 4337 (1999); 10.1063/1.370360 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.94.16.10 On: Fri, 19 Dec 2014 18:01:48Biased switching in an interacting pair of magnetic particles C. Xu and P. M. Hui Department of Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong J. H. Zhou Department of Physics, Suzhou University, Suzhou, 215006, People’s Republic of China Z. Y. Li CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, People’s Republic of China and Department of Physics, Suzhou University, Suzhou, 215006, People’s Republic of China ~Received 22 August 2001; accepted for publication 6 February 2002 ! The switching dynamics in a pair of coupled magnetic particles is studied numerically via the Landau-Liftshitz-Gilbert form of a set of damped gyromagnetic equations. The effects of dipolarinteraction between the particles, anisotropy energy, switching and a small bias field are included.Initial conditions in which the moments are parallel and antiparallel to each other are considered.The presence of a small transverse bias field and inter-particle coupling assists in the switching ofa moment. Due to the complicated interplay among different factors, the switching in one particlemay lead to a corresponding precession of moment in a neighboring particle.We show that differentfinal configurations may be attained depending on the inter-particle separation and the initialconfiguration. © 2002 American Institute of Physics. @DOI: 10.1063/1.1465511 # I. INTRODUCTION Modern techniques of thin film growth are capable of producing high quality arrays of small magnetic particles.1–3 These granular magnetic systems have attracted much atten-tion due to their possible applications as high density mag-netic recording medium and magnetic sensing devices. Oneof the important problems in these systems concerns themagnetization reversal in fine magnetic particles. An under-standing of the switching behavior in small magneticparticles, 4–6for example, is of crucial importance in the de- velopment of devices for which the working mechanism de-pends on the dynamical response of the magnetic dipole mo-ments. The behavior of fast switching in a single domain mag- netic particle was studied about 40 years ago. 7,8Kikuchi7 considered a single-domain spherical particle and used theLandau-Lifshitz-Gilbert ~LLG!form of the damped gyro- magnetic equation to analyze the shortest time for magneti-zation reversal. The LLG equation can be expressed as dm dt5gm3H2a mm3dm dt, ~1! where gis the gyromagnetic ratio, ais a phenomenological damping constant, mis the magnetic moment of the particle, andHis the total effective field. When Hincludes only the external field and the demagnetizing field, Kikuchi7showed that there is an optimal value of the damping constant ( a 51) for which the magnetization reversal is fastest. Re- cently, Stamps and Hillebrands5investigated the switching behavior of an isolated, single domain, uniaxial magneticparticle in terms of the transition rates controlled by a smalltransverse bias field. They found the interesting and signifi-cant result that a small bias field, which is an order of mag-nitude smaller than the effective anisotropy field, could beused to achieve a faster magnetization reversal with the switching time decreases linearly as the bias field increases. In an array of small magnetic particles, the particles are typically densely arranged so that the coupling between theparticles should be taken into account when one studies thedynamical processes of the particles. In this work, we studythe switching behavior of a pair of identical uniaxial aniso-tropic dipoles. We aim to study qualitatively the effects ofinter-particle interaction and see how the interaction may al-ter the single-particle results in Ref. 5. Bertram andMallinson 9investigated the switching of a dipole pair and found that different initial states of the dipole pair ~i.e., dif- ferent initial angles between the magnetic moments !may lead to different final orientations in which the moments areeither parallel or antiparallel. Here, we study the switchingbehavior of a coupled pair of moments taking into accountthe effects of inter-particle dipolar interaction, anisotropy en-ergy, switching and a small transverse bias field. Initial con-ditions corresponding to parallel and antiparallel momentsare studied. It is shown that the inter-particle coupling leadsto highly nontrivial dynamics in the pair of moments. Thecharacteristic time of switching of one moment and the finalconfiguration of the two moments depend sensitively on theinterplay among the dipolar interactions, the strength of theswitching and bias fields, and the anisotropy energy. II. MODEL OF CALCULATION We consider two coupled magnetic dipole moments m1 andm2with magnitudes m15m25m. The orientations (u1,w1) and ( u2,w2)o fm1andm2, respectively, are de- fined in Fig. 1. The particles are chosen, without loss ofgenerality, to be on the yaxis with a separation r sbetween m1andm2. The magnetic moments are treated as point di- poles. We choose the anisotropy axes5of the two particles toJOURNAL OF APPLIED PHYSICS VOLUME 91, NUMBER 9 1 MAY 2002 5957 0021-8979/2002/91(9)/5957/5/$19.00 © 2002 American Institute of Physics [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.94.16.10 On: Fri, 19 Dec 2014 18:01:48be parallel to each other and along the zˆdirection. An array of particles with parallel anisotropy axes can be fabricatedwith the axes perpendicular to the film. 3The switching be- havior of the two moments can be described by the coupledLLG equations dm 1 dt5gm13H12a m1m13dm1 dt, ~2! dm2 dt5gm23H22a m2m23dm2 dt, ~3! whereH1(H2) is the effective field acting on m1(m2) which includes the external magnetic fields ~switching field h and bias field h8), the dipolar field, and the effective anisot- ropy field given by5kmz5(2K/m2)mz, whereKis the an- isotropy energy and mis the magnitude of the magnetic mo- ment. The fields H1andH2are, therefore, given by H15~h11kmz1!zˆ1h18xˆ1h12, ~4! H25~h21kmz2!zˆ1h28xˆ1h21, ~5! whereh12(h21) is the dipolar field due to m2(m1) acting on m1(m2). The dipolar fields can be written as h1253~m2rs!rs rs52m2 rs3, ~6! h2153~m1rs!rs rs52m1 rs3, ~7! whererspoints from dipole moment 1 to moment 2. The parameter kmcharacterizes the strength of the anisotropy field. For convenience, we write t85tmg(11a2)21and ex- press Eqs. ~1!and~2!as m12dm1 dt85m1m13H12am13~m13H1!, ~8! m22dm2 dt85m2m23H22am23~m23H2!. ~9! More explicitly, the orientation of moment mias a function of time can be found by solving the following equations:dui dt85sinwiHix m2coswiHiy m2aF2cosuicoswiHix m 2cosuisinwiHiy m1sinuiHiz mG, ~10! dwi dt85cosuicoswi sinuiHix m1cosuisinwi sinuiHiy m2Hiz m 2aFsinwi sinuiHix m2coswi sinuiHiy mG, ~11! wherei51,2 andHx,Hy,Hzare the three components of H. It is, in general, difficult to solve the coupled Eqs. ~10! and~11!analytically. These equations, however, can be readily solved numerically for the switching behavior of apair of magnetic dipole moments. III. RESULTS AND DISCUSSION A. Parallel initial configuration First we consider the effect of the presence of only an inter-particle dipolar field between m1andm2. The initial condition is that both magnetic moments are along the 1zˆ direction. Only taking dipolar interaction into account, the two moments will finally relax to a state in which they are aligned along the yˆdirection. The relaxation is characterized by a reduced relaxation time trelaxwhich can be found by solving Eqs. ~10!and~11!numerically. Figure 2 shows the dependence of trelaxon the inter-particle spacing rs.I n Gaussian units, the separation can be measured in units ofr 05(4pm/3)1/33103nm. In Fig. 2, rsis given as a dimen- sionless quantity using r0as the reference separation. In the calculation, we set the magnitude of the magnetic moment m to unity.As rsincreases, trelaxincreases rapidly as the dipolar FIG. 1. The system considered consists of two single-domain magnetic particles modeled as a pair of magnetic moments m1andm2. The angles u1,2andw1,2specify the orientation of m1andm2, which are separated by a distance rs. The switching field is labeled hand the small transverse field is labeled h8. FIG. 2. Reduced relaxation time trelaxas a function of rs, which is given as a dimensionless quantity. The circles are the numerical results from Eqs. ~10!and~11!. The solid line is a fit of the numerical results to a rs3depen- dence.5958 J. Appl. Phys., Vol. 91, No. 9, 1 May 2002 Xuet al. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.94.16.10 On: Fri, 19 Dec 2014 18:01:48field becomes weaker.As the dipolar field falls off as 1/ rs3,i t is thus expected that trelax;rs3. The solid line in Fig. 2 is a fit of the numerical data to a rs3dependence. When the effects of the effective anisotropic field are included, the parallel initial configuration with both moments aligned along the zˆdirection may be stable when the aniso- tropic field is stronger than the dipolar field. Now we con-sider the effect of an external switching field happlied on one of the moments, say m 1, in the 2zˆdirection so as to flip its alignment. If his smaller than the anisotropy field char- acterized by k,m1does not switch. For h.k, the energy barrier against switching can be overcome and m1switches. The reduced switching time, tswitch, is small for k/h!1 and increases rapidly as k/h!1, as shown in Fig. 3. For a single isolated magnetic particle, a small trans- verse bias field assists in the switching in that it shortens thecharacteristic time to achieve switching. 5Here we apply a small bias field h8onm1only along the xˆdirection.The bias fieldh8and the dipolar interaction due to a neighboring par- ticle are useful in assisting the switching of a moment. In thepresence of h 8,m1may be switched even when k/h.1. Figure 4 shows the results for the switching time tswitchob- tained numerically from Eqs. ~10!and~11!. The choice of parameters is h/m50.8, g/a510, which follows that in Ref. 5,k/h51.1 andrs53. In general, introducing a small bias field h8shortens the switching time. For a given small h8, the presence of a neighboring moment helps to lower the switching time. It turns out that the final configuration of thetwo moments depends on the initial configuration, thestrength of the interaction, the switching and the bias field.This sensitivity on different variables leads to a ‘‘phase dia-gram’’ with rich features as shown in Fig. 5. Different re-gions in Fig. 5 with h 8/hagainstrsand other parameters kept fixed are labeled by the final configuration of the twomoments.The xaxis represents the strength of the interaction and theyaxis gives the ratio of the bias to the switching field. As discussed, for small r s(rs,0.9 for our choice of parameters !where dipolar interaction dominates, the finalstate, labeled region ~I!in Fig. 5, corresponds to configura- tions in which the two moments are aligned ~or nearly aligned !in theyˆdirection. For slightly larger rs, the dipolar field diminishes, while the switching and bias field lead to a fast switching of m1and leave m2still aligned in the 1zˆ direction. The dipolar field assists in the switching in that it destabilizes the initial configuration. In this regime, switch-ing is achieved for an arbitrarily small bias field @see region ~II!#. Forr slarger than a critical value, the dipolar field fur- ther reduces and becomes increasingly irrelevant. The com-petition is now between the combined switching and biasfields, which tend to switch the moment, and the anisotropyeffect, which tends to keep the moment from switching.Thusfor a fixed r s, there exists a critical value of the bias field above which switching is achieved.This competition leads tothe line separating the final configurations labeled ~II!and ~III!in Fig. 5. FIG. 3. Reduced switching time tswitchas a function of k/hin the absence of a bias field ( h850). Other parameters are rs53 and a51.0. FIG. 4. The dependence of reduced switching time tswitchonh8/h. The open and closed circles correspond to results for initially parallel and antiparallel configurations, respectively, with rs53. Other parameters are k/h51.1 and g/a510. Results for the switching of a single isolated moment are included ~triangles !for comparison. FIG. 5. Aplot of h8/hagainstrsshowing the final configuration of the two moments in different regions of the plot. Initially, both moments are aligned along the 1zˆdirection. Other parameters are k/h51.1 and g/a510.5959 J. Appl. Phys., Vol. 91, No. 9, 1 May 2002 Xuet al. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.94.16.10 On: Fri, 19 Dec 2014 18:01:48B. Antiparallel initial configuration It is also interesting to consider the initial condition in which the dipole moments are antiparallel such that m1is aligned in the 2zˆdirection and m2is aligned in the 1zˆ direction.The switching field hacting on m1is now in the 1 zˆdirection. We again fix the other parameters as k/h51.1, h/m50.8, and g/a510. As before, the dynamics involve complicated precessions of the two moments, with the finalconfiguration depending sensitively on h 8/handrs. The de- pendence of the characteristic time tswitchfor antiparallel ini- tial condition on h8/his also shown in Fig. 4 ~see closed circles !.Alarger bias field is needed to reduce the character- istic time for initially antiparallel moments. The results alsoshow that for the parameters considered, a small bias field(h 8;0.03h) could reduce the switching time for both paral- lel and antiparallel initial configurations as well as for anisolated moment. Further increase in the bias field does notshorten the switching time significantly. It is illustrative to look into the complicated dynamics in the switching for both initial conditions. Figure 6 shows thecomponents of the two moments as a function of time forparallel @Figs. 6 ~a!and 6 ~b!#and antiparallel @Fig. 6 ~c!and 6~d!#initial configurations with a bias field h 850.05h. Thezˆ component of m1indicates the switching and the xˆandyˆ components indicate the precession of m1. Form2, it pre- cesses around the zˆaxis, with the deviation of m2zfromm approaching a maximum when the switching of m1takes place. Since the dynamics is caused by the change in theeffective magnetic field as a result of the dynamics of m 1, the deviation of m2from the zˆdirection depends on the strength of the bias field and the inter-particle separation. Similar to Fig. 5, we show in Fig. 7 the ‘‘phase diagram’’ in a plot of h8/hagainstrsfor the antiparallel initial condi- tion. Region ~I!corresponds to the regime where the inter-particle dipolar field dominates. Region ~II!corresponds to a narrow range of parameters in which the dipolar field is suf-ficiently effective to induce a simultaneous switching of m 2 asm1switches. For large separations where the dipolar in- teraction is negligible, a critical value of h8is needed to overcome the energy barrier for switching in an isolated par-ticle. This leads to the separation of regions ~III!and~IV!in Fig. 7 for large r s. We have checked that the results for large rsfor both initial configurations are consistent. For interme- diate separations where the inter-particle interaction is stilleffective, the complicated dynamics of the two moments, asillustrated in Fig. 6, leads to nontrivial ‘‘phase boundary’’separating regions ~III!and~IV!. IV. SUMMARY The switching dynamics of a dipole moment in a pair of magnetic particles is studied numerically via the Landau-Liftshitz-Gilbert form of a set of damped gyromagneticequations. The effects of dipolar interaction between the par-ticles, anisotropy energy, switching and a small bias field areincluded. Initial conditions in which the moments are paralleland antiparallel to each other are considered. The dynamicsof the two moments are nontrivial due to the coupling be-tween the two moments. The switching in one particle maylead to corresponding precession of moments in a neighbor-ing particle. We show that different final configurations mayresult depending on the inter-particle separation, which de-termines the dipolar interaction. Practically, when arrays ofparticles are to be used as a storage devices, care should betaken in the process of switching the moment of a particle assuch switching may affect the orientation of moments inneighboring particles. The present work has been confined tothe ideal case of two identical particles with the switchingand biased fields applied locally on one of the two particles.More realistically, the effects of shape anisotropy and local-ized fields with a finite extent should also be considered. The FIG. 6. The dynamics of m1andm2when a switching and bias field are applied to m1in a system in which h8/h50.05,rs53,k/h51.1, and g/a 510. The components of mx,my,a n dmzas a function of time are shown for both moments. Both parallel initial configuration @~a!and~b!#and anti- parallel initial configuration @~c!and~d!#are considered. FIG. 7. Aplot of h8/hagainstrsshowing the final configuration of the two moments in different regions of the plot. Initially, m1is aligned in the 2zˆ direction and m2is aligned in the 1zˆdirection. Other parameters are k/h 51.1 and g/a510.5960 J. Appl. Phys., Vol. 91, No. 9, 1 May 2002 Xuet al. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.94.16.10 On: Fri, 19 Dec 2014 18:01:48present formalism can be readily generalized to include these complications as well as to extend the calculation to an arrayof particles. ACKNOWLEDGMENTS Work at CUHK was supported in part by a grant from the Research Grants Council ~RGC!of the Hong Kong SAR Government through Grant No. CUHK4129/98P. One of us~P.M.H. !also acknowledges the support from RGC Grant No. HKUST6142/00P. Work at Suzhou University was sup-ported by the National Natural Science Foundation of Chinaunder Grant No. 19774042.1M. Hehn, K. Ounadjela, J. P. Bucher, F. Rousseaux, D. Decanini, B. Bartenlian, and C. Chappert, Science 272, 1782 ~1996!. 2C. Stamm, F. Marty, A. Vaterlaus, V. Weich, S. Egger, U. Maier, U. Ramsperger, H. Fuhrmann, and D. Pescia, Science 282,4 4 9 ~1998!. 3C. Haginoya, S. Heike, M. Ishibashi, K. Nakamura, K. Koike, T. Yoshimura, J. Yamamoto, and Y. Hirayama, J. Appl. Phys. 85,8 3 2 7 ~1999!. 4W. K. Hiebert, A. Stankiewicz, and M. R. Freeman, Phys. Rev. Lett. 79, 1134 ~1997!. 5R. L. Stamps and B. Hillebrands, Appl. Phys. Lett. 75, 1143 ~1999!. 6M. Bauer, J. Fassbender, B. Hillebrands, and R. L. Stamps, Phys. Rev. B 61, 3410 ~2000!. 7R. Kikuchi, J. Appl. Phys. 27, 1352 ~1956!. 8D. O. Smith, J. Appl. Phys. 29, 264 ~1958!. 9H. N. Bertram and J. C. Mallinson, J. Appl. Phys. 41,1 1 0 2 ~1970!.5961 J. Appl. Phys., Vol. 91, No. 9, 1 May 2002 Xuet al. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.94.16.10 On: Fri, 19 Dec 2014 18:01:48

1.4798669.pdf

Nonlinear effects contributing to hand-stopping tones in a horna) Takayasu Ebiharab)and Shigeru Y oshikawac) Graduate School of Design, Kyushu University, 4-9-1 Shiobaru, Minami-ku, Fukuoka, 815-8540, Japan (Received 11 October 2012; revised 8 March 2013; accepted 13 March 2013) Hand stopping is a technique for playing the French horn while closing the bell relatively tightly using the right hand. The resulting timbre is called “penetrating” and “metallic.” The effect of handstopping on the horn input impedance has been studied, but the tone quality has hardly ever been considered. In the present paper, the dominant physical cause of the stopped-tone quality is dis- cussed in detail. Numerical calculations of the transmission function of the stopped-horn model andthe measurements of both sound pressure and wall vibration in hand stopping are carried out. They strongly suggest that the metallicness of the stopped tone is characterized by the generation of higher harmonics extending over 10 kHz due to the rapidly corrugating waveform and that the asso-ciated wall vibration on the bell may be responsible for this higher harmonic generation. However, excitation experiments and immobilization experiments performed to elucidate the relationship between sound radiation and wall vibration deny their correlation. Instead, the measurement resultof the mouthpiece pressure in hand stopping suggests that minute wave corrugations peculiar to the metallic stopped tones are probably formed by nonlinear sound propagation along the bore. VC2013 Acoustical Society of America . [http://dx.doi.org/10.1121/1.4798669] PACS number(s): 43.75.Fg, 43.25.Cb, 43.40.At, 43.40.Ga [JW] Pages: 3094–3106 I. INTRODUCTION This paper deals with hand stopping, a technique in which the horn player closes the bell relatively tightly with the right hand. The placement of the player’s right hand inthe bell is so important in playing the horn that practical techniques on the way how to use the right hand are explained in detail in Farkas’ textbook. 1From a physical viewpoint, the input impedance measurement of the horn carried out by Backus2and Benade3demonstrated that the right hand in the bell in normal playing raised the cutoff fre-quency of the bell (about 500 Hz of the open bell without the hand) up to about 800 Hz and lowered the frequencies of the higher resonances. On the other hand, if the hand is placed to close the bell almost completely and the horn is strongly blown, a drastic change in the tonal characteristic is produced. This techniqueis referred to as “hand stopping” (or “gestopft” in German). 1 The timbre of the stopped tone is described as “metallic brit-tle and rough.” 4,5The spectrum of the stopped tone (note F 4) analyzed by Meyer indicates the following characteristics:5 (1) The third to the fifth harmonics (corresponding to the fre-quencies from about 1 kHz to about 2 kHz) are noticeablyweakened; (2) a spectral peak is formed around 3 kHz; (3) strong higher harmonics produced above 10 kHz. It follows that frequency analysis with an upper limit of 10 kHz orabove is necessary to consider the metallic stopped-tone color, which is characterized by the higher harmonics above 3 kHz. The purpose of the present paper is to discover the physical mechanisms responsible for these tonal peculiarities(referred to hereafter as metallicness , in contrast to brassi- ness) in hand-stopping the horn. Because the brassiness is quantitatively measured using the brassiness potential Bby Meyer et al. , 6some quantitative measure to indicate the met- allicness should be hopefully explored. Backus’ paper2first discussed the effects of the hand stopping from the physical viewpoint as well as those of the right hand put in the bell normally. Concerning the stopped- tone frequencies of the F horn, Backus reported that all theresonance frequencies except the first and second moved down, and each of those resonances (order n) ended up a semitone above the frequency where the neighboring lower resonance (order n/C01) had been originally located. In other words, the third resonance falls near where the second was, the fourth near the third, and so on; but the resulting frequen-cies of all resonances move up a semitone compared with those of resonances of the original normally played (not stopped) tone as indicated in Fig. 1. This moving down of the resonance mode is due to the interstices formed around the hand (or the fingers thrust hard into the bell) that termi- nate the horn with a bigger inertance. 2 In spite of his study on the input impedance in hand stop- ping, Backus2hardly mentioned the tone color of the stopped tone, saying that the change in the stopped-tone quality was due to the same cause as that produced by trumpet mutes: The channels between the fingers acted as high-pass filters andincreased the amplitudes of higher harmonics relative to those of lower harmonics. However, the validity of his high-pass fil- ter theory on the stopped-tone color has not been examined experimentally. Moreover, although there are a few studies on the stopped tones or the muted tones of the horn, they investi-gate those tones from phenomenological viewpoints or based on input-impedance measurement methods. 7,8Hence, it may be stated that there are no studies on the decisive physical cause which creates the stopped-tone quality.a)Dedicated to Yoshinori Ando, 1928–2013. b)Present address: YAMAHA Corporation, 203 Matsunokijima, Iwata, Shizuoka, 438-0192 Japan. c)Author to whom correspondence should be addressed. Electronic mail:shig@design.kyushu-u.ac.jp 3094 J. Acoust. Soc. Am. 133(5), May 2013 0001-4966/2013/133(5)/3094/13/$30.00 VC2013 Acoustical Society of America The physical cause of the stopped tone timbre is there- fore discussed in the present paper in more detail. First, the validity of Backus’ high-pass filter theory should be exam- ined. A comparison of the transmission function (defined asthe ratio of the sound pressure in the mouthpiece and that at the bell in the frequency domain) between the open horn without the hand in the bell and the stopped horn will be anappropriate approach for the purpose. Here, the transmission function must be considered in a wide frequency range reaching 10 kHz as explained above. So probably it is notsuitable to discuss the transmission function of the stopped horn based on an experimental approach because there is an upper frequency limit when the multiple microphone methodis used to measure impedance, and this may be as low as 4-6 kHz. 9,10 Possible causes other than Backus’ theory on the timbre of the stopped tone also should be explored by changing the viewpoint. The steepening of the waveform (or the formation of the shock wave) due to nonlinear propagation through astraight tube in the trumpet and trombone 11,12comes to the mind. According to the latest study by Myers et al. ,6the French horn investigated by them indicated the brassinesspotential Bof 0.51, and a saxhorn basse (a bass instrument in B-flat included in saxhorn family) the brassiness potential of which was also 0.51 demonstrated nonlinear sound propaga-tion along the bore. Additionally, Stevenson 13concluded that the brassy timbre of the horn was derived from the shock-wave formation in the pipe. Thus this possibilityseems to be worth discussing in the present paper, too. On the other hand, Wachter 7calculated the internal pressure in the horn bore when a stopping mute was inserted,and he illustrated that internal pressures on the 4th and 5th modes were about five times larger at the narrow mute bellneck than those at the open horn bell without the right hand. Then it may be assumed that the same strong internal pres- sure is also caused by hand stopping at the inner position ofthe inserted right hand. Moreover, Kausel et al. 14have shown the model illustrating the wall vibration of brass instruments and have indicated that the displacement of thewall vibration is proportional to the internal pressure. Therefore it is expected that the wall vibration at the stopped-horn bell is several times larger than that at the nor-mal horn bell and could therefore contribute the metallic stopped-tone timbre. Hence, the historical question concerning the tonal effect due to the wall vibration of wind instruments, which has been investigated and debated for a long time, 15–18 should be considered again. Kausel et al.14review the his- tory of research on the wall effects. The present authors add some comments. It is variously confirmed that a cylindrical wall section can vibrate in response to the acoustic pressure in the pipe but cannot radiate appreciable sound.16–18For example, if an organ pipe is immersed in water and driven by the water jet,a significant wall vibration can be easily confirmed from the measurement of the radiation pattern. 18However, such a confirmation is almost completely impossible in air becausethe wall effect is much weaker than the acoustic radiation from the pipe end. On the other hand, the flaring bell section seems to behave differently. Moore et al. 19visualized the modal structure of trumpet bells using electronic speckle- pattern interferometry and found that the mode frequencies followed a generalized Chladni’s law. Later, Moore et al.20 found that when the wall vibrations at the trumpet bell sec- tion were damped, the acoustic pressure radiated from the trumpet increased by more than 3 dB in the fundamental buthardly changed in the higher harmonics. They showed that the change in the fundamental seemed to be caused by the feedback of the wall vibration to the player’s lips throughthe air column. Nief et al. 21regarded the antinodes of the wall vibration modes over a trombone’s bell as monopoles and calculated their radiation efficiency. They reported thatthe radiation efficiency of each vibration mode indicated a high-pass filter behavior with its own cutoff frequency. Kausel et al. 22mounted the horn in a box filled with sand to damp the horn’s pipe and bell and analyzed the sound radiated from the horn, which was played using the ar- tificial lip. As a result, the spectral centroid of the radiatedsound decreased by about 150 Hz and the amplitudes of higher harmonics reduced by 27 dB at most. Although they suggested that this result was caused by a change in the radi-ation impedance due to the boundary conditions given by the sand around the flaring part of the bell, 22the sand might have prevented the bell wall from directly radiating thehigher harmonics. In the light of these recent researches mentioned in the preceding text, it is interesting to know whether the wall vibration due to the hand stopping can sig-nificantly create the metallic stopped-tone or not. Some mechanisms in which the wall vibration is excited are also discussed recently. Whitehouse 23denoted that the FIG. 1. Pitch change by hand stopping. The right column shows the 1st to 6th mode of the tone in normal playing with the right hand in the bell; the left column shows the corresponding mode of the “stopped” tone. For exam- ple, C 3(the 3rd mode of the normally played tone) moves down to F# 2(the 2nd mode of the stopped tone) by hand stopping. This F# 2is a semitone above F 2, which corresponds to the 2nd mode of the normally played tone. Note that the 1st mode (the pedal tone) is not affected, and the 2nd mode F 2 is almost lost by hand stopping. J. Acoust. Soc. Am., Vol. 133, No. 5, May 2013 T. Ebihara and S. Y oshikawa: Nonlinear effects in horn hand-stopping tones 3095player’s lip oscillation was the dominant source to generate the wall vibration in playing the trumpet. Meanwhile, Kauselet al. 14presented their wall vibration model by assuming that the displacement of the bell wall was axially symmetric and caused by the internal modal pressure. Both mechanismsare based on linear wave transfer to the bell wall from the player’s lip or the resonant air column. In contrast to their viewpoints, nonlinear vibrations excited by the large-amplitude external force are essential to investigate the vibrations of percussion instruments. 24,25 Thus it may be appropriate to examine the possibility of non- linear wave excitation to study the wall vibration of the stopped horn if internal pressure in hand stopping is large enough to excite nonlinear wall vibration. The nonlinearvibration of shells and panels has been eagerly studied in the past few decades in the field of mechanical engineering, and these studies were reviewed in detail by Amabili andPa€ıdoussis. 26In particular, when the shells or panels are oscillated by the sinusoidal driving force, the higher harmon- ics called superharmonics are excited from the geometrical nonlinearity on the strain-displacement relations of the shells or panels.27 The present paper consists of the following five sections. In Sec. II, the input impedance and transmission functions of the French horn are calculated both with and without a sim- ple geometric model in place of the player’s hand and com-pared with Backus’ high-pass filter theory. The measurement of the radiated acoustic pressure and the associated wall vibration is carried out in Sec. IIIto know how significantly the wall vibration affects the hand stopping. The results are discussed from two viewpoints. Is the wall vibration in hand stopping in agreement with a model proposed by Kauselet al. ? 14Is the wall vibration efficiently radiated into the air? An experiment based on large-amplitude excitation upon the bell wall is carried out in Sec. IVby supposing that a nonlin- ear mechanism can be responsible for the wall vibration in hand stopping. In Sec. V, the effect of wall vibration is stud- ied by immersing the bell and horn sections in sand, as car-ried out by Moore et al. 20and Kausel et al.22Also, contributions of nonlinear propagation and wave steepening to metallic hand-stopped tones are investigated by measuringthe acoustic pressure in the mouthpiece in Sec. V. Finally, the conclusions are given in Sec. VI. II. CALCULATIONS BASED ON SIMPLE MODELINGS In horn playing, player’s right hand is used in three ways: Not placed in the bell (the horn in this case is calledanopened horn in the present paper, although this way is not usually used in actual playing); placed in the bell as usual (called a normal horn ); inserted into the bell almost completely (called a stopped horn ). Both input impedances and transmission functions of these three kinds of the hornare calculated in this section, using a simple geometric model in place of the player’s hand. The purpose of these calculations is to examine Backus’ high-pass theory on thestopped tone. Besides, the pressure distribution along the horn is calculated to make clear the pitch change caused by hand stopping.A. Input impedances of the horns 1. Opened horn The geometrical data of the bore shape of the horn were provided by the Lawson Brass Instruments Inc., the manu- facturer of our horn.28The cross-sectional figure of the horn is shown in Fig. 2(a). The total length is 385 cm: 145 cm of the flaring bell portion, 182 cm of the cylindrical portion, 51.6 cm of the lead pipe portion, and 6.4 cm of the mouth- piece. Also the diameter of the bell is 30.5 cm and that of thecylindrical portion is 1.2 cm. Following the method by Causs /C19eet al. , 29the bore is first divided into a series of small sections (cylinders and trun-cated cones) to apply simple acoustic theory. In the nth sec- tion from the bell, input pressure P n,in, input volume velocity Un,in, output pressure Pn,out, and output volume velocity Un,outare related by the transmission matrix Tnas follows: Pn;in Un;in/C18/C19 ¼TnPn;out Un;out/C18/C19 ¼AnBn CnDn/C18/C19Pn;out Un;out/C18/C19 :(1) Each element of Tnis given by equations developed by Mapes-Riordan30by including the effects of visco-thermal losses (see Table II of Ref. 30). From the continuity of Pand Uat boundaries between element pipes, the following rela- tion is finally obtained: Pm Um/C18/C19 ¼Y nTnPbell Ubell/C18/C19 ¼AB CD/C18/C19 Pbell Ubell/C18/C19 ; (2) where PmandUmare the pressure and the volume velocity at the mouthpiece input end, and PbellandUbellare those at the bell output end. Thus, the input impedance Zin¼Pm/Um of the horn is yielded as Zin¼APbellþBU bell CPbellþDU bell¼AZbellþB CZbellþD; (3) where Zbell¼Pbell/Ubellcorresponds to the radiation imped- ance at the bell. On the assumption that the horn bell is set inan infinite plane baffle 31thisZbellis calculated as follows: Zbell¼qc pa21/C0J1ð2kaÞ kaþjH1ð2kaÞ ka/C20/C21 ; (4) where J1is the first-order Bessel function of the first kind, H1the first-order Struve function, qthe air density, kthe wave number, and athe bell radius. The reason for calculat- ing the radiation impedance with an infinite baffle is that aformula to yield the radiation impedance at the bell without the baffle 29is not available for calculating the radiation im- pedance at sufficiently high frequency due to the upper fre-quency limit imposed by the approximation. The presence of the baffle seems to have a relatively small effect. 31 The magnitude of the input impedance Zinof the natural horn is illustrated in Fig. 2(d) with the thick line. Moreover, Table Isummarizes the resonance frequency frof the input impedance peaks and the frequency interval Dfrbetween the neighboring peak frequencies. The resonance frequencies have nearly equal intervals except for the interval between 3096 J. Acoust. Soc. Am., Vol. 133, No. 5, May 2013 T. Ebihara and S. Y oshikawa: Nonlinear effects in horn hand-stopping tonesthe first and second peaks, and this result is consistent with Backus’ experiment [see Fig. 10(a) in Ref. 2]. 2. Normal-horn model The shape of the right hand is intricate, so strict calcula- tion of the input impedance of the normal horn by the pre- ceding method is almost impossible. Instead the right hand ismodeled here by a thin plate with an orifice set near the posi- tion of right hand. The area of the orifice corresponds to that of the interstice between the bell wall and the player’s hand.For our calculation, it is supposed that the distance from the bell to the plate is 7.6 cm, the plate diameter is 8.4 cm (equal to the internal diameter of the bell there), the plate thickness is 3.8 cm, and the orifice diameter is 5 cm. These dimensionsare defined so that the input impedance yielded by the calcu- lation is consistent with measurement results by Dell et al. , 8 cited in the following text. The enlarged view of the cross section of this model is illustrated in Fig. 2(b). The magnitude of the input impedance of the normal- horn model is demonstrated in Fig. 2(d), and its phase is in Fig. 2(e) with the dashed line, respectively. Dell et al.8has TABLE I. Resonance frequencies frand their intervals Dfof the input impedance of the opened horn and the stopped-horn model. The frequency rations R between the resonance frequencies at the closest corresponding modes of the opened horn and the stopped-horn model are also given. Opened horn Stopped-horn model Frequency ratio R mode tone fr(Hz) Dfr(Hz) ! mode tone fr(Hz) Dfr(Hz) mode R(cent) I (Pd.) 29 – ! I0(Pd.) 27 – II F 2 84 55 ! II0(Almost disappeared) III C 3 131 47 ! III0F# 2 95 – III0/II 213 IV F 3 175 44 ! IV0C# 3 141 46 IV0/III 127 VA 3 219 44 ! V0F# 3 185 44 V0/IV 96 VI C 4 259 40 ! VI0A# 3 231 46 VI0/V 92 VII D# 4 300 41 ! VII0C# 4 277 46 VII0/VI 116 VIII F 4 344 44 ! VIII0E4 322 45 VIII0/VII 123 IX G 4 387 44 ! IX0F# 4 366 44 IX0/VIII 107 XA 4 431 43 ! X0G# 4 411 45 X0/IX 104 FIG. 2. (Color online) (a) The cross section of the natural horn used to calculate the input impedance. (b) The enlarged view of the cross section of the normal-horn model. (c) The enlarged view of the cross section of the stopped-horn model. (d) Magnitudes of input impedance Zinof the opened horn (thick solid line), the normal-horn model (dashed line) and the stopped-horn model (thin solid line). (e) Phase angle of Zinof the normal-horn model (dashed line) and the stopped-horn model (thin solid line). J. Acoust. Soc. Am., Vol. 133, No. 5, May 2013 T. Ebihara and S. Y oshikawa: Nonlinear effects in horn hand-stopping tones 3097reported that the player’s hand lowers the frequencies of the impedance peaks above the 6th (corresponds to C 5in their Fig. 8) by about 20 cents as shown in their Fig. 8and increases their magnitudes by about 6 M Xabove cutoff fre- quency (around 500 Hz) as shown in their Fig. 4. Using the parameters listed in the preceding text, our calculation agrees with these measurements. Thus the proposed method for modeling the player’s hand seems to be an appropriateapproximation. 3. Stopped-horn model The input impedance of a stopped-horn model is calcu- lated here by a similar model but with the plate set furtherfrom the outer end of the bell and with a smaller orifice. The distance from the bell to the plate is set at 15.2 cm, the plate diameter at 6.4 cm, thickness at 2.5 cm, and the orifice diam-eter at 6 mm for our calculation. The orifice diameter is equal to that of a narrow bell neck of a stopping mute. The enlarged view of the cross section of this model is illustratedin Fig. 2(c). The calculated input impedance of our stopped-horn model is shown in Figs. 2(d) and2(e) by the thin solid line.The arrows in Fig. 2(d) indicate the direction of the change in the input impedance peaks. In addition, the resonance fre-quencies and the frequency intervals are summarized in Table Ialong with the frequency ratio Rbetween the reso- nance frequencies at the same mode number of the openedhorn and the stopped-horn model. The input-impedance cal- culation on the stopped-horn model shows almost complete consistency with Watcher’s measurement, 6andRindicates the value around a semitone (except for Ron III0/II) just as Backus’ study made clear.2 B. Transmission functions of the normal- and stopped-horn models From reasonable results in the preceding text on the input impedance of the normal- and stopped-horn models, it seems to be appropriate to calculate the transmission func- tion defined as H¼Pbell/Pmbased on these models. The cal- culation result of the respective transmission function is illustrated in Fig. 3(a)with the black thin line (for the opened horn), the gray line (for the normal-horn model), and theblack thick line (for the stopped-horn model). The resonance peaks at 3.5 kHz found in all results are probably generated by the mouthpiece (the peaks at 3.5 kHz shifted to 3.4 kHzwhen the volume of the mouthpiece cup was enlarged by about 20% with the rim diameter unchanged). Besides, the shoulder at around 4.5 kHz found in the result of the normal-horn model and that at 6.8 kHz in the result of the stopped- horn model are caused by the respective configuration of the plate and orifice. It should be noted that the transmissionfunction of the stopped-horn model does not indicate a spe- cific high-pass filter characteristic except around 6.8 kHz. Thus Backus’ explanation that the stopped tone is caused bythe high-pass filter of transmission function does not explain all the observed features. According to our model, Backus’ estimation is qualita- tively correct but cannot quantitatively account for the cutoff frequency. The resonance at 6.8 kHz found in the transmis- sion function of the stopped-horn model corresponds to theresonance frequency of an opened pipe of 2.5 cm, i.e., the orifice length of the modeled plate. If the plate thickness (orifice length) is longer than that in our model, the reso-nance due to the orifice should appear at a lower frequency just as the orifice with a length of 3.8 cm in the normal-horn model affects the transmission function at around 4.5 kHz.Such a transmission function might form a high-pass filter above 3.5 kHz. Backus possibly expected the transmission function of the stopped horn to show such a low cutoff fre-quency. But this may not be the case in actual player’s right hand in hand stopping as inferred from Fig. 3(a). C. Pressure distribution along the horn In this subsection, the pressure distribution along the horn is calculated to obtain a qualitative understanding of the pitch change caused by hand stopping. Although Backus has described that the pitch change by hand stopping iscaused by the inductive termination at the bell, 2the explana- tion seems to be unsatisfactory. When Eq. (2)is applied, the internal sound pressure Pkand the volume velocity Ukon the FIG. 3. (a) Calculated transmission functions of the opened horn (dotted line), the normal-horn model (gray solid line) and the stopped-horn model (black solid line). (b) Pressure distribution along the horn. The 4th mode of the opened horn, F 3under Pbell¼5 Pa (solid line) and the 3rd mode of the stopped-horn model, C# 3(originally F 3) under Pbell¼1 Pa (dashed line). The internal pressure in hand stopping at 370 cm from the mouthpiece is 12 times larger than that in the opened horn at the same position. 3098 J. Acoust. Soc. Am., Vol. 133, No. 5, May 2013 T. Ebihara and S. Y oshikawa: Nonlinear effects in horn hand-stopping tonesoutput end of the kth short pipe counted from the bell ( k¼1, 2,…) is given as Pk Uk/C18/C19 ¼Yk/C01 nTnPbell Ubell/C18/C19 ¼AkBk CkDk/C18/C19 Pbell Ubell/C18/C19 :(5) Thus Pk¼PbellAkþBk Zbell/C18/C19 ; (6) where a radiated sound pressure Pbellis arbitrarily given. The calculations of the pressure distributions of F 3(the 4th mode of the opened-horn, supposing Pbell¼5 Pa) and C# 3 (the 3rd mode of the stopped-horn model, originally F 3, sup- posing Pbell¼1 Pa) are illustrated in Fig. 3(b). From the result, it is evident that the pressure mode pat- tern along the horn in hand stopping changes from that corre-sponding to the “closed-opened” pipe to more closely approximating a “closed-closed” pipe. In other words, both pressures at the mouthpiece end and at the bell end are themaxima in hand stopping. This effect means a larger inert- ance at the bell; this makes the wavelength in the horn longer than the original one and shifts the corresponding originalmode (order n) in the “closed-opened” pipe down to the next lower mode (order n/C01) in the “closed-closed” pipe. That is the essential cause of pitch change by hand stopping.Note that this conclusion is consistent with Backus’ ex- planation indicated in the preceding text. However, our ex- planation based on the shift of the pressure-distributionpattern demonstrates much more explicitly how the hand stopping causes the puzzling pitch descent shown in Fig. 1. Its dominant cause is regarded as the shift of the mode pat-tern from the nth “closed-opened” pipe mode to the ( n/C01)th “closed-closed” pipe mode if our hand-modeling method is relevant. III. MEASUREMENTS OF SOUND RADIATION AND WALLVIBRATION In the previous section, it is argued that Backus’ theory about the metallicness of the stopped tone has limitations. In this section, experimental measurements of the radiated sound pressure and the wall vibration in hand stopping areconsidered. Kausel et al. 14already reported that the local amplitude of wall displacement scould be modeled by s¼pr2 Ehcos3/; (7) where pwas the maximum positive instantaneous sound pressure in the pipe with radius of rand flare angle of /,E was the Young’s modulus of the material, and hthe wall thickness. According to this equation, it is expected that the FIG. 4. Examples of the measured waveforms and their spectra (bottom frames) from three players A, B, and C. (a) the radiated sound pressure in normal pl ay- ing of C 4; (b) the radiated sound pressure in hand stopping of C# 4; (c) the vibration velocity of the bell wall in normal playing of C 4; (d) the vibration velocity of the bell wall in hand stopping of C# 4. J. Acoust. Soc. Am., Vol. 133, No. 5, May 2013 T. Ebihara and S. Y oshikawa: Nonlinear effects in horn hand-stopping tones 3099amplitude of wall vibration grows large enough to affect the tone color in hand stopping. This is because the sound pres-sure pat the right hand position in hand stopping is about 12 times higher than that in normal playing, as shown in Fig. 3(b). The purpose of this section is to examine whether or not the wall vibration at the bell contributes to the metallic timbre in hand stopping. A. Experimental setup The radiated sound and the wall vibration were meas- ured in an anechoic chamber (length 4.5 m /C2width 5.5 m /C2height 4.5 m) of our university using both normal playing (with a player’s right hand putting in the bell as usual) and hand stopping. The natural horn used for calcula-tion in Sec. IIwas played by three amateur horn players A, B, and C. Players A and B have played the horn for 12 years, and player C has played for 8 years. A microphone (B&K4191) calibrated with a piston-phone (B&K 4228) was placed at the center of the bell exit surface, and a small pie- zoelectric charge accelerometer (B&K 4374, 5 mm in diame-ter, 6.7 mm in height, and 0.75 g in weight) was attached with double-sided adhesive tape at a position of 1 cm distant from the edge of the bell. See Fig. 8in Sec. IVconcerning the setup of a microphone and an accelerometer. The result- ing data were recorded on a computer with the sampling rate of 44.1 kHz after amplifying by a measuring amplifier (B&K2636 for the microphone, and B&K 2609 for the accelerome- ter). The recorded data were then analyzed. The tones meas- ured in normal playing were F 3,C 4, and F 4, and those in hand stopping were F# 3,C# 4, and F# 4. These tones were played 10 times for 4 s each in p,mf,fandff, respectively. B. Experimental result The experimental measurements for C 4and C# 4played atmezzo forte are shown in Fig. 4. The waveforms of the radiated sound pressure and the vibration velocity (yielded by numerically integrating the measured acceleration) areshown for typical waveforms observed 2 s after the attack. The spectra illustrated at the bottom of Fig. 4are given by averaging the peak amplitude of the Fourier transforms of 10samples measured from 1 to 2 s after the attack. The error bars shown in the spectra denote the double standard error of the mean, i.e., the 95% confidence limits. It is evident that the waveforms of the radiated stopped tone shown in Fig. 4(b) indicate rapidly corrugating changes (minutely indented waveforms), while those of the normaltone shown in Fig. 4(a) do not. Furthermore, the spectra of the stopped tone given in the bottom of Fig. 4(b) indicate the same characteristics as have been denoted by Meyer, 5that is, as already shown in Sec. I, an amplitude reduction of the harmonics from 1 to 2 kHz and an emphasis of higher har- monics held up to 10 kHz. This strongly suggests that thepenetrating metallic stopped tone can be caused by the rap- idly corrugating changes observed in the waveforms. Another characteristic noted by Meyer 5is the spectral peak at 3 kHz. In the present study, this seems to be weak on play- ers A and B, while it seems to be at 2 kHz on player C. This difference may depend on the players’ hand positions orplaying style and skills. Also the radiated pressure ampli- tudes in hand stopping are reduced to about 1/5 of thenormal-tone amplitudes. This reduction is probably due to the inserted right hand that obstructs the sound radiation from the bell. Moreover, the amplitude ratio of five timesbetween the normal and stopped tones is consistent with the supposition on their radiated sound pressures given in Sec. II C. Rapidly corrugating changes are also found in the wave- forms of the wall vibration velocity in hand stopping as shown in Fig. 4(d). Although the waveform is not obvious for player B, the spectra of wall vibration velocity demon- strate that the amplitudes on the higher frequencies are nearly 40 dB larger in hand stopping. Thus the rapidly corru-gating changes observed in both the radiated pressure wave- form and the wall-vibration velocity waveform are related: The wall vibration directly generates the radiated soundpressure with the rapidly corrugating changes or, conversely, the rapidly corrugating changes produced in the internal sound pressure generate the corresponding wall vibration.The former possibility suggests direct radiation from the bell wall and the latter suggests other mechanisms such as non- linear wave steepening along the bore and lip vibration. Another experimental result helps decide between the possibilities. The rapidly corrugating changes are found in the waveform in loud normal playing as well as in hand stop-ping. Figures 5(a) and5(b) illustrate the waveforms of the radiated sound pressure and the vibration velocity on F 4 played in ffby player C, respectively. Figures 5(c) and5(d) are the spectra of the respective waveforms. The radiated sound pressure waveform indicates the rapidly corrugating changes similar to the waveforms of the stopped tone,although such changes are limited to the region of the wave trough. Moreover, the waveform of Fig. 5(a) shows a tend- ency of wave steepening, which is caused by the nonlinearwave propagation in a long cylindrical pipe 6,11,12or by other means such as nonlinear response of the lips.20The detected wave steepening appears to be in agreement withStevenson’s study 13in which he has reported the develop- ment of the shock wave in the horn pipe and its contribution to brassy timbre. Furthermore, Stevenson13has concluded that the mouth- piece pressure and the lip motion in the brassy playing did not indicate any peculiarities compared with the normalplaying. If we follow his study, it may be anticipated that the mouthpiece pressure and nonlinear wave steepening along the horn are irresponsible for the rapidly corrugating changesin metallic hand-stopped tones. Of course, this prediction should be experimentally examined (see Sec. VB). The measured sound pressure with rapidly corrugating changes (minute indented waveforms) seems to be generated by a superposition of the radiated tone from the air column and that from the wall vibration. The sound pressure radiatedfrom the air column in hand stopping is decreased to about 1/5 of the normal tone due to the inserted right hand as al- ready shown above. However, the wall vibration on the bellseems to be increased by the right hand because the inserted hand generates the strong internal sound pressure compared with the normal playing. Thus it is expected that the 3100 J. Acoust. Soc. Am., Vol. 133, No. 5, May 2013 T. Ebihara and S. Y oshikawa: Nonlinear effects in horn hand-stopping tonesminutely corrugating waveform radiated from the bell wall can be emphasized in hand stopping if the radiation condition is satisfied31,32and that the metallic stopped tones result in. C. Discussion The measured wall vibrations at the bell edge are com- pared with the model proposed by Kausel et al.14to discuss how the wall vibrations in hand stopping differ from those in normal playing. It is notable that the amplitudes of the veloc- ity in hand stopping are significantly larger than those in nor-mal playing as already shown in Figs. 4(c)and4(d). It seems difficult to explain these large acceleration amplitudes in hand stopping from Kausel’s model 14presented by Eq. (7). If Eq. (7)is valid, the acceleration amplitude of the wall vibration in hand stopping must be smaller than that in nor- mal playing as the acoustic pressure measured at the centerof the bell end (almost proportional to the inner pressure p) in hand stopping is much smaller than that in normal playing [see Figs. 4(a)and4(b)]. In Fig. 6, a comparison of wall vibration displacement between the estimation by Kausel’s local model and our measurement is illustrated on F 4and F# 4to discuss this con- tradiction. Equation (7)is calculated at the bell end by using the measured pressure pand the values of r¼15.3 cm, E¼110 GPa, h¼0.63 mm, and /¼67/C14from our natural horn. On the other hand, the measured wall displacement is obtained by two numerical integrations of the measured acceleration. The result shown in Fig. 6indicates that Kausel’s local model14can be only applied to the normal playing. The displacements given by two methods are almost equivalent to each other in normal playing as shown in Fig.6(a). However, in hand stopping, the displacement from our measurement is about three times larger than that calculated from Kausel’s model as shown in Fig. 6(b). This disagreement in hand stopping may be attributed to the localized formulation of Kausel’s model where the prop- agation of the wall vibration along the bell surface isneglected. However, the wall vibration generated at the right-hand position in hand stopping can be spread through- out the bell wall. The wall vibration at the bell edge meas-ured in hand stopping seems to be the vibration propagated from the right-hand position, which has not been considered in Kausel’s modeling. 14As a result, the wall vibration throughout the horn bell in hand stopping may be several times larger than that in normal playing. IV. NONLINEAR EXCITATION OF THE BELL WALL Our experimental result and discussion in Sec. IIIpossi- bly suggested that the stopped tones and the loud normal tones characterized by the rapidly corrugating waveforms were substantially derived from the bell wall vibration. Thefollowing question should be then clarified: How is the higher-frequency wall vibration generated in hand stopping and loud normal playing? Because the internal pressure atthe right-hand position is large, this pressure seems to drive the bell wall strongly. This section reports an excitation experiment on the bell wall is carried out to determine themechanism responsible. A possible candidate responsible for strong higher- harmonic wall vibration in hand stopping is nonlinear vibra-tions of the bell itself. Wachter 6demonstrated that when a stopping mute was inserted into the bell, the internal pres- sure at the bell neck was about five times larger than that ofan opened horn with the same mouthpiece pressure. Therefore it may be expected that the similar large internal pressure is also generated in hand stopping and such a largepressure strongly drives the bell wall. When thin shells such as the horn bell are oscillated by a periodic driving force of such a large amplitude, the higher harmonic frequencies,called superharmonics , can be excited because of geometri- cal nonlinearity from the strain-displacement relations of the shells. 27In hand stopping, the internal sound pressure at the right hand position may be regarded as the external force exciting the bell structure. The magnitude of this force is large enough to generate nonlinear vibration. Thus it is FIG. 5. Measured waveforms and their spectra in ffplaying of F 4by player C. (a) the radiated sound pressure; (b) the vibration velocity of the bell wall; (c) the spectra of the radiated sound pressure; (d) thespectra of the vibration velocity. J. Acoust. Soc. Am., Vol. 133, No. 5, May 2013 T. Ebihara and S. Y oshikawa: Nonlinear effects in horn hand-stopping tones 3101anticipated that the vibration with higher frequencies is gen- erated at the bell and that the vibration radiates the penetrat- ing sound contributing to the metallic timber of the stoppedtone into the air. Figure 7(a) illustrates the internal sound-pressure distri- bution of F 4(the 8th mode of the opened-horn model, sup- posing Pbell¼20 Pa) and F# 4(the 9th mode of the stopped- horn model, supposing Pbell¼5 Pa), where the method shown in Sec. II Cis employed [cf. Fig. 4(b) for the value of Pbellin hand stopping]. It is verified that the internal pressure Pintat the bell neck of F# 4is about six times larger than that of F 4as indicated by the curves between 3.6 and 3.7 m dis- tant from the mouthpiece [see the enlarged view of Pint shown in Fig. 7(b)]. The pressure distribution gives the force amplitude Fwhich drives the bell wall as follows:F¼ð3:7 3:6PintðxÞ/C22rðxÞpdx/C2512 N ; (8) where r(x) is the radius of the bore at the distance xfrom the mouthpiece. On the other hand, Fat the bell neck of the opened-horn model is calculated as 2.8 N. This result demonstrates that the horn bell is more strongly oscillated by the internal pressure in hand stopping than in normalplaying. The wall excitation experiment was therefore carried out to yield the bell-wall vibration and the sound radiationunder such a large exciting force using the setup illustrated in Fig. 8. An impedance head (B&K 8001) attached to a mini-shaker (B&K 4810) was suspended with a tripod andfixed on the bell neck of the natural horn used in the previous experiment. The sinusoidal wave of the frequency of 280 Hz was sent from a function generator (NF 1930) to a poweramplifier (B&K 2718), and the resulting amplified signal drove the shaker. The exciting force was measured by the impedance head, the radiated sound from the bell wall by themicrophone, and the wall vibration at the bell edge by the accelerometer, as described for the experiment reported in Sec. III. When the amplitude of the sinusoidal exciting force of 280 Hz (corresponding to the sounding frequency of C 4#) was increased to approximately 5 N, an extremely metallicsound was heard. The measured waveforms of the exciting force F ex, the wall vibration velocity at the bell end Vend, and the radiated sound pressure Pradin that condition are illus- trated in Figs. 9(a),9(b), and 9(c), respectively. The waveform of Fexshown in Fig. 9(a) is almost sinu- soidal although it includes a slight distortion due to its largeamplitude, while the waveform of V endshown in Fig. 9(b) appears a little corrugated. The waveform of Vendseems to resemble that observed in hand stopping by player B shownin Fig. 4(d). Furthermore, the waveform of P rad, the ampli- tude of which is about one-sixth to 1/10 of the radiated pressure shown in Fig. 4(b), contains apparent rapidly corru- gating changes similar to the waveform of the stopped tone FIG. 6. A comparison between the wall displacement calculated from Kausel’s model (dashed line) and that given from our measurement (solidline). (a) normal playing of F 4; (b) hand stopping of F# 4. FIG. 7. Acoustic pressure distributions along the horn. (a) the 8th mode of the opened horn, F 4under Pbell¼20 Pa (solid line) and the 9th mode of the stopped-horn model, F# 4under Pbell¼5 Pa (dashed line); (b) the enlarged view around the bell neck. 3102 J. Acoust. Soc. Am., Vol. 133, No. 5, May 2013 T. Ebihara and S. Y oshikawa: Nonlinear effects in horn hand-stopping tonesgiven in Fig. 4(b). These rapidly corrugating changes of Prad observed in the large-amplitude excitation experiment are extremely nonlinear because their amplitudes depend strongly on the amplitude of the sinusoidal exciting force.Amplitude levels of 20log 10j^Vendjand 20log 10j^Pradjare illustrated in Fig. 9(d). The frequency points of Fig. 9(d) correspond to the harmonic frequencies the fundamental fre-quency of which is 280 Hz. These characteristics may allow the generation of superharmonics. It is proposed that if the radiated metallic sound is superposed on the tone generatedacoustically and radiated from the bell bore in hand stop- ping, the metallic timbre of the stopped tone is generated. Additionally, radiation efficiency L rad defined as 20log 10j^Prad/^Vendjis demonstrated in Fig. 9(e) to show the radiation characteristics of the nonlinear wall vibra- tion. The values of Lradtend to increase as the harmonic frequencies are increased. Moreover, it is notable that the value of Lradat 280 Hz is about 25 dB lower than that at560 Hz (the second harmonic frequency). The result may suggest that there is a threshold frequency (which deter-mines whether the vibration is efficiently radiated from the bell wall) between 280 and 560 Hz. It seems plausible that the wall vibrations of sufficiently high frequency can beradiated into the air efficiently if we assume wall radiation from a cylindrical pipe. 32 However, the vibration amplitude of Vend¼20 mm.s/C01 shown in Fig. 9(b) is not appropriate when comparing with the velocity amplitude in horn playing shown in Fig. 4(d). This discrepancy may arise because the vibrating force wasgiven by a point source. The effect of axially symmetrical in- ternal pressure should be properly considered in hand stop- ping. Thus it is difficult to say definitely that thesuperharmonics are generated in hand stopping and that they affect the timbre of the stopped and fortissimo tones. V. DIRECT EXAMINATIONS ON THE ORIGIN OF METALLIC STOPPED TONES A. Immobilization experiment of the horn body Let us directly confirm whether the wall vibration con- tributes the timbre of the stopped tones by strongly dampingthe horn body. If the metallic stopped tone is generated by the radiation of the superharmonic wall vibration, the tonal metallicness should be removed when the horn bell and thepipes are completely damped. This immobilization experi- ment was carried out by applying a method employed in Kausel’s demonstration. 22The bell of the natural horn used in the preceding text was mounted in a wooden enclosure, and the pipe of the horn was placed in a box. If these two boxes were filled with a quantity of sand, wall vibrations ofthe horn body should be strongly damped. The horn was blown by player A in hand stopping in the anechoic room. The radiated sound pressure and the wall vibration at the belledge were measured in the same way as shown in the previ- ous experiments. The measurements were carried out under four conditions: No sand in the boxes (called the freecondi- tion), with the sand poured into the bell section only (the bell-damped condition), with the sand into the pipe section only (the pipe-damped condition), and with the sand into both sections (the fully damped condition). The player was asked to keep his right hand in the same position and to play at the same volume in each condition. The amplitude of the vibration velocity at the bell is about 20 times smaller in the fully damped condition than that in the free condition as shown in Figs. 10(a) and10(b) . The amplitude of the spectral envelope of the velocity is also reduced by 20–40 dB in the frequency range up to 10 kHz due to the damping [see Figs. 10(c) and10(d) ]. Therefore if the wall vibration radiates the metallic stopped tones, some clear differences in stopped tones should be observed between damping conditions. Figure 11illustrates the radiated sound pressures of the stopped tones and their spectral envelopes in four damping conditions in the preceding text. All the measured soundwaveforms still remain characteristic minute wave corruga- tions, even though the horn body is strongly damped by the sand. Also, their spectral envelopes do not show definite FIG. 8. Block diagram used in our wall-excitation experiment. FIG. 9. The results of the wall-excitation experiments. (a) almost sinusoidal excitation force Fex; (b) vibration velocity Vendat the bell end; (c) sound pressure radiated from the bell-wall vibration, Prad; (d) peak amplitudes of spectra of the radiated sound pressure (black line) and the vibration velocity (gray line); (e) radiation efficiency Lradof the bell wall. J. Acoust. Soc. Am., Vol. 133, No. 5, May 2013 T. Ebihara and S. Y oshikawa: Nonlinear effects in horn hand-stopping tones 3103differences regardless of whether the horn is damped or not. Some peaks in the damped conditions are smaller than those in free condition, e.g., the frequencies over 4.5 kHz in the damped condition. These reductions of the high-frequencycomponents may be caused by the damping of wall vibration and then slightly affect the tone color. However, the differ- ence between conditions with and without sand damping isnot so decisive. Although the prime characteristic of the me- tallic stopped tone is the peak around 3 kHz as suggested by Meyer, 5the immobilization of the horn body cannot remove that peak. The experimental results in fortissimo playing are somewhat similar to those shown here. Therefore it may be inferred that the wall vibration does not primarily affect the stopped and fortissimo sounds even if it emphasizes their high-frequency harmonics.B. Measurement of the mouthpiece pressure Other mechanisms responsible for the stopped-tone gen- eration must be considered because the hypothesis that thewall vibration contributes the metallic timbre is inconsistentwith the results reported in the preceding text. It should beexamined whether nonlinear propagation along the bore gen-erates the rapidly corrugating waveform in hand stopping,although this possibility has been excluded in Sec. III B, based on Stevenson’s study. 13In the context of nonlinear propagation or wave steepening, the corrugating waveform(or change) may be adequately replaced with the wave cor- rugation . The Burgers equation predicts that the shock wave is generated if the length of the cylindrical pipe is longerthan the critical distance: 11,12,33 FIG. 10. Changes of wall vibrations in our immobilization experiment. (a) the waveform of wall vibration velocity in free condition; (b) that in bell-damped condition; (c) spectralenvelope of wall vibration velocity in free condition; (d) that in bell- damped condition. FIG. 11. Measured waveforms andspectral envelopes of the stopped tones under four damping condi- tions. (a) free; (b) bell-damped; (c) pipe-damped; (d) fully damped. 3104 J. Acoust. Soc. Am., Vol. 133, No. 5, May 2013 T. Ebihara and S. Y oshikawa: Nonlinear effects in horn hand-stopping tonesxc¼2cPatc ðcþ1Þ½dPm=dt/C138max; (9) where c¼1.4 is the Poisson ratio, Patthe mean atmospheric pressure, cthe sound speed, and Pmthe mouthpiece pressure. Even if the length of cylindrical pipe of the instrument isshorter than x c, the wave steepening is occurred and the higher-frequency component of the radiated tone is increased.12,33 The sound pressure in the mouthpiece Pmwas measured using the method presented by Norman et al.33to estimate the effect of the nonlinear propagation by calculating thecritical length x cin hand stopping. A pressure transducer (PCB 106B) was attached to a mouthpiece (YAMAHA 30D4). The signals from the transducer that was powered bya conditioning amplifier (B&K 2693) were amplified with a measuring amplifier (B&K 2636) and recorded on a com- puter with the sampling rate of 44.1 kHz. The sound pressureradiated from the horn bell was measured using a micro- phone (B&K 4191) in the same way as the above experi- ment. The natural horn was blown by two skilled amateurplayers D and E in normal playing of F 4and in hand stop- ping of F 4#. The waveforms of Pmand Pbellin hand stopping are illustrated in Figs. 12(a) and12(c) , respectively. The wave- form of Pmin normal playing indicates flat peaks and deep troughs as Stevenson reported.13However, that in hand stop- ping is completely different and shows rough and arched peaks. The roughness of the peak seems to resemble that of Pmin almost fortissimo playing,33and the arched peak is rather similar to the waveform measured in the trombone mouthpiece in loud playing.12 The rate of change of the mouthpiece pressure dPm/dtis illustrated in Fig. 12(b) . This plot indicates that themaximum of dPm/dtin hand stopping is much larger than that in normal playing, although dPm/dtin normal playing is not shown. Equation (9) gives xc/C251 m when dPm/ dt¼40 MPa/s. Thus the temporal change of Pmin hand stop- ping is large enough to generate the shock wave.Furthermore, other smaller values of dP m/dtat different peaks can cause the wave steepening. Therefore it may be suggested that the nonlinear propagation along the bore char-acterizes not only the brassiness of the fortissimo tones but also the metallicness of the stopped tones. Particularly, the wave corrugation characterizing the metallic stopped tonesis possibly formed by a combination of many minute wave steepenings as a result of nonlinear propagation. This antici- pation will be confirmed by relevant simulations 12,34in near future. Also, some quantitative measure to indicate the met- allicness will be proposed through a more detailed investiga- tion of the wave corrugation in near future. VI. CONCLUSIONS The purpose of our investigation is to clarify the physi- cal cause of the metallic tone quality generated by the hand stopping in the horn. This tonal metallicness is discriminated from the conventional brassiness characterizing brassy tones in brass instruments. It is finally suggested that the primary mechanism responsible for the metallic stopped tones is thepeculiar sound pressure in the mouthpiece and its nonlinear propagation along the bore. The time derivative of the mouthpiece pressure involves several secondary peaks [seeFig.12(b) ], which seem to make up wave corrugations in the radiated pressure as the primary peak produces dominant wave steepening. In other words, many successive wavesteepenings associated with secondary peaks are combined with each other to form wave corrugations characterizing the metallic hand-stopped tones. Before reaching to the final result in the preceding text, acoustical characteristics of the stopped-horn were first calcu- lated based on the stopped-horn model. A simplified geomet-ric model of the hand gave results quantitatively different from the high-pass filter explanation proposed by Backus. 2 This simplified model also suggested that the modes corre- sponded approximately to those of a “closed-closed” pipe. This explains how strong internal pressures could be pro- duced at the right hand position and that those pressures couldin turn induce bell wall vibration with large amplitudes. Based on this prediction, the radiated sound pressure and the associated wall vibration in hand stopping weremeasured. The higher harmonics of the metallic stopped tone were obtained up to 10 kHz, and they seemed to be derived from rapidly corrugating changes found in the soundpressure waveform. This minute indented waveform respon- sible for the metallic stopped tone seemed to be generated by the bell-wall vibration because strong higher harmonicswere included in the wall vibration velocity in hand stopping and the similar rapidly corrugating changes were also observed in loud normal playing. The excitation experiment and the immobilization experiment of the horn bell were carried out to consider how the higher harmonic wall vibration was generated in hand FIG. 12. Typical results of the mouthpiece-pressure measurement. (a) mouthpiece pressure in hand stopping; (b) rate of temporal change of the mouthpiece pressure; (c) radiated sound pressure of the stopped tone. J. Acoust. Soc. Am., Vol. 133, No. 5, May 2013 T. Ebihara and S. Y oshikawa: Nonlinear effects in horn hand-stopping tones 3105stopping. When the amplitude of the sinusoidal exciting force upon the bell wall was increased to 5 N, the superhar-monics due to nonlinear vibration of shell structures were produced on the bell, and the radiated sound pressure dem- onstrated the extremely indented waveform as well as thestopped tone. However, the vibration velocity at the bell edge was too large compared with that generated in real hand stopping. It was thus doubtful that the superharmonicsof wall vibration were generated in hand stopping and that they affected the timbre of the stopped and fortissimo tones. Actually, the immobilization experiment using a quantity ofsand indicated that the rapidly corrugating changes in the stopped tones remained even though the horn was strongly damped by the sand. Also their spectral envelopes did notshow definite differences between a fully damped and a not- damped (free) horn. Finally, the mouthpiece pressure in hand stopping was measured and the critical distance for shock-wave generation was estimated. The waveform of the mouthpiece pressure seemed to be slightly corrugated, and the calculated criticaldistance was less than 1 m. Thus it may be safely suggested that the metallicness in the stopped tones is generated by the nonlinear propagation along the horn bore. Although theshock-wave generation seemed to be responsible for the wave corrugation in the sound pressure of the stopped tones, both waveforms of the mouthpiece pressure and the radiated pres-sure were pretty different from those of the fortissimo tones. Hence numerical simulations of nonlinear propagation in the time domain should be applied to the hand stopping in a horn. ACKNOWLEDGMENTS The authors deeply thank Yuki Kamimura, Ryohei Akanabe, Yu Nobara (Kyushu University), and Aki Jinno (Seinan Gakuin University) for their devoted participation inthe experiments as the horn players. 1P. 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Rossing, The Physics of Musical Instruments , 2nd ed. (Springer-Verlag, New York, 1998), Secs. 7.6–7.8. 32K. Saijyou and S. Yoshikawa, “Analysis of flexural wave velocity andvibration mode in thin cylindrical shell,” J. Acoust. Soc. Am. 112, 2808–2813 (2002). 33L. Norman, J. P. Chick, D. M. Campbell, A. Myers, and J. Gilbert, “Player control of ‘brassiness’ at intermediate dynamic levels in brass instruments,” Acta Acust. Acust. 96, 614–621 (2010). 34J. Gilbert, L. Menguy, and M. Campbell, “A simulation tool for brassiness studies,” J. Acoust. Soc. Am. 123, 1854–1857 (2008). 3106 J. Acoust. Soc. Am., Vol. 133, No. 5, May 2013 T. Ebihara and S. Y oshikawa: Nonlinear effects in horn hand-stopping tonesCopyright of Journal of the Acoustical Society of America is the property of American Institute of Physics and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. 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Inverse spin-Hall effect induced by spin pumping in metallic system K. Ando, S. Takahashi, J. Ieda, Y. Kajiwara, H. Nakayama et al. Citation: J. Appl. Phys. 109, 103913 (2011); doi: 10.1063/1.3587173 View online: http://dx.doi.org/10.1063/1.3587173 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v109/i10 Published by the American Institute of Physics. Related Articles Identifying the roles of the excited states on the magnetoconductance in tris-(8-hydroxyquinolinato) aluminum Appl. Phys. Lett. 102, 113301 (2013) Identifying the roles of the excited states on the magnetoconductance in tris-(8-hydroxyquinolinato) aluminum APL: Org. Electron. Photonics 6, 47 (2013) Chemically functionalized graphene for bipolar electronics Appl. Phys. Lett. 102, 103114 (2013) Tunable magnetic and transport properties of p-type ZnMnO films with n-type Ga, Cr, and Fe codopants Appl. Phys. Lett. 102, 102407 (2013) Ferromagnetism and electronic transport in epitaxial Ge1−xFexTe thin film grown by pulsed laser deposition Appl. Phys. Lett. 102, 102402 (2013) Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 19 Mar 2013 to 129.25.131.235. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsInverse spin-Hall effect induced by spin pumping in metallic system K. Ando,1,a)S. Takahashi,1,2J. Ieda,2,3Y . Kajiwara,1H. Nakayama,1T. Y oshino,1K. Harii,1 Y . Fujikawa,1M. Matsuo,3,4S. Maekawa,2,3and E. Saitoh1,2,3 1Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 2CREST, Japan Science and Technology Agency, Sanbancho, Tokyo 102-0075, Japan 3The Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan 4Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan (Received 24 December 2010; accepted 4 April 2011; published online 23 May 2011) The inverse spin-Hall effect (ISHE) induced by the spin pumping has been investigated systematically in simple ferromagnetic/paramagnetic bilayer systems. The spin pumping driven byferromagnetic resonance injects a spin current into the paramagnetic layer, which gives rise to an electromotive force transverse to the spin current using the ISHE in the paramagnetic layer. In a Ni 81Fe19=Pt film, we found an electromotive force perpendicular to the applied magnetic field at the ferromagnetic resonance condition. The spectral shape of the electromotive force is well reproduced using a simple Lorentz function, indicating that the electromotive force is due to the ISHE induced by the spin pumping; extrinsic magnetogalvanic effects are eliminated in thismeasurement. The electromotive force varies systematically by changing the microwave power, magnetic-field angle, and film size, being consistent with the prediction based on the Landau–Lifshitz–Gilbert equation combined with the models of the ISHE and spin pumping. Theelectromotive force was observed also in a Pt =Y 3Fe4GaO 12film, in which the metallic Ni 81Fe19 layer is replaced by an insulating Y 3Fe4GaO 12layer, supporting that the spin-pumping-induced ISHE is responsible for the observed electromotive force. VC2011 American Institute of Physics . [doi:10.1063/1.3587173 ] I. INTRODUCTION There has been rapidly growing interest in the field of spintronics, which is a new device technology promising ef- ficient magnetic memories and computing devices based onelectron spins. 1–5In this field, the generation, manipulation, and detection of a spin current, a flow of electron spins in a solid, are essential techniques that realize the active controland manipulation of the spin degree of freedom in solid state systems. 6In this stream, intense theoretical and exper- imental interests have been focused on the direct spin-Halleffect (DSHE), which refers to the generation of spin cur- rents from charge currents via the spin-orbit interaction [see Fig. 1(a)]. 7–26The spin-orbit interaction is an outcome of the relativistic quantum theory. While the magnitude of the spin-orbit interaction in the vacuum is quite small, the effects are enhanced in semiconductors and metals depend-ing on the electronic structures. 27The first observations of t h eD S H Ea r ea c h i e v e db yo p t i c a lm e a n si na n n-doped bulk semiconductor15and in 2D hole gas.16The DSHE cou- ples a spin current with a charge current in a solid, and thus will be essential for the integration of spin-current technol- ogy into conventional electronics based on a charge current. The spin-orbit interaction responsible for the DSHE is also expected to cause the inverse process of the DSHE: the inverse spin-Hall effect (ISHE), a process that converts a spincurrent into an electric voltage as shown in Fig. 1(b). 17,28–36 In a solid, existence of a spin current can be modeled as thattwo electrons with opposite spins travel in opposite directions along the spin-current spatial direction Js,a ss h o w ni n Fig.1(b).H e r e , rdenotes the spin polarization vector of the spin current. The spin-orbit interaction bends trajectories of these two electrons in the same direction and induces an electric field EISHEtransverse to Jsandr, which is the ISHE. The relation among EISHE,Js,a n d ris therefore given by28 EISHE/Js/C2r: (1) By measuring EISHE, the ISHE can be used for the direct and sensitive detection of a spin current. Here, note that the direction of the spin-polarization vector rof a spin current can be readily obtained from the magnitude and sign of EISHEbecause of Eq. (1), which is an essential function for a spin-current-detection technique. The ISHE was recently observed using a spin-pumping method operated by ferromagnetic resonance28,34,37(FMR) and by a nonlocal method in metallic nanostructures.17,29,31 The strong spin-orbit interaction in metals, e.g., Pt, allows the observation of the ISHE at room temperature. Since the ISHE enables the electric detection of a spin current, it willbe useful for exploring spin currents in condensed matter. In particular, the spin pumping method requires only a simple ferromagnetic/paramagnetic bilayer film, making it a keytechnique to investigate spin currents in a wide range of sam- ple systems. The spin pumping refers to the generation of spin cur- rents from precessing magnetization MðtÞ; 38–46in a ferro- magnetic/paramagnetic bilayer system, a spin current is pumped out of the ferromagnetic layer into the paramagnetica)Author to whom correspondence should be addressed. Electronic mail: ando@imr.tohoku.ac.jp. 0021-8979/2011/109(10)/103913/11/$30.00 VC2011 American Institute of Physics 109, 103913-1JOURNAL OF APPLIED PHYSICS 109, 103913 (2011) Downloaded 19 Mar 2013 to 129.25.131.235. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionslayer in the FMR condition as shown in Fig. 2. In the model of the spin pumping,39the dc component of a generated spin current density jsis expressed as js¼x 2pð2p=x 0/C22h 4pg"# r1 M2 sMðtÞ/C2dMðtÞ dt/C20/C21 zdt; (2) where x,/C22h,g"# r, and Msare the angular frequency of magnet- ization precession, the Dirac constant, the real part of the mixing conductance, and the saturation magnetization. Here,½MðtÞ/C2dMðtÞ=dt/C138 zis the zcomponent of MðtÞ/C2dMðtÞ=dt. The zaxis is directed along the magnetization-precession axis. In this paper, we investigate the ISHE induced by the spin pumping systematically in ferromagnetic/paramagnetic bilayer systems. This paper is organized as follows. InSecs. IIandIII, we demonstrate the electric detection of spin currents generated by the spin pumping using the ISHE. In Sec. IV, we show the microwave power, film size, and mag- netic field angle dependence of the ISHE signal induced by the spin pumping. In Secs. VandVI, the phenomenologicalformalism of the spin pumping in a thin film system is derived using the model of the spin pumping and the Lan- dau–Lifshitz–Gilbert (LLG) equation. In Sec. VII, we esti- mate the spin-Hall angle of Pt using the model of the spin pumping and the experimental results. The ISHE induced by the spin pumping in a ferrimagnetic insulator/paramagneticmetal system is shown in Sec. VIII. The last Sec. IXis devoted to conclusions. II. EXPERIMENTAL PROCEDURE Figure 3(a) shows a schematic illustration of a sample used in this study. The sample is a Ni 81Fe19=Pt bilayer film comprising a 10-nm-thick ferromagnetic Ni 81Fe19layer (a 0.4/C21.2 mm2rectangular shape) and a 10-nm-thick para- magnetic Pt layer (a 0.4 /C22.2 mm2rectangular shape). These layers were patterned using metal masks. The Pt layer was fabricated by sputtering on a thermally oxidized Si substrateand then the Ni 81Fe19layer was evaporated on the Pt layer in a high vacuum. The thickness of the oxidized layer of the Si substrate is /C24100 nm. Two electrodes are attached to both ends of the Pt layer. For the measurement, the sample system is placed near the center of a TE 011cavity at which the magnetic-field component of the microwave mode is maximized while the electric-field component is minimized [see Fig. 3(c)]. During the measure- ment, a microwave mode with frequency f¼9.44 GHz exists in the cavity and the external magnetic field Halong the film plane is applied perpendicular t o the direction across the elec- trodes, as illustrated in Fig. 3(a). Since the magnetocrystalline anisotropy in Ni 81Fe19is negligibly small, the magnetization in the Ni 81Fe19layer is uniformly aligned along the magnetic field direction. When Handffulfill the FMR condition, a pure spin current with a spin polarization rparallel to the magnet- ization-precession axis in the Ni 81Fe19layer is resonantly injected into the Pt layer by the spin pumping [see Fig. 3(b)].28 This injected spin current is converted into an electric voltage using the strong ISHE in the Pt layer25a ss h o w ni nF i g . 3(b). By measuring the electric voltage, we can detect the ISHEinduced by the spin pumping. We measured the FMR signal and the electric potential difference Vbetween the electrodes attached to the Pt layer. All the measurements were performedat room temperature. III. OBSERVATION OF INVERSE SPIN-HALL EFFECT INDUCED BY SPIN PUMPING IN METALLIC SYSTEMS Figure 4(a)shows the FMR spectra dIðHÞ=dHmeasured for the Ni 81Fe19=Pt film and a Ni 81Fe19film where the Pt layer is missing. Here, Idenotes the microwave absorption intensity. The spectral width W[see the inset to Fig. 4(a)] for the Ni 81Fe19film is clearly enhanced by attaching the Pt layer. This result shows that the magnetization-precession relaxation is enhanced by attaching the Pt layer, since the spectral width Wis proportional to the Gilbert damping con- stant47a. This spectral width enhancement demonstrates the emission of a spin current from the magnetization precession induced by the spin pumping; since a spin current carriesspin-angular momentum, this spin-current emission deprives the magnetization of the spin-angular momentum and thus FIG. 2. (Color online) A schematic illustration of the spin pumping. MðtÞ denotes magnetization. Jsandrare the spatial direction and the spin-polar- ization vector of a spin current, respectively. FIG. 1. (Color online) (a) A schematic illustration of the direct spin-Halleffect. J c,Js, and rdenote a charge current, the spatial direction of a spin current, and the spin-polarization vector of the spin current, respectively. (b) A schematic illustration of the inverse spin-Hall effect. EISHE,Js, and r denote the electromotive force due to the inverse spin-Hall effect, the spatial direction of a spin current, and the spin-polarization vector of the spin cur- rent, respectively.103913-2 Ando et al. J. Appl. Phys. 109, 103913 (2011) Downloaded 19 Mar 2013 to 129.25.131.235. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsgives rise to additional magnetization-precession relaxation, or enhances a. Figure 4(b) shows the dc electromotive force signal V measured for the Ni 81Fe19=Pt film under the 200 mW micro- wave excitation. In the Vspectrum, an unconventional signal appears around the resonance field HFMR. This unconven- tional signal is relevant to the ISHE. It is notable that thespectral shape of this electromotive force is well reproduced using a Lorentz function as shown in Fig. 4(b). The symmetric Lorentz shape of the electromotive force shows that extrinsic electromagnetic effects are eliminated in this measurement; the electromotive force observed here is due entirely to the ISHE induced by the spin pumping. Thein-plane component of a microwave electric field may induce a rectified electromotive force via the anomalous-Hall effect (AHE) in cooperation with FMR [see Figs. 3(d),5(a), and 5(b)]. 28,48–50The electromotive force due to the ISHE and AHE can be distinguished in terms of their spectral shapes.37 Since the magnitude of the electromotive force due to the ISHE induced by the spin pumping, VISHEðHÞ, is propor- tional to the microwave absorption intensity, VISHEðHÞismaximized at the FMR condition. In contrast, the sign of the electromotive force due to the AHE [ /eðtÞ/C2mðtÞ], VAHEðHÞ, is reversed across the ferromagnetic resonance field HFMRas shown in Figs. 5(a) and5(b), where eðtÞand mðtÞare the microwave electric field and the magnetization component perpendicular to eðtÞ, respectively. VAHEinduced byeðtÞ/sinxtandmðtÞ/sinðxtþuÞis proportional to cosu/C0cosð2xtþuÞ. The dc component of this electromo- tive force changes its sign across HFMR, since the magnetiza- tion-precession phase shifts by pat the resonance [see Figs. 5(a)and5(b)].38Therefore, the electromotive force due FIG. 4. (Color online) (a) Field ( H) dependence of the FMR signals dIðHÞ=dHfor the Ni 81Fe19=Pt film and the Ni 81Fe19film. Here, Idenotes the microwave absorption intensity. HFMRis the resonance field. The inset shows the definition of the spectral width Win the present study. (b) Field dependence of the electric-potential difference Vfor the Ni 81Fe19=Pt film under the 200 mW microwave excitation. The open circles are the experi- mental data. The curve in red shows the fitting result using a Lorentz func- tion for the Vdata. (c) The spectral shape of the electromotive force due to the inverse-spin Hall effect (ISHE) and the anomalous-Hall effect (AHE). (d) An atomic force microscope image of the surface of the Ni 81Fe19layer, where the surface roughness Ra ¼1:9/C210/C01nm. FIG. 5. (Color online) A schematic illustration of the anomalous-Hall effect (AHE) in the Ni 81Fe19=Pt film. (a) The dc voltage generation induced by the AHE when H<HFMR. (b) The dc voltage generation induced by the AHE when H>HFMR.x¼2pfandfis the microwave frequency. FIG. 3. (Color online) (a) A sch ematic illustration of the Ni 81Fe19=Pt film used in the present study. His the external magnetic field. (b) A schematic illustration of the spin pumping and the inverse spin-Hall effect in the Ni 81Fe19=Pt film. MðtÞis the magnetization in the Ni81Fe19layer. EISHE,Js,a n d rdenote the electromotive force due to the inverse spin-Hall effect, the spatial direction of a spin current, and the spin-polarization vector of the spin current, respectively. (c) The sample configuration in a TE 011microwave cavity. At the center of the cavity, the magnetic-field component hof the microwave is maximized while the electric field component eis minimized. An external dc mag- netic field Hwas applied perpendicular to the direction across the elec- trodes along the film plane. (d) Schematic illustrations of the sample placed along the longitudinal axis of the cavity at the cavity canter (top panel) and off the cavity center (bottom panel).103913-3 Ando et al. J. Appl. Phys. 109, 103913 (2011) Downloaded 19 Mar 2013 to 129.25.131.235. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsto the ISHE and AHE are of the Lorentz shape and the dis- persion shape, respectively, as shown in Fig. 4(c). IV. MICROWAVE POWER, FILM SIZE, AND MAGNETIC FIELD ANGLE DEPENDENCE OF ELECTROMOTIVEFORCE DUE TO INVERSE SPIN-HALL EFFECTINDUCED BY SPIN PUMPING Figures 6(a) and6(b) show the microwave power PMW dependence of the FMR signal dIðHÞ=dHand the electromo- tive force signal Vfor the Ni 81Fe19=Pt film, respectively. The electromotive force shown in Fig. 6(b) increases with the microwave power, being consistent with the prediction of the model of the spin pumping. Here, the FMR intensityI FMR, defined as the total amplitude of dIðHÞ=dHshown in Fig. 6(a), is proportional to P1=2 MWas shown in the inset to Fig.6(c), indicating that the microwave power ( <200 mW) is lower than the saturation of the ferromagnetic resonance absorption for the present system. In Fig. 6(c), we show the microwave power PMWde- pendence of the electromotive force VISHE, where VISHE is estimated as the peak height of the resonance shape in the V spectrum as shown in Fig. 4(b). Figure 6(c)shows that VISHE increases linearly with the microwave power, which is con- sistent with the prediction of a direct-current-spin-pumping model.39Equation (2)indicates that the dc component of a spin current generated by the spin pumping is proportional to the projection of MðtÞ/C2dMðtÞ=dtonto the magnetization- precession axis. This projection is proportional to the squareof the magnetization-precession amplitude. In this case, therefore, the induced spin current or the electromotive force due to the ISHE is proportional to the square of the magnet-ization-precession amplitude, i.e., the microwave power P MW[see Eq. (12)in Sec. V]. The magnitude of the electromotive force observed in the Ni 81Fe19=Pt film varies systematically by changing the film size. The film size dependence of VISHE for the Ni81Fe19=Pt film is shown in Figs. 7(a)and7(b). The size of the film is characterized by wandlof the Ni 81Fe19layer as shown in the inset to Figs. 7(a)and7(b), respectively. Figure 7(a)shows that the ISHE signal VISHEincreases linearly with w. In contrast, VISHE shows no variation by changing las shown in Fig. 7(b). These results are consistent with the prediction of the spin-pumping induced ISHE, since wandl are the width of the NiFe layer parallel and perpendicular to EISHE, respectively. This demonstrates the spatial uniformity of the spin pumping in such macroscopic systems. The external magnetic field angle dependence of the electromotive force is also consistent with the prediction of the combination of the ISHE and the spin pumping. In Figs. 8(a) and8(b), we show the in-plane magnetic field angle uHdependence of the FMR signal dIðHÞ=dHand the electromotive force signal Vfor the Ni 81Fe19=Pt film, respec- tively. Here, the in-plane angle uHis defined as shown in the inset to Fig. 8(c). The FMR intensities and the resonance fields are almost identical for all the uHvalues, showing that the change in uHminimally affects the resonance condition of the Ni 81Fe19layer. In contrast, the electromotive force sig- nal in the Vspectra decreases as uHdecreases, and vanishes atuH¼90/C14.In Fig. 8(c), the normalized ISHE signal VISHE=Vmaxis plotted as a function of the in-plane magnetic field angle uH. With increasing the magnetic field angle uHfrom uH¼0, VISHE decreases monotonically and changes its sign when 90/C14<uH<180/C14. Notably, this variation is well reproduced using cos uH, being consistent with the model of the ISHE described in Eq. (1); since the spin polarization rof the dc component of a spin current generated by the spin pumping is directed along the magnetization-precession axis, or theexternal magnetic field direction, Eq. (1)predicts V ISHE /jJs/C2rjx/cosuH. Here, jJs/C2rjxdenotes the xcompo- nent of Js/C2r[see the inset to Fig. 8(c)]. Figure 9(c)shows the out-of-plane magnetic field angle hHdependence of the electromotive force Vfor the Ni81Fe19=Pt film. In this measurement, the external mag- netic field was applied at an angle of hHto the normal vec- tor of the film plane, as shown in Fig. 9(a). Here, note that FIG. 6. (Color online) (a) The microwave power dependence of the FMR signal dIðHÞ=dHfor the Ni 81Fe19=Pt film. (b) The microwave power de- pendence of the electromotive force Vfor the Ni 81Fe19=Pt film. (c) The microwave power PMWdependence of the electromotive force VISHEfor the Ni81Fe19=Pt film. VISHE is estimated as the peak height of the resonance shape in the Vspectrum as shown in Fig. 4(c). The inset shows the P1=2 MWde- pendence of the FMR intensity IFMR=Imax FMR, where IFMRis estimated as the total amplitude of dIðHÞ=dHshown in Fig. 6(a).103913-4 Ando et al. J. Appl. Phys. 109, 103913 (2011) Downloaded 19 Mar 2013 to 129.25.131.235. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsthe spin polarization rof a spin current generated by the dc spin pumping is directed along the magnetization-precession axis [see Fig. 9(b)]. With changing hH, the ISHE signal disap- pears at hH¼0, which is consistent with the prediction of the ISHE and the spin pumping. When hH¼0, the precession axis of the magnetization in the Ni 81Fe19layer is directed along the normal axis of the film plane. In this situation, the spin polarization vector of a spin current ris parallel to the flow direction of the spin current Js, and thus Eq. (1)predicts EISHE/Js/C2r¼0, which is consistent with the experimen- tal result. To quantitatively describe the variation of the ISHE signal shown in Fig. 9(c), in the following section, we derive the phenomenological formulation of the spin pumping in thin film systems based on the LLG equation combined with the model of the spin pumping. V. PHENOMENOLOGICAL FORMULATION OF SPIN PUMPING IN THIN FILM SYSTEMS The dynamics of magnetization MðtÞin a ferromagnetic film under an effective magnetic field Heffis described by the LLG equation dMðtÞ dt¼/C0cMðtÞ/C2Heffþa MsMðtÞ/C2dMðtÞ dt: (3) Here, c,a, and Msare the gyromagnetic ratio, the Gilbert damping constant, and the saturation magnetization, respec-tively. Firstly, we consider Eq. (3)in an equilibrium condi- tion, where the equilibrium magnetization direction Mis directed to the zaxis [see Fig. 10(a) ]. Here, we consider a soft ferromagnetic thin film, e.g., Ni 81Fe19, and we neglect the magnetocrystalline anisotropy. The external magnetic fieldHand the static demagnetizing field HMinduced by M are taken into account as the effective magnetic field Heff: Heff¼HþHM; (4) where FIG. 7. (Color online) (a) The Ni 81Fe19-layer width wdependence of the ISHE signal VISHE.wis defined as shown in the inset. The solid circles are the experimental data. The solid line shows the linear fit to the data. (b) The Ni81Fe19-layer width ldependence of the ISHE signal VISHE.lis defined as shown in the inset. The solid circles are the experimental data. The solid lineshows V ISHE=Vl¼0:8mm ISHE ¼1:0. (c) The Pt layer thickness dNdependence of the magnitude of the ISHE signal VISHEwhen the thickness of the Ni 81Fe19 layer is dF¼10 nm. Here, /C22VISHE¼VISHE=VdN¼dF¼10nm ISHE andVdN¼dF¼10nm ISHE is the magnitude of the ISHE signal when dN¼dF¼10 nm. The solid line shows the linear fit to the data. (d) The Ni 81Fe19layer thickness dFdepend- ence of the magnitude of the ISHE signal VISHEwhen the thickness of the Pt layer is dN¼10 nm. FIG. 8. (Color online) (a) The in-plane magnetic field angle uHdependence of the FMR signal dIðHÞ=dHfor the Ni 81Fe19=Pt film. (b) The in-plane mag- netic field angle uHdependence of the electromotive force Vfor the Ni81Fe19=Pt film. Here, the in-plane angle uHis defined as shown in the inset to Fig. 8(c). (c) The in-plane magnetic field angle uHdependence of the ISHE signal measured for the Ni 81Fe19=Pt film. VISHE=Vmaxis the nor- malized spectral intensity extracted by a fitting procedure using Lorentz functions from the measured electromotive-force spectra. uHis the in-plane magnetic field angle. The filled circles are the experimental data. The solidcurve shows cos u H.103913-5 Ando et al. J. Appl. Phys. 109, 103913 (2011) Downloaded 19 Mar 2013 to 129.25.131.235. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsH¼H0 sinðhM/C0hHÞ cosðhM/C0hHÞ0 B@1 CA; HM¼/C04pMscoshM0 sinhM coshM0 B@1 CA:(5) hHandhMare the external-magnetic-field angle and the magnetization angle to the normal axis of the film plane,respectively [see Fig. 10(a) ]. The static equilibrium condition, namely, M/C2H eff¼0, yields an expression, which relates hH andhM,a s 2HsinðhH/C0hMÞþ4pMssin 2hM¼0; (6) where His the strength of the external magnetic field. We then consider the magnetization MðtÞprecession around the zaxis, where MðtÞ¼MþmðtÞas shown in Fig.10(a) .MandmðtÞare the static and the dynamic com- ponents of the magnetization, respectively. We take into account the external magnetic field H, the static demagnetiz- ing field HMinduced by M, the dynamic demagnetization fieldHmðtÞinduced by mðtÞ, and the external ac field hðtÞas the effective magnetic field Heff: HeffðtÞ¼HþHMþHmðtÞþhðtÞ; (7) where HmðtÞ¼/C0 4pmyðtÞsinhM0 sinhM coshM0 B@1 CA; hðtÞ¼heixt 0 00 B@1 CA:(8)A small precession of the magnetization mðtÞ¼(mxeixt; myeixt;0) around the equilibrium direction Mis assumed as a solution of Eqs. (3)and(7).H e r e , x¼2pf,w h e r e fis the microwave frequency. The resonance condition is readily obtained by neglecting the external ac field and the damping term and by finding the eigenvalue of xof Eq. (3).B yi g n o r - ing the second order contribution of the precession amplitude, mxandmy, we find the ferromagnetic resonance condition FIG. 9. (Color online) (a) A schematic illustration of the measurement setup for the out-of-plane magnetic field angle dependence of the ISHE signal. hH is the external-magnetic-field angle to the normal vector of the film plane. (b) A schematic illustration of the inverse spin-Hall effect induced by the spin pumping. hMis the magnetization angle to the normal vector of the film plane. (c) The out-of-plane magnetic field angle hHdependence of the elec- tromotive force Vfor the Ni 81Fe19=Pt film. FIG. 10. (Color online) (a) A schematic illustration of the coordinate system used for describing a ferromagnetic film. MandmðtÞare the static and the dynamic components of the magnetization MðtÞ.His the external magnetic field. hHandhMare the external-magnetic-field angle and the magnetization angle to the normal vector of the film plane, respectively. (b) The magnetic- field-angle hHdependence of the ferromagnetic resonance field HFMRmeas- ured for the Ni 81Fe19=Pt film. The filled circles represent the experimental data. The solid curve is the numerical solution of Eqs. (6)and(9)using pa- rameters 4 pMs¼0:745 T and x=c¼0:319 T. (c) The field-angle hHdepend- ence of the magnetization angle hMfor the Ni 81Fe19=Pt film estimated using Eq.(6). (d) The out-of-plane magnetic-field-angle hHdependence of the ISHE signal VISHEmeasured for the Ni 81Fe19=Pt film. VISHE=Vmaxis the normalized spectral intensity extracted by a fitting procedure using Lorentz functions from the measured electromotive-force spectra. hHis the external magnetic field angle to the normal axis of the film plane. The filled circles are the exper- imental data. The solid curve is the theoretical curve calculated from Eq. (13).103913-6 Ando et al. J. Appl. Phys. 109, 103913 (2011) Downloaded 19 Mar 2013 to 129.25.131.235. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsx c/C18/C192 ¼HFMRcosðhH/C0hMÞ/C04pMscos 2hM ½/C138 /C2HFMRcosðhH/C0hMÞ/C04pMscos2hM/C2/C3 :(9)Here, we used Eq. (6). The dynamic components of the mag- netization mðtÞin the FMR condition are obtained from Eqs. (3)and(7)using Eqs. (6)and(9): mxðtÞ¼4pMshc2axcosxtþ4pMscsin2hMþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4pMs ðÞ2c2sin4hMþ4x2q /C20/C21 sinxt/C26/C27 8paxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4pMs ðÞ2c2sin4hMþ4x2q ; (10) myðtÞ¼/C04pMshccosxt 4paffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4pMs ðÞ2c2sin4hMþ4x2q : (11) Using Eqs. (2),(10), and (11), we obtain spin current density jsas js¼g"# rc2h2/C22h4pMscsin2hMþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð4pMsÞ2c2sin4hMþ4x2q /C20/C21 8pa2ð4pMsÞ2c2sin4hMþ4x2hi : (12)Here, the spin-polarization vector rof the spin current is directed along the magnetization-precession axis, since the dc component of MðtÞ/C2dMðtÞ=dtis directed along the z axis. The phenomenological expression of the out-of-plane magnetic field angle dependence of the ISHE signal, VISHE, is obtained using Eqs. (1)and(12). Since the spin-polariza- tion vector rof a spin current is directed along the zaxis [see Fig. 10(a) ], in our sample system, the dc electromotive force VISHEdue to the ISHE induced by the spin pumping is proportional to JssinhM:VISHE/JssinhMbecause of the relation EISHE/Js/C2r. Therefore, we find the ISHE signal VISHEas VISHE/g"# rc2h2/C22hsinhM4pMscsin2hMþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð4pMsÞ2c2sin4hMþ4x2q /C20/C21 8pa2ð4pMsÞ2c2sin4hMþ4x2hi : (13) We can compare the experimental result shown in Fig. 9(c) with Eq. (13)using the values of the saturation magnetization 4pMsand the magnetization angle hMfor the Ni 81Fe19=Pt film. These parameters are readily obtained from the mag-netic field angle h Hdependence of the ferromagnetic reso- nance field HFMRusing Eqs. (6)and(9). In Fig. 10(b) ,w e show the magnetic field angle hHdependence of the reso- nance field HFMR. Substituting the measured HFMRvalues at hH¼0 and 90/C14into Eq. (9), we obtain 4 pMs¼0:745 T and x=c¼0:319 T. Substituting 4 pMs¼0:745 T and the meas- ured HFMRvalues into Eq. (6), we obtain the hHdependence ofhMas shown in Fig. 10(c) . In Fig. 10(d) , we show the out-of-p lane magnetic field angle hHdependence of the normalized ISHE signal VISHE=Vmax. Notable is the drastic variation observed in VISHE=Vmaxacross hH¼0, which is reminiscent of the varia- tion of hMs h o w ni nF i g . 10(c) . We compared the experimental result shown in Fig. 10(d) (solid circles) with Eq. (13) and found that the experimentally me asured field-angle dependence ofVISHEis well reproduced by Eq. (13)a ss h o w ni nF i g . 10(d) (solid curve), demonstrating t he validity of the models of the ISHE and the spin pumping described in Eqs. (1)and(2).VI. SPIN PUMPING AND MAGNETIZATION- PRECESSION TRAJECTORY When the external magnetic field is applied oblique to the film plane as shown in Fig. 9(a), the magnetiza- tion-precession trajectory, th e trajectory of the point of a magnetization vector is distorted due to a demagnetization fi e l da ss h o w ni nF i g . 11(b) . Since the spin pumping gen- erates spin currents from magnetization precession, the precession trajectory is expected to strongly affect the amplitude of the generated spin currents. In this section,we discuss the amplitude of the spin current generated by the spin pumping in terms of a magnetization-precession trajectory. The amplitude of a spin current generated by the spin pumping is maximized when the magnetization-pre- cession axis is oblique to the film plane. We define thenormalized spin current amplitude ~J sas~Js/C17Js=JhH¼hM¼0 s , where JhH¼hM¼0 s is the spin-current amplitude when the magnetization precesses in a c ircular orbit (the external magnetic field is applied perpendicular film plane: hH¼hM¼0) [see Fig. 11(a) ]. Using Eq. (12),w efi n d ~Js as103913-7 Ando et al. J. Appl. Phys. 109, 103913 (2011) Downloaded 19 Mar 2013 to 129.25.131.235. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions~Js¼2x4pMscsin2hMþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4pMs ðÞ2c2sin4hMþ4x2q /C20/C21 4pMs ðÞ2c2sin4hMþ4x2: (14) In Fig. 12(a) , we show the contour plot of the normalized spin current amplitude ~Jsas a function of the magnetization angle hMand the saturation magnetization 4 pMs, where x¼5:93/C21010s/C01andc¼1:86/C21011T/C01s/C01. Figure 12(a) shows that the magnetization angle hMat which the amplitude of a spin current is maximized depends strongly on the saturation magnetization 4 pMs. Here, the magnetiza- tion angle hMat which ~Jsis maximized is given as sinhM¼3/C01=4ffiffiffiffiffiffiffiffiffiffiffiffiffi 2x 4pMscs : (15) In order to characterize a magnetization-precession trajec- tory, we define the ellipticity Aof a magnetization-preces- sion trajectory as A/C17jmyj=jmxj, where jmxjandjmyjare the major and minor radiuses of the trajectory, respectively [see Fig.11(b) ]. Using Eqs. (10) and(11), we obtain the elliptic- ity of a magnetization-precession trajectory as A¼2x 4pMscsin2hMþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4pMs ðÞ2c2sin4hMþ4x2q :(16) The relation between the spin current amplitude ~Jsand the ellipticity, A, of a magnetization-precession trajectory is obtained by combining Eqs. (14) and (16), which is expressed as ~Js¼4A 1þA2 ðÞ2: (17) This simple expression indicates that the amplitude of a spin current is maximized when the precession trajectory is dis- torted: A¼1=ffiffiffi 3p . A magnetization-precession trajectory is characterized also by the elliptical area of a magnetization-precession tra- jectory, which is defined as S/C17pjmxjjmyj[see Fig. 11(b) ].We define the dimensionless area of a magnetization-preces- sion trajectory as ~S/C17S=ShH¼hM¼0, where ShH¼hM¼0is the area of the magnetization-precession trajectory when the magnetization precesses in a circular orbit [see Fig. 11(a) ]. Using Eqs. (10)and(11), we obtain ~Sas ~S¼2x4pMscsin2hMþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4pMs ðÞ2c2sin4hMþ4x2q /C20/C21 4pMs ðÞ2c2sin4hMþ4x2: (18) This expression of ~Sis exactly the same as that of the spin current amplitude ~Jsin Eq. (14); the spin current amplitude ~Jsis determined by the elliptical area of a magnetization-pre- cession trajectory ~S. The spin-pumping amplitude ~Jsis maxi- mized when the area of magnetization-precession trajectory ~Sis maximized. To demonstrate the validity of these results, we investi- gated the out-of-plane magnetic field angle dependence of the spin-pumping amplitude using the ISHE. In Fig. 12(c) , we show the magnetization angle hMdependence of ~VISHE¼VISHE=sinhMfor the Ni 81Fe19=Pt film and a Ni/Pt FIG. 11. (Color online) (a) The magnetization-precession trajectory when the magnetization-precession axis is perpendicular to the film plane. (b) The magnetization-precession trajectory when the magnetization-precession axis is oblique to the film plane. (c) The magnetization-precession trajectory when the magnetization-precession axis is parallel to the film plane. FIG. 12. (Color online) (a) The contour plot of the normalized spin currentamplitude ~J sobtained from Eq. (14)as a function of the magnetization angle hMand the saturation magnetization 4 pMs. (b) The magnetic field angle hH dependence of the resonance field HFMRfor the Ni 81Fe19=Pt (solid circles) and Ni/Pt films (open circles). The inset shows the hHdependence of the mag- netization angle hM. (c) The magnetization angle hMdependence of the spin pumping efficiency ~VISHE=VhM¼90/C14 ISHE for the Ni 81Fe19=Pt (solid circles) and Ni/ Pt films (open circles), where ~VISHE¼VISHE=sinhM.VhM¼90/C14 ISHE is the ISHE sig- nal measured when hH¼90/C14. The solid curve shows the theoretical curve proportional to Eq. (14). Here, we assume that ais independent of hM.T h e parameters used in the calculations are 4 pMs¼0:745 T and x=c¼0:319 T for the Ni 81Fe19=Pt film and 4 pMs¼0:171 T and x=c¼0:312 T for the Ni/Pt film.103913-8 Ando et al. J. Appl. Phys. 109, 103913 (2011) Downloaded 19 Mar 2013 to 129.25.131.235. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsfilm. ~VISHE is proportional to the amplitude of spin currents Jsgenerated by the spin pumping because of VISHE /JssinhM. The polar angle of the magnetization hMis estimated from the external magnetic field angle hHand the resonance field HFMRusing Eqs. (6)and(9). The hHdepend- ence of HFMRandhMare shown in Fig. 12(b) and the inset to Fig.12(b) , respectively. The experimentally measured mag- netization-angle hMdependence of the spin current ampli- tude ~VISHE is well reproduced using Eq. (14) as shown in Fig.12(c) . Here, note that in contrast to the simple variation of~VISHE observed in the Ni/Pt film, ~VISHE for the Ni81Fe19=Pt film is maximized when the magnetization pre- cession axis is oblique to the film plane, which means that the spin-pumping amplitude is maximized when a magnet- ization-precession trajectory is distorted. VII. ESTIMATION OF SPIN-HALL ANGLE OF PLATINUM The spin-pumping-induced ISHE observed in the Ni81Fe19=Pt film enables the estimation of the spin-Hall angle hSHE/C17rSHE=rNof the Pt layer, where rSHEandrN are the spin-Hall conductivity and the electrical conductivity of the Pt layer, respectively. In this section, we quantify the spin-Hall angle of Pt based on the phenomenological modeldiscussed in Sec. IIIand the experimental results shown in Figs. 4(a)and4(b). In the Ni 81Fe19=Pt film, the spin current injected into the Pt layer decays along the ydirection [see Fig. 13(a) ] due to spin relaxation as jsðyÞ¼sinh½ðdN/C0yÞ=kN/C138 sinhðdN=kNÞj0 s; (19) where jsðyÞdenotes the spin current density. dNandkNare the thickness and the spin-diffusion length of the Pt layer,respectively. The spin current density j 0 sat the interface (y¼0) is obtained using Eq. (12)withhM¼p=2a s j0 s¼g"# rc2h2/C22h4pMscþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð4pMsÞ2c2þ4x2q /C20/C21 8pa2ð4pMsÞ2c2þ4x2hi : (20) The spin current jsðyÞdescribed in Eq. (19) is converted into an electromotive force VISHEusing the ISHE in the Pt layer: VISHE¼½RFRN=ðRFþRNÞ/C138Ic¼w½rNþðdF=dNÞrF/C138/C01hjci. An equivalent circuit model for the Ni 81Fe19=Pt film is shown in Fig. 13(b) . Here, RFandRNare the electrical resist- ance of the Ni 81Fe19and Pt layer, respectively. Ic/C17ldNhjci is the charge current generated by the ISHE. wandlare the width and length of the Ni 81Fe19layer, respectively [see the inset to Figs. 7(a)and7(b)].rFis the electrical conductivity of the Ni 81Fe19layer. dFis the thickness of the Ni 81Fe19 layer. Using Eq. (19), we obtain the averaged charge current density defined as hjci¼ð 1=dNÞÐdN 0jcðyÞdy: hjci¼hSHE2e /C22h/C18/C19kN dNtanhdN 2kN/C18/C19 j0 s; (21)since the ISHE converts a spin current jsðyÞinto a charge current jcðyÞasjcðyÞ¼hSHEð2e=/C22hÞjsðyÞ. Using Eq. (21),w e find the electromotive force due to the ISHE induced by the spin pumping as VISHE¼whSHEkNtanhðdN=2kNÞ dNrNþdFrF2e /C22h/C18/C19 j0 s: (22) Here, the magnitude of the electromotive force VISHE decreases with increasing the thickness of the Pt and Ni81Fe19layers as shown in Figs. 7(c) and7(d), supporting the validity of Eq. (22). The real part of the mixing conduct- ance g"# ris given by51,52 g"# r¼2ffiffiffi 3p pMscdF glBxðWF=N/C0WFÞ; (23) where gandlBare the gfactor and the Bohr magneton, respectively. We assumed that the Pt layer is a spin sink and abackflow spin current is negligibly small. 39WF=NandWFare the FMR spectral width [see the inset to Fig. 4(a)] for the Ni81Fe19=Pt film and the Ni 81Fe19film, respectively. Using the parameters g¼2:12, 4 pMs¼0:745 T, dF¼10 nm, c¼1:86/C21011T/C01s/C01,x¼5:93/C21010s/C01,WF=N ¼7:58 mT ;andWF¼5:34 mT, we find the real part of the mixing conductance at the Ni 81Fe19=Pt interface as g"# r¼2:31/C21019m/C02. Using Eq. (22) with the parameters g"# r¼2:31/C21019m/C02,dN¼10 nm, kN¼10 nm,53 a¼0:0206, h¼0:16 mT, w¼1:2m m , rN¼2:0/C2106 ðXmÞ/C01,rF¼1:5/C2106ðXmÞ/C01,a n d VISHE¼37lV, we obtain the spin-Hall angle of the Pt layer as hISHE¼0:04. The spin-Hall angle hISHE is given by the sum of the side jump and the skew scattering contributions. The side jump contribution is given by hISHE¼ð3=8Þ1=2ðkFkNÞ/C01,54where kFis the Fermi momentum. Using kF/C241010m/C01and kN/C2410 nm, we have hISHE/C240:006 for Pt. This value is one order of magnitude smaller than the experimental value, sug-gesting that the skew scattering is the main mechanism of the spin-Hall effect in the Pt layer. VIII. INVERSE SPIN-HALL EFFECT INDUCED BY SPIN PUMPING IN FERRIMAGNETICINSULATOR/PARAMAGNETIC METAL SYSTEM In this section, we demonstrate that the ISHE voltage appears also in a Pt =Y3Fe4GaO 12film, in which the metallic FIG. 13. (Color online) (a) A schematic illustration of the coordinate system used for describing a spin current jsðyÞ.j0 sis the injected spin current density. dis the thickness of the Pt layer. (b) An equivalent circuit for the Ni81Fe19=Pt film. RFandRNare the resistance of the Ni 81Fe19and Pt layer, respectively. Icis the charge current generated by the ISHE.103913-9 Ando et al. J. Appl. Phys. 109, 103913 (2011) Downloaded 19 Mar 2013 to 129.25.131.235. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsNi81Fe19layer is replaced by an insulating Y 3Fe4GaO 12 layer. This result strongly supports that the ISHE induced by the spin pumping is responsible for the electromotive forceobserved in ferromagnetic/paramagnetic bilayer systems. Figure 14(a) shows a schematic illustration of a Pt=Y 3Fe4GaO 12bilayer film. Here, Y 3Fe4GaO 12is a ferri- magnetic insulator. A polycrystal 100-nm-thick Y 3Fe4GaO 12 film was grown on a 0.7-mm-thick Gd 3Ga5O12(111) single- crystal substrate by metal organic decomposition. Then, a10-nm-thick Pt layer was sputtered on the Y 3Fe4GaO 12layer. Immediately before the sputtering, the surface of the Y3Fe4GaO 12film was cleaned by Ar-ion bombardment in a vacuum. The surface of the Y 3Fe4GaO 12l a y e ri so fa 1.0/C24.0 mm2rectangular shape. Two electrodes are attached to both ends of the Pt layer as shown in Fig. 14(a) . For the measurement, the Pt/Y 3Fe4GaO 12film is placed near the cen- ter of a TE 011cavity. During the measurements, the micro- wave mode with a frequency of 9.44 GHz is excited in thecavity, and an external static magnetic field is applied along the film. The microwave power is 200 mW, a value lower than the saturation of the ferromagnetic resonance absorptionfor the present sample [see the inset to Fig. 14(b) ]. Figure 14(b) shows the microwave absorption signal dIðHÞ=dHand the electric-potential difference Vmeasured for the Pt/ Y 3Fe4GaO 12film when uH¼0. In the Vspec- trum, an electromotive force signal appears at the resonance field. This indicates that the electromotive force is inducedin the Pt layer concomitant with FMR in the Y 3Fe4GaO 12 layer. This electromotive force is found to disappear in aCu/ Y 3Fe4GaO 12film [see Fig. 14(b) ], where the Pt layer is replaced by a Cu layer in which the spin-orbit interaction is very weak,25indicating that the spin-orbit interaction in the Pt layer is responsible for the voltage generation. The in-plane magnetic field angle u Hdependence of the electro- motive force VISHE is shown in Figs. 14(c) and14(d) . The variation of VISHE is consistent with the prediction of the ISHE induced by the spin pumping; VISHE disappears at uH¼90/C14and changes its sign at 90/C14<uH<180/C14. This indicates that the electromotive force observed in thePt=Y 3Fe4GaO 12film is attributed to the ISHE induced by the spin pumping due to the finite mixing conductance of the conduction electrons in the Pt layer.55This result shows that the dynamical exchange between local spins and conduction electron spins exists not only in a Pt/single crys- tal Y 3Fe4GaO 12interface but also in a Pt/polycrystal Y3Fe4GaO 12interface. The appearance of the electromotive force in the Pt =Y3Fe4GaO 12film is the direct evidence that the electromotive force observed in ferromagnetic/paramag-netic bilayer films is attributed to the ISHE induced by the spin pumping; electromagnetic artifacts are irrelevant in the measurement. IX. SUMMARY The inverse spin-Hall effect (ISHE) induced by the spin pumping was investigated systematically in simple ferro- magnetic/paramagnetic bilayer systems. The electromotiveforce due to the ISHE induced by the spin pumping was observed in a Ni 81Fe19=Pt film. The microwave power andthe magnetic field angle dependence of the electromotive force is well reproduced by a model calculation based on the Landau–Lifshitz–Gilbert equation combined with the models of the ISHE and spin pumping. This model calculation showsthat the spin current amplitude is determined by the area of the magnetization-precession trajectory, which is maximized when the external magnetic field is applied oblique to thefilm plane. The ISHE induced by the spin pumping was observed also in a Pt =Y 3Fe4GaO 12film, in which the metal- lic Ni 81Fe19layer is replaced by an insulating Y 3Fe4GaO 12 layer. This result is the direct evidence that the electromotive force observed in ferromagnetic/paramagnetic bilayer sys- tems is attributed to the ISHE induced by the spin pumping.Since the spin pumping enables the spin-current injection into a wide range of systems, this method will be essential for exploring spin currents in condensed matter. ACKNOWLEDGMENTS This work was supported by a Grant-in-Aid for Scientific Research in Priority Area “Creation and control of spin FIG. 14. (Color online) (a) A schematic illustration of the Pt =Y3Fe4GaO 12 film used in the present study. His the external magnetic field and uHis the in-plane magnetic field angle. (b) Field ( H) dependence of the microwave absorption signal dIðHÞ=dHand the electric-potential difference Vfor the Pt=Y3Fe4GaO 12(Pt/YIG) film (blue curve) and the Cu =Y3Fe4GaO 12(Cu/ YIG) film (black curve) under the 200 mW microwave excitation. The inset shows the P1=2 MWdependence of the FMR intensity IFMR=Imax FMR. (c) The in- plane magnetic field angle uHdependence of Vfor the Pt =Y3Fe4GaO 12film. (d) The in-plane magnetic-field-angle uHdependence of the ISHE signal measured for the Pt =Y3Fe4GaO 12film. VISHE=Vmaxis the normalized spec- tral intensity. The filled circles are the experimental data. The solid curve shows cos uH.103913-10 Ando et al. J. Appl. Phys. 109, 103913 (2011) Downloaded 19 Mar 2013 to 129.25.131.235. 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1.4967006.pdf

Dynamics of antiferromagnetic skyrmion driven by the spin Hall effect Chendong Jin , Chengkun Song , Jianbo Wang , and Qingfang Liu, Citation: Appl. Phys. Lett. 109, 182404 (2016); doi: 10.1063/1.4967006 View online: http://dx.doi.org/10.1063/1.4967006 View Table of Contents: http://aip.scitation.org/toc/apl/109/18 Published by the American Institute of Physics Dynamics of antiferromagnetic skyrmion driven by the spin Hall effect Chendong Jin,1Chengkun Song,1Jianbo Wang,1,2and Qingfang Liu1,a) 1Key Laboratory for Magnetism and Magnetic Materials of the Ministry of Education, Lanzhou University, Lanzhou 730000, People’s Republic of China 2Key Laboratory for Special Function Materials and Structural Design of the Ministry of the Education, Lanzhou University, Lanzhou 730000, People’s Republic of China (Received 3 August 2016; accepted 22 October 2016; published online 2 November 2016) Magnetic skyrmion moved by the spin-Hall effect is promising for the application of the generation racetrack memories. However, the Magnus force causes a deflected motion of skyrmion, whichlimits its application. Here, we create an antiferromagnetic skyrmion by injecting a spin-polarized pulse in the nanostripe and investigate the spin Hall effect-induced motion of antiferromagnetic skyrmion by micromagnetic simulations. In contrast to ferromagnetic skyrmion, we find that theantiferromagnetic skyrmion has three evident advantages: (i) the minimum driving current density of antiferromagnetic skyrmion is about two orders smaller than the ferromagnetic skyrmion; (ii) the velocity of the antiferromagnetic skyrmion is about 57 times larger than the ferromagneticskyrmion driven by the same value of current density; (iii) antiferromagnetic skyrmion can be driven by the spin Hall effect without the influence of Magnus force. In addition, antiferromagnetic skyrmion can move around the pinning sites due to its property of topological protection. Ourresults present the understanding of antiferromagnetic skyrmion motion driven by the spin Hall effect and may also contribute to the development of antiferromagnetic skyrmion-based racetrack memories. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4967006 ] Spin Hall effect (SHE) has attracted large interest due to its potential applications in spintronic devices. 1–4In the heavy-metal/ferromagnet bilayer structure, when an in-plane current is injected into the heavy-metal, pure spin currents are generated and flow across the multilayer interface due tothe SHE, these pure spin currents exert spin-transfer torque (STT) on the ferromagnet layer and modify the magnetiza- tion dynamics. 2,4,5Lately, some studies have demonstrated that SHE-induced STT can drive the motion of skyrmion.6,7 The skyrmion-based racetrack memory is promising for building the next-generation magnetic memories owing to its small size, low driving current density and ability of moving around the pinning sites.8–11However, the presence of the Magnus force causes a deflected motion of skyrmion12,13and then leads the skyrmion annihilated at the edge of the nanostripe.8,14 Recently, antiferromagnetic (AFM) skyrmions have been predicted in the AFM materials due to the presence of bulk Dzyaloshinskii-Moriya interaction (DMI).15–19The AFM skyrmion is comprised of two coupled topological spin configurations with opposite topological numbers.15,20When the AFM skyrmion is driven by spin-polarized current, the Magnus forces with opposite direction cancel each other out due to the opposite topological number. Therefore, the AFMskyrmion can move along the injected current without deflection. 15,20In this work, we first create an AFM sky- rmion in the nanostripe by the injection of a current pulse by micromagnetic simulations, and then compare the motion of AFM skyrmion and FM skyrmion driven by the SHE.Moreover, we investigate the SHE-induced AFM skyrmion motion when the pinning site exists.The AFM skyrmion can be considered as two coupled topological spin configurations (skyrmions) with oppositetopological numbers due to the presence of AFM exchangeinteraction as shown in Fig. 1(a). Therefore, the total topo- logical number of the AFM skyrmion is zero. In order to bet- ter understand the structure of the AFM skyrmion, we give the spatial profiles of m x,my, and mzacross the AFM sky- rmion as shown in Fig. 1(b). The results show that the mag- netic moments present alternatively positive and negativearrangement. By micromagnetic simulations, we initially created an AFM skyrmion in the middle of the AFM nanostripe by localinjection of a spin-polarized current pulse with the DMIstrength fixed at 1.2 J/m 3as shown in Fig. 2(a). A perpendicu- lar spin-polarized current (1 /C21013A/m2) is injected into the nanostripe (the initial state is the AFM ground state) in a circleregion with diameter of 20 nm at 0 ns. Under continuous injec-tion of spin-polarized current, a magnetic structure similar to vortex is appeared in the current injection region as shown in the inset at 0.05 ns. The perpendicular spin-polarized currentis turned off at 0.1 ns and followed with 0.9 ns long relaxationof the nanostripe. A stably AFM skyrmion is nucleated in thenanostripe. Following, we investigated the ground states of the nanostripe in the presence of different strength DMI, as shown in Fig. 2(b). For D/C200.7 mJ/m 2, the ground state is an AFM state. For 0.7 mJ/m2<D/C201.2 mJ/m2, the ground state is a circle AFM skyrmion with its size increasing with theincrease of D. For 1.2 mJ/m 2<D/C201.9 mJ/m2, the circle AFM skyrmion is stretched to the edge of the nanostripe. For D>1.9 mJ/m2, the ground state becomes a multidomain structure. Considering that we are investigating the racetrackbased on the AFM skyrmion, Dvaries from 0.8 to 1.2 mJ/m 2 in the following simulations (see supplementary material If o r the model and magnetic simulation details).a)Author to whom correspondence should be addressed. Electronic mail: liuqf@lzu.edu.cn. Tel.: þ86-0931-8914171. 0003-6951/2016/109(18)/182404/5/$30.00 Published by AIP Publishing. 109, 182404-1APPLIED PHYSICS LETTERS 109, 182404 (2016) FM skyrmion can be categorized into Bloch FM sky- rmion and N /C19eel FM skyrmion as the type of DMI: bulk DMI or interfacial DMI.9While, for the negative value of exchange constant as in AFM materials, the AFM skyrmionis classified into Bloch AFM skyrmion and N /C19eel AFM sky- rmion. In this section, we demonstrate that SHE can drivefour types of skyrmion moving along þxdirection of the nanostripe by injecting different directions of in-plane cur-rent j aas shown in Fig. 3. In the Pt/antiferromagnet (ferro- magnet) bilayer system, due to the SHE, pure spin currentsare generated and exert a torque similar to STT on the AFM(FM) materials when injecting an in-plane current into the Ptlayer. The orientation of spins injected into the AFM (FM)materials can be described as 4,6 ~r¼/C0sgnhSHð~z/C2~jaÞ; (1) where hSHis the spin-Hall angle5,21of Pt with the value of 0.07. As shown in Fig. 3(a), we inject an in-plane current ja into Pt along – yaxis. Equation (1)indicates that the pure spin currents in – xdirection flow across the interface and drive the Bloch AFM skyrmion moving along þxdirection. For N /C19eel AFM skyrmion and Bloch FM skyrmion shown inFigs. 3(b) and3(c), the current jais needed injecting in – x direction into Pt, then the skyrmion moves along þxdirec- tion driven by the pure spin currents in þydirection. For N/C19eel FM skyrmion shown in Fig. 3(d),aþydirection in- plane current jais injected into Pt, the pure spin currents þx direction drive the N /C19eel FM skyrmion moving along þx direction. Fig.4(a)shows the velocity of the Bloch AFM skyrmion and Bloch FM skyrmion as a function of ja. The Bloch FM skyrmion starts to move in the nanostripe with the velocityof 0.9 m/s when driven by the current density of 1 /C210 10A/ m2, and the velocity of the Bloch FM skyrmion linearly increases to 88 m/s when jalinearly increases to 1 /C21012A/ m2. For the Bloch AFM skyrmion, its velocity is also a linear function of ja. Interestingly, the Bloch AFM skyrmion moves with the velocity of 0.5 m/s when driven by the current den-sity of 1 /C210 8A/m2. In addition, when jaincreases to 1/C21012A/m2, the Bloch AFM skyrmion can still move in the nanostripe with the velocity of 5046 m/s. Here, we also investigate the N /C19eel skyrmions (FM and AFM skyrmions) driven by the SHE and the results are similar to Bloch sky-rmions (FM skyrmion and AFM skyrmion). Let us comparethe velocity of the AFM skyrmion and FM skyrmion, the FIG. 1. (a) Top-views of a Bloch AFM skyrmion in a region with the size of 100/C2100/C21n m3(x/C2y/C2z), the orange, white, and green represent where the zcomponent of the magneti- zation is positive, zero, and negative, respectively. The small black arrows indicate the in-plane magnetizationdistribution of the skyrmion (b) Spatial profiles of M x/Ms(mx),My/Ms(my), andMz/Ms(mz) across the Bloch AFM skyrmion correspond to the black dot- ted line which is marked in (a). FIG. 2. (a) Creation of an AFM sky- rmion in a 100 nm wide nanostripe (D¼1.2 J/m3) by injection of a 0.1 ns long spin-polarized pulse with the value of 1 /C21013A/m2, the injection region is a circle region with diameter of 20 nm in the middle of the nano- stripe. The snapshots of the nanostripe are shown at the time of 0 ns, 0.05 ns, and 1 ns. (b) The ground state of anAFM skyrmion in the nanostripe in the presence of a different D.182404-2 Jin et al. Appl. Phys. Lett. 109, 182404 (2016) critical driving current density of the AFM skyrmion is about two orders smaller than that of the FM skyrmion and the velocity of the AFM skyrmion is about 57 times larger than that of the FM skyrmion driven by the same value of currentdensity. The velocity of the skyrmion depends strongly on the material parameters. We calculated the influence of the material parameters on the velocity of the skyrmion andfound that the skyrmion velocity is inversely proportional to M s,A, and Ku. Here, the values of Ms,A(absolute value) and Kuof the AFM skyrmion are all much smaller than that of the FM skyrmion. Therefore, when driven by the same value of the current density, the velocity of the AFM skyrmion is far larger than that of the FM skyrmion. Moreover, in therecent literatures, which also reported that the antiferromag- nets are more promising for spintronics than ferromagnets due to the fast magnetic dynamics. 15,22,23The Bloch sky- rmion motion can be described by reducing the Landau- Lifshitz-Gilbert equation to the following equation:6,24 G/C2v/C0aD$/C1vþ4pBR$/0¼p 2/C18/C19 /C1ja¼0; (2) where G¼(0 0 G) is called the gyrocoupling vector repre- senting the Magnus force, G¼4pQrepresents the Magnus force proportional to the skyrmion number, and the skyrmion number can be expressed as Q¼1 4pÐÐ~m/C1@~m @x/C2@~m @y/C16/C17 dxdy,v is the velocity of the skyrmion, ais the Gilbert damping, D$is the dissipative tensor, Bis linked to SHE, R$is the in-plane rotation matrix, /0¼p 2for Bloch skyrmion and /0¼0f o rN /C19eel skyrmion. Fig. 4(b) shows the positions ofthe Bloch FM skyrmion in the nanostripe with the current density of 2 /C21012A/m2at different times. The Bloch FM skyrmion moves closer to the edge of the nanostripe when moving along the þxdirection of the nanostripe due to the Magnus force. The AFM skyrmion is composed of two cou- pled topological objects with opposite topological numbers. Therefore, the Magnus forces with opposite direction canceleach other out. As shown in Fig. 4(c), the Bloch AFM sky- rmion moves along the þxdirection of the nanostripe with- out deflection with j a¼2/C21012A/m2, and the shape of the Bloch AFM skyrmion changes to ellipse because of the large velocity of the Bloch AFM skyrmion. Further increasing the current density to 3 /C21012A/m2as shown in Fig. 4(d), the deformed Bloch AFM skyrmion is stretched to two domain walls at 0.02 ns, and keep the domain walls structure stable moving in the nanostripe. In addition, we investigated the velocity of the AFM skyrmion as a function of the DMI strength with current den- sity fixed at 1 /C21011A/m2as shown in Fig. 5(a). The veloc- ity of the AFM skyrmion increases linearly from 216 to 514 m/s with Dincreasing from 0.7 to 1.2 mJ/m2. Fig. 5(b) shows the sketches of the AFM skyrmion motion with FIG. 3. Four different designs of skyrmion motion driven by the SHE. (a) The motion of Bloch AFM skyrmion driven by the SHE. (b) The motion of N/C19eel AFM skyrmion driven by the SHE. (c) The motion of Bloch FM sky- rmion driven by the SHE. (d) The motion of N /C19eel FM skyrmion driven by the SHE. The small black arrows indicate the magnetization distribution of skyrmions. The color map of the component of magnetization is given in the right of the FIG. For (a) and (b), the orange, white, and green indicate where thezcomponent of the magnetization is positive, zero, and negative, respec- tively. For (c) and (d), the red, white, and blue indicate where the zcompo- nent of the magnetization is positive, zero, and negative, respectively. The yellow arrows indicate the moving direction of skyrmion. The hollow black arrows indicate the direction of injected current ja. FIG. 4. Comparison of the motion of AFM skyrmion and FM skyrmion driven by the SHE. (a) The velocity of Bloch AFM skyrmion and Bloch FM skyrmion as a function of current density ja. For the AFM skyrmion case, Ms¼376/C2103A/m, A¼/C06.59/C210/C012J/m, Ku¼1.16/C2105J/m3, D¼1.2/C210/C03J/m3; For the FM skyrmion, Ms¼580/C2103A/m, A¼1.5/C210/C011J/m, Ku¼8/C2105J/m3, DMI strength D¼3.5/C210/C03J/m3 (see supplementary material I). (b) Top-views of the Bloch FM skyrmion position in the nanostripe with the current density of 2 /C21012A/m2at different times. (c) Top-views of the Bloch AFM skyrmion position in the nanostripe with the current density of 2 /C21012A/m2at different times. (d) Top-views of the Bloch AFM skyrmion position in the nanostripe with the current density of 3 /C21012A/m2at different times.182404-3 Jin et al. Appl. Phys. Lett. 109, 182404 (2016) different Dcorresponding to Fig. 5(a). Comparing the posi- tions of AFM skyrmion at 0 ns, 0.5 ns, and 1.0 ns, we canfind that the velocity of the AFM skyrmion is proportional to theDwhen the AFM skyrmion is driven by the same current density. Many works in the literature show that the skyrmion is topologically protected and can pass around pinning sites. 9,25 In this section, we investigate whether the AFM skyrmion has the ability to move around pinning sites. Fig. 6(a) dis- plays the depinning of the SHE-induced AFM skyrmion motion with an isosceles right triangular notch of zero satu-ration magnetization. The notch depth is in the range 20 to 45 nm with a step size of 5 nm, and the current density is in the range from 1 /C210 10to 15/C21010A/m2. The black square indicates that the AFM skyrmion is blocked and stops in the front of the notch, like the case of notch depth of 30 nm and current density of 5 /C21010A/m2as shown in Fig. 6(b). The green circle indicates that the AFM skyrmion passes thenotch, like the case of notch depth of 30 nm and current den- sity of 8 /C210 10A/m2as shown in Fig. 6(c). The AFM sky- rmion moves around the notch by reducing its size, and then restores to its original size and moves along its initial trajec- tory after passing the notch. The red triangle indicates thatthe AFM skyrmion annihilates when running into the notch as shown in Fig. 6(d), like the case of notch depth of 35 nm and current density of 9 /C21010A/m2. The AFM skyrmion reduces its size before passing the notch, and then it ispinned and annihilated by the notch. In addition, we investi-gate the SHE-induced AFM skyrmion motion with a notch of stronger perpendicular anisotropy (the perpendicular anisotropy constant is 2 /C210 5J/m3for the notch and 1.16/C2105J/m3for the rest of the nanostripe) (see supple- mentary material II). The results show that the annihilation of the AFM skyrmion do not appear for large notch depth,the AFM always passes the notch with a large current den- sity. These results demonstrate that the AFM skyrmion has the property of topological protection. In summary, an AFM skyrmion is first created in the nanostripe by a vertical injection of spin-polarized pulse, and it can stably exist in the nanostripe with DMI strength0.8–1.2 mJ/m 2. Furthermore, we analyze four types of sky- rmions (Bloch AFM skyrmion, N /C19eel AFM skyrmion, Bloch FM skyrmion and N /C19eel FM skyrmion) motion driven by the SHE and compare the SHE-induced motion of FM and AFM skyrmions. The results show that the AFM skyrmion can be driven by a low current density which is about two orderssmaller than that of the FM skyrmion. When skyrmions aredriven by the same value of current density, the velocity ofthe AFM skyrmion is about 57 times larger than that of theFM skyrmion. Moreover, the AFM skyrmion can be driven by the SHE without the influence of Magnus force because it is composed of two coupled topological spin configurationswith opposite topological numbers. At last, we investigate FIG. 5. (a) The velocity of the AFM skyrmion as a function of DMI strength with current density fixed at 1 /C21011A/m2. (b) Top-views of the Bloch AFM skyrmion motion driven by the current density of 1 /C21011A/m2in the presence of a different D. FIG. 6. Depinning of the SHE-induced motion of AFM skyrmion with a tri- angular notch of zero saturation magnetization. Black square, green circle, and red triangle represent that the AFM skyrmion is blocked by the notch, the AFM skyrmion passes the notch, and the AFM skyrmion moves out the nanostripe from the notch, respectively.182404-4 Jin et al. Appl. Phys. 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1.4759313.pdf

Magnetization states of a spin-torque oscillator having perpendicular polarizer and planar free layer Daisuke Saida and Shiho Nakamura Citation: Journal of Applied Physics 112, 083926 (2012); doi: 10.1063/1.4759313 View online: http://dx.doi.org/10.1063/1.4759313 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/112/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Observations of thermally excited ferromagnetic resonance on spin torque oscillators having a perpendicularly magnetized free layer J. Appl. Phys. 115, 17C740 (2014); 10.1063/1.4868494 A generalized tool for accurate time-domain separation of excited modes in spin-torque oscillators J. Appl. Phys. 115, 17D108 (2014); 10.1063/1.4861212 Magnetization dynamics of a MgO-based spin-torque oscillator with a perpendicular polarizer layer and a planar free layer J. Appl. Phys. 112, 083907 (2012); 10.1063/1.4758308 Macrospin and micromagnetic studies of tilted polarizer spin-torque nano-oscillators J. Appl. Phys. 112, 063903 (2012); 10.1063/1.4752265 Nanocontact spin-transfer oscillators based on perpendicular anisotropy in the free layer Appl. Phys. Lett. 91, 162506 (2007); 10.1063/1.2797967 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.212.109.170 On: Tue, 09 Dec 2014 13:18:32Magnetization states of a spin-torque oscillator having perpendicular polarizer and planar free layer Daisuke Saida and Shiho Nakamura Storage Materials and Devices Laboratory, Corporate Research and Development Center, 1, Komukai-Toshiba-cho, Saiwai-ku, Kawasaki 212-8582, Japan (Received 28 September 2011; accepted 25 September 2012; published online 25 October 2012) The dynamics and magnetization structures of spin-torque oscillators (STOs) consisting of a planar free layer and a perpendicular polarizer and having diameters of 10–100 nm are investigated bymicromagnetic and macrospin simulations. For models having a diameter of 50 nm, the current- dependent frequency exhibited three oscillation modes: uniform oscillation, continuous oscillation with edge-localized core of the z-component of the magnetization ( M zcomponent), and non- continuous rotation. The uniform oscillation mode and edge-localized oscillation mode are distinguished from each other by observing the frequencies of the Mzcomponents. Further, we found that the oscillation frequency of the edge-localized mode changed in a step-like fashionunder an external magnetic field, which was not observed in the uniform oscillation mode. The frequency in the edge-localized mode became saturated as the current increased toward the non- continuous mode, with the trajectory of the core gradually moving toward the center of the freelayer. Finally, a vortex was formed and the oscillation stopped. The above behavior was observed in STOs having a diameter of not less than 30 nm. VC2012 American Institute of Physics . [http://dx.doi.org/10.1063/1.4759313 ] I. INTRODUCTION Oscillations of magnetization at the free layer in spin valves and magnetic tunnel junctions caused by spin- polarized d.c. current have attracted much attention due tothe oscillations being in the gigahertz frequency range, 1–7 raising the possibility of utilization in radio-frequency (RF) devices.4The useful features of these spin-torque oscillators (STOs), such as typical diameters of less than 100 nm and wide-range frequency tunability, are promising for opening up new applications for RF devices. STOs consisting of a perpendicular polarizer and a pla- nar free layer2–7oscillate in both directions of current in the absence of an external magnetic field and thus offer advan-tages for practical applications. The polarization direction of injected electrons is initially perpendicular to the free layer. The flow of out-of-plane spin-polarized electrons tends topush the magnetization of the free layer slightly out of plane. This yields the onset of a large perpendicular demagnetizing field around which the magnetization of the free layerprecesses. The features of the oscillations in these STOs vary with current intensity. There exist three regions in the current-dependent frequency. 4,7In the low-current region (region (I)), the magnetization of the free layer oscillates uniformly. The oscillation continuously maintains a domain structure in themiddle-current region (region (II)). A vortex is generated and the oscillation stops in the large-current region (region (III)). There is a discontinuity in the frequency when the currentchanges between region (I) and region (II). The dependence of the frequency on current exhibits saturation in region (II). STOs that have a diameter larger than the exchange length ( l ex¼/H20881(2A/Ms2)), where Ais the exchange stiffness constant and Msis the saturation magnetization, exhibit avortex state or a multidomain state in order to reduce magne- tostatic energy.8In one of the multidomain states, the Mz component of magnetization has a maximum intensity at the edge of a circle and decreases toward the circle center.9 Novosad et al. called this oscillation mode the edge- localized mode. There have been no other discussions of theedge-localized mode in pillar structures. In this paper, the dynamics in an STO consisting of a planar free layer with a perpendicular polarizer are calcu-lated using a micromagnetic simulation. Stray fields having a spatial distribution from the perpendicular polarizer are con- sidered, whereas in-plane external fields are beyond thescope of consideration. The discontinuity in the frequency between region (I) and region (II) appeared to be related to a shift in the oscillation state from the uniform oscillationmode toward the edge-localized mode. The latter mode exhibited quantized behavior against the external field along the perpendicular direction. In the current range betweenregion (II) and region (III), the edge-localized mode changed to a vortex state. This kind of mode shift was observed in STOs having a diameter greater than about 30 nm. II. EXPERIMENTAL Numerical simulations were carried out using a Landau- Lifshitz-Gilbert (LLG) micromagnetics simulator thatincludes spin-transfer torque. 10STOs having a planar free layer and a perpendicular polarizer were modeled as shown in Fig. 1(a). The polarizer consists of a material that has a saturation magnetization ( Ms) of 1.0 /C2106A/m and uniaxial magnetic anisotropy along the z-axis with a value of 8.0/C2105J/m. The thickness of the polarizer was 8 nm in all models. A field with an exchange bias of 8.0 /C2106A/m3was applied to the top surface of the polarizer in order to exclude 0021-8979/2012/112(8)/083926/9/$30.00 VC2012 American Institute of Physics 112, 083926-1JOURNAL OF APPLIED PHYSICS 112, 083926 (2012) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.212.109.170 On: Tue, 09 Dec 2014 13:18:32magnetization fluctuations in the polarizer. Cu with thickness of 8 nm was utilized as a spacer. A material having Ms¼8.0/C2105A/m and an x-axis uni- axial anisotropy of 5.0 /C2102J/m3was used for the free layer.6The typical diameter and thickness of the free layer were 50 nm and 3 nm, respectively. Diameters of between 10 nm and 100 nm were also calculated for comparison.Spin-polarized current flowing through the pinned layer affected the free layer. Positive current corresponds to elec- trons flowing from the free layer to the polarizer. The spinpolarization ( P) and damping constant ( a) were assumed to be 0.6 and 0.01, respectively. 3 The mesh cell size used in the simulations was 2n m/C22n m/C21 nm. Since the exchange length is lex¼5.7 nm, the configured unit cell element sizes are smaller than the exchange length. A mesh cell size of1n m/C21n m/C21 nm was adopted to calculate the magnetiza- tion distributions for diameters of 10 nm and of 20 nm. The M zcomponents of the free layer were extracted from each mesh cell. Values of Mztilting toward the positive and nega- tive current directions correspond to /C01 and þ1, respectively. Macrospin models were also calculated for making a comparison with the dynamics in the micromagnetics. In the macrospin models, STOs of the same size having only onemagnetic moment in each layer were considered by using similar parameters as in the micromagnetics. The effect of a demagnetization field was included to take into account thesize in the free layer. Each simulation was started from an initial magnetiza- tion state consisting of the equilibrium state without anyexternal magnetic field or current. Fast Fourier transforms (FFTs) were performed to obtain the spectrum of the oscilla-tions in the free layer. 2 13or 214data points, corresponding to the time variation of the magnetic moment over a period of 10 ns, were used in the FFT. III. RESULTS AND DISCUSSION A. Current dependence of frequencies and magnetization states Figure 2shows the current-dependent frequency of oscillations in the Mxcomponent. The plot shows the oscilla- tion frequencies that had the largest intensity at each current. The magnetization of the free layer was parallel to the uniax-ial magnetic field ( H k) in the initial state. Gray and white circles correspond to results calculated by the micromag- netics and macrospin models, respectively. The critical cur-rents, from which the STO starts to oscillate and maintains the oscillation for 10 ns, are /C021lA(I c/C0) and 28 lA(Icþ). At currents from /C021 to /C041lA (1.1 to 2.1 /C2106A/cm2) and from 28 to 55 lA (1.4 to 2.8 /C2106A/cm2), the frequency increases with current at a gradient that is almost the same as in the macrospin model. Black arrows indicate discontinu-ities in the frequency. For example, under the negative applied current, a discontinuity was observed in the micro- magnetics at around /C040lA. At currents from /C056 to /C088lA (2.9 to 4.9 /C210 6A/cm2), the frequency saturated in the micromagnetics model whereas the frequency continued to increase in the macrospin model. At I</C090lA, the oscil- lation stopped. These general features are similar to those described by Houssameddine et al.4and Firastrau et al.7 Three regions are defined in the current-dependent fre- quency. Region (I) is the small current range in which the frequency increases with current. A discontinuity in the fre- quency is observed between region (I) and region (II) forFIG. 1. (a) Schematic of a spin-torque oscillator having a perpendicular po- larizer and a planar free layer. (b) Axes of anisotropies (i) with twofold rota- tional symmetry, (ii) with fourfold rotational symmetry, and (iii) with both twofold and fourfold rotational symmetries. FIG. 2. Oscillation frequency versus current. The plot shows the largest peak frequency at each current. Black circles and white circles indicate the results calculated by micromagnetics and by macrospin, respectively. The relationship is divided into three regions denoted (I), (II), and (III). Black arrows indicate discontinuities in the frequency.083926-2 D. Saida and S. Nakamura J. Appl. Phys. 112, 083926 (2012) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.212.109.170 On: Tue, 09 Dec 2014 13:18:32both current polarities. In region (III), the vortex core moves toward the center of the free layer and then the oscillation stops. In terms of the magnetization states in each region, the magnetizations were almost parallel and rotated uniformly in region (I). The characteristics of the spectra of fluctuations in theMzcomponent are similar to the spectra calculated using the macrospin model as discussed in Sec. III B. In region (II), the Mzcomponent exhibited a maximum at the edge of the free layer. An example of the magnetization profile in region (II) is shown in Fig. 3(a). Figure 3(a)(i) shows a snap- shot of the Mzcomponent at time 5 ns at a current of /C060lA (3.1/C2106A/cm2, region (II)). A large Mzcomponent was observed around the left side of the structure. This region rotated in a clockwise direction along the edge of the freelayer. At this current, the magnetization oscillated continu- ously. Figure 3(a)(ii) shows a top view of the magnetization distribution. The magnetization distribution exhibits a fanshape. A region with a large M zcomponent is observed around the y-axis at x¼0 in Fig. 3(a)(ii) . Since the Mzcom- ponent takes a maximum at one point on the edge of thecircle and decreases toward the opposite edge, the oscillation state in region (II) is considered to be an edge-localized mode. 9We call the peak in the Mzcomponent an edge- localized core. As the current increases toward region (III), the frequency saturates. In region (III), a vortex is generated. The core initially rotates around the edge of the free layerand then moves toward its center. Figure 3(b)(i) shows the M zcomponent at time 10 ns at a current of /C0110lA (5.6/C2106A/cm2, region (III)). The vortex core is clearly identifiable at the center of the free layer. We have confirmed that this vortex formation is not related to a current-induced magnetic field, since the vortex also occurred when the fieldwas not considered. The frequency saturation in region (II) and stabilization of the vortex core at the center in region (III) are characteris-tic of STOs having a perpendicular polarizer and a planar free layer in the absence of magnetic fields in the x-yplane. 7We found that regions (II) and (III) can be distinguished from each other at one point in the spectra of the Mzcompo- nents. We also discovered the reason for the frequency satu- ration in region (II). The oscillation behavior before andafter the discontinuity and the change from the fan-like state with edge-localized mode to the vortex mode in Secs. III B andIII C will be discussed later. B. Behavior of oscillations before and after discontinuity Figure 4(a) shows the time variations in the average Mz components. The gray and black lines correspond to currents of/C040lA (region (I)) and /C050lA (region (II)), respec- tively. The black dashed line shows the result of the macro-spin simulation at /C050lA. The inset shows an enlargement of the average M zcomponent during the period from 7 to 8 ns, which exhibits slight fluctuations. Figure 4(b) shows the spectra of these fluctuations from FFT analysis. Although the frequencies are twice as high as those from the Mxcom- ponents in region (II), the fluctuations in region (II) aresmaller than in region (I). The peak frequencies below 40 GHz observed in the M xandMzcomponents are summarized in Table I. Simulation results for Ku¼0 J/m3are also given for comparison. In the macrospin simulation with Ku¼0J / m3, there is no peak in the Mzcomponent spectrum. When Ku¼5.0/C2102J/m3is applied, a frequency two times higher than in the Mxcomponent is observed in the Mzcomponent. In the micromagnetics, the STOs with Ku¼0 J/m3had fre- quencies in the Mzcomponent four times higher than the Mx component at currents of /C040lA (region (I)) and /C050lA (region (II)). Frequencies existed at both two times and four times the frequencies of the Mxcomponent at /C040lA (region (I)). At /C050lA (region (II)), other frequencies were also identified including one almost the same as the Mx component. Figure 1(b)(i) shows anisotropy with twofold rotational symmetry. Owing to the introduction of uniaxial anisotropy FIG. 3. (a)(i) Snapshots of the magnetiza- tion in the free layer at a current of/C060lA after 5 ns. (a)(ii) Top view of the magnetization distribution when the STO is oscillating continuously. The direction of each magnetic moment is indicated by triangles. (b)(i) Snapshot of the vortex mode in the free layer and (b)(ii) top view at a current of /C0110lA after 10 ns.083926-3 D. Saida and S. Nakamura J. Appl. Phys. 112, 083926 (2012) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.212.109.170 On: Tue, 09 Dec 2014 13:18:32along the x-direction, the Mzcomponent is affected by the anisotropy twice in each precession, giving the two times frequency. Figure 1(b)(ii) shows anisotropy with fourfold rotational symmetry. In the calculations, the circular struc-ture of the free layer is described using a square mesh. Since flat surfaces that generate magnetic charges exist in the diag- onals in the x-yplane, anisotropy occurs in these directions. The M zcomponent is affected by anisotropy four times in each precession, giving the four times higher frequency. Figure 1(b)(iii) shows anisotropy having both twofold rota- tional symmetry and fourfold rotational symmetry. In the macrospin calculations, the oscillation frequency of the Mz component was not observed to modulate in the case of Ku¼0 J/m3but was observed to modulate in the case of Ku¼5.0/C2102J/m3owing to the twofold rotational symme- try. Since anisotropy due to the mesh structure does not existin the macrospin simulation, the fourfold rotational symme- try does not occur. However, in the micromagnetics, fourfold rotational symmetry was observed at a current of /C040lAi n the case of K u¼0 J/m3. Both the twofold rotational symme- try and fourfold rotational symmetry were obtained at the two currents of /C040lA and /C050lA in the case of Ku¼5.0/C2102J/m3. The major difference between the cases of/C040lA and /C050lA is the appearance of peaks having frequencies almost the same as the Mxcomponent. These peaks are thought to have arisen from the edge-localizedcore having rotated around the edge of the free layer. The os- cillation modes in the cases of region (I) and region (II) are obviously different. Discontinuities in the current-dependent frequency were observed for both current polarities. Figure 5(a) shows the time variations of the Mzcomponent around the discontinuity under positive current. The black and gray lines correspondto the time variations at currents of 55 lA (region (I)) and 56lA (region (II)), respectively. In the 55 lA case, the mag- netization was tilted in the z-direction up until 1.6 ns, after which the M zcomponent exhibited saturation. At a current of 54 lA, just below 55 lA, the Mzcomponent exhibited almost the same behavior. In the 56 lA case, however, the Mzcomponent varied after 1.6 ns. The inset shows an enlargement of the Mzcomponent during the period from 7 to 8 ns. The oscillation frequencies of the Mzcomponent included frequencies that were two times and four times higher than in the Mxcomponent at 55 lA. At 56 lA, how- ever, frequencies almost the same as in the Mxcomponent were also found. This indicates a change in the oscillation mode. Differences in the behavior of the Mzcomponent between region (II) and region (III) were also observed in the demagnetization energies ( Ed). Figure 5(b) shows the time variations of Ed. The inset shows an enlargement of Ed during the period from 7 to 8 ns. The black and grey lines TABLE I. Frequencies of the Mzcomponent for different Kuin region (I) and region (II). The results of the macrospin simulation are also shown. Current ( lA) Ku(J/m3) Peak frequency of Mxcomponent (GHz) Frequencies of Mzcomponent (GHz) Macrospin /C050 0 6.96 … /C050 5.0 /C21027.08 14.2 Micromagnetics /C040 0 4.15 16.6/32.7 5.0/C21024.15 8.3/16.4/32.7 /C050 0 6.47 25.6 5.0/C21026.35 5.13/6.59/12.7/18.8/25.4/31.7FIG. 4. (a) Time variation of the Mzcomponent. The gray and black lines correspond to the Mzcomponent at currents of /C040lA (region (I)) and /C050lA (region (II)), respectively, as calculated by micromagnetics simulation. The black dashed line shows the average value of the Mzcomponent at /C050lA as cal- culated by macrospin simulation. The inset shows an enlargement of the Mzcomponent during the period from 7 to 8 ns. (b) Spectra of the Mzcomponent cal- culated from the time variations in (a).083926-4 D. Saida and S. Nakamura J. Appl. Phys. 112, 083926 (2012) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.212.109.170 On: Tue, 09 Dec 2014 13:18:32correspond to the time variations at currents of 55 lA (region (I)) and 56 lA (region (II)), respectively. Up until 1.6 ns, Ed exhibited the same behavior in the 55 lA and 56 lA cases. After 2 ns, however, the amplitude of the oscillation in Ed remained constant in the 55 lA case but decreased in the 56lA case. In addition, the average value of Edincreased, and the Mzcomponent tilted even further along z-direction, resulting in higher oscillation frequencies. The change between modes was also observed in the distribution of magnetizations. The Mzcomponent in the free layer at time 10 ns is shown in Fig. 6(a). The minimum, max- imum, and averaged Mzvalues are also indicated together with the magnetization according to the macrospin model.An abrupt change in the maximum and minimum values oftheM zcomponent is observed between region (I) and region (II). Figure 6(b) shows typical histograms of the distribution ofMzcomponents in each region. In region (I), the Mzcom- ponent is somewhat spread out, although each individualmagnetization oscillates uniformly. The distribution becomes broader in region (II). At the boundary between regions (I) and (II) at a current of 54 lA, the distribution nar- rows and the shape changes transiently. The change in the distribution of the M zcomponent is attributed to the abrupt change in the maximum and minimum Mzvalues. The response to a magnetic field was different between region (I) and region (II). Figure 7shows the peak frequency in the Mxcomponent versus the magnetic field when a mag- netic field is applied along the z-direction. White rectanglesFIG. 5. (a) Time variation of the average value of the Mzcomponent. The black and gray lines correspond to the average value of the Mzcomponent at currents of 55 lA (region (I)) and 56 lA (region (II)), respectively, as calculated by micromagnetics simulation. The inset shows an enlargement of the Mzcomponent during the period from 7 to 8 ns. (b) Time variation of the demagnetization energy ( Ed). The gray line is superimposed on the black line. The inset shows an enlargement of the Edcomponent during the period from 7 to 8 ns. FIG. 6. (a) Current dependence of the Mzcomponent. Gray circles, white circles, and triangles correspond to minimum, maximum, and average values of Mz in the free layer. Rectangles indicate the macrospin simulation. (b) Histograms of typical distributions of the Mzcomponent in each region.083926-5 D. Saida and S. Nakamura J. Appl. Phys. 112, 083926 (2012) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.212.109.170 On: Tue, 09 Dec 2014 13:18:32indicate the response of the uniform oscillation mode at a current of /C040lA (region (I)). White circles correspond to the response of the edge-localized mode at a current of /C050lA (region (II)). The magnetization of the free layer tilts toward the þzdirection under an external magnetic field with positive polarity. The oscillation frequency does not change with the external magnetic field in the case of the uniform oscillation mode. This feature is also observed in the macrospin model. In contrast, two features were observed in the edge-localized mode. First, the oscillationfrequency increases with increasing magnetic field along z-direction. Second, the frequency changes in a staircase pat- tern. For example, under an external field in the range of/C01.7/C210 4A/m to /C02.4/C2104A/m, the oscillation frequency remains constant at 6.10 GHz. Although the frequency does not change, the phase of the oscillation was found to vary.These features were observed throughout all of region (II). Although we have not determined the reason for the appear- ance of this quantized behavior, quantization effects havebeen predicted to appear when the size of the free layer becomes comparable to the wavelength of a spin wave. 11 The observed effect may relate to this prediction. C. Frequency saturation and generation of the vortex In the edge-localized mode (region (II)), the current- dependent frequency exhibits saturation at currents from /C055lAt o/C088lA and from 80 lAt o9 9 lA. The average value of the Mzcomponent also saturated when the fre- quency became saturated.4,7We examined the trajectories of the edge-localized core, with typical trajectories illustrated in Fig. 8(a). The black line indicates the edge of the free layer. The gray line shows the trajectory of the edge- localized core at a current of /C060lA (region (II)). The core rotates along the edge of the free layer. The gray dashed lineslightly inside the edge of the free layer represents the trajec- tory at a current of /C080lA (region (II)). We confirmed that the angular velocities of the cores rotating along the edge ofthe free layer corresponded to the frequencies shown in Fig.2. The radii of the trajectories of the edge-localized cores versus current are plotted in Fig. 8(b) as rectangles. The radii decreased throughout the current range where the frequencywas saturated. In this current range, E dwas also found to sat- urate. Since the frequency is determined by the effectivemagnetic field, which is dominated by E d, the oscillation fre- quency was also expected to saturate. However, the magne- tostatic energy ( Em) decreased when the edge-localized core moved toward the center of the free layer giving a curling magnetization distribution. Figures 9(a) and9(b) show the evolutions of the Mxand Mzcomponents at a current of /C0110lA. The black and gray lines correspond to the evolutions calculated by micromag- netics and macrospin, respectively. In the macrospin calcula-tions, the oscillation of the M xcomponent decreased gradually up until 3 ns, and after 3 ns, the magnetization became parallel to the z-direction. However, the MxandMz components exhibited different behavior in the micromag- netics calculations after 0.5 ns. During the period fromFIG. 7. Dependence of the peak frequency in the Mxcomponent on the mag- netic field at currents of /C040lA (open squares) and /C050lA (open circles). FIG. 8. (a) Trajectories of the edge-localized cores and vortex cores. The outer black circle indicates the edge of the free layer. The gray line, the gray dashed line, and the black line correspond to the trajectories at currents of /C060lA,/C080lA, and /C0110lA, respectively, (b) Radii in the trajectory of the edge-localized core and vortex core at different currents at time 2 ns.The vortex cores indicated by triangle markers moved to the center of the free layer after 2 ns.083926-6 D. Saida and S. Nakamura J. Appl. Phys. 112, 083926 (2012) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.212.109.170 On: Tue, 09 Dec 2014 13:18:32approximately 1.6 ns to 3.0 ns, the oscillations in the Mx component maintained virtually the same amplitude but the frequency varied. A vortex was generated with a core that rotated slightly inside the edge of the free layer. The radius of the trajectory during the first 2 ns is plotted in Fig. 8(b). Although we do not define the frequency precisely owing to the dissipation process, it seems to oscillate at about 4 GHz. The trajectory of the vortex core after 3 ns is plotted in Fig.8(a) as a black line. It describes a circular orbit with decreas- ing trajectory radius until finally the vortex core is becomes fixed at the center of the free layer, at around 10 ns. At thispoint, the magnetization at the center of the core is perpen- dicular to the plane, whereas the magnetizations around the core are parallel to the plane. The M zcomponent, therefore, exhibits the wide distribution shown in Fig. 6(b). The mode shift between region (II) and region (III) can be clarified by examining the minimum values of the Mzcomponent as shown in Fig. 6(a). Vortex behavior is demonstrated in Ref. 12. The steady trajectory of the vortex is determined by a balance betweenfour components: spin-transfer torque, damping force, restor- ing force ( F R), and gyro force ( FG). When the vortex core is rotating around close to the center of the free layer, it has asmall angular velocity. This state is called a translation mode. 12,13The typical oscillation frequency is below 2 GHz, although it depends on the ratio between the radius ( R) and the thickness ( L) of the free layer. In Fig. 9(a), the angular velocity decreased after 3 ns at a frequency of less than 1 GHz. This behavior may be related to the translation mode.Since F Ris larger than FGafter 3 ns in our calculation, the vortex core gradually moves toward the center. The high- frequency oscillation in the vortex mode has been reportedto be a radial eigenmode. 8If the radial eigenmode had occurred in our sample, the frequency would be estimated to be 10 GHz.9Although the frequency is not the same in Fig. 2, the edge-localized mode may vary due to the radial eigen- mode at 3 ns. D. Requirement for oscillation As shown in Fig. 2, the STO did not oscillate at currents of/C020lAo r2 0 lA (1.0 /C2106A/cm2). Since the response oftheMxcomponent to the current was almost the same in the micromagnetics and macrospin models near the critical cur-rent, the critical current density ( J c) for the oscillation can be described by the macrospin model. Taking the LLG equation into account,14the time variation in the magnetization in the free layer can be described as follows: 1þa2 c/C1d~M dt¼/C0 ~M/C2ð~Hef fþaaJ~pÞþa Ms~M /C2~M/C2ð /C0 ~Hef fþaJ a~pÞno ; (1) where cis the gyromagnetic ratio, aJis the amplitude of the spin-transfer torque,14and~pand~Hef fcorrespond to the unit vectors of the pinned layer and effective magnetic field, respectively. The first term in Eq. (1)represents a precession of the magnetization. The second term describes a damping effect. The magnetization shifts toward the direction of /C0~Hef fþaJ a~p. In the steady-state oscillation, the balance between the spin-transfer torque that pushes the magnetiza- tion out-of-plane and the damping torque that pulls the mag- netization in-plane satisfies the following equation: /C0~Hef fþaJ a~p¼0: (2) The same forms as used by Liu et al.5are adopted for ~p and~Hef f ~p¼~z;~Hef f¼HkMx Ms~xþðHpin/C04pNzMzÞ~z; (3) where Hkis a uniaxial anisotropy magnetic field, Hpinis the stray field from the pinned layer at the center of the freelayer, and 4 pN zMzis a demagnetization field due to the Mz component of the free layer. In the case of Fig. 1(a),Hpin takes a negative value. Considering the Mzcomponent, the critical current of the STO having a stray field from the pinned layer is given by Jc¼2e /C22h/C18/C19Ms/C1t gðhÞ/C20/C21Hk 2þaðHpin/C04pNzMzÞ/C18/C19 :(4)FIG. 9. Evolution of (a) the average Mxcomponent and (b) the average Mzcomponent as calculated by micromagnetics during 10 ns at a current of /C0110lA. The gray lines correspond to the results calculated by the macrospin model.083926-7 D. Saida and S. Nakamura J. Appl. Phys. 112, 083926 (2012) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.212.109.170 On: Tue, 09 Dec 2014 13:18:32Here, gis the spin-torque efficiency factor from Slonc- zewski’s model, Jcis a function of the angle ( h) between the magnetizations of the free layer and of the pinned layer, andtcorresponds to the thickness of the free layer. Equation (4) shows that J cdepends on the condition of h. Using Eq. (4), Jcis estimated to be /C021lA, which is the same value as that observed from the micromagnetics calculation for an STO with a diameter of 50 nm. For the STO with a diameter of 24 nm, Jcfrom Eq. (4)also coincides with the value obtained by micromagnetics. This indicates that the stray field from the pinned layer affects the oscillation in the STO. Equation (4)shows that Jcdepends on Hk. From the micromagnetics simulation, the critical currents at Ku¼1.0 /C2103J/m3and Ku¼2.0/C2103J/m3were estimated to be 40lA and 60 lA, respectively. It is important to note that no oscillations were obtained at Ku¼5.0/C2103J/m3in the micromagnetics simulations. When the value of Kuwas increased, the current range of oscillation became narrower.In order to obtain continuous oscillations, Eq. (2)needs to be satisfied. Since the maximum amplitude of the demagnetiza- tion field is 4 pN zMs, the contribution from the spin-transfer torque needs to be smaller than this. aJ a/C204pNzMs: (5) Therefore, 1 a/C22h 2e/C1gðhÞ MstJ/C26/C27 /C204pNzMs: (6) Rearranging Eq. (6)and substituting MsforMz, gives the following relationship: Ku/C20aMsð8pNzMs/C0HpinÞ: (7) To obtain an oscillation, the value of Kuneeds to satisfy Eq.(7), resulting in an estimate of 1.3 /C2104J/m3as the maxi- mum value of Kuthat gives an oscillation. Since the demag- netization varies spatially in the free layer where the vortex exists, the intensity is expected to be less than 4 pNzMs.T h i s might generate the difference between the maximum valuesofK uthat is needed for oscillation from analytical estimates and micromagnetics simulations. From Eqs. (4)and(7),w e understand that the selection of a small Kuis one of the key issues concerning operation of the oscillator at lower currents. E. Phase diagram For the STO with a diameter of 20 nm, the current- dependent frequency exhibited the same behavior as the macrospin model in the current range from /C03lAt o/C09lA (0.95 – 2.9 /C2106A/cm2) and from 3 lAt o1 5 lA (0.95 – 4.6/C2106A/cm2). Discontinuities in the frequency were not observed and the frequency did not exhibit a steppedresponse to an applied magnetic field along z-direction. These results indicate that the dynamics of the magnetization oscillation are size-dependent. A phase diagram of the oscillation modes in models having different diameters (10–100 nm) and thicknesses(1–5 nm) was examined. For disk-shaped magnetic nano- structures, the magnetization modes, which appear in the static state, depend on the size of the free layer normalizedby the exchange length. 15The results are summarized in Fig. 10, in which Dandtcorrespond to the diameter and thick- ness of the free layer, respectively. Circles show the configu-rations of the free layer where both the edge-localized mode and the vortex mode are observed. Another example of a pla- nar free layer with a perpendicular polarizer 4is also plotted in the region of the circles. Rectangles indicate configura- tions of the free layer where only the uniform oscillation mode is obtained. The solid gray line indicates the boundarythat distinguishes between the single domain and the vortex with no current. 15In our results, the boundary where the vor- tex exists differs compared to the solid gray line. This isbecause we considered the spin-transfer torque and the stray field from the polarizer. IV. CONCLUSIONS The magnetization structures of STOs consisting of a planar free layer and a perpendicular polarizer and having a diameter of 50 nm and thickness of 2 nm were investigatedby micromagnetic and macrospin simulations. Three oscilla- tion modes were observed in the current-dependent fre- quency. In the uniform oscillation mode (region (I)), themagnetization of the free layer oscillated uniformly. In the edge-localized mode (region (II)), the M zcomponent exhib- ited maximum intensity at the edge of the free layer anddecreased toward the opposite edge, and in the vortex mode (region (III)), a vortex occurred and the oscillation stopped. These mode shifts were observed in STOs of size greaterthan approximately 30 nm. When the oscillation changed from the uniform oscilla- tion mode to the edge-localized mode, a discontinuityoccurred in the frequency. The mode shift from region (I) to region (II) could also be distinguished by examining theFIG. 10. Phase diagram of dynamics in free layers of different sizes. Rectan- gles correspond to cases that oscillated similarly to the macrospin model. Circles represent cases that formed a vortex core in region (III). The gray line corresponds to the boundary above which the vortex mode is stable in the static state.15083926-8 D. Saida and S. Nakamura J. Appl. Phys. 112, 083926 (2012) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.212.109.170 On: Tue, 09 Dec 2014 13:18:32maximum and minimum values of the Mzcomponent in the free layer. In the edge-localized mode, the current-dependent frequency exhibited saturation with the saturation corre-sponding to a change in the trajectory of the edge-localized core. The frequency in this mode also exhibited a stepped response to a magnetic field applied along the z-direction. Inregion (III), a vortex core occurred around the edge of the free layer. The mode shift between region (II) and region (III) could be identified by examining the minimum valuesof the M zcomponent. During the first several ns after the vortex occurred, the core of the vortex rotated with a high angular velocity, and then the core gradually moved towardthe center with decreasing angular velocity. The behavior suggests a mode shifts from the radial eigenmode to the translation mode. Since the structure consisting of a planar free layer and a perpendicular polarizer maintains oscillations in the giga- hertz range at ultralow current densities, it is suitable forapplication in RF devices. 1S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J. Schoel- kopf, R. A. Buhrman, and D. C. Ralph, Nature 425, 380 (2003).2O. Redon, B. Dieny, and B. Rodmacq, U.S. patent 6,532,164 (March 11, 2003). 3A. Kent, B. Ozyilmaz, and E. Barco, Appl. Phys. Lett. 84, 3897 (2004). 4D. Houssameddine, U. Ebels, B. Delaet, B. Rodmacq, I. Firastrau, F. Pon- thenier, M. Brunet, C. Thirion, J. Michel, L. Prejbeanu-Buda, M. Cyrille, O. Redon, and B. Dieny, Nature Mater. 6, 447 (2007). 5K. J. Lee, O. Redon, and B. Dieny, Appl. Phys. Lett. 86, 022505 (2005). 6Y. Liu, H. He, and Z. Zhang, Appl. Phys. Lett. 91, 242501 (2007). 7I. Firastrau, D. Gusakova, D. Houssameddine, U. Ebels, M. Cyrille, B. Delaet, B. Dieny, O. Redon, J. Toussaint, and L. Buda-Prejbeanu, Phys. Rev. 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1.4975694.pdf

Effect of nanostructure layout on spin pumping phenomena in antiferromagnet/ nonmagnetic metal/ferromagnet multilayered stacks A. F. Kravets , Olena V. Gomonay , D. M. Polishchuk , Yu. O. Tykhonenko-Polishchuk , T. I. Polek , A. I. Tovstolytkin , and V. Korenivski Citation: AIP Advances 7, 056312 (2017); doi: 10.1063/1.4975694 View online: http://dx.doi.org/10.1063/1.4975694 View Table of Contents: http://aip.scitation.org/toc/adv/7/5 Published by the American Institute of Physics Articles you may be interested in Spintronics of antiferromagnetic systems (Review Article) AIP Advances 40, (2014); 10.1063/1.4862467 Spin transport in antiferromagnetic NiO and magnetoresistance in Y3Fe5O12/NiO/Pt structures AIP Advances 7, 055903055903 (2016); 10.1063/1.4972998 Ferromagnetic resonance and interlayer exchange coupling in magnetic multilayers with compositional gradients AIP Advances 7, 056307056307 (2017); 10.1063/1.4974282 Non-volatile spin wave majority gate at the nanoscale AIP Advances 7, 056020056020 (2017); 10.1063/1.4975693 Stable tetragonal phase and magnetic properties of Fe-doped HfO2 nanoparticles AIP Advances 7, 056315056315 (2017); 10.1063/1.4976583AIP ADV ANCES 7, 056312 (2017) Effect of nanostructure layout on spin pumping phenomena in antiferromagnet/nonmagnetic metal/ferromagnet multilayered stacks A. F. Kravets,1,2,aOlena V. Gomonay,3,4D. M. Polishchuk,1,2 Yu. O. Tykhonenko-Polishchuk,1,2T. I. Polek,1A. I. Tovstolytkin,1 and V. Korenivski2 1Institute of Magnetism, National Academy of Sciences of Ukraine, 03680 Kyiv, Ukraine 2Nanostructure Physics, Royal Institute of Technology, 10691 Stockholm, Sweden 3Institut f ¨ur Physik, Johannes Gutenberg Universit ¨at Mainz, D-55099 Mainz, Germany 4National Technical University of Ukraine “KPI”, 03056 Kyiv, Ukraine (Presented 3 November 2016; received 23 September 2016; accepted 7 November 2016; published online 6 February 2017) In this work we focus on magnetic relaxation in Mn 80Ir20(12 nm)/Cu(6 nm)/Py( dF) antiferromagnet/Cu/ferromagnet (AFM/Cu/FM) multilayers with different thickness of the ferromagnetic permalloy layer. An effective FM-AFM interaction mediated via the conduction electrons in the nonmagnetic Cu spacer – the spin-pumping effect – is detected as an increase in the linewidth of the ferromagnetic resonance (FMR) spectra and a shift of the resonant magnetic field. We further find exper- imentally that the spin-pumping-induced contribution to the linewidth is inversely proportional to the thickness of the Py layer. We show that this thickness dependence likely originates from the dissipative dynamics of the free and localized spins in the AFM layer. The results obtained could be used for tailoring the dissipative proper- ties of spintronic devices incorporating antiferromagnetic layers. © 2017 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). [http://dx.doi.org/10.1063/1.4975694] Antiferromagnets (AFMs) are attractive materials for spintronic applications. They operate at high frequencies and thus have the potential to functionally fill the “terahertz gap” in electronics. Due to their lack of a macroscopic magnetic moment, AFMs produce no stray fields and therefore potentially can provide higher scalability for magnetic memory devices. High typical values of the spin-flop fields prevent AFMs from spontaneous thermally-induced switching and increase the data retention times. In addition, recent experimental1and theoretical investigations2have shown that AFMs are sensitive to spin-polarized currents and can be used as active elements in spintronic devices. Direct observation of spintronic effects in AFMs is challenging due precisely to the same reasons that make AFMs competitive with their ferromagnetic counterparts: the magnetoresistance in AFM- based devices is low due to the absence of net magnetization in AFM, and the dynamics require very high excitation frequencies, beyond the capabilities of microwave circuits. An alternative technique to detect the spin dynamics of AFM films was recently implemented by a number of groups.3–7This technique is based on the spin pumping effect, which is reciprocal to the spin-transfer torque effect.8,9 A metallic ferromagnetic layer (FM) is excited at its resonance frequency (FMR) and pumps spin current into a neighbouring non-magnetic layer interfaced with an antiferromagnetic film (AFM) at the other surface. The linewidth of the FMR spectrum increases due to the presence of the AFM layer and thereby provides information about the interaction of the nonequilibrium conduction-electron spins and the localized AFM moments. aElectronic mail: anatolii@kth.se 2158-3226/2017/7(5)/056312/6 7, 056312-1 ©Author(s) 2017 056312-2 Kravets et al. AIP Advances 7, 056312 (2017) The interpretation of such experiments is not quite straightforward, however, as different processes contribute to the effective damping in a multilayered sample: spin-dependent scattering at the interfaces10and in the bulk, energy exchange between the free and localised spins, spin-diffusion, etc. An efficient theoretical approach to this problem, based on nonequilibrium thermodynamics, was proposed in Ref. 11 for ferromagnetic (FM)/nonmagnetic (NM) bilayers, and was further generalized for FM/NM/FM systems.12Spin-pumping from an AFM layer was recently predicted in Refs. 13 and 14. In this paper we focus on the dissipative response, expressed via the FMR linewidth, of MnIr/Cu/Py multilayers with different thickness of the Py layer. We generalize the Onsager for- malism for the case of the discrete system AFM/NM/FM and calculate the effective Gilbert damping of the FM layer, taking into account the spin-pumping and spin-accumulation effects in both the FM and AFM layers. While the previous experiments3,4have studied the damping dependence vs thickness of the AFM layer, we focus on the properties of the FM layer and especially the FM/NM interface. Our experiments reveal an inverse dependence of the additional, AFM-induced damping on the thickness of the FM layer, in agreement with our theoretical predictions. Our results should be useful for tailoring dissipation in spintronic devices. For the experiments we use multilayers Substrate/Ta(5)/Py(3)/Mn 80Ir20(12)/Cu(6)/Py( dF)/Al(4), hereinafter AFM/Cu/FM( dF), with the FM layer of variable thickness, dF= 3, 6, 9, 12, 15 nm. The numbers in parenthesis denote thickness in nanometers of the corresponding layers; Py = Ni80Fe20. In these multilayers, Mn 80Ir20(12), Cu(6) and Py( dF) form the functional combination of the AFM/NM/FM stack, while the other layers are auxiliary. The top Al layer is a protective cap- ping layer. The bottom layers facilitate the formation of the optimal crystalline and magnetic structure of Mn 80Ir20(12). We also fabricated a set of reference samples with identical structure but without Py(3)/Mn 80Ir20(12) layers. The multilayers were deposited at room temperature (295 K) on thermally oxidized silicon substrates using magnetron sputtering in an AJA Orion 8-target system.15The base pressure in the deposition chamber was 5 108Torr and the Ar pressure used during deposition was 3 mTorr. The exchange pinning between Py(3) and Mn 80Ir20(12) layers was set in during the deposition of the multilayers using an in-plane magnetic field of 1 kOe. We use an X-band ELEXSYS E500 spectrometer equipped with an automatic goniometer to measure the out-of-plane and in-plane angular dependencies of the FMR spectra. The operating frequency is 9.85 GHz, the temperature is 295 K. The spectra show no signal from the Py(3) buffer layer, while the signal from Py( dF) is clearly visible. We record the magnetic-field derivative of the microwave absorption and fit each spectrum by a Lorentzian function to obtain the resonance field Hr and the linewidth
in the in-plane and the out-of plane geometries [Fig. 1(b)]. Typical FMR spectra measured for the in-plane orientation are shown in the inset to Fig. 2(a). When FMR is excited, a moving magnetization in the FM pumps a spin current into the NM and AFM layers.16The spin current is proportional to the effective field HF, which determines the FIG. 1. (a) Schematic view of the energy and spin exchange within a trilayer system FM/NM/AFM. The magnetic layers (FM and AFM) are symbolically separated into subsystems of localized (coloured area) and free (white area) spins. Wide arrows show the fluxes that originate from different mechanisms. Vertical arrows correspond to spin exchange between the localized and free spins inside the FM and AFM layers. (b) Schematic view of the FMR experiment, where HkxOy is the in-plane geometry and HkxOzis the out-of-plane geometry.056312-3 Kravets et al. AIP Advances 7, 056312 (2017) FIG. 2. (a) In-plane resonance field Hr(triangles) and linewidth
(circles) vs thickness dFof the Py layer for AFM/Cu/FM( dF) multilayers (bold symbols, solid lines) and for reference samples (open symbols, dashed lines). Inset shows typical FMR spectra for AFM/Cu/FM( dF) samples with dF= 3 and 15 nm. (b) Py-thickness dependence of spfor AFM/Cu/FM( dF). Inset shows spMFproduct as a function of d1 F. The solid line is guide to the eye. magnetic dynamics in the FM layer. The spin current can induce exchange of angular momentum between the different subsystems of the conduction and localized electrons in the NM and AFM layers. Moreover, it can stimulate additional spin pumping from the AFM layer induced by the dynamic magnetization MAF, which follows the motion of the localized AFM moments.13,17,18In addition, free conduction-electron spins in our metallic AFM can interact with the dynamic magnetization MAFand also accumulate, similar to that in the NM layer. While the spin polarization in FM is so strong that spin accumulation in it can be neglected, in the metallic AFM spin accumulation and spin polarization by the localized moments are comparable. Therefore, the transport of spins through the AFM/NM/FM system and the corresponding dissipative phenomena within the trilayer depend upon the balance between the free and localized spins within all three layers of the structure. Treating the AFM/NM/FM as a discrete system, one can distinguish between five subsystems, shown schematically in Fig. 1(a): three reservoirs of free spins in FM (spin density sF), NM (spin density sN), and AFM (spin density sAF), and localized FM (macroscopic magnetization MFMFmF) and AFM moments (characterized with the N ´eel order parameter L=MAFIand macroscopic mag- netization MAFMAFmAF). Here we introduce the saturation magnetizations MFandMAFof the FM and AFM layers, respectively. In equilibrium, free spins in the FM are mostly parallel to the FM magnetization, sFkMF. In the NM and AFM layers, the population of free spin-up and spin-down electrons is balanced, sN=sAF= 0, since MAF= 0. In the framework of linear nonequilibrium thermodynamics, spin densities sF,sN,sAF, and magnetizations mF,mAFcan be treated as thermodynamic variables aj,j=1:::5. The conjugated thermodynamic forces are calculated as the derivatives of free energy:19Xj=@F=@aj(we assume that the temperature is constant). The thermodynamic forces for the free spins coincide with the spin accumulation potentials (s) F(in FM), (s) N(in NM), and (s) AF(in AFM). For the localized moments the corresponding forces are proportional to the effective fields MFVFHF(in FM) and MAFVAFHAF (in AFM). Thermodynamic currents Jj˙ajare related to the thermodynamic forces via the Onsager coefficients ˆL: (˙mAF,˙mF,˙sAF=e,˙sF=e,˙sN=e)T=ˆL MAFVAFHAF,MFVFHF,(s) AF,(s) F,(s) NT, (1) where eis electron charge. Using the Onsager reciprocity principle and the symmetry considerations, one can reduce relations (1) to the following form: ˙mAF= AFHAF ~ e2MAFVAFl GAF b(s) AF+GAF S(s) FmF l, ˙mF= FHF ~GF S e2MFVFmF(s) AFmF, ˙sAF= ~ eGAF bHAF+1 eGAF 0(s) AF,˙sF= ~ eGF bHF+1 eGF 0(s) FmF, ˙sN= ~ eGAF SHAF ~ eGF SHF+1 eGN((s) F(s) AF),(2)056312-4 Kravets et al. AIP Advances 7, 056312 (2017) where is the gyromagnetic ratio and ~is the Planck constant. We neglect spin accumulations in the NM layer, since the spin-diffusion length in the NM layer is relatively long. We also set (s) N=0 and take into account strong spin polarization in the FM layer, so, that (s) F=(s) FmF. In the second equation of (2) we use Landau-Lifshitz representation of the magnetic damping in FM ( _FHF), as it is consistent with the Onsager’s concept of conjugated currents ( ˙mF) and forces ( HF). Conversion to the standard Gilbert form can be obtained from equations of motions for FM as HF=mF˙mF= . The interpretation of the coefficients in Eq. (2) is schematically shown in Fig. 1(a). Diagonal coefficientsLjjfor the localized spins are related with the internal damping in the FM (damping parameterF) and AFM (damping parameter AF) layers. Diagonal coefficients Ljjfor the free spins are proportional to the corresponding conductances, GF 0andGAF 0. The nondiagonal coefficients responsible for the cross-coupling effects between the AFM and FM layers are of two types. First, the spin-mixing conductances GF SandGAF Soriginate from the dephasing of the free electrons at the FM/NM and NM/AFM interfaces,8and are responsible for the spin-pumping phenomena. The free electrons in NM reflecting from the FM/NM and NM/AFM interfaces acquire additional nonequi- librium spin polarization, which is related to the dynamic magnetization of the AFM and FM films. Second, the bulk conductivities, GAF bandGF bdescribe the exchange of angular momentum between the subsystems of the localized and free spins in the FM and AFM layers. In our case of strong polar- ization inside the FM layer, the term with GF bcan be neglected. Lastly, GNis the spin conductivity in the NM layer. The first of Eqs. (2) reproduces the well-known result of AFM spintronics:2,20the spin-torque induced by a spin-polarized current (last term in the r.h.s.). It is clear from Eq. (2) that this torque originates not only from the current polarized by the FM layer, but also from the spin accumulation inside the AFM layer. The second of Eqs. (2), for the FM magnetization is similar to the corresponding equation for FM/NM bilayers,4,9,21,22with the only difference that spin accumulation (s) AFtakes place in the AFM layer. This fact reflects the “duality” of the metallic AFM, which manifests the prop- erties of FM (non-zero magnetization of the localized spins) as well as NM (has free spins that can accumulate). There is one principal difference, however, between spin accumulation (s) Fin FM/NM systems, and (s) AFin AFM/NM/FM trilayers. The FM magnetization is large and fully defines the orientation of the spin accumulation in a FM/NM bilayer. In the AFM/NM/FM sys- tem, the spin accumulation (s) AFis defined by the interplay between the magnetic dynamics in the AFM and the spin flow between the FM, NM, and AFM layers, with the result that its spin ori- entation is defined by a non-trivial interplay of a number of factors and points essentially in any direction. To describe the magnetic dynamics of an AFM/NM/FM trilayer one must start from the balance equations for the localized moments in the FM and AFM layers, and take into account the spin flows through the interfaces and the dissipative terms given by Eq. (2). In particular, the equation for the FM moments can be written as ˙mF= mF(HF+H)1 eMFVFmF˙sNmF + FHF ~GF S e2MFVFmF(s) AFmF. (3) The first term in Eq. (3) corresponds to the standard Landau-Lifshitz dynamics in the presence of external magnetic field H. The second term describes a spin flux through the interface, which coincides with the spin current, ˙sN, from the adjacent NM layer. Cross products with mFreflect the fact that only transverse (with respect to mF) spin component flows out of FM. Last two terms correspond to the Onsager forces, according to Eq. (2). For the AFM/Cu/FM( dF) system used in our FMR-induced spin pumping experiment, we can set (s) F=0 as no electric voltage is applied across the structure. We further assume no spin accumulation inside the AFM layer, (s) AF=0, since the spin-diffusion length in AFM (0.3 nm for Cu/IrMn23) is much shorter than the AFM thickness. Then, from Eq. (1) and (3) we obtain the effective dynamic056312-5 Kravets et al. AIP Advances 7, 056312 (2017) equation for the FM layer: ˙mF= mF(HF+H)+ * ,F+ ~GF S e2MFVF+ -HF+ 2~2GAF S e2MFVFmFHAFmF. (4) The second term in the r.h.s. of Eq. (4) points to an increase of the effective damping due to the presence of the FM/NM interface, which leads to a corresponding increase in the FMR linewidth
. In addition, the last term in Eq. (4) predicts a field-like contribution to the FM dynamics, which results exclusively from the spin pumping by the AFM layer, as the direct exchange between the FM and AFM is fully suppressed by the Cu spacer. This field, _HAFmF, can contribute to the value of the resonant field Hr, and the contribution can be estimated as follows. The typical AFMR frequencies are much larger than the FMR frequency of the FM layer, so the dynamics of the AFM is driven solely by the FM, and HAF_HF. The additional field is then _GAF SHFmF=MFVF. According to Eq. (4), both spin-pumping-induced corrections to the linewidth and the resonant field are inversely proportional to MFVF_MFdF. Fig. 2(a) illustrates this tendency of
(dF) and Hr(dF) measured for our samples. To confirm the thickness dependence of the effective damping predicted by Eq. (4), we calculated the incremental change in the AFM-induced linewidth as
sp=

inhom
ref, where
refis the linewidth of the FMR of the reference sample. Contribution
inhom , which originates from a possible inhomogeneity of the sample, is calculated according to the procedure described in Refs. 24 and 25. This contribution is below 2 Oe for the samples with dF6 nm and equals to 8 Oe for dF= 3 nm. It should be noted that for the multilayer with the thickest Py layer ( dF= 15 nm), the FMR linewidth (55 Oe) well agrees with the values reported by other research groups for well-characterized high- quality Py films.3,26 Figure 2(b) shows the thickness dependence of the spin-pumping-induced contribution to Gilbert damping obtained from
sp. In agreement with the theory, Eq. (4), spMFgrows linearly with d1 F. We believe that the observed thickness dependence of the damping parameter points to the important role of free spins in the magnetic dynamics of the AFM/Cu/FM trilayer. We also conclude that the observed Hr(dF) dependence indicates that the localized AFM moments affect the dynamics of the FM layer through the dynamic exchange via conduction electrons in the system. However, this contribution from the localized moments can be partially masked by the exchange bias due to the second Py layer and thus requires further analysis. In summary, we observe spin-pumping effect in AFM/NM/FM multilayers as an increase in the linewidth of the FMR and shift of the resonant magnetic field. Basing on Onsager formalism, we calculate additional damping and field-like torque on FM moments due to the presence of AFM layer. The inverse dependence of damping and resonant field vs the thickness of FM layer supports the hypothesis of AFM influence on FM dynamics. The contribution from the spin-pumping effect to the FMR linewidth is separated and shown be affected by the changes in the thickness of the ferromagnetic layer. The physical mechanisms of the observed
spvs.dFbehaviour are analyzed and show a rich interplay of the conduction-vs-lattice spins in the five effective sub-systems of the structure. These results provide a deeper understanding of the spintronic effects in nanostructures containing antiferromagnets and can prove useful for designing future spintronic devices. ACKNOWLEDGMENTS Support from the Swedish Stiftelse Olle Engkvist Byggm ¨astare, the Swedish Research Council (VR grant 2014-4548), and the National Academy of Sciences of Ukraine (project 0115U00974) are gratefully acknowledged. OG acknowledges support of the Fundamental research program of the National Academy of Sciences of Ukraine “Fundamental problems of creation of new nanoma- terials and nanotechnologies”, the ERC Synergy Grant SC2 (No. 610115), and the Transregional Collaborative Research Center (SFB/TRR) 173 SPIN+X. 1A. H. MacDonald and M. Tsoi, Phil. Trans. R. Soc. A 369, 3098 (2011). 2E. V . Gomonay and V . M. Loktev, Low Temp. Phys. 40, 17 (2014). 3P. Merodio, A. Ghosh, C. Lemonias, E. Gautier, U. Ebels, M. Chshiev, H. B ´ea, V . Baltz, and W. E. Bailey, Appl. Phys. Lett. 104, 032406 (2014).056312-6 Kravets et al. AIP Advances 7, 056312 (2017) 4L. Frangou, S. Oyarz ´un, S. Auffret, L. Vila, S. Gambarelli, and V . Baltz, Phys. Rev. Lett. 116, 077203 (2016). 5H. Wang, C. Du, P. C. Hammel, and F. Yang, Phys. Rev. Lett. 113, 097202 (2014). 6C. Hahn, G. de Loubens, V . V . Naletov, J. Ben Youssef, O. Klein, and M. Viret, Europhys. Lett. 108, 57005 (2014). 7Z. Qiu, J. Li, D. Hou, E. Arenholz, A. T. N0Diaye, A. Tan, K.-i. Uchida, K. Sato, S. Okamoto, Y . Tserkovnyak, Z. Q. Qiu, and E. Saitoh, Nat. Commun. 7, 12670 (2016). 8Y . Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. B 66, 224403 (2002). 9A. Brataas, Y . Tserkovnyak, G. E. W. Bauer, and P. J. Kelly, in Spin Current , edited by S. Maekawa, S. O. Valenzuela, E. Saitoh, and T. Kimura (Oxford University Press, Oxford, 2012), p. 87. 10S. M. Rezende, M. A. Lucena, A. Azevedo, F. M. de Aguiar, J. R. Fermin, and S. S. P. Parkin, J. Appl. Phys. 93, 7717 (2003). 11A. Brataas, G. E. W. Bauer, and P. J. Kelly, Phys. Rep. 427, 157 (2006). 12T. Taniguchi and W. M. Saslow, Phys. Rev. B 90, 214407 (2014). 13R. Cheng, J. Xiao, Q. Niu, and A. Brataas, Phys. Rev. Lett. 113, 057601 (2014). 14W. Zhang, M. B. Jungeisch, W. Jiang, J. E. Pearson, A. Hoffmann, F. Freimuth, and Y . Mokrousov, Phys. Rev. Lett. 113, 196602 (2014). 15A. F. Kravets, A. N. Timoshevskii, B. Z. Yanchitsky, M. A. Bergmann, J. Buhler, S. Andersson, and V . Korenivski, Phys. Rev. B 86, 214413 (2012). 16Y . Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin, Rev. Mod. Phys. 77, 1375 (2005). 17I. V . Baryakhtar and B. A. Ivanov, Sov. J. Low Temp. Phys. 5, 361 (1979). 18H. Gomonay and V . Loktev, J. Magn. Soc. Jpn. 32, 535 (2008). 19H. B. Callen, Phys. Rev. 73, 1349 (1948). 20R. Cheng and Q. Niu, Phys. Rev. B 89, 081105 (2014). 21A. Brataas, Y . Tserkovnyak, and G. E. W. Bauer, Phys. Rev. B 84, 054416 (2011). 22Y . Ohnuma, H. Adachi, E. Saitoh, and S. Maekawa, Phys. Rev. B 89, 174417 (2014). 23R. Acharyya, H. Y . T. Nguyen, W. P. Pratt, and J. Bass, J. Appl. Phys. 109, 07C503 (2011). 24S. Mizukami, Y . Ando, and T. Miyazaki, Jpn. J. Appl. Phys. 40, 580 (2001). 25A. F. Kravets, D. M. Polishchuk, Y . I. Dzhezherya, A. I. Tovstolytkin, V . O. Golub, and V . Korenivski, Phys. Rev. B 94, 064429 (2016). 26E. Montoya, P. Omelchenko, C. Coutts, N. R. Lee-Hone, R. H ¨ubner, D. Broun, B. Heinrich, and E. Girt, Phys. Rev. B 94, 054416 (2016).

1.4977983.pdf

Vortex-antivortex pairs induced by curvature in toroidal nanomagnets Smiljan Vojkovic , Vagson L. Carvalho-Santos , Jakson M. Fonseca , and Alvaro S. Nunez Citation: Journal of Applied Physics 121, 113906 (2017); doi: 10.1063/1.4977983 View online: http://dx.doi.org/10.1063/1.4977983 View Table of Contents: http://aip.scitation.org/toc/jap/121/11 Published by the American Institute of PhysicsVortex-antivortex pairs induced by curvature in toroidal nanomagnets Smiljan Vojkovic,1Vagson L. Carvalho-Santos,2Jakson M. Fonseca,3and Alvaro S. Nunez4 1Instituto de F /C19ısica, Pontificia Universidad Cat /C19olica de Chile, Campus San Joaqu /C19ın Av. Vicu ~na Mackena, 4860 Santiago, Chile 2Instituto Federal de Educac ¸~ao, Ci ^encia e Tecnologia Baiano - Campus Senhor do Bonfim, Km 04 Estrada da Igara, 48970-000 Senhor do Bonfim, Bahia, Brazil 3Universidade Federal de Vic ¸osa, Departamento de F /C19ısica, Avenida Peter Henry Rolfs s/n, 36570-000 Vic ¸osa, MG, Brasil 4Departamento de F /C19ısica, Facultad de Ciencias F /C19ısicas y Matem /C19aticas, Universidad de Chile, Casilla 487-3, Santiago, Chile (Received 27 October 2016; accepted 21 February 2017; published online 21 March 2017) We show that the curvature of nanomagnets can be used to induce chiral textures in the magnetization field. Among the phenomena related to the interplay between the geometry and magnetic behavior of nanomagnets, an effective curvature-induced chiral interaction has beenrecently predicted. In this work, it is shown that magnetization configurations consisting of two struc- tures with opposite winding numbers (vortex and antivortex) appear as remanent states in hollow toroidal nanomagnets. It is shown that these topological configurations are a result of a chiral interac-tion induced by curvature. In this way, the obtained results present a new form to produce stable vor- tices and antivortices by using nanomagnets with variable curvature. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4977983 ] I. INTRODUCTION Proper control over magnetization textures is at the heart of a large variety of technological applications. This makes any new tool to control the fate of magnetization patterns a very interesting avenue for research. In this context, topological spin configurations such as chiral skyrmions, magnetic bubble domains, vortices, and antivor tices have been widely studied due to the possibility to produce devices based on magnonics and spintronics technology.1–12For example, skyrmion based “race-track” memory devices4,13and vortex based transistors14 have been recently proposed. Chiral skyrmions are two- dimensional topological solit ons localized in nanoscale cylin- drical regions.15–17The understanding of the physics of magnetic skyrmions and their appl ications in potential spin- tronic devices requires the know ledge of the properties of an isolated skyrmion. In this context, an analytical solution for such isolated skyrmions has been recently obtained and com- pared with experimental results.18Despite vortices being also characterized by a winding number (vorticity) that belongs to the first class of the first homotopy group,19in systems in which the magnetization field is free to rotate in any direction and the exchange energy is described by isotropic Heisenberg Hamiltonian (nonlinear sigma model), the stability of vorticescannot be ensured by arguments from the homotopy theory because its (skyrmionic) topological charge, formally defined asQ¼ 1 4pÐdr2m/C1@xm/C2@ym/C0/C1, is not an integer (a vortex is topologically equivalent to a meron and Q¼1/2). Thus, the sta- bilization of vortices in magnetic systems is ensured by subtle competition between the dipolar and exchange energies.20In this context, vortices can appear as the groundstate of submi- cron magnetic elements of different shapes.21–23In addition, vortex domain walls are stable c onfigurations in cylindrical nanotubes and, in the last decade, their static and dynamic properties are under intense investigation.24–33Unlikeskyrmions and vortices, the preparation of a nanostructure that contains a stable antivortex is a challenging task. Indeed,antivortices can appear as an unstable magnetization configu-ration during a vortex core reversal. 34Nevertheless, experi- mental and theoretical works have reported that a stablesingle antivortex can appear in asterisk 9,35and cross-like11 shaped permalloy particles. In this letter, we provide an addi- tional mechanism for the nucleation, control, and transport ofchiral textures of the magnetization by appealing to the geo-metrical curvature of the system. Curvature has become a cornerstone in the modern description of the behavior of magnetic systems. In fact, a lotof effort has been dedicated to the understanding of the effectsthat the geometrical properties of a system have on the staticand dynamic behavior of the magnetization. 36For instance, the presence of curvature and torsion in curved wires yieldsthe appearance of a domain wall pinning and a Walker limit 37 during the domain wall motion.38–40T h eb a s i c ss p i nw a v e physics behind the geometrical confinement of magnetic tex-tures has been investigated in cylindrical nanomagnets and acurvature-induced asymmetric spin wave has been predictedto appear in the spin wave dispersion along ferromagneticnanotubes. 41As a last example, the coupling between an external magnetic field strength and the surface curvature canlead to the appearance of a 2 p-skyrmion excitation on mag- netic surfaces described by the Heisenberg Hamiltonian. 42,43 Another very interesting phenomenon related to the geometry of the nanomagnet that has been discussed inthe last few years is the curvature-induced magnetochiraleffect. This effect consists of changes occurring on the mag-netic properties of curved nanomagnets that depend on thechirality of the magnetization configuration. Such curvature-induced magnetochiral effects have been studied in the con-text of a vortex domain wall displacing along a magnetic 0021-8979/2017/121(11)/113906/7/$30.00 Published by AIP Publishing. 121, 113906-1JOURNAL OF APPLIED PHYSICS 121, 113906 (2017) nanotube.28In this case, a difference in the magnetostatic energy was noted when the magnetization of the vortexdomain wall is pointing inward and outward of the tube sur- face, in such way that before the Walker breakdown regime is reached, the domain wall must overcome an energy barrierthat depends on its initial chirality. 28Furthermore, by consid- ering a bending in a magnetic film, it was shown that geome- try can break the inversion symmetry and give rise to chiraleffects, leading to a handedness in the magnetization, which becomes more pronounced as the curvature of the film increases. 44In this context, approximating the full extent of the dipolar interaction by a suitably chosen shape anisotropy, Gaididei et al.45have obtained a functional to calculate the exchange energy for magnetic shells with an arbitrary geome-try as a function of its Gaussian and mean curvatures. They have shown that an effective anisotropy and a Dzyaloshinskii- Moriya-like interaction (DM-like interaction) 46,47are induced by curvature. Pylypovskyi et al showed that this effective DM-like interaction is responsible for a chiral symmetry breaking of a domain wall motion in magnetic helices48and M€obius ring.49Based on the above and on the fact that the torus presents a curvature varying from negative (internal bor- der) to positive (external border), with a relatively easy geo-metrical description, we propose that this shape is ideal to analyze the predicted chiral effects induced by the curvature. In this work, it is shown that the variable curvature of thetorus yields chiral magnetization patterns with opposite wind- ing numbers (vortex and antivortex), and so, a new way to sta- bilize vortices and antivortices in magnetic nanoparticles byusing curvature effects is proposed. The magnetic properties of toroidal nanomagnets have been previously studied. 50,51 However, the geometrical parameters considered so far do not allow the appearance of more complex structures during the reversal process that could evidence a curvature-induced DM- like interaction. Here, we study hollow toroidal nanoparticlesthat are expected to display the qualitative features renderedby the curvature-induced DM-like interaction. 45 II. THEORETICAL MODEL In general, in a curved surface parametrized by the coor- dinates ( q1,q2), we can write the magnetization in an orthog- onal curvilinear basis, ( ^q1;^q2;^n¼^q1/C2^q2), in the form ~M¼MSm, with MSbeing the saturation magnetization and m¼^ncosHþ^q1sinHcosUþ^q2sinHsinU;(1) whereH/C17Hq1;q2ðÞ andU/C17Uq1;q2ðÞ describe the angles of the magnetization vector field in a curvilinear background. In this way, the exchange energy density for an arbitrary curved magnetic shell is explicitly written below45,52 Eex A¼sinHrU/C0X ðÞ /C0cosH@CUðÞ @U/C20/C212 þrH/C0CUðÞ ½/C1382; (2) where Ais the stiffness constant, CUðÞis a matrix depending on the Gauss and mean curvatures of the nanomagnet, and X is a modified spin connection, defined as45X¼ ^q1/C1@^q2 @q1/C18/C19 ^q1þ ^q1/C1@^q2 @q2/C18/C19 ^q2: (3) A torus with genus 1 embedded in a 3D-space can be parametrized by ~r¼Rþrsinh ðÞ ^xcosuþ^ysinu ðÞ þ ^zrcosh;(4) where Rand rare, respectively, the toroidal and poloidal radii and hplays the role of the polar angle describing the torus surface (See Fig. 1). In this case, its Gaussian curva- ture is evaluated as KhðÞ¼sinhrRþrsinh ðÞ½/C138/C01, and then, the curvature of the torus varies from a positive to a nega- tive value along the polar-like angle, that is, Kp=2ðÞ ¼ rRþrðÞ½/C138/C01and K3p=2ðÞ ¼/C0 rR/C0rðÞ½/C138/C01.F r o mE q . (3), the modified spin connection for the toroidal geometry can be calculated from the geometrical parameters defined inEq.(4),l e a d i n gt o X thðÞ¼/C0 ^ucosh Rþrsinh: (5) It can be noted that the spin connection varies as a function ofh, pointing along /C0^uforh2/C0 p=2;p=2 ðÞ andþ^u forh2p=2;3p=2 ðÞ . In this context, we have that Xt0ðÞ¼ /C0^u=RandXtpðÞ¼^u=R. According to Gaididei et al.,45if magnetostatic energy is approximated by an easy-tangential anisotropy (this is a good approximation for thin shells) anda tangential magnetization configuration ( H¼p=2) is con- sidered, the magnetic energy can be split into three compo- nents: (i) the “standard” exchange interaction E E¼ArUðÞ2, which homogenizes the spatial distribution of the magnetiza- tion vector, minimized for U¼constant ; (ii) an effective anisotropy, given by EA¼AC2; and (iii) an effective DM- like interaction, given by ED¼/C02ArU/C1X ðÞ . The latter contribution is minimized when the magnetization displays inhomogeneous distribution textures. The magnetizationground state can be interpreted as a result of the interplay among dipolar, a curvature-induced magnetic field pointing along the normal direction and these three curvature-inducedinteractions. 45 If we consider a general magnetization distribution on a toroidal shell, the curvature-induced DM-like energy densityis given by 52 FIG. 1. Left figure presents a torus section showing the adopted coordinate system describing the torus external surface. Right figure shows a zoom of the region highlighted by a circle with the magnetization parametrization as a function of the toroidal unitary vectors ( r;h;u).113906-2 Vojkovic et al. J. Appl. Phys. 121, 113906 (2017)ED A¼2 Rþrsinh ðÞ2/C2 coshsin2H@uU /C0sinhðcos2HsinU@uH/C0sinHcosHcosU@uUÞ/C3 þ1 r2cosU@hH/C0cosHsinHsinU@hU ðÞ : (6) From Eq. (6), one can estimate the effective DM-like interaction strength induced by curvature. Indeed, by using adimensional analysis, we have that D=a 2/C24A=R¼J=Raand thus,D/C24 Ja=RðÞ , where ais the lattice constant and Jis the exchange constant. From Eqs. (1)and(6), it can be noted that the effective DM-like interaction term to the energy of ain-surface vortex state, given by H¼p=2 and U¼p=2, and a single domain state, represented by H¼hþp=2 and U¼p=2 (pointing along x-axis direction), is 0 and then, no chiral effects are present for these magnetization fields.Nevertheless, due to its dependence on h, chiral effects com- ing from the curvature-induced DM-like interaction must beevident when other magnetization configurations appearingin a toroidal nanomagnet are analyzed. Therefore, to studymetastable states in a toroidal nanomagnet and analyze the possibility of the appearance of chiral interactions induced by variable curvature, we have performed micromagneticsimulations for hollow Permalloy nanotori and solved theLandau-Lifshitz-Gilbert equation 53,54 @m @t¼c0 l0Mm/C2dE dmþam/C2@m @t; (7) where c0¼l0jcj¼l0gjlBj=/C22h, with cthe gyromagnetic ratio, MSis the saturation magnetization, Eis the free energy density, and ais the dimensionless Gilbert damping parame- ter. The first term of Eq. (7)describes the precession of the magnetization under the influence of an effective mag-netic field H eff¼/C0 1=l0MS ðÞ dE=dm, and the second term accounts for the relaxation mechanisms that dissipate energy by making a torque towards the effective field. The effectivemagnetic field that each magnetic moment experiments iscreated by the exchange, dipolar and anisotropy interactions,as well as the external magnetic field. III. RESULTS The LLG equation was solved by using the 3D Object Oriented MicroMagnetic Framework (OOMMF)55package using the computational facilities available in nanohub.56The simulations were run using an exchange constant A¼1:3 /C210/C011J/m, a saturation magnetization MS¼860 kA/m, a cubic mesh with size 1 nm, and a damping a¼0:5. The con- sidered geometric parame ters are toroidal radius R¼52 nm, internal poloidal radius ri¼20 nm, and external poloidal radius re¼26 nm, resulting in a 6 nm thickness torus. In the simula- tions, we have saturated the m agnetization with an external magnetic field in such way that the initial state consists of asingle domain along the x-axis direction. Then, we have dimin- ished the magnetic field until 0 and analyzed the remanent state of the magnetization.Fig. 2(a) presents the obtained configuration, and the appearance of a vortex at the external torus border is evident.To analyze the magnetization at the internal borders, we have done four cutoffs highlighted in the planes shown in Fig. 2(a). Fig. 2(b) shows a representation of the obtained magnetiza- tion configuration, given by the in-plane VA ansatz,U¼argrcoshþiRþrsinh ðÞ cosu7p=2 ðÞ/C2/C3 . The signals /C0þðÞ are related to the interval of the azimuthal angle, that is, – for u2/C0p=4;p=4 ½/C138 andþforu23p=4;5p=4 ½/C138 .F r o m Figs. 2(c)–2(f) , the formation of the two vortices state with FIG. 2. Snapshot of the remanent state of the magnetization. (a) evidences the presence of a vortex at the external border of the torus. (b) shows a repre-sentation of the magnetization as a function of handuaiming to show the in-surface component of the VA pair. (c)–(f) show a front view of the planes represented in (a). These highlighted planes evidence the appearance of a vortex at the external border of the torus ((c) and (f)) and the anti-vortex at the internal border ((d) and (e)). The horizontal bar shows the color scale of the magnetization along the x-axis.113906-3 Vojkovic et al. J. Appl. Phys. 121, 113906 (2017)opposite chiralities at the external borders (positive curvature) of the torus and antivortices, also with opposite chiralities, atthe internal borders (negative curvature) of the torus can beobserved. Such opposite circulation at the ends of the torus could be associated with the symmetry properties of the LLG equation. 57That is, if the static magnetic structure is gener- ated by starting from saturation in the in-plane direction andreducing the external field to zero, vortices with opposite chir-alities form on opposite ends of a sample. This effect is known, for example, for cylinders or whiskers 58with flat ends that are originally magnetized along their symmetry axis. Onthe other hand, we have performed micromagnetic simula-tions for a half torus section with the same previouslydescribed geometrical parameters. The formation of only one VA pair has been observed, and there are no two vortices with opposite chiralities in this case. That is, the two vorticesappearing at the ends of the integer torus would also appear ifthe torus would cut on two halves and each half was under theaction of an in-plane magnetic field. Then, such pair of vorti- ces at the opposite borders of the torus cannot be associated with the symmetry properties of the LLG equation.Nevertheless, the symmetry of the LLG equation continues toexist because each VA pair presents opposite chiralities. These VA pairs can be interpreted as minimal versions of a cross-tie domain wall, 59,60separating two domains that are formed along the poloidal angle: One domain pointingalong /C0^u(/C0p=6/C20h/C20p=6) and another along þ^u (5p=6/C20h67p=6). The formation of one domain in the upper and down regions of the toroidal nanoparticle is aresult of the interplay between the shape anisotropy (magne-tostatic) and exchange interactions or, in an equivalent way,from the interplay between dipolar, DM-like, and curvature- induced effective anisotropy interactions. Indeed, if we con- sider an in-surface magnetization configuration ( H¼p=2) and the previous definitions of E DandEA, we have that ED¼2Acosh Rþrsinh ðÞ2@uU (8) and EA¼A1 r2cos2Uþsin2h Rþrsinh ðÞ2sin2U"# : (9) From the analysis of Eqs. (8)and(9), it can be noted that for h¼0o r h¼p;ED/C242=R2andEA/C241=r2. Since R>r, the solution that minimizes the energy depends on the relation R/r. By taking the parameters described in the simu- lations, we have that R¼2rand then EAdominates, making the in-surface solution U¼p=2 more favorable. On the other hand, for h!p=2, the first term of Eq. (9)dominates and it favors the solution U¼0. Therefore, from the adopted parameters, EDeven plays the role in such way that spatially inhomogeneous distributions must take place. These spa- tially inhomogeneous states lead to the formation of observed upper and down domains, evidenced in Fig. 2(a). The main consequence of the appearance of these twodomains is the formation of the VA pair, and in this case,curvature-induced chiral states are observed and DM-likeinteraction determines the region where the vortex and the antivortex must appear. In fact, by assuming q¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rcosh ðÞ 2þRþrsinh ðÞ sinu/C2/C32q , the ansatz61 H¼arccos 1 /C0q qc/C18/C192"#48 < :9 = ;;q/C20qc (10) can be used to describe the vortex ( þ) and antivortex ( /C0) core with radius qc, appearing at h¼p=2/C0p=2ðÞ . In this context, the winding number of the metastable state is deter-mined by @ uUappearing into the second term of Eq. (6). Indeed, the signal of second term in Eq. (6)presents a depen- dence on hin such way that ED2 6p=2¼7sinHcosHcosU@uU: (11) Therefore, from a direct analogy with the planar case, the ansatz U¼6arctan rcosh=Rþrsinh ðÞ sinu/C2/C3 can be adopted to describe vortex ( þ) or antivortex ( /C0) configura- tions. In this case, the term of the energy given in Eq. (11) is minimized for @uU<0 (vortex) around h¼p=2 and @uU>0 (antivortex) around h¼/C0p=2. Then, the DM-like interaction is responsible for the appearance of a vortex in the external border of the torus and an antivortex in the inter- nal border of the torus. Due to the fact that an antivortex is found at the internal border of the torus, one can conjecture that the antivortex is for a negative curvature as the vortex is for a positive curvature.62,63Nevertheless, the study on the magnetic groundstate of particles with negative curvature is still lacking. The antivortex is the topological counterpart of a vortex, having opposite (skyrmionic) topological charges, and so,the appearance of two VA pairs preserves the topological charge of the initial state ( Q¼0 for a single domain). In fact, while a magnetic vortex (topologically equivalent to a meron) has a topological charge Q v¼þ1=2 (the spin vector field wraps half of the sphere in the physical space), an anti- vortex (equivalent to an antimeron) presents Qa¼/C01=2. Vortices and antivortices are indeed expected to become merons and antimerons when the separation between them becomes larger than the diameter of their non-coplanar cores.64The conservation of the topological charge occurs because two configurations belonging to different homotopy classes cannot be continuously deformed one to another,65 and thus, isolated vortices (antivortices) or two pairs of vorti-ces (antivortices) are not possible in this case. Since such pair is not topologically stable, its stability is ensured by the subtle competition between exchange and dipolar energies. In addition, the two VA pair state demands lower energy than a two vortices pair state since if two pairs of vortices would be the remanent state, four antivortices must appear at h¼0pðÞ/¼0pðÞin order to smoothly connect such vorti- ces pairs, increasing the dipolar and exchange energy of the system. Connecting the VA pairs, an in-surface state along the torus can be noted, that is, the remanent state consists of a quasi-tangential state connecting two VA pairs. From Eq. (6)and the described geometrical parameters ( R¼52 nm and113906-4 Vojkovic et al. J. Appl. Phys. 121, 113906 (2017)r¼26 nm), one can estimate the curvature-induced DM-like interaction effective strength in D/C24 0:005/C00:012 meV for the external ( h¼p=2) and internal ( h¼/C0p=2) borders of the torus, respectively. The formation of VA on a toroidal shell is a very inter- esting result, both from the fundamental point of view andfrom the applied point of view. From the fundamental point of view, the appearance of a configuration with a positive winding number at the external border and a configurationwith negative winding number at the internal border of the torus evidence the intrinsic relationship between the curva- ture and magnetization properties of ferromagnetic nanopar-ticles. Then, one can conclude that in consequence of a curvature-induced DM-like interaction, there is the possibil- ity of the appearance of a curvature-induced topologicalmetastable state in curved nanomagnets. From the applied point of view, if it is possible to generate and guide the VA state from a curved section of a nanowire to a straight sectionwithout annihilate them, this pair could be thought as a can- didate to compose devices based on the concept of spin logic operations 6or “race-track” memory.66Another interesting possibility is to study the interaction of this pair vortex- antivortex with oscillating magnetic fields aiming to use these chiral states as nano-emitter/nano-collector (nanoan-tenna) devices. 67,68 To highlight the role of the curvature in the generation of the VA pair, we have performed micromagnetic simulationsfor a hollow cylinder with the same geometrical dimensions of the torus. In fact, differences must be evident when we con- sider the two geometries, since by parametrizing the cylinderin the natural cylindrical coordinate system ( ^n¼^q;^q 1 ¼^u;^q2¼^z), the modified spin connection for the cylindrical geometry is evaluated as Xc¼0 and thus, there is not a curvature-induced DM-like interaction term to the magnetic energy in this case (consequently no chiral configuration must be noted). Indeed, by observing the remanent state given inFig.3, it can be noted that it consists of a single vortex turning around the ring hole and chiral metastable states do not appear in this case. The absence of a VA pair in the sample withexactly the same dimensions but with vanishing curvature is a strong indicator that the origin of the chiral texture lies on the curvature of the hollow torus. On the other hand, if the magnetic field is pointing along thez-axis direction, a different behavior must be expected for a hollow cylindrical nanomagnet. In this case, the param-etrization is given by ( ^n¼^z;^q 1¼^q;^q2¼^u) and conse- quently Xc¼/C0 ^u=q. Therefore, a chiral effect must be noted. In this context, we have also studied the behavior ofthe magnetization in hollow toroidal and cylindrical nano- magnets by analyzing the remanent state when an external magnetic field is applied along the z-axis direction. As expected, due to the symmetry of the LLG equation, in rema- nence, the formation of a double vortex with opposite chiral- ities for both geometries can be noted (see Fig. 4). However, it can be noted that due to the smooth curvature of the torus, the vortex state is still present at its internal neck while due to the high exchange energy cost to support a vortex,the internal neck of the hollow cylindrical ring presents a single domain state pointing along the z-axis direction (SeeFig. 3(b)). For both cases, the formation of a two opposite vortex state can be explained by the need to reduce the dipo-lar energy (vortices with the same chirality would lead to larger dipolar energy) and the nanomagnets behave as an array of nanorings separated by a vertical distance d. 69This remanent state is also very interesting for applications in FIG. 3. Snapshot of the remanent state for a hollow cylindrical nanoparticle when the magnetic field is pointing along the x-axis direction. The highlighted planes evidence that no chiral states appear in a hollow cylindri- cal magnetic particle. FIG. 4. Remanent states of hollow toroidal (a) and cylindrical (b) nanopar-ticles when an out-of-plane magnetic field is applied. The vertical bar shows the color scale of the magnetization along the x-axis.113906-5 Vojkovic et al. J. Appl. Phys. 121, 113906 (2017)spintronic and magnonic devices, due to the possibility to control the chirality of the vortex on one side of the nano- magnet by controlling the other one. The experimental evidence of the obtained metastable states can be found by analyzing the reversal process of the magnetization during a hysteresis cycle. To understand themechanism behind the nucleation and annihilation of the observed VA pair ( H x) and two opposite vortices ( Hz) and to show how these states can be experimentally observed, wehave studied the hysteresis curves appearing from each rever- sal process. The hysteresis curves describing such reversal processes are shown in Fig. 5. It is observed that when the magnetic field is applied along the x-axis direction, the hyster- esis curve presents a reduction in the magnetization at rema- nence. This reduction evidences the formation of the two VApairs. From the analysis of Fig. 5(a), it is observed that as the magnetic field increases in the opposite direction, each VA pair annihilates giving place to a transient state formed by twoin-surface domain walls at the opposite sides of the torus. These domain walls join themselves and the hysteresis curve presents a typical neck associated with the nucleation of a vor-tex state. For jH xj/C21100 mT, the vortex configuration gives place again to a double domain wall, which diminishes their lengths when the magnetic field continues to increase, disap-pearing for H x/C25250 mT. On the other hand, Fig. 5(b)describes the hysteresis curve when the magnetic field is applied along the z-axis direction. A fast decrease in the mag- netization for H/C20500 mT can be noted, evidencing the for- mation of the opposite vortices with a small region in whichmagnetization points along the z-axis. These vortices remain at remanence and are annihilated at jH zj/C25100 mT. It is also noted that in both cases (in-plane and out-of-plane magneticfields), the magnetic field strength that annihilates the rema-nent state is small (in the order of 200 mT), and thus, theobtained metastable states are easily nucleated and annihi- lated. Thus, the creation and annihilation of a VA state in toroidal nanotubes could be used in data storage devices. IV. CONCLUSIONS In conclusion, chiral topological interactions induced by curvature can take place in a hollow nanomagnet with vari- able curvature. Due to its variable curvature, the remanent state of the magnetization in a hollow torus consists of a con-figuration in which vortices appear at the external borders andantivortices are present at the internal border of the torus.Qualitative analysis supports the fact that a VA state has lower energy than a state in which the torus presents two pairs of vortices. From a direct comparison of the results obtainedfor a geometry with variable curvature with the remanent stateof a hollow cylindrical nanomagnet (Gaussian curvature is 0),we showed that the VA state is a result of an effective DMI-like interaction induced by curvature. In this case, we have shown the new possibility to stabilize vortices and antivortex in magnetic nanoparticle by using curvature. This chiral statecould be used in devices working under the concept of spin-tronic, race-track memory, and nanoantennas. Finally, theremanent state when the external magnetic field is pointingalong the z-axis direction consists of a two vortex state with opposite chirality. In this case, main differences between toroidal and cylindrical cases live in the vortex structure inthe internal border of both geometries. The reversal processesfor both cases ( Hpointing along xandzdirections) were ana- lyzed. The mechanism behind the appearance and annihilation of the VA pairs has been described by analyzing the hystere- sis curves and it was shown that the VA pairs annihilate them-selves and the reversal process is followed by the nucleationof a single vortex state. ACKNOWLEDGMENTS We thank the Brazilian agencies CNPq (grant No. 301015/2015-5), Fapesb (grant No. JCB0063/2016),and Fapemig for financial support. A.S.N. would like toacknowledge funding from Grant No. Fondecyt 1150072.A.S.N. also acknowledges support from Financiamiento Basal para Centros Cient /C19ıficos y Tecnol /C19ogicos de Excelencia, under Project No. FB 0807 (Chile). We are gratefull to J. Ot /C19alora and A. Bogdanov for their valuable comments on our work. 1S. Woo, K. Litzius, B. Kr €uger, M.-Y. Im, L. Caretta, K. Richter, M. Mann, A. Krone, R. M. Reeve, M. Weigand, P. Agrawal, I. Lemesh, M.-A.Mawass, P. Fischer, M. Kl €aui, and G. S. D. Beach, Nat. Mater. 15, 501 (2016). 2C. Moreau-Luchaire, C. Moutafis, N. 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1.5104313.pdf

AIP Advances 9, 085205 (2019); https://doi.org/10.1063/1.5104313 9, 085205 © 2019 Author(s).Low damping magnetic properties and perpendicular magnetic anisotropy in the Heusler alloy Fe 1.5CoGe Cite as: AIP Advances 9, 085205 (2019); https://doi.org/10.1063/1.5104313 Submitted: 26 April 2019 . Accepted: 30 July 2019 . Published Online: 06 August 2019 Andres Conca , Alessia Niesen , Günter Reiss , and Burkard Hillebrands ARTICLES YOU MAY BE INTERESTED IN Electrical generation and propagation of spin waves in antiferromagnetic thin-film nanostrips Applied Physics Letters 114, 232403 (2019); https://doi.org/10.1063/1.5094767 Spintronic terahertz-frequency nonlinear emitter based on the canted antiferromagnet- platinum bilayers Journal of Applied Physics 125, 223903 (2019); https://doi.org/10.1063/1.5090455 Angle-resolved broadband ferromagnetic resonance apparatus enabled through a spring- loaded sample mounting manipulator Review of Scientific Instruments 90, 076103 (2019); https://doi.org/10.1063/1.5113773AIP Advances ARTICLE scitation.org/journal/adv Low damping magnetic properties and perpendicular magnetic anisotropy in the Heusler alloy Fe 1.5CoGe Cite as: AIP Advances 9, 085205 (2019); doi: 10.1063/1.5104313 Submitted: 26 April 2019 •Accepted: 30 July 2019 • Published Online: 6 August 2019 Andres Conca,1,a)Alessia Niesen,2 Günter Reiss,2 and Burkard Hillebrands1 AFFILIATIONS 1Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universität Kaiserslautern, 67663 Kaiserslautern, Germany 2Center for Spintronic Materials and Devices, Physics Department, Bielefeld University, 100131 Bielefeld, Germany a)conca@physik.uni-kl.de ABSTRACT We present a study of the dynamic magnetic properties of TiN-buffered epitaxial thin films of the Heusler alloy Fe 1.5CoGe. Thickness series annealed at different temperatures are prepared and the magnetic damping is measured, a lowest value of α= 2.18 ×10−3is obtained. The perpendicular magnetic anisotropy properties in Fe 1.5CoGe/MgO are also characterized. The evolution of the interfacial perpendicular anisotropy constant K⊥ Swith the annealing temperature is shown and compared with the widely used CoFeB/MgO interface. A large volume contribution to the perpendicular anisotropy of (4.3 ±0.5)×105J/m3is also found, in contrast with vanishing bulk contribution in common Co- and Fe-based Heusler alloys. ©2019 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/1.5104313 .,s The need for strong perpendicular magnetic anisotropy (PMA)1–5and low damping properties6–9in next-generation spin- transfer-torque magnetic memory (STT-MRAM) generates a large interest towards Heusler alloys. In addition, large tunneling magne- toresistance (TMR) ratios with MgO tunneling barriers have been reported for several of them.10,11For the application in devices based on STT switching, a low damping parameter αis important since the critical switching current is proportional to αM2 S12for in- plane magnetized films, where MSis the saturation magnetization. With perpendicular magnetization, the critical current is further reduced and it is proportional to αMS.13Therefore, a large effort is directed to study the PMA properties of Heusler alloys with low damping. For the PMA of thin Heusler films, the interface-induced per- pendicular anisotropy is essential and its strength is given by the per- pendicular interfacial anisotropy constant K⊥ S. The interface proper- ties, and therefore the value of K⊥ S, are strongly influenced by the conditions of the annealing, which is required to improve the crys- talline order of the Heusler and MgO layers and to achieve large TMR values.Here, we report the evolution of the PMA properties with annealing of the PMA properties in Fe 1.5CoGe with a MgO inter- face, by measuring different thickness series and a comparison is made with the well-known CoFeB/MgO interface. The Gilbert damping parameter αchanges with varying thickness and annealing temperature is also discussed. The films were grown by sputtering, Rf-sputtering was used for the MgO deposition and dc-sputtering for the rest. For Fe 1.5CoGe, the layer stack is MgO(S)/TiN(30)/Fe 1.5CoGe( d)/MgO(7)/Si(2) with d= 80, 40, 20, 15, 11 and 9 nm. Four series were deposited and three of them annealed for one hour at 320○C, 400○C and 500○C. For CoFeB, the layer stack structure is MgO(s)/Ta(5)/Ru(30)/ Ta(10)/MgO(7)/CoFeB( d)/MgO(7)/Ta(5)/Ru(2) d= 80, 40, 20, 15, 11, 9, 7 and 5 nm. The annealing was performed at 325○C and 360○C. The dynamic properties and material parameters were stud- ied by measuring the ferromagnetic resonance using a strip-line vector network analyzer (VNA-FMR). For this, the samples were placed facing the strip-line and the S 12transmission parameter was recorded. AIP Advances 9, 085205 (2019); doi: 10.1063/1.5104313 9, 085205-1 © Author(s) 2019AIP Advances ARTICLE scitation.org/journal/adv Crystallographic properties of the CFA thin films were deter- mined using x-ray diffraction (XRD) measurements in a Philips X’Pert Pro diffractometer with a Cu anode. The XRD data corresponding to two 40 nm thick samples in the as-deposited state and annealed at 500○C are shown in Fig. 1(a). The (002) superlattice and the fundamental (004) peak of Fe 1.5CoGe can be observed already for the as-deposited state but they experi- ence a strong intensity increase with the thermal treatment. The TiN layer acts as a seed layer and its role in improving growth has been reported also for other alloys.1,14Due to the similar lattice constant of TiN and MgO, the TiN film diffraction peaks are close to the sub- strate reflections and therefore difficult to separate. The films are B2-ordered, the presence of the (111) is not proven and therefore L21order cannot be confirmed. Figure 1(b) shows X-ray reflectometry (XRR) data for the same films as in the (a) panel. The large number of oscillations prove the low roughness of the interfaces. This is due to the low roughness below 1 nm of the TiN buffer.14The similarity between both data sets also proves that the topology of the interfaces do not vary in the studied temperature range. Figure 2 shows the dependence of the field linewidth ΔHof the FMR peak on the resonance frequency fFMR for the sample series with no thermal treatment (as-deposited) and for the series annealed at 320○C and 400○C. In order to prevent poor visibility due to FIG. 1 . (a) X-ray diffraction patterns of 40 nm thin Fe 1.5CoGe layers as-deposited, and annealed at 500○C. The (002) superlattice and the fundamental (004) peak of the Fe 1.5CoGe are clearly visible, confirming the partial B2 crystalline order. (b) X-ray reflectometry data corresponding to the samples in (a). FIG. 2 . Linewidth dependence on the frequency for Fe 1.5CoGe thin films with a thickness of 20 nm for different annealing temperatures. The data sets have a vertical offset to improve visibility (+0.5 and +2 mT for the 400○C and as-deposited series, respectively). The lines correspond to a linear fit to extract the damping parameter α. The hollow points are not considered for the fits. data overlap, the sets are shifted in the vertical axis, except the one corresponding to 320○C. The actual linewidth at 6 GHz is in the range 3.25 ±0.15 mT. The lines represent the result to a linear fit to Eq. 1 to extract the damping parameter α: μ0ΔH=μ0ΔH0+4παfFMR γ. (1) Here, ΔH0is the inhomogeneous broadening and is related to film quality, and γis the gyromagnetic ratio. A deviation from this simple linear behavior is observed for the lower frequency range and these points have not been considered for the fit (hollow circles). This faster increase of linewidth with fre- quency is common in fully epitaxial Heusler layers7,15and has been related with an increased anisotropic two-magnon scattering in the thin films for low frequency values resulting in an anisotropic ΔH. This anisotropy is not exclusive to Heusler alloys but it is expected in any epitaxial ferromagnetic film.16–18The exact conditions for observation, however, depend on the material parameters and the spin-wave dispersion. For instance, in epitaxial Fe films, the low frequency ΔHbehavior deviates only from a linear behavior when magnetic dragging due to crystalline anisotropy is dominant.19 An additional sample set annealed at 500○C showed no visi- ble FMR peak pointing to a degradation of the magnetic properties of Fe 1.5CoGe for high annealing temperature. This is in constrast with Co-based Heusler alloys where large temperatures are typically required for optimal properties. For instance, for Co 2FeAl, lowest damping is achieved at 600○C7and for Co 2MnSi, very low damping is still present at 750○C.20 The results for the damping parameter αobtained from the lin- ear fits are summarized in Fig. 3. A reduction of damping is observed when comparing the as-deposited samples to the annealed ones but the samples annealed at 400○C show larger damping than the ones annealed at 320○C. Combined with the absence of an FMR peak for 500○C, this reinforces the conclusion that the optimal anneal- ing temperature for good dynamic properties of Fe 1.5CoGe is low. The lowest damping is 2.18 ±0.03×10−3for the 20 nm thick film annealed at 320○C. For Co-based Heusler alloys, the lowest reported damping is achieved to be 7 ×10−4in Co 2MnSi.20For AIP Advances 9, 085205 (2019); doi: 10.1063/1.5104313 9, 085205-2 © Author(s) 2019AIP Advances ARTICLE scitation.org/journal/adv FIG. 3 . Dependence of the Gilbert damping parameter αon the thickness dfor three sample series: as-deposited, annealed at 320○C, and annealed at 400○C. Co2FeAl, values around 1-3 ×10−3are reported depending on the annealing conditions.7,8Concerning Fe-based alloys, values in the range of 1.2-1.9 ×10−3are reported for Fe 1+xCo2−xSi,9for Fe2Cr1−xCoxSiαvaries between 9 ×10−3with the lowest value of 8 ×10−4for Fe 2CoSi.21Therefore, the obtained value for the damping parameter in our alloy is in the lower range of previously reported ones and slightly reduced compared to those reported for the related alloy CoFeGe.22It is also smaller than the ones reported for widely used polycrystalline CoFeB23,24and permalloy.25–28 The thickness dependence of αshows a minimum around 20 nm and an increase for larger and smaller thicknesses. This behavior has been already observed for Co 2FeAl,8and the reasons are similar to that alloy and different for the two thickness ranges. For soft magnetic thin films, a strong damping increase with increas- ing thickness is expected starting at a certain value. An example can be found for NiFe in the literature.29The reason is a non- homogeneous magnetization state for thicker films which opens new loss channels via increased magnon scattering and other effects.30,31 An increase due to eddy current losses, which scale with d2, may also contribute to this behavior.46For the thinner films, the damp- ing increase is due to two reasons. When the thickness is reduced and the effect of the interface anisotropy is becoming larger the magnetization state is becoming more inhomogeneous due to thecounterplay between the demagnetization field and the anisotropy field.32In addition, other effects related to an increased role of surface roughness with decreasing film thickness play also a role. The effective magnetization Meffis extracted using a fit to Kit- tel’s formula33to the dependence of the resonance field HFMRon the resonance frequency fFMR. For a more detailed description of the FMR measurement and analysis procedure see Ref. 34. Meffis related to the saturation magnetization of the film by36–38 Meff=Ms−H⊥ K=Ms−1 μ0Ms(K⊥ S d+K⊥ V) (2) where K⊥ SandK⊥ Vare the perpendicular surface (or interfacial) and the bulk anisotropy constants, respectively. Figure 4 shows the dependence of M effon the inverse thickness 1/dfor the three sample series: as-deposited, annealed at 320○C and annealed at 400○C. The slope provides the value of K⊥ S. The con- stant shows a positive value for the as-deposited series, 0.41 ±0.12 mJ/m2, i.e. favouring a perpendicular orientation of the magneti- zation. However, the value is small in absolute value and it grows only slightly upto 0.51 ±0.17 mJ/m2when the samples are treated at 320○C. The annealing at 400○C changes the situation drastically. The value of K⊥ Sis much larger, −1.36±0.14 mJ/m2, but it also suffers a change of sign which implies that the interface induces an in-plane orientation of the magnetization. It is remarkable that this change of the magnetic properties of the interface developes without a large modification of the morphology, as proven by the XRR data shown in Fig. 1(b). The inset in Fig. 4 summarizes the dependence of K⊥ Son the annealing temperature. The evolution of K⊥ S, including the sign change, is caused by a rearrangement of atoms at the immediate interface and is not con- nected to a roughness modification. Theoretical studies45for the Heusler/MgO interface show that for the interface originated PMA properties, the termination of the Heusler film and the strength of the hybridization of certain orbitals in ordered interfaces are criti- cal. By locally improving the crystalline order, the hybridization of the orbitals is modified and also the termination can be changed due to the fact that thermal energy is required for formation of a certain termination. Taking into account the saturation magnetization Msobtained by alternating gradient magnetometer (AGM), 1100 ±120 kA/m, it FIG. 4 . Dependence of Meffextracted from the Kittel fit on the inverse thickness 1/ dfor three Fe 1.5CoGe sample series: as-deposited, annealed at 320○C and annealed at 400○C. The lines are a fit to Equation 2. The inset shows the evolution of K⊥ Swith the annealing temperature. AIP Advances 9, 085205 (2019); doi: 10.1063/1.5104313 9, 085205-3 © Author(s) 2019AIP Advances ARTICLE scitation.org/journal/adv is possible to determine also the volume contribution to the perpen- dicular anisotropy to be (4.3±0.5)×105J m3. This large value ensures, that for a 1.5 nm thin film and even for the 320○C case, the PMA properties are dominated by the bulk contribution. Recently, we reported the evolution of the PMA properties of the Co 2FeAl/MgO interface8and the situation is very different for that alloy. First, no interface-generated PMA is present in the as-deposited samples and it only appears after annealing. Second, K⊥ Sis always positive and larger than for Fe 1.5CoGe/MgO and the absence of a remarkable volume contribution makes the PMA there controlled only by the interface. Also in the related alloy Co 20Fe50Ge30there is no bulk contribution to the perpendicular anisotropy and a interface con- tribution (0.9 mJ/m2) larger than that obtained here.35The most probable reason for the difference is the lower Fe content in our case. The presence of this strong bulk contribution to PMA is quite remarkable since it is absent in common Co- and Fe-based Heusler alloys and only observed in tetragonally distorted MnGa or MnGe related Heusler alloys.39–41 For comparison, the PMA properties of the widely used CoFeB/MgO interface were also measured. The data is shown in Fig. 5 for annealing temperatures of 325○C and 360○C. An as- deposited series was not characterized since CoFeB is amorphous in that state. The lines are a fit to Eq. 2 with a prefactor 2 before K⊥ S to account for the presence of two interfaces since a trilayer system MgO/CoFeB/MgO was used. The CoFeB/MgO interface shows a robust interface perpendic- ular anisotropy, three times larger than for Fe 1.5CoGe/MgO, and slightly decreases with temperature. The bulk contribution is zero or too small to be detectable. These results are comparable to the FIG. 5 . Dependence of Meffextracted from the Kittel fit on the inverse thickness 1/dfor two MgO/CoFeB/MgO sample series: annealed at 325○C and annealed at 360○C. The lines are a fit to Equation 2 with a prefactor 2 (see text).literature.42–44We conclude that while the Co 2FeAl8and CoFeB /MgO are very similar in absolute values, thermal evolution and relative weight of interface and bulk contribution to PMA, Fe1.5CoGe/MgO differ strongly. The exact meaning of the concept of inhomogeneous magne- tization used to describe our films, and of the counterplay between demagnetizing field and anisotropy field need to be described with more detail: in an ideal thin film with smooth interfaces and in the case of K⊥ S=0, the demagnetizing field induces ideally a perfect in- plane orientation of the magnetization and a homogeneous state with an external applied field. For the case of a large enough K⊥ S>0 and for a thickness below a critical value ( d<dmin) the magnetiza- tion is again fully homogeneous but oriented perpendicularly and ford>dmaxan homogeneous in-plane state is expected. In between, for a transition region dmin<d<dmaxdifferent inhomogeneous states can be formed. Some of them can be modelled by a simple analytical model or by micromagnetic simulations as for instance in Ref. 32 but in most cases they will be also influenced by defects and magnetic history and would be difficult to model. It has to be noted that an inhomogeneous magnetization state near the interface is always present due to the fact that the interface anisotropy acts only locally. The effective anisotropy field commonly used for com- paring the strength does not imply that it is there is a field applied homogenously on the film. In summary, the damping properties of the Heusler alloy Fe1.5CoGe and the perpendicular magnetic anisotropy of the Fe1.5CoGe/MgO system have been studied. From the thickness dependent magnetic properties for as-deposited and annealed series we obtained a minimum value for αof 2.18 ±0.03×10−3for a 20 nm thick film. The evolution of the interface perpendicular anisotropy constant on the annealing temperature is shown and compared with the standard interface CoFeB/MgO. We found a large and dominant volume contribution to the PMA, which differs from CoFeB or other well studied alloys as Co 2FeAl and the inter- face contribution suffers a sign change depending on the annealing temperature. We explained the increase on damping with decreas- ing thickness in terms of a counterplay between demagnetizing field and interface PMA and correlate it with the obtained values forK⊥ S. Financial support by M-era.Net through the HEUMEM project is gratefully acknowledged. REFERENCES 1A. Niesen, J. Ludwig, M. Glas, R. Silber, J.-M. Schmalhorst, E. Arenholz, and G. Reiss, J. Appl. Phys. 121, 223902 (2017). 2Y. Takamura, T. Suzuki, Y. Fujino, and S. Nakagawa, J. Appl. Phys. 115, 17C732 (2014). 3T. Kamada, T. Kubota, S. Takahashi, Y. Sonobe, and K. Takanashi, IEEE Trans. on Magn. 50, 1 (2014). 4B. M. Ludbrook, B. J. Ruck, and S. Granville, J. Appl. Phys. 120, 013905 (2016). 5B. M. Ludbrook, B. J. Ruck, and S. Granville, Appl. Phys. Lett. 110, 062408 (2017). 6M. 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B 87, 184431 (2013). 39A. Niesen, N. Teichert, T. Matalla-Wagner, J. Balluf, N. Dohmeier, M. Glas, C. Klewe, E. Arenholz, J.-M. Schmalhorst, and G. Reiss, J. Appl. Phys. 123, 113901 (2018). 40A. Niesen, C. Sterwerf, M. Glas, J.-M. Schmalhorst, and G. Reiss, IEEE Trans. on Magn. 52, 1 (2016). 41K. Z. Suzuki, A. Ono, R. Ranjbar, A. Sugihara, and S. Mizukami, IEEE Trans. on Magn. 53, 1 (2017). 42D. C. Worledge, G. Hu, D. W. Abraham, J. Z. Sun, P. L. Trouilloud, J. Nowak, S. Brown, M. C. Gaidis, E. J. O’Sullivan, and R. P. Robertazzi, Appl. Phys. Lett. 98, 022501 (2011). 43X. Liu, W. Zhang, M. J. Carter, and G. Xiao, J. Appl. Phys. 110, 033910 (2011). 44T. Liu, Y. Zhang, J. W. Cai, and H. Y. Pan, Sci. Rep. 4, 5895 (2014). 45R. Vadapoo, A. Hallal, H. Yang, and M. Chshiev, Phys. Rev. B 94, 104418 (2016). 46M. A. W. Schoen, J. M. Shaw, H. T. Nembach, M. Weiler, and T. J. Silva, Phys. Rev. B 92, 184417 (2015). AIP Advances 9, 085205 (2019); doi: 10.1063/1.5104313 9, 085205-5 © Author(s) 2019

1.332450.pdf

Wall oscillations of domain lattices in underdamped garnet films B. E. Argyle, W. Jantz, and J. C. Slonczewski Citation: Journal of Applied Physics 54, 3370 (1983); doi: 10.1063/1.332450 View online: http://dx.doi.org/10.1063/1.332450 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/54/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Magnetooptical analysis of domainwall oscillations in garnet films J. Appl. Phys. 61, 4210 (1987); 10.1063/1.338477 Coupled oscillations of domaindomain wall system in garnet films J. Appl. Phys. 57, 3701 (1985); 10.1063/1.334995 Domain lattice FMR in magnetic garnet thin films J. Appl. Phys. 53, 2098 (1982); 10.1063/1.330710 Magnetic domain wall waves, parametrically excited in a stripe domain lattice in a garnet film Appl. Phys. Lett. 38, 930 (1981); 10.1063/1.92186 Abstract: Continuouswave domain wall oscillation in bubble garnet films (invited) J. Appl. Phys. 52, 2353 (1981); 10.1063/1.328928 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.174.21.5 On: Thu, 18 Dec 2014 14:09:00Wall oscillations of domain lattices in underdamped garnet films B. E. Argyle, W. Jantz,a) and J. C. Slonczewski IBM T. J. Watson Research Center, Yorktown Heights, New York 10598 (Received 1 November 1982; accepted for publication 19 January 1983) This report describ~sa recentl~ d~veloped domain-waH-oscillation spectrum analyzer and the results of some sensitive quantitative measurements of domain wall motion response in un~erdamped garnet films. Objects of study include periodic bubble domain lattices and periodic stripe patterns. Wall ~ispJacemen~s as small as 5 A (only a few percent of the wall width) are detec~able, thus allowmg observatIOn of resonance multiplets due to wall flexure not seen by other techn.lques. The.dat~ pre~ented here are taken in the linear region at low drive (well below onset of velocl.ty saturatIOn) m th!n fil~ samples ofGd, Ga:YIG grown epitaxially on Gd3Ga5012. The domam wal.l resonance hn~wldth determination of the Gilbert damping parameter agrees with monodomam fe.rromagne~lc resonance, resolving previous discrepancies with respect to effective mas~ and dampmg coefficient found in free oscillations following pulsed excitations in thin films and III c?~plex susceptibility spectra in thick single crystal plates. Losses and nonlinearities due to coe:clvlty appear supp.ressed in dense domain configurations by the presence of strong restoring forces. Companson of our theory and experiments show that the rigid-wall model acco~nts well for the first resonance, including its dependence on lattice period and external field applIed ?ormal to the film. The Doring mass, however, needs correction by numerical caIcul~tlOns for effects of stray fields. Empirical and a priori restoring-force coefficients agree very well with our observed resonance frequencies. PACS numbers: 75.70.Kw, 76.90. + d, 78.20.Ls I. INTRODUCTION The importance to computer storage applications of magnetic bubble domains 1 provided the initial impetus for investigation of the physics of bubble walls. Subsequent physical studies of bubbles in garnet films have revealed many interesting phenomena and stimulated new theoretical concepts connected with the solitonic behavior of walls and domains.2-4 These dual incentives of relevance and curiosity have spurred advances in experimental techniques of observ ing fine details of wall motion. Broadly speaking, the experiments fall into two cate gories, impulse and continuous wave excitation. The pulsed field technique is familiar in the forms of pulsed bubble col lapse, pulsed gradient-field bubble propagation, and pulsed field applied perpendicular to the film to excite damped wall oscillations. 5-11 Continuous wave experiments6. 1 2-16 have in cluded: measurements of complex initial susceptibility on stripe domains in bulk single crystalsl2; work on epitaxial thin films detecting magnetooptically the wall oscillations of stripe domains,6,13-15 bubbles, 16.17 and one isolated wallIS; and the recent studies of Wigen, Dotsch, and co workers 19-21 using a slot line rftechnique22 to investigate the dependence of bubble latticel9 and stripe wall reson ances20,21 on dc fields. Much of present day knowledge of bubble dynamics has derived from experiments of the first category applying an impulse to the bubble and then monitoring the time evolu tion of the motion. In principle, bubble dynamics is governed by the nonlinear Landau-Lifshitz equation which regards the magnetic medium as a continuous distribution of cou- ") Present address: Fraunhofer Institute ftir Angewandte Festkiirperphysik, Eckerstr. 4, D-7800, Federal Republic of Germany. pled gyroscopes. For practical interpretation of experimen tal results it helps to transform these equations into a form which involves a single effective displacement coordinate whose response to these rapidly changing fields depends upon internal micromagnetic features within the wall such as Bloch lines and Bloch points, whether present statically or induced transiently during the motion, The motion is also known to depend on internal influences such as the stray magnetic fields of magnetic poles at the free surface and at the film-substrate interface which can cause the wall mass to be unevenly distributed through the thickness of the film even when Bloch lines are absent.23.24 There also exist tor ques on the wall magnetization due to damping. It is natural, therefore, to expect the decaying free wall oscillations to contain frequency information useful for gaining basic un derstanding of wall dynamics. However, interpretation of impulse experiments has been limited to a single resonance frequency of one bubble,8.9,11 an array of domains,'-7 or an isolated straight wall. 10 Moreover, the results were restricted to such large amplitudes, in order to obtain sufficient signal strength, that the frequency often depended on amplitUde, However, the most fundamental, elementary, first-or der excitations occur at small amplitude, whose precise ob servation requires a continuous wave method. We have, therefore, constructed a wall oscillation spectrum analyzer described in Sec. II to monitor magneto-optically the ampli tude of oscillations over a range of frequency from about 0,1-150 MHz.25 The dynamic range encompasses four de cades in drive field and more than three decades in re sponse.26 This paper focuses on the linear responses at low drive fields. Additional work in progress describes dynamic behavior in the nonlinear regime, some aspects of which were recently summarized,26 Our instrumentation can also 3370 J. Appl. Phys. 54 (6). June 1983 0021-8979/83/063370-17$02.40 © 1983 American Institute of Physics 3370 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.174.21.5 On: Thu, 18 Dec 2014 14:09:00be configured to investigate finite wavelength excitations, e.g., by means of spatial Fourier transform magneto-op tics.27 Isolated bubble responses28 due to Bloch lines and Bloch points, isolated charged wall responses,29 and stand ing waves in confined stripe domain arrays,30 have also been measured with this instrumentation. The descriptions of ap paratus and techniques in these references were brief. One purpose of this report is to describe fully the appa ratus and techniques. A second purpose is to present new results and conclusions about domain wall dynamics. These results bear on (1) the validity of the well-known simple har monic oscillator model embodying assumptions of a rigid wall with a single effective mass and a single restoring force coefficient (Studies of frequency versus static external fields provide tests of this model ); (2) the role of wall-flexure, par ticularly its dependence on dc fields; (3) the relationship of wall-resonance line width and FMR linewidth measure ments of the viscous-damping parameter; (4) the presence (or absence) of effects due to coercivity. The physics presented in this paper includes that part of our current research which deals with well-behaved small amplitude resonances in perfect stripe and bubble lattices. We defer to the future publication of effects due to in-plane dc fields, to nonlinearity at high amplitude, and to perturba tions of domain-lattice vibrations caused by Bloch lines and by bubble-lattice imperfections. Section III describes how periodic lattices of bubbles are generated and how a predetermined lattice spacing can be obtained reproducibly. Characterization of the sample films for their static properties and investigation of domain walls for their dynamic properties such as effective mass, viscous damping and coercivity are described in Sec. IV. In Sec. V collective resonances of periodic bubble and stripe lattices are measured and discussed in terms of flexural modes. II. EXPERIMENTAL PROCEDURE Domain wall oscillations are excited with a spatially homogeneous, continuous wave sinusoidal magnetic field hz in the frequency range 0.1-150 MHz. They are sensed by the Faraday magneto-optic effect using laser light and high speed photomultiplier (PMT). A. Apparatus A diagram illustrating the magneto-optic domain-wall oscillation spectrometer is shown in Fig. 1. The setup in cludes auxiliary equipment needed for visual observation and for photographic and video recording. The polarizing microscope is an inverted Zeiss Axiomat. The sample is illu minated with either an incoherent Hg arc lamp for the visual inspection of domain patterns or with a coherent cw Ar+ laser for enhancing dynamic PMT signals. The laser may be directed for either reflection or transmission illumination. In opaque films, reflection is the only method. Otherwise, transmission is preferred because it reduces back reflections into the PMT oflight from any optical surfaces not contrib- 3371 J. Appl. Phys., Vol. 54, No.6, June 1983 uting to the magneto-optic signals. Various optical diaph ragms and measuring scale comparators (not shown) are in sertable at intermediate focal planes. Magnetic domain contrast is made visible through nearly crossed polarizers. Binocular inspection, TV monitoring, photographing, and PMT detection are available. Dividing optics allow simulta neous viewing of the domain pattern while its dynamic re sponse to the rf magnetic field is being recorded by the PMT. A nest ofthree orthogonal pairs of Helmholtz coils pro vides static field components Hz normal to the film and Hp at any desired angle if> within the plane of the film. The Helm holtz construction allows optical access to the sample and convenient interchange of sample and sample holders. The inverted microscope arrangement allows top loading of these Helmholtz coils. The microscope's infinity-corrected optics enables the objective lens to be displaced along the optic axis to focus at the common center of the Helmholtz coils where the sample film is located. Specific domain patterns-parallel stripe patterns (PSP), amorphous and periodic bubble lattices (ABL, PBL), or isolated bubbles (!B)-are created by techniques de scribed in Sec. III. A nine-turn pancake coil having an inner diameter of about I.S mm and placed below the samplel4 is used to generate the nearly spatially homogeneous rf field hz (t ) for inducing domain wall oscillations. B. Signal processing Magneto-optic signals produced by small oscillations of magnetic domain walls in thin garnet films are expected to be very weak. To ensure adequate signal-to-noise ratio (SNR), we find that intense laser illumination, a high gain PMT detector, and high performance electronic signal processing are essential. The Ar+ laser provides selectable wavelengths in the sample's spectral region of high magneto-optic figure of merit. Both the real image of the domain's magneto-optic contrast pattern and the spatial-Fourier-transform diffrac tion image are available in our setup as will be discussed and compared below. In both cases, frequency tuneable signal processing is performed by heterodyne detection and narrow band filtering at the IF by an rf spectrum analyzer (SA), the Tektronix 7L13. A time base for sweeping frequency and XY recorder voltages for plotting amplitudes versus fre quency are provided by a Tektronix 7613 storage scope. The SA controls and phase locks to a Tektronix TR 502 tracking generator (TG), whose amplified output feeds the pancake coil. The SA, therefore, monitors only signals at the excita tion frequency generated by the TG, i.e., only the fundamen tal amplitude is measured by the SA. However, with its nar row-band detection, we obtain a substantial suppression of the large shot noise which is present inherently in Faraday signals from thin film samples. In our measurements using an RCA PF 10 12 PMT, the residual noise, using the narrow est resolution bandwidth filter (30 Hz) available in the SA, is typically between -90 and -110 dBm. The internal noise of the SA is approximately -140 dBm. The magneto-optic signals from wall oscillations are typically between -60 and -100 dBm, depending on the strength of hz excitation and the laser intensity. Argyle, Jantz, and Slonczewski 3371 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.174.21.5 On: Thu, 18 Dec 2014 14:09:00DC FIELDS Hz (COILS NOT SHOWN) I PULSE GENERATOR ~--------~~ ..:rn---~: _____ m I W POLARIZER t ...,Hy ~Hx -60 TO 0 dBm RF SPECTRUM ANALYZER SWEPT FREQ~ENCY AND PHASE LOCKING STORAGE OSCILLOSCOPE VARIABLE ATTENUATOR I I I I : : (RE,Ga):YIG FILM I ~~~ I ~~~~f---PANCAKE COIL ~ OBJECTIVE LENS BACK FOCAL PLA~E FOURIER TRANSFORM AUXILIARY LENS OPTICS BINOCULAR L1------TI~ POL I , I -, ---~ - SIT +t ANALYZER: i ,', --------VIDICON I I I I :O:~~-:::J g ~8~~!=-' ·· .. 0 r~'1 I If,' II , I ' I II I I I : I :/ I! 2.5 ns APERTURE MIRROR AT INTERMEDIATE IMAGE PLANE \ THUMB WHEEL APERTURE SELECTOR PMT DETECTOR FIG. I. Diagram of experimental apparatus using a polarizing microscope and photomultiplier (PMT) detector to monitor domain wall oscillations in thin garnet films. A laser illuminates in either reflection (solid light path labeled I) or transmission (dashed path labeled 2). Insertion of an auxiliary lens to focus on the back focal plane of the objective lens images the Fraunhofer diffraction pattern, thus forming an optical Fourier transform of the domain magneto-optic contrast pattern. Half-silvered and front-surface mirrors allow an image of the PMT aperture to be superimposed on the sample image (real or transform). Position and size of the PMT aperture are adjustable relative to the real or transform optical pattern. A tracking generator and amplifier are connected to the pancake coil generating an rf magnetic field that excites wall oscillations. PMT signals are measured with the spectrum analyzer. C. The optic detection modes The light intensity modulation resulting from the do main wall oscillation can be obtained in either the real or diffracted image. In the direct observation mode, a real im age of the domain area under investigation is formed in the plane of an exit diaphragm that apertures the light path to the PMT. The alternating up and down magnetic domains are made visible as contrasting bright and dark areas (photo graphed in Fig. 2) by sending the transmitted light through a sheet polarizer oriented at nearly 900 with respect to the po larization of the incident laser. Since wall oscillations result in alternating size variations of the domains, the amount of light impinging onto the PMT will be modulated at the wall oscillation frequency. It is less well known that wall motions can also be mea sured using magneto-optic light diffraction. This technique comes to mind when one considers that static magnetic do mains produce well-defined Fraunhofer diffraction spot pat terns32-37 when the arrangement of domains is periodic and the illumination is well collimated. Figure 2 shows the dif- 3372 J. Appl. Phys., Vol. 54, No.6, June 1983 (a) •••••••••• ••••••••••••• ............. •••••••••••• •••••••••••• •••••••••••• •••••••••••• • • • • • • •••••••••••• "' .......... . . ~ ......... . . , ." ....... . (el (b) • • • • .... :r-•••• ••••• • • • •• • . ' . -. . . ... ~ .. • • (d) FIG. 2. Photographs of the real and the Fraunhofer diffraction image of magnetic domain patterns at zero applied fields. A parallel stripe pattern (PSP), shown in (a) produces light scattering pattern in (b), and the periodic bubble lattice (PBL) produces corresponding photos in (c) and (d). Argyle, Jantz, and Slonczewski 3372 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.174.21.5 On: Thu, 18 Dec 2014 14:09:00fraction patterns ofa PSP (b) and a PBL (d). These images are obtained by inserting an auxiliary lens to focus on the back focal plane of the objective, as indicated in Fig. 1. This ar rangement constitutes Fourier-transform optics. Modula tion of wall positions produces modulated spot intensities as will be shown below. While detailed analysis oflight diffrac tion from static PBL domains has been given by Kuhlow,38 our discussion presented here illustrates features of the in formation to be gained both from static and dynamic mea surements in the diffraction image as contrasted with the real image. The diffraction pattern [Fig. 2 (b)] of a static PSP in the xy plane with the walls aligned along y consists of a line of spots at positions (xn' Yn) in the Fourier transform plane f 21Tn Xn =----, Yn =0, kopt P (2.1) determined by the stripe array period P, the effective focal length f of the diffraction optics, and the magnitude of the wave vector kort of the diffracted light. The spot intensities depend not only on the period P but also upon the difference in widths d1 -d2 = E (d1 + d2 = P) of the alternating up down domains. Only those components of light rotated through Faraday-effect angles ± f3 are diffracted. Because f3 for bubble films is usually small, the polarizer and analyzer must be nearly crossed to remove most of the unrotated light. Setting the analyzer to 90°-<5 with respect to the polar izer produces diffraction intensities3? in the undiffracted beam I~SP = K [(dP)2 sin2 f3 cos2 8 -(l/2)(dP )sin(2 f3 )sin(28) + sin2 8 cos2 f3] (2.2) and in the nth order diffracted beam I~sP = K ~~2 sin2 f3 cos2 8 sin2 [ n; (1 + EIP)], (n 1=0) (2.3) with a constant K depending on illumination (intensity and wavelength) and optical transmission efficiency. The nth or der spot is extinguished when dP=(21In)-1, with IdPI<l, (n1=O), where I is an integer. For example, at Hz = ° we have E = ° and all even spots are extinguished as demonstrated in the photograph of Fig. 2(b). For static diffraction spots originating from a hexagon al PBL the intensity of nth order is38 (2.4a) where r is the bubble radius, a is the lattice parameter, i.e., the shortest distance between bubble centers, Pn is given by Pn = 2/3, 2, 4/3, 28/3, 613 for integers n = 1-5, (2.4b) and J1 is the first-order Bessel function. The series of photo graphs in Fig. 3 illustrates the influence of Hz, by reducing r, on the diffraction pattern. Dependencies reproducing Eq. (1.4) may be obtained experimentally by measuring the in tensity of the various orders as a function of Hz. However, r(Hz) may be measured at discrete points over a broad range of Hz, without changing the lattice parameter (see Sec. III) 3373 J. Appl. Phys., Vol. 54, No.6, June 1983 BIAS FIELD (Oe) 37 20 13 o -26 REAL IMAGE FRAUNHOFER DIFFRACTION FIG. 3. Variation of the real and the Fraunhofer diffraction image ofa PBL with the strength of bias field Hz oriented normal to the magnetic film. Note the disappearance of certain diffraction orders at certain fields. Spot intensi ty variations with H, compared with magneto-optic diffraction theory" give bubble radii ro(H,) as in Fig. 9. by recording discrete values Hz where specific diffraction orders are extinguished at the characteristic values of ria according to the known zeros of the Bessel function J1• Data so acquired are included in Fig. 9. The variations of diffraction spot intensities with do main size also result in light modulation signals in the event of domain wall oscillations. For Hz = 0, the rf field hz(t) applied to a PSP modulates the domain widths so that the ~~r-Ow CL...J (J)« wu Ct:(J) ~(!) wO ~...J (J)~ r (J) o 25 rf FIELD AMPLITUDE hz t I 50 75 100 125 FREQUENCY (MHz) FIG. 4. Frequency dependence of the PMT detector sensitivity and the rf field used to oscillate the domain walls (See text, Sec. II 0). Argyle, Jantz, and Slonczewski 3373 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.174.21.5 On: Thu, 18 Dec 2014 14:09:00factor c = d 1 -d2 producing magneto-optic signals be comes C = Cl"COS OJ! , (2.5) where OJ is the rf excitation frequency times 21T. Substitution in Eq. (2.2) shows that the zero-order intensity 10 contains components at OJ, 2w, and dc I :;sp = K [( l/2)(cIIP)2 sin2 (J cos2 8 cos(2wt) - ( 112)(£ ziP )sin(2 (J )sin(28 )"cos(OJt ) + sin2 8 cos2 {J + (112)(£11 P)2 sin2 (J cos2 8 ]. (2.6) Thus, the un diffracted intensity modulation at OJ as mea sured selectively by the SA provides a signal linearly related to the wall-motion fundamental amplitude £1 averaged over the area illuminated by the laser. On the other hand, combining Eqs. (2.3) and (2.5) shows that the higher order spots have no intensity modulation at the fundamental OJ for Hz = O. Furthermore, if the analyzer is set to 8 = 0, even the zero-order spot contains no compo nent at OJ. With a static bias creating a nonzero, static E in addition to the rf variation, all spot intensities generally are modulated at 2w and OJ. Because the optical amplitude in a Fraunhofer diffrac tion pattern is the spatial Fourier transform of the optical amplitude pattern in the real image, the diffraction pattern formed by light transmitted through a bubble film portrays reciprocal or k space, and provides information about the collective behavior of a multidomain pattern. In addition, signals from local deviations from regular periodic domain behavior, e.g., due to local defects in the lattice or due to various wall states, are suppressed by the spatial integration of the Fourier transform. Such studies are particularly con venient with the instrumentation described here. Having the capability to image the back focal plane of the objective lens, using an auxiliary lens, a selected part of the diffraction pat tern, in particular the central spot for the present study, can be apertured and sent to the PMT. Other diffracted spots due to wavelike excitations with wave vector k =j:. 0 have recently been generated dynamically and studied also by this tech nique.27 By contrast, signals from the real image can measure domain wall oscillations very locally, so that lattice imper fections,34 isolated bubbles,28 or even small segments of a bubble wall3! and a charged wa1l29 may be inspected selec tively with suitable control of size and position of the PMT aperture. D. System calibration A substantial portion of the results presented here deal with resonance frequencies depending on external fields, amplitude shapes, and other parameters such as the geome try of the domain array. For these investigations, absolute calibration of neither the drive nor the oscillation amplitude is required. Hence, in these cases, the system adjustments are made to achieve best performance for each experiment. Gen erally, the signal [second term in Eq. (2.6), for example] in creases with the analyzer offset angle 8. However, increasing 8 also increases the dc background intensity [third and fourth terms in Eq. (2.6)] and hence PMT noise, eventually 3374 J. Appl. Phys., Vol. 54, No.6, June 1983 saturating the PMT. The value of 8 for best signal-to-noise ratio (SNR) depends upon laser intensity and non ideal fea tures of the instrument such as a nonzero polarizer-analyzer extinction ratio, stray reflected light or blooming in the op tics, and PMT dark current.39 A suitable setting of 8 is readi ly established by maximizing SNR while observing the PMT signal with the spectrum analyzer. For the measurements presented here, 8 was typically 0.5-1.5". The intensity and focusing of the laser were also adjusted depending on the desired area for investigation and on limitations such as PMT saturation current or sample heating as detected, for example, in changes of bubble collapse field. For some investigations, particuls.r1y the nonlinear ef fects such as saturation velocity,26 calibration of both the drive amplitudes, and the wall oscillation amplitudes is re quired. Calibration of the drive field hz relative to current in the pancake coil is straightforward.40 Since the dc resistance of the coil is very small (about In), it is connected to ground with a 50-n coaxial feed through type termination, and the rf voltage across this terminator is monitored by a high fre quency oscilliscope. Because this terminator is purely resis tive over a broad frequency range, the current through the coil and, thereby, hz can be determined as a function offre quency, as shown in Fig. 4, bottom curve. The rolloff of hz with increasing frequency for fixed rf power settings of the TG and amplifier is due mostly to increasing inductive reac tance of the coil, whereas the superimposed oscillatory vari ation indicates imperfect matching. The low frequency cali bration factor for the coil, typically 25 Del A, was obtained by comparing the known dc field for collapse of a bubble with the amplitude of a wide trapezoidal current pulse just sufficient for collapse in the presence of a static Hz of suffi cient strength to stabilize the bubble some known increment below the collapse field. The trapezoidal shape avoids errors due to the difference between static and dynamic collapse,2 and due to dynamic overshoot effects. 2 One procedure to calibrate PMT signal against actual domain wall displacements is to establish a relation between dc output voltage Vofthe PMT and, for instance, the size of a bubble,8 measured directly with a microscope filar micro meter. After correction for a dc background (Vsn) due to shot noise, V = (V ~MT -V;n) 1/2 should be proportional to ?, hence one expects that r and V 1/2 data exhibit the same de pendence on Hz. This is indeed the case, as demonstrated in Fig. 5, where for further corroboration a calculation of r(Hz) using Callen's and Joseph's formula41 is also displayed. One thus obtains a calibration between a change..::1 r and the corre sponding change ..::1 V in PMT voltage. This calibration car ries over to dynamic ac measurements by replacing the dc field Hz with the rf field hz inferred from the current in the pancake coil, and by replacing the dc voltage V with the rf PMT signal measured with the SA. All system parameters must be held constant and the frequency dependence of hz and of the PMT sensitivity (discussed below) taken into ac count. A less time-consuming procedure14 for the PMT cali bration is to utilize the nonresonant low-frequency (-I MHz) oscillatory displacement of domain walls. Once this linear PMT signal versus drive has been calibrated against Argyle, Jantz, and Slonczewski 3374 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.174.21.5 On: Thu, 18 Dec 2014 14:09:004r-SAMPLE A E ~x -.:!-3r-a:: ·~x rJ) ;2 0 « 2- ....... 1 a:: ............ -THEORY x .... .) W -1 • DIRECT OBSERVATION III III II-PMT MEASUREMENT =:J x III I I I ~ 50 55 60 65 70 BIAS FIELD Hz (Oel FIG, 5, Static radius of isolated bubble vs bias field Hz, Calculation and direct measurement by filar optical micrometer serve to establish calIbra tion of PMT sensitivity to changes in bubble size, the quasistatic wall displacement, this (low frequency) cali bration is used as an intermediate reference that can be mea sured with the SA itself. This procedure has the advantage that the bandwidth of the measuring and the calibrating in strument are the same, so that correction for background effects of shot noise are the same. A typical calibration coef ficient is about 1 nm of wall amplitude per microvolt PMT signal. The frequency dependence of the PMT sensitivity and thus the frequency dependence of the system's wall ampli tude calibration is determined by measuring PMT noise ver sus frequency with the SA. In our magneto-optic detection scheme shot noise dominates other types such as Johnson noise. Although shot noise current from the photocathode is independent of rffrequency, the current gain ofthe PMT can vary with frequency, for example, due to the inter-dynode capacitances. Thus, by performing a spectral analysis of PMT noise, under illumination and load conditions used in our wall oscillation measurements, but without drive (hz = 0), we have determined the frequency variation of the system (PMT + SA) sensitivity. The top curve of Fig. 4 shows such a spectrum. Consequently, wall oscillation spec tra as recorded, exhibit the same roll off with frequency and may be corrected accordingly. The PMT sensitivity rolloff obtained from the shot noise versus frequency was also confirmed by comparing with the swept frequency response of a paramagnetic single crystal slab of Tb3AIs012 under similar conditions as are used in measuring domain wall oscillations. This material is an ideal oscillator for such a test because, being paramagne tic, it has no domains and, far away from paramagnetic reso nance, its magnetic susceptibility is independent of rf fre quency. Having the largest Verdet constant of all the paramagnetic garnets,42 it produces good magneto-optic sig nals. These two independent measurements of PMT fre quency response agreed. Prominent features of the apparatus include high sensi tivity, wide dynamic range and versatility. Wall motions as small as a few angstroms have been detected. The dynamic range is typically three orders of magnitude in wall ampli- 3375 J. Appl. Phys., Vol. 54, No, 6, June 1983 tude and at last four orders in hz for bubble films -5-Jlm thick containing -5-Jlm domains. Features giving it the ver satility described above are demonstrated in results given throughout this paper and earlier publications?5-31 The ver satility again relies essentially on the magneto-optic tech nique which gives remarkable freedom of choice with re spect to domain configurations and excitations to be studied . Finally, the structures generating the rf drive fields may be chosen arbitrarily to provide for spatially homogeneous or, if desired, controlled inhomogeneous excitation. III. CONTROLLED GENERATION AND STATIC PROPERTIES OF BUBBLE LATTICES A. Amorphous lattice nucleation and density Amorphous bubble lattices (ABL) of varying densities are used for controlled generation of periodic bubble lattices (PBL) having nearly arbitrary spacing. The ABL is first gen erated using in-plane demagnetization. Upon removal of a saturating in-plane field H p, a particular domain configura tion or phase is nucleated, e.g., bubbles, stripes, or a mixture of both. The nucleation phase of choice may be controlled by two parameters, the fixed bias field Hz applied before, dur ing, and after the in-plane saturation, and the orientation ¢J of the saturating in-plane field, Hp. Depending on proper choices of these nucleation parameters, bubble lattices of specified lattice constant are produced by transforming the initially generated phase into a periodic arrangement. The underlying physics of the nucleation has been dis cussed by Hubert et ai.,43 giving special attention to the "ho mogeneous" nucleation (defined below). To demonstrate our experimental control of the nucleation phase the sequence of domain patterns shown in Fig. 6 were nucleated with fixed bias fields Hz noted in the photographs and with a single fixed orientation ¢J of the saturating in-plane field H p. At the high bias field Hz = -45 Oe of picture (a), a dilute pattern of only stripe domains is generated after removal of H p. The magnetization of the white domains is opposite to Hz. Re peating the process of saturation and nucleation at a low Hz results in the mixed pattern of bubbles and stripes as in pic ture (b). The ratio of bubbles to stripes continues to increase as Hz is reduced, and eventually at a sharply defined value, Hz = -34.2 ± 0.1 Oe, a pure ABL is produced. The ABL phase (bubbles only) continues to be nucleated down to a lower boundary-again sharply defined-at 8.6 ± 0.1 Oe where (picture e) stripe domains begin to reappear. With further reduction of Hz , stripes become more abundant, and near Hz = 0 the "homogeneous" phase occurs which Hu bert et al.43 have defined as a roughly balanced arrangement of oppositely oriented domains. Upon increasing the now inverted Hz, the above sequence of different remanent con figurations is repeated but with domain orientations invert ed [picture (i) through (011. The characteristic boundaries for pure ABL generation are now Hz = + 10.7 Oe and Hz = + 36.0 Oe. The symmetry of these positive and negative nucleation boundaries about Hz = 0 as well as the occur rence of the homogeneous nucleation (Fig. 6, h near Hz = 0 occurred as an accidental consequence of the particular choice of ¢J as will be seen in Sec. IIIB below. Argyle, Jantz, and Slonczewski 3375 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.174.21.5 On: Thu, 18 Dec 2014 14:09:00, ' ~""'~ B. Nucleation phase and magnetocrystalline anisotropy Data in Fig. 7 show the dependence on ¢ of the four characteristic Hz values defining the intervals or phase boundaries of pure ABL generation. The threefold symme try exhibited by these data is due to the cubic magnetocrys talline anisotropy as discussed by Hubert et al.43 For a quali tative understanding, assume that the sample is saturated parallel to [112], i.e, with Hp parallel to the projection of an oblique easy [111] axis onto the (Ill) film plane. Let + Hz point along [111] normal to the film. Upon gradual removal 0000 OOOO{ 50 0° ° 0 00 ° ° 0° I ° ,0 ° 40 ' : oooooL 00 00°0 0 0 °0 30 -0° 0 AZIMUTHAL ANGLE cP FIG. 7. Upper and lower threshold bias fields H, that allow nucleating amorphous bubble patterns (defined in text). The threefold variation with direction of the in-plane saturating field is due to the influence of the cubic magnetocrystalline anisotropy on the nucleation process. 3376 J. Appl. Phys., Vol. 54, No.6, June 1983 FIG. 6. Domain patterns nucleated after applying and removing a strong saturating in-plane field while different perpendicu lar bias fields Hz, as indicated, are being held fixed. The orientation of in-plane sa turating field is also fixed. Note the well defined threshold bias fields that separate pure and mixed domain configurations. of Hp, the cubic anistropy tends to tilt the magnetization towards the easy [11 I] axis; hence supporting the tilting ef fected by a positive bias field. IfHp is reversed, i.e., if satura tion is along [TT2], then the anisotropy would tend to tilt the magnetization towards [TT 1], hence opposing the torque of the steady positive bias field. Therefore, in this case, a higher + Hz is required to produce the same effective upward tilt- ing and consequently the same domain pattern. In our exper iment, Hp parallel to [112] corresponds to ¢ = 108° (modulo 120°) in Fig. 7, whereas the photographs of Fig. 6 have been taken at ¢ = 22°, which is approximately an intermediate orientation where there is almost no preferential tilting due to cubic anisotropy. Note that the cos 3¢ angular depen dence found within experimental accuracy is not superim posed with a cos ¢ variation, characteristic of a tilted anisot ropy, as may occur in films on slightly misaligned substrates.43 We use the partially controlled domain nucleation of ABL's to generate PBL's having specified lattice parameters a over a substantial interval 8-25 f.lm. While ABL's of differ ent density may be nucleated as in Fig. 6(d) and 6(e) even more dilute ABL's may be obtained starting from mixed pat terns such as Fig. 6(c) and rising Hz after Hp has been re moved, causing the short stripe domains to run back into the bubble shape. Starting from ABL's of different bubble densi ties, PBL's with correspondingly different a are obtained by applying a small pulsed or sinusoidal "annealing" field.44 Therefore, by simply varying Hz, before removal of Hp > 41TM, one can easily control the lattice parameter. This enabled us to study its influence on resonance frequencies (see Sec. V) and to regenerate specified lattices in order to reproduce previously obtained data. Argyle, Jantz, and Slonczewski 3376 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.174.21.5 On: Thu, 18 Dec 2014 14:09:00(b) ..... ~.;. . ........... ••••••••••• ••• • • • • . .......... . •••••••••• ••••••••••• •••••••• _il"-_ ••••.•.• (d) FIG. 8. Some bubble lattice defects. In analogy to crystal lattices, they may be characterized as (a) interstitial, (b) vacancy, (c) dislocation line, and (d) grain boundary. C. Removal of lattice imperfections While the annealing process above eventually produces large lattice areas with a perfect hexagonal structure, inter mediately, a number of more or less localized lattice de fects4s•46 are observed. These defects provide interesting analogies to crystal lattice defects. For illustration, some de fects are displayed in Fig. 8. By far the most common defect is the combination of a fivefold and a sevenfold coordinated bubble, terminating a surplus bubble column. This distur bance may be translated by splitting the sevenfold coordinat ed bubble, increasing the length of the surplus column by one bubble. The consecutive repetition of this process is compar able to weaving an additional thread into a fabric. Alternati vely, the surplus column may link with a neighboring col umn, so part of the latter becomes the surplus column which is thus laterally displaced. Both processes have been ob served experimentally and result in removal of the defect out of the region under investigation. D. Lattice spacing and radial compliance Once a lattice has been generated and annealed, the PBL lattice parameter a does not change over a wide vari ation of Hz. (See Fig. 3.) This is obviously due to the fact that a changing Hz does not increase nor (below collapse) de crease the number of bubbles present in the film. By contrast, theoretical energy minimization46--48 would require a vari ation of a with Hz. This experimental finding is taken into account in our theoretical calculations (Sec. V) that invoke the measured value of a as a fixed independent input param eter. Figure 9 shows the dependence of the bubble radius R on Hz for different lattice parameters. The data were ob tained using the diffraction technique associated with zeros of the Bessel function J1 as described in Sec. II C. Also shown are our theoretical calculations (described in Sec. V A) containing no adjustable parameters. The agreement is very satisfactory. For comparison, the field dependence of r for an isolated bubble between run-out and collapse is also given to illustrate how in a lattice the stray field of the neigh boring bubbles tends to reduce r and to stabilize the domain pattern upon decreasing Hz. 3377 J. Appl. Phys., Vol. 54, No.6, June 1983 4.0 E 3-3.5 /:, ~=IO.9fLm £:; ~ 0 £:; ; + 9.9 <l: I-{}-I ~ 30 ----...... ~ 915 <l: ~84 ~ ~ o 25 w ...J [D [D 15 2.0 -THEORY T I o 10 20 SAMPLE D PBL \ ISOLATED \ BUBBLE \ \ \ \ \ ~o~~~" \ ~~/:, \ I~~I \ T 30 40 50 60 BIAS FIELD HzeOe) PIG. 9. Dependence of the radius r of bubble domains on the bias field Hz in PBL's having fixed lattice spacings, a. The data were obtained using the diffraction technique described in Sec. II C and Eq. (2.4). The solid lines are calculated without adjustable parameters using the theory in Sec. V A. The theoretical r(Hz) dependence for an isolated bubble is shown for compari son. Slopes drldH, give radial compliance used in determining effective wall mass from domain wall resonance (DWR) in Sec. IV B. From the data given in Fig. 9 other useful quantities may be deduced, in particular the bubble compliance dr/dHz as a function of Hz and a. In the following section it will be elaborated how the compliance data may represent effective stiffness constants which, in conjunction with the domain wall resonance frequency (Sec. V) allow determina tion of the effective mass of stripe and bubble domain walls. IV. SAMPLE AND DOMAIN WALL PARAMETERS Our bubble films are characterized by static material and dynamic domain wall parameters. We have determined static material parameters by standard techniques, as de tailed below in Sec. IV A. The behavior of dynamic wall motion in the low drive region may be described by a linear equation of motion mij + bi; + Cq = 2M(hz cos rut -He sgn i;) (4.1) for domain wall displacements q driven sinusoidally to oscil late in a potential well characterized by a force constant C (dyne/ cm 3), inertial mass density m, viscous damping coeffi cient b, and coercive field He. Results for these quantities (or related ones) obtained by our domain wall resonance method will be compared with conventional methods and with theo retical calculations in Sec. V. A. Material parameters Table I lists, in standard notation, values for composi tion coefficients x and y (in Gdx Y3 _xGayFe s _yOd, thick ness h, material length parameter 1= ~AKu /1rM;, wall thickness parameter Ll = ~ A / K u , uniaxial anisotropy field Hu = 2KJM, , magnetic saturation 41TMs' anisotropy ratio parameter Q = H u /41TMs' and exchange stiffness coefficient A. This material was chosen for this work because of the small viscous damping. The two samples are labelled A and Argyle, Jantz, and Slonczewski 3377 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.174.21.5 On: Thu, 18 Dec 2014 14:09:00TABLE I. Static material parameters and gyromagnetic ratio of garnet bubble films (see text). h Po I Hu 41TM Sample x y ( 11) ( 11) 111) (Oe) (G) Q A 0.45 1.1 4.5 8.6 0.48 347 132 2.63 0 0.60 1.6 6.2 8.4 0.37 170 105 1.62 D following previous publications.7 The uniaxial anisotropy field Hu was determined with ferromagnetic resonance (FMR) at 9.2 GHz, while I and 41TM, were determined from measurements of the equilibrium stripe domain period Po and the bubble collapse field by the method of Fowlis and Copeland49 which is only valid for infinite Q. Adjustment for finite Q was made according to the procedure of Blake et al. 50 This correction for specimen D, for example, decreased 41TM by 3.5% while the correction for I was found to be negligible. The parameters Q ( = Hu/41TM,), .,1 ( = I /2Q), and A ( = 2/2ff2M;/ Hu) were calculated (as indicated). Since A is very sensitive to uncertainties in I and 41TMs' we also measured the Curie temperatures Tc and estimated the A from T c by the molecular field procedure of Malozemoff and Slonczewski.51 Additional estimates for A are possible by interpolation among the spin wave resonance values mea sured by Henry and Heinz52 in YIG doped with Ga. These values are also listed under the columns marked A (T c) and A(SWR), respectively. B. Effective wall mass density Because a domain wall responds to an applied pressure with finite acceleration, it theoretically possesses an inertial mass density. The classical mass density derived by Doring53 mD = (21TrL1 )-1 results from spin precession about the wall's demagnetizing field, and is only valid for an infinite flat wall. However, when bounded in a thin film, the wall's effective mass can increase as a result of surface demagnetiz ing fields because these allow the wall spins near the film surface to be more labile. This surface stray field effect leads to a nonuniform distribution m(z), with greater mass density near each surface. Finally, domain wall structures (Bloch lines and points) introduce significant modifications to the local mass. The resulting nonuniform loading of the wall has profound influence on resonance frequencies and results in spatially inhomogeneous wall oscillation amplitudes, as cal culated below in Sec. V Band elsewhere.4 Clearly, under these circumstances an empirical determination can only present an average description of the effective domain wall mass density. Nevertheless, because a proper theory can take into account the film surface effects, it is interesting to com pare with experimental findings. The wall mass meff is deter mined empirically using the resonance condition for rigid (unflexed) oscillation (2MsdHJdr)112 21TVn = , meff (4.2) where dHz/dr is the restoring force coefficient in a domain within the lattice, evaluated quasistatically, e.g., from in- 3378 J. Appl. Phys., Vol. 54, No.6, June 1983 A A(T,) A(SWR) r .:1 (10-' Tc (10-' (10' (10' ( 11) erg/em) (OK) erg/em) erg/cm) sOe) 0.086 1.34 405-435 1.7-2.2 1.4--1.8 1.866 0.114 1.05 345 0.80 0.87 1.725 verse slopes of the data of Fig. 9, where Vn is one of the fundamental radial oscillation mode frequencies as mea sured in Sec. V. Mode number n = 0 approaches the rigid condition when film thickness h is small, as in our samples. However, mode n = 2 is more appropriate for large h such as used in early experiments on thick single crystal platelets. 12 Sec. V B gives further discussion of this point. We consider only n = 0 in the remainder of this section. The numerical results valid for Hp = 0 giving meff and the reduced mass m* = meff/mD are collected in Table II where mD is calculated using sample parameters from Table I. Both theory and experiment yield a significant increase in wall mass with a consistency better than expected in view of the neglect of mass distribution effects. Quite satisfactorily, Va and dHJdr are found to have the same dependence on a, hence meff is independent of a, as expected from considera tion of the fact the wall thickness 1TL1 = 0.1 J.Lm is small com pared with wall separations on the order of 5 /i-m. Further discussion of various systematic approaches to determine meff are given in Ref. 63. Our experimental determination of meff neglects the well known frequency pulling effects of viscous damping and other possible line distortions such as coercivity. The follow ing discussion in Secs. IV C and IV D will justify this by showing that in a dense lattice the coercive force can be ne glected and any shifts in peak response due to damping are small, at least in the low-drive linear region. TABLE II. Effective domain wall mass density meff obtained experimental ly (e) by observing resonance frequency Vo and restoring force coefficient dH,/dq, or determined theoretically (t) by method similar to Hubert'S'· or (e/t) by combining observed Vo with theoretical dH,Idq calculated from theory ofKooy and Enz"' for the PSP, Hofelt4" for the PBL, and Callen and Josephs41 for the lB. Theory (t) includes effects of surface stray fields but neglects effects of wall flexure. Flat wall mass due to Doring" mv = (21TY.:1 r-I considers only the effect of wall demagnetizing field in an unbounded Bloch wall. Calculated mv = 0.53 for Sample A and 0.47 for 0 (in units of 10-10 g/cm') are combined with meW giving reduced mass mO. Domain pattern PSP PBL IB Sample D D D D D D D A A meW m*= Method (10-10 g/cm') mew/mv e 0.95 2.0 eft 1.04 2.2 0.89 1.9 e 0.77 1.6, eft 0.87 1.8, e 0.66 1.4 t 0.75 1.6 e 0.70 1.3 0.80 1.5 Argyle, Jantz, and Slonczewski 3378 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.174.21.5 On: Thu, 18 Dec 2014 14:09:0040 > E30 UI Ul Z o 0-20 Ul UI It: Z ~ 10 u o ~ STRI PE HEAD vv -JOOO',-" )OODOC OO~OO )OO~OOC noooc \'0 0.5 1.5 2 2.5 20 Hz FIELD AMPLITUDE (Oe) FIG. 10. Domain wall response of a PBL and a stripe head (held in place by surrounding bubble domains) to a low frequency field modulation, showing that the PBL wall oscillation is much less (if at all) influenced by coercivity. Dashed circle shows area of measurement defined by aperture in front of PMT detector. C. Coerclvlty The coercivity He [last term in Eq. (4.1)] describes the lower threshold for wall motion and has received particular attention for device applications. For a bubble domain, He may be inferred from the minimum drive required to trans late the bubble. With respect to domain wall oscillations, the presence of He would mean that, for a drive h. below some minimum h r;in~He' wall motion vanishes. Within the pres ent experimental sensitivity of detecting ~ O.S-nm motions at drives hz ;:;: 1.0 mOe (i.e., Figs 10 and 11) the minimum drive field for lattice oscillations lies imperceptibly close to zero. We presume this absence of a noticeable influence of coercivity on both PBL and PSP oscillations is closely relat ed to previous observations44.45 that exercising a dense do main pattern undervalues He in regard to motions in a dilute pattern44 or an isolated bubble.45 The effect arises because the mutual stabilization of neighboring walls dominates the pinning effects of coercivity. To demonstrate this effect in our observations of oscillatory response, we applied a low frequency (20 Hz) bias-field modulation to avoid resonance enhancement and measured the response of two different domain configurations, at the same position in the sample. The results in Fig. 10 show that amplitudes of PBL oscilla tion are strictly proportional to the drive amplitude (no ap parent He effect), whereas the response of the more isolated stripe head clearly extrapolates to zero amplitude at nonzero drive. Figure 10 thus demonstrates that He is nonzero in our samples, but that this is inconsequential for dense lattice PBL and PSP resonances. Vella-Coleiro et al.55 recently showed that stripe heads or "finger domains" are indeed suitable for measuring coerci vity values adequate to describe the threshold for bubble mo tion. To stabilize stripe heads within the field of view of the microscope their technique requires applying a static gradient 3379 J. Appl. Phys., Vol. 54, No. 6,June 1983 ~ Z LIJ :I LIJ o '" ..J 0- f/) o ..J ..J ; I LIJ 0- il: t; LIJ C) '" a: LIJ ~ 40 45 SAMPLE 0 PSP Hz·O Hp· ~ Oe 50 FREOUENCY (MHz) 55 FIG. II. Domain wall resonance oscillation of a parallel stripe pattern (PSP) recorded using a very low drive field hz = 12 mOe to avoid nonlinear line distortions and using smallest spectrum analyzer bandwidth (30 Hz) to suppress noise ofPMT. A small in-plane field Hp = 5 Oe serves to remove possible vertical Bloch lines from the area of measurement and to inhibit horizontal Bloch line formation. This domain wall resonance linewidth, de termined after correcting for noise background (dashed curve), gives Gilbert damping coefficient aDWR = 0.0088 in agreement with FMR (Table III). field dH. / dx that is unnecessary in our approach. Instead, we simply create a bubble array dilute enough to be unstable with respect to stripe out upon reduction of a uniform bias field Hz . D. Viscous damping The Gilbert phenomenological term for damping in the Landau-Lifshitz equation contains a strength coefficient a. As is well known, a may be obtained in uniform procession ferromagnetic resonance (FMR) from the peak-to-trough linewidth,1H of a derivative spectrum according to vJ ,1H aFMR =---, 2 Hr (4.3) where Hr > 41TMs is the field for resonance. Two values ob tained at 9.18 GHz, a1 and all' are listed in Table III, be cause different linewidths are observed when Hr is applied perpendicular and parallel to the film. Since domain wall motion also involves spin precession, wall oscillations are also subject to damping. However, it cannot be assumed a priori that the same damping parameter applies in the two forms of resonance, because the spin precession frequencies differ by more than two orders of Argyle, Jantz, and Slonczewski 3379 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.174.21.5 On: Thu, 18 Dec 2014 14:09:00TABLE III. g values from FMR and damping parameters a and'{ '( = aM,Iy) from linewidths in monodomain FMR and domain wall resonance (DWR) of parallel stnpe p~ttern. Column for Impulsed free oscillation (IFO) was measured by Shaw et al. 7 utilizing large pulsed bias drive ( -3 Oe) in contrast with 0.0 I Oe-peak sinusOIdal dnve for our DWR. Column,{ ~c is based on sample composition and the,{ 'obtained with FMR tabulated by Vella-Coleiro's et at. 12 Sample g value I aF\1R 'I aFMR A 2.12 0.012 0.0084 D 1.96 0.010 0.0075 magnitude. Earlier studies by Vella-Coleiro et al. 12 showed equivalence for the case oflarge a as encountered in materi als containing non-S-state "relaxer" ions. However, that work and later studies by deLeew et at. 56 indicated disagree ment for underdamped compositions as are of interest here (unless a large in-plane field is applied). We conclude from our measurements however, that the discrepancy is only due to the investigative methods previously employed. To evalu ate a correctly, the oscillator must be in the linear region of response versus drive. From our measurements we know that this requirement, with respect to domain wall reso nance, is by no means trivial, but demands very low drive and corresponding sensitive recording techniques. The spectrum in Fig. 11 represents the amplitudes of wall motion resonance averaged over the laser illumination area containing about 10 stripe domain periods. They were detected using the zero-order diffraction technique de scribed in Sec. II C. The sensitivity of the technique is dem onstrated by the good signal-to-noise ratio even when the peak response is only ..1x~2.5 mm. We estimate this peak response by multiplying together the drive field hrr = 10- mOe peak, the compliance dx/dHz = 27 nm/Oe, and the resonant enhancement factor Q = v /..1 v~ 10. The com pliance value was determined from initial susceptibility us ing a magneto-optic loop tracer and was also theoretically computed.26 The linewidth..1v = 4.5 MHz at half maximum ampli tude from domain wall resonance (DWR) in Fig. 11 leads to a determination of damping coefficient aOWR proportional to b in Eq. (4.1). From an assumed Lorentzian line shape _ by..1 21T..1V aOWR= 2M = v1~' Y elf (4.4) where y is the gyromagnetic ratio and where the effective field in the denominator requires special attention. It has been shown2 that Heff = HI + H2, with and Hz=..1dHJdq. (4.5) (4,6) (4.7) HI is the stiffness field acting on the wall moment due to the wall demagnetization, an applied field and the exchange, while H2 measures the restoring force acting on the wall of thickness..1 undergoing displacements q of arbitrary wave vector k. We find for our samples using Table I that the dominant terms, for k~O, are 3380 J. Appl. Phys .. Vol. 54, No.6, June 1983 aDWR 0.0088 and aIFO 0.024 0.034 1T HI =41TM+-H 2 p' ,{ ~.c A ;)WR 0.034 0.043 0.042 (4.8) (4.9) Thus, for sample D having an applied field H = 5 Oe as in • p Fig. 11, we find HI = 105 Oe and Hz = 2.7 Oe. Hence, the linewidth..1v = 4.5 MHz in Fig. 11 yields aOWR = 0.0088 which compares favorably with the values a1 = 0.010 and all = 0.0075 obtained in FMR. TheH p applied here parallel to the walls tends to displace vertical Bloch lines to the ends of the stripes outside the area of measurement and to sup press horizontal Bloch line formation. That aOWR falls between the two aFMR values may relate to the fact that in a thin film the wall contains nearly all moment orientations. The dependence of..1 v on Hp inherent in Eqs. (4.4) and (4.5) is largely verified experimentally as shown in Fig. 12. We attribute this agreement between DWR and FMR damping (Table III) to the fact that the large dynamic range available in our instrumentation allows recording the DWR spectra in the linear region. Earlier reported discrepancies56 are probably related to the fact that this linear region has such limited extent in drive field. We find this limit is only a few tens of milli-Oersted.25•57 An example is the fourfold higher value, designated in Table III as aIFO' having been obtained earlier (in these same samples) from impulsed free 9r-----------------------------~ N PSP I SAMPLE D ~ ~/ 8 n=O MODE I I- Po =8.2fLm / Q ,.i// ~ / W 7 / ~ 1/~ ..J w J/ u 6 z ~/ <{ z 0 / (/) / W 5 / a:: V~ ./3 .". Hp ..J .6Z1=aOWR 2.". Y 4.".M(I+'2 4.".M) ..J ~ 50 Hp (Oe) FIG. 12. Resonance linewidth obtained as in Fig. II vs in-plane field Hp applied parallel to the walls of a PSP. Solid line is from equation shown (theory described in Section IV 0) using Gilbert damping parameter deter mined as in Fig. 11 at Hp = 5 Oe. Argyle, Jantz, and Slonczewski 3380 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.174.21.5 On: Thu, 18 Dec 2014 14:09:00oscillation (IFO) of stripe domains and their time decay. 7 The amplitude of pulsed field that was used (~3 Oe) far exceeded the region for linear response according to our results in the frequency domain where, referring to Fig. 2 of Ref. 25, one sees that a drive field of even 0.1 Oe results in a strong asym metric broadening of the resonance line, thus producing fic titiously large effective a values. Impulse techniques are probably limited to the high drive region because the wide frequency band necessary for undistorted measurements in the time domain does not allow equally powerful SNR en hancement as is possible when measuring in the frequency domain. That impulse techniques have not detected the exis tence of a higher mode may then also be more readily under stood. V. COLLECTIVE RESONANCES AT LOW DRIVE A significant new finding of our study of domain wall dynamics under continuous wave excitation is the observa tion of mUltiple domain wall resonances. We present here a detailed quantitative study of the associated frequencies, de pending on a number of parameters such as Hz, the domain configuration, the material constants, wall structures, and finally the drive amplitude. The first three of these topics will be addressed here. In addition, it is borne out by experiment and corroborated by theory that a variety of collective oscil lation modes do exist that differ in characteristic features such as wall flexing along z, and spatial phase variation. To measure small amplitude collective resonances of PSP's and PBL's, we chose the optical Fourier transform mode of observation (see Sec. II C) which records the aver age dynamic behavior of the domains being illuminated (about 30 bubbles or an equivalent area of parallel stripe domains). To ensure linear response, it is necessary to main tain hz at very low values on the order of 10 mOe to obtain data adequate for comparison with small signal theory. As described in Sec. IV nonlinear frequency shifts and line shape distortions are observed at astonishingly low drive25•26 and become very drastic at elevated power, where nonlinear phenomena such as velocity saturation 14.26 tend to dominate the oscillatory behavior. This is true, in particular, for sam ple 0, but less so for sample A. Figure 13 shows typical traces of SA output versus the driving TG frequency of hz, both for a PBL and a PSP, with parameters given in the caption. For the purpose of demon stration, the drive amplitude was chosen large enough to produce sizeable amplitudes of domain oscillation, already markedly affected by nonlinear distortion. In order of in creasing frequency, a broad shoulder and two resonance peaks, well separated by an interval of very small response, are discernible. The origin of the broad low frequency shoulder is com plex and will be discussed in a later pUblication. Here we address the two resonances, labelled Vo and V2 for reasons to become clear below, and we show that there are collective oscillations that differ with respect to wall bending, along the z direction, but are both characterized by having a spa tially uniform phase. By spatially uniform we mean that, for the case of a PBL, all lattice bubbles exhibit the same ins tan- 3381 J. Appl. Phys., Vol. 54, No.6, June 1983 o SAMPLE D Hz=Hp=O FIG. 13. Swept frequency response of domain waH oscillations (not correct ed for rollolfin drive field and detector sensitivity shown in Fig. 4). Spectra show two strong resonances and a broad low frequency shoulder in both PSP and PBL and reveal line-shape distortions occurring at intermediate drive "",0.1 Qe. The lower curves taken with zero drive measure PMT noise. Noise is larger here than in Fig. 11 because oflarger stray background light occurring in the reflection mode of detection and larger (I kHz) spectrum analyzer IF bandwidths. Both noise and signal roll-olfwith frequency con tain distortions due to diminishing PMT sensitivity. (See Sec. II DJ. taneous displacement r -ro, whereas for the case of a PSP the phase of x -Xo is equal for every other stripe wall. Hence, these oscillation modes are very similar in the sense that all bubbles and all stripe domains with the same orienta tion of Ms expand and shrink in phase. This behavior is equi valent to an excitation with wave vector k = 0, or infinite wavelength. Stressing the analogy to crystal lattices, we may identify this mode as being of the "pseudo-optic phonon" type. The assignment of the two resonances to collective k = 0 or uniform mode oscillations is supported by the fact that the drive field hz being spatially homogeneous couples most effectively to constant phase oscillations. This behavior is well known for FMR, where in general the uniform preces sion (all spins in the sample precess at the same phase) is by far the strongest resonance, whereas finite wavelength exci tations, i.e., magnetostatic modes and spin waves, as a rule, can be generated linearly only by virtue of some spatial inho mogeneity provided by the microwave field or the sample (e.g., spin pinning at the sample27 surface). Indeed, k #0 do main wall excitations have been observed in sample 0 in the vicinity of strong sample inhomogeneities, i.e., parallel cracks. These k #0 excitations occur near in frequency but at smaller amplitude compared to the k = 0 mode discussed here. Further corroboration that Vo and v2 are uniform mode excitations is obtained by comparing their frequencies with theoretical predictions for k = 0 oscillation modes. A. Fundamental resonance We discuss first the lower frequency Vo' (Note that the subscript does not refer to the k = 0 wave property.) Figure 14 shows the dependence of Vo on Hz for a PBL. A calcula tion, with the effective mass parameter fitted to the data, is shown as a solid line. The theory is essentially that of Ho feIt48 and Tomas58 except that the lattice parameter a is tak en as a constant independent of Hz, as observed experimen tally (see Sec. III D). Argyle, Jantz, and Slonczewski 3381 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.174.21.5 On: Thu, 18 Dec 2014 14:09:0080~ 70· N 60- ~ ~-~50" ~ ~ 40 w => &3 30 0:: LL 101 0' SAMPLE D Hp=O -40 -20 o 20 BIAS FIELD Hz (Oe) 40 60 FIG. 14. Fundamental mode resonance vs bias field Hz at Hp = O. The the ory (solid curve) reproduces the decrease of va with Hz observed (circles) for the PBL. For a PSP the resonance frequency (squares) is almost constant, as emphasized by the dashed line (not calculated). In the Hofelt -Tomas theory, the harmonic oscillator resonant frequency is Vo = (C /metr)I/2/21T, (5.1) where C, the restoring force coefficient [see also Eq. (4.1 )], is computed numerically from an exact two-dimensional Fourier analysis of the three-dimensional stray field energy E per unit wall area for the periodic lattice of circular cylin ders. In our own notation C is defined by C=d2E/d?, (5.2) where the equilibrium radius ro is given by ° = dE/dr. (5.3) In our experiments, it appears that conditions at the edges of the lattice prevent readjustment of the bubble-lattice param eter a with changes in Hz. We, therefore, use measured val ues of a instead of invoking Hofelt's equation48 correspond ing to dE Ida = ° in our notation. We recapitulate, using our own notation, the Hofelt Tomas relations needed for the main resonance mode of a regular triangular lattice of bubbles having lattice parameter a (see Fig. 15). The effective spring constant for oscillation of the bubble radius is obtained from the total energy of the domain structure calculated by the usual methods of domain statics.2•59 The energy U per unit volume of the magnetic film has three terms: (0) 000 QQO GGO (b) Kyf \ L·_~ o K, (5.4) FIG. 15. Pictures of a bubble lattice and its reciprocal for defining variable parameters a, r, and k. 3382 J. AppL Phys., Vol. 54, No.6, June 1983 The Zeeman energy due to the bias field Hz is Uz = _4_1TM_v'3..:..sH--=-z (; r The wall energy is (41TMsflr Uw =--~ v'3a2 (5.5) (5.6) The demagnetizing energy (1I2)ffSH2dxdydz is ob tained conventionally by Fourier series expansion over the two dimensions parallel to the film plane. The well-known resultS9 is written conventionally in the form + 64~ I K -3(1 -ehK) [rJ1(KrW} , (5.7) 3ha K#D where In(z) (n = 0,1,2'00') is the Bessel function. Here the indicated summation is carried over all of the distinct vec tors K of the reciprocal lattice defined by K = n1al + n2a2, (5.8) where the integers n 1 and n2 are not simultaneously zero. By substituting the above relations, one finds that the condition of static equilibrium d U / dr = 0 reduces to a func tion Hz(r) given by .i + [1 _ 41T(.!....)2] 2r v'3 a -81T~ I K -2(1 -ehK)Jo(Kr)JdKr). v'3a~h K#O (5.9) The restoring-force coefficient C on unit of wall area is v'3a2 d 2U C=--. 41Tr d? One finds the result 2MHz 81TM2 { ( r)2 C = --r-+ --r- -1 + 4v'31T -;; + 81Tr I K -2(1 -e -hK) v'3ha2 K#D (5.10) X [KrJ~(Kr) + Jo(Kr)JI(Kr) -Krn(Kr)]}. (5.11) In making computations, it is convenient to select a val uefor r, then obtain Hz from Eqs. (5.9) and C from Eq. (5.11). The convergence of the sums in Eqs. (5.9) and (5.11) is made somewhat faster by introducing the factor (1 + eblK -K,))-I into the summand. The value of C is thereby unchanged in the limit K 1-+ 00. The truncated sum, including all terms satisfying K < K 2 is stable over appreciable ranges of the computational constants b, KI' andK2 as long as the inequal ities bKI~l and O<.hKI<.hK2 are well satisfied. Choosing hK2 in the neighborhood of 100 thus produces sufficient ac curacy for plotting the theoretical curves r versus Hz in Fig. 9 and Vo versus Hz in Fig. 14 (PBL). The agreement with the PBL data of Fig. 14 is very satisfactory with respect to the shape of the curve. As men tioned, one adjustable parameter is used to induce a fit at Hz Argyle. Jantz, and Slonczewski 3382 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.174.21.5 On: Thu, 18 Dec 2014 14:09:00= 0; it is the effective mass density obtained from the har monic oscillator relation (5.1). Good agreement occurs for meff = 0.85 X 10-10 g/cm2, corresponding to a reduced mass m * = 1. 8, in reasonable agreement with m * = 1. 65 ob tained in Sec. IV B by a purely experimental procedure. We conclude that the assignment of the Vo resonance to the spa tially uniform "breathing" mode is well justified. The down ward slope dvofdHz may be understood qualitatively as the combined influence of the restoring force that stabilizes the radius of an isolated bubble59 and the restoring force due to the repulsive dipole interaction between the lattice bubbles. The former increases slightly with Hz almost up to the col lapse field, whereas the latter decreases because the bubbles shrink and hence their dipole moment decreases. Data have been taken for the whole range of positive and negative Hz in which the PBL is stable. The stability range for positive Hz (opposite to the magnetization of the bubbles) is limited by the spontaneous collapse of a fraction (-1/3) of the bubbles. Theoretically, it has been predicted60 that a spatially regular collapse pattern eliminating every third bubble should prevail, whereafter the PBL should rear range to form a PBL with larger a. One instance of experi mental confirmation has also been reported.61 We observe a somewhat irregular collapse pattern, possibly due to sample inhomogeneities. At negative Hz, the expanding bubbles mutually distort themselves, forming a honeycomb lattice.46 Even then, we do not observe any variation of a, unless an irreversible deformation of the PBL occurs by the merging of adjacent domains. We note that the agreement between our experimental and theoretical results in Fig. 14 persists at negative Hz> in contrast to previous results. 19 We attribute this, at least in part, to our calculation based on constant a, as observed experimentally, whereas in Ref. 19 the data were compared with the theory as given in Ref. 48 that predicts a rapid increase of a upon honeycomb deformation of the lat tice of bubbles. While in Fig. 13 the Vo resonance frequency is almost the same for the PBL and the PSP domain patterns, this is somewhat accidental, because, as shown in Sec. V D the fre quencies observed in a PBL depend distinctly on a. In turn, for a PSP, the resonance frequencies depend on the stripe lattice period Po. We did not investigate this dependence quantitatively, because the PSP is less stable compared with the PBL and Po cannot be adjusted arbitrarily for given field settings. The strong decrease of Vo with Hz observed for a PBL (Fig. 14) does not occur for a PSP (Fig. 16) nor for an isolated bubble (Fig. 17). Indeed, in the case of the PSP, symmetry with respect to sign reversal of Hz demands that the slope dvoldH z vanish at Hz = O. Roughly the same weak depen dence on Hz is obtained by measuring the width of the up and down stripe domains and then calculating the spring constant using the minimum energy formulation by Kooy and Enz.62 An effective mass density may then be calculated by fitting the two sets of data. One obtains m* = 2.2 in units of the Doring mass, and this result is somewhat higher than the purely experimental value evaluated in Sec. IV and enu merated in Table II. 3383 J. Appl. Phys., Vol. 54, No.6, June 1983 4O x 0 ~ 30 :I: >-u Z 20 lJJ ~ 0 lJJ a: LL. 10 r e:: ---~ o o x SAM PLE 0 Hp'O measured 0 9 x r ~.1. ·~TL, ... ........ ~TL, calculated from L" L2 10 20 30 BIAS FIELD (De) 0 50 FIG. 16. Resonance frequency of PSP vs bias field H,. The calculated data are obtained from measured stripe widths, L, and L,. B. Higher order flexural modes The interpretation of the Vo mode given so far, as well as the respective theories, has not considered possible varia tions of the wall amplitude along the film thickness dimen sion, z. In other words, it has been assumed that the wall motion retains the straight cylindrical geometry. This as sumption is in fact an over-simplification except for frequen cies well below resonance. In the first place, the fundamental mode does exhibit a significant wall flexure due to nonuni form mass loading and demagnetizing fields. Secondly, in addition to the fundamental mode, now more appropriately characterized by the property that the oscillation amplitude has the same sign throughout the film, there is a series of calculated and partially observed higher order modes of flexed membrane oscillations. These are distinguished by and ordered according to-an increasing number of nodal lines in the case of stripe walls, or nodal circles in the bubble walls. N I ~ 301- >-251- ~ o W 0:: LL. 2Qr- I 56 SAMPLE A ISOLATED BUBBLE Hp=O 0 0 0 0 I I 58 60 BIAS FIELD (Oe) Dif 0 0 0 0 I I I 62 64 66 FIG. 17. Resonance frequency of isolated bubble (IB) vs Hz over range where bubble restoring force dH,ldr measured in Fig. 5 is approximately constant. Combining the average frequency (24.7 MHz) observed here with dHJdr = 8.0X 104 Oe/cm from data of Fig. 5 and M, = 10.5 G (Table I) leads to bubble wall's effective mass mell' = O.70X 10-10 g/cm2(Table II). Argyle, Jantz, and Slonczewski 3383 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.174.21.5 On: Thu, 18 Dec 2014 14:09:00r f I .... -0.,. ____ -\ y ____ --W "" STRAY FIELD FIG. 18. Schematic illustration of periodic lattice of stripe domains with twisted wall structure induced by surface stray field. Figure 13 clearly exhibits a higher resonance frequency Vz both for the PBL and the PSP. We interpret it to be the particular higher order flexed membrane oscillation that ex hibits two nodal lines (PSP) or circles (PBL) between the film surfaces. A theory, fully described elsewhere, 54 derives the effects of wall flexure and stray fields on the resonances for the PSP. We use known equations of wall motion which result from carrying the Landau-Lifshitz equations for the magnetization vector M (y, z, t) to the limit of an infinitely thin 1800 wall? Referring to the stripe lattice illustrated in Fig. 18 consider the particular 1800 wall separating domains satisfying Mz = M and Mz = -M, and resting at the plane y = O. We let x(z, t ) be its instantaneous displacement from the plane y = 0 toward the domain satisfying Mz = -M. The azimuthal angle I/!(z, t ) of the moment M evaluated at the middle of the wall is also a dynamical degree of freedom. After some unessential terms are neglected, the equations of motion are2 81/! _ H YUo a2x 8t -y z + 2M az2 (5.12) and Ll 0-I ax = 21TyM sin 21/! _ 1TY H cos I/! _ 2yA a2 1/! . at 2 y Maz2 (5.13) Here, -y is the gyromagnetic ratio, A is the exchange stiff ness [exchange-energy density = AM -2(VM)2], 0'0 = 4(A n=O _~2 /' " l' . / " .-.. w > 1.0 1= Ku )112 is wall energy density with uniaxial anisotropy K ·sin2 8, and 1TLlo = 1T(A1Ku )112 is the effective wall thicknes;' The terms on the right hand side of Eq. (5.12) take into ac count wall pressures due to the magnetic field and wall-sur face tension, respectively. The terms on the right hand side ofEq. (5.13) take into account wall-moment torques due to local demagnetization, long-range-demagnetizing field com ponent Hy' and exchange stiffness, respectively. Damping effects are neglected . In the absence of external fields, symmetry dictates stat ic wall positions at equal distances w = PoI2. The alternat ing magnetic charges distributed on the film surfaces pro duce the twisted static equilibrium wall structure I/! = I/!o(z) illustrated schematically in Fig. 18. To find the normal modes of oscillation, Eqs. (5.12) and (5.13) are linearized for small departures from the static solu tions x = 0, I/! = I/!o(z). For Hy (z) it is sufficient to take the stray field of the static domain configuration evaluated by spatial Fourier analysis.54 The component Hz(z) is written effectively as a linear integral functional Hz(z) = Su(z, Z')x(ZI)dz' of x(z), ultimately Fourier-transformed for compu tational convenience. 54 Numerical computation produces eigenmodes Xn (z) and eigenfrequencies Vn• Calculated Xn (z) for samples A and Din the case Hz = 0, shown in Fig. 19, display the amplitude distribution through the film thickness of the normal flexed membrane modes n = 0, 1,2,3, and 4. The n = 0 mode is assigned to the Va resonance discussed above and Fig, 19 shows that the amplitUde depends substantially on z, de creasing towards the center of the film. The predicted shape for the n = 1 mode has odd symmetry and thus should neither be excited by our uniform field hz(t) nor detected magneto-optically in the limit of weak optical absorption. The n = 2 mode may, however, couple to a homogeneous hz drive field and produce a net magneto-optic signal, because the amplitude distribution does not integrate to zero over the film thickness. To support the assignment that the observed V2 resonance is the n = 2 mode, experimental and theoretical PSP mode frequencies and relative amplitudes are collected in Table IV A. The agreement for mode frequencies is quite satisfactory considering that the theory is valid only for high Q, whereas for our samples, especially sample D, the param- -1 -------r- T~~------'------ n=O '.. /'. \ . /'. '\ Y / '. \ ::s w Q: 0.5 '-' f- Z w ::::E w ci u ::s a.. In Ci -0.5 ...J ...J -< ~ -1.0 0 3384 :, / '. \ .. 1 ". \2 3.: j', .... \ 'I .. ,: / , ~ \. r-_.---:-L .. ---':". -.\.: j / ,-:~~ \ V ~; \ /' SAMPLE A'.. .:, " ~ .' , ~ r~, ~ ___ L ____ l __ ----1 ___ 1. ___ ~_ l __ ~ . 'j 3: ,'. \. // " ,"'~'" ~\ .::,: l I • ----'>0-- : :1 .... ',., / ...... ', ./\ j ? SAMPLE D ". '.>~ "- .' , ~_------L--1-- I I ---------l-_ 0.2 0.4 0.6 O.B 1.0 0 0.2 0.4 0.6 O.B 1.0 Z .;. FILM THICKNESS Z .;. FILM THICKNESS J. Appl. Phys., Vol. 54, No.6, June 1983 FIG. 19. Calculated amplitude distribu tion through the film thickness of the fundamental and three higher order flexed membrane resonance oscillations. The modes n = 0 and n = 2 are observed experimentally, as in Fig. 13 and Ref. 25. Argyle, Jantz, and Slonczewski 3384 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.174.21.5 On: Thu, 18 Dec 2014 14:09:00TABLE IV. Experimental and theoretical results for the resonance frequencies and relative amplitudes of the two lowest order symmetric flexed membrane resonances. (A) PSP domains. (B) PBL domains (experimental values only). Sample D' has thickness 30% greater than Sample D. Sample A D Po(llm) lattice period 8.5 7.0 expt. 48 45 vo(MHz) theor. 45 38 (A)PSP expt. 112 76 theor. 125 84 expt. 0.01 0.27 theor. 0.01 0.08 (B)PBL Sample A D D' ao(llm) lattice spacing 8.7 10.5 11.6 vo(MHz) expt. 52 43 42 eter is not very different from unity (see Table I). Only fair agreement with experiment is found in relative mode intensi ties X (v2)IX (vo) obtained by integrating numerical results xn(z) in Fig. 19 through the film thickness, z. For the PBL only experimental results are reported (Ta ble IV B), as the theory has not been extended to the case of curved walls. It is clear, however, from the theory for straight PSP walls that the amplitude ratio X (v2)IX (vo) grows with increasing thickness. While such a trend is indi cated in the comparison (Table IV A) between samples A and 0 having thicknesses 4.5 and 6.2 /-lm, respectively, further verification is found in PBL data of Table IV B in cluding an additional sample (0') grown from the same melt as 0 but for longer time to give a thickness (8.1 /-lm) 30% larger. C. Effective-mass correction Having found that the n = 0 mode is fairly flat suggests that it should be representable by the rigid wall model if the effective mass were properly corrected for the effect of sur face stray fields. We have computed this correction using the same principles as Hubert,24 but employing our simpler equation of motion [Eq. (5.13)]. One considers the abstract problem of a wall in a constant state of motion with velocity aqlat = V, independent of z and t. The variable tfr{z) is now independent of t. The second-order differential Eq. (5.13) is then integrat ed numerically with respect to z. We find it effective to divide the intervaIO.;;;z.;;;h into 100 discrete steps. We then solve Eq. (5.13) by the Newton-Raphson method for convergence to an exact zero, given an approximate zero. We begin essen tially with the solution tfr = 0 valid for the case A = 00, V = O. Then we diminish A in steps, solving the equation for each value of A, iterating from the previous solution, until the desired value of A is attained and the corresponding solu tion tfr(O, z) imbedded. Vis varied similarly to imbed the solu tion set tfr( V, z) for a range of V including V = O. The general effective-mass expression meff = 2M{d [f dztfr( V, Z)]} V = 0, yh dV (5.14) 3385 J. Appl. Phys., Vol. 54. No.6, June 1983 v2(MHz) X (v2)/X(vo) expt. expt. 118 0.07 70 0.13 65 0.63 based on Hamilton's equation of motion, then gives us the desired result. The numerical results are given on lines 3, 7, and 9 in Table II labeled t. D. Dependence on lattice parameter The method described in Sec. III A allows us to gener ate PBL's of specified lattice parameter reproducibly within a range 8.4 :S a :S 20/-lm. We can, therefore, study systemati cally t4e influence of this parameter on the resonance fre quencies and thereby test some theoretical formulations for the lattice breathing mode response. Figures 20 and 21 show Vo data versus a-I for a wide range of bubble lattice param eters. A high Hz bias field was chosen in Fig. 20 to prevent bubble stripeout in dilute lattices. This data reveals instructi vely how the contribution of the lattice neighbors to the wall stabilizing potential and hence to the resonance frequency gradually decreases with increasing lattice parameter. It is also seen that Vo for the lattice approaches monotonically and levels off at-the resonance for an isolated bubble as indicated by the data point at a-I = O. Figure 21 gives Vo data observed at Hz = 0 together with our theoretical dependence. The slope of the theoretical line was obtained from first principles derived in Sec. V B, 4o,------------------------------------. -;:.30 :I: ~ ~ ~ -------•.. -. 520 lJJ [ Hp=O Hz = 560e SAMPLE 0 PBL 100~------------~5~-------------~10~----~ (BUBBLE LATTICE PERIODf'(},--'KIOO) FIG. 20. Observed dependence of the Vo resonance on the PBL bubble den sity. The open circle refers to an isolated bubble. Argyle, Jantz. and Slonczewski 3385 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.174.21.5 On: Thu, 18 Dec 2014 14:09:00~ 50 :r: :::E >-48 () Z w => a ~ 46 lL. w () Z 44 « z o (/) w 0::: 42 • SAMPLE 0 Hz=O P8l m*=1.89 • • 0.09 0.10 0.11 0.12 • RECIPROCAL BUBBLE SPACING Cum-1) FIG. 21. Comparison of experimental and theoretical dependence of Vo on the PBL density for the case of zero bias field. Domain wall mass is the only fitting parameter. i.e., without recourse to adjustable parameters. The scale of the frequencies was obtained with a fitted value m* = 1.89 for reduced effective mass. This value compares favorably with the value m* = 1.80 obtained in Sec. V B from a fit to Vo versus Hz. The average of these two determinations (m*) = 1.85 is reported in Table II where comparison can be made with m* = 1.65 obtained by a purely experimental technique reported in Sec. V A. ACKNOWLEDGMENTS We gratefully acknowledge R. E. Mundie and R. La Maire for helpful technical assistance in setting up our appa ratus, and R. W. Shaw of Monsanto Corporation for loaning us samples and describing earlier measurements. We are also indebted to J. H. Spreen who collaborated in some of this research but was unable to participate in writing this paper. 'A. H. Eschenfelder, Magnetic Bubble Technology (Springer, New York, 1980). 2 A. P. Malozemoff and J. C. Slonczewski, Magnetic Domain Walls in Bub ble Materials (Academic, New York, 1979). 'T. H. O'Dell, Ferromagnetodynamics (Wiley, New York, 1981). 'J. e. Slonczewski, in Physics of Dejects, Les Houches, Session XXXV, (1980)INorth-Holland, New York, 1981), p. 634. 'B. A. Calhoun, E. A. Giess, and L. Rosier, App!. Phys. Lett IS, 287 (1971). 0B. E. Argyle and A. P. Malozemoff, AlP Conf. Proc. 10, 344 (1972). 7R. W. Shaw, J. W. Moody, and R. M. Sandfort, J. Appl. Phys. 45, 2672 (1974). "G. P. Vella-Coleiro, Appl. Phys. Lett. 29,144 (1976). "B. R. Brown, G. R. Henry, R. W. Koepcke, and e. E. Wieman, IEEE Trans. Magn. MAG·H, 1391 (1975). '°F. H. deLeeuw, Physica B 86-88, 1320 (1976). liB. E. McNeal and F. B. Humphrey, 1. Appl. Phys. 50, 1010 (1979); 50, 1020 (1979) "G. P. Vella-Coleiro, P. H. Smith, and 1. G. Van Uilert, J. App!. Phys. 43, 2428 (1972); App!. Phys. Lett. 21, 36 (1972). "D. C. Fowlis and J. A. CopeJand, AlP Conf. Proc. 10,393 (1972). 3386 J. Appl. Phys., Vol. 54, No.6, June 1983 14B, E. Argyle and A. Halperin. IEEE Trans. Magn. MAG-9, 28311973). "G. R. Woolhouse and P. Chaudhari, Philos. Mag. 31,161 (1975). 161. Tomas, Phys. Status. Solidi A 30,587 (1975). 17B. A. Calhoun (private communication). 18F. H. DeLeeuw and J. M. Robertson. J. App\. Phys. 46,3182 (1975). 19H. Dotsch and P. E. Wigen, Solid State Commun. 30, 611 \1979). 20p. E. Wigen, H. Dotsch, and J. Morkowski, Proceedings of the Interna tional Conference on Physics of Magnetic Materials (1980), edited by Z. Kaczkowski and H. K. Lachowicz, p. 196. 2tA. A. Parker, M. A. Parker, and P. E. Wigen, J. App\. Phys. 52, 2347 (1981). 22H. Dotsch, H. 1. Schmitt, andJ. Muller, App\. Phys. Lett. 23, 639 (1973) . HE. Schlomann, AlP Conf. Proc. 18, 183 (1974). 24A. Hubert, J. Appl. Phys. 46, 2276 (1975). "w. Jantz, B. E. Argyle, and J. C. Slonczewski, IEEE Trans Magn. MAG- 16,657 (1980). 'oJ. e. Slonczewski, B. E. Argyle, and J. H. Spreen, IEEE Trans. Magn. MAG-l7, 2760 (1981). 27J. H. Spreen and B. E. Argyle, App!. Phys. lett. 38, 930 (198 1); B. E. Ar gyle, J. e. Slonczewski, W. Jantz, 1. H. Spreen, and M. H. Kryder, IEEE Trans. Magn. MAG-IS, 1325 (1982). '"w. Jantz, J. e. Slonczewski, and B. E. Argyle, J. Magn. Magn. Mater. 23, 8 (1981). 29M. H. Kryder and B. E. Argyle, J. Appl. Phys. 53,1664 (1982). JOJ. H. Spreen and B. E. Argyle, J. Appl. Phys. 53, 4315 (1982). "w. Jantz, 1. Appl. Phys. 53, 2543 (1982). "J. F. Dillon and J. P. Remeika, J. App!. Phys. 34, 637 (1963). HR. V. Telesnin, A. G. Shishkov, E. N. I1icheva, N. G. Kanavina, and N. A. Ekonomov, Phys. Status Solidi A 12, 303 (\972). "G. R. Woolhouse and P. Chaudhari, AlP Conf. Proc. 18, 247 (1973). "T. R. Johansen, D. I. Norman, and E. J. Torok, J. App!. Phys.42, 1715 (1971). ·'"B. E. Argyle and P. Chaudhari, IBM Tech. DiscI. Bull. 18,603 (\975). 17J. H. Spreen, Technical Report 39, Microwave and Quantum Optics Group, Massachusetts Institute of Technology, Cambridge, Massachu setts, June 1977. '"B. Kuhlow, Optik 53,115 (1979); 53.149 (1979), and references cited ther- ein. J"B. R. Brown, IBM J. Res. Dev. 16, 19 (1972). 4°B. E. Argyle and A. Halperin, IEEE Trans. Magn. MAG-9, 238 (1973). 4'H. Callen and R. M. Josephs, J. App!. Phys. 42,1977 (1971). 42c. B. Rubenstein. 1. G. Van Vitert, and W. H. Grodkiewicz, J. App!. Phys. 35,3069 (1964). 4'A. Hubert, A. P. Malozemoff, and J. C. Deluca, J. Appl. Phys. 45, 3562 (1974). 44B. E. Argyle, J. C. Slonczewski, and O. Voegeli, IBM J. Res. Dev. 20, \09 (1976). 45G. R. Woolhouse and P. Chaudhari, Philos. Mag. 31, 161 (1975) 4hF. A. De Jange and W. F. Druyvesteyn, AlP Conf. Proe. 5, 130 (1971); Festkiirperprobleme XII, 531 (1972). 47J. A. Cape and G. W. Lehman, J. Appl. Phys. 42,5732 (1971). 4KM. H. H. Hofelt, IEEE Trans. Magn. MAG-9, 621 (1973). 44D. e. Fowlis and J. A. Copeland, AlP Conf. Proe. 5, 240 (1971). "'T. G. W. Blake, C. C. Shir, Y. O. Tu, and E. Della·Torre, IEEE Trans. Magn. MAG·18, 970 (1982). 51A. P. Malozemoff, 1. e. Slonezewski, and E. H. Giess, AppJ. Phys. Lett. 24,396 (1974). "R. D. Henry and D. M. Heinz, AlP Conf. Proc. 18, 194 (1973). "w. Doring, Z. Naturforsch. 3A, 373 (1948). '4J. C. Slonczewski, J. Magn. Magn. Mater. 23, 305 (1981). "G. p. VelJa-Coleiro, W. P. Venard, and R. Rolfe, IEEE Trans. Magn. MAG-16, 625 (1980). "'F. H. de leeuw, R. van den Doe!, and U. Enz, Rep. Prog. Phys. 43, 689 (1980). '7See data shown in Ref. 26, Fig. 10. '"I. Tomas, Phys. Stat. Status Solidi A 30,329 (1975). '"A. A. Thiele, J. Appl. Phys. 41,1139 (1970). h"V. G Bar'yakhtar, V. V. Gann, and Y. l. Gorobets, Sov. Phys. Solid State 18,1158 (1976) [Fiz. Tverd. Tela-(Leningrad) 18, 1990 (1976).] O'V. G. Bar'yakhtar, Y. 1. Gorobets, O. V. Il'chishin, and M. V. Petrov, Sov. Phys. Solid State 19, 1658 (1977) [Fiz. Tverd. Tela (Leningrad) 19, 2829 0971)]. ole. Kooy and U. Enz, Philips Res. Rep. 15,711960). O'W. Jantz, B. E. Argyle, and J. e. Slonczewski, J. Magn. Magn. Mater. 28, 285 (1982). Argyle, Jantz, and Slonczewski 3386 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.174.21.5 On: Thu, 18 Dec 2014 14:09:00

1.5140673.pdf

J. Appl. Phys. 127, 073904 (2020); https://doi.org/10.1063/1.5140673 127, 073904 © 2020 Author(s).Generation of magnetic skyrmions by focused vortex laser pulses Cite as: J. Appl. Phys. 127, 073904 (2020); https://doi.org/10.1063/1.5140673 Submitted: 29 November 2019 . Accepted: 30 January 2020 . Published Online: 19 February 2020 O. P. Polyakov , I. A. Gonoskov , V. S. Stepanyuk , and E. K. U. Gross ARTICLES YOU MAY BE INTERESTED IN Hyperbolic metamaterials: From dispersion manipulation to applications Journal of Applied Physics 127, 071101 (2020); https://doi.org/10.1063/1.5128679 Modeling of magnetic thermal noise in stable magnetic sensors Journal of Applied Physics 127, 073902 (2020); https://doi.org/10.1063/1.5134883 Transition regime in the ultrafast laser heating of solids Journal of Applied Physics 127, 073101 (2020); https://doi.org/10.1063/1.5143717Generation of magnetic skyrmions by focused vortex laser pulses Cite as: J. Appl. Phys. 127, 073904 (2020); doi: 10.1063/1.5140673 View Online Export Citation CrossMar k Submitted: 29 November 2019 · Accepted: 30 January 2020 · Published Online: 19 February 2020 O. P. Polyakov,1,2,3 ,a) I. A. Gonoskov,1,4,b)V. S. Stepanyuk,1,c)and E. K. U. Gross1,5,d) AFFILIATIONS 1Max-Planck-Institut für Mikrostrukturphysik, Weinberg 2, 06120 Halle, Germany 2Faculty of Physics, M. V. Lomonosov Moscow State University, Leninskie Gory, Moscow 119991, Russia 3V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, 65 Profsoyuznaya Street, Moscow 117997, Russia 4Institute of Physical Chemistry, Friedrich Schiller University Jena, Helmholtzweg 4, 07743 Jena, Germany 5The Fritz Haber Research Centre for Molecular Dynamics, Institute of Chemistry, The Hebrew University of Jerusalem, Safra Campus, Jerusalem 91904, Israel a)Electronic mail: o_polyakov@physics.msu.ru b)Electronic mail: ivan.gonoskov@uni-jena.de c)Author to whom correspondence should be addressed: stepanyu@mpi-halle.mpg.de d)Electronic mail: hardy@mpi-halle.mpg.de ABSTRACT We propose a method to generate magnetic skyrmions by intense laser pulses optimally focused on a magnetically ordered 2D-layer. In par- ticular, we consider few-cycle intense pulses with the magnetic vortex structure near the focus region on the layer. The spin dynamics ismodeled using the Landau –Lifshitz –Gilbert equation and includes the Dzyaloshinskii –Moriya interaction. We demonstrate that skyrmions can be observed within a few picoseconds after the end of the laser pulse. We analyze the physical picture of this process and work out which laser pulse and 2D-layer parameters are required for the generation. Published under license by AIP Publishing. https://doi.org/10.1063/1.5140673 I. INTRODUCTION Magnetic skyrmions and their fascinating properties have been the subject of intense research in recent years. They are local- ized magnetic topological structures that can be observed invarious magnetic materials. 1–6Due to their integer topological charge, they show strong stability under various internal and exter-nal perturbations. 7–9This makes them promising candidates for novel memory devices and skyrmion-based computing.10–16Also, skyrmions can be used for reservoir17,18and neuromorphic19,20 computing. Hence, the creation, erasure, and accurate control of a single skyrmion and a thorough understanding of the underlying microscopic processes are of extraordinary importance. Since the typical size of a skyrmion ranges from a few tens up to a few hundred nm, a possible tool to control a single skyrmion istightly focused femtosecond laser pulses with optical wavelengths.The enormous growth of the laser technology in the past decade has shown significant progress in two directions: (i) the generation ofultra-short laser pulses consisting of a few optical cycles with the controllable carrier-envelope-phase (CEP) 21and (ii) the very tight focusing of laser radiation close to the fundamental limit22based on achievements in spatial light modulators (SLMs) and/or ultraprecise adaptive or conventional optics.23,24All these features potentially are beneficial to the creation/erasure and control of a single skyrmion with ultrahigh resolution, both temporal and spatial.25–30 In this paper, we theoretically demonstrate the possibility of generating a single skyrmion by an intense few-cycle laser pulse focused to a magnetic 2D-layer. In particular, we use laser pulses with the vortex magnetic field structure near the focus region. The field distribution we use corresponds to an exact solution ofMaxwell ’s equations and is called e-dipole pulse, 22which can be generated experimentally in different ways. Such pulses, having a very special temporal and spatial shape, can locally excite themagnetic system and, as we demonstrate, lead later to the forma- tion of a single skyrmion.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 073904 (2020); doi: 10.1063/1.5140673 127, 073904-1 Published under license by AIP Publishing.II. MATERIALS AND METHODS The required laser pulses, i.e., the electromagnetic field configu- rations must have specific properties in order to excite the magneticmaterial locally and in a proper way. Before we analyze the behavior of the magnetic material under such fields, we discuss the general properties that the possible class of focused laser needs to be: (1)tightly focused (the localization of the magnetic field must be on theorder of a few hundred nm or less) and (2) the effect of the magneticfield of the laser pulse on the material must be sizable, while the effect of the electric field should be much less or negligible. The first condition can be easily achieved by using various tightly focused laser pulses, multiple focused pulse configurations,or 4 πfocusing of laser radiation (see Ref. 22and references therein). In terms of spatial focusing efficiency (the theoretically most localized field configuration for a given main wavelength) and the related energy focusing efficiency (theoretically highest peakintensity/amplitude under a given average power of incoming radi-ation), the optimal class of pulses is dipole pulses. 22Dipole pulses are exact closed-form solutions of Maxwell ’s equation in vacuum and can be generated experimentally by different techniques; see Ref.22for full details. In particular, magnetic dipole pulses provide the highest pos- sible magnetic field in the focus with the vortex electric fieldaround, while electric dipole pulses give the vortex magnetic field structure around the focus point. As we will demonstrate later, the electric dipole (e-dipole) pulses are preferably used for the genera-tion of a single skyrmion in a 2D magnetic layer. The closed-form expression for the magnetic field of e-dipole pulse is the following: 22 ~B(~r,t)¼/C0~n/C21 c2r€~dþ(t,r)þ1 cr2_~d/C0(t,r)/C20/C21 , (1) where r¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2þy2þz2p ,~n¼~r=ris a normal vector, and ~d+(t,r)¼~dt/C0r c/C0/C1 +~dtþr c/C0/C1 . The vector function ~d(τ) is called virtual dipole moment, and it can be an arbitrary smooth function in the general dipole-pulse solution of the Maxwell equations. For our calculations, we choose: ~d(τ)¼~zd0e/C0τ2=σ2sin (ωτþf), where ~zis a unit vector along the zaxis, ω¼2πc=λis the characteristic frequency of the radiation with main wavelength λ,fis an arbitrary given phase that determines the pulse CEP, σdetermines the pulse duration, and d0is a constant that depends on the total energy of the pulse. It can be expressed also as a function of average power of the incoming radiation ( P), namely, d0¼ffiffiffi 3p c3=2ffiffiffi Pp =ω2.T h e vortex structure of the magnetic field at some fixed point in time is shown in Fig. 1 . The second conditions related to the pulse CEP need to be considered in detail. The effect of the quasi-monochromatic mag-netic field with the near zero field integral (Ð~B(~r,t)dt/C250) can lead to some magnetic excitations due to nonlinearities or specific conditions. However, short tightly focused laser pulses can provideadditional options to excite the magnetic material due to a directlinear effect of the magnetic field. As it was theoretically predicted and directly measured, short laser pulses can have in some regions a non-zero magnetic fieldintegral (which can be, in principle, arbitrary high), ~M(~r)¼ð pulse~B(~r,t)dt=0: (2) In our case, the few-cycle e-dipole pulse with controllable CEP has this property for the points around the focus with high mag-netic field amplitudes, while the analogous integral for the electricfield (Ð~E(~r,t)dt) is exactly zero for the focus point (0, 0, 0) (this is shown exactly in Ref. 22). This not only gives suitable conditions for the single skyrmion generation, but also can prevent material destruction in the focus point. To summarize, e-dipole pulses w ith suitably chosen parame- ters have a strong and very localized magnetic field while theelectric field is very small in the same region of space. The maximum electric field amplitude is achieved in the focus point (0, 0, 0), but in this point, the corresponding field integral over the pulse durationÐ~E(~r,t)dt;0 due to the fundamental mathe- matical properties of the exact solution. This makes e-dipole pulses ideal candidates for creating localized magnetic excitations. To find out whether topological structures like skyrmion can begenerated in this way will be the objective of the simulationsdescribed below. Among the different pulse parameters, the crucial one for the process is the magnetic field integral M¼j~M(~r 0)j,a si t is a good measure for the induced local magnetic excitation in our case. In the calculations described below, with realistic parametersfor both the material and the laser pulse, we demonstrate a singleskyrmion generation with M¼30 T ps. However, with some addi- tional optimization for the material and the external interactions, this value can be reduced to M/difference1 T ps. Such values can be achieved by using a sequence of short pulses with M=0 generated by a table top laser system with proper pulse-splitting and focusing.The principal possibility for the generation of short tightly focused intense laser pulses with similar vortex field structures near the focus point was discussed in Ref. 31. FIG. 1. Distribution of the magnetic field of the e-dipole pulse of Eq. (1)in the plane z¼0. The typical size of the vortex is less than λ.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 073904 (2020); doi: 10.1063/1.5140673 127, 073904-2 Published under license by AIP Publishing.We describe the magnetic material as a 2D-layer consisting of ordered magnetic moments (magnetization vectors μsSi), which are separated in space and interact with each other. The dynamics of each magnetic moment in our model is described by the Landau – Lifshitz –Gilbert (LLG) equation, @Si @t¼/C0γSi/C2Hi effþα μsSi/C2@Si @t, (3) where Si;μs=μsis the unitary magnetization vector (related to the ith“atom ”) with magnetic moment μs,γis the gyromagnetic ratio,andαis the damping parameter. The interaction with other mag- netic moments is given by the effective magnetic field Hi effacting toith magnetic moment (see also Refs. 3and 32–36for the description of the model), Hi eff¼@ @SiX j(j=i)Jij μsSiSjþX j(j=i)Dij μsSi/C2Sj/C2/C3 þSiHiþKi μs(Siea)22 43 5, (4) where Jijis the exchange interaction energy between the ith and jth magnetic moments ( Jij.0 for antiferromagnetic coupling and Jij,0 for ferromagnetic coupling), Kiis the anisotropy energy of theith magnetic moment, eais a unit vector representing the direc- tion of the easy magnetization axis, Hiis an external magnetic field acting to the ith magnetic moment. Vector Dij;Drijdefines the Dzyaloshinskii –Moriya (DM) interaction between ith and jth mag- netic moments. In our case, Dis a constant with dimension of an energy and rijis a unit vector. For magnetization vectors located in the considered 2D-layer, rijis perpendicular to the line connecting siteiand site j(origins of magnetic moments), and it is parallel to the plane ( x,y, 0). For modeling the system, we consider a 2D-structure consist- ing of 900 initially ordered magnetic moments (30 /C230), as shown inFig. 2 . The parameters of the considered magnetic structure are as follows: J¼0:5 meV, D¼0:12 meV, K¼0:1 meV, μs¼2μ0, and α¼0:5. These numbers correspond to well-studied cases in the literature (see, e.g., Ref. 35). The size of the structure is 600/C2600 nm2. The laser spot size depends both on the main wave- length of the radiation and on the efficiency of focusing. We used avery common laser wavelength of 800 nm and nearly ideal focusing FIG. 2. Initial state of the considered system of magnetic moments (the frozen boundary condition case). FIG. 3. The snapshots of the considered magnetic system after irradiation by an e-dipole pulse within the first ps (the open boundary condition case).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 073904 (2020); doi: 10.1063/1.5140673 127, 073904-3 Published under license by AIP Publishing.so that the size can be estimated as a few hundred nm, which is quite achievable in experiments.22 III. RESULTS Starting from this initial condition, the system is irradiated by the sequence of the above-described e-dipole pulses for an effective total duration of 1 ps, yielding a total effective magnetic integral M¼30 T ps. The further dynamics of the system is shown in the series of snapshots given in Fig. 3 . As it is clearly seen from Fig. 3 , the behavior of the magnetic system is much more slower in comparison with the duration of the few-cycle optical wavelength laser pulse. This confirms in first approximation, our assumption that the instantaneous magneticfield effect of the few-cycle e-dipole pulse to the material, mainlydue to linear field-material interaction, and thus it is proportionalto the pulse field integral Mdefined in Eq. (2). The particular choice of e-dipole pulse is beneficial for the fast skyrmion genera- tion due to special features of the effective magnetic field spatialdistribution: (i) the field is localized in a few hundred nm or less,which provides the relatively rapid formation of a single skyrmion in certain location; (ii) it has a vortex structure that “activates ”a crucial role of the DM-interaction responsible for the furtherskyrmion formation (see below more detailed consideration of theDM-interaction impact). These features of using e-dipole pulsesunderline the difference between our proposal and other discussed methods to create skyrmions by external fields (see, for example, Ref. 37). Among others, our proposed e-dipole field configuration can be obtained by realistic focused laser pulses, which keep mag-nificent features of spatial and temporal resolution. The formation of a single skyrmion (which is related in our topology to the appearance of the magnetization vectors anti paral- lel to the zaxis) can be detected between 5 ps and 7 ps. More precise analysis gives the time /difference5:5 ps for the formation of a well- pronounced skyrmion with unit topological charge. In particular,we calculated the topological charge Qon the discrete lattice in the same way, as it was proposed in Ref. 38. We obtained a value of Q/difference0:93 and Q/difference1:0 (open and frozen boundary conditions, respectively) for the final skyrmion, which is conserved duringseveral hundred picoseconds after the formation. Thus, we may conclude that the considered skyrmion is stable at least under the interactions with borders, small magnetic perturbations caused bylaser pulse in periphery, and considerably long free evolution. IV. DISCUSSION Now, we discuss the physical reasons that lead to the dynam- ics shown in Fig. 3 and, in particular, the skyrmion formation within first 10 ps. For convenience, we assume that the magneticfield of the e-dipole pulse excites the system within first 0 :1p s . I n order to demonstrate the dynamics of the whole system, let us consider three particular magnetic moments located near the focus of the e-dipole pulse. The physical characteristics of the firstone, closest to the focus, are marked by gray color, and for thesecond and third by red and blue colors, respectively. In particu- lar, in Fig. 4 , we demonstrate the anisotropy-, exchange-, and DM-energy difference (between current instantaneous value andinitial equilibrium one) as a function of time for the considered three magnetic moments. As one can conclude from Fig. 4 , the behavior of the consid- ered magnetic moments differs significantly from one another. We see that after rapid nearly linear excitation, the system demonstratescomplex dynamics: the first magnetic moment anisotropy andexchange energy differences (marked by gray color) grow rapidlyup to the time moment /difference5 ps. This indicates that the interaction between neighboring magnetic moments (close to the focus) increases as the angle increases between them. After the formationof the skyrmion (appearance of the magnetization vectors antipar-allel to the zaxis), we can observe its growth: according to the exchange energy dependence in Fig. 4 , the skyrmion boundary lies close to the third magnetic (blue line). FIG. 4. The anisotropy-, exchange-, and DM-energy difference (between current instantaneous value and initial equilibrium one) as a function of time for the considered three magnetic moments marked by gray, red, and blue colors (first, second, and third magnetic moments from the skyrmion “center, ”respec- tively). The gray spline-line, red dashed line, and blue dotted line are the energycurves for the corresponding moments.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 073904 (2020); doi: 10.1063/1.5140673 127, 073904-4 Published under license by AIP Publishing.We note here the significant role of the DM-interaction (see, for example, Ref. 39) for the formation of the skyrmion. This interaction is not only responsible for the direct turn of the mag-netic moments, which leads to the skyrmion formation, but also,as it is shown in Fig. 4 , is responsible for the formation of magne- tization vortex-precursor of the skyrmion. According to the DM-interaction energy dependence, the initial vortex structure of magnetic moments (which formed after the end of laser pulse)starts to squeeze to the center up to some moment before theskyrmion formation. Thus, the presence of the DM-interaction iscrucial for the skyrmion generation in our system. V. CONCLUSIONS In this paper, we propose and discuss a new method of the single skyrmion generation based on the interaction of short opti- mally focused laser pulses with a magnetic material. For the gen- eration, we considered a special optimally focused laser pulsecalled the e-dipole pulse, which can directly excite the magneticmedium close to the focus. Due to the special spatial structure of magnetic field in e-dipole pulse, it induces vortex-like excitation, which further evolves and transforms to a single skyrmion.Although we demonstrate this behavior for the specific parametersof laser pulses and magnetic material, we presume that this effectcan occur for a wide range of parameters. This constitutes our conception for the controlled single skyrmion generation in a magnetic material by proper focused laser pulses. Our conception could also give some other possibilities to control the skyrmions, i.e., to erase, move, and measure the skyr-mions by focused laser pulses. In this way, the special advantage of using tightly focused laser pulses is that they provide con- trolled localized excitation to the system with exceptional spatialand time resolution. SUPPLEMENTARY MATERIAL See the supplementary material for the topological charge of the system as a function of time for different boundary conditions. ACKNOWLEDGMENTS O.P.P. thanks S. Zheltoukhov for helpful advice about the visualization of the results. REFERENCES 1A. N. Bogdanov and D. A. Yablonskii, “Thermodynamically stable vortices in magnetically ordered crystals, ”Sov. Phys. JETP 68, 101 –103 (1989). 2S .M ü h l b a u e r ,B .B i n z ,F .J o n i e t z ,C .P f l e i d e r e r ,A .R o s c h ,A .N e u b a u e r ,R .G e o r g i i , and P. Böni, “Skyrmion lattice in a chiral magnet, ”Science 323,9 1 5 –919 (2009). 3S. Heinze, K. Von Bergmann, M. Menzel, J. Brede, A. Kubetzka, R. Wiesendanger, G. Bihlmayer, and S. Blügel, “Spontaneous atomic-scale magnetic skyrmion lattice in two dimensions, ”Nat. Phys. 7,7 1 3 –718 (2011). 4A. K. Nayak, V. Kumar, T. Ma, P. Werner, E. Pippel, R. Sahoo, F. Damay, U. K. Rößler, C. Felser, and S. S. Parkin, “Magnetic antiskyrmions above room temperature in tetragonal Heusler materials, ”Nature 548, 561 –566 (2017). 5A. Fert, N. Reyren, and V. 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A 86, 053836 (2012). 23G. Leuchs, K. Mantei, A. Berger, H. Konermann, M. Sondermann, U. Peschel, N. Lindlein, and J. Schwider, “Interferometric null test of a deep parabolic reflec- tor generating a Hertzian dipole field, ”Appl. Opt. 47, 5570 –5584 (2008). 24Y. Takei and H. Mimura, “Effect of focusing flow on stationary spot machining properties in elastic emission machining, ”Nanoscale Res. Lett. 8, 237 (2013). 25M. Finazzi, M. Savoini, A. R. Khorsand, A. Tsukamoto, A. Itoh, L. Duò, A. Kirilyuk, T. Rasing, and M. Ezawa, “Laser-induced magnetic nanostructures with tunable topological properties, ”Phys. Rev. Lett. 110, 177205 (2013). 26H. Fujita and M. Sato, “Ultrafast generation of skyrmionic defects with vortex beams: Printing laser profiles on magnets, ”Phys. Rev. B 95,1–12 (2017). 27X. Zhang, Y. Zhou, K. M. Song, T.-E. Park, J. Xia, M. Ezawa, X. Liu, W. Zhao, G. Zhao, and S. Woo, “Skyrmion-electronics: Writing, deleting, reading and pro- cessing magnetic skyrmions toward spintronic applications J. Phys. Condens. Matter 32, 143001 (2020). 28W. Koshibae and N. Nagaosa, “Creation of skyrmions and antiskyrmions by local heating, ”Nat. Commun. 5, 5148 (2014). 29G. Berruto, I. Madan, Y. Murooka, G. M. Vanacore, E. Pomarico, J. Rajeswari, R. Lamb, P. Huang, A. J. Kruchkov, Y. Togawa, T. LaGrange, D. McGrouther, H. M. Rønnow, and F. Carbone, “Laser-induced skyrmion writing and erasing in an ultrafast cryo-Lorentz transmission electron microscope, ”Phys. Rev. Lett. 120, 117201 (2018).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 073904 (2020); doi: 10.1063/1.5140673 127, 073904-5 Published under license by AIP Publishing.30S. G. Je, P. Vallobra, T. Srivastava, J. C. Rojas-Sánchez, T. H. Pham, M. Hehn, G. Malinowski, C. Baraduc, S. Auffret, G. Gaudin, S. Mangin, H. Béa, and O. Boulle, “Creation of magnetic skyrmion bubble lattices by ultrafast laser in ultrathin films, ”Nano Lett. 18, 7362 –7371 (2018). 31Ó. Martínez-Matos, P. Vaveliuk, J. G. Izquierdo, and V. Loriot, “Femtosecond spatial pulse shaping at the focal plane, ”Opt. Express 21, 25010 (2013). 32O. Polyakov and V. Stepanyuk, “Tuning an atomic switch on a surface with electric and magnetic fields, ”J. Phys. Chem. Lett. 6, 3698 (2015). 33W. Wang, M. Beg, B. Zhang, W. Kuch, and H. Fangohr, “Driving magnetic skyrmions with microwave fields, ”Phys. Rev. B 92, 020403(R) (2015). 34L. Rózsa, K. Palotás, A. Deák, E. Simon, R. Yanes, L. Udvardi, L. Szunyogh, and U. Nowak, “Formation and stability of metastable skyrmionic spin structures with various topologies in an ultrathin film, ”Phys. Rev. B 95, 094423 (2017).35C. Heo, N. S. Kiselev, A. K. Nandy, S. Blugel, and T. Rasing, “Switching of chiral magnetic skyrmions by picosecond magnetic field pulses via transient topological states, ”Sci. Rep. 6, 1 (2016). 36W. Koshibae and N. Nagaosa, “Theory of antiskyrmions in magnets, ”Nat. Commun. 7, 10542 (2016). 37V. Flovik, A. Qaiumzadeh, A. K. Nandy, C. Heo, and T. Rasing, “Generation of single skyrmions by picosecond magnetic field pulses, ”Phys. Rev. B 96, 140411(R) (2017). 38B. Berg, “Definition and statistical distribution of a topological number in the lattice O(3) s-model, ”Nucl. Phys. B 190, 412 –424 (1981). 39L. M. Sandratskii, “Insight into the Dzyaloshinskii-Moriya interaction through first-principles study of chiral magnetic structures, ”P h y s .R e v .B 96, 024450 (2017).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 073904 (2020); doi: 10.1063/1.5140673 127, 073904-6 Published under license by AIP Publishing.

1.1846411.pdf

Cell size corrections for nonzero-temperature micromagnetics M. Kirschner, T. Schrefl, F. Dorfbauer, G. Hrkac, D. Suess, and J. Fidler Citation: Journal of Applied Physics 97, 10E301 (2005); doi: 10.1063/1.1846411 View online: http://dx.doi.org/10.1063/1.1846411 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/97/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Simulations of magnetic hysteresis loops at high temperatures J. Appl. Phys. 116, 123910 (2014); 10.1063/1.4896582 Effect of longitudinal degree of freedom of magnetic moment in body-centered-cubic iron J. Appl. Phys. 113, 17E112 (2013); 10.1063/1.4794136 Stripe-vortex transitions in ultrathin magnetic nanostructures J. Appl. Phys. 113, 054312 (2013); 10.1063/1.4790483 Influence of the magnetic dipole interaction on the properties of magnetic vortices in particles of small size Low Temp. Phys. 30, 70 (2004); 10.1063/1.1645157 Quantum spin liquid in the 2D anisotropic Heisenberg model with frustrated next nearest neighbor exchange Low Temp. Phys. 24, 572 (1998); 10.1063/1.593639 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.3.43.136 On: Tue, 02 Dec 2014 18:20:38Cell size corrections for nonzero-temperature micromagnetics M. Kirschner,a!T. Schrefl, F. Dorfbauer, G. Hrkac, D. Suess, and J. Fidler Solid State Physics, Vienna University of Technology, A-1040 Vienna, Austria sPresented on 8 November 2004; published online 6 May 2005 d Micromagnetic calculations at nonzero temperatures depend on the computational cell size. This paper shows that the spontaneous magnetization MSof exchange-coupled moments has to be scaled by a Bloch-like law, which is similar to the well-known temperature dependence of MS. Using this scaling law, nonatomistic Metropolis Monte Carlo and stochastic Landau–Lifshitz–Gilbertsimulations are performed in an external field of 0.1 T. The error of the equilibrium magnetizationat a temperature of T/T C=0.38 and a cell size of 1.5 nm is then 0.9% as compared with atomistic calculations. In contrast, a cell size-independent MSleads to an overestimation of the temperature of 3.2%. © 2005 American Institute of Physics .fDOI: 10.1063/1.1846411 g I. INTRODUCTION The decreasing dimensions of magnetic devices make the use of nonzero-temperature micromagnetics essential.Normally, the materials in micromagnetic simulations aresupposed to be magnetized to the saturation magnetizationfor large cells, M S,‘, at each point,1independent of the com- putational cell size ssee Fig. 1 d. This common assumption is valid only if the computational cell contains enough atomis-tic spins to justify the use of statistics and the laws ofthermodynamics. 2But between the atomistic level with the atomistic saturation magnetization MS,0and the thermody- namic region with MS,‘, the use of cell size-independent spontaneous magnetizations leads to wrong numerical resultseven at temperatures far below the Curie temperature T C. The knowledge of temperature- and cell size-dependent cor-rections of the system parameters has become crucial, sinceIgarashiet al. 3pointed out that small computational cells are necessary to find correct results due to the importance ofhigh-frequency spin waves. Dobrovitsky et al. 4,5and Grinstein and Koch6suggested two different methods of statistical coarse-graining andrenormalization for magnetic systems. Contrary to theirmathematical treatment we propose an approximate andstraightforward coarse-graining procedure. In this paper, we study the equilibrium value of the mag- netization of exchange-coupled moments in an isotropic ex-ternal field of 0.1 T as a function of temperature and com-putational cell size. We perform nonatomistic MetropolisMonte Carlo sMCdand stochastic Landau–Lifshitz–Gilbert sLLG dsimulations and figure out that a cell size-dependent spontaneous magnetization M S,cellobtained from atomistic MC calculations sufficiently improves the numerical results.The Bloch-like scaling law of M S,cellas a function of cell size and the behavior of the small but systematic remaining de-viations between atomistic and nonatomistic calculations arediscussed. Using a cell size-dependent M S,cellgives differ- ences of less than 1% between atomistic and nonatomisticcalculations even at a temperature of T/T C=0.38 and a cell size of 1.5 nm. Calculations with a cell size-independentsaturation magnetization yield an error of 3.2% which can be linearly mapped onto a higher temperature. Our results showan excellent agreement between MC and stochastic LLGsimulations for all cell sizes and temperatures and point outthat both methods work well down to a cell size of 1.5 nm. II. MODEL In this work a three-dimensional s3Dddiscrete Heisen- berg model of exchange-coupled magnetic moments in anisotropic external field H=Hzˆis used.The total energy of the system is given by E=ADxo i=1N3 o kijls1−mi·mjd−m0MSDx3H·o i=1N3 mi.s1d HereAis the exchange constant, Dxis the computational cell size sregular cubic lattice d, andNdenotes the number of moments in one direction.The sum kijlis carried out over all nearest-neighbor pairs, mi=Mi/MSrepresents a unit vector on lattice site iwith the cartesian components mi,a, andMSis the spontaneous magnetization. We carry out MC simulations on an atomistic level sDx=a, whereais the atomistic lattice constant dwith the atomistic MS,0and calculate mean values of the n3moments within averaging cubes in the center of the magnetic systemsFig. 2 d. In this paper we focus on the spontaneous magne- tization per moment in the averaging cell adElectronic mail: markus.kirschner@tuwien.ac.at FIG. 1. Number of atoms in the computational cell and the required satura- tion magnetization MSfor micromagnetic simulations as function of the discretization length. A cell size-independent saturation magnetization MS,‘ is valid only for large computational cells. Between the atomistic level with MS,0and the thermodynamic region with MS,‘, the saturation magnetization MS,celldepends on the cell size.JOURNAL OF APPLIED PHYSICS 97, 10E301 s2005 d 0021-8979/2005/97 ~10!/10E301/3/$22.50 © 2005 American Institute of Physics 97, 10E301-1 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.3.43.136 On: Tue, 02 Dec 2014 18:20:38MS,cell=Ko a=13 Ma2L1/2 s2d with Ma=MS,0 n3o i=1n3 mi,a, s3d andkMzl, the averaged magnetization per moment parallel to the external field. The mean values of the other components vanish. kMzlshould be independent of the averaging cell size Dx=nasincemi,zis linear in Eq. s3d. However, MS,cellde- creases with increasing Dxbecause of thermal fluctuations. The idea is to perform nonatomistic calculations with onemacrospin per averaging cube with the appropriate cell size-dependent M S,celland look at the mean value kMzlof the macrospin in the center of the system. III. NUMERICAL METHODS Metropolis Monte Carlo and Langevin dynamics simu- lations are the most commonly used methods to numericallytreat large spin systems. Both methods were applied to cal-culate equilibrium properties of the described magnetic sys-tem. The properly chosen system sizes ensure that none ofthe equilibrium quantities depend on the total number of mo-ments. Periodic boundary conditions are assumed in all di-rections. A. Metropolis Monte Carlo For the MC simulations the well-tested Metropolis algorithm7is used. The program starts with a completely ordered ferromagnetic configuration parallel to the externalfield. To be sure that the results are independent of the start-ing configuration, we verified our calculations with the ini-tially frustrated moments. The MC algorithm randomly picks out a magnetic mo- mentm i, selects a new direction mi8within a cone around mi, and computes the energy difference DE=E8−E. This trial step is then accepted according to Boltzmann’s factorexps−DE/k BTd. The new directions have to be chosen carefully. We fol- low the suggestions of Serena et al.8to achieve an isotropic and homogeneous sampling probability inside the cone. Theopening angle of the cone is increased sdecreased dby 2° whenever the acceptance rate exceeds sfalls below d50%. One MC step is defined as N3trial steps as described above, i.e., on average every spin is considered once per MCstep. For every temperature we discard the first 10 4MC steps and use the next 105steps for averaging. The system size is mainly based on the exchange length9and thus on the ex- change constant A. For the atomistic MC calculations we use N3=603, whereas the size of the central averaging cube runs fromn3=23to 303. For the nonatomistic simulations N3 =153turned out to be sufficient for Ał10−11J/m since only the mean value kMzlof the central macrospin is of interest. B. Langevin dynamics The Langevin dynamics simulations are based on the stochastic LLG equation with multiplicative white-noiseterm in Stratonovich interpretation. 10For the stochastic time integration the Heun method is applied with a time step ofDt=0.1 ps. We execute 2 310 6time steps for every tempera- ture and exclude the first 105steps from the averaging pro- cedure. The nonatomistic LLG simulations are carried outwithN 3=133moments, and kMzlof the central macrospin is evaluated. IV. NUMERICAL RESULTS AND DISCUSSION The material parameters are m0MS,0=1T,A=1 310−12J/m, and a=0.376 nm. Since the damping constant ain the LLG equation does not affect the equilibrium prop- erties of magnetic systems, the relatively large value of a =1 is used to reduce the time until the system reaches equi-librium. The material is exposed to an external field of m0H=0.1 T. Temperatures are given in units of the critical temperature TC=78.4 K obtained from mean-field theory. Since the results are normalized to TCandMS,0, they are found to be valid for a large range of the system parameters. Figure 3 shows the spontaneous magnetization MS,cellas a function of the cell size Dxfor different temperatures re- sulting from the atomistic MC calculations. MS,cellas a func- tion of Dxdecays according to the Bloch-like scaling law FIG. 2. Schematic depiction of the applied coarse-graining procedure. Ato- mistic MC simulations sleft-hand side dyield kMzland the cell size- dependent MS,cellas function of the averaging cube size. MS,cellis then used for nonatomistic calculations sright-hand side d. FIG. 3. The spontaneous magnetization is a function of the averaging cell size,MS=MS,cellsDx,Td.MS,cellwas computed according to the procedure in Fig. 2 as a function of temperature Tand cell size Dx.10E301-2 Kirschner et al. J. Appl. Phys. 97, 10E301 ~2005 ! [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.3.43.136 On: Tue, 02 Dec 2014 18:20:38MS,cellsDx,Td=MS,‘sTd+fMS,0−MS,‘sTdgSa DxD3/2 ,s4d whereMS,‘sTdrepresents the experimentally found sponta- neous magnetization for large cells, which obviously de- pends on A. This scaling law with the exponent 3/2 is at least valid for temperatures less than T/TC<0.75 and for exchange constants within 10−13J/młAł10−11J/m. The results show that the external field strength has only a weakimpact on M S,cell. For example, the difference of MS,cellbe- tween simulations in zero field and 0.5 T is less than 0.2%forDx=1.5 nm. M S,‘as function of Tis given in Table I and is found to agree well with Bloch’s T3/2law up to T/TC <0.75. The atomistic MC simulations yield cell size- independent equilibrium values of kMzl, as expected. Table I presents the results for different temperatures. Figure 4 summarizes the results for kMzlat equilibrium obtained via nonatomistic MC and LLG simulations at T/TC=0.38, normalized to the correct value kMzl/MS,0=0.8487. Both methods coincide excellently for all cell sizes and temperatures. Calculations with a constant MS=MS,‘ lead to systematic errors which can be decreased by the use of a cell size-dependent MS,cell. For instance, the deviation of 3.2% at 1.5 nm can be reduced to 0.9%, i.e., by a factor of3.6. We conclude that the use of cell size-dependent sponta- neous magnetizations sufficiently reduces errors due to dif-ferent computational cell sizes. In contrast to the renormal-ization group methods, the suggested coarse-grainingprocedure allows other parameters, such as the exchangeconstant or the external field, unaffected and makes astraightforward implementation possible. Only a very highdemand on the accuracy requires renormalization of othersystem parameters, as presented in other works. 4–6 V. SUMMARY Based on the results of this paper we suggest a simple coarse-graining procedure for nonzero-temperature micro-magnetics: s1dDetermine the atomistic spontaneous magnetization M S,0=MSsT=0d, the spontaneous magnetization for large cells MS,‘sTdat the desired temperature, and the atomic lattice constant a. s2dApply Eq. s4dto calculate the cell size-dependent MS,cell for nonatomistic simulations. ACKNOWLEDGMENT This work was supported by the Austrian Science Fund sY132-N02 d. 1W. F. Brown, Phys. Rev. 58, 736 s1940 d. 2G. Bertotti, Hysteresis in Magnetism sAcademic Press, New York, 1998 d. 3M. Igarashi, M. Hara, Y. Suzuki, A. Nakamura, and Y. Sugita, IEEETrans. Magn.39,2 3 0 3 s2003 d. 4V. V. Dobrovitsky, M. I. Katsnelson, and B. N. Harmon, J. Magn. Magn. Mater.221, 235 s2000 d. 5V. V. Dobrovitsky, M. I. Katsnelson, and B. N. Harmon, Phys. Rev. Lett. 90, 067201 s2003 d. 6G. Grinstein and R. H. Koch, Phys. Rev. Lett. 90, 207201 s2003 d. 7N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, J. Chem. Phys. 21,1 0 8 s1953 d. 8P. A. Serena, N. García, and A. Levanyuk, Phys. Rev. B 47, 5027 s1993 d. 9V. Tsiantos, W. Scholz, D. Suess, T. Schrefl, and J. Fidler, J. Magn. Magn. Mater.242–245, 999 s2002 d. 10W. Scholz, T. Schrefl, and J. Fidler, J. Magn. Magn. Mater. 233,2 9 6 s2001 d.TABLE I. Results of the atomistic MC simulations for different tempera- tures. T/TC MS,‘/MS,0 kMzl/MS,0 0.13 0.9535 0.9532 0.26 0.9037 0.90340.38 0.8491 0.84870.51 0.7878 0.7871 FIG. 4. Mean value of the magnetization parallel to the field axis as a function of cell size Dx, resulting from nonatomistic Monte Carlo sMCdand Landau–Lifshitz–Gilbert sLLG dsimulations. No cell size corrections were used for hsMCdand1sLLG d. Simulations with cell size-dependent MS,cell accordingto Eq. s4dresultedin nsMCdand3sLLG d.Then, the error is less than 1% at 1.5 nm.10E301-3 Kirschner et al. J. Appl. Phys. 97, 10E301 ~2005 ! [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.3.43.136 On: Tue, 02 Dec 2014 18:20:38

1.1485971.pdf

A linearized Eulerian sound propagation model for studies of complex meteorological effects Reinhard Blumricha)and Dietrich Heimannb) Institut fu¨r Physik der Atmospha ¨re, DLR-Oberpfaffenhofen, D-82234 Weßling, Germany ~Received 3 January 2001; revised 4 April 2002; accepted 22 April 2002 ! Outdoor sound propagation is significantly affected by the topography ~including ground characteristics !and the state of the atmosphere. The atmosphere on its part is also influenced by the topography. A sound propagation model and a flow model based on a numerical integration of thelinearized Euler equations have been developed to take these interactions into account. The outputof the flow model enables the calculation of the sound propagation in a three-dimensionallyinhomogeneous atmosphere. Rigid, partly reflective, or fully absorptive ground can be considered.ThelinearizedEulerian ~LE!soundpropagationmodelhasbeenvalidatedbymeansoffourdifferent scenarios. Calculations of sound fields above rigid and grass-covered ground including ahomogeneous atmosphere deviate from analytic solutions by <1 dB in most parts of the computed domain. Calculations of sound propagation including wind and temperature gradients above rigidground agree well with measured scale model data. Calculations of sound propagation over a screenincluding ground of finite impedance show little deviations to measured scale model data which areprobably caused by an insufficient representation of the complex ground impedance. Furthercalculations included the effect of wind on shading by a screen. The results agree well with themeasured scale model data. © 2002 Acoustical Society of America. @DOI: 10.1121/1.1485971 # PACS numbers: 43.28.Fp, 43.28.Gq, 43.28.Js @LCS# I. INTRODUCTION The propagation of sound outdoors is influenced by the predominant topography and the state of the atmosphere.Thetopography includes ground characteristics, ground-basedobstacles, and terrain features which cause reflection and dif-fraction. The atmosphere affects sound propagation by gra-dients of wind and temperature which lead to refraction andby turbulent fluctuations which cause scattering.Temperatureand humidity determine the degree of absorption of theacoustic wave energy. Overviews of outdoor sound propaga-tion were given, for example, by Piercy et al. 1and Embleton.2Ground characteristics were discussed, for ex- ample, by Attenborough.3 Topography and atmosphere, however, do not act inde- pendently. Terrain features, obstacles, and roughness ele-ments modify the exchange of momentum, energy ~sensitive and latent heat, radiation !, and mass ~of water !between ground and air and thus determine the behavior of atmo- spheric parameters. Hence, the topography exerts a directand indirect ~via the atmosphere !influence on the sound field. Several models exist for a frequency-dependent calcula- tion of outdoor sound propagation. Almost all are based onthe acoustic wave equation. The various algorithms use ei-ther the parabolic wave equation, ray-tracing methods, or aHankel transform of the Helmholtz equation. They considertopographic and atmospheric influences to some extent. Dif-ferent algorithms for refraction in the presence of flat, evenground were reviewed in 1995. 4Beyond that, more compli-cated situations of sound propagation have been discussed, as for example multiple reflection and diffraction in a non-refracting atmosphere in 1997, 5or diffraction and refraction due to inhomogeneous atmosphere above plane ground in1998. 6The combined effect of a barrier and wind gradients has been investigated by means of measured wind speedprofiles. 7,8 Recent studies use input data from either meteorological mesoscale models or fluid dynamics models which simulatethe atmospheric environment influenced by the topography.These models provide consistent atmospheric data for thewhole considered volume. The atmosphere is seen to be sta-tionary while the sound propagates through the model do-main. The acoustic models, however, still approximate theatmospheric influences on the sound waves. The particlemodel of Heimann et al. 9based on ray-tracing methods, for instance, requires us to parametrize diffraction.The approachof Holeet al. 10based on a fast-field program assumes range- independent acoustical properties of the atmosphere. The acoustic models based on the parabolic equation ~PE!need to combine the effects of wind and temperature gradients in oneparameter, viz., the vertical gradient of the effective speed ofsound. As a consequence, the three-dimensionality of thewind vector is neglected. 11–13 Our investigations aim at a simulation of sound propa- gation through an inhomogeneous atmosphere including asfew approximations as possible. For this, a grid-based algo-rithm derived from the Euler equations was chosen as a basisfor a sound propagation model. The model uses input datafrom a meteorological mesoscale model or a flow model. Afrozen atmospheric state is considered, as in the case of therecent studies mentioned above.The linearized Eulerian ~LE! sound propagation model is able to consider flow features a!Electronic mail: reinhard.blumrich@dlr.de b!Electronic mail: d.heimann@dlr.de 446 J. Acoust. Soc. Am. 112(2), August 2002 0001-4966/2002/112(2)/446/10/$19.00 © 2002 Acoustical Society of America Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 193.0.65.67 On: Wed, 17 Dec 2014 10:01:40and other acoustical properties of the atmosphere in three dimensions and dependent on the range. Diffraction behindobstacles is explicitly simulated, hence, a parametrization isnot necessary. The calculations can be carried out for two orthree dimensions, for single or multiple frequencies. Our approach allows determining the contribution of single atmospheric influences to the whole effect on soundwaves. Thus, the error of neglecting specific influences, e.g.,the vertical wind speed component, can be estimated andrequirements for simplified, efficient algorithms can be de-fined. The error of neglecting specific influences will be dis-cussed in a separate paper. The present article introduces the flow model and the LE sound propagation model and discusses the validation. Re-sults of calculations are compared with either analytical cal-culations or measured data and other sound propagationmodels. The measured data were derived from scale modelmeasurements. Inhomogeneous atmosphere is either simu-lated by the flow model ~here: 2D !or prescribed by constant wind and temperature gradients. II. MODEL DESCRIPTION The flow model and the sound propagation model are based on the governing equations of a compressible andadiabatic gaseous medium in a nonrotating system.These arethe equation of motion, the equation of continuity, and thefirst law of thermodynamics for adiabatic processes @Eqs. ~1!–~3!# ~gravity is neglected !: ]u ]t1~u„!u521 r„p, ~1! ]r ]t1u„r52r„u, ~2! cp rdr dt5cv pdp dt. ~3! We usex5xi1yj1zkas the position vector and u5ui 1vj1wkas the three-dimensional velocity vector. Here i, j, andkare the unity vectors in the x,y, andzdirections, respectively. kpoints to the vertical. With t,r, andpwe denote the scalar quantities time, density, and pressure, re-spectively. c pandcvare the specific heats of air at constant pressure and constant volume, respectively. With the help of Eq. ~3!the density rcan be eliminated from Eq. ~2!, resulting in a new equation for the pressure p: ]p ]t1u„p52kp„u ~4! with k5cp/cv. In the following the atmospheric variables f5(u,p,r) are split up into their meteorological, turbulent, and acousticparts: f5f¯1f81f9. ~5! The overbar variables f¯denote mean wind u¯, air pressure p¯,and air density r¯. The single prime variables f8indicate the turbulent deviations from the mean meteorological values.Finally, the double prime variables f9signify the deviations from f¯according to acoustic waves, in particular sound pressurep9and particle velocity u9. The time scale of the acoustic fluctuations is smaller than that of the turbulent fluc-tuations. A. Flow model The flow model is based on the nonlinear Eqs. ~1!and ~4!. It provides numerical solutions for u¯andp¯in a neutrally stratified atmosphere in which buoyancy does not play a role~note: gravity is neglected !. Since subgrid-scale turbulent fluctuations cannot be explicitly calculated, their effect onthe mean meteorological variables is parametrized. For thisreason a further equation is introduced by which the turbu- lent kinetic energy E ¯50.5uu8u2is calculated. Following Yamada,14the temporal derivative of E¯is given by ]E¯ ]t1u¯„E5KMu„u¯u22cEE¯3/2 ,. ~6! The two terms on the right side describe production and dissipation of turbulent kinetic energy. The production termonly accounts for shear production, because buoyant accel-erations are not described by Eqs. ~1!and~4!anyway.c E 51.2 is a constant. ,is a mixing length which is calculated in analogy to Blackadar15by ,5kd 11kd/,‘~7! withdbeing the distance to the closest rigid surface. k 50.4 is the von Karman constant. ,‘is set to 25 m. KMis the turbulent diffusion coefficient for momentum. It is deter-mined from the local turbulent kinetic energy by thePrandtl–Kolmogorov relation 16 KM5,A0.2E¯. ~8! The equation of motion of the flow model is completed by a turbulent diffusion term which accounts for the turbulenceeffects on the mean flow field: ]u¯ ]t1~u¯„!u¯521 r¯„p¯1~„KM„!u¯. ~9! The pressure equation of the flow model reads ]p¯ ]t1u¯„p¯52kp¯„u¯. ~10! Equations ~6!,~9!, and ~10!are numerically integrated in time on a staggered ~Arakawa C-type !17orthogonal grid with grid cells of the size Dx3Dy3Dz. The velocity components u¯,v¯, andw¯are defined on the side walls of the grid cells which are perpendicular to the respective component. Thescalarsp ¯andr¯are defined in the center of the grid cells. The time step Dtmust obey the Courant–Friedrich– Levi criterion18Dt<0.5Dxc21wherecdenotes the fastest 447 J. Acoust. Soc. Am., Vol. 112, No. 2, August 2002 R. Blumrich and D. Heimann: Eulerian sound propagation model Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 193.0.65.67 On: Wed, 17 Dec 2014 10:01:40signal velocity. In the framework of Eqs. ~9!and~10!this is the speed of sound, i.e., c5Akp¯/r¯. In general, explicit forward-in-time and centered-in-space differences are em-ployed. As an exception, the diffusion term is solved by animplicit time-scheme. Ground and obstacles are defined by ‘‘solid’’ grid cells, i.e., the velocity components ~normal and tangential !on their side walls are set to zero. This corresponds to a no-slipboundary condition at material surfaces and prevents the airfrom penetrating either ground or obstacles. For special stud-ies, however, model options can be invoked which accountfor free-slip boundary conditions or semi-permeable ob-stacles such as, for example, hedges. The semi-permeabilitycan be simulated by a fractional damping of the flow velocitywhich acts as a momentum sink. 19 At the time of initialization, flat ground and permeable obstacles are assumed. Accordingly, the initial flow is hori-zontal and uniform. The friction at ground is accounted forby a logarithmic vertical wind speed profile: uu¯u5u* klnSz1zo zoD ~11! with the friction velocity u*, the von Karman constant k 50.4, and the roughness length zo. Friction velocity and roughness length have to be prescribed. After initialization the permeability of elevated ground and obstacles is successively reduced to zero within a certainspan of time. This procedure helps to avoid the generation ofspurious distortions due to initial inconsistencies betweenflow and rigid boundaries. Towards the upper and lateral inflow boundaries of the numerical grid domain the solution for u ¯is forced to adapt to the corresponding values of the initial flow with increasingemphasis ~nudging technique !. 20 B. LE sound propagation model Although the physical formulation of the flow model includes sound waves, numerical reasons suggest the use of aseparate acoustic model. The problem originates from thelarge difference in magnitude between the speed of the meanairflow ( ’10 1m/s) and the particle velocity associated with sound waves ( !1022m/s). As a consequence, numerical noise that inevitably emerges from the air flow solution su-perimposes and distorts the wave solution. In the following we describe a sound propagation model which is based on prognostic equations of u 9andp9. The model equations are deduced from Eqs. ~1!and~4!withu 5u¯1u9~velocity !,p5p¯1p9~pressure !, and r215a5a¯ 1a9~specific volume !. Since the atmosphere is seen to be stationary, the turbulent parts f8are disregarded here. Tur- bulence effects on the mean flow field u¯are included ~see previous subsection !. Taking the stationarity of the mean state into account ~u¯, p¯, and a¯are constant in time !and after linearization with respect to the mean state, i.e., neglecting the nonlinear terms(u 9„)u9andu9„p9, the model equations read as]u9 ]t1~u¯„!u91~u9„!u¯52a¯„p92a9„p¯1n„2u9, with a¯51 r¯anda9521 kp9 p¯1 r¯,~12! ]p9 ]t1u¯„p91u9„p¯52kp¯„u92kp9„u¯. ~13! The last term on the right side of Eq. ~12!, which is a diffu- sion term, was added in order to simulate the effect of atmo-spheric absorption. It can be understood as an artificial vis-cosity which takes into consideration classical absorption aswell as molecular absorption. The diffusion coefficient nis empirically determined as a function of temperature and rela-tive humidity such that the frequency-dependent attenuationdue to air absorption is simulated according to atmosphericattenuation coefficients given by ISO 9613-1. 21These attenu- ation coefficients cannot be used directly because the lengthof the propagation path ~including refraction and reflections ! is not given explicitly by the simulation. The prognostic model equations are numerically solved on an orthogonal staggered grid. The numerical scheme con-forms to that of the flow model except that the explicitforward-in-time scheme is also used for the diffusion term.The spatial distribution of u ¯,p¯, and r¯is taken from the results of the flow model. The values of u9andp9are ini- tially set to zero. Sound waves are triggered by a prescribed harmonic os- cillation of the pressure p9at the location of the source psource9~t!5( i51nf ~pi9cos@2pfit#!, ~14! withpi9being the pressure amplitude at frequency fiandnf the total number of frequencies. In order to better represent the directivity of a sound source, grid points neighboring tothe source can be defined to be solid. In order to avoid spurious reflections from the lateral and top boundaries an impedance boundary condition is ap-plied. The relation between particle velocity and sound pres-sure is given by uu9u5p9 Za, ~15! whereZa5r¯cdenotes the impedance of air. At the bound- aries, only the perpendicular component of the particle ve-locity has to be determined. For example, the component u 9 ~note:u95u9i1v9j1w9k) is calculated at the lateral bound- aries in the xdirection by u95ASp9 ZaD2 2v922w92. ~16! III. VALIDATION The quality of the LE sound propagation model has been proved by comparisons of model results with analytical so-lutions or experimental data from scale model measure-ments. The comparisons focus on the reproduction of groundcharacteristics, refraction due to wind and temperature gra- 448 J. Acoust. Soc. Am., Vol. 112, No. 2, August 2002 R. Blumrich and D. Heimann: Eulerian sound propagation model Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 193.0.65.67 On: Wed, 17 Dec 2014 10:01:40dients, shading effect of obstacles, and the influence of wind on the shading effect. These scenarios have been chosen tovalidate the LE model especially with respect to short- andmedium-range propagation including topographic and atmo-spheric influences. An overview of the considered scenariosis given in Table I. The details are discussed in the followingsubsections. In case of a comparison with scale model mea-surements, all numbers given refer to the full-scale context. The investigated applications are restricted to frequen- cies lower than 1.5 kHz and ranges shorter than 110 m. Inthese cases, atmospheric absorption remains below 1 dB ~ac- cording to ISO 9613-1, 21temperature 20°C, relative humid- ity 70% !. Therefore, absorption is neglected and the diffu- sion coefficient nwas consequently set to zero. Turbulent scattering of sound waves is not considered here. All pre-sented calculations were carried out in two dimensions, i.e.,thex-zplane. The grid spacing Dxused in the LE model usually ranges between 1 10and1 12of the considered wave length l.I n the case of ground of finite impedance, however, the groundlayer ~see Sec. IIA !required a minimum spatial resolution. Thus, for low frequencies a grid spacing smaller than 1 12l has to be used ~e.g.,1 24lwithf5100 Hz !. The time step of the numerical integration was set to Dt’0.45Dx/c. In the case of multi-frequency runs @i.e.,nf.1 in Eq. ~14!#, grid spacing and time steps were adapted to the center or themaximum frequency.The respective values of DxandDtare given in Table I. A. Reproduction of ground characteristics The reproduction of specific ground characteristics was investigated on the basis of reflection from a rigid, i.e., to-tally reflecting, surface and a representation of grass-coveredground. To be able to attribute effects on the sound field unambiguously to the ground response, the calculations werecarried out without obstacles and atmospheric influences.This configuration allows us to compare the numerical cal-culations with analytical solutions based on the Weyl–vander Pol equation ~a general description is given by Suther- land and Daigle, 22the specific implementation of the equa- tion is described in a previous paper23!. Rigid ground was reproduced simply by setting the ver- tical component of the particle velocity at the ground surfaceto zero. The presented sound field above rigid ground ( f 5100 Hz) shows the interference pattern due to the super- position of direct and reflected waves ~Fig. 1 !.The numerical results agree well with the analytic solution. Over a largerange the deviations are lower than 1 dB ~Fig. 1 !. Near the interference minima, the deviations are higher since their po-sitions are slightly shifted. The reproduction of ground of complex, finite imped- ance is not straightforward within the LE model. Since thecalculations are based on real numbers, a complex imped-ance cannot be explicitly prescribed. Instead, a ground layerof specific acoustic properties is required. The density rgof the ground layer is set to be higher than the density of the air r¯. This corresponds to a lower sound speed within the layer. Additionally, the wave is attenuated exponentially with depthwithin the layer. Every nth grid-column was set to be rigid, in order to enforce local reaction. By varying the parameters rg,n, and layer thickness, the ground response can be tuned to reproduce complex ground impedances. As a reference for the characteristics of grass-covered ground, the best fit of a two-parameter impedance model24to measured data23was taken. The best fit was found for anTABLE I. Cases chosen for validation of the LE sound propagation model.The grid spacing Dxranges between 1 10–1 12l, except for the reproduction of ground characteristics ~1 24l!. The time step amounts to Dt’0.45 Dx/c. With multi-frequency runs, grid spacing and time steps were adapted to the center or the maximum frequency. Scenario Obstacle Ground WindTemperature ~°C!Frequency ~range ! ~Hz!Validation dataGrid spacing Dx ~m!Time step Dt ~ms! Ground characteristicsnone rigid none 15 100 analytic results0.14 200 none grass- coverednone 15 100 analytic results0.14 200 Wind and temperaturegradientsnone rigid gradient gradient 42–990 measured data0.03 35 none rigid gradient 20 112–990 measured data0.03 35 Shading by screenscreen cotton and feltnone 15 120–1260 measured data0.028 37.2 Influence of windon shadingeffectscreen rigid gradient 15 185–375 measured data0.10 140 screen rigid gradient 15 370–750 measured data0.05 70 screen rigid gradient 15 740–1500 measured data0.025 35 449 J. Acoust. Soc. Am., Vol. 112, No. 2, August 2002 R. Blumrich and D. Heimann: Eulerian sound propagation model Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 193.0.65.67 On: Wed, 17 Dec 2014 10:01:40effective flow resistivity se5100 kPasm22and an effective decrease of porosity with depth ae59m21. The set of pa- rameters rg,n, and layer thickness which reproduces such a ground the best was determined empirically by a comparisonwith the analytic solution. The interference pattern of the sound field above grass- covered ground ~f5100 Hz, Fig. 2 !is less pronounced com- pared to the case of total reflection.The relatively high soundpressure level close to the ground is caused by the groundwave. The ground wave appears especially at frequenciesbelow approximately 200 Hz ~for the considered ground and geometry !.Asurface wave, which also raises the sound pres- sure at low frequencies close to the ground, appears usuallyonly at larger distances. 2According to an estimation @Eq. ~17!in Sutherland and Daigle22#, for the given ground im- pedance and source height surface waves should arise at dis-tances larger than 150 m. The deviations from the analyticsolution are again less than 1 dB over a large range ~Fig. 2 !. In the area above the sound source and around the interfer-ence minima larger deviations appear. However, soundpropagation simulations normally aim at the far field up to 15 m above the ground. In this area, the numerical and theanalytic solution agree almost as well as in the case withtotal reflection. B. Wind and temperature gradients Atmospheric inhomogeneities were considered in the form of two different configurations including wind and tem-perature gradients above rigid, plane ground. The results ofthe calculations were compared with data from scale modelmeasurements which were carried out by Gabillet et al. 25 Additionally, Gabillet et al.provide Gaussian beam solu- tions. Since the experimental data were available for a whole frequency range, multi-frequency runs were carried out inorder to compare frequency spectra. The frequencies rangedfromf542 to 990 Hz in steps of 14 Hz ( n f569) in the first case and from f5112 to 990 Hz ~steps: 14 Hz, nf564!in the second case. The frequency spectra were determined FIG. 1. Top: Sound pressure level above rigid ground ~dBre20mPa,f5100 Hz, source at x50m ,z 51.2 m !calculated by the LE model. Bottom:Absolute deviation of the calculated sound field from the corre-sponding analytic solution in dB.The vertical scales arestretched by a factor of 2. 450 J. Acoust. Soc. Am., Vol. 112, No. 2, August 2002 R. Blumrich and D. Heimann: Eulerian sound propagation model Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 193.0.65.67 On: Wed, 17 Dec 2014 10:01:40from the time domain data at the receiver position by means of a fast Fourier transform. Source and receiver positions aregiven below. The first case considers downwind propagation includ- ingatemperaturegradient.Windspeed ~inm/s !andtempera- ture~in °C !profiles above z50.4 m are given by Eq. ~13!in Ref. 25: u ~z!51.826lgSz mD14.16,T~z!50.744lgSz mD120 ~17! ~lg denotes the logarithm to base 10 !. Belowz50.4 muand Tare extrapolated linearly down to the ground by means of the gradient at z50.4 m. Source and receiver are located 2 and 5 m above the ground, respectively. The separation dis-tance amounts to 100 m. The LE model result is comparedwith the measured data and a Gaussian beam solution ~Fig. 3!. The minimum sound pressure level of the LE result ap- pears at the same frequency as measured, the level itself, FIG. 2. Top: Sound pressure level above grass-covered ground ~dBre20mPa,f5100Hz, source at x50m , z51.10 m, ground layer thickness 51.68m, calculated by LE model !. Bottom: Absolute deviation of the cal- culated sound field from the analytic solution ~two- parameter impedance model, se5100 kPasm22,ae 59m21!in dB. FIG. 3. Frequency spectra of downwind propagation including a tempera- ture gradient above rigid, plane ground ~sound pressure level refree field !. hs52m ,hr55m ,d5100 m. Dashed line: measured data, dotted line: LE result, solid line: Gaussian beam solution. Gaussian beam solution, mea-sured data, and graph adapted from Ref. 25, Fig. 11 ~a!. 451 J. Acoust. Soc. Am., Vol. 112, No. 2, August 2002 R. Blumrich and D. Heimann: Eulerian sound propagation model Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 193.0.65.67 On: Wed, 17 Dec 2014 10:01:40however, is overestimated. Over the whole spectrum, thea- greement of the LE result with the measured data, and theGaussian beam solution as well, is satisfactory. The second case considers upwind sound propagation in an isotherme atmosphere. The wind speed profile ~in m/s !is given by Eq. ~15!in Ref. 25: u ~z!524.85Sz 3mD0.14 ~18! Source and receiver are locate d 1 m above the ground at a distanceof80m.Figure4showsthefrequencyspectraofthemeasured data, a Gaussian beam solution, and the LE result.Between 300 and 700 Hz, the LE result agrees perfectly withthe measured data. Both, LE result as well as the Gaussianbeam solution, deviate from the measured data by around 1 dB, on average, over the whole frequency range. C. Shading effect of a screen The shading effect of a screen was first calculated for a homogeneous resting atmosphere. The results of the calcula-tions were compared with data from a scale model measure-ment which was carried out by Rasmussen. A principal de-scription of the measurement is given by Rasmussen andArranz. 7Geometry and ground impedance differed slightly from this description. The setup of the measurement, andthus for the calculations as well, is presented in Fig. 5. The ground consisted of two different layers of cotton and felt of which the acoustical characteristics are describedby a two-parameter impedance model. 26The area in front of the screen ~source side !is described by an effective flow resistivity of se5150 kPasm22and an effective rate of po- rosity decrease with depth of ae5350 m21. Behind the screen ~receiver side !the values amount to se 57kPasm22andae5125 m21, respectively ~note: these numbers refer to the full-scale frequencies !. A multi-frequency run was carried out with frequencies ranging from f5120 to 1260 Hz in steps of 30 Hz ( nf 539). The spectrum was calculated for the position of the receiver. However, only one set of ground parameters can beused for one run, thus, the representation of the ground im-pedances is optimized for only one ~mean !frequency. FIG. 4. Frequency spectra of upwind propagation above rigid, plane ground ~no temperature gradient, sound pressure level refree field !.hs5hr51m , d580 m. Dashed line: measured data, dotted line: LE result, solid line: Gaussian beam solution. Gaussian beam solution, measured data, and graphadapted from Ref. 25, Fig. 13. FIG. 5. Setup of the measurements of Rasmussen ~full scale, not true to scale!and the corresponding calculations ~see Figs. 6 and 7 !. The vertical line in the middle represents the totally reflecting screen. FIG. 6. Calculated sound pressure level ~dBre20mPa! in the presence of a screen ~f5120–1260Hz, steps: 30 Hz, source at x50m, setup see Fig. 5, ground imped- ance source side: se5150 kPasm22,ae5350 m21, receiver side: se57kPasm22,ae5125 m21!. Atmo- spheric influences have been excluded. 452 J. Acoust. Soc. Am., Vol. 112, No. 2, August 2002 R. Blumrich and D. Heimann: Eulerian sound propagation model Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 193.0.65.67 On: Wed, 17 Dec 2014 10:01:40In the presence of a screen the shading effect is evident ~Fig. 6 !. The remaining sound behind the screen is caused by diffraction. In front of the ~totally reflecting !screen, interfer- ences appear due to reflections. Figure 7 presents the calcu-lated and the measured frequency spectrum, both normalizedto free-field propagation. The calculated spectrum generallyresembles the measured one.The double dip structure around280 and 480 Hz is not exactly matched in terms of dip fre-quencies and levels. There are also positive and negativebiases at higher frequencies. The deviations are probably dueto the simplified representation of the ground, which wasoptimized for one frequency only. D. Effect of wind on the shading effect The last considered scenario deals with the presence of a screen in an inhomogeneous moving atmosphere. Again, theresults of the LE model were compared with results from scale model measurements, in this case measurements bySalomons. 11The measurements focused on the effect of screens on wind gradients which in turn impair the efficiencyof screens.The setup is given in Fig. 8.The ground surface isassumed to be rigid and plane. The wind fields were calculated by the flow model de- scribed in Sec. IA. The initial, undisturbed wind speed pro-file is given by u ~z!5u9 ln~11z/z0!lnS11z9 z0D ~19! withz959 m and the roughness length z050.001 m ~see Ref. 11 !. The wind speed at a height of 9 m u9was varied from 0 to 14 m/s in steps of 2 m/s. The flow model wasintegrated over 30 s until a steady state was achieved. Figure 9 shows a representation of the wind field for u 9510 m/s. The deformation of the wind field due to the screen is clearly visible.Above the screen the wind gradientsare much stronger than in the undisturbed case. In the wake FIG. 7. Comparison of calculated ~solid line !and measured ~dashed line ! frequency spectrum ~dBrefree field, same calculation as in Fig. 6 !at the receiver position ~see Fig. 5 !. FIG. 8. Setup of the measurements of Salomons ~full scale, not true to scale ! and the corresponding calculations ~see Figs. 9 and 10 !. Both receivers are not in line of sight to the source, the ground surface is rigid and plane. FIG. 9. Example of a wind field @u9510 m/s, see Eq. ~19!#, calculated by the flow model ~top: horizontal component u¯, bottom: vertical component w¯!.T h e combination of positive and negative ~dashed lines !val- ues of both components indicates a vortex behind thescreen. 453 J. Acoust. Soc. Am., Vol. 112, No. 2, August 2002 R. Blumrich and D. Heimann: Eulerian sound propagation model Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 193.0.65.67 On: Wed, 17 Dec 2014 10:01:40of the screen a vortex forms as the negative values of the horizontal component indicate. The simulations were performed for eight different val- ues ofu9and three different octave bands. Each octave band was represented by 20 equidistant frequencies ~250-Hz band: 185 to 375 Hz, 500-Hz band: 370 to 750 Hz, 1000-Hz band:740 to 1500 Hz !. From each simulation, the sound pressure levels at the receiver positions ~Fig. 8 !were taken and nor- malized with respect to the sound pressure level of the no-wind case ( u 950). The results are presented in Fig. 10 together with the measurements of Salomons. In addition, the results of a para- bolic equation ~PE!model from Salomons are shown. The left column shows the relative sound pressure level at a re-ceiver height o f 6 m for the three different octave bands, the right column at a receiver height of 12 m. At 6-m height the results of the LE model are very simi- lar to the PE model results by Salomons. They also fit themeasured data well, except for the 250-Hz case. In the lattercase, both models yield similar deviations. The calculationsfor the receiver height of 12 m also show satisfactory results.Especially for 250 Hz, the results of the LE model agreeperfectly with the measured data. For 500 Hz the relativesound pressure level is slightly too high, which means theeffect of the wind is slightly overestimated. In the case of1000 Hz, at lower wind speeds ( u 9<6 m/s) the LE modelagrees well with the measured data. For u9>8 m/s the wind effect is predicted to be higher than measured. IV. CONCLUSIONS The LE model turns out to be a suitable, fairly accurate numerical tool for calculating sound propagation under vari-ous topographic and atmospheric conditions. Input data frommeteorological mesoscale models or flow models can beused to allow for a three-dimensionally inhomogeneous at-mosphere. Single- or multi-frequency runs are possible intwo or three dimensions. The comparison of model results with analytic solutions for rigid and grass-covered ground shows that such groundcharacteristics can be reproduced. The representation of afinite ground impedance, however, requires a costly determi-nation of ground parameters. The calculations of soundpropagation above rigid, plane ground including wind andtemperature gradients result in frequency spectra whichagree satisfactorily with the measured results and a Gaussianbeam solution. LE model results for shading by a screen above ground of finite impedance show reasonable agreement with mea-surements. The calculated frequency spectrum resemblesthe measured one; however, deviations are visible. Afrequency-dependent representation of the ground character-istics would presumably improve the reasonable agreementwith the measurements. This would require one calculationfor each frequency. The calculations concerning the effect ofwind on diffraction behind a screen agree well with the mea-sured data except for the 250-Hz octave band and 6-m re-ceiver height. In the latter case, the calculations of two dif-ferent sound propagation models show the same deviations.This suggests that the deviations are not caused by errors ofthe algorithms. The crucial treatment of finite ground impedance and the time-consuming calculations ~e.g., NEC-SX4 single proces- sor: around 10000 cpu-seconds for a 4600 31200 point grid and a integration time of 2.3 s including atmospheric influ-ences !are the two main disadvantages of the LE model. On the other hand, this model enables a simulation of soundpropagation through inhomogeneous atmosphere withoutmajor approximations. The wind vector can be considered inthree dimensions, diffraction behind obstacles is explicitlyincluded, and the representation of topography is limitedonly by the spatial resolution of the grid.Thus, the LE modelcan be used as a tool for specific situations which can beconsidered by other models only in a simplified way. Theeffect of approximations and isolated atmospheric influencescan be estimated and hereby specific requirements for sim-plified, efficient algorithms can be defined. ACKNOWLEDGMENTS The authors would like to thank K. B. Rasmussen for providing experimental data of his scale model measure-ments and answering patiently our questions about the data.Thanks go also to E. M. Salomons for helpful discussionsand to E. Premat for providing information about measure-ment data. FIG. 10. Sound pressure level as a function of u9relative to the no-wind case (u950) at a receiver height of 6 m ~left column !and 12 m ~right column !for the three different octave bands. Dashed line: LE model; other lines: PE model for different wind gradients; dots with error bars: measure-ments. PE model results, measurements, and graph adapted from Ref. 11. 454 J. Acoust. Soc. Am., Vol. 112, No. 2, August 2002 R. Blumrich and D. Heimann: Eulerian sound propagation model Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 193.0.65.67 On: Wed, 17 Dec 2014 10:01:401J. E. Piercy, T. F. W. Embleton, and L. C. Sutherland, ‘‘Review of noise propagation in the atmosphere,’’ J. Acoust. Soc. Am. 61, 1403–1418 ~1977!. 2T. F. W. Embleton, ‘‘Tutorial on sound propagation outdoors,’’ J. Acoust. Soc. Am. 100,3 1 –4 8 ~1996!. 3K. Attenborough, ‘‘Review of Ground Effects on Outdoor Sound Propa- gation from Continuous Broadband Sources,’’Appl. Acoust. 24, 289–319 ~1988!. 4K. Attenborough, S. Taherzadeh, H. E. Bass, X. Di, R. Raspet, G. R. Becker, A. Gu ¨desen, A. Chrestman, G. A. Daigle, A. L’Espe ´rance, Y. Ga- billet, K. E. Gilbert, Y. L. Li, M. J. White, P. Naz, J. M. Noble, and H.A.J. M. Hoof, ‘‘Benchmark cases for outdoor sound propagation models,’’J.Acoust. Soc. Am. 97, 173–191 ~1995!. 5E. M. Salomons, ‘‘Sound Propagation in Complex Outdoor Situations with a Non-RefractingAtmosphere: Model Based onAnalytical Solutionsfor Diffraction and Reflection,’’Acust. Acta Acust. 83, 436–454 ~1997!. 6K. M. Li, V. E. Ostashiev, and K. Attenborough, ‘‘The Diffraction of Sound in a Stratified Moving Atmosphere Above an Impedance Plane,’’Acust. Acta Acust. 84,6 0 7 –6 1 5 ~1998!. 7K. B. Rasmussen and M. G. Arranz, ‘‘The insertion loss of screens under the influence of wind,’’ J. Acoust. Soc. Am. 104, 2692–2698 ~1998!. 8N. Barriere and Y. Gabillet, ‘‘Sound Propagation Over a Barrier with Realistic Wind Gradients. Comparison of Wind Tunnel Experiments withGFPE Computations,’’Acust. Acta Acust. 85, 325–335 ~1999!. 9D. Heimann and G. Gross, ‘‘Coupled simulation of meteorological param- eters and sound level in a narrow valley,’’ Appl. Acoust. 56, 73–100 ~1999!. 10L. R. Hole and H. M. Mohr, ‘‘Modeling of sound propagation in the atmospheric boundary layer:Application of the MIUU mesoscale model,’’J. Geophys. Res. D 104, 11891–11901 ~1999!. 11E. M. Salomons, ‘‘Reduction of the performance of a noise screen due to screen-induced wind-speed gradients. Numerical computations and wind-tunnel experiments,’’ J. Acoust. Soc. Am. 105, 2287–2293 ~1999!. 12M.West andY. Lam, ‘‘Prediction of sound fields in the presence of terrain features which produce a range dependent meteorology using the genera-lised terrain parabolic equation ~GT-PE !model,’’ Proc. Internoise, 2/943 ~2000!.13E. M. Salomons and K. B. Rasmussen, ‘‘Numerical computation of sound propagation over a noise screen based on an analytic approximation of thewind speed field,’’Appl. Acoust. 60, 327–341 ~2000!. 14T. Yamada, ‘‘Simulations of nocturnal drainage flows by a q2lturbulence closure model,’’ J. Atmos. Sci. 40, 91–106 ~1983!. 15A. K. Blackadar, ‘‘The vertical distribution of wind and turbulent ex- change in a neutral atmosphere,’’J. Geophys. Res. 67, 3095–3102 ~1962!. 16See, for example, R. B. Stull, An Introduction to Boundary Layer Meteo- rology ~Kluwer Academic, Dordrecht, 1988 !, Chap. 6. 17A.Arakawa andV. R. Lamb, ‘‘Computational design of the basic dynamic process of the UCLAgeneral circulation model,’’Methods Comput. Phys. 17, 173–265 ~1977!. 18D. R. Durran, Numerical Methods for Wave Equations in Geophysical Fluid Dynamics ~Springer, New York, 1999 !, Chap. 2.2.3. 19N. R. Wilson and R. H. Shaw, ‘‘A higher order closure model for canopy flow,’’ J. Appl. Meteorol. 16, 1197–1205 ~1977!. 20A principal description of the technique is given by H. C. Davies and R. H. Turner, ‘‘Updating prediction models by dynamical relaxation: an ex-amination of the technique,’’ Q. J. R. Meteorol. Soc. 103, 225–245 ~1977!. 21International Standards Organization, ‘‘Acoustics-Attenuation during propagation outdoors—Part I: Method of Calculation of theAttenuation ofSound by Atmospheric Absorption,’’ ISO/DIS 9613-1, September 1990. 22L. C. Sutherland and G. A. Daigle, ‘‘Atmospheric sound propagation,’’inEncyclopedia of Acoustics , edited by M. J. Crocker ~Wiley, New York, 1997!, Sec. 4.3, Vol. I, Chap. 32, pp. 341–365. 23R. Blumrich and J. Altmann, ‘‘Ground Impedance Measurement Over a Range of 20 m,’’Acust. Acta Acust. 85, 691–700 ~1999!. 24K. Attenborough, ‘‘Acoustical Impedance Models for Outdoor Ground Surfaces,’’ J. Sound Vib. 99, 521–544 ~1985!,E q . ~31!. 25Y. Gabillet, H. Schroeder, G. A. Daigle, and A. L’Esprance, ‘‘Application of the Gaussian beam approach to sound propagation in the atmosphere:Theory and experiments,’’ J. Acoust. Soc. Am. 93, 3105–3116 ~1993!. 26K.Attenborough, ‘‘Ground parameter information for propagation model- ing,’’ J. Acoust. Soc. Am. 92, 418–427 ~1992!,E q . ~12!. 455 J. Acoust. Soc. Am., Vol. 112, No. 2, August 2002 R. Blumrich and D. Heimann: Eulerian sound propagation model Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 193.0.65.67 On: Wed, 17 Dec 2014 10:01:40

1.2432403.pdf

Ultrahigh frequency magnetic susceptibility of Co and Fe cylindrical films deposited by pulsed laser ablation J. Vergara, C. Favieres, and V. Madurga Citation: Journal of Applied Physics 101, 033907 (2007); doi: 10.1063/1.2432403 View online: http://dx.doi.org/10.1063/1.2432403 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/101/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Combination of ultimate magnetization and ultrahigh uniaxial anisotropy in CoFe exchange-coupled multilayers J. Appl. Phys. 97, 10F910 (2005); 10.1063/1.1855171 Influence of ion implantation on the magnetic properties of thin FeCo films J. Appl. Phys. 97, 073911 (2005); 10.1063/1.1875737 Determination of the thickness of pulsed laser deposited cylindrical Co films by their magnetoelastic effects J. Appl. Phys. 96, 1850 (2004); 10.1063/1.1768611 Microstructure and damping in FeTiN and CoFe films J. Appl. Phys. 93, 6671 (2003); 10.1063/1.1556099 First-principles theory of magnetoelastic coupling and magnetic anisotropy strain dependence in ultrathin Co films on Cu(001) J. Appl. Phys. 83, 7258 (1998); 10.1063/1.367752 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.115.103.99 On: Fri, 22 Aug 2014 03:59:02Ultrahigh frequency magnetic susceptibility of Co and Fe cylindrical films deposited by pulsed laser ablation J. Vergara,a/H20850C. Favieres, and V. Madurga Departamento de Física, Universidad Pública de Navarra, Campus de Arrosadía, E-31006 Pamplona, Spain /H20849Received 23 February 2006; accepted 22 November 2006; published online 5 February 2007 /H20850 Soft magnetic Co and Fe films were deposited by pulsed laser ablation on Cu wires. The circular magnetic susceptibility of the cylindrical films was measured by applying an ac circular magneticfield at frequencies up to 6 GHz. The measured ferromagnetic resonance frequency of the Co filmswas 2.5 GHz, and the real part of the magnetic susceptibility at the lowest frequency of ourmeasurements was 150. These results were explained as solutions of the Landau-Lifshitz-Gilbertequation, taking into account that the easy magnetization direction of the Co films was along thewire axis. Further measurements with an axial external magnetic field were consistent with thisview. In contrast, for the Fe films, significant values of the circular magnetic susceptibility weremeasured only in the presence of external magnetic fields applied along the wire axis. This resultindicated that the easy magnetization direction of the Fe films was not along the Cu wire axis. Themagnetic anisotropy of both Co and Fe cylindrical films might originate from magnetoelasticeffects, resulted from built-in stress. The different behaviors of the Co and Fe cylindrical filmswould then be attributable to the different signs of their magnetostriction constants /H9261 s. ©2007 American Institute of Physics ./H20851DOI: 10.1063/1.2432403 /H20852 I. INTRODUCTION Recently soft magnetic films have been used in magnetic inductors with applications in devices working at ultrahighfrequencies /H20849uhf /H20850. Planar solenoids and sandwich strips have been developed to improve the quality factor and frequencyresponse of integrated magnetic inductors. In addition, softmagnetic films have been tested for their use as magneticcores in the former planar devices. 1–3In these tests, planar deposition geometry has routinely been used. Also worth investigating are the magnetic properties of cylindrical soft magnetic films deposited on wires. Magneticflux closure may be achieved in particular situations in whichthe magnetization lies along the circular direction. In such a situation, because of the absence of magnetic poles, the re-sponse of the films to circular magnetic fields will increase ina similar way to that reported for ferromagnet/metal/ferromagnet trilayers fabricated with lateral magneticflanges, cf. Refs. 2and4. This cylindrical geometry is also suitable for measuring the magnetoelastic properties of the deposits, particularly themagnetostriction constant of the films. This magnitude wasdetermined by measuring the inverse Wiedemann effect/H20849IWE /H20850in cylindrical deposits. 5,6In addition, IWE measure- ment also enables determination of the thickness of cylindri-cal films to an accuracy of 0.3 nm. 7 The cylindrical geometry of our deposits also permitted a good estimation of the high frequency magnetic suscepti-bility of the films as detailed in this work. To perform theformer measurements, a coaxial transmission line wasfabricated. 8–11Losses in the coaxial waveguide, less than0.1 dB at 6 GHz, were considerably smaller than those mea- sured in planar waveguides /H20849strip loop /H20850used to measure pla- nar deposits.12 Previous results on samples with a cylindrical geometry have also been reported in the literature. These reports de-tailed the magnetic properties of soft magnetic wires /H20849with typical diameters of 125 /H9262m/H20850and microwires /H20849with typical diameters of 8 /H9262m/H20850, mainly focusing on the giant magne- toimpedance /H20849GMI /H20850effect they measured. To compare these results with the results of our work, we concentrated on re-ports in which the measurements were performed in wires ormicrowires at ultrahigh frequencies. It has been reported thatthe GMI effect is equivalent to the ferromagneticresonance, 13and that therefore, the circular magnetic suscep- tibility of the samples could be deduced from impedancemeasurements in the uhf regime. 14The longitudinal uhf mag- netic susceptibility has also been measured both for soft Co-based magnetic wires with a small negative magnetostrictioncoefficient and for Fe-based wires with a positive magneto-striction coefficient 15to frequencies up to 10 GHz.16In all these reports, the thickness of the soft wires and microwiresexceeds the value of the penetration length /H20849roughly around 1 /H9262m at 1 GHz /H20850. To compare the theoretical predictions with the experimental values of the GMI or the magnetic suscep-tibility, the frequency dependence of the penetration depthmust be taken into account within the model. Yet, a discrep-ancy between the theoretical and experimental values of the permeability was found at frequencies below 3 GHz, 17as- suming a model of uniform magnetization within the wholesample volume. On the other hand, in the frequency range ofmegahertz, far below the ferromagnetic resonance /H20849FMR /H20850 frequencies, simulated results for the GMI were in agreementwith the experimental results. 18 a/H20850Electronic mail: jvergara@unavarra.esJOURNAL OF APPLIED PHYSICS 101, 033907 /H208492007 /H20850 0021-8979/2007/101 /H208493/H20850/033907/6/$23.00 © 2007 American Institute of Physics 101 , 033907-1 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.115.103.99 On: Fri, 22 Aug 2014 03:59:02In this work we present the measurements of the uhf circular magnetic susceptibility of cylindrical Co and Fe de-posits fabricated by pulsed laser ablation deposition /H20849PLAD /H20850. Our deposits were roughly 30 nm thick, much smaller thanthe penetration depth of the electromagnetic waves in thefrequency range of our measurements; thus the analysis ofour experimental data was less complicated than the analysisof soft magnetic wires. Previous studies on planar samples revealed that PLAD Co films exhibited good characteristics for their applicationsat ultrahigh frequencies. These samples exhibited an amor-phous structure related to their particular magneticproperties. 6,7The saturation magnetization of the PLAD Co films was roughly 1.4 T and they exhibited soft magneticproperties. The coercivity was 1 Oe. In addition, in the as-deposited state, the room-temperature resistivity was100 /H9262/H9024cm, which was one order of magnitude larger than the value measured in bulk crystalline Co.12,19Deposition of the Co films at an oblique angle of incidence allowed obser-vation of an easy magnetization direction perpendicular tothe incidence plane. The anisotropy field could be changedfrom 10 to 700 Oe by changing the deposition angle. 20 The structure of the PLAD Fe films was also either nanocrystalline or amorphous in a way similar to that foundfor Co. This particular nanostructure of the PLAD Fe filmswas related to their soft magnetic properties. The coercivefield of planar films was roughly 5 Oe for PLAD Fe sampleswith a typical thickness of 40 nm. The value of the saturationmagnetization of the PLAD Fe films was 2 T, which wasvery close to the value measured in bulk crystalline Fe. Inplanar PLAD Fe films, an in-plane easy magnetization direc-tion was also induced in the samples by oblique deposition.Changing the deposition angle could vary the value of theanisotropy fields from 15 to 200 Oe. In addition, a relativelyhigh resistivity was measured at room temperature in theas-deposited Fe films /H20849also on the order of 100 /H9262/H9024cm/H20850. These results of planar PLAD Fe films will be reported else-where. All of these properties make both the Co and the FePLAD films good candidates for use in the uhf range. Thus,the purpose of this work is to investigate the magnetic prop-erties of such cylindrical films in the gigahertz range. II. EXPERIMENT The PLAD technique was used to fabricate pure Co and pure Fe cylindrical films. The light beam from a pulsedNd:YAG laser /H20849YAG denotes yttrium aluminum garnet; /H9261 =1064 nm, 20 Hz repetition rate, and a pulse duration of5n s /H20850was focused on the surface of the targets. The average energy during the pulses on the target, per unit time and area,was 0.2 GW/cm 2. The base pressure in the deposition cham- ber /H20849Neocera /H20850was 10−6mbar. Cu wires, 250 /H9262m in diameter, were used as substrates. The substrates were rotated alongthe wire axis during the deposition process to obtain homo-geneous cylindrical deposits. The rotation speed was 20 rpm,and the distance between the target and the substrate was75 mm. The deposition time was 60 min for each film. The final thickness was 30 nm for the Co films and 35 nm for the Fefilms. The thickness values of the films were not obtained by direct measurements but from the ratio of the magnetic mo-ment of the samples to the value of the saturation magneti-zation of the Co and Fe films The saturation magnetizationwas determined by measuring the Hall effect of the films, 21 and the magnetic moment was measured with a vibratingsample magnetometer /H20849VSM, EG&G /H20850. The complex magnetic susceptibility of the samples was measured with a Hewlett-Packard 8753 network analyzer ina frequency range from 30 kHz to 6 GHz and also with anAgilent 8712ET network analyzer in the range from300 kHz to 1.3 GHz. In both frequency ranges, we used acoaxial line to measure the circular susceptibility of thesamples. As indicated above, the 250 /H9262m diameter Cu sub- strate, with the magnetic film deposited on it, was used as theinner conductor of the coaxial line. The outer conductor ofthe transmission line was a brass tube with an inner diameterof 1 mm and an outer diameter of 1.4 mm /H20849cf. Fig. 1/H20850.A Teflon® tube was fitted between the inner and outer conduc-tors. The former geometry was chosen in order to adjust thevalue of the impedance of the coaxial line /H20849Z 0/H20850to 50 /H9024.22 With this coaxial geometry, it was expected that only the TEM mode of the electromagnetic wave would propagate inthe transmission line. Thus, in the line, the uhf electric fieldwas radial while the uhf magnetic field was circular. Both ends of the coaxial transmission line were con- nected to SubMiniature version A /H20849SMA /H20850terminals. The transmission line was short circuited at one of its ends, andthe other end was connected to the network analyzer. For zeroing, we first measured the reflection coefficient of the transmission line, in the presence of a dc magneticfield, perpendicular to the transmission line. This constructinitiated a response of the PLAD films to the uhf magnetic field in the line applied along the circular direction, whichwe have considered to be negligible. Thus, in this case, themeasured reflection coefficient was taken as the reference ofa null magnetic susceptibility. The reflection coefficient ofthe line was then measured after the removal of the trans-verse dc magnetic field. Then we were able to use the estab-lished zero signal reference as the context for measuring thecylindrical film response to the circular ac magnetic field. Adc magnetic field was subsequently applied in a directionparallel to the axis of the transmission line. The propagation constant of the electromagnetic wave in the coaxial line was estimated from the measurements of thereflection coefficient. The magnetic susceptibility of the cy- FIG. 1. Drawing of the geometry of the PLAD cylindrical samples and coaxial waveguide.033907-2 Vergara, Favieres, and Madurga J. Appl. Phys. 101 , 033907 /H208492007 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.115.103.99 On: Fri, 22 Aug 2014 03:59:02lindrical films was then calculated from the values of the propagation constant, following the procedure detailed inRef. 23. To increase the value of the measured signals, we used samples for the susceptibility measurements that consisted offour portions of cylindrical deposit. The length of each por-tion was 8 mm, and the separation between them, to allowassessment of a segment of Cu substrate without deposit,was 4 mm; thus, the total length was 52 mm including aninitial and final segments of Cu wire for the contacts with theSMA connectors /H20849Fig.1/H20850. These samples were deposited on a single Cu wire by the use of shadow masks. An 8 mm longportion of the cylindrical deposit on the Cu wire was alsoused for the VSM measurements. III. EXPERIMENTAL RESULTS The characterization of the magnetic properties of our samples was done both in dc and in the range of uhf. In dc,in the cylindrical geometry, PLAD Co films exhibited aneasy magnetization direction along the wire axis according tothe room-temperature magnetic hysteresis loops measured byVSM /H20849cf. Fig. 2/H20850. The anisotropy field was on the order of 80 Oe according to the former hysteresis loops. In the uhf range, Fig. 3shows the results of the measure- ments of the real and imaginary parts of the circular mag-netic susceptibility of the Co cylindrical films as a functionof the frequency. In addition, the figure shows results fordifferent values of the applied dc longitudinal magnetic fieldB. The curves plotted in this figure correspond to typical resonance phenomena in which the value of the FMR fre-quency increased with increasing values of the applied dclongitudinal magnetic field. On the other hand, the value ofthe real part of the circular magnetic susceptibility at thelowest frequency of our measurements /H208490.2 GHz /H20850fell from 150 to 85 with an increase in value to 50 Oe of the applied dc longitudinal magnetic field. PLAD cylindrical Fe films also exhibited anisotropic be- havior. Room-temperature magnetization hysteresis loopsmeasured by VSM along the longitudinal and transverse di-rections /H20849parallel and perpendicular to the axis of the cylin- der respectively /H20850revealed this anisotropy /H20849cf. Fig. 4/H20850. In thiscase, the value of the anisotropy field H kwas 40 Oe. How- ever, the longitudinal direction /H20849parallel to the wire axis /H20850was not the easy magnetization direction, in contrast to what wasfound in PLAD Co cylindrical films, instead the easy direc-tion was the circular direction or another one close to it. The complex circular susceptibility of the PLAD cylin- drical Fe films was also determined following a proceduresimilar to that used for the Co film. The values of the realand imaginary parts of the circular susceptibility are plottedin Fig. 5as a function of the frequency of the ac magnetic field in the coaxial line and for different values of the applieddc longitudinal magnetic field. FIG. 2. /H20849Color online /H20850Room-temperature magnetization curves of the PLAD Co cylindrical film along the easy /H20849longitudinal /H20850and hard /H20849trans- verse /H20850magnetization directions. Dashed lines are guides to the eyes. FIG. 3. /H20849Color online /H20850Real /H20849a/H20850and imaginary /H20849b/H20850parts of the circular magnetic susceptibility of the cylindrical Co films for different values of theapplied longitudinal dc magnetic field. FIG. 4. /H20849Color online /H20850Room-temperature magnetization curves of the PLAD Fe cylindrical film along the hard /H20849longitudinal /H20850and easy /H20849transverse /H20850 magnetization directions. Dashed lines are guides to the eyes.033907-3 Vergara, Favieres, and Madurga J. Appl. Phys. 101 , 033907 /H208492007 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.115.103.99 On: Fri, 22 Aug 2014 03:59:02Resonance processes were also observed in the measure- ments of the circular magnetic susceptibility. The value ofthe FMR frequency increased with an increasing value of theapplied dc magnetic field; however, the value of the real partof the magnetic susceptibility at the lowest frequencies ofour experiments presented a maximum when the appliedmagnetic field was on the order of 40 Oe. This result differedfrom that found for the Co film, in which the circular sus-ceptibility at the lowest frequencies of our measurements de-creased with an increasing value of the longitudinal dc mag-netic field /H20849cf. Fig. 3/H20850. IV. DISCUSSION The results of the uhf circular magnetic susceptibility of our PLAD cylindrical Co film have been analyzed as solu-tions of the Landau-Lifshitz-Gilbert /H20849LLG /H20850equation. We have assumed that our Co film consisted of a single magneticdomain whose magnetic moment remained parallel to theeasy magnetization axis in this case, the axis of the Cu sub-strate, according to the VSM results shown in Fig. 2. Thus, the magnetic moments of the single magnetic domain of ourCo sample precessed coherently under the influence of theaxial dc field and the applied uhf circular magnetic field. Inaddition, because the thickness of our PLAD films is fourorders of magnitude smaller than the radius of the wirewhere they were deposited, for computational effects wehave considered our cylindrical films like planes. Conse-quently, we have fitted the experimental values of the real and imaginary parts of the circular magnetic susceptibility ofthe Co cylindrical films at frequencies up to 6 GHz to the theLLG equation solutions for planar films with an easy mag-netization axis perpendicular to the direction of the uhf mag-netic field. 24In this computation, the fitting parameters were the damping coefficient /H9251and the FMR frequency /H9275r. The value of the saturation magnetization /H20849/H92620Ms/H20850of the Co films was 1.4 T. Thus, Fig. 6displays the result of the former fit when the applied longitudinal dc magnetic field was 12 Oe.A good agreement between experimental and theoretical re-sults was found at frequencies above 1 GHz. To confirm the validity of the present approach, we plot- ted in Fig. 7the experimental values of the FMR frequencies of the Co film for different values of the external magneticfield applied along the axis of the cylinder: these values wereobtained from Fig. 3. We have fitted the former values to theoretical ones obtained from the solutions of the LLGequation for films. 24The only fitting parameter in this case was the value of the anisotropy field Hk. The value of Hkof the best fit was 90 Oe, which was similar to the experimentalvalue found from VSM measurements as displayed in Fig. 2 /H20849roughly 80 Oe /H20850. Thus, from these results, it is reasonable to conclude that the solutions of the LLG equation for films with a uniaxialmagnetic anisotropy provided an accurate description of themagnetization dynamics of the Co cylindrical films at fre-quencies above 1 GHz. In the low frequency range, somediscrepancies arose between the experimental and theoreticalvalues of the magnetic susceptibility, which could be theresult of a nonuniform magnetization of our samples, similarto that found for soft magnetic wires. 17 In spite of this quantitative disagreement at frequencies below 1 GHz, we did find a qualitative agreement betweenthe experimental and theoretical values of the circular mag-netic susceptibility at the lowest frequency of our experi-ments. According to Fig. 3/H20849a/H20850at frequencies below 1 GHz, the experimental values of the circular magnetic susceptibil-ity of the Co films decreased with application of an externaldc longitudinal magnetic field. On the other hand, at low FIG. 5. /H20849Color online /H20850Real /H20849a/H20850and imaginary /H20849b/H20850parts of the circular magnetic susceptibility of the cylindrical Fe films for different values of theapplied longitudinal dc magnetic field. FIG. 6. /H20849Color online /H20850Real /H20849a/H20850and imaginary /H20849b/H20850parts of the circular magnetic susceptibility of the cylindrical Co films in an applied longitudinaldc magnetic field of 12 Oe. Solid lines are the fits to the theoretical valuesobtained from Ref. 24. The values of the fitting parameters /H20849resonance fre- quency /H9263rand damping coefficient /H9251/H20850are explicitly indicated.033907-4 Vergara, Favieres, and Madurga J. Appl. Phys. 101 , 033907 /H208492007 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.115.103.99 On: Fri, 22 Aug 2014 03:59:02frequencies, the magnetic susceptibility calculated from the solution of the LLG equation was inversely proportional tothe sum of the anisotropy field and the externally appliedmagnetic field. 25Thus with an increasing value of the axial external magnetic field, the theoretical value of the magneticsusceptibility is expected to decrease too. For the PLAD cylindrical Fe films both the VSM and the uhf magnetic susceptibility measurements indicate that therewas no uniaxial anisotropy along the wire axis, unlike theoutcome for Co. Thus, the uhf magnetic susceptibility cannotbe described as a solution of the LLG equation. We did find, however, a qualitative agreement between the experimental measurements and the solutions of the LLGequation for films with a uniaxial magnetic anisotropy in adirection parallel to that of the applied uhf magnetic field.Particularly, the theoretical value of the real part of the mag-netic susceptibility at relatively low frequencies was ex-pected to be at maximum in a bias longitudinal magneticfield close to the value of the anisotropy field H k.26Thus, Fig.5/H20849a/H20850indicates that this maximum value of the real part of the magnetic susceptibility was achieved in an appliedfield of 40 Oe, similar to the value of H kdetermined by VSM measurements. On the other hand, the dependence of the FMR fre- quency of the Fe cylindrical films on the dc longitudinallyapplied magnetic field also qualitatively agreed with the so-lutions of the LLG equation for films, with uniaxial magneticanisotropy almost parallel to the direction of the uhf mag-netic field. These results suggested that a uniaxial magneticanisotropy along the circular direction or close to it would bepresent in the cylindrical Fe films. To explain these differences in the magnetic anisotropy between Co and Fe films, the different possible origins of themagnetic anisotropy were considered. The magnetic aniso-tropy of our films might be attributable to /H208491/H20850shape effects, /H208492/H20850to the particular crystalline structure of the films, or fi- nally /H208493/H20850to magnetoelastic effects. In our case, the geometry of our PLAD Co and Fe films was similar. The small differ- ence in the thickness of the deposits /H2084930 nm for Co films and 35 nm for Fe films /H20850was not sufficient to explain the different magnetic anisotropy observed in our Fe and Co films. Be-cause the structure of both the Co and Fe films was either amorphous or nanocrystalline according to the x-ray diffrac-tion measurement, 12,27a magnetocrystalline anisotropy also was not a viable explanation. Thus, the origin of the ob-served magnetic anisotropy of our PLAD Co and Fe filmscan probably be attributed to magnetoelastic effects. Mea-surements of the IWE showed that the magnetostriction con-stant /H20849/H9261 s/H20850was negative for Co films and positive for Fe films.27We concluded that this difference in the sign of the measured value of /H9261sfor the PLAD Co and Fe films could be responsible for their difference in the magnetic anisotropy. As a result of the deposition process, residual stresses could be present in the ferromagnetic films, giving rise tostrained films. It seemed reasonable to assume that a similardistribution of stresses /H9268/H20849probably, in this case, positive ra- dial stresses giving rise to an effective circular traction /H20850 would affect both Co and Fe films. This characteristic wouldfavor the magnetization of the Co film to point in the longi-tudinal direction /H20849parallel to the wire axis /H20850because of the negative value of its magnetostriction constant. In contrast,the magnetization of the Fe films would tend to lie close tothe circular direction because of both the presence of residualstresses and the positive sign of the magnetostriction con-stant of Fe films. The magnetic anisotropy of the Fe films could be mini- mized by applying a longitudinal stress in order to createalso in the cylindrical Fe films an easy magnetization direc-tion along the wire axis. In this situation and as a result ofthe high saturation magnetization of the Fe films, a highmagnetic susceptibility at uhf would be observed in the Fefilms. The induction of a longitudinal anisotropy in the cy-lindrical Fe films will be the focus of future work. V. CONCLUSIONS Cylindrical Co and Fe films have been fabricated by the pulsed laser ablation deposition technique. These cylindricalfilms exhibited soft magnetic properties as demonstratedfrom the measurement of room-temperature magnetic hyster-esis loops and good behavior in the uhf range. The measuredcircular magnetic susceptibility was relatively large up to theFMR frequency, roughly around 3 GHz. The value of theFMR frequency could be tailored by applying a longitudinaldc magnetic field. These characteristics made the films suit-able for applications in devices working at uhf. Nevertheless,Co and Fe cylindrical films showed significant differences:for Co films the low frequency value of the real part of thecircular magnetic susceptibility decreased with application ofa dc longitudinal magnetic field. In contrast, for the PLADFe cylindrical films, the circular susceptibility measured atthe lowest frequencies of our experiments showed a maxi-mum at 40 Oe. The uhf circular magnetic susceptibilities of the Co and Fe cylindrical films were analyzed as solutions to theLandau-Lifshitz-Gilbert equation. Both the ac and the dcmeasurements revealed that the orientation of the easy mag-netization direction of the Co film /H20849parallel to the axis of the cylindrical Cu substrate /H20850differed from that observed in the Fe films /H20849close to the circular direction /H20850. For the Co cylindri- FIG. 7. /H20849Color online /H20850FMR frequencies of the Co cylindrical film in the presence of a dc longitudinal magnetic field. The solid line indicates thetheoretical value.033907-5 Vergara, Favieres, and Madurga J. Appl. Phys. 101 , 033907 /H208492007 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.115.103.99 On: Fri, 22 Aug 2014 03:59:02cal films a good agreement between the experimental results and the theoretical model from the LLG equation was foundabove 1 GHz. For the Fe films, we found only a qualitativeagreement between the predictions of the LLG model andboth the experimental values of the susceptibility at the low-est frequencies of our measurements and the values of theFMR frequencies. For our Fe cylindrical films the easy mag-netization direction was not perpendicular to the direction ofthe applied uhf magnetic field. This difference in the magnetic anisotropy observed be- tween Co and Fe films was explained as a consequence ofmagnetoelastic effects and as a result of the fact that themagnetostriction constant of the Co films was negative, incontrast to the positive value of the Fe films. ACKNOWLEDGMENT The authors acknowledge Professor Mario Sorolla /H20849Uni- versidad Pública de Navarra /H20850for providing one of the net- work analyzers, the HP 8753 one, and for help with the uhfmeasurements. 1Y. Hayakawa, A. Makino, H. Fujimori, and A. Inoue, J. Appl. Phys. 81, 3747 /H208491997 /H20850. 2V. Korenivski and R. B. van Dover, J. Appl. Phys. 82, 5247 /H208491997 /H20850. 3T. Sato, Y. Miura, S. Matsumura, K. Yamasawa, S. Morita, Y. Sasaki, T. Hatanai, and A. Makino, J. Appl. Phys. 83, 6658 /H208491998 /H20850. 4A. Sukstanskii, V. Korenivski, and A. Gromov, J. Appl. Phys. 89, 775 /H208492001 /H20850. 5C. Favieres and V. Madurga, J. Non-Cryst. Solids 287, 390 /H208492001 /H20850. 6C. Favieres and V. Madurga, J. Phys.: Condens. Matter 16, 4725 /H208492004 /H20850. 7C. Favieres and V. Madurga, J. Appl. Phys. 96, 1850 /H208492004 /H20850. 8T. P. Kehler and R. L. Coren, J. Appl. Phys. 41, 1346 /H208491970 /H20850.9T. P. Kehler and R. L. Coren, J. Appl. Phys. 42, 1433 /H208491971 /H20850. 10O. Acher, J. L. Vermeulen, P. M. Jacquart, J. M. Fontaine, and P. Baclet, J. Magn. Magn. Mater. 136, 269 /H208491994 /H20850. 11A.-L. Adenot, O. Acher, D. Pain, F. Duverger, M.-J. Malliavin, D. Dami- ani, and T. Taffary, J. Appl. Phys. 87, 5965 /H208492000 /H20850. 12V. Madurga, J. Vergara, and C. Favieres, Mater. Res. Soc. Symp. Proc. 674, T3.2 /H208492001 /H20850. 13A. Yelon, D. Ménard, M. Britel, and P. Ciureanu, Appl. Phys. Lett. 69, 3084 /H208491996 /H20850. 14L. G. C. Melo, P. Ciureanu, and A. Yelon, J. Magn. Magn. Mater. 249,3 3 7 /H208492002 /H20850. 15O. Acher, P. M. Jacquart, and C. C. Boscher, IEEE Trans. Magn. 30,4 5 4 2 /H208491994 /H20850. 16O. Acher, A. L. Adenot, and S. Deprot, J. Magn. Magn. Mater. 249,2 6 4 /H208492002 /H20850. 17A. Yelon, L. G. C. Melo, P. Ciureanu, and D. Ménard, J. Magn. Magn. Mater. 249, 257 /H208492002 /H20850. 18J. L. Muñoz, J. M. Barandiarán, G. V. Kurlyandskaya, and A. García- Arribas, J. Magn. Magn. Mater. 249,3 1 9 /H208492002 /H20850. 19V. Madurga, J. Vergara, R. J. Ortega, I. P. de Landazábal, and C. Favieres, Mater. Res. Soc. Symp. Proc. 562, 283 /H208491999 /H20850. 20V. Madurga, J. Vergara, and C. Favieres, J. Magn. Magn. Mater. 272–276 , 1681 /H208492004 /H20850. 21E. D. Dahlberg, K. Riggs, and G. A. Prinz, J. Appl. Phys. 63, 4270 /H208491988 /H20850. 22D. M. Pozar, Microwave Engineering /H20849Addison-Wesley, Reading, 1990 /H20850, p. 166. 23V. Bekker, K. Seemann, and H. Leiste, J. Magn. Magn. Mater. 270,3 2 7 /H208492004 /H20850. 24J. Ben Youssef, P. M. Jacquart, N. Vukadinovic, and H. Le Gall, IEEE Trans. Magn. 38, 3141 /H208492002 /H20850. 25D. Pain, M. Ledieu, O. Acher, A. L. Adenot, and F. Duverger, J. Appl. Phys. 85, 5151 /H208491999 /H20850. 26B. A. Belyaev, A. V. Izotov, and S. Ya. Kiparisov, JETP Lett. 74,2 2 6 /H208492001 /H20850. 27V. Madurga, C. Favieres, and J. Vergara, 49th MMM Conference, Jack- sonville, Florida /H20849unpublished /H20850; V. Madurga, C. Favieres, and J. Vergara, J. Non-Cryst. Solids /H20849in press /H20850.033907-6 Vergara, Favieres, and Madurga J. Appl. Phys. 101 , 033907 /H208492007 /H20850 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.115.103.99 On: Fri, 22 Aug 2014 03:59:02

5.0016340.pdf

Appl. Phys. Lett. 117, 082407 (2020); https://doi.org/10.1063/5.0016340 117, 082407 © 2020 Author(s).Ultrafast coherent control of higher-order spin waves in a NiFe thin film by double- pulse excitation Cite as: Appl. Phys. Lett. 117, 082407 (2020); https://doi.org/10.1063/5.0016340 Submitted: 04 June 2020 . Accepted: 14 August 2020 . Published Online: 27 August 2020 Makoto Okano , Tomohiro Takahashi , and Shinichi Watanabe ARTICLES YOU MAY BE INTERESTED IN Enhancement of the spin–orbit torque efficiency in W/Cu/CoFeB heterostructures via interface engineering Applied Physics Letters 117, 082409 (2020); https://doi.org/10.1063/5.0015557 Concurrent magneto-optical imaging and magneto-transport readout of electrical switching of insulating antiferromagnetic thin films Applied Physics Letters 117, 082401 (2020); https://doi.org/10.1063/5.0011852 Skyrmion Brownian circuit implemented in continuous ferromagnetic thin film Applied Physics Letters 117, 082402 (2020); https://doi.org/10.1063/5.0011105Ultrafast coherent control of higher-order spin waves in a NiFe thin film by double-pulse excitation Cite as: Appl. Phys. Lett. 117, 082407 (2020); doi: 10.1063/5.0016340 Submitted: 4 June 2020 .Accepted: 14 August 2020 . Published Online: 27 August 2020 Makoto Okano,a) Tomohiro Takahashi, and Shinichi Watanabea) AFFILIATIONS Department of Physics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, Kanagawa 223-8522, Japan a)Authors to whom correspondence should be addressed: okano@phys.keio.ac.jp and watanabe@phys.keio.ac.jp ABSTRACT By using a double-pulse excitation scheme, we demonstrate the optical control of higher-order spin waves in a ferromagnetic permalloy thin film. Three spin precession modes are observed in the film under single pulse excitation. Based on a theoretical calculation, these spin preces- sion modes are assigned to the fundamental Kittel mode and the first- and second-order perpendicular standing spin-wave (PSSW) modes. In order to excite the first-order PSSW mode selectively, we use double-pulse excitation at 400 nm. We show that, by choosing an appropriatetime interval between the two pump pulses, it is possible to obtain a relatively large amplitude of the first-order PSSW mode while theamplitudes of the other two modes are almost completely suppressed. By analyzing the spin dynamics, it is found that the spin precessionmotion under the double-pulse excitation condition can be explained by the interference between the spin waves that have been induced by the first and second pulses. Our finding indicates that the selective excitation of a spin precession mode with a high precession frequency in a ferromagnetic metal may be realized by a multiple-pump pulse excitation scheme with more than two pulses. Published under license by AIP Publishing. https://doi.org/10.1063/5.0016340 The use of spin waves has several advantages over the use of elec- trical charges in conventional electronic devices, and thus the develop-ment of the so-called spintronic and magnonic devices has attracted considerable interest. 1–3One of the most prominent characteristics of magnonic devices is a very low energy dissipation; the propagation ofa spin wave is not accompanied by Joule heating. 1Because the coher- ent and collective spin precession motions play a key role in spintronic and magnonic devices, the manipulation of spin dynamics is still a hot topic in this research field. In particular, the optical control of spindynamics with ultrashort pulses is considered to be a good solution forultrafast spin-based devices. Therefore, numerous studies regardingthe optical control of spin dynamics have been performed on various magnetic materials. 4–21Ferromagnetic metals are one of the most sig- nificant materials in terms of the development of spin-based devices,because these metals have several advantages over antiferromagneticmaterials. For example, their microfabrication is easier and their cost is lower. However, the frequencies of the fundamental spin precession modes in ferromagnetic metals are lower than those in antiferromag-netic materials. 4,6–10,12,13,15–21Because a high spin precession fre- quency is needed for high-speed spintronic devices, it is necessary tocircumvent this disadvantage. One possible way is the excitation of a higher-order spin-wave mode, such as the perpendicular stand-ing spin-wave (PSSW) mode. 21–29However, the utilization of spin precession motions with higher frequencies is still difficult, becausethe fundamental and the higher-order modes are excited simulta-neously by a single optical pulse and the signal of the former modeis larger than that of the latter modes. Thus, for the effective utili- zation of a spin precession motion with a higher precession fre- quency, the selective optical excitation of the higher-order mode isrequired. In various magnetic materials, it has been shown that optical double-pulse excitation can be used to selectively generate certain spin precession motions. 5–9,16,19In the antiferromagnetic material YFeO 3, the selective excitation of spin precession mode with a frequencyhigher than the fundamental mode has been achieved. 8On the other hand, such a selective excitation of a higher-order mode in a ferromag-netic metal by femtosecond laser pulses has not yet been reported.Recently, we found that the double-pulse excitation is useful for themanipulation of the spin precession motion in a ferromagneticNi 0.8Fe0.2(Py) thin film.19Therefore, this excitation scheme may be Appl. Phys. Lett. 117, 082407 (2020); doi: 10.1063/5.0016340 117, 082407-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/apluseful for the selective excitation of a spin precession mode with a high precession frequency in ferromagnetic metals. Here, we report on the ultrafast coherent control of higher-order spin waves in a Py thin film by double-pulse excitation. Using single- pulse excitation, we first identify three spin precession modes, i.e., the fundamental Kittel mode and two PSSW modes, in the time-resolved magneto-optical Kerr effect (TR-MOKE) signals. Then, by using double-pulse excitation, it is found that the amplitudes of these spin precession modes strongly depend on the time when the second pump pulse arrives. We show that it is possible to select only the first-order PSSW mode by choosing an appropriate time delay between the two pulses. To understand the mechanism of this selective excitation, the dependence of the TR-MOKE signals on the time delay is evaluated by a fitting procedure. The obtained characteristic oscillations of the amplitudes of the spin precession modes with respect to the time delay can be interpreted in terms of the interference between the spin wavesthat have been induced by the first and second pulses. The results evi- dence that the ultrafast, coherent, selective excitation of a spin preces- sion mode with a higher frequency than that of the Kittel mode can be realized in a ferromagnetic metal by multiple optical pump pulses. The Py film was deposited on a silicon substrate by electron beam evaporation. To easily excite the spin precession modes with higher frequencies by optical pumping, we chose a film thickness ofapproximately 70 nm. 22,23For the observation of the temporal profiles o ft h es p i np r e c e s s i o nm o t i o n si nt h eP yt h i nfi l m ,w ep e r f o r m e dT R - MOKE measurements in the polar Kerr configuration. Here, the change in the polarization state of the probe light reflected from the sample surface is proportional to the change in the magnetization component normal to the surface.30As a light source for the TR- MOKE measurements, we used a Ti:Sapphire pulsed laser with a repe- tition rate of 80 MHz and a pulse duration of 90 fs. The experimental setup was based on the typical pump-probe configuration. The laser output beam at a wavelength of 800 nm was divided into the pump and probe paths by a polarization beam splitter (BS). The pump pulse was focused on a barium borate crystal for frequency doubling, and then the 400-nm pump beam was divided by a BS to generate twopump pulses. By using an optical delay line, the second pump pulse was delayed by Dt>0. A third BS was used to align the first and sec- ond pump pulses coaxially. Using an objective lens with a numerical aperture of 0.8, the pump and probe pulses were tightly focused on the sample surface. The beam diameters of the pump and probe pulses were approximately 3.1 and 1.6 lm, respectively, as determined by the knife-edge method. Using these beam diameters, we evaluated excita- tion fluences of /C250.09 and 0.28 mJ/cm 2for the two pump pulses and the probe pulse, respectively. In order to improve the signal-to-noise ratio, the fluence of the probe pulse was set to be larger than those of the pump pulses within the linear response regime. We used a perma- nent magnet to apply a magnetic field Hextequal to 4.2 kOe at the sam- ple position as estimated by a Gauss meter. The field direction was slightly tilted with respect to the sample surface normal. More details of the experimental system are described elsewhere.19All measure- ments were carried out at room temperature. We first measured the TR-MOKE signal of the Py film under the single-pulse excitation condition, which allows us to understand the fundamental properties of the optically induced spin precession motions. Figure 1(a) shows the results, where the time tcorresponds to the probe delay time. At t¼0, a large sudden change in theTR-MOKE signal appears. This change is attributed to the thermally induced ultrafast demagnetization.31Fort>0, the TR-MOKE signal shows a damped oscillatory behavior. The oscillation reflects thechange in the magnetization component normal to the sample surfacedue to the spin precession motions. 22Therefore, we interpret the com- plicated temporal evolution of the TR-MOKE signal in Fig. 1(a) as a simultaneous excitation of multiple spin precession modes by theultrashort optical pulse. To identify the spin precession modes inthe frequency domain, we calculated the Fourier transform of theTR-MOKE signal within the time range from 150 to 650 ps. This time range was chosen to exclude the sudden change around the time ori- gin. The squared amplitude of the Fourier transform is plotted as afunction of frequency in Fig. 1(b) . Peak structures can be clearly observed at around 10, 18, and 27 GHz. This indicates that althoughmany higher spin precession modes would be simultaneously excited by the ultrafast optical pulse, three spin precession modes were only observed with sufficient amplitudes. To further understand the magnetization dynamics that occur in the Py thin film after single-pulse excitation, we evaluated the oscilla-tion parameters of each spin precession mode by fitting the function 11 IKerrtðÞ¼X2 n¼0Ancos 2 pfntþ/n ðÞ exp/C0t sn/C18/C19 þBexp/C0t sB/C18/C19 (1) to the experimental curve. Here An;fn;/n;andsnare the ampli- tude, the precession frequency, the initial phase, and the decay time of the spin precession mode with mode number n. Here, we d e fi n e dt h em o d en u m b e r nin ascending order. BandsBare the amplitude and the decay time of the thermal background. In thefitting procedure, we only utilized the TR-MOKE signal fromt¼150 to 650 ps. The TR-MOKE signal is well reproduced by theFIG. 1. (a) Probe delay time dependence of the TR-MOKE signal of the Py thin film under single-pulse excitation. The black solid curve represents the fitting result. (b) Frequency spectrum showing the squared amplitude of Fourier transform of the time-domain signal in (a). The black arrows indicate the peak frequencies of thespin precession modes. (c) Mode number dependence of the precession frequency.The uncertainty is smaller than the plotted symbols. The fitting result is shown with the solid curve.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 082407 (2020); doi: 10.1063/5.0016340 117, 082407-2 Published under license by AIP Publishingfitting curve. Note that Breflecting the effect of thermal back- ground is negligibly small in this work. The frequencies of the spinprecession modes obtained from the experiment by this method are tabulated in Table I and plotted in Fig. 1(c) . To clarify the origins of the spin precession modes, we calculated the precession frequencies of the nth order PSSWs in our sample. The spin precession frequency of the nth order PSSW mode, f n, can be written as3,32 2pfn l0c/C18/C192 ¼ Hextcosh0/C0b ðÞ /C0Mscos2h0þ2Aex l0Msnp d/C18/C192() /C2Hextcosh0/C0b ðÞ /C0Mscos2h0þ2Aex l0Msnp d/C18/C192() ; (2) where candl0are the gyromagnetic ratio and the permeability in a vacuum, respectively. dis the sample thickness. The variables Msand Aexare the saturation magnetization and the exchange stiffness con- stant of Py. h0andbare the angles of the magnetization and Hextwith respect to the normal direction of the film surface. These two variables satisfy the following equation:32 2Hextsinh0/C0b ðÞ /C0Mssin 2h0¼0: (3) We fitted Eq. (2)to the experimental result in Fig. 1(c) using the fitting parameters bandAex. The other parameters were fixed to the measured values of Hext¼4.2 kOe and d¼70 nm and to the previ- ously reported value of Ms¼11.8 kOe.33The fitting curve shown with the solid curve in Fig. 1(c) reproduces the experimental trend well. In addition, we obtained Aex¼2.6/C210/C011J/m, which is on the same order of magnitude as the exchange stiffness constant reported in Ref.34. We summarized the theoretically calculated precession fre- quencies of the three modes in Table. I . The good agreement between theory and experiment strongly supports the validity of the following assignments: the spin precession mode with the lowest frequency (n¼0) is attributed to the fundamental Kittel mode. The spin preces- sion modes with n¼1 and 2 are assigned to the first- and second- order PSSW modes, which correspond to out-of-plane standing waves. Figure 2 s h o w st h eT R - M O K Es i g n a l st h a tw e r eo b t a i n e db y double-pulse excitation. The probe delay time twas measured relative to the instant of excitation by the first pump pulse. The excitation by the second pump pulse was delayed by Dtwith respect to the first one. To identify the effect of Dton the TR-MOKE signal, we varied Dt from 0 to 133 ps. Note that the period of the Kittel mode is about 100 ps. The colored data points in Fig. 2 evidence that the spin dynamics in the range t>Dtstrongly depend on the value of Dt.In particular, forDt¼53.3 ps, the TR-MOKE signal in the range t>Dtappears toconsist of a single oscillation component. This result implies that the selective excitation of a spin precession mode is possible by double- pulse excitation. To gain insights into the TR-MOKE signals that were induced by double-pulse excitation, each TR-MOKE signal was fitted by Eq. (1). We fitted the TR-MOKE signals in the range from t¼150 to 650 ps to extract the influence of the second pulse on the spin dynamics. Tominimize the uncertainty in the fitting procedure, a global fitting wasperformed with the shared parameters, f nandsn.Figure 2 shows that the fitting curves (black solid curves) reproduce the experimental results well. Therefore, we use the fitting results of the amplitudes forfurther analysis. The amplitudes of the spin precession modes after the excitation by the second pulse are plotted as a function of DtinFig. 3 . The data show an oscillatory behavior with respect to Dt.As the mode number nincreases, the maximum value of the amplitude decreases and the oscillation period becomes shorter. Moreover, at Dt¼53.3 ps, only the first-order PSSW mode (red data) has a relatively large ampli- tude. This corresponds to the selective excitation observed in Fig. 2 for Dt¼53.3 ps. To understand the mechanism of the selective excitation of the first-order PSSW mode, we need to discuss the origin of the character- istic oscillations of the amplitudes observed in Fig. 3 .I nt h ep r e v i o u s work, we found that the TR-MOKE signal due to the Kittel mode under double pulse excitation is well explained by a simple model with the Landau–Lifshitz–Gilbert equation that accounts for the laserheating. 19In the model, the spin precession motion is driven by the laser-induced modification of an effective magnetic field, leading to the precession motion of the magnetization of the Py film. The mostTABLE I. Experimentally obtained and theoretically calculated spin precession fre- quencies for Kittel and PSSW modes in the Py thin film. n Experiment (GHz) Calculation (GHz) Assignment 0 9.9 60.1 11.0 Kittel mode 1 17.5 60.1 15.7 First PSSW 2 26.6 60.3 27.2 Second PSSWFIG. 2. TR-MOKE signals of the Py thin film under double-pulse excitation for differ- ent delay times Dt. The first pump pulse excited the sample at t¼0 ps. Black solid curves represent the fitting results. The signals are vertically offset for clarity.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 082407 (2020); doi: 10.1063/5.0016340 117, 082407-3 Published under license by AIP Publishingessential point of this model is that the direction of magnetization of the Py film, corresponding to the phase of the TR-MOKE signal, at thetime when the second pulse arrives determines the amplitude of theTR-MOKE signal after second pulse irradiation. More details aredescribed in Ref. 19. On the basis of this knowledge that the phase of corresponding oscillation when the second pulse illuminates the sam-ple governs the amplitude of the corresponding spin waves, we con-sider a simple phenomenological model based on the interferencebetween the spin waves that have been induced by the first- andsecond-pump pulses. Since the excitation fluences of the two pump pulses were identi- cal, the TR-MOKE signal after excitation by the second pulse, I 0 KerrtðÞ, can be written as I0 KerrtðÞ¼IKerrtðÞþHt/C0Dt ðÞ IKerrt/C0Dt ðÞ ; (4) whereHxðÞis the Heaviside step function. Because Dtis shorter than the decay times snthat were on the order of several hundreds of pico- seconds, we neglect the influence of the damping term in Eq. (1). Under this assumption, the amplitude of the nth mode after excitation by the second pulse as a function of Dt,Cn,i se x p r e s s e db y CnDtðÞ¼A0 nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2þ2cos 2 pfnDt ðÞp ; (5) where A0 nis the constant that is proportional to Anin Eq. (1).T h es o l i d curves in Fig. 3 represent the fitting results, where we used the spin precession frequencies fnobtained from the data in Fig. 2 .T h eDt dependences in Fig. 3 are well reproduced by the fitting curves. This good agreement between experiment and calculation proves that thewave interference model can be used to interpret the characteristicbehavior of the double-pulse-induced spin dynamics in Py thin films. By using the wave interference model, the selective excitation of the first-order PSSW mode for Dt¼53.3 ps can be interpreted straightforwardly. According to Eq. (5), the local maxima (minima) of a spin precession mode with precession frequency f nare located atDt¼m=2fn,w h e r e mis an even (odd) integer. As described in Table I , the precession frequencies of the first- and second-order PSSW modes are almost two and three times larger than f0(the Kittel mode), respectively. Therefore, the Dt-values of the local minima of the Kitteland second-order PSSW modes are almost the same, and only the first- order PSSW mode is significantly excited when Dt¼53.3 ps. Our finding suggests that the wave interference model can be applied to multiple-pulse excitation techniques including the other excitation schemes.35,36In analogy with Fig. 3 , there may be other sys- tems where it is possible to enhance the amplitude of a certain spin precession mode by choosing the correct time intervals of multiple pump pulses and simultaneously suppress other spin precession motions. In such a case, one can selectively utilize a single mode with a higher precession frequency even if many modes are simultaneously excited by a single optical pulse. In addition, the precession frequency of each spin precession mode can be modified by changing parameters such as sample thickness and external magnetic field22and by chang- ing the excitation schemes.35,36Because the excitation of the higher- order spin precession mode with a frequency over 1 THz in the ferromagnetic-normal metal multilayers has been reported,35,36we expected that it should be possible to realize a ferromagnetic- metal–based spintronic device with a high precession frequency as well as antiferromagnetic materials. In summary, we have demonstrated the selective generation of the first-order PSSW in a ferromagnetic Ni 0.8Fe0.2thin film by double- pulse excitation. By choosing an appropriate time delay between the two pump pulses, the amplitude of the first-order PSSW mode can be enhanced relative to the amplitudes of the Kittel and second-order PSSW modes. The observed selective generation can be simply described by taking into account the interference between the spin waves that have been induced by the first and second pump pulses. Our finding may be extended to the selective generation of spin pre- cession motion with a high precession frequency by three or more pump pulses. We believe that our results will facilitate the develop- ment of ultrafast spintronic devices based on ferromagnetic metal thin films. AUTHORS’ CONTRIBUTIONS M.O. and T.T. contributed equally to this work. This work was partially supported by JSPS KAKENHI Grant No. JP18H02040 and JST CREST Grant No. JPMJCR19J4, Japan. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1S. Maekawa, Concepts in Spin Electronics (Oxford University Press, 2006). 2I./C20Zutic´, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). 3D. D. Stancil and A. Prabhakar, Spin Waves: Theory and Applications (Springer, New York, 2009). 4A. V. Kimel, A. Kirilyuk, P. A. Usachev, R. V. Pisarev, A. M. Balbashov, and T.Rasing, Nature 435, 655 (2005). 5F. Hansteen, A. Kimel, A. Kirilyuk, and Th. Rasing, Phys. Rev. B 73, 014421 (2006). 6T. Satoh, N. P. Duong, and M. Fiebig, Phys. Rev. B 74, 012404 (2006). 7A. V. Kimel, A. Kirilyuk, and T. Rasing, Laser Photonics Rev. 1, 275 (2007). 8K. Yamaguchi, M. Nakajima, and T. Suemoto, Phys. Rev. Lett. 105, 237201 (2010). 9T. Kampfrath, A. Sell, G. Klatt, A. Pashkin, S. M €ahrlein, T. Dekorsy, M. Wolf, M. Fiebig, A. Leitenstorfer, and R. Huber, Nat. Photonics 5, 31 (2011).FIG. 3. The amplitudes of the spin precession modes derived from the fitting of the double-pulse-induced TR-MOKE signals shown in Fig. 2 . The error bars represent the uncertainties of the fitting results obtained using Eq. (1). The solid curves corre- spond to the fitting curves obtained using Eq. (5).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 082407 (2020); doi: 10.1063/5.0016340 117, 082407-4 Published under license by AIP Publishing10N. Kanda, T. Higuchi, H. Shimizu, K. Konishi, K. Yoshioka, and M. Kuwata- Gonokami, Nat. Commn. 2, 362 (2011). 11J. Kisielewski, A. Kirilyuk, A. Stupakiewicz, A. Maziewski, A. Kimel, Th. Rasing, L. T. Baczewski, and A. Wawro, Phys. Rev. B 85, 184429 (2012). 12T. Satoh, Y. Terui, R. Moriya, B. A. Ivanov, K. Ando, E. Saitoh, T. Shimura, and K. Kuroda, Nat. Photonics 6, 662 (2012). 13S. Baierl, M. Hohenleutner, T. Kampfrath, A. K. Zvezdin, A. V. Kimel, R. Huber, and R. V. Mikhaylovskiy, Nat. Photonics 10, 715 (2016). 14J. Stigloher, M. Decker, H. S. K €orner, K. Tanabe, T. Moriyama, T. Taniguchi, H. Hata, M. Madami, G. Gubbiotti, K. Kobayashi, T. Ono, and C. H. Back, Phys. Rev. Lett. 117, 037204 (2016). 15T. Cheng, J. Wu, T. Liu, X. Zou, J. Cai, R. W. Chantrell, and Y. Xu, Phys. Rev. B 93, 064401 (2016). 16I. 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1.1565503.pdf

Tilted media in a perpendicular recording system for high areal density recording Y. Y. Zou, J. P. Wang, C. H. Hee, and T. C. Chong Citation: Applied Physics Letters 82, 2473 (2003); doi: 10.1063/1.1565503 View online: http://dx.doi.org/10.1063/1.1565503 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/82/15?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Ferromagnetic resonance analysis of internal effective field of classified grains by switching field for granular perpendicular recording media J. Appl. Phys. 111, 07B722 (2012); 10.1063/1.3679466 Size-dependent reversal of grains in perpendicular magnetic recording media measured by small-angle polarized neutron scattering Appl. Phys. Lett. 97, 112503 (2010); 10.1063/1.3486680 Changes in switching fields of CoCrPt – SiO 2 perpendicular recording media due to Ru intermediate layer under low and high gas pressures J. Appl. Phys. 105, 013926 (2009); 10.1063/1.3065525 Fast reversal dynamics in perpendicular magnetic recording media with soft underlayer J. Appl. Phys. 91, 8662 (2002); 10.1063/1.1450832 High frequency effects in perpendicular recording media J. Appl. Phys. 87, 4990 (2000); 10.1063/1.373225 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.123.44.23 On: Thu, 18 Dec 2014 22:32:26Tilted media in a perpendicular recording system for high areal density recording Y. Y. Zou Data Storage Institute Singapore, 5 Engineering Drive 1, NUS, Singapore 117608, and Electrical and Computer Engineering, National University of Singapore, 119620, Singapore J. P. Wang,a)C. H. Hee, and T. C. Chong Data Storage Institute Singapore, 5 Engineering Drive 1, NUS, Singapore 117608 ~Received 9 September 2002; accepted 7 February 2003 ! Toovercomethehighsensitivityofthesignal-to-noiseratio ~SNR!totheswitchingfielddistribution in the perpendicular media, we proposed to use tilted media in the perpendicular recording systemin this letter. It was found that a much better tolerance of the easy-axis distribution could beachieved when tilted media were used instead of perpendicular media in a perpendicular recordingsystem.We then analyzed the range of the switching field at the freezing points, and found that highK umagnetic material is feasible in tilted media.The results indicated that the areal density could be more than 62% higher than that of perpendicular media when 45° easy-axis tilted media with Ku 57.03106erg/cm3were used. Finally, simulation of the switching dynamics revealed that a much faster magnetization switching could be achieved in 45°-tilted media than in perpendicularmedia. © 2003 American Institute of Physics. @DOI: 10.1063/1.1565503 # Perpendicular recording was proposed about 20 years ago. 1It is always considered as a promising candidate for ultra-high-density magnetic recording. The recording arealdensity as high as 1 Tbit/in. 2was forecasted.2However, there are still many open issues for perpendicular recording.Among them, the high sensitivity of the signal-to-noise~SNR!on the switching field distribution ~SFD!of media during the recording process is one of the main concerns.Based on the Stoner–Wohlfarth model, 3different switching fields exist for single-domain particles with different angles abetween the head field and the easy axes. Unfortunately, for perpendicular media in a perpendicular recording systemwith a keepered single-pole head, ais quite small where the switching field and its sensitivity to aare near to their maxi- mum values.As an example, when aequals 0°, a change of aas small as 1° will lead to a significant drop of switching field from Hkto 0.92Hk. Therefore, in such a recording system, a broad SFD will always exist even when easy-axisdistribution ~EAD!is small. Use of tilted media is a possible way to solve the above problem. By adjusting the easy-axis direction in the tiltedmedia, acan be tuned to the points where the sensitivity of switching field to ais near to its minimum. For example, when ais tuned to 45°, a change as big as 10° will only lead to a slight increase of switching field from 0.5 to 0.51 Hk. Therefore, a better tolerance of EAD, and consequently, a small SFD can be achieved. Moreover, as the switching fieldis also near to its minimum at these points, high K umedia can be used to increase the recording density and thermalstability. Tilted media was initially used in magnetic taperecording, 4–6in which better recording performance has been achieved. Recently, tilted magnetic thin film was fabri-cated with the combined effect of the tilted columnar struc-ture and tilted easy axis, 7while use of tilted film media to extend the longitudinal magnetic recording8was proposed. In this letter, we first compared EAD tolerance of tilted me-dia to that of perpendicular media in both qualitative andquantitative ways. We then studied the feasibility of usinghighK umaterial in tilted media by analyzing the range of switching field at the freezing points. Simulation results forimproving areal density by using tilted media with higher K u were provided. Finally, we compared the switching dynamics for 45°-tilted media with perpendicular media. In our micromagnetic simulation, the magnetic layer was modeled as a two-dimensional ~2D!hexagonal array9,10with 256364 grains. Each grain represented a single domain par- ticle. For each grain, the magnetization dynamic obeyedLandau–Lifshitz–Gilbert ~LLG!equation and the easy axis was tilted out of the film plane. The out-of-plane componentof anisotropy field was Gaussian distributed with mean value uand standard deviation s1, while the in-plane component of anisotropy field of each grain was uniformly distributed.The media parameters used were as follows: grain size ~i.e., diameter of grain !57 nm, magnetic film thickness 513 nm, and M s5400 emu/cm3.Kuwas varied from 4.0 3106to 7.5 3106erg/cm3. The total effective field consid- ered included external field, anisotropy field, and magneto-static field. No exchange coupling between the grains wasassumed. A single-pole head and tilted film with the soft under- layer were used during the writing process. The permeabilityof the soft underlayer was assumed to be infinity. The effectof this assumption is equivalent to have an image pole at theother side of the soft underlayer symmetrically. With such akeepered single-pole head, highly oriented field can be gen-erated. The method of Iwasaki et al. 11was used to calculatea!Current address: Department of Electrical and Computer Engineering, The Center for Micromagnetics and InformationTechnologies ~MINT !, Univer- sity of Minnesota, Minneapolis, MN55455-0154; author to whom corre-spondence should be addressed; electronic mail: jpwang@ece.umn.eduAPPLIED PHYSICS LETTERS VOLUME 82, NUMBER 15 14 APRIL 2003 2473 0003-6951/2003/82(15)/2473/3/$20.00 © 2003 American Institute of Physics This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.123.44.23 On: Thu, 18 Dec 2014 22:32:26the generated field. In this method, the Karlqvist ring head model was used to calculate the head field from the single-pole head based on the approximately complementary rela-tionship between the head fields generated by these twoheads. The head parameters were set as follows: polethickness 5100 nm, track width 5110 nm, deep gap field (H g)515 kOe, and magnetic spacing 58 nm. During the read-back process, the leakage flux was sensed by a magne-toresistive ~MR!head. Then, the noise power, signal power, and SNR of the read-back voltage were calculated in fre-quency domain. The thermally induced reversal was simu-lated using a Monte Carlo method. 12 To compare the recording performance when tilted and perpendicular media are used in the perpendicular recordingsystem, we first simulated the recording patterns. Figure 1showed the recorded magnetization patterns for the perpen-dicular media and 45°-tilted media. From Figs. 1 ~a!and 1 ~b!, we can clearly see that the magnetization fluctuates drasti-cally in perpendicular media even when a small easy-axisdistribution ( s151°) was included. While in the existing literature, the easy-axis distribution of the obtained perpen-dicular media is always larger than 3° if the polycrystallineor amorphous substrates are used. Therefore, more magneti-zation fluctuation can be expected in a practical case. Perfor-mance is much better in 45°-tilted media. As indicated inFigs. 1 ~c!and 1 ~d!, no much change of magnetization pat- terns was observed even when a large easy-axis distribution( s155.15°) was included. To analyze the above difference of recording perfor- mance in a quantitative way, we calculated the noise andsignal power of read-back voltage. Figure 2 showed the re-sults for the perpendicular media and 45°-tilted media. From the viewpoint of noise power analysis, significant increase ofnoise power can be observed in the perpendicular mediawhen EAD is included ~i.e., s1Þ0.0°). Specifically, as indi- cated in Fig. 2 ~a!, such a noise increase reaches its maximum of 22.1 dB at Ku54.03106erg/cm3and its minimum of 13.2 dB at Ku54.33106erg/cm3. This is because a large SFD always exists in the perpendicular media as pointed outearly. Thus, the magnetic particles cannot be reversed uni-formly, causing a large noise power. However, with the in-crease of K u, the difference of noise power between the cases of s150 and s1Þ0 is reduced. This is reasonable.As Kuis increased to a certain extent, the writing field can no longer overcome the switching field.As a result, the effect ofeasy-axis distribution is reduced. In the tilted media, perfor-mance becomes much better, especially when the easy axesare tilted to 45°. As shown in Fig. 2 ~b!, the increase of noise power due to inclusion of EAD is marginal and reaches itsmaximum of 3.2 dB at K u54.03106erg/cm3even when the standard deviation of EAD is as large as 5.15°. From the viewpoint of signal power analysis, we can see that at the start, the signal degrades uniformly and slowlywith increase of K ufor both perpendicular media and 45°-tilted media. After Kureaches certain values ~i.e., around 4.5 3106erg/cm3for perpendicular media and around 7.5 3106erg/cm3for 45°-tilted media !, the value of s1has totally different effect on the signal power. In the perpendicular media, signal becomes stronger when s1is increased and Kuis fixed. However for the 45°-tilted media, the situation is reversed. This is because in the former case,the switching field becomes smaller when ais away from 0°, while the switching field becomes a little bigger whenthe ais away from 45° for the latter case. Overall, as the noise power increases rapidly in the per- pendicular media, while the increase of noise power in the45°-tilted media and the decrease of signal power in bothmedia are slowly, much better SNR can be achieved for45°-tilted media when EAD is taken into consideration. Asan example, the SNR for perpendicular and tilted media are22.32 and 29.86 dB, respectively, with K u54.0 3106erg/cm3ands155.15°. We also calculated the range of aand the corresponding switching field at the freezing points, which were defined asthe points where head field is larger than the switching fieldfor the last time. 4Table I showed that the avalues at the freezing points are around 0°–15° when the easy axes aretilted to 90° ~i.e., perpendicular media !. Compared to those a FIG. 1. Magnetization patterns for films with easy axes tilted to ~a!90° 60.0°, ~b!90°61.0°, ~c!45°60.0°, and ~d!45°65.15°. FIG. 2. Noise and signal power for films with different EADs and Kuand tilted to ~a!90° and ~b!45°.2474 Appl. Phys. Lett., Vol. 82, No. 15, 14 April 2003 Zouet al. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.123.44.23 On: Thu, 18 Dec 2014 22:32:26values for the media tilted to 45° or 60°, they are much smaller.As a consequence, the minimum field ~i.e., switching field!at which irreversible magnetization rotation can occur becomes much larger in the perpendicular media accordingto the Stoner–Wohlfarth model, which was discussed in de-tail previously. The switching field values with K u54.0 3106erg/cm3are listed in Table I, in which Hkis the an- isotropy field and Hgis the magnitude of the deep gap field. From the listed switching field values, we can infer thatmuch higher K umaterial can be used in the tilted media ~e.g., at 45° or 60°) than in the perpendicular media under the same head field. By taking advantage of the feasibility of using high Ku material in the tilted media, high recording density can be achieved. We carried out a series of simulation for 45°-tiltedmedia with different K uand grain size. The head field strength was kept unchanged in order to make a fair com-parison. Our goal was to push K uas high as possible. Then by making use of achieved high Ku, we tried to reduce the grain size as much as possible while still maintained a goodSNR ~i.e., above 20 dB !and thermal stability. Our results indicated that K ucan reach 7.0 3106erg/cm3, while the grain size can be reduced to 5.5 nm in 45°-tilted media. TheSNR and its thermal decay results are shown in Fig. 3. Com-pared with the results in Fig. 3 for perpendicular media withgrain size 57 nm and K u54.23106erg/cm3, we can clearly see that the recording areal density can be increased 62% inthe 45°-tilted media while the recording and thermal decayperformance is still much better than that in perpendicularmedia. This is reasonable. As the demagnetization field intilted media 8decreases with the increase of tilted angle from 0° to 90°, the demagnetization field in 45°-tilted media isstronger than that in regular perpendicular media under samemedia parameters. However, based on the decay estimation method 13for fully decoupled grains, using high Kumedia will reduce the effect of the demagnetization field due to theincrease of the anisotropy field H k, and consequently result in much better thermal stability, as shown in Fig. 3. There-fore, we can safely say that under similar recording and ther-mal decay performance, more than 62% increase of arealdensity can be achieved if 45°-tilted media with high K uare used instead of perpendicular media in a perpendicular re-cording system. Finally, we simulated the switching dynamics of the tilted media and perpendicular media. In the simulation, thedamping constant and the step time were set to be 0.15 and0.01 ps, respectively. The normalized magnetization was ini-tially in its remnant state. The external field of 1.5 H kwas applied in the 2zdirection. Clearly, in 45°-tilted media, the switching speed is much faster than that in the perpendicularmedia, as shown in Fig. 4. It only takes 46.88 ps for magne-tization to reach the x–yplane and another 94.8 ps to spiral down to the point where M z/Msis20.95. However, in the perpendicular media ( u589.9°), it takes 1456.83 ps for magnetization to reach the x–yplane and another 111.48 ps to spiral down to the point at which Mz/Msis20.95. Such a drastic reduction of reversal time is due to the large rever-sal torque in 45°-tilted media, which results from the largeangle between the magnetization and total effective field. As an alternative, we can tilt the head instead of the media. With tilted head, acan also be adjusted to the opti- mum values, while the excellent thermal stability in perpen-dicular media can be maintained. But design of a head ca-pable of generating uniform and tilted field could bechallenging. 1S. Iwasaki and Y. Nakamura, IEEE Trans. Magn. 13, 1272 ~1977!. 2R. Wood, IEEE Trans. Magn. 36,3 6~2000!. 3E. C. Stoner and E. P. Wohlfarth, Philos. Trans. R. Soc. London 1-240,7 4 ~1948!. 4H. J. Richter, IEEE Trans. Magn. 29, 2258 ~1993!. 5N. H. Yeh, J. Magn. Soc. Jpn. 21, 269 ~1997!. 6S. R. Cumpson, B. K. Middleton, and J. J. Miles, J. Magn. Magn. Mater. 154, 382 ~1996!. 7Y. F. Zheng, J. P. Wang, and V. Ng, J. Appl. Phys. 91, 8007 ~2002!. 8C. H. Hee, Y. Y. Zou, and J. P. Wang, J. Appl. Phys. 91, 8004 ~2002!. 9H. N. Bertram and J. G. Zhu, Solid State Phys. Rev. 46, 271 ~1992!. 10C. H. Hee, J. P. Wang, H. Gong, and T. S. Low, J. Appl. Phys. 87, 5535 ~2000!. 11S. Iwasaki, Y. Nakamura, and K. Ouchi, IEEE Trans. Magn. 17, 2535 ~1981!. 12Y. Kanai and S. H. Charap, IEEE Trans. Magn. 27, 4972 ~1991!. 13D. Weller and A. Moser, IEEE Trans. Magn. 35, 4423 ~1999!.TABLE I. Writing angles and switching fields for films with easy axes tilted to different angles at the freezing points ( Hk520 kOe;Hg515 kOe). Tilted angle ~degree !Angle between easy axis and applied field ~degree !Switching field ( Hk51.33Hg) 90 0–15 Hk;0.61Hk 60 30–45 0.52 Hk;0.5Hk 45 45–60 0.5 Hk;0.52Hk FIG. 3. SNR and thermal decay performance for 45°-tilted media with Ku 57.03106erg/cm3and grain size 55.5 nm perpendicular media with Ku 54.23106erg/cm3and grain size 57nm . FIG.4. Switchingbehaviorsforfilmtiltedto ~a!89.9° and ~b!45°. Thedots were plotted every 1 ps.2475 Appl. Phys. Lett., Vol. 82, No. 15, 14 April 2003 Zouet al. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.123.44.23 On: Thu, 18 Dec 2014 22:32:26

1.4944777.pdf

Spin current transport in ceramic: TiN thin film Hongyu An , Yusuke Kanno , Takaharu Tashiro , Yoshio Nakamura , Ji Shi , and Kazuya Ando, Citation: Appl. Phys. Lett. 108, 121602 (2016); doi: 10.1063/1.4944777 View online: http://dx.doi.org/10.1063/1.4944777 View Table of Contents: http://aip.scitation.org/toc/apl/108/12 Published by the American Institute of Physics Spin current transport in ceramic: TiN thin film Hongyu An,1Yusuke Kanno,1Takaharu Tashiro,1Y oshio Nakamura,2JiShi,2 and Kazuya Ando1,3,a) 1Department of Applied Physics and Physico-Informatics, Keio University, Yokohama 223-8522, Japan 2Department of Metallurgy and Ceramics Science, Tokyo Institute of Technology, 2-12-1, Ookayama, Meguro-ku, Tokyo 152-8552, Japan 3PRESTO, Japan Science and Technology Agency, Kawaguchi, Saitama 332-0012, Japan (Received 20 January 2016; accepted 11 March 2016; published online 22 March 2016) The spin current transport property in a ceramic material TiN has been investigated at room temperature. By attaching TiN thin films on Ni 20Fe80with different thicknesses of TiN, the spin pumping experiment has been conducted, and the spin diffusion length in TiN was measured to be around 43 nm. Spin-torque ferromagnetic resonance has also been taken to investigate the spin Hall angle of TiN, which was estimated to be around 0.0052. This study on ceramic material provides apotential selection in emerging materials for spintronics application. VC2016 AIP Publishing LLC . [http://dx.doi.org/10.1063/1.4944777 ] Spintronics is a booming technology which can control both the spin and charge degrees of freedom of electrons. This technology is a promising candidate for fabricating lowpower, high speed, and non-volatile memory and logic devi-ces. 1,2To realize the practical application, the generation, manipulation, and detection of a spin current, a flow of spins, are essential techniques to be developed.3 The spin current transport can be measured by spin pumping through the inverse spin Hall effect (ISHE)4–6and, oppositely, the spin-torque ferromagnetic resonance (ST-FMR) through the spin Hall effect (SHE). 7–10The former one injects pure spin currents from ferromagnet (FM) into attached nonmagnetic metals (NM) under ferromagnetic res- onance condition. The injected spin current is subsequentlyconverted into an electric voltage through the ISHE. Whilefor the latter one, a microwave-frequency charge current is applied in the NM layer, which can generate an oscillating transverse spin current by the SHE, and then is injected intothe adjacent FM layer. The spin current drives magnetizationprecession through spin torque, leading to an oscillation ofthe resistance due to the anisotropic magnetoresistance of FM layer. Recently, spin current transport has been observed in various materials, such as metals, 4,6,11–17semiconduc- tors,18–25conducting polymers,26graphene,27and antiferro- magnets.28–30These studies drastically extend the emerging class of materials to be used in spintronics and push forward the practical applications of spintronics. On the other hand, spin current transport in ceramic materials has barely beenstudied yet by now. As a ceramic material, titanium nitride(TiN) is widely used as diffusion barrier in microelectronicsdue to its high thermal stability. 31It can well block the diffu- sion of the metals into the silicon substrates, but with a high electric conductivity to allow a good electric connectionbetween them. If the spin current transport can be detected inTiN, it will further extend the candidate materials applied for spintronics. Moreover, it will give a possibility for the inte- gration between spintronics and the traditional electronics.Therefore, to study the spin current transport in such a mate- rial is important in both fundamental and practical aspects ofspintronics. In this letter, we study the spin current transport in TiN films by attaching a FM film at room temperature. The spindiffusion length in TiN has been investigated by usingNi 20Fe80/TiN bilayer through spin pumping, and the spin Hall angle of TiN has been investigated by using Ni 81Fe19/ TiN through ST-FMR, respectively. The reason we useNi 20Fe80in the spin pumping experiment is that Ni 20Fe80has a larger saturation magnetization, which can generate alarger signal in FMR absorption spectral. On the other hand,Py (Ni 81Fe19) was used in ST-FMR experiment due to its smaller shape anisotropy. Figure 1(a) shows the schematic illustration of the Ni20Fe80(20 nm)/TiN( dNnm) samples ( dN¼10, 20, 30, 40, and 50 nm) used in spin pumping. First, a TiN single layerwith 0.5 mm /C23 mm rectangular shape (patterned by metal mask) was deposited on thermally oxidized Si substrates bydc magnetron sputtering. The base pressure in the chamberbefore deposition was better than 5 /C210 /C05Pa, and the depo- sition pressure was 0.2 Pa. Nitrogen and argon gases withfixed flow ratio of 1:8.7 were applied during sputtering.Then, a Ni 20Fe80single layer (0.5 mm /C21.5 mm rectangular shape) with thickness of 20 nm was deposited on the TiNfilms at room temperature. The deposition pressure was0.73 Pa with argon gas flow rate of 10 sccm. For the fabrica-tion of a Ni 81Fe19/TiN bilayer used in ST-FMR, the sub- strates were patterned into 10 lm/C2130lm rectangular shape by standard photolithography before deposition, andlift-off technique was used to take off the rest part of thefilms after deposition. For Ni 81Fe19deposition, RF power was used at the deposition pressure of 0.2 Pa with argon gasflow rate of 10 sccm. The film thickness was controlled by the deposition time with a pre-calibrated deposition rate. Atomic force microscopy (AFM) was utilized to directlyobserve the surface of the films and x-ray diffraction (XRD)profiles were taken on Bruker D8 Discover diffractometerby applying Cu K aradiation. For the spin pumping, FMR absorption spectra were measured using a coplanara)ando@appi.keio.ac.jp 0003-6951/2016/108(12)/121602/4/$30.00 VC2016 AIP Publishing LLC 108, 121602-1APPLIED PHYSICS LETTERS 108, 121602 (2016) transmission waveguide in a 6–8 GHz range. During the measurement, an in-plane external magnetic field Hwas applied, as shown in Fig. 1(a). For the ST-FMR, as shown in Fig.1(b), a RF microwave current was applied to the sample and an external magnetic field in the x-yplane is applied at an angle of 45oto the x-axis (the longitudinal direction of the strip). By using a bias tee, a dc voltage signal across the sam-ple from the mixing of the RF current and oscillating resist- ance can be measured simultaneously during the microwave current application. All the measurements were conducted atroom temperature. Since the microstructure of the NM layer, especially the surface morphology, can significantly affect the spin current transport property from FM to NM layers, the microstructureof the TiN single layer films with different thicknesses havebeen investigated. Figures 2(a)–2(d) show the AFM images of the surface morphology for the TiN films by changing the thickness from 10 to 100 nm. All the films exhibit continuoussurfaces, and the surface root-mean-square roughness R RMS measured by AFM (Fig. 2(e)) in all the films is lower than 1 nm, indicating the quite smooth surface morphology for the TiN film in the thickness range of 10 to 100 nm. The corre- sponding XRD profiles are shown in Fig. 2(f). All the films exhibit TiN (200) peak and the intensity increases with theTiN thickness, besides which, a weak TiN (111) peak beginsto appear when the thickness increases to 100 nm. From the above study, it can be seen that the surface morphology and the microstructure of the TiN films are well controlled, whichis considered to have minor effect on the spin current trans-port property in the thickness range of 10 to 100 nm. The spin diffusion length k N, which reflects the distance that a spin cur- rent can transfer in TiN, can be determined by measuring TiNthickness d Ndependence of the Gilbert damping constant afor the Ni 20Fe80/TiN films. The Gilbert damping constant a can be quantified by measuring microwave frequency fde- pendence of the FMR spectral width l0DH32 l0DH¼l0DHextþ2pa cf; (1) where l0DHextandcare the extrinsic contribution to the spec- tral width and the gyromagnetic ratio, respectively. Figure 3(a) shows the dependence of the FMR spectral width l0DHon the microwave frequency fby changing the TiN thickness. The FMR spectral width increases linearly with microwave fre- quency f, consistent with Eq. (1). By fitting the experimental data shown in Fig. 3(a)with Eq. (1), the TiN layer thickness dNdependence of the Gilb ert damping constant ais obtained. The measured Gilbert damping c onstant is related to the effec- tive spin mixing conductance g"# effas33,34 g"# eff¼l0MNi20Fe80s dF glBaF=N/C0aF ðÞ ; (2) where l0MNi20Fe80s ;dF,g, and lBare the saturation magnet- ization, the thickness of the Ni 20Fe80layer, the gfactor, and the Bohr magneton, respectively. aF=Nand aFare the Gilbert damping constant for the Ni 20Fe80/TiN bilayer and a Ni20Fe80film ( dN¼0), respectively. In Fig. 3(b), we show the TiN thickness dNdependence of g"# eff. Here, the effective spin mixing conductance g"# effis expressed as34 FIG. 1. (a) Schematic illustration of the setup for the spin pumping. HandhRF denote the external magnetic field and th e Oersted field generated by the micro- wave. (b) Schematic illustration of the ST-FMR experiment. M,sH,a n d sST denote the magnetization, the torque due to the Oersted field induced by the charge current in TiN and the torque due to spin current induced by the SHE. FIG. 2. AFM images of the surface morphology for the TiN single layerswith different thicknesses d N: (a) 10 nm, (b) 30 nm, (c) 50 nm, and (d) 100 nm. (e) The surface roughness and (f) the XRD profiles of the TiN single layers with different thicknesses.121602-2 An et al. Appl. Phys. Lett. 108, 121602 (2016) g"# eff¼g"# r1 1þ2ffiffiffiffiffiffiffiffi n=3p tanh dN=kN ðÞ/C16/C17/C01; (3) where g"# randnare the real part of the intrinsic spin mixing conductance and the element-dependent factor, respectively.Therefore, by fitting the experimental data using Eq. (3),t h e spin diffusion length in TiN is obtained. As shown in Fig. 3(b), the experimental result is well reproduced using Eq. (3).T h e obtained spin diffusion length in TiN is k N¼4363:7n m , which is three times larger than the spin diffusion length in Ti(/C2413.3 nm). 35 In order to quantify the spin Hall angle of TiN, we have conducted the ST-FMR experiment on a Ni 81Fe19(8 nm)/ TiN(50 nm) bilayer. With passing a microwave currentthrough the longitudinal direction of the device, an external magnetic field His swept in the in-plane direction with an angle of 45 /C14between Hand the longitudinal direction of the device. The magnetization of the Ni 81Fe19layer is influenced by two torques generated from the RF charge current, anin-plane torque s STfrom the oscillating spin current and an out-of-plane torque sHfrom the oscillating magnetic field (Fig. 1(b)). The equation of motion for the magnetization in the Ni 81Fe19layer can be expressed by the Landau-Lifshitz- Gilbert equation including the ST component36 dm dt¼/C0cm/C2Heffþam/C2dm dt þc/C22h 2l0MstJSHm/C2r/C2m ðÞ /C0cm/C2hRF;(4)where Heff;/C22h,t,JSH, and hRFare the sum of external mag- netic field and the out-of-plane demagnetization field, theDirac constant, the thickness of the Ni 81Fe19layer, spin cur- rent density generated by the SHE, and the Oersted field gen- erated by the RF charge current in the TiN layer. The thirdand fourth terms on the right side in Eq. (4)are the result of the in-plane and out-of-plane torques generated by themicrowave current, respectively. Figure 4(a)shows the measured ST-FMR spectra for the RF frequencies from 4 to 8 GHz. The dc voltage signal V mix can be generated from the mixing of the RF charge current and the oscillating resistance under the magnetization pre- cession. As can be seen, Vmixsignificantly changes under the resonant conditions. The detected voltage signal containingthe in-plane and out-of-plane torques in ST-FMR measure- ment can be fitted by 7,10 Vmix¼SW2 l0H/C0l0HFMR ðÞ2þW2 þAWl0H/C0l0HFMR ðÞ l0H/C0l0HFMR ðÞ2þW2; (5) where S,A,W, and HFMRare the magnitude of a symmetric component, the magnitude of an antisymmetric component, the spectral width, and the FMR field. As shown in Fig. 4(a), the experimental results are well reproduced using Eq. (5). From the ST-FMR results shown in Fig. 4(a), the spin Hall angle hST SHof TiN, the ratio between the spin current density and the charge current density, can be calculatedusing 7,10 hST SH¼S Ael0MPy stdN /C22hffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þl0MPy s l0HFMRs : (6)FIG. 3. (a) The dependence of the FMR spectral width l0DHon the micro- wave frequency ffor the Ni 20Fe80/TiN ( dNnm) films ( dN¼0, 10, 20, 30, 40, and 50 nm). (b) The dependence of the effective spin mixing conduct- ance g"# effon the TiN thickness dN. FIG. 4. (a) ST-FMR spectra of Ni 81Fe19(8 nm)/TiN(50 nm) versus l0H.T h e RF power is 300 mW, and the RF frequency varies from 4 to 8 GHz. The open circles are the experimental data, and the solid curves are the fitted curves by using Eq. (5). (b) The dependence of resonance frequency fon the FMR field HFMR. The solid line is fitted from Kittel’s formula. (c) The spin Hall angle hST SHof TiN calculated from ST-FMR spectra at a different frequency f.121602-3 An et al. Appl. Phys. Lett. 108, 121602 (2016) Here, the saturation magnetization Msis obtained from the microwave frequency fdependence of the FMR field HFMR shown in Fig. 4(b)using the Kittel formula 2pf c¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l0HFMR l0HFMRþl0MPy s/C16/C17r ; (7) where c¼1:61/C21011T/C01s/C01and l0MPy s¼0:607 T are obtained. Using the measured parameters with Eq. (5),w e obtain the spin Hall angle hST SHfor different resonance fre- quencies f, as shown in Fig. 4(c). As expected, hST SHshows no significant dependence on the frequencies f. By averaging all the measured values, the spin Hall angle of TiN is obtained ashST SH¼0.005260.0008. This value is ten times larger than the reported value of Ti ( /C243:6/C210/C04).35 In summary, we have investigated the spin current trans- port property in TiN thin films. In order to investigate thespin diffusion length in TiN, the spin pumping experimenthas been conducted by attaching TiN on Ni 20Fe80with changing the thickness of TiN. Furthermore, the ST-FMRexperiment has been taken to determine the spin Hall angleof TiN. This study on the TiN ceramic material provides apotential selection of materials for the spintronicsapplication. This work was supported by JSPS KAKENHI Grant Nos. 26220604, 26103004, 26600078, PRESTO-JST“Innovative nano-electronics through interdisciplinarycollaboration among material, device and system layers,” theMitsubishi Foundation, the Asahi Glass Foundation, theMizuho Foundation for the Promotion of Sciences, and theCasio Science Promotion Foundation. 1S. Wolf, D. Awschalom, R. Buhrman, J. Daughton, S. Von Molnar, M. Roukes, A. Y. Chtchelkanova, and D. Treger, Science 294, 1488 (2001). 2C. Chappert, A. Fert, and F. N. Van Dau, Nat. Mater. 6, 813 (2007). 3F. J. Jedema, A. Filip, and B. Van Wees, Nature 410, 345 (2001). 4E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Lett. 88, 182509 (2006). 5S. O. Valenzuela and M. Tinkham, Nature 442, 176 (2006). 6K. Ando, S. Takahashi, J. Ieda, Y. Kajiwara, H. Nakayama, T. Yoshino, K. Harii, Y. Fujikawa, M. Matsuo, S. Maekawa et al. ,J. Appl. Phys. 109, 103913 (2011). 7L. Liu, T. Moriyama, D. Ralph, and R. Buhrman, Phys. Rev. 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1.587272.pdf

Biological materials studied with dynamic force microscopy D. Anselmetti, M. Dreier, R. Lüthi, T. Richmond, E. Meyer, J. Frommer, and H.J. Güntherodt Citation: Journal of Vacuum Science & Technology B 12, 1500 (1994); doi: 10.1116/1.587272 View online: http://dx.doi.org/10.1116/1.587272 View Table of Contents: http://scitation.aip.org/content/avs/journal/jvstb/12/3?ver=pdfcov Published by the AVS: Science & Technology of Materials, Interfaces, and Processing Articles you may be interested in Dielectric constant microscopy for biological materials AIP Conf. Proc. 1512, 520 (2013); 10.1063/1.4791140 Magnetic dissipation force microscopy studies of magnetic materials (invited) J. Appl. Phys. 83, 7333 (1998); 10.1063/1.367825 Scanning Force Microscopy in Biology Phys. Today 48, 32 (1995); 10.1063/1.881478 Atomic force microscopy of biological samples at low temperature J. Vac. Sci. Technol. B 9, 989 (1991); 10.1116/1.585442 Dynamic Testing of Biological Materials J. Acoust. Soc. Am. 52, 161 (1972); 10.1121/1.1981991 Redistribution subject to AVS license or copyright; see http://scitation.aip.org/termsconditions. Download to IP: 155.97.178.73 On: Wed, 26 Nov 2014 01:39:13Biological materials studied with dynamic force microscopy D. Anselmetti, M. Dreier, R. Luthi, T. Richmond, E. Meyer, J. Frommer, and H.-J. Guntherodt Institute or PhY.vics. Unin'r.litr of' Basel. Klinl(elhcrMstrasse 82. CH-4056 Basel. SH'it~erland (Received 9 August 1993: accepted 10 December 1993) Biological materials such as hexagonal packed intermediate (HPJ) layers, DNA, tobacco mosaic virus and collagen deposited on various substrates with noncontact dynamic force microscopy under ambient conditions were investigated. This method is highly suited for the investigation of soft organic matter where a minimized interaction between tip and sample is needed for nondestructive and reliable operation. Hence, additional anchoring of the biological specimens was no longer found to be crucial. The vertical and lateral resolution limits of this gentle method were determined to be <D. I 11m and 1-2 nm. respectively, allowing very stable and high resolution results on all investigated systems. By taking approach curves and monitoring the dynamic properties of the cantilever (resonance frequency and Q value) during the experiment, the interaction mechanism between tip and sample was found to be dominated by attractive van der Waals interaction and capillary forces. Furthermore, initial results from a HPJ layer imaged with noncontact dynamic force microscopy in a water environment are presented. I. INTRODUCTION The imaging of biological matter is diflicult because of its nonconducting and soft character. The reasons for investigat ing these materials are manifold. Two types of experiments \vhich appear pfllmising can be addressed by scanning probe microscopy (SPM): (I) Structural information about the bio logical specimens. measured in direct space. is obtainable without intense or indirect sample preparation and sample stabilization techniques. (2) In situ investigation of mol ecules and their biochemical functions. especially in liquid environment. could lead to a more complete understanding of biochemical processes and structures. Scanning tunneling microscopy (STM) and repulsive scanning force microscopy (AFM) pioneered the use of SPM techniques in investigating hiological objects. I .1 The first breakthrough in the investigation of soft materials was the implementation of AFM in liquid environment to (a) mini mize the strong repUlsive interaction forces and (b) to keep the biological specimens in their native conformation and environment..J-6 In contrast to conventional AFM where the sample is measured by repulsive short-range interaction forces. non contact scanning force microscopy 7.)\ or dynamic force mi croscopy (DFM)'i·1O provides the possibility of measuring all types of samples nondestructively by longer range interac tion forces. In this work, we report on DFM investigation under am bient conditions of biological matter. such as HPJ layer, DNA. tobacco mosaic virus (TMV) and collagen. We discuss resolution limits and possible interaction mechanism for this new promising technique. Furthermore. we present initial re sults where a HPI layer was investigated with DFM in water environment. II. EXPERIMENT In contrast to conventional AFM where quasi static tip ex cursions of a soft cantilever are measured. in DFM a stiff cantilever, mounted onto a piezoelectric bimorph, is dynami cally driven by an ac voltage close to its mechanical reso nance frequency fl) with an amplitude of 0.5-2.0 nm (Fig. I). The cantilever motion is detected by either a lever deflection II or an interferometri 2 readout. Force gradients resulting from van der Waals, electrostatic or magnetic inter action forces detune the mechanical resonance system ac cording to I ~'+F' f' 0= 27T //letT' ( I) where F', c. and //leff denote force gradient. cantilever spring constant. and effective mass of cantilever, respectively. The resulting shift in resonance frequency can be measured by either frequency or amplitude detection methods. Depending on the quality value Q of the resonance the former method is mostly applied in UHV (Q>50 000), whereas the latter is highly suited for DFM experiments in air (Q~500). For the investigation of biological matter under ambient conditions. we used lock-in technique or rms-to-dc conversion for am plitude detection. In contrast to lock-in technique where am plitudes are measured at a certain reference frequency, in rms-to-dc conversion 13 all amplitude contributions in a cer tain frequency window (in our setup: 30 kHz-2 MHz) are integrated. Although both methods are inherently different we found no evidence of different results from measuring soft biological matter. One qualification is that the cantilever amplitude is not exclusively affected by force gradients, but that damping ef fects resulting from, e.g., hydrodynamic interaction have also to be taken into account.l~ All data presented are raw data and were taken using mi crofabricated Si cantilevers with integrated tips 15 (lo~400 kHz, ('=20-80 N/m). In some cases, we used carbon whis ker tips made by a focussed electron beam in a poor vacuum.lh 1500 J. Vac. Sci. Techno!. 8 12(3), May/Jun 1994 0734-211 Xl94112(3)11500141$1.00 ©1994 American Vacuum Society 1500 Redistribution subject to AVS license or copyright; see http://scitation.aip.org/termsconditions. Download to IP: 155.97.178.73 On: Wed, 26 Nov 2014 01:39:131501 AnselmeHi et al.: Biological materials studied with DFM Displacement Sensor Feedback Loop FIG. I. Scheme of the experimental setup for noncontact DFM. The canti lever is mounted onto a piclOelectric bimorph and is driven by an ac voltage near its mechanical resonance frequency with an amplitude of ~: =0.5 -2.0 nm. 1501 III. RESULTS AND DISCUSSION We investigated the HPI layer. a hexagonal packed inter mediate layer of deinococclIs radiodllrans forming two dimensional protein sheets. Figure 2(a) shows a DFM image of these protein sheets adsorbed on glass where the hexago nal structure of the protein array with a periodicity of 18 nm is clearly visible. The measured height of the layers of ~5 nm is in good agreement with theory. Beside the periodicity of the HPI. we were able to resolve the hollow core of the protein rings. suggesting a lateral resolution of ~2 nm with DFM10 as can be seen in the insert of Fig. 2(a). In previous works. we estimated the vertical resolution of this method to be <0.1 nm. by imaging monoatomic steps of an InGaAs surface.1o•17 DNA strands 18 were deposited onto mixed organic Langmuir-Blodgett (LB) films on siliconl !) without addi tional anchoring or stabilization. In Fig. 2(b). the DNA strands are clearly visible in this DFM image (0.5XO.5 tLm2). Interestingly. we found no evidence that the molecules were displaced during the experiment suggesting minimal lateral FIG. 2. (a) DFM image of HPI layers deposited on glass. The sheet structure of this hexagonal packed protein array with a periodicity of I H nm and the hollow core structure of the protein rings (inset) can clearly be identified. indicating high lateral resolution power. The scale bar corresponds to 200 nm. (bl Linear double stranded DNA molecules deposited on a mixed organic LB film on silicon and imaged with DFM. DNA strands are clearly visible sometimes forming loops. The scale bar corresponds to 200 nm. (c) Tobacco mosaic viruses (TMV) depmited onto organic LB films of phase separated fluorocarbon (dark areas) and hydrocarbon (white circular areas) domains. The 0.3 JLm long and 18 nm thick viruses can be imaged vcry reliably. The scale bar corresponds to 500 nm. (dl DFM image of mutated procollagen I sprayed onto mica. The 300 nm long triple-helix molecule with its two peptide terminals. can clearly he identified. The scale bar corresponds to 200 nm. JVST B -Microelectronics and Nanometer Structures Redistribution subject to AVS license or copyright; see http://scitation.aip.org/termsconditions. Download to IP: 155.97.178.73 On: Wed, 26 Nov 2014 01:39:131502 Anselmetti et al.: Biological materials studied with DFM forces. Closer examination of the DNA strands in Fig. 2(b) even reveals structure along the DNA strands with a corru gation of 0.1-0.3 nm on a length scale of 6-8 nm. Although the measured height of ~ 2 nm is in agreement with theory, the measured width of 8 -12 nm suggests contribution from . 10 tIP structure. The same type of organic LB substrate was used to adsorb tobacco mosaic viruses (TMV). Figure 2(c) represents a DFM image (1.5 X 1.5 ,urn") where isolated rod-shaped TMVs, homogeneously distributed over the surface, can be identified. Closer scrutiny of the TMVs reveals preferential adsorption at the juncture between domain boundaries of hy drocarbon and fI uorocarbon domains.2o Last, we investigated natural human mutated procol\agen I sprayed onto a mica substrate.21.22 The 300 nm long and 1.3 nm thick triple helix molecules can be clearly identified in Fig. 2(d). The measured molecule height and width is 0.3 and 13 nm, respectively, indicating an instrumental resolu tion vertically of <0.1 nm as determined in previous works.IO.17 The measured reduced height compared to the thickness of the molecule is attributed to a nesting of the amino acids to the mica surface whereas the increased width is partly attributed to tip convolution effects. The direction of the amino acid sequence can be determined because the col lagen amino-terminal (N)-and the carboxyl-terminal (C) peptide ends can clearly be identified by their size. Mutated collagens often exhibit kinks in their folded structure which are subject of ongoing research2J and can now be studied by DFM. In contrast to a recently developed tapping mode,24 where the cantilever is oscillated close to its resonance frequency with an amplitude of 50-100 nm and where the tip interacts repulsively, hitting the sample surface, we work with much lower amplitudes (0.5-2.0 nm) in the noncontact mode. We performed approach experiments where the cantilever was excited with white noise by ~ I nm and where the rms-to-dc converter was used to stabilize and control the tip with re spect to the sample surface. The dynamic properties of the cantilever, resonance frequency and Q value, were extracted by Fourier analyzing the cantilever signal and calculating a fit resonance function (Fig. 3). In this figure, we show how resonance frequency and Q-value change upon the tip's ap proach to the sample surface. In (a) the "free" cantilever (~I ,urn away from the sample surface) exhibits a resonance frequency and a Q value of 10= 149320 Hz and 500, respec tively, which was was extracted by the calculated fit curve (a'). By approaching the cantilever tip to a working distance of 2-3 nm the upper peak slightly shifts to a lower frequency (most probably due to attractive van der Waals interaction) and a pronounced second peak appears in the spectrum at a lower frequency (b). Curve (c) was taken where the tip was placed in a working distance of 1-2 nm where the cantilever spectra could be fitted with a lower frequency of 10= 147280 Hz and a Q value of \35 (c'). High resolution images could be obtained in both working distances of curve (b) as well as of curve (c). The second peak in curves (b) and (c) is difficult to explain by a pure van der Waals interaction. We interpret this result as owing to an additional small liquid meniscus which forms due to the ever present water layer under am- J. Vac. Sci. Technol. B, Vol. 12, No.3, May/Jun 1994 1502 134800 frequency [Hz) 154530 FIG. 3. Three cantilever spectra taken at three different working distances. Curves (a), (b) and (c) are experimental data whereas curves (a') and (e') are calculated to fit the corresponding curves (a) and (e). Curve (a) represents a spectrum taken of a "free" cantilever at a tip-sample separation of ~ I Jim (fn= 149320 Hz, Q=500). Upon approaching the tip to a working distance of 2-3 nm a pronounced second peak appears at a lower frequency. Upon further approaching to a working distance 1-2 nm. the peak of the "free" cantilever motion disappears nearly completely ending up with a cantilever spectra characterized by a lower resonance frequency of f~ = 147 280 Hz and a smaller Q value of 135. For details see text. bient condition. According to Eq. (1), we deduce a force interaction gradient of -0.68 N/m for this capillary interac tion with the measured frequency shift of 2040 Hz and the spring constant of the cantilever of c = 25 N/m. The mea sured low Q value at very close working distances leads us to the conclusion that in absence of electrostatic and mag netic forces, the interaction is dominated by van der Waals, capillary and hydrodynamic interaction. The obtained DFM images can therefore be interpreted mostly as constant damp ing contours. In a further experiment, we immersed the whole cantile ver in water.25 III situ experiments in a liquid environment with a noncontact SPM method are crucial for investigation of biomolecules with respect to their natural conformation and function. We chose a HPI layer which was crosslinked to a glass substrate. In Fig. 4, we present a first DFM image taken ill situ under water, where a 5-6 nm height protein sheet of HPI can clearly be identified showing the 18 nm periodicity of the protein array (inset of Fig. 4). Although the image quality is a little poorer compared to our data taken in air, we believe that ongoing work will show the power and reliabilty of ill situ noncontact DFM with respect to the in vestigation of structure and function of biomaterials. IV. SUMMARY In summary, we applied DFM to biomaterials such as HPI layer, DNA, TMV, and collagen. The vertical and lateral resolution limits of this method were determined to be <0.1 Redistribution subject to AVS license or copyright; see http://scitation.aip.org/termsconditions. Download to IP: 155.97.178.73 On: Wed, 26 Nov 2014 01:39:131503 Anselmetti et 81.: Biological materials studied with DFM FIG. 4. DFM image of a HPI layer taken in situ under water (I x I J.Lm2). In this differentiated representation the sheet structure of this protein array with a periodicity of 18 nm (inset) can ciearly be identified. and -2 nm, respectively. We showed that this gentle and reliable technique allows careful and highly reproducible in vestigation of these materials with a very high resolution. Under ambient conditions, we found the interaction domi nated by van der Waals and capillary forces. In situ DFM investigation of a HPJ layer under water shows the possiblity of applying this gentle method also in a more realistic envi ronment for biology and biochemistry. ACKNOWLEDGMENTS We would like to thank Ch. Gerber and B. Michel from IBM Research Laboratory in Ruschlikon for support. We ac knowledge sample preparation of the HPI layer from S. Karrasch, F. Schabert and A. Engel as well as manufacturing of the carbon whisker tips by Andreas Hefti from the Mau rice E. Muller Institute at the Biocenter of the University of Basel. The mixed LB films for adsorption of TMV and DNA were provided by M. Fujihira at the Tokyo Institute of of Technology. Collagen extraction and preparation was pro vided by A. M. Romanic, L. D. Spotila, E. Adachi, Y. Hojima, D. J. Prockop, Ch. Fauser and J. Engel from the Jefferson Institute of Molecular Medicine, Philadelphia, and from the Biocenter of the University of Basel. Support from H. R. Hidber, A. Tonin and M. Monfreda in building elec tronics is gratefully acknowledged. This work was supported by the Swiss National Science Foundation and the "Komis sion zur Forderung der wissenschaftlichen Forschung." JVST B -Microelectronics and Nanometer Structures 1503 1M. Amrein. R. Diirr, S. Stasiak. H. Gross. and G. Travaglini. Science 243. 1708 (1989). 2R. Guckenberger, W. Wiegrabe. A. Hillebrand. T. Hartmann. Z. Wang. and W. Baumeister. Ultramicroscopy 31, 1,27 (1989). 3S. A. c. Gould. B. Drake. C. B. Prater. A. L. Weisenhorn. S. Manne. H. G. Hansma. P. K. Hansma. J. Massie. M. Longmire. V. Elings. B. Dixon. B. Mukergee, C. M. Peterson. W. Stoeckenius. T. R. Albrecht. and C. F. Quate, J. Vac. Sci. Techno!. A 8.369 (1990). 4B. Drake, C. B. Prater. A. L. Weisenhorn. S. A. C. Gould. T. R. Albrecht. C. F. Quate, D. S. Cannell. H. G. Hansma. and P. K. Hansma. Science 243. 1586 (1989). 5H. G. Hansma, J. Vesenka, C. Siegrist. G. Kelderman. H. Morett. R. L. Sinsheimer, V. Elings. C. Bustamante. and P. K. Hansma. Science 256. 1180 (1992) . • y L. Lyubchenko, P. I. Oden, D. Lampner. S. M. Lindsay. and K. A. Dunker. Nuc!. Acids Res. 21. 1117 (1991,1. 7y Martin. C. C. Williams. and H. K. Wickramansinghe. J. App!. Phys. 61, 4723 (1987). iM. Nonnenmacher. J. Greschner. O. Wolter. and R. Kassing. J. Vac. Sci. Techno!. B 9. 1358 (1991). 9Noncontact dynamic force microscopy data from DNA. TMV. and HPI layer were first presented at the workshop on "STM and AFM and Bio logical Objects" held in Paris. October 1992 at the Fondation Fourmentin-Gilbert by J. Frommer and E. Meyer. University of Base!. 1OD. Anselmetti. R. Luthi. E. Meyer. T. Richmond. M. Dreier. J. Frommer. and H.-J. Guntherodt. Nanotechnology (in press). "G. Meyer and N. M. Amer. App!. Phys. Lett. 53, 1045 (1988). 120. Anselmetti. Ch. Gerber. B. Michel, H.-J. Guntherodt. and H. Rohrer, Rev. Sci. Instrum. 63, 3003 (1992). I1We use a commercial rms-to-dc converter (AD 536) to convert all ampli tude signals, independently of signal shape. to a dc output voltage by taking the absolute value of the input signal. squaring it and dividing it by a feedback output. The dc output signal is fed into the DFM feedback electronics. The operating bandwidth was set to 30 kHz-2 MHz. which also allows working with very stiff force sensors. 14M. Nonnenmacher, Ph.D. thesis. Kassel. Germany. 1990. 150. Wolter, Th. Bayer. and J. Greschner. J. Vae. Sci Techno!' B 9. 1353 (1991). 16D. J. Keller and C. Chih-Chung. Surf. Sci. 268. 333 (19921. 17R. Liithi, E. Meyer, L. Howald. H. Haefke. D. Anselmetti. M. Dreier. M. Ruetschi. T. Bonner, R. M. Overney. J. Frommer. and H.-J. Guntherodt. Proceedings of Scanning Tunneling Microscopy 1993 (STM '93), Beijing, August 1993. IRThe DNA strands were supplied by the Institute Gustave Roussy. 94805 Villejuif Cedex. France. 19R. Overney, E. Meyer. J. Frommer. D. Brodbeck. R. Luthi. L. Howald. H.-J. Guntherodt, M. Fujihira. H. Takano. and Y Gotoh. Nature 359. 133 (1992). 20J. Frommer, R. Luthi. E. Meyer. D. Anselmetti. R. Overney. H.-J. Guntherodt. and M. Fujihira. Nature 364. 198 (19931. 21The investigated collagen is a natural mutation of procollagen I (human) [a2(1) G661--'S). 22A. M. Romanie, L. D. Spotila. E. Adachi. J. Engel. Y Hojima. and D. J. Prockop, 1. Bio!. Chern. (in press). 23J. Engel and D. J. Prockop. Annu. Rev. Biophys. Biophys. Chern. 20. \37 (1991). 24Digital Instruments, Santa Barbara. CA. 25M. Dreier and D. Anselmetti (in preparation). Redistribution subject to AVS license or copyright; see http://scitation.aip.org/termsconditions. Download to IP: 155.97.178.73 On: Wed, 26 Nov 2014 01:39:13

5.0007603.pdf

Appl. Phys. Lett. 116, 222404 (2020); https://doi.org/10.1063/5.0007603 116, 222404 © 2020 Author(s).Enhancing the soft magnetic properties of FeGa with a non-magnetic underlayer for microwave applications Cite as: Appl. Phys. Lett. 116, 222404 (2020); https://doi.org/10.1063/5.0007603 Submitted: 20 March 2020 . Accepted: 14 May 2020 . Published Online: 02 June 2020 Adrian Acosta , Kevin Fitzell , Joseph D. Schneider , Cunzheng Dong , Zhi Yao , Yuanxun Ethan Wang , Gregory P. Carman , Nian X. Sun , and Jane P. Chang ARTICLES YOU MAY BE INTERESTED IN The accurate measurement of spin orbit torque by utilizing the harmonic longitudinal voltage with Wheatstone bridge structure Applied Physics Letters 116, 222402 (2020); https://doi.org/10.1063/1.5145221 Chiral-anomaly induced large negative magnetoresistance and nontrivial π-Berry phase in half-Heusler compounds RPtBi (R=Tb, Ho, and Er) Applied Physics Letters 116, 222403 (2020); https://doi.org/10.1063/5.0007528 Magnetization switching by nanosecond pulse of electric current in thin ferrimagnetic film near compensation temperature Applied Physics Letters 116, 222401 (2020); https://doi.org/10.1063/5.0010687Enhancing the soft magnetic properties of FeGa with a non-magnetic underlayer for microwave applications Cite as: Appl. Phys. Lett. 116, 222404 (2020); doi: 10.1063/5.0007603 Submitted: 20 March 2020 .Accepted: 14 May 2020 . Published Online: 2 June 2020 Adrian Acosta,1 Kevin Fitzell,1 Joseph D. Schneider,2 Cunzheng Dong,3 ZhiYao,4 Yuanxun Ethan Wang,4Gregory P. Carman,2Nian X. Sun,3 and Jane P. Chang1,a) AFFILIATIONS 1Department of Chemical and Biomolecular Engineering, University of California, Los Angeles, California 90095, USA 2Department of Mechanical and Aerospace Engineering, University of California, Los Angeles, California 90095, USA 3Department of Electrical and Computer Engineering, Northeastern University, Massachusetts 02115, USA 4Department of Electrical and Computer Engineering, University of California, Los Angeles, California 90095, USA a)Author to whom correspondence should be addressed: jpchang@ucla.edu ABSTRACT An ultra-thin ( /C242.5 nm) non-magnetic Cu underlayer was found to have a significant effect on the microstructure, magnetic softness, and magne- tostriction of sputter-deposited Fe 81Ga19(FeGa) thin films. Compared to the experimental control where FeGa was deposited directly on Si with- out an underlayer, the presence of Cu increased the in-plane uniaxial anisotropy of FeGa and reduced the in-plane coercivity by nearly a factor offive. The effective Gilbert damping coefficient was also significantly reduced by a factor of four, between FeGa on Si and FeGa on a Cu underlayer. The FeGa films on Cu also retained a high saturation magnetostriction comparable to those without an underlayer. The enhancement of the desir- able magnetic properties for microwave applications is attributed to the Cu underlayer, promoting the (110) film texture and increasing the com-pressive film strain. The results demonstrated that the structural control is viable to simultaneously achieve the necessary magnetic softness andmagnetostriction in FeGa for integration in strain-mediated magnetoelectric and microwave devices. Published under license by AIP Publishing. https://doi.org/10.1063/5.0007603 The efficient control of magnetism at the nanoscale via voltage in magnetoelectric (ME) composites has a potential impact in severaltechnologically important areas, such as next-generation memory andlogic devices as well as microscale antenna devices. 1–3One of the major material design challenges in the coupling and efficiency of strain-mediated ME devices is the lack of ferromagnetic materials that exhibit both high magnetostriction and magnetic softness to achieve ahigh magnetomechanical coupling. Fe xGa1/C0xis a ferromagnetic material that exhibits fairly high magnetostriction in bulk and polycrystalline alloys, which makes itpromising for strain-mediated ME devices. 4–8One of the barriers for high frequency applications of FeGa films has been their large ferro-magnetic resonance (FMR) linewidth, which is typically observed inthe range of /C24620–700 Oe at the X-band for the [100] easy axis. 9,10 However, more recent works have shown that the fabrication of high quality epitaxial films can exhibit greatly enhanced high frequencyproperties achieving narrower FMR linewidths in the range of/C2480–220 Oe at the X-band along the [100] easy axis. 11,12Indeed, thedeposition parameters can have a significant influence on the struc- ture, magnetic softness, and magnetostrictive properties of FeGa thinfilms and have been well documented. 13–17Along that line, researchers have also explored the addition of C, B, and N to FeGa thin films topromote soft magnetic properties by reducing their grain size anddiminishing their magnetocrystalline anisotropy. 4,18,19 A number of studies have previously explored the effect of under- layers (such as Ru, NiFe, Cu, Co, etc.) between the substrate and sput-tered FeCo thin films on their soft magnetic properties. 20–27The observed enhancement in the soft magnetic properties of FeCo hasbeen attributed to the effect of the underlayer on the interfacial magne-toelastic energy. 20,28–30Similarly, in this work, the strategy of using a non-magnetic underlayer (Cu) was explored to study its effect on themicrostructure and texture of the FeGa film, which dictates the attain-able magnetic softness and magnetostriction. The FeGa and Cu films in this study were grown via DC magne- tron sputter deposition using an ULVAC JSP 8000 sputter system with a base pressure of 2 /C210 /C07Torr at room temperature. Si (100) Appl. Phys. Lett. 116, 222404 (2020); doi: 10.1063/5.0007603 116, 222404-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplsubstrates were used for all the depositions without any initial removal of the native oxide. The FeGa films were formed using a target with an Fe to Ga ratio of 80/20 at a DC bias power of 200 W and an Ar pres- sure of 0.5 mTorr; the Cu underlayer was sputtered at a DC bias power of 100 W and an Ar pressure of 0.5 mTorr. Scanning electron micros- copy (SEM) imaging was used to calibrate the growth rate of the films. The thickness of the FeGa films was measured to be 100.3 61.7 nm (see Fig. S1); the nominal value is henceforth referred to as 100 nm. The relative composition of the FeGa films was measured at Fe:Ga (81.463.0):(18.6 60.5), which was determined via x-ray photoelec- tron spectroscopy (XPS) with a mono-chromated Al K asource (see Fig. S2). The structures of the films were characterized via x-ray dif- fraction (XRD) using a Panalytical X’Pert Pro X-ray Powder Diffractometer with a Cu K asource and Fityk software package.31 The room temperature magnetic properties of FeGa thin films were measured via superconducting quantum interference device (SQUID) magnetometry using a Quantum Design MPMS3. The high- frequency FMR linewidth was measured using a short-circuited strip line connected to a vector network analyzer (VNA). For these mea- surements, the samples were placed facing the strip line and a large saturating magnetic field was first applied parallel to the strip line to establish a baseline for the measurement. The reflection coefficient (S11) was then measured as a function of bias magnetic field (0–600 Oe) and frequency (100 MHz to 6 GHz).32 Magnetostrictive characterization was performed by depositing FeGa, with and without a Cu underlayer, on thin Si cantilevers (100lm thickness). An MTI-2000 fiber-optic sensor was used to detect the deflection of the cantilever tip due to changes in the internal stress of the FeGa thin films, with details described elsewhere.18 In this work, thin films of Fe 81Ga19(100 nm) were deposited either directly onto Si substrates or with a thin 2.5 nm Cu underlayer. Figure 1 shows the in-plane magnetic hysteresis (MH) loops for the 100 nm FeGa films deposited on Si with and without a Cu underlayer normalized to the saturation magnetization (see Fig. S3 for the full MH loops at high fields). The FeGa film deposited directly onto a Sisubstrate, without the Cu underlayer, shows a coercivity of 84 Oe. For an FeGa film deposited onto a 2.5 nm Cu underlayer, a much smaller coercivity of 17 Oe was achieved. These results are consistent with those previously observed for FeCo films in which a Cu underlayerpromotes a large decrease in in-plane coercivity. 20,24,33Additionally, the FeGa films deposited with a Cu underlayer displayed an enhanced uniaxial anisotropy, as observed from the increase in remnant magnetization. The high-frequency characteristics of FeGa films deposited on Si with and without a Cu underlayer were studied using broadband FMR spectroscopy. Figure 2 shows the S 11absorption as a function of mag- netic bias field (0–600 Oe) at a fixed frequency of 6 GHz. These S 11 absorption spectra are cross sections of the entire FMR spectra overthe frequency range of 100 MHz–6 GHz (inset). For a 100 nm FeGa film deposited without an underlayer, the FMR spectra are characterized by a very low peak absorption and very broad FMR linewidth ( >600 Oe at 6 GHz) that extends beyond the maximum magnetic field applied. In contrast, the FeGa film deposited on a Cu underlayer is characterized by a significant enhancement in the FMR response with a narrow linewidth of /C24190 Oe at 6 GHz. The effective Gilbert damping coefficient, a eff, was calculated by fitting the FMR linewidth of the absorption as a function of frequency for the FMR spectra from 3 GHz to 6 GHz to the equation: DH¼2aeffx=cþDH0,w h e r e xis the frequency, cis the gyromag- netic ratio ( /C252.8 MHz/Oe), and DH0is the frequency-independent linewidth broadening (see Fig. S4). The value of aefffor FeGa on Si was 0.2160.11 but decreased to 0.05 60.01 when grown with a Cu underlayer. The enhanced soft magnetic properties of the FeGa film grown on an underlayer originated from the effect of the underlayer on its microstructure. Structural characterization of the FeGa films was first FIG. 1. In-plane magnetic hysteresis loops of 100 nm FeGa sputtered on a Si sub- strate with and without a Cu underlayer. FIG. 2. S11absorption spectra as a function of magnetic bias field at 6 GHz for 100 nm FeGa films sputtered with and without a Cu underlayer. The insets showFMR spectra as a function of both frequency (100 MHz–6 GHz) and magnetic bias(0–600 Oe).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 116, 222404 (2020); doi: 10.1063/5.0007603 116, 222404-2 Published under license by AIP Publishinginvestigated using XRD. Figure 3 shows the spectra highlighting the bcc (110) diffraction for a 100 nm FeGa film sputtered with and with- out a Cu underlayer. The films deposited onto a Cu underlayer display a large shift of the (110) diffraction line position, which is caused byr e l a t i v ec h a n g e si nt h es t r a i no ft h efi l m s .T h es h i f ti nt h ep e a kp o s i t i o nrepresents a relative increase of 0.28% compressive strain for the FeGa film on Cu compared to FeGa deposited directly onto a Si substrate. The FeGa film deposited with a Cu underlayer also showed an increase ( /C2430%) in the intensity of the (110) diffraction peak as com- pared to that without an underlayer, indicating an increased (110) polycrystalline texture. This enhancement can be attributed to the close lattice match of the FeGa (110) film texture (d ¼2.05860.001 A ˚) during growth to the underlying Cu (111) film texture (d ¼2.087 60.001 A ˚), which is highlighted in supplemental Fig. S5. This is con- sistent with a previous study that used Cu as a buffer layer to achieve epitaxial growth of FeGa films by encouraging a (110) crystalline tex-ture along the growth direction. 34However, a broader comparison of t h ei m p a c to fC ua sab u f f e rl a y e rf o rs p u t t e r e dF e G afi l m so nt h e i rs o f t magnetic and magnetostrictive properties has not been studied, which is the focus of this work. Furthermore, it is expected that there is areduction in the grain width for the FeGa films with a Cu underlayer(evidenced by the underlayer studies for FeCo films 20,25–27); however, while the trends in the full-width at half-maximum of the peaks can generally provide insight into the grain size, the XRD data only capture the lattice spacing for out-of-plane diffraction and would not be appro-priate for examining changes in the width of the grains that would bein-plane to the film. In order to obtain magnetostriction measurements for FeGa films, a perpendicular AC magnetic field is applied along the short axis of thesilicon cantilever during the measurement, while a 100 Oe bias field isinitially applied and held constant in the long axis in order to saturate the magnetization and assess the full magnetostriction. The magnetic field-induced stress, b, is calculated from the deflection at the cantilevertip using the following relation from Ref. 35:b¼/C0 dt 2 sEs=3tfl2ð1þvsÞ,where dis the deflection, tsandtfare the substrate and film thicknesses (100lm and 100 nm, respectively), lis the distance between the clamp- ing edge and the probe location (27 mm), and Esand/C23sare the Young’s modulus and Poisson ratio of the Si substrate (169 GPa and 0.069,respectively, along the [110] in-plane direction for a Si (100) substrate 36). Note that the elastic properties of the Cu underlayer are neglected from the calculation as the behavior of the underlayer is dominated by the bulk of the Si substrate. For thin films, the magnetic field-induced stress is considered the more relevant parameter to describe magnetostrictive effects since thelateral deformation is blocked by the substrate, and one can measure only the stress. This also avoids the need to measure the elastic proper- ties of thin films, which can be difficult. For comparison to other litera-ture studies on magnetostrictive thin films, the magnetostriction interms of strain can be assessed from the relation of k¼/C0 2 31þ/C23f Ef/C16/C17 /C2b,w h e r eE fand v fare the Young’s modulus and Poisson ratio of the FeGa film, which are approximated following the convention thatEf 1þ/C23f/C16/C17 ¼50 GPa.37 From the data in Fig. 4 , it is found that the FeGa film deposited without an underlayer reaches a maximum magnetic field-inducedstress of 7.4 MPa, corresponding to a magnetostriction of 99 ppm. Thefilm grown on the Cu underlayer largely maintains a comparable level of magnetostriction (95 ppm), displaying a maximum magnetic field- induced stress of 7.2 MPa. The importance of the results here is tohighlight that the soft magnetic properties of the FeGa films can beenhanced without trade-off of the high magnetostriction values. In summary, in order to enhance the soft magnetic properties of sputtered FeGa thin films, a strategy of using a thin 2.5 nm Cu under-layer between FeGa and the Si substrate was explored. It was foundthat an 80% decrease in coercivity and a 75% decrease in the effectiveGilbert damping coefficient can be achieved by using Cu as an under- layer. It is observed that an underlayer serves to influence the FIG. 3. XRD spectra of the main bcc(110) FeGa peak when grown with and with- out a Cu underlayer. Solid lines are the best voigt fit of the data in circles. Verticaldashed lines are used to highlight the shift in the (110) peak across samples. FIG. 4. (Left axis): calculated magnetostriction for 100 nm FeGa sputtered with and without a Cu underlayer as a function of the AC magnetic field (along the short axis of the cantilever sample). (Right axes): directly measured cantilever deflection and corresponding calculated stress. An initial bias field of 100 Oe was applied to satu-rate the magnetization along the long axis of the cantilever sample and held con-stant during the measurement.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 116, 222404 (2020); doi: 10.1063/5.0007603 116, 222404-3 Published under license by AIP Publishingmicrostructure of the FeGa films, resulting in an increased (110) poly- crystalline texture and an increase in compressive film strain for the FeGa films. The saturation magnetostriction is largely retained for anFeGa film grown with a Cu underlayer (95 ppm) compared to the FeGa film without an underlayer (99 ppm). These results demonstrate that the underlayer strategy is effective at providing structural control to simultaneously enhance the mag-netic softness of FeGa while retaining its desirable magnetostrictive properties. High magnetostriction, low effective Gilbert damping, and low coercivity of FeGa grown on a lattice-matched underlayer such asCu make it an attractive material for strain-mediated ME and other microwave device applications. See the supplementary material for SEM imaging, XPS scans, full scale in-plane MH loops for the FeGa films, and complete XRD spec-tra of FeGa, Cu, and Si substrate materials. We acknowledge the use of the fabrication facility at the Integrated Systems Nanofabrication Cleanroom (ISNC) and the Molecular Instrumentation Center (MIC) at the California NanoSystems Institute (CNSI) at UCLA. This work was alsosupported by the NSF Nanosystems Engineering Research Center for Translational Applications of Nanoscale Multiferroic Systems (TANMS) under the Cooperative Agreement Award (No. EEC-1160504). Ryan Sheil is acknowledged for performing the XRD and XPS measurements. 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