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1.5087227.pdf	AIP Advances 9, 065002 (2019); https://doi.org/10.1063/1.5087227 9, 065002 © 2019 Author(s).Magnetic anisotropy of half-metallic Co2FeAl ultra-thin films epitaxially grown on GaAs(001) Cite as: AIP Advances 9, 065002 (2019); https://doi.org/10.1063/1.5087227 Submitted: 30 December 2018 . Accepted: 23 May 2019 . Published Online: 04 June 2019 Bolin Lai , Xiaoqian Zhang , Xianyang Lu , Long Yang , Junlin Wang , Yequan Chen , Yafei Zhao , Yao Li , Xuezhong Ruan , Xuefeng Wang , Jun Du , Wenqing Liu , Fengqiu Wang , Liang He , Bo Liu , and Yongbing Xu ARTICLES YOU MAY BE INTERESTED IN Reducing virtual source size by using facetless electron source for high brightness AIP Advances 9, 065001 (2019); https://doi.org/10.1063/1.5098528 Design and development of a test rig for the performance evaluation of automotive exhaust thermoelectric generator AIP Advances 9, 065004 (2019); https://doi.org/10.1063/1.5093587 Experimental and numerical study on the drainage performance and fluid flow of Venturi tubes AIP Advances 9, 065003 (2019); https://doi.org/10.1063/1.5099420AIP Advances ARTICLE scitation.org/journal/adv Magnetic anisotropy of half-metallic Co 2FeAl ultra-thin ﬁlms epitaxially grown on GaAs(001) Cite as: AIP Advances 9, 065002 (2019); doi: 10.1063/1.5087227 Submitted: 30 December 2018 •Accepted: 23 May 2019 • Published Online: 4 June 2019 Bolin Lai,1 Xiaoqian Zhang,1Xianyang Lu,1 Long Yang,1Junlin Wang,2 Yequan Chen,1 Yafei Zhao,1Yao Li,1Xuezhong Ruan,1Xuefeng Wang,1 Jun Du,3Wenqing Liu,1,4 Fengqiu Wang,1 Liang He,1,a) Bo Liu,5 and Yongbing Xu1,2,a) AFFILIATIONS 1Jiangsu Provincial Key Laboratory of Advanced Photonic and Electronic Materials, Collaborative Innovation Center of Advanced Microstructures, School of Electronic Science and Engineering, Nanjing University, Nanjing 210093, China 2York-Nanjing Joint Centre (YNJC) for Spintronics and Nano Engineering, Department of Electronics, The University of York, YO10 3DD, United Kingdom 3National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China 4Department of Electronic Engineering, Royal Holloway University of London, Egham TW20 0EX, United Kingdom 5Zhejiang Hikstor Technology Co., Ltd., Hangzhou 311305, China a)Authors to whom correspondence should be addressed: heliang@nju.edu.cn and ybxu@nju.edu.cn ABSTRACT Single crystalline Co 2FeAl ﬁlms with different thicknesses varying from 3.6 to 10.6 nm have been grown on GaAs (001) using Molecule Beam Epitaxy (MBE). The magnetic characteristics were investigated by in-situ magneto-optical Kerr effect (MOKE). For all the samples, the angle dependent magnetization energy has a relatively high and steep peak around [110] direction which is the hard axis, and a wide basin from [1¯10] to [100] which is the range of the easy axis. More interestingly, the magnetic anisotropy includes a strong uniaxial component due to the Co 2FeAl/GaAs interface, a cubic one from Co 2FeAl crystalline structure, and an unexpected localized anisotropy term around the [110] direction. All the three anisotropy components overlap their own hard axis around [110] direction resulting in a steep energy barrier, which leads to unusual inverted hysteresis loops around [110]. Our ﬁndings add a building block for using half-metallic Co 2FeAl thin ﬁlms in the application of magnetic storage devices. ©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/10.1063/1.5087227 I. INTRODUCTION Half metallic ferromagnets (HMFs) are ideal materials to be employed in spintronic devices since they possess 100% spin polar- ization at the Fermi level.1Among HMFs, Co 2FeAl (CFA) as one kind of Heusler alloys at present is one of the most promising spintronic materials because of its very low Gilbert damping con- stant2and high Curie temperature ( ∼1000K).3Recently, a giant tunnel magnetoresistance (TMR) ratio of 360% at room temper- ature has been reported on the magnetic tunnel junctions (MTJs) of Cr/CFA(30)/MgO(1.8)/CoFe(0.5)/CFA(5) (unit: nm).4However, the use of CFA as a ferromagnetic (FM) electrode in devices needs precise knowledge and control of its magnetic properties. In this sense, one of the key parameters is the magnetic anisotropy,which is inﬂuenced by the substrate, orientation, growth tempera- ture and thickness.5,6 Until now, the magnetic anisotropy of CFA was mainly studied on thick CFA ﬁlms.6–9However, CFA as a free layer in MTJ is usually of only several nanometers thickness.4,10–12Therefore, the magnetic anisotropy of such ultra-thin CFA ﬁlms needs further investigation. II. EXPERIMENTAL DETAILS Single crystalline CFA thin ﬁlms with thicknesses ranging from 3.6 to 10.6 nm grown on GaAs (001) substrates have been pre- pared using Molecule Beam Epitaxy (MBE). GaAs (001) is chosen as the substrate because of the very small lattice mismatch between GaAs (001) (a=5.653 Å) and CFA (a=5.730 Å).13Prior to deposition, AIP Advances 9, 065002 (2019); doi: 10.1063/1.5087227 9, 065002-1 © Author(s) 2019AIP Advances ARTICLE scitation.org/journal/adv the substrates were annealed in-situ at 540○C to remove the oxida- tion layer. During deposition, the substrates were kept at 300○C, and the growth rate of CFA was 0.08 Å/s. The base pressure in the deposition chamber was less than 3 ×10-10mbar. Reﬂection high energy electron diffraction (RHEED) was used to monitor the in-situ growth dynamics with the electron beam along the [1 ¯10] and [100] directions. Upon the deposition of the CFA layer, in-situ lon- gitudinal magneto-optical Kerr effect (MOKE) was applied to mea- sure the magnetic properties of the ﬁlms as soon as the substrates were cooled down to room temperature. This in-situ measurement avoids the inﬂuence of the capping layer and/or surface contamina- tion and gives us the chance to measure the intrinsic properties of the CFA ﬁlms. Before being taken out of the growth chamber, all ﬁlms were capped with a 2 nm Al layer to avoid oxidation. Atomic force microscope (AFM) was employed to characterize the surface morphology. The crystal structure of the samples was measured by X-ray diffraction (XRD). III. RESULTS AND DISCUSSIONS Figure 1(a) and (b) display the RHEED patterns of a 6.4 nm CFA ﬁlm along [1 ¯10] and [100] respectively. Streaky diffraction lines show high quality single crystal ﬁlms with a quasi-2D surface. The 2×1 peaks appearing in Figure 1(a) also imply the well-ordered FIG. 1 . RHEED patterns of a 6.4 nm Co 2FeAl (001) ﬁlm along (a) [ 1¯10] and (b) [100] respectively, demonstrating a quasi-2D growth mode. The 1/2 peaks in a) also imply the well-ordered surface. (c) An AFM image of the 6.4 nm Co 2FeAl sample capped by a 2 nm Al ﬁlm. The roughness is 0.88nm, demonstrating a smooth surface morphology.atoms on the surface.14Root mean square (RMS) roughness obtained from the AFM image as shown in Figure 1(c) is 0.88 nm, demonstrating a smooth surface of the 6.4 nm CFA sample after Al capping. XRD results suggest a B2 structure without any phase sep- aration (not shown here). Detailed XRD results are included in our previous reports.15 In-plane magnetic hysteresis loops have been obtained along different orientations by the in-situ longitudinal MOKE measure- ment. Figure 2 shows some representative hysteresis loops of sam- ples with different thicknesses. The external magnetic ﬁeld was applied along the angle ', which is deﬁned between the ﬁeld direction and the [1 ¯10] direction within the (001) plane. For all the samples, square hysteresis loops appear from 0○to 45○, while “hard” loops lie between 90○to 105○. More interestingly, when ' approaches to 90○and 105○the thin samples (3.6 nm and 5.0 nm) demonstrate unusual inverted hysteresis loops, characterized by a negative remanence and negative coercive ﬁeld. The inverted hys- teresis loops are also observed in the samples of 6.4 nm, 7.0 nm and 10.6 nm along 105○direction. Figure 3(a) displays the normalized remanent magnetization (M r/M S) extracted from the hysteresis loops as a function of 'for the 3.6 nm sample. The M r/M Scurve has a wide plateau from 0○to 60○which demonstrates a range of easy axes. Due to the inverted hysteresis loop, remanent magnetization becomes negative at '= 90○(270○) and 105○(285○), suggesting the hard axis lies in this range. To explore the origin of this observed negative remanent magnetization, the normalized magnetization energy ( εM) as a func- tion of 'for the 3.6 nm sample is plotted in Figure 3(b), which is determined by numerical integration of the hysteresis loops:16 εM=/integral.dispMS MH=0H dM . (1) where MSis the saturation magnetization and MH=0represents the magnetization when the external ﬁeld H is zero. MSat room tem- perature was determined by vibrating sample magnetometer (VSM) and is almost a constant of ∼1000 emu/cm3for all the ﬁlms.15A high asymmetric energy barrier rises around 90○(270○). It has a sheer cliff from 75○to 90○yet a relatively gentle slope on the other side. And there is an energy basin from 0○(180○) to 45○(225○). Similar asymmetric εM(') or M r/M S(') has also been reported in ultra-thin magnetic metal ﬁlms such as Fe(001)/Au(001),17 Fe(001)/GaAs(001),18,19and CoFeB/GaAs(001).20But the physics behind the asymmetric barrier around the hard axis was not fully discussed. Here a quantitative analysis of the asymmetric barrier has been carried out. The crystalline structure of CFA is cubic, belong- ing to the Fm¯3mspace group. Therefore, an in-plane fourfold mag- netic anisotropy are expected in the (001) epitaxial ﬁlms.21However, a dominant in-plane uniaxial magnetic anisotropy has often been reported in CFA22,23or other Heusler alloy20,24–26ﬁlms grown on GaAs(001) substrates. A uniaxial anisotropy term along with a cubic one was used to ﬁt the εM(')16,17,27but this asymmetric εM(') can- not be ﬁtted well because of the steep asymmetric potential barrier. Therefore, a third phenomenological anisotropy term is introduced to describe this potential barrier and is localized in a limited range around 90○(270○): εM(φ)=−1 4Keff csin2(2(φ−θc))+Keff usin2(φ−θu) + Asin(5(φ−θa)). (2) AIP Advances 9, 065002 (2019); doi: 10.1063/1.5087227 9, 065002-2 © Author(s) 2019AIP Advances ARTICLE scitation.org/journal/adv FIG. 2 . Hysteresis loops of Co 2FeAl thin ﬁlms with different thicknesses measured by in-situ MOKE. The angle 'is deﬁned between the external ﬁeld direction and [ 1¯10] direction (as shown in Figure 3(a)). '= 90○represents the [110] direction. The blue (red) lines represent the MOKE signals while the external ﬁeld decreases (increases), respectively. For all the samples, square hysteresis loops appear around 0○to 45○. Unusual inverted hysteresis loops appear along 90○for the ﬁlms of 3.6 nm and 5.0 nm and along 105○for 6.4 nm, 7.0 nm and 10.6 nm samples. Keff candKeff urepresent the effective cubic and uniaxial anisotropy coefﬁcients respectively. A is the magnitude of the third localized term and is nonzero only when θa−36○≤φ≤θa+ 36○or θa+ 144○≤φ≤θa+ 216○.θcand θuaccount for small rotationsof the anisotropy axes and are conﬁned to values ≤10○as shown in Figure 3(e), which is probably due to experimental error. The red solid line shown in Figure 3(b) is the ﬁtted result while the uniax- ial, the cubic and the localized components are plotted here in blue, FIG. 3 . (a) The normalized remanent magnetization (M r/MS) as a function of the angle 'in polar coordinate for the 3.6 nm Co 2FeAl sample. Remanent magnetization is negative at '= 90○(270○) and 105○(285○). (b) Normalized magnetization energy ( εM) for the 3.6 nm Co 2FeAl sample. Red squares are experimental data. The red line is the ﬁtted result, and other three lines are the uniaxial (blue), the cubic (orange), and the localized (purple) components. (c) Normalized magnetization energy for all the Co2FeAl samples with different thicknesses. The symbols are experimental data and the solid lines represent ﬁtted results. (d)-(e) The thickness dependent ﬁtted results. As the thickness increases Keff uandAdecreases, while Keff cstays constant. AIP Advances 9, 065002 (2019); doi: 10.1063/1.5087227 9, 065002-3 © Author(s) 2019AIP Advances ARTICLE scitation.org/journal/adv orange and purple lines respectively. The uniaxial anisotropy has its easy axis along around 0○(180○) direction, while the easy axes of the cubic one is along 45○(225○) and 135○(315○). The hard axes of these three anisotropies overlap on around 90○, leading to the steep energy barrier. Figure 3(c) demonstrates the normalized εM(') curves of all CFA samples. They all exhibit a steep barrier and a wide basin and can be ﬁtted well by Equation (2), as indicated by the solid lines. The wide basin of εM(') between 0○to 45○implies a wide range of easy axes, which means the magnetic moment can rotate freely within these angles. This property is potentially useful in the Spin Transfer Torque Magnetic Random Access Memory (STT-MRAM) devices utilizing the spin transfer torque of a spin polarized current to ﬂip the free layer. The torque is deﬁned as /uni20D7τ=η 2µ0eMstFJm→×(m→×M→), where M→is the unit magnetization vector of the free layer and M→ is the spin polarization direction of the incoming current from the ﬁxed layer.28But at the beginning of the ﬂip, the effective torque is zero when m→is antiparallel with the ﬁxed layer. And the device relies on the thermal activation to rotate the magnetic moment away from the easy axis. With the easy axis basin, the required ther- mal activation energy is lowered, facilitating the magnetic moment rotation by STT.Figure 3(d)–(e) show Keff u,Keff c,A,θc,θuandθaextracted from the ﬁtted data as a function of ﬁlm thickness. Keff cis nearly constant within the error range, which is consistent with the bulk-related cubic anisotropy. On the other hand, Keff uand Adecrease as the ﬁlm thickness increases, suggesting the uniaxial and the third local- ized anisotropy are interface related properties. But the third term is weaker than the uniaxial anisotropy and can only be observed in ultra-thin ﬁlms. They are probably due to the preferable stack- ing of Co on GaAs along [110] with Co-Ga bonding, similar to Co-Ga bonding at Co 2CrAl/GaAs interface.29The demonstration of the CFA/GaAs interface remains a future issue which requires in-plane XRD measurements or cross-sectional transmission elec- tron microscopy (TEM). With the angle dependent magnetization energy εM('), the observed unusual inverted hysteresis loop can be explained as fol- lows. Figure 4(a) shows the hysteresis loop of the 3.6 nm CFA sam- ple with the ﬁeld applied along 90○. The rotation of the magnetic moment at different ﬁeld value is also illustrated in the insets of Figure 4(a) and the magnetization energy is marked in Figure 4(b) correspondingly. At position A, the magnetization lies at 90○along with the external ﬁeld +H max. When the ﬁeld decreases, due to the high energy barrier along the extremely hard axis at about 85○ FIG. 4 . (a) The unusual inverted hysteresis loop of the 3.6 nm Co 2FeAl sample with the external ﬁeld applied along 90○, or [110] direction. The insets demonstrate the mag- netic moment (black arrows) at different external ﬁelds (blue arrows). The orange arrows are the projection of the mag- netic moment along the external magnetic ﬁeld direction (90○). Position B and F demonstrate the negative remnant magnetization at zero external magnetic ﬁeld. (b) Schematic diagram of the anisotropic energy of the 3.6 nm sample. Black circles are experimental data and the purple line is to guide the eye. The arrows demonstrate the movement of the magnetization in the potential space as the external magnetic ﬁeld sweeps. At around 85○(265○), the potential barrier is so high, that the magnetization jumps from G to H (C to D). AIP Advances 9, 065002 (2019); doi: 10.1063/1.5087227 9, 065002-4 © Author(s) 2019AIP Advances ARTICLE scitation.org/journal/adv (as shown in Figure 4(b)), the magnetization moment rotates towards 180○rather than towards 0○. When the external ﬁeld decreases to zero, the magnetic moment reaches the center of the easy axis basin range, that is, position B (also shown in Figure 4(b)). And its projection to the external ﬁeld is pointing to 270○, thus leaving a negative remnant magnetization at position B. As the ﬁeld further decreases, the magnetic moment keeps rotating towards 270○direction continually up to position C. After that, the mag- netic moment jumps to the other side of the barrier (about 265○ in Figure 4(b)) and reaches position D. This is because it requires much more energy for the moment to rotate continually towards the high energy barrier. With the external ﬁeld decreasing to –H max, the moment ﬁnally approaches around 270○, position E. Position E-J shows the reverse process of the magnetization moment rotation with the external ﬁeld increasing from –H maxto +H max. The high energy barrier around 85○and the wide energy basin together lead to the inverted hysteresis loops. As the ﬁlm thickness increase, the barrier becomes weak and the inverted hysteresis loop vanishes. IV. CONCLUSIONS In conclusion, we have studied the angle dependent magnetiza- tion energy of ultra-thin CFA ﬁlms epitaxially grown on GaAs (001) substrates. A strong uniaxial magnetic anisotropy, a weak cubic one, and a third localized one around [110] direction have been found in all the ﬁlms. These three components overlap their own hard axis around [110] direction leading the steep magnetization energy bar- rier at [110] and a wide energy basin from [1 ¯10] to [100]. This leads to the observation of unusual inverted hysteresis loops around [110] direction. Our ﬁndings add a building block for using half-metallic CFA thin ﬁlms in the application of magnetic storage devices. SUPPLEMENTARY MATERIAL See supplementary material for the conﬁrmation of the inverted hysteresis loops. ACKNOWLEDGMENTS This work is supported by the National Key Research and Development Program of China (No. 2016YFA0300803, 2017YFA0206304), the National Basic Research Program of China (No. 2014CB921101), the National Natural Science Foundation of China (No. 61427812, 11774160, 11574137, 61474061, 61674079, 51771053), Jiangsu Shuangchuang Program, the Natural Science Foundation of Jiangsu Province of China (No. BK20140054), and UK EPSRC EP/S010246/1. REFERENCES 1R. A. de Groot, F. M. Mueller, P. 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1.4870291.pdf	Modelling current-induced magnetization switching in Heusler alloy Co2FeAl-based spin-valve nanopillar H. B. Huang, X. Q. Ma, Z. H. Liu, C. P. Zhao, and L. Q. Chen Citation: Journal of Applied Physics 115, 133905 (2014); doi: 10.1063/1.4870291 View online: http://dx.doi.org/10.1063/1.4870291 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/115/13?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Simulation of multilevel cell spin transfer switching in a full-Heusler alloy spin-valve nanopillar Appl. Phys. Lett. 102, 042405 (2013); 10.1063/1.4789867 Micromagnetic simulation of spin-transfer switching in a full-Heusler Co2FeAl0.5Si0.5 alloy spin-valve nanopillar J. Appl. Phys. 110, 033913 (2011); 10.1063/1.3619773 Spin-transfer switching in an epitaxial spin-valve nanopillar with a full-Heusler Co 2 FeAl 0.5 Si 0.5 alloy Appl. Phys. Lett. 96, 042508 (2010); 10.1063/1.3297879 Micromagnetic simulations of current-induced magnetization switching in Co ∕ Cu ∕ Co nanopillars J. Appl. Phys. 102, 093907 (2007); 10.1063/1.2800999 Current-induced magnetic switching in nanopillar spin-valve systems with double free layers J. Appl. Phys. 101, 09A512 (2007); 10.1063/1.2714314 [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: 130.113.86.233 On: Tue, 17 Feb 2015 12:23:30Modelling current-induced magnetization switching in Heusler alloy Co2FeAl-based spin-valve nanopillar H. B. Huang,1,2X. Q. Ma,2Z. H. Liu,2C. P . Zhao,2and L. Q. Chen1 1Department of Materials Science and Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802, USA 2Department of Physics, University of Science and Technology Beijing, Beijing 100083, China (Received 6 February 2014; accepted 21 March 2014; published online 2 April 2014) We investigated the current-induced magnetization switching in a Heusler alloy Co 2FeAl-based spin-valve nanopillar by using micromagnetic simulations. We demonstrated that the elimination of the intermediate state is originally resulted from the decease of effective magnetic anisotropy constant. The magnetization switching can be achieved at a small current density of 1.0 /C2104A/cm2 by increasing the demagnetization factors of x and y axes. Based on our simulation, we found magnetic anisotropy and demagnetization energies have different contributions to the magnetization switching. VC2014 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4870291 ] I. INTRODUCTION In the past decades, spin transfer torque (STT)1,2has attracted considerable attention due to its application in high density magnetic random access memory (MRAM).3–7Spin polarized electrons carried spin angular momenta from theﬁxed layer to the free layer. It causes free layer to switch when the current density exceeds a critical current density J c. However, the critical current density required to induce mag-netization switching in the spin-valves is as high as 10 6–108A/cm2, and it is challenging to reduce J cto achieve the compatibility with highly scaled complementary metal-oxide-semiconductor technology while maintaining thermal stability. 8–12Recently, Heusler alloys13with lower saturation magnetization M s, smaller Gilbert damping constant a,a n d higher spin polarization constant gare demonstrated to be excellent candidates for reducing J ccompared to the normal metal and metallic alloys, i.e., Fe,14,15Co,16–18CoFe,19,20 Py,21–23and CoFeB.24–27 Existing experimental work demonstrated that J cof Co2MnGe, Co 2FeSi, and Co 75Fe25spin-valves were 1.6/C2107,2 . 7/C2107, and 5.1 /C2107J/cm2, respectively.28Spin transfer torque switching was also achieved experimentally in Co2FeAl 0.5Si0.5(CFAS)-based spin valve, exhibiting a two-step magnetization switching.29In our previous work, we demonstrated that the two-step switching was resulted from the four-fold magnetic anisotropy of CFAS and asymmetricspin transfer torque. 30Based on the two-step switching, we continued to develop a multilevel bit spin transfer multi-step magnetization switching by changing the magnetic anisot-ropy. 31Recently, Sukegawa et al.32reported the spin transfer switching of Heusler Co 2FeAl (CFA)-based spin valve, where the intermediate state was not found in the current-inducedmagnetization switching. However, there has been no expla- nation for the absence of the intermediate state in full-Heusler CFA spin valve nanopillar. Furthermore, the critical currentdensity of 2.9 /C210 7A/cm2due to the enhancement of the Gilbert damping constant of CFA is too large for the applica- tion. Therefore, a theoretical understanding of spin transferswitching of CFA-based nanopillar is necessary to reduce the critical current density. In this paper, we investigated the effects of magnetic anisotropy and demagnetization in the spin transfer torque switching of a Heusler alloy CFA-based spin-valve nanopil- lar by using micromagnetic simulations. We demonstratedthat the elimination of the intermediate state results from the decrease of effective magnetic anisotropy constant. In addition, the critical current density of magnetizationswitching can be reduced to 1.0 /C210 4A/cm2by increasing the demagnetization factors along x and y axes. We also discussed the effects of magnetic anisotropy and demagnet-ization ﬁelds. II. MODEL DESCRIPTION Figure 1(a) shows the geometry of spin-valve CFA (30 nm)/Ag (4 nm)/CFA (2 nm) and the elliptical cross sec- tion area is 250 /C2190 nm2. We employ a Cartesian coordi- nate system where the x-axis is the long axis of the ellipsealong the CFA [110] direction (easy axis) and the y-axis is along the short axis ([ /C22110]). The two CFA layers are sepa- rated by a thin Ag layer, and the bottom CFA layer is thefree layer whose magnetization dynamics is triggered by a spin-polarized current. The top CFA layer is the pinned layer with its magnetization vector Pﬁxed in the direction along the positive x axis. The initial magnetization vector Mof the layer is along the negative or positive x axis. The positive current is generally deﬁned as electrons ﬂowing from thepinned layer to the free layer. In this paper, we focus on the effect of the magnetic anisotropy and demagnetization ener- gies on magnetization switching. As shown in Figure 1(b), we observed a four-fold magnetic anisotropy ﬁeld along x and y axes, and the demagnetization ﬁeld along x, y, and z axes. Due to the ultrathin ﬁlm, the thickness of the free layeris much more lesser than its lateral dimensions. The presence of an out-of-plane component of magnetization leads to a large demagnetization ﬁeld perpendicular to the plane of thelayer. This demagnetization ﬁeld forces the magnetization 0021-8979/2014/115(13)/133905/5/$30.00 VC2014 AIP Publishing LLC 115, 133905-1JOURNAL OF APPLIED PHYSICS 115, 133905 (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: 130.113.86.233 On: Tue, 17 Feb 2015 12:23:30vector of the free layer to precess along a direction normal to the ﬁlm plane and impede the magnetization switching. The magnetization dynamics is described by using a generalized Landau-Lifshitz-Gilbert-Slonczewski (LLGS) equation,1,2which can be written as dM dt¼/C0c0M/C2Hef f/C0ac0 MsM/C2ðM/C2Hef fÞ /C02lBJ ð1þa2ÞedM s3gðM;PÞM/C2ðM/C2PÞ þ2lBaJ ð1þa2ÞedM s2gðM;PÞðM/C2PÞ; (1) where Heffis the effective ﬁeld, c0¼c/(1þc2),cis the elec- tron gyromagnetic ratio, and ais the dimensionless damping parameter. The effective ﬁeld includes the magnetocrystalline anisotropy ﬁeld, the demagnetization ﬁeld, the external ﬁeld,and the exchange ﬁeld, namely H eff¼HkþHdþHextþHex. In addition, regarding STT term, lB,J ,d ,e ,M s, are the Bohr magneton, the current density, the thickness of the free layer,the electron charge, and the saturation magnetization, respec- tively. The scalar function 1,2g(M,P) is given by g( M,P) ¼[/C04þ(1þg)3(3þM/C1P/Ms2)/4g3/2]/C01, where the angle between MandPish.M/C1P/Ms2¼cosh.The magnetic parameters are adopted as followed: satu- ration magnetization M s¼9.0/C2105A/m, exchange constant A¼2.0/C210/C011J/m, Gilbert damping parameter a¼0.01, and spin polarization factor g¼0.76.29The initial magnetiza- tions of free and pinned layers are along the /C0x axis and þx axis, respectively. The dynamics of magnetization was inves-tigated by numerically solving the time-dependent LLGS equation using the Gauss-Seidel projection method 33–36with a constant time step Dt¼0.0238993 ps. The samples were discrete in computational cells of 2 /C22/C22n m3. III. RESULTS AND DISCUSSION We investigated the effects of magnetic anisotropy and demagnetization ﬁelds in current-induced magnetization switching in a full-Heusler CFA-based spin-valve nanopillarof 250 /C2190 nm 2by using numerical simulations. Figure 2(a) shows the temporal magnetization component evolutions of hmxiat a constant current density of 4.0 /C2106A/cm2. There are three lines representing magnetization evolutions with dif- ferent magnetic anisotropy constants: /C01.0/C2104J/m3(black), /C01.0/C2103J/m3(red), and 2.8 /C2103J/m3(blue). In the experi- ment,29three states were obtained: the parallel (P), antiparallel (AP), and intermediate (I: perpendicular to P) states. The inter- mediate state appears at a constant current density, and themagnetization switching is called 90 /C14switching. We attributed this 90/C14switching to the balance between STT and the four- fold in-plane magnetocrystalline anisotropy of Heusler-basedfree layers. 30The 90/C14switching (half switching) behavior, as observed experimentally, was obtained at the large magnetic anisotropy constant of /C01.0/C2104J/m3. However, the interme- diate state disappears when the magnetic anisotropy constant K1decreases to /C01.0/C2103J/m3. Therefore, 180/C14switching can be achieved at the same current due to the decease of themagnetic anisotropy constant. Furthermore, we also observe 180 /C14magnetization switching under the positive magnetic ani- sotropy constant of 2.8 /C2103J/m3. It is concluded that the elimination of 90/C14switching is resulted from the decrease of FIG. 1. Model geometry deﬁnition of CFA-based spin valve in Cartesian coordinates (left). Different contributions of magnetocrystalline anisotropicﬁeldH k, demagnetization ﬁeld Hdin the free layer (right). FIG. 2. (a) The temporal magnetization component evolutions of hmxiat the constant current density of 4.0 /C2106A/cm2with different magnetic anisotropy constants of /C01.0/C2104J/m3(black), /C01.0/C2103J/m3(red), and 2.8 /C2103J/m3(blue). (b) The temporal magnetization component evolutions of hmxiwith dif- ferent current densities of 1.0 /C2106A/cm2(black), 2.0 /C2106A/cm2(red), and 3.0 /C2106A/cm2(blue) at the same magnetic anisotropy constant of 2.8/C2103J/m3.133905-2 Huang et al. J. Appl. Phys. 115, 133905 (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: 130.113.86.233 On: Tue, 17 Feb 2015 12:23:30effective magnetic anisotropy constant \|K 1\|. Figure 2(b) shows the temporal magnetization component evolutions of hmxiat the same magnetic anisotropy constant of 2.8 /C2103J/m3. There are three lines representing magnetization evolutionswith different current densities: 1.0 /C210 6A/cm2(black), 2.0/C2106A/cm2(red), and 3.0 /C2106A/cm2(blue). It is observed that there is no 90/C14switching in the magnetization switching with the increase of current density. Therefore, we provide the evidence for the elimination of the intermediate state in full-Heusler CFA-based spin valve. Figure 3shows the temporal magnetization component evolutions with different demagnetization factors N x,y (Nx¼Ny). The magnetization is driven by a small current density of 1.0 /C2104A/cm2. Magnetization switching cannotbe accomplished if the demagnetization factors N xand N yare equal to 0.02. However, we observe 90/C14and 180/C14magnetiza- tion switching at a small current density of 1.0 /C2104A/cm2 when N x(Nx¼Ny) increases to 0.10 and 0.20, respectively. In our simulation, the size of the free layer in z direction (2 nm) is signiﬁcantly smaller than those of x and y directions(250 nm /C2190 nm), resulting in much stronger demagnetiza- tion ﬁelds in z direction. The higher demagnetization ﬁeld in z direction impedes the development of hm zi. Therefore, the demagnetization ﬁeld along z axis is a barrier prohibiting the magnetization switching from the initial /C0x direction to the ﬁnal x direction. However, by decreasing the z axisdemagnetization factor and increasing the x or y axe demag- netization factors, the magnetization can be switched easily at a small current. This provides an effective method to decreasethe critical current density of spin transfer switching of CFA-based nanopillar. As shown in Figure 4, the corresponding magnetization distributions of the 250 /C2190 nm 2ellipse under different demagnetization factors present different magnetization switching behavior. The colors represent different domainarea, orange /C0x, yellow þx, green þy axis, and dark green /C0y axis, respectively. In the ﬁrst row, the initial magnetiza- tion is along /C0x axis with a single domain at 1.195 ps. After applying a constant current density of 1.0 /C210 4A/cm2, the multi-domain could be found at 1.673 ns. The magnetization will become the single domain along /C0x axis again since STT input energy can overcome the energy barrier. This multi-domain evolution process can be explained by the large current input energy. The energy per unit time pumpedinto the nanopillar by the current is so large that the FIG. 3. The temporal magnetization component evolutions of hmxiat the constant current density of 1.0 /C2104A/cm2with different demagnetization factors N x,y. FIG. 4. Snapshots of magnetization distribution of the 250 /C2190 nm2ellipse under different demagnetization factors. The orange represents the magnetization along /C0x axis, yellow þx axis, green þy axis, and dark green /C0y axis.133905-3 Huang et al. J. Appl. Phys. 115, 133905 (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: 130.113.86.233 On: Tue, 17 Feb 2015 12:23:30formation of magnetic excitations with the wavelength is much shorter than the element size, leading to the formationof multi-domains. If the demagnetization factors of N xand Nyincrease to 0.10, the magnetization switching will show the 90/C14switching (half switching). Finally, we observe that the magnetization is switched from the initial /C0x direction to the ﬁnal þx direction when the demagnetization factors of Nxand N yincrease to 0.20. Thus, we also attribute the elimi- nation of the intermediate state to the increase of the demag- netization factors along x and y axes. To study the role of magnetic anisotropy and demagnet- ization energies, we simulated the switching dynamics with- out the magnetic anisotropy energy E aniand demagnetization energy E dem. Figures 5(a) and5(b) show the temporal evolu- tions of hmxi(black), hmyi(red), and hmzi(blue) at the con- stant current density of 8.0 /C2106A/cm2without taking into account magnetic anisotropy and demagnetization energies.Figure 5(c) shows the magnetization evolutions at the same current density including all the energetic contributions. We observe 180 /C14magnetization switching in Figures 5(a) and 5(b), and 90/C14switching in the Figure 5(c). The magnetic ani- sotropy energy impedes 180/C14magnetization switching at the beginning of magnetization oscillation. The magnetizationswitching time is 3.3 ns in Figure 5(a) without the magnetic anisotropy energy, and the switching time decreases signiﬁ- cantly to 1.2 ns with the magnetic anisotropy energy.Therefore, the anisotropy energy ﬁrst impedes, and then accelerates the magnetization reversal after the magnetiza- tion component hm xiis equal to 0. Furthermore, the 90/C14 magnetization switching under the current density of 8.0/C2106A/cm2in Figure 5(c) becomes the 180/C14switching after removing the demagnetization energy in Figure 5(b). We also observe the small magnetization oscillation after the reversal in Figure 5(b), and it indicates that the demagnetiza- tion energy makes magnetization oscillation stable in theeasy axis. IV. CONCLUSIONS We investigated the effects of magnetic anisotropy and demagnetization energies on spin transfer torque switching of a Heusler CFA-based alloy spin-valve nanopillar usingmicromagnetic simulations. It is demonstrated that the elimi- nation of the intermediate state is resulted from the decease of effective magnetic anisotropy constant. The magnetizationswitching can be achieved by increasing the demagnetization factors of x and y axes even with a small current density of 1.0/C210 4A/cm2, which is 100 times smaller than the normal critical current of 106–108A/cm2. Both magnetic anisotropy and demagnetization energies impede 180/C14magnetization switching, however, the anisotropy energy signiﬁcantlyreduces magnetization switching time and the demagnetiza- tion energy stabilizes magnetization oscillation along the easy axis. ACKNOWLEDGMENTS This work was sponsored by the US National Science Foundation under the Grant No. DMR-1006541 (Chen andHuang), and by the National Science Foundation of China (11174030). The computer simulations were carried out on the LION and Cyberstar clusters at the Pennsylvania StateUniversity. 1L. Berger, Phys. Rev. B 54(13), 9353 (1996). 2J. C. Slonczewski, J. Magn. Magn. Mater. 159(1–2), L1–L7 (1996). 3J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers, and D. C. Ralph, Phys. Rev. Lett. 84(14), 3149 (2000). 4B.€Ozyilmaz, A. Kent, D. Monsma, J. Sun, M. Rooks, and R. Koch, Phys. Rev. Lett. 91(6), 067203 (2003). 5E. B. Myers, Science 285(5429), 867–870 (1999). 6M. Tsoi, A. G. M. Jansen, J. Bass, W. C. Chiang, M. 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5.0020852.pdf	Appl. Phys. Lett. 117, 122403 (2020); https://doi.org/10.1063/5.0020852 117, 122403 © 2020 Author(s).All-optical probe of magnetization precession modulated by spin–orbit torque Cite as: Appl. Phys. Lett. 117, 122403 (2020); https://doi.org/10.1063/5.0020852 Submitted: 04 July 2020 . Accepted: 08 August 2020 . Published Online: 21 September 2020 Kazuaki Ishibashi , Satoshi Iihama , Yutaro Takeuchi , Kaito Furuya , Shun Kanai , Shunsuke Fukami , and Shigemi Mizukami COLLECTIONS This paper was selected as Featured ARTICLES YOU MAY BE INTERESTED IN Robust spin–orbit torques in ferromagnetic multilayers with weak bulk spin Hall effect Applied Physics Letters 117, 122401 (2020); https://doi.org/10.1063/5.0011399 Nano-second exciton-polariton lasing in organic microcavities Applied Physics Letters 117, 123302 (2020); https://doi.org/10.1063/5.0019195 Strong interface-induced spin-charge conversion in YIG/Cr heterostructures Applied Physics Letters 117, 112402 (2020); https://doi.org/10.1063/5.0017745All-optical probe of magnetization precession modulated by spin–orbit torque Cite as: Appl. Phys. Lett. 117, 122403 (2020); doi: 10.1063/5.0020852 Submitted: 4 July 2020 .Accepted: 8 August 2020 . Published Online: 21 September 2020 Kazuaki Ishibashi,1,2Satoshi Iihama,3,4,a) Yutaro Takeuchi,5Kaito Furuya,5Shun Kanai,4,5,6,7 Shunsuke Fukami,2,4,5,6 and Shigemi Mizukami2,4,6 AFFILIATIONS 1Department of Applied Physics, Graduate School of Engineering, Tohoku University, 6-6-05, Aoba-yama, Sendai 980-8579, Japan 2WPI Advanced Institute for Materials Research (AIMR), Tohoku University, 2-1-1, Katahira, Sendai 980-8577, Japan 3Frontier Research Institute for Interdisciplinary Sciences (FRIS), Tohoku University, Sendai 980-8578, Japan 4Center for Spintronics Research Network (CSRN), Tohoku University, Sendai 980-8577, Japan 5Laboratory for Nanoelectronics and Spintronics, Research Institute of Electrical Communication (RIEC), Tohoku University, Sendai 980-8577, Japan 6Center for Science and Innovation in Spintronics (CSIS), Core Research Cluster (CRC), Tohoku University, Sendai 980-8577, Japan 7Frontier Research in Duo (FRiD), Tohoku University, Sendai 980-8577, Japan a)Author to whom correspondence should be addressed: satoshi.iihama.d6@tohoku.ac.jp ABSTRACT Laser-induced magnetization precession modulated by an in-plane direct current was investigated in a W/CoFeB/MgO micron-sized strip using an all-optical time-resolved magneto-optical Kerr effect microscope. We observed a relatively large change in the precession frequency, owing to a current-induced spin–orbit torque. The generation efﬁciency of the spin–orbit torque was evaluated as /C00.3560.03, which was in accordance with that evaluated from the modulation of damping. This technique may become an alternate method for the evaluation ofspin–orbit torque. Published under license by AIP Publishing. https://doi.org/10.1063/5.0020852 Spin–orbit torque (SOT) has attracted signiﬁcant attention as it allows simple, reliable, and fast manipulation of magnetization in thin ﬁlms. 1–3Conventionally, SOT has been investigated using electrical means such as spin-torque ferromagnetic resonance4–7and second harmonic Hall effect measurement.8–11In the spin-torque ferromag- netic resonance, magnetization precession is excited by the SOT gener- ated from an injected in-plane RF current. Subsequently, thegeneration efﬁciency of the SOT, i.e., the effective spin-Hall angle in nonmagnetic heavy metals, can be evaluated by analyzing its spectrum amplitude and shape. In the second harmonic Hall effect measure- ment, the magnetization angle is adiabatically changed by the SOT induced by a low-frequency in-plane alternating current. This change in the magnetization angle is detected through the planar Hall effect or an anomalous Hall effect voltage. Even though these techniques are widely utilized, parasitic electrical voltages are induced by spin-charge conversion as well as the thermoelectric effect. 5,10 A direct observation of magnetization precession modulated by the SOT is free from such parasitic effects, and thus, it is a promisingapproach. The time-resolved measurement of magnetization preces- sion modulated by the SOT has been previously reported.12–14These studies mainly focused on the change in the relaxation time of magne- tization precession by the SOT.12,13However, no study has focused on the change in precession frequency due to SOT. In general, the evalua- tion of frequency is more precise than that of the precession relaxation time. Therefore, it is intriguing to observe and understand the effect ofSOT on frequency. In this Letter, we report, for the ﬁrst time, an obser- vation of the modulation of magnetization precession frequency owing to the SOT, from which the generation efﬁciency of the SOT was obtained. Thin-ﬁlm stacks of W(5)/CoFeB(2.4)/MgO(1.3)/Ta(1) (thickness in nm) were fabricated via DC/RF magnetron sputtering on Si/SiO 2 substrates. Here, we used a W underlayer, which is reported to showlarge SOT efﬁciencies. 15–17The fabrication condition was similar to that for samples exhibiting a high effective spin-Hall angle.18The thickness of the W layer was determined from our previous ﬁndings on the W/CoFeB structure in which a 5-nm-thick W layer showed Appl. Phys. Lett. 117, 122403 (2020); doi: 10.1063/5.0020852 117, 122403-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/apllarge SOT efﬁciency.18The samples were patterned into rectangular strips with a width wof 10lm and a length Lof 40lm by photoli- thography and Ar ion milling. The magnetization precession dynamicswere investigated using an all-optical time-resolved magneto-opticalKerr effect (TRMOKE) microscope. The setup was similar to thatreported previously. 19–21The wavelength, pulse duration, and pulse repetition rate of the emission laser were approximately 800 nm, 120 fs, and 80 MHz, respectively. The wavelength of the pump laser waschanged to 400 nm using a BaB 2O4(BBO) crystal, and its intensity was modulated using a mechanical chopper at a frequency of 370 Hz.The pump and probe beams were focused on the sample surface usingan objective lens, as shown in Fig. 1(a) . The diameter of the probe beam spot was approximately 1 lm. The pump ﬂuence was less than 2.0 mJ/cm 2, and the probe ﬂuence was /C241 mJ/cm2.T h ep u m p - induced change in the Kerr rotation angle of the reﬂected probe beam was detected using a balanced photodiode detector, as a function ofpump–probe delay time. Figures 1(b) and1(c)show the schematic dia- grams of the coordinate system and the experimental geometries. Anexternal magnetic ﬁeld H extwas applied with an out-of-plane angle hH and an in-plane-angle /¼90/C14. The angle of the magnetization direc- tion hin an equilibrium state was determined through the external magnetic ﬁeld and magnetic anisotropy. We tested two geometries, as shown in Figs. 1(b) and1(c), to clarify these differences. One geometry is that the sample strip was parallel to the x-axis [ Fig. 1(b) ](Jck^x), where Jcdenotes the charge current density. The other geometry is that the sample strip was parallel to the y-axis [ Fig. 1(c) ](Jck^y). Note that only the former case was reported previously.12,13All measure- ments were performed at room temperature. Figure 2(a) shows the typical time-domain measurement for the sample with various electrical currents Ifor the Jck^ygeometry. Here, DhKis the pump laser-induced change in the Kerr rotation angle, which is proportional to the z-component of the magnetization. At a delay time of a few hundred femtoseconds, ultrafast demagnetization was induced through pump laser illumination. Subsequently, the mag- netization recovery and the damped precession of magnetization wereobserved. The magnetization precession triggered by ultrafast demag-netization is well documented in a previous study. 22The relaxation time sand frequency fof the precession were evaluated by the least-squares ﬁtting of the damped sinusoidal function23to the TRMOKE signal DhK DhK¼AþBexp/C0/C23tðÞ þCexp/C0t s/C18/C19 sin 2 pftþ/0 ðÞ : (1) Here, the ﬁrst two terms represent the change due to the recovery from the demagnetization and are characterized by the amplitudes ofAand Band the recovery rate /C23. The last term in Eq. (1)represents the change due to the damped magnetization precession. Cand/ 0 denote the precession amplitude and initial phase, respectively. The dashed black curves in Fig. 2(a) denote the ﬁtting curve, calculated using Eq. (1).Figure 2(b) shows the typical normalized signals with I¼0;65 mA applied to the x-axis at hH¼25/C14andl0Hext ¼352 mT ( Jck^x). In this ﬁgure, the remagnetization background sig- nals, i.e., the ﬁrst and second terms in Eq. (1), were subtracted. The modulation of the relaxation time of the precession was clearlyobserved. This trend was similar to that observed in previous stud- ies. 12,13Figure 2(c) shows the typical normalized signals with I¼0; 65 mA applied to the y-axis at hH¼4/C14andl0Hext¼270 mT (Jck^y). In contrast to Fig. 2(b) , a distinct change was observed in the precession frequency. Note that the magnetization is not saturatedalong the magnetic ﬁeld direction, namely, h6¼h HinFig. 1 ,b e c a u s e the applied magnetic ﬁeld is smaller than the out-of-plane demagnetiz- ing ﬁeld of the sample. The reason for the choice of the ﬁeld angle fortwo different experimental geometries is discussed later. Figures 3(a) and3(b)show the modulation of the inverse relaxa- tion time 1 =swith Iunder J ck^xatl0Hext¼6352 mT. The reversal ofl0Hextchanges the sign of the slope of the inverse relaxation time vs FIG. 1. (a) Schematic of the sample stacking structure and the optical setup. Schematic of the coordinate system and the experimental geometries with a directcurrent applied along the x-axis (b) and y-axis (c). FIG. 2. (a) Typical time-domain data with various direct currents Ialong the y-axis at a ﬁxed ﬁeld angle hH¼4/C14and the external magnetic ﬁeld l0Hext¼270 mT for the W/CoFeB/MgO/Ta ﬁlms for the geometry of Jck^y. The dashed curves in (a) represent data calculated using Eq. (1)and were ﬁtted to the experimental data. (b) Normalized time-domain data ^DhKwith I¼0;65 mA applied to the x-axis at hH¼25/C14andl0Hext¼352 mT. (c) ^DhKwith I¼0;65 mA applied parallel to they-axis at hH¼4/C14andl0Hext¼270 mT.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 122403 (2020); doi: 10.1063/5.0020852 117, 122403-2 Published under license by AIP Publishingcurrent, indicating that the observed change in the relaxation time is caused by the SOT. Figures 3(c) and3(d) show the modulation of fre- quency fwith Iunder Jck^yatl0Hext¼6270 mT. The frequency shift in both Figs. 3(c) and3(d) exhibited negative slopes, in contrast to the data of 1 =svsI[Figs. 3(a) and3(b)]. To understand the frequency and relaxation time modulated by the SOT, we describe our analysis using the Landau–Lifshitz–Gilbert(LLG) equation, which includes the SOT, dm dt¼/C0cl0m/C2Heffþam/C2dm dt/C0cl0Hsm/C2ðm/C2rÞ; (2) where mis the unit magnetization vector, Heffis the effective magnetic ﬁeld, ais the Gilbert damping constant, ris the spin polarization vec- tor, and cis the gyromagnetic ratio. Here, we consider only the damping-like SOT in Eq. (2). Assuming that the SOT is caused by the spin-Hall spin current generated from the W layer, the SOT effective ﬁeld Hsis expressed as Hs¼/C22hngI 2el0MstCFBtWw; (3) where /C22h,e,Ms;tCFB,a n d tWare the Dirac constant, electron charge, saturation magnetization, CoFeB layer thickness, and W layer thick-ness, respectively. grepresents the fraction of the electrical current ﬂowing into the W layer. nis the generation efﬁciency of the SOT. It should be noted that nis determined not only by the spin-Hall angle, but also by the spin transparency at the interface as well as the damping-like SOT generated at the interface; therefore, nis termed as the effective spin-Hall angle. In the absence of the electrical current,the precession frequency f, the inverse relaxation time 1 =s,a n dt h e i r ﬁeld components H 1,H2can be expressed by the following equations:24 1 s¼1 2l0caðH1þH2Þ; (4) f¼l0c 2pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃH1H2p; (5) H1¼Hextcosðh/C0hHÞ/C0Meffcos2h; (6)and H2¼Hextcosðh/C0hHÞ/C0Meffcos 2 h; (7) where Meffis the effective demagnetizing ﬁeld. The magnetization angle hwas determined based on the balance between the Zeeman energy and the effective demagnetizing energy, 2Hextsinðh/C0hHÞ/C0Meffsin 2h¼0: (8) In the presence of an electrical current, Hsaffected the precession frequency and relaxation time, depending on the geometries, as dis-cussed below. First, we present our analysis on the geometry of J ck^x(rk^y). The SOT behaves as a damping-like torque, and the additional termproportional to Iwas added to Eq. (4), in which the geometry [ Fig. 1(b)] is similar to that of the anti-damping SOT switching in so-called “Type-y” devices. 3Consequently, the theoretical slope ds/C01=dIwas obtained using the LLG equation [Eq. (2)], ds/C01 dI¼/C0c/C22hng 2eMstCFBtWwsinh: (9) Equation (9)indicates that the linear modulation of the inverse relaxa- tion time was caused by the SOT. The generation efﬁciency of theSOT nwas evaluated as /C00.3560.07 using Eqs. (8)and(9)with the experimental ds /C01=dIvalue obtained from the data shown in Fig. 3(a) . We performed a least-squares ﬁtting to the data of 1 =svsI using a quadratic polynomial /cþbIþaI2with adjustable parame- tersa,b,a n d c[the curves shown in Figs. 3(a) and3(b)] to extract the slope b/C17ds/C01=dIfrom the experimental data. It is to be noted that the parabolic term aI2was negligibly small and originated from Joule heating. The other parameters used were c¼185 Grad/s/T, Meff¼512 kA/m, and Ms¼1051 kA/m. Additionally, gof 0.57 was used, which was evaluated from the measured resistivity of the W ﬁlm,205lXcm, and the CoFeB ﬁlm, 128 lXcm. Next, we present the analysis of the geometry of J ck^y(rk^x). The direction of the SOT term in Eq. (2)is in the y–z plane in the Jck^xgeometry [ Fig. 1(b) ], while the direction of the SOT is parallel to^xin the Jck^ygeometry [ Fig. 1(c) ]. Those SOT terms induce addi- tional effective ﬁelds and change the equilibrium magnetization angle. In the Jck^y(Jck^x) geometry, the SOT term induces the effective ﬁeld in the y–z(x–y) plane and changes the h(/). The his determined by the balance of the torque stemming from the Zeeman and effective demagnetizing energies. Hence, the linear change in frequency as a function of Iwas induced via a change in honly for the Jck^ygeome- try, because the change in frequency caused by the change in /is inde- pendent of the polarity Ifor the Jck^xgeometry. The relationship between the magnetization angle hand electrical current Iwas derived from Eq. (2)under an equilibrium condition as follows: 2Hextsinðh/C0hHÞ/C0Meffsin 2h/C02Hs¼0: (10) This equation is identical to Eq. (8)when no electrical currents are applied. We differentiated Eq. (10) with respect to IatI/C250, and the relationship between handIis obtained as follows: @h @I¼/C22hng 2el0MstCFBtWwH 2: (11) The theoretical slope df/dIwas derived from Eqs. (5)–(7) and(11), FIG. 3. Modulation of the inverse relaxation time 1 =swith direct currents Iapplied along the x-axis under hH¼25/C14; an external magnetic ﬁeld l0Hextof (a) 352 mT and (b) /C0352 mT. Modulation of the frequency with direct currents Iapplied along they-axis under hH¼4/C14; an external magnetic ﬁeld l0Hextof (c) 270 mT and (d) /C0270 mT. Curves denote quadratic polynomials cþbIþaI2ﬁtted to experimental data.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 122403 (2020); doi: 10.1063/5.0020852 117, 122403-3 Published under license by AIP Publishingdf dI¼l0c 4pﬃﬃﬃﬃﬃﬃ H2 H1r @H1 @hþﬃﬃﬃﬃﬃﬃ H1 H2r @H2 @h ! /C2/C22hng 2el0MstCFBtWwH 2:(12) We did not take into account the effects of the Oersted ﬁeld, Joule heating, and the ﬁeld-like SOT in Eq. (2); however, these effects were negligible at around I/C250o n df/dIin this geometry. This is because the possible changes in fcaused by the three above-mentioned factors should be independent of the polarity of I.M e a n w h i l e ,t h eO e r s t e d ﬁeld and the ﬁeld-like SOT are small in this study,18and the parabolic change in fvsImostly originated from Joule heating, as mentioned earlier. Therefore, we interpreted a linear change in the frequency asthe effect of the damping-like SOT, as described in Eq. (2).T h i si sc o n - trary to the case of the J ck^xgeometry, in which the Oersted ﬁeld and the ﬁeld-like SOT induce frequency modulation. Here, it should be noted that the frequency modulation df/dIcalculated by using Eq. (12) is increased when the magnetic ﬁeld direction is close to the ﬁlm nor-mal, whereas the damping modulation ds /C01=dIexhibits a maximum when the magnetization direction is parallel to the ﬁlm plane [Eq. (9)]. We evaluated the value of df/dIfrom the data of fvsIusing the qua- dratic polynomial ﬁt, as similarly performed for the data of 1 =svsI [the curves in Figs. 3(c) and3(d)]. Subsequently, the generation efﬁ- ciency of the SOT nwas evaluated as /C00.3560.03 using Eqs. (6)–(8) as well as (12)and the experimental df/dIvalue obtained from the data is shown in Fig. 3(c) .T h e nevaluated by the frequency modulation is in accordance with the nevaluated by the modulation of damping and in agreement with a previous study.17 Figure 4(a) shows the values of df/dIevaluated from the experi- mental data measured at various external magnetic ﬁelds. The curves denote the values calculated using Eq. (12).Figure 4(b) shows the gen- eration efﬁciency of the SOT nevaluated using Eq. (12) at various external magnetic ﬁelds. The nvalues were independent of the external magnetic ﬁelds. Moreover, the nvalue obtained from the frequency modulation was agree well with that from the inverse relaxation timemodulation within the experimental errors. Therefore, the generationefﬁciency of the SOT was precisely determined from the frequencymodulation. In summary, we performed time-resolved measurements of mag- netization precession modulated by the SOT in two different geome-tries, i.e., J ck^xandJck^y, in the W/CoFeB/MgO structure. In the ﬁrst case, Jck^x, the modulation of the relaxation time for magnetiza- tion precession was observed, which was consistent with the previous studies. In the second case, Jck^y, the modulation of the frequency for magnetization precession induced by the SOT was clearly observed.The generation efﬁciency of the SOT was estimated from the analysisof the change in precession frequency, which was almost independentof the external magnetic ﬁeld. This study suggests that all-optical TRMOKE measurements of the precession frequency shift are effective tools for evaluating the generation efﬁciency of the SOT. This study was partially supported by KAKENHI (Nos. 19K15430 and 19H05622), the ImPACT Program of CSTI, Advanced Technology Institute Research Grants, and the Centerfor Spintronics Research Network (CSRN). DATA AVAILABILITY The data that support the ﬁndings of this study are available from the corresponding author upon reasonable request. REFERENCES 1I. M. Miron, K. Garello, G. Gaudin, P.-J. Zermatten, M. V. Costache, S. Auffret, S. Bandiera, B. Rodmacq, A. Schuhl, and P. Gambardella, Nature 476, 189 (2011). 2L. Liu, O. J. Lee, T. J. 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Ando, and S. Mizukami, Phys. Rev. B 94, 020401(R) (2016). 20Y. Sasaki, K. Suzuki, A. Sugihara, A. Kamimaki, S. Iihama, Y. Ando, and S. Mizukami, Appl. Phys. Express 10, 023002 (2017). 21A. Kamimaki, S. Iihama, Y. Sasaki, Y. Ando, and S. Mizukami, Phys. Rev. B 96, 014438 (2017). 22M. van Kampen, C. Jozsa, J. T. Kohlhepp, P. LeClair, L. Lagae, W. J. M. de Jonge, and B. Koopmans, Phys. Rev. Lett. 88, 227201 (2002). 23S. Iihama, S. Mizukami, H. Naganuma, M. Oogane, Y. Ando, and T. Miyazaki, Phys. Rev. B 89, 174416 (2014). 24S. Mizukami, F. Wu, A. Sakuma, J. Walowski, D. Watanabe, T. Kubota, X. Zhang, H. Naganuma, M. Oogane, Y. Ando, and T. Miyazaki, Phys. Rev. Lett. 106, 117201 (2011). FIG. 4. External magnetic ﬁeld Hextdependence of (a) df/dIand (b) the generation efﬁciency of the SOT ninJck^ygeometry under hH¼4/C14. Curves in (a) denote the calculated values of df/dIusing Eq. (12).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 122403 (2020); doi: 10.1063/5.0020852 117, 122403-4 Published under license by AIP Publishing
5.0049912.pdf	J. Appl. Phys. 129, 193902 (2021); https://doi.org/10.1063/5.0049912 129, 193902 © 2021 Author(s).Method to suppress antiferromagnetic skyrmion deformation in high speed racetrack devices Cite as: J. Appl. Phys. 129, 193902 (2021); https://doi.org/10.1063/5.0049912 Submitted: 10 March 2021 . Accepted: 28 April 2021 . Published Online: 19 May 2021 P. E. Roy COLLECTIONS This paper was selected as an Editor’s Pick ARTICLES YOU MAY BE INTERESTED IN Application concepts for ultrafast laser-induced skyrmion creation and annihilation Applied Physics Letters 118, 192403 (2021); https://doi.org/10.1063/5.0046033 Native oxide reconstructions on AlN and GaN (0001) surfaces Journal of Applied Physics 129, 195304 (2021); https://doi.org/10.1063/5.0048820 Skyrmion propagation along curved racetracks Applied Physics Letters 118, 172407 (2021); https://doi.org/10.1063/5.0045969Method to suppress antiferromagnetic skyrmion deformation in high speed racetrack devices Cite as: J. Appl. Phys. 129, 193902 (2021); doi: 10.1063/5.0049912 View Online Export Citation CrossMar k Submitted: 10 March 2021 · Accepted: 28 April 2021 · Published Online: 19 May 2021 P. E. Roya) AFFILIATIONS Hitachi Cambridge Laboratory, Hitachi Europe Limited, Cambridge CB3 0HE, United Kingdom a)Author to whom correspondence should be addressed: per24@cam.ac.uk ABSTRACT A method for enhancing the stability of high speed antiferromagnetic skyrmions in racetrack devices is proposed and demonstrated numeri- cally. Spatial modulation of the Dzyaloshinskii –Moriya interaction via a patterned top heavy metal gives rise to a strong confining potential. This counteracts skyrmion deformation perpendicular to the direction of propagation and the subsequent annihilation on contact with theracetrack ’s horizontal boundaries. An achievable increase in the maximum driving current density of 135%, enabling higher velocities of 28%, is predicted. Furthermore, an extended saturating behavior of the mobility relation due to the imposed confinement is also found at large driving amplitudes, further enhancing skyrmion stability at high velocities. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0049912 I. INTRODUCTION Magnetic skyrmions, first theoretically predicted, 1–3and later experimentally verified4are small topological swirls or defects in the magnetization field of a magnet. Like domain walls and vortices, theyare a type of magnetic texture. Reasons for interest in skyrmions are their nm-sized footprints, low threshold driving current density, and free particle-like behavior. 5This has prompted proposals for applica- tions such as information carriers in racetrack memory and logicdevices, 6–10elements in memristors for artificial synapses,11in spintronics-based transistor concepts,12and as constituents in mag- nonic crystals for spinwave-based computing and logic devices.13 To date, most works consider ferromagnetic skyrmions.5–10,12,13 However, with the emergence of antiferromagnetic (AFM) spin- t r o n i c sa n di t sa d v a n t a g e ss u c ha su l t r a f a s t( T H z )d y n a m i c s ,insensitivity to external magnetic fields, and the absence of a skyrmion Hall angle, the AFM skyrmion is proposed to replace its ferromagnetic counterpart. 14–20Skyrmions in synthetic AFMs initially proposed by simulations21were recently stabilized exper- imentally,22and current-driven skyrmion bubbles in synthetic AFMs have also been demonstrated.23Observations of skyrmions in intrinsic AFMs have now recently been reported.24,25This encourages the study of AFM skyrmion dynamics and means tocontrol it for future applications. A propagating AFM skyrmion laterally deforms with increasing velocity. 14–16,26–28This deformation is dominated by an elongationperpendicular to the direction of propagation. In a finite system such as a racetrack, the elongated AFM skyrmion can come in contact with the horizontal boundaries. On contact, the AFM skyrmion breaks up into a domain wall pair,15,16,28destroying the skyrmion structure. This imposes a critical drive current, limiting the maximum velocity. Therefore, ideas on how to counteract thisbehavior are of importance for future device implementation. Works focusing on methods to suppress the deformation are to date scarce. 26,28An experimentally feasible proposal, applica- ble to ultrathin AFM racetrack devices, was put forth by Huang et al.28who considered the effect of a nisotropic interfacial Dzyaloshinskii –Moriya interaction. Such anisotropy distorts the equilibrium shape of the AFM skyr mion from circular to ellipti- cal. This anisotropy can be induced via strain to squeeze the sky- rmion along the racetrack width, counteracting elongation duringpropagation. They demonstrated an increase in the critical driving current density, after which it breaks up into a domain wall pair, of up to 20% and an associated maximum velocity increase of 7% for their considered system. In this work, a racetrack design strategy for suppression of AFM skyrmion elongation and stable propagation at high velocities is proposed. An illustration of the device is shown in Fig. 1(a) . The racetrack consists of a heavy metal/AFM/heavy metal (HM/AFM/HM) heterostructure. The bottom HM (HM1) stabilizes a skyrmion in the AFM layer and facilitates driving via the spin-Hall effect.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 193902 (2021); doi: 10.1063/5.0049912 129, 193902-1 Published under an exclusive license by AIP PublishingThe patterned top HM (HM2) acts to locally reduce the interfacial Dzyaloshinsii-Morya interaction (iDMI) strength, leading to aniDMI profile along the racetrack ’s short axis, as shown in Fig. 1(b) . The reduced iDMI gives rise to a potential well, whose barriers counteract AFM skyrmion elongation, schematically shown inFig. 1(c) . An efficient suppression of the AFM skyrmion elongation is demonstrated with a maximum enhancement of 135% and 28%for the critical current density and maximum velocity, respectively. Further, a saturating behavior of the mobility relation is found, enhancing stability at high velocities. II. WORKING PRINCIPLE AND DEVICE STRUCTURE The working principle is to counteract the skyrmion elonga- tion by imposing a lateral potential barrier. A potential barrier forthe AFM skyrmion can be formed by local modulation of the magnetic parameters. 29Whereas a region of reduced iDMI inter- acts repulsively with the AFM skyrmion, a region of reduced mag-netocrystalline anisotropy (MA) acts attractively, counteractingthe scattering effect of the reduced iDMI. 29It is experimentally known that the iDMI can be tuned nearly independently with respect to other material parameters in HM1/Ferromagnet/HM2 trilayers via the properties of the top HM (HM2)30,31as the anisotropy comes mostly from the bottom HM (HM1) interface.30 Although shown for ferromagnets, this should readily be applica-ble to AFM systems. Therefore, modulation of the iDMI is consid- ered here as the means of forming a potential barrier. Alteration of the iDMI via HM2 is in its simplest form based on the factthat, for a given HM, the sign of the iDMI depends on whetherthe interface to the HM is on the top or the bottom of the magnet ’s surfaces, i.e., dependent upon the direction of mirror symmetry breaking. 32This, in conjunction with a dependence ofthe iDMI on the HM thicknesses, in principle enables the possi- bility to continuously modulate the iDMI. In this work, a reduc-tion of the net iDMI in selected regions is required. The HMs are high spin –orbit materials such as Pt, W, Ta, and Ir 33,34and an e x a m p l es t a c kf o rt h i sw o r kc o u l db eP t / A F M / P t .T h em a t e r i a l sfor HM1 and HM2 need not be the same though, provided aHM2 material is chosen such that the iDMI contribution from itdoes not positively add to the iDMI contribution from HM1 due to an intrinsic iDMI constant of opposite sign canceling the required iDMI sign change governed by the opposite symmetrybreaking direction mechanism. 35The easiest way to ensure a sub- tractive contribution from HM2 is of course if HM1 and HM2 are of the same material. In the ideal case, equal thicknesses of HM1 and HM2 (where HM1 and HM2 are of the same material)should effectively cancel the net iDMI as has been experimentallydemonstrated. 30,36In effect, a natural AFM skyrmionic device structure considered here consists of a patterned trilayered race- track as shown in Fig. 1(a) . Here, HM1 mediates the spin-Hall effect induced torques, to drive the skyrmion along the AFMracetrack while the patterned HM2 is used to locally modulatet h en e ti D M I .T h i si sd e p i c t e di n Fig. 1(b) , with the modulated and unmodulated net iDMI strengths denoted by D 0and D, respectively. Imposing D0,Denhances the confinement with a potential barrier exclusively along the y-direction as sketched in Fig. 1(c) . This suppresses the AFM skyrmion elongation along y. The degree of iDMI reduction is conveniently characterized by the dimensionless parameter η¼D/C0D0 D. For the purpose of this work, 0 /C20η/C201 is considered, with η¼1 corresponding to com- plete cancellation of the iDMI in the regions under HM2 result-ing in the strongest confinement for the skyrmion. The range of ηconsidered herein should be experimentally feasible as its bounds comprised of the cases no HM2 present ( η¼0) and a FIG. 1. (a) Portion of the racetrack with a traveling AFM skyrmion. An AFM layer hosting the skyrmion is coupled to a HM layer below (HM1) and a patterned HM layer above (HM2). A current density Jflows in HM1, driving the skyrmion in the AFM. HM2 is of such thickness and/or material that it cancels or partially cancels the net iDMI in the regions below the patterned HM2. The computational domain has dimensions, L¼2000 nm, w¼100 nm, and t¼0:4 nm. (b) Cross section in the yz-plane showing the spatial modulation of the iDMI strength, D0under HM2 (in all other regions, the iDMI strength is D). (c) Cross section depicting a raising of the repulsive poten- tialVyðÞnear the horizontal boundaries. Two illustrative cases are qualitatively shown; when D0¼D, the only contribution to confinement is a weak repulsive interaction with the physical boundaries of the AFM layer. With HM2, D0,D, the confinement is enhanced, counteracting skyrmion elongation along y. The degree of iDMI modula- tion is parametrized by η¼D/C0D0 D.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 193902 (2021); doi: 10.1063/5.0049912 129, 193902-2 Published under an exclusive license by AIP Publishingcomplete cancellation of the iDMI under HM2 ( η¼1), which can be achieved with HM2 being of the same material and thick- ness as HM1. Intermediate values of ηmay be realized via thick- ness variations of HM2. Although structurally and conceptuallysimple, fabrication of the narrow HM2 tracks along the horizon-tal boundaries of the device may pose some challenges. However, these challenges may be surmountable by using state of the art fabrication techniques as employed for spin-transfer torque mag-netoresistive random access memories (STT-MRAM). 37 For fast AFM dynamics, a material with low magnetic dissipation/ damping is desirable.38This could be achieved by using semiconduct- ing or insulating AFMs such as NiO, MnO, FeO, and CoO, where spin-scattering is more suppressed.38It was theoretically predicted that the AFM skyrmion is subject to diffusive motion at finite tem-peratures and that the diffusion coefficient is inversely proportionalto the AFMs dissipation constant (Gilbert damping). 14If this is the case, the AFM skyrmion will be subject to an increasing thermal dif- fusive behavior with decreasing dissipation constant. However, thechoice of an insulating material for the AFM also means that nocurrent flows directly through it and as such, with HM1 connected toa good heat sink, thermal diffusion of the AFM skyrmion due to Joule heating in the adjacent HM1 could be kept under control. III. MODELING PROCEDURE A fully compensated AFM is considered, i.e., each magnetic moment is antiferromagnetically coupled to its nearest neighbors. The AFM is modeled on a uniform 2D mesh with equal node spacings, Δin each direction. At each node, the time evolution of the normalized (unit) magnetization ^m¼M=M S,w h e r e MSis the saturation magnetization, is solved via the Landau –Lifshitz – Gilbert equation with added spin-Hall torques.39Denoting a site on the mesh by the pair of integer indices i,j,t h ed y n a m i c so f ^mi,jis as follows:39 d^mi,jðÞ dt¼/C0γ^mi,jðÞ/C2Hi,jðÞ eþα^mi,jðÞ/C2d^mi,jðÞ dtþτi,jðÞ: (1) In Eq. (1),γis the gyromagnetic ratio, Hi,jðÞ eis the effective field arising from the magnetic interaction energies, αis the Gilbert damping accounting for dissipation, and τi,jðÞis the torque due to the spin accumulation arising from the spin-Hall effect when a current density Jis present in HM1. The spin-Hall torque can be written in terms of a spin-Hall field Hi,jðÞ SHsuch that τi,jðÞ¼/C0γ^mi,jðÞ/C2 ^mi,jðÞ/C2Hi,jðÞ SH/C16/C17 with Hi,jðÞ SH¼/C22hθSHjJj 2jejμ0MSt/C0^n/C2^ji,jðÞ/C0/C1 .40Here, /C22his Planck ’s reduced constant, θSHis the spin-Hall angle, μ0is the mag- netic permeability in vacuum, and eis the electron charge. The unit vector ^nis directed from HM1 toward the AFM and ^ji,jðÞis the unit direction of Jat site i,jðÞ. The interaction energies within the AFM taken into account are exchange interaction, iDMI and MA. Within a discrete (atomistic) representation, these are15,27,32 Eex¼/C0 JexX i,j^mi,jðÞ/C1^miþ1,j ðÞþ^mi,jþ1ðÞ/C16/C17 , (2)EDM¼dX i,j/C0^y/C1^mi,jðÞ/C2^miþ1,j ðÞ/C16/C17 þ^x/C1^mi,jðÞ/C2^mi,jþ1ðÞ/C16/C17 hi , (3) Ek¼kX i,j1/C0^mi,jðÞ/C1^z/C16/C172/C20/C21 : (4) Equation (2)describes the nearest neighbor exchange interaction energy, Eexwith strength Jex,E q . (3)represents the iDMI energy, EDMof strength d, and Eq. (4)is the MA energy, Ekwith anisotropy constant k. All interaction strengths are in units of Joules. The dis- crete effective field at site i,jis obtained from E¼EexþEDMþEk via the relation Hi,jðÞ e¼/C01 μ0μsδE δ^mi,jðÞ,w h e r e μ0is the magnetic perme- ability in vacuum and μsis the saturation magnetic moment. The resulting discrete effective field is Hi,jðÞ e¼J μ0μs^miþ1,j ðÞþ^mi,jþ1ðÞþ^mi/C01,j ðÞþ^mi,j/C01ðÞ/C16/C17 þd μ0μsh /C0^y/C2^miþ1,j ðÞ/C0^mi/C01,j ðÞ/C16/C17 þ^x/C2^mi,jþ1ðÞ/C0^mi,j/C01ðÞ/C16/C17 i þ2k μ0μs^mi,jðÞ/C1^z/C16/C17 ^z, (5) where the first term is the nearest-neighbor exchange field, the second term is the iDMI field, and the third term is the anisot-ropy field. Although ^mis not a continuous variable in space owing to AFM coupling between all magnetic moments, there is a direct correspondence between a finite difference implementation of the micromagnetic (continuum) formalism and that of the dis-crete (atomistic) approach described above. This correspondenceis valid for the usual interaction terms with the exception of the demagnetizing field (dipole –dipole interactions). However, for fully compensated intrinsic AFMs , the demagnetizing energy is normally neglected (due to magnetic flux closure on the atomicscale). As such, standard micromagnetic codes for the study ofcompensated AFMs have gained ground in the literature. 14–17,41 In this work, the commercial LLG Micromagnetic Simulator45 package was used. The justification and applicability of finite dif- ference micromagnetics to intrinsic AFMs is as follows: Withinthis micromagnetic representation, the energy densities (Joulesper unit volume) are 32,42,43 εex¼Aex∇^mðÞ2, (6) εDM¼Dm x@mz @x/C0mz@mx @x/C18/C19 þmy@mz @y/C0mz@my @y/C18/C19 /C20/C21 , (7) εK¼Ku1/C0^m/C1^zðÞ2/C2/C3 , (8) where εexis the exchange energy density with exchange stiffness Aex,εDMis the iDMI energy density with strength D,a n d εkthe MA energy with anisotropy constant Ku. The resulting effective field is obtained from ε¼εexþεDMþεKviaHe¼/C01 μ0Msδε δ^m. Out of the stated interactions, the form of the on-site MA terms isJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 193902 (2021); doi: 10.1063/5.0049912 129, 193902-3 Published under an exclusive license by AIP Publishingindependent of a discrete or continuum representation; the anisot- ropy field stemming from Eq. (8),Hi,jðÞ K¼2Ku μ0MS^mi,jðÞ/C1^z/C0/C1^zwhich is in direct correspondence to the last term in Eq. (5). Also, the spin-Hall field Hi,jðÞ SH, being an on-site interaction, is independent of the representation. Therefore, it suffices to show correspondence of Eqs. (2)and (3)to the finite difference implementation of Eqs. (6)and (7). We may do this by considering their respective effective fields entering Eq. (1). Considering the uniform 2D mesh with equal spacing Δalong both spatial dimensions, the contin- uum representations of the exchange field HEx¼2A μ0MS∇2^m43 and iDMI field HDM¼2D μ0MS∇/C1^mðÞ ^z/C0∇mz ½/C138 .32By using second central differences for the Laplacian and central differences for the gradients (i.e., second order accurate expansions), the finite differ-ence representations of the interaction fields at a site i,jare Hi,jðÞ Ex¼2A μ0MS∇2^mi,jðÞ/C252A μ0MSΔ2[^miþ1,j ðÞþ^mi,jþ1ðÞ þ^mi/C01,j ðÞþ^mi,j/C01ðÞ/C04^mi,jðÞ] (9) and Hi,jðÞ DM¼2D μ0MS∇/C1^mi,jðÞ/C16/C17 ^z/C0∇mi,jðÞ zhi /C25D μ0MSΔh /C0^y/C2^miþ1,j ðÞ/C0^mi/C01,j ðÞ/C16/C17 þ^x/C2^mi,jþ1ðÞ/C0^mi,j/C01ðÞ/C16/C17 i : (10) In Eq. (9), the term 4 ^mi,jðÞdoes not contribute to the torques in Eq.(1)because ^mi,jðÞ/C2^mi,jðÞ¼0. Comparison of Eq. (9)to the first term in Eq. (5)shows that they are of the same form.Similarly, comparing Eq. (10) to the second term in Eq. (5)shows a direct correspondence also for the iDMI field. This correspon-dence justifies using standard finite difference micromagnetic packages where second order accurate finite difference implemen- tations of the spatial derivatives are commonly employed. The parameters used in the simulations for easily stabilization of an AFM skyrmion are D¼0:5 mJ/m 2,D0is varied in the range 0,D½/C138 such that 0 /C20η/C201. All other material parameters are con- sidered unaltered by the presence of HM2; Ku¼0:1 MJ/m3, MS¼300 kA/m, A¼/C02:5 pJ/m, α¼0:005,Δ¼0:4 nm and the spin-Hall angle of HM1, θSH¼0:07 typical for Pt.44 In the first step, static configurations of a single AFM sky- rmion for each considered value of ηare stabilized toward the left vertical boundary ( x,L=2) of the racetrack to provide enough track length for steady state propagation. The narrow track widthensures preferential stabilization of the skyrmion centered aty¼w=2 even in the absence of HM2. The magnetization field of a relaxed skyrmion is shown in Fig. 2(a) . These relaxed states are used as starting configurations for subsequent dynamical simula- tions. To simplify data-extraction and visualization, the continuousNéel field representation is used. The Néel vector field is con-structed by a linear combination of four surrounding ^mvectors by a tetramerization procedure. 27,46The Néel field representation of Fig. 2(a) is shown in Fig. 2(c) . In the second step, AFM skyrmion propagation for different values of η, under the action of spin-Hall torque due to a current- density Jin HM1 is simulated (no current flows in HM2). From the dynamical simulations, mobility relations, i.e., steady state velocity vvs driving amplitude Jare established and the degree of skyrmion deformation during propagation is extracted. Steadystate velocities are computed by tracking the position of then z-profile [see Fig. 2(c) ]. Skyrmion deformation at steady state propagation is characterized by the corresponding ΔSk xandΔSk y according to their definitions in Fig. 2(c) . FIG. 2. (a) 3D vector plot of magnetization field ( ^m) of a static, relaxed radial AFM skyrmion for η¼0(D0¼D). Within the width of the ring like formation, ^mrotates toward the xy-plane and is completely in-plane in the center. Outside and inside the ring, ^mk+^z. (b) Vector plot along a cross section ( xz-plane) through the center of the AFM skyrmion in (a). (c) Color plots of the spatial distribution of the Néel vector components nx,ny, and nz. The shown distributions are calculated from the ^m-distribution in (a). Corresponding line-profiles are also shown through the center of the skyrmion along cross sections marked out in the color plot s by dashed lines. The width parameters ΔSk xandΔSk yare indicated on each line-profile.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 193902 (2021); doi: 10.1063/5.0049912 129, 193902-4 Published under an exclusive license by AIP PublishingIV. RESULTS AND DISCUSSION First, to demonstrate efficient suppression of AFM skyrmion elongation, a comparison is made between the two extreme cases ofη¼0 (weak confinement) and η¼1 (maximum confinement). Snapshots of the propagating skyrmion at steady state as a function ofJare shown in Fig. 3 .I nFig. 3(a) , i.e., for η¼0, the AFM sky- rmion elongates rapidly with increasing J. The highest Jsustaining the skyrmion structure is J Cη¼0 ðÞ ¼ 4:375/C21011A/m2with an associated vmaxη¼0 ðÞ ¼ 14:8 km/s. At higher J, the skyrmion hits the horizontal boundaries and breaks up into a DW pair [last snap- shot in Fig. 3(a) ]. In contrast, for η¼1, the rate of elongation with increasing Jappears greatly reduced as shown in Fig. 3(b) with JCη¼1 ðÞ ¼ 1:031/C21012A/m2and vmaxη¼1 ðÞ ¼ 18:8 km/s. In addition, as the elongation slows down, a contraction along the x-direction appears to take place. The consequences of this are dis- cussed later. To check that the AFM skyrmion structure remainsintact even under unsteady conditions, a simulation was runwhereby Jis instantaneously turned off for case η¼1 at steady state propagation with J¼1:031/C210 12A/m2. The skyrmion safely returned to its static equilibrium structure. It is concluded thatefficient suppression of the elongation can be achieved, allowing higher driving amplitudes and higher propagating velocities. Next, the dynamical behavior of the AFM skyrmion as a func- tion of ηis studied by computing mobility ( vvsJ) relations and correlating their behaviors to skyrmion deformation. Results areshown in Fig. 4 . Mobility relations are presented in Fig. 4(a) . The last point on each mobility curve gives the corresponding J Cand vmaxfor each η. Both Jcandvmaxas a function of ηare summarized inFig. 4(b) . The dependence of ΔSky yandΔSky xonηis plotted in Figs. 4(c) and4(d), respectively. Two main observations in Fig. 4(a) are made. First, a saturat- ing behavior of the mobility relation becomes increasingly promi- nent with increasing η. Second, from Fig. 4(b) , both JCand vmax increase with increasing η. This extended region of saturation in v is beneficial for stability and reliability at high operational speeds asit suppresses an abrupt breaking of the AFM skyrmion within a wide range of high driving amplitudes. By comparing J Cη¼0 ðÞ to JCη¼1 ðÞ inFig. 4(b) , the maximum achievable increase in JCis approximately 135%. The associated increase in vmaxis found to be 28%. These values compare favorably to results by Huang et al. ,28 who reported achievable enhancements of 20% and 7% in JCand vmax, respectively. Thus, the method proposed in this work achieves skyrmion stability over a wider range of drive currents and higherpropagation speeds. The emergence of an increasing degree of mobility saturation with increasing ηshould be connected to the rate of skyrmion deformation in the high velocity regime. This is reasonable, sincethe velocity under the action of spin-Hall torques depends on theskyrmion size, 26,47in contrast to spin transfer torque.26Velocity saturation of an AFM skyrmion was predicted by Salimath et al. under spin-Hall torque driving.26They showed that v/J/C1ΔSk x, meaning that a saturation behavior in the mobility is expected ifthere is at some point, a trend of contraction in Δ Sk X. However, they found that it was not reached in practice as skyrmion elonga-tion and subsequent annihilation on the racetrack ’s boundaries occurred before a saturation in vcould take place. In Fig. 4(c) ,i t is clear that a rapid increase in Δ Sky yin the high velocity regime is increasingly suppressed with increasing η.T h i sm e a n ss k y r m i o n annihilation is delayed to occur at higher velocities. In Fig. 4(d) , ΔSky xinitially grows with increasing v, attains a maximum, and then decreases. With increasing η, the velocity range where con- traction takes place is extended. Therefore, we are able to achievea saturating behavior in vand the region of mobility saturation behavior extends with increasing η. Two secondary observations in Fig. 4(a) need clarification. First, there is a sharp increase in vforη¼0a tJ/C253:3/C210 11A/m2, deviating from an expected quasilinear behavior. This is attributedto the rapid increase in the AFM skyrmion lateral dimensions athigh velocities 26[seeFigs. 4(c) and4(d)]. In fact, ΔSky xhas a diver- gent behavior at high enough velocities.26,28Second, in the driving range J/C20JCη¼0 ðÞ , mobilities for η.0 are slightly lower com- pared to when η¼0. This may be connected to a small reduction in the skyrmion size due to the imposed potential even at low J [indicated in Figs. 4(c) and4(d)]; for a given J, a smaller skyrmion moves slower than a larger one.47However, a slightly lower mobil- ity at low and moderate driving amplitudes is outweighed by thedemonstrated benefits of enhanced stability, larger applicable FIG. 3. Suppression of AFM skyrmion elongation at high velocities by lateral modulation of iDMI. The figure shows steady state maps of the AFM skyrmion ’s nz-component with increasing driving current density J. Superimposed vector distributions correspond to the Néel field ’s in-plane components, nxandny. The driving current density for each snapshot is written under each plot in units of A/m 2. (a) In the absence of HM2, η¼0(D0¼D). (b) Maximum confinement, η¼1(D0¼0). Dashed red horizontal lines indicate the boundaries between regions of iDMI strengths Dand D0.F o r η¼0, confinement along the track width is weak, leading to rapid skyrmion elongation with increasing Jand a break up into DWs on contact with the physical boundary. For η¼1, strong confinement along the track width suppresses AFM skyrmion elongation and thus contact with the boundaries occur at much higher J. This also results in higher maximum velocities. For η¼1,vmax¼18:8 km/s and for η¼0, vmax¼14:8 km/s.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 193902 (2021); doi: 10.1063/5.0049912 129, 193902-5 Published under an exclusive license by AIP Publishingdriving J, and higher achievable velocities. From a technological point of view, this means that one can safely propagate the AFMskyrmion with driving amplitudes large enough to induce veloc-ities v.v maxη¼0 ðÞ . V. CONCLUSIONS In summary, a design strategy for AFM skyrmion racetracks with enhanced stability at high propagation velocities is proposedand numerically demonstrated. Lateral patterning of a top HM layer modulates the iDMI providing a strong confinement within the track width. This counteracts AFM skyrmion elongation athigh velocities preventing contact with the racetrack ’s horizontal boundaries and subsequent skyrmion annihilation. Increases in thecritical current density J Cand maximum propagation velocity vmax of 135% and 28%, respectively, are predicted. Furthermore, efficient suppression of the elongation allowed for the emergence of anextended saturating regime in the mobility relation, enabling stablepropagation in the high velocity regime. It is possible that combin- ing the approach adopted in this work with anisotropic iDMI, pro- posed by Huang et al. 28may further enhance AFM skyrmionstability at high velocities. 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1.5115266.pdf	APL Mater. 7, 101120 (2019); https://doi.org/10.1063/1.5115266 7, 101120 © 2019 Author(s).Microwave magnon damping in YIG films at millikelvin temperatures Cite as: APL Mater. 7, 101120 (2019); https://doi.org/10.1063/1.5115266 Submitted: 16 June 2019 . Accepted: 29 September 2019 . Published Online: 24 October 2019 S. Kosen , A. F. van Loo , D. A. Bozhko , L. Mihalceanu , and A. D. Karenowska APL Materials ARTICLE scitation.org/journal/apm Microwave magnon damping in YIG films at millikelvin temperatures Cite as: APL Mater. 7, 101120 (2019); doi: 10.1063/1.5115266 Submitted: 16 June 2019 •Accepted: 29 September 2019 • Published Online: 24 October 2019 S. Kosen,1,a) A. F. van Loo,1,2 D. A. Bozhko,3,4,5 L. Mihalceanu,3 and A. D. Karenowska1 AFFILIATIONS 1Clarendon Laboratory, Department of Physics, University of Oxford, Oxford OX1 3PU, United Kingdom 2Center for Emergent Matter Science, RIKEN, Wako, Saitama 351-0198, Japan 3Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universitaet Kaiserslautern, 67663 Kaiserslautern, Germany 4School of Engineering, University of Glasgow, Glasgow G12 8LT, United Kingdom 5Department of Physics and Energy Science, University of Colorado at Colorado Springs, Colorado Springs, Colorado 80918, USA a)sandoko.kosen@physics.ox.ac.uk ABSTRACT Magnon systems used in quantum devices require low damping if coherence is to be maintained. The ferrimagnetic electrical insulator yttrium iron garnet (YIG) has low magnon damping at room temperature and is a strong candidate to host microwave magnon excitations in future quantum devices. Monocrystalline YIG films are typically grown on gadolinium gallium garnet (GGG) substrates. In this work, comparative experiments made on YIG waveguides with and without GGG substrates indicate that the material plays a significant role in increasing the damping at low temperatures. Measurements reveal that damping due to temperature-peak processes is dominant above 1 K. Damping behavior that we show can be attributed to coupling to two-level fluctuators (TLFs) is observed below 1 K. Upon saturating the TLFs in the substrate-free YIG at 20 mK, linewidths of ∼1.4 MHz are achievable: lower than those measured at room temperature. ©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/10.1063/1.5115266 .,s Microwave magnonic systems have been subject to extensive experimental studies for decades. This work is motivated not only by an interest in their rich basic physics but also by their poten- tial application as information carriers in beyond-CMOS electron- ics.1,2Recently, enthusiasm has grown for the study of magnon dynamics at millikelvin (mK) temperatures, the temperature regime in which solid-state microwave quantum systems operate.3–12This work offers the possibility to explore the dynamics of microwave magnons in the quantum regime and to study novel quantum devices with magnonic components.13–15 Arguably the most important material in the context of room- temperature experimental magnon dynamics is the ferrimagnetic insulator yttrium iron garnet (Y 3Fe5O12, YIG). Pure monocrys- talline YIG has the lowest magnon damping of any known material at room temperature16and is produced in the form of bulk crys- tals and films. Films suitable for use as waveguides in conjunctionwith micron-scale antennas are grown by liquid-phase epitaxy to a thickness of between 1 and 10 μm on gadolinium gallium gar- net (Gd 3Ga5O12, GGG) substrates. The use of GGG is motivated by the need for tight lattice matching to assure a high crystal qual- ity. Recently, YIG films were recognized as promising media for the study of magnon Bose-Einstein condensation and related macro- scopic quantum transport phenomena.17–20In the context of quan- tum measurements and information processing, YIG films hold noteworthy promise; however, if they are to be practical, they must be shown to exhibit the same (or better) dissipative properties at millikelvin temperatures as they do at room temperature. Magnon linewidths in YIG at millikelvin temperatures have thus far only been characterized in bulk YIG resonators (specifi- cally, spheres).4,6,7,10,21Bulk YIG has been shown to retain its low magnon damping at millikelvin temperatures. However, in the case of YIG films grown on GGG, the story is more complex. GGG is APL Mater. 7, 101120 (2019); doi: 10.1063/1.5115266 7, 101120-1 © Author(s) 2019APL Materials ARTICLE scitation.org/journal/apm TABLE I . Comparing results at 300 K and at 20 mK. YIG/GGG Substrate-free YIG Size 2 mm ×3 mm×10μm ∼1 mm×1 mm×30μm w/d 1.7 mm/70 μm 0.9 mm/540 μm 300 Kα1a= (22±4)×10−5α2a= (8.9±0.5)×10−5 Δf○,1a= (0.7±0.4) MHz Δf○,2a= (0.9±0.1) MHz 20 mK α1b= (74±5)×10−5α2b= (2.3±0.7)×10−5 Pb=−65 dBm Δf○,1b= (1.7±0.6) MHz Δf○,2b= (1.1±0.1) MHz 20 mK α1c= (85±6)×10−5α2c= (9.3±1.0)×10−5 Pc=−100 dBm Δf○,1c= (2.6±0.6) MHz Δf○,2c= (2.0±0.1) MHz a geometrically frustrated magnetic system,22and it has long been known that at temperatures below 70 K, it exhibits paramagnetic behavior that has been reported to increase damping in films grown on its surface.23–25The behavior of GGG at millikelvin temperatures is yet to be thoroughly characterized,26–28but recent results at mil- likelvin temperatures have suggested that magnon damping in YIG films grown on GGG is higher than expected if the properties of the YIG system alone are considered.9,11,29In this work, we report a comparative set of experiments on YIG films with and without GGG and move toward a more complete understanding of the damping mechanisms involved. We present data from the measurement of two YIG samples: a 11μm-thick film and a substrate-free 30 μm-thick film. Both sam- ples are grown using liquid phase epitaxy with the surface normal of the YIG film (and the substrate) parallel to the ⟨111⟩crystallo- graphic direction. The substrate-free YIG is obtained by mechani- cally polishing off the GGG until a 30 μm-thick pure YIG film is obtained.25The corresponding lateral size of each sample can be found in Table I. We measure the damping in both films using the microstrip- based technique illustrated in Fig. 1(a).30The sample is positioned above a microstrip and magnetized by an out-of-plane magnetic field ( B). Continuous-wave microwave signals transmitted through the microstrip probe the ferromagnetic resonance of the sample. In the room-temperature experiments, the transmitted signals are mea- sured by connecting the two ends of the microstrip directly to a commercial network analyzer. In our low-temperature experiments, the sample is mounted on the mixing-chamber plate of a dilution refrigerator, as shown in Fig. 1(b), similar to that used in Ref. 11. A microwave source is used to generate the input microwave sig- nal. At the input line, three 20 dB attenuators are used to ensure an electrical noise temperature that is comparable to the temperature of the sample. The output signals then pass through two circulators, a bandpass filter, and an amplifier, before they are downconverted to a 500 MHz signal at room temperature. A data acquisition (DAQ) card then digitizes the transmitted signal at a 2.5 GHz sampling frequency. Signals are usually averaged about 50 000 times before being digitally downconverted in order to obtain a signal similar to the one shown in Fig. 1(a). The magnon linewidth is given by the full-width at half maximum (FWHM) of the Lorentzian fit to thetransmitted signal. The measurement frequency range is between 3.5 GHz and 7 GHz. The low-frequency limit is imposed by the limited bandwidth of the circulators used for our low-temperature measurement setup, and the top of the measurement band is determined by the maximum external field that can be applied by our magnet. The measured damping comprises contributions from the sam- ple and from radiation damping caused by its interaction with the microstrip. In our experiments, radiation damping originates from eddy currents excited in the microstrip by the magnetic field of the magnons31,32and can be decreased by increasing the separation between the sample and the microstrip ( d) at the expense of reduc- ing the measured absorption signal strength ( A). There is therefore a tradeoff to be made between being able to measure linewidths very FIG. 1 . (a) The measurement configuration used to characterize the sample’s damping. The sample and the microstrip (signal line) are separated from each other by a spacer. When the microwave drive is resonant with the magnons in the sample, a decrease in the transmitted signal is observed. (b) The low-temperature setup and its corresponding data acquisition system at room temperature. APL Mater. 7, 101120 (2019); doi: 10.1063/1.5115266 7, 101120-2 © Author(s) 2019APL Materials ARTICLE scitation.org/journal/apm close to the intrinsic linewidth of the sample (thick spacer, negligible radiation damping) and being able to achieve a sufficient signal-to- noise (SNR) ratio (thin spacer, non-negligible radiation damping). Table I lists the microstrip-sample spacings ( d) in our experiments. Since earlier experiments suggested that the YIG/GGG linewidth would be higher at low temperature, the YIG/GGG sample is closer to the microstrip to maintain a sufficient SNR. Within the YIG film itself, the primary contributions to magnon damping are intrinsic processes,33,34temperature-peak pro- cesses,35,36two-level fluctuator (TLF) processes,4and two-magnon processes.35,36Intrinsic processes are due to interactions with optical phonons and magnons; they are expected to decrease with reducing temperature. Temperature-peak processes originating from inter- actions with rare-earth impurities are only significant at low tem- perature (above 1 K). TLF processes are due to damping sources that behave as two-level systems; they are dominant below 1 K. Two-magnon processes have their origins in inhomogeneities in the material; in our experiments, they are minimized by magnetizing the sample out of plane.37,38 Figure 2 compares the magnon linewidth ( Δf) of each sample at 300 K (room temperature) and at 20 mK, as a function of the fer- romagnetic resonance frequency ( f○). Results at 300 K are obtained by sweeping the input microwave frequency under constant B-field. Results at 20 mK are obtained by sweeping the B-field at constant input microwave frequency. In the latter case, the linewidths are measured in terms of magnetic field ( ΔB) and converted to units of frequency ( Δf) via the relation Δf= (γ/2π)ΔB, whereγis the gyromagnetic ratio. Note that there is no conversion factor other thanγ/2πthat is used to translate the low-temperature field-domain data into the frequency domain. A linear fit to Δf= 2αf○+Δf○ gives the characteristic Gilbert damping constant α(unitless) and the FIG. 2 . Magnon linewidths ( Δf) vs resonance frequency ( f○) for a YIG/GGG film and a substrate-free YIG film. Datasets at room temperature (300 K, •) are obtained with an input power of −25 dBm. Datasets at 20 mK are obtained for two input powers: Pb=−65 dBm (▲) and Pc=−100 dBm (∎). Each error bar repre- sents the standard deviation of the linewidth values obtained from repeated mea- surements. Dashed lines are linear fits; the details of these fits are summarized in Table I. Note the different scaling of the vertical axes of the plots.inhomogeneous broadening contribution Δf○. Table I summarizes the results of linear fits to data in Fig. 2. We first compare the results at 300 K and 20 mK obtained at a relatively high input drive level ( Pb=−65 dBm). The substrate-free YIG shows a measured linewidth decreasing from the room temper- ature value to approximately 1.4 MHz at 20 mK. The reduction in damping is as anticipated by existing models that describe the intrin- sic damping of YIG.33–35The radiation damping contribution to the linewidth for the substrate-free YIG is small due to the large spacing from the microstrip ( d= 540μm). The YIG/GGG sample is substantially closer ( d= 70μm) to the microstrip, and its measured αtherefore includes a non-negligible radiation damping contribution αr. In our raw data, uncorrected for this effect, we measure a damping constant at 20 mK ( α1b) that is 3.4 times larger than the room temperature value ( α1a). Following Ref. 31, the radiation damping can be modeled with an equivalent Gilbert damping constant αr=CgMs, where Cgdepends on the geometry of the system and Msis the saturation magnetization of the sample. As both 300 K and 20 mK measurements are performed with identical sample geometry, it is reasonable to expect that the change in αras the temperature is lowered is due to the change in Ms. Therefore, the increase in αrbetween 20 mK and 300 K is determined by the ratio of the saturation magnetization, i.e., Ms(20 mK)/ Ms(300 K) ≈1.4.39The fact that we see a significantly larger damping increase (α1b/α1a≈3.4) in the YIG/GGG and a decrease ( α2b/α2a≈0.26) in the substrate-free YIG indicates that the GGG plays an important role in increasing the magnon linewidth of the YIG/GGG sample at 20 mK. The parameters αandΔf○in both samples increase as the input drive level ( Pc) reduces, as shown in Table I. This behavior can be explained by the TLF model upon which we shall elaborate later. Figure 3 shows the temperature dependence of the magnon linewidths for both samples measured at low input power ( Pc=−100 dBm). For the YIG/GGG results in Fig. 3, the radiation damping contribution ( αr=CgMs31) across the examined temperature range can be considered to be an approximately constant vertical shift to FIG. 3 . Temperature ( T) dependent magnon linewidths ( Δf) for both YIG/GGG film and substrate-free YIG measured with input power Pc=−100 dBm. Note the different scaling of the vertical axes of the plots. APL Mater. 7, 101120 (2019); doi: 10.1063/1.5115266 7, 101120-3 © Author(s) 2019APL Materials ARTICLE scitation.org/journal/apm each dataset. This is due to the small change (less than 0.07%) in Ms of YIG between 20 mK and 9 K.39 Above 1 K, linewidths of both samples increase as the temper- ature is increased up to 9 K. In this temperature range, damping is dominated by temperature-peak processes caused by the presence of rare-earth impurities in the YIG.25,35,39–41When temperature-peak processes are dominant, the linewidth of the sample peaks at a char- acteristic temperature ( Tch) determined by the damping mechanism and the type of impurity. Temperature-peak processes at low temperatures fall into two categories:35,36those associated with (1) rapidly relaxing impurities and (2) slowly relaxing impurities. Rapidly relaxing impurities pro- duce a Gilbert-like damping and a characteristic temperature Tch that is independent of the magnon resonance frequency f○. Slowly relaxing impurities produce a non-Gilbert-like damping and a corre- sponding characteristic temperature that decreases as the resonance frequency f○is lowered. The behavior observed in Fig. 3 at 9 K, with the linewidth for the f○= 4 GHz being higher than that at f○= 7 GHz, indicates that impurities of slowly relaxing type dominate in this temperature range. As the temperature is decreased below 1 K, linewidths for the substrate-free YIG start to increase and eventually saturate, as shown in Fig. 3. This can be explained by the presence of two-level fluctua- tors (TLFs) and has been previously observed in a bulk YIG.4In the TLF model, the damping sources are modeled as an ensemble of two- level systems with a broad frequency spectrum.42,43The linewidth contribution can be expressed as ΔfTLF=CTLFωtanh(̵hω/2kBT)√ 1 +(P/Psat), (1) where CTLFis a factor that depends on the TLF and the host material properties. The power-dependent term can be rewritten as P/Psat= Ω2 rτ1τ2, where Ω ris the TLF Rabi frequency, and τ1andτ2are respectively the TLF longitudinal and transverse relaxation times.44 At high temperatures ( kBT≫hfTLF), thermal phonons satu- rate the TLFs and therefore the material behaves as if the TLFs were not present. At low temperatures ( kBT≪hfTLF) and low drive levels (P≪Psat), most of the TLFs are unexcited. Under these conditions, the TLFs increase the damping of the material by absorbing and re-emitting magnons or microwaves at rates set by their lifetimes, coupling strength, and density. When the drive level is increased past a certain threshold ( P≫Psat), the TLFs are once again saturated and therefore do not contribute to the damping. Evidence for the presence of the TLFs is shown in Figs. 2 and 4. The datasets for 20 mK in Fig. 2 show that the linewidths for both samples are lower when the drive level is higher ( PbvsPc). Figure 4(a) shows a similar behavior in the substrate-free YIG. Above 1 K, linewidths for both drive levels are similar: an indica- tion that the relevant TLFs are saturated by thermal phonons. The differences in linewidths for the two drive levels begin to appear as the temperature is lowered below 1 K. Figure 4(b) shows the linewidths of the substrate-free YIG as a function of drive level ( P) at three different temperatures (1 K, 300 mK, and 20 mK). At 1 K, there is no observable power depen- dence as the relevant TLFs have been saturated by the thermal phonons. At 20 mK and 300 mK, the linewidth increases as the power decreases, saturating at millikelvin temperatures. This is in FIG. 4 . Magnon linewidths ( Δf) in the substrate-free YIG film for various temper- atures ( T) and input powers ( P). (a) Temperature dependent linewidths for two different input powers Pb=−65 dBm and Pc=−100 dBm. (b) Power dependent linewidths at 20 mK, 300 mK, and 1 K. The dashed lines are the fits to the data. agreement with the theory previously articulated and the fits shown by dashed lines in Fig. 4(b). The data are fitted using Eq. (1) with an additional y-intercept to account for non-TLF linewidth contributions. For the f○= 7 GHz dataset in Fig. 4(b), Psatat 300 mK is clearly larger than at 20 mK. This is in-line with expectations: τ1andτ2 are anticipated to decrease as the temperature is increased, leading to a higher Psat(recall that Psat∝1/τ1τ2).44–46The exact temper- ature dependence of 1/ τ1τ2is not clear; in previous experiments, a phenomenological model was suggested with the quantity 1/ τ1τ2 varying from T2toT4.45This places the ratio Psat(300 mK)/ Psat(20 mK) in the range of 23.5 dB–47 dB. The fitted Psatvalues from our data correspond to a ratio of approximately 22.5 dB, suggestive of a T2behavior. It should be noted that the f○= 4 GHz, T= 300 mK dataset shows a very weak TLF effect since there are sufficient thermal phonons to saturate the TLFs with central frequencies around 4 GHz; this is not the case for higher frequency datasets taken at the same temperature. A higher Psatis also observed at 300 mK for f○= 5 GHz and f○= 6 GHz (data not shown). Figure 4(a) shows that the input power Pb=−65 dBm used in our experiments is not enough to saturate the relevant TLFs for temperatures between 100 mK and 1 K. The datasets obtained with high drive level ( Pb) in Fig. 4(a) show that the linewidth difference δf= \|Δf(f○= 7 GHz) −Δf(f○= 4 GHz)\| broadens as the temperature is increased from 100 mK to 300 mK, narrowing back APL Mater. 7, 101120 (2019); doi: 10.1063/1.5115266 7, 101120-4 © Author(s) 2019APL Materials ARTICLE scitation.org/journal/apm as the temperature reaches 1 K. If a higher drive level is used, δfis expected to be smaller at temperatures between 100 mK and 1 K. In conclusion, the substrate GGG on which typical YIG films are grown significantly increases the magnon linewidth at mil- likelvin temperatures. However, if the substrate is removed, it is possible to obtain YIG linewidths at millikelvin temperatures that are lower than the room-temperature values. Measured linewidths of both YIG/GGG and substrate-free YIG systems above 1 K are con- sistent with the temperature-peak processes, typically observed in YIG containing rare earth impurities. Damping due to the presence of unsaturated two-level fluctuators is observed in both YIG/GGG and substrate-free YIG films below 1 K. We observe the TLF satu- ration power to be higher at higher temperatures in agreement with the existing literature. We further verify that using a high drive level reduces the linewidths of the substrate-free YIG films down to ∼1.4 MHz ( f○= 3.5 GHz to 7.0 GHz) at 20 mK. Looking forward, our measurements suggest that—in the con- text of the development of magnonic quantum information or mea- surement systems—it may be worthwhile to investigate the pos- sibility of growing YIG films on substrates other than GGG, or techniques which circumvent the use of a substrate entirely.47–50It should be emphasized that the current experimental configuration does not allow us to pinpoint the origin of the TLFs; further investi- gations into TLFs in YIG would be useful in obtaining high-quality YIG magnonic devices that operate in the quantum regime. Note added in proof . A preprint by Pfirrmann et al.21recently reported experiments concerning the effect of two-level fluctuators on the linewidth of bulk YIG. This work helpfully complements our investigations into the behavior of YIG films. We thank A. A. Serga for helpful discussions and J. F. Gregg for the use of his room-temperature magnet. 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1.5121265.pdf	Low Temp. Phys. 45, 935 (2019); https://doi.org/10.1063/1.5121265 45, 935 © 2019 Author(s).Ultrafast spin dynamics and spintronics for ferrimagnets close to the spin compensation point (Review) Cite as: Low Temp. Phys. 45, 935 (2019); https://doi.org/10.1063/1.5121265 Published Online: 27 September 2019 B. A. Ivanov Ultrafast spin dynamics and spintronics for ferrimagnets close to the spin compensation point (Review) Cite as: Fiz. Nizk. Temp. 45,1 0 9 5 –1130 (September 2019); doi: 10.1063/1.5121265 View Online Export Citation CrossMar k Submitted: 23 July 2019 B. A. Ivanov1,2,a) AFFILIATIONS 1Institute of Magnetism of the NAS of Ukraine and MES of Ukraine, 36b Vernadsky Blvd., Kiev 03142, Ukraine 2Taras Shevchenko National University, 2 Glushkov Ave., Kiev 03127, Ukraine a)Email: bor.a.ivanov@gmail.com ABSTRACT The possibilities of applying magnets with full or partial magnetic moment compensation in various spin groups to improve the performance of magnetic electronic devices using spin current (spintronics) are discussed. The e ﬀects of an exchange enhancement of the spin dynamics in antiferromagnets are well known. Over the past few years, antiferromagnetic spintronics has turned into an independent, rapidly developing ﬁeld of applied physics of magnetism. This article provides for a detailed analysis of the possibility of using another class of magnetic materials, such as ferrimagnets close to the spin compensation point, in which the indicated acceleration e ﬀects are also detected. A comparative analysis of these two classes of magnets is conducted. The nonlinear spin dynamics of ferrimagnets are examinedusing a nonlinear sigma-model for the antiferromagnetic vector, describing the di ﬀerence in spin densities of various spin groups. The simple conclusion derived based on this model is presented, and its real parameters for popular ferrimagnets, amorphous alloys of iron, and rare earth elements, are discussed. The di ﬀerent nonlinear e ﬀects of spin dynamics, ranging from homogeneous spin vibrations in small particles to the dynamics of solitons, domain walls, ferrimagnetic skyrmions, and vortices, are analyzed. The possibility of exciting suchdynamic modes using spin torque, and their application in ultrafast spintronics is considered. Published under license by AIP Publishing. https://doi.org/10.1063/1.5121265 TABLE OF CONTENTS 1. Introduction 1 2 The fundamental concepts of the phenomenological theory of magnets3 2.1. Types of magnetic order: antiferromagnetic features 3 2.2. Spin dynamics (Landau-Lifshitz equation) 5 3. Spin dynamics of ferrimagnets based on the generalized sigma-model6 4. Nonlinear homogeneous spin oscillations 9 5. Precession solitons (magnon droplets) in uniaxial ferrimagnets 12 6. The domain wall dynamics in biaxial ferrimagnets 14 6.1. General considerations and model formulation 146.2. The structure and limiting velocity of the domain wall 156.3. Forced motion of the domain wall 16 7. Features of topological solitons, skyrmions, and vortices in ferrimagnets20 7.1. Static structure and gyroscopic dynamics 207.2. Vortices in small ferrimagnetic particles 21 7.3. Skyrmions —stability and dynamics 22 8. Conclusion 23 1. INTRODUCTION Modern progress in the fundamental and applied physics of magnetism is largely associated with the development of nanotech- nology, i.e. the creation and use of submicron magnetic particles(nanoparticles). This progress is associated with the prospect ofcreating new magnetic systems for recording and processinginformation with increased recording density and speed. 1,2 The development of nanomagnetism has given rise to a number of new e ﬀects. The ﬁrst that should be noted is the dis- covery of giant magnetoresistance,3,4which manifests itself when a current with polarized electron spins ﬂows through a ferro- magnetic metal. In this case, the idea of the spin current is the fundamental concept. For example, spin current can beLow Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 45,000000 (2019); doi: 10.1063/1.5121265 45,000000-935 Published under license by AIP Publishing.generated by passing an ordinary electric current through a layer of hard magnetic material (polarizer), or by using the spin Hall eﬀect (see Appendix 1 ). Another remarkable e ﬀect of the spin current was predicted shortly after the discovery of giant magne-toresistance, and consists of the fact that the spin currentﬂowing through a magnet can create a speci ﬁc“anti-damping ” torque and compensate for the natural attenuation of the spin dynamics 5,6(see also reviews Refs. 7–10). As a result of such anti-damping, the stable state of a magnetic nanoparticle with acertain direction of the magnetic moment can become unstablewhen exposed to a spin current. This instability can progress in two ways: under the action of the spin current, the magnetic moment of the nanoparticle either ﬂips over (reverses in a direc- tion that turns out to be stable), or the nanoparticle can go intoa state with considerable magnetic moment ﬂuctuations that are stable. These e ﬀects make it possible to create various magnetic electronic devices with submicron dimensions. The ﬁrst example presents the possibility of recording information by currentpulses, while reading can be done using the giant magnetoresis-tance e ﬀect. The implementation of the second mode makes it possible to create a solid-state nanogenerator, the so-called spin-torque oscillator (see Appendix 1 ). Options for creating spin analogues of typical electronic devices, diodes, have alsobeen proposed, 11–15and even spin transistors have been dis- cussed.16These discoveries gave rise to the idea of a new ﬁeld of applied physics of magnetism, known as spintronics (SPINelecTRONICS), in which the main role is played not bythe charge but by the electron spin, and not by the electric, butby the spin current. 8–10 The general pattern of developing various solid-state electronic devices is determined by both an increase in the density of active ele- ments and an increase in the speed of the device. In particular, formemory systems it is important to both reduce the size of one bitand increase the speed of writing and reading information. For gen-erators, signi ﬁcantly increasing the operating frequency is essential, and the possibility of restructuring it is also important. At present, it is believed that both of these problems can be solved by using anti-ferromagnets instead of standard ferromagnetic materials. 17–19 The possibilities of implementing spintronics based on anti- ferromagnets became clear after the publication of Ref. 20,i n which it was shown that the spin current acts e ﬀectively even on materials with a compensated magnetic moment. In order toincrease the density of the magnetic elements in a magneticmicrocircuit, it is important to reduce the magnetic moment of an individual particle and the magnetic ﬁelds it creates, while also maintaining the “rigidity ”of the magnetic order, i.e. its stability with respect to thermal ﬂuctuations and ﬂuctuations of external ﬁelds. Long-range dipole interaction for su ﬃciently dense arrays of magnetic particles prevents the magnetic state bistability of an individual particle from being used for record- ing. In this case, magnetic particles in the vortex state could bepromising, but the critical size of such particles exceedshundreds of nm 21,22and reducing it poses a serious challenge (see Refs. 21–23). It is clear that antiferromagnetic particles with a high Néel temperature, a small (or zero) magnetic moment, and low sensitivity to the action of an external magnetic ﬁeld, could provide a solution to this problem.It is fundamentally important that the eigenfrequencies of antiferromagnet spin vibrations are much higher than those of standard ferromagnetic materials. These frequencies, due to theso-called exchange enhancement e ﬀect, range from hundreds of gigahertz to several terahertz, see E.A. Turov et al. 24and recent reviews.25,26The characteristic times associated with the nonlin- e a rd y n a m i c so fa n t i f e r r o m a g n e ts, such as the switching time between di ﬀerent spin states, are several orders of magnitude shorter than the corresponding values for ferromagnets (fororthoferrites, the switching speed is of the order of picosec-onds). 27,28For antiferromagnets, the e ﬀect of magnetic state switching by a spin current was experimentally discovered in Refs. 29and 30. Antiferromagnets can e ﬃciently conduct spin current at distances no smaller than that of ferromagnets.31,32The eﬀects of spin current ampli ﬁcation,32and even some spin current superﬂuidity analogues were discussed Refs. 33–37. In principle, the presence of two branches of magnons allows for the creation of more ﬂexible devices for logic and information processing.38 Particularly attractive is the idea of creating a spin-torque auto-oscillator with an antiferromagnetic active element thatwill operate in the terahertz frequency range. Terahertz radia- tion includes (somewhat conditionally) electromagnetic waves with frequencies from 300 GHz to 3 THz, which corresponds towavelengths from 1 mm to 100 microns, see Fig. 1 .I nr e c e n t years, the need to master the terahertz wave range has been increasing sharply, but there is a clear de ﬁcit of compact sources of terahertz radiation. This is formulated as “the problem of ﬁlling the terahertz gap ”, see Refs. 39–42.P o s s i b l e uses of terahertz waves include applications in 4G and 5G tele-communication systems and space communications, the search and security of prohibited materials, biology and medicine, information technology, ultrafast data processing, see the recentdiscussion in Ref. 42. At present, several speci ﬁc ideas for such auto-oscillators operating in the terahertz range have been proposed, based on dielectric antiferromagnets. 43–48The performed calculations, taking FIG. 1. A diagram of various electromagnetic wave ranges and the most typical coherent sources of these waves. Abbreviations: IR (infrared); VL (visible light); UV (ultraviolet), x-ray radiation. The acronym SR (synchrotron radiation) is used to denote x-ray sources based on synchrotron radiation of charged particles inaccelerators.Low Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 45,000000 (2019); doi: 10.1063/1.5121265 45,000000-936 Published under license by AIP Publishing.into account the real parameters of the utilized materials, showed their high e ﬃciency, however, to date, no generation e ﬀects have been observed and none of these ideas have been experimentallyimplemented. The author of this review does not have the slightest doubt that antiferromagnets will ﬁnd their application in future spin- tronics devices, especially terahertz spintronics. However, the antiferromagnetic ordering of s pins has a number of characteris- tic features that are absent in standard ferromagnets (or amor-phous ferrimagnets, which will be discussed in this review). Thesefeatures are caused by the high sensitivity of the antiferromag- netic order to the presence of defects that violate the sublattice structure of the crystalline sample; they will be discussed in thenext section. It is currently di ﬃcult to assess how important these e ﬀects will be. Therefore, it is of interest to search for alternative materi- als that could be used in spintronic devices that operate in the terahertz range, but without creating the problems inherent inantiferromagnets. In a way, the solution is hiding in plain sight:the fact that the exchange enhancement e ﬀects of dynamic parameters also occur in ferrimagnets directly near the sublattice spin compensation point has been known for a long time. 49To be speci ﬁc, this is the point at which the limiting velocity of domain walls and other solitons increases (and becomes purelyexchange in nature). 49It is surprising that this feature of spin dynamics went experimentally undiscovered for many decades, despite the direct theoretical predictions of the “antiferromag- netic”frequency behavior and ultrafast soliton dynamics, as well as the observation of such e ﬀects for antiferromagnets.50–52It is only recently that a study experimentally documented a sharp increase in the velocity of a domain wall in an amorphous GdFeCo alloy as the spin compensation point was approached,53 and this increase was interpreted in terms of exchange accelera-tion. This work demonstrates the reality of creating ferrimagnetichigh-speed spintronic devices, and the possibility of manufactur- ing the necessary high-quality nanostructures. Recently, ultrafast magnetic moment switching in ferrimagnetic garnets close to thecompensation point was demonstrated in Ref. 54. It is theoreti- cally shown that the eigen vibration frequencies of magnetic vor-tices in ferrimagnet particles can reach tens of GHz, which is signiﬁcantly higher than that of standard ferromagnetic vortices (less than a gigahertz). 55 Note that ferrimagnets are much more convenient than anti- ferromagnets from the point of view of application in nanophysics, and especially spintronics. For example, amorphous alloys of rare-earths with elements of the iron group (such as the famousGdFeCo, which has an ultrafast change in the sign of the magneticmoment, about equal to a picosecond, during heating by a femto-second laser pulse 56–58) are typical ferrimagnets, but standard nanotechnologies can be used for them, the same as for the classi- cal materials of nanomagnetism, iron, nickel or permalloy.Switching e ﬀects have been observed not only for ﬁlms, but also for microparticles 57and nanoparticles58of this material. In addition, these alloys, like many other ferrimagnets, have metal conductivity, which allows the use of standard magnetoresistance e ﬀects to read the information system signals or to convert the energy ofspin vibrations into an alternating electric current. It is equallyimportant that their magnetic anisotropy can be varied, which is important for high-frequency applications. However, the exact speciﬁcs of spintronic e ﬀects in ferrimagnets have gone practically unstudied. This article is dedicated to the description of spin dynamics in ferrimagnets that are in the “antiferromagnetic ”parameter region, when exchange enhancement e ﬀects are present and their dynamic properties are close to those of antiferromagnets. Particular atten-tion is paid to amorphous ferrimagnetic alloys. A comparative anal-ysis of the spin order for ferromagnets or ferrimagnets versusantiferromagnets is carried out. The generalized sigma-model equa- tion, known for “pure”antiferromagnets, is derived for the case of real, almost compensated ferrimagnets, taking into account thedissipative processes and the impact of the spin current. Based onthe obtained equations, various nonlinear spin dynamics modesthat can be used in terahertz spintronics devices are investigated. Although most of the attention is devoted to the theory (the review was written by a theoretician), some (a few) currently availableexperimental data that testify to the exchange acceleration of spindynamics, including nonlinear, near the spin compensation pointof ferrimagnets, are discussed. 2. THE FUNDAMENTAL CONCEPTS OF THE PHENOMENOLOGICAL THEORY OF MAGNETS 2.1. Types of magnetic order: antiferromagnetic features Magnetic ordering is usually associated with the appearance of a nonzero average magnetization value Mor spin density s. The magnetization Mand the spin density vector sare related by equa- tionM=–gμ Bs, where gis the Landé factor ( g-factor); hereinafter, μBis understood to be the magnitude of the Bohr magneton (the magnetic moment of the electron is antiparallel to its spin). For ferromagnets, the average values of all spins are parallel, andthe description of their spin system is limited to one vector charac-teristic, which is the magnetization M. This property characterizes not only the simplest single-element ferromagnets, but also ferro- magnetic alloys such as the Ni xFe1-xpermalloy. The magnetization is also nonzero for those ferrimagnets, in which it is possible todistinguish several di ﬀerent groups of magnetic ions with antipar- allel spins. In the simplest case, there are two such groups, corre- sponding to the magnetization M 1,M2and spin densities s1,s2. For ferrimagnets, the spontaneous magnetization M=M1+M2or the spin density s=s1+s2,a r en o n z e r oa sar u l e ,a l t h o u g ht h e y can vanish at some selected externa l parameters (at the so-called compensation points, see below f or more details). Note that even for single-element ferromagnets containing transition element ions in the s-state, the value of the g-factor can noticeably di ﬀer from the value g= 2.0023 …for a free electron. For example, for “classical ” ferromagnets, iron and nickel with a face-centered cubic lattice, thevalues g= 2.2 and 2.09 are accepted, for cobalt with a hexagonal close-packed lattice, g= 2.14. The g-factors for di ﬀerent groups of magnetic ions in a ferrimagnet can vary, which leads to thediﬀerence in the conditions of magnetization and spin density compensation. In the language of the theory of symmetry, the appearance of magnetization means that there is a spontaneous symmetryLow Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 45,000000 (2019); doi: 10.1063/1.5121265 45,000000-937 Published under license by AIP Publishing.breaking of time re ﬂection ^T. In this transformation, the magne- tization vector changes its sign, M→−Mast→–t. This property is inherent not only to the simplest single-element ferromagnet,but also to many other magnets, in particular, ferromagneticalloys such as permalloy Fe 1-xNix. It is also characteristic to ferri- magnets. It is important to note that for all these materials, the time re ﬂection operation must be considered independently of the symmetry operations associated with the crystal structure ofthe magnet. This is manifested in th e possibility that ferromagne- tism exists in amorphous ferromagnets (such as permalloy) oramorphous ferrimagnets. For example, in the latter alloys of tran- sition and rare-earth metals that are currently popular, such as Gd x(FeCo) 1-x, the magnetizations of these two subsystems are antiparallel. Antiferromagnets are a fundamentally di ﬀerent class of magnets in which spontaneous magnetization can be strictly equal to zero, but the symmetry with respect to time inversion is sponta- neously broken.24This situation can be imagined by assuming that the crystal lattice of the antiferromagnet contains two magneticsublattices, the magnetizations of which M 1andM2are exactly equal in length and are antiparallel, M1+M2= 0. The assumption of the exact equality of M1andM2implies that these sublattices are necessarily crystallographically equivalent, i.e. there is at leastone element of the crystal symmetry group (the so-called oddelement ^g (–), see Ref. 24) that transfers one sublattice to the other. In particular, the presence of such equivalence guarantees the equality of the sublattice g-factors and the simultaneous compensa- tion of both the total spin density and the magnetization overa wide range of changes in the external conditions. The antiferro-magnetism vector L=M 1–M2is the order parameter for an antifer- romagnet. The same characteristic is also introduced for ferrimagnets, but for an antiferromagnet the vector L=M1–M2 changes its sign not only with respect to time re ﬂection ^T, but also during some crystal symmetry group transformations ^g(–). The symmetry aspect is extremely important to the physics of antiferro- magnetism, because it is the existence of odd elements of the crystal group ^g(–)that is a strict criterion for antiferromagnetism.24 This criterion must also be ful ﬁlled for so-called antiferromagnets with weak ferromagnetism, or canted antiferromagnets, in which,when the vector Lis oriented along certain directions, a spontane- ous weak magnetic moment M weakappears proportional to L, Mweak i=dikLk, and the form of the tensor dikis determined by the magnetic symmetry of the crystal. It is important to note that themagnetization of ferrite near the spin compensation point can be small and, in principle, be comparable to the weak moment of anti- ferromagnets, i.e. both of these classes of materials have similarstatic behavior. However, the role of the weak magnetic momentM weakin“ideal”antiferromagnets and of the uncompensated moment of ferrimagnets in spin dynamics is fundamentally diﬀerent, see the review in Ref. 59. It should be noted that amor- phous antiferromagnets also exist, see Ref. 60, but their spin struc- ture is fundamentally di ﬀerent from the standard case of a magnet with several sublattices. Because the antiferromagnetic order is so sensitive to the crystal lattice, working with nanosystems that include antiferro- magnetic active elements could prove problematic. Not only is thepresence of local defects important in this case, but so is the non-ideal particle shape of the antiferromagnet, see the review in Ref. 61for details. Strictly speaking, pure antiferromagnetism with completely compensated magnetic moments of the sublatti-ces does not exist for real nanoparticles: it is di ﬃcult to expect that for a real system of N=N 1+N2spins, where N1,N2is the number of sites in each sublattice, it is possible to get the exact equality N1=N2atN∼105–109. For atomic clusters that include hundreds or thousands of spins with antiferromagnetic interac-tion, the role of the surface becomes very noticeable. Even for anatomically smooth surface, the number of particles in the sublatti-ces can di ﬀer. As a result, the decompensation of the total spin in an antiferromagnetic particle occurs. Let us give consider ferritin particles, which play an important role in the life of mammals.These particles are used as a model object for the experimentalstudy on the properties of antiferromagnet nanoparticles, espe-cially the e ﬀects of macroscopic quantum tunneling, see Refs. 62 and 63. The ferritin particle contains approximately 4500 Fe 3+ iron ions with S= 5/2 spin, coupled by antiferromagnetic interac- tion and ordered in an almost ideal magnetic structure of atypical crystalline antiferromagnet —hematite. At the same time, the uncompensated moment of the ferritin particle is not small, and amounts to about 200 μ B, or about 1% of the maximum possible value. As will be argued below in the main part ofthe review, it is at this value of decompensation that thefundamental di ﬀerences between ferrimagnets and antiferromag- nets appear. However, on the other hand, this value is optimal for some applications of ferr imagnets in spintronics —see Section Four. Thus, from the point of view of spin dynamics,ferritin should be considered as a ferrimagnet close to the com-pensation point. The presence of decompensation fundamentally changes the properties both classical 49,64,65and quantum66–68 dynamics of magnetic nanoparticles with antiferromagnetic interaction. A similar pattern emerges for another important example of an antiferromagnetic nanosample: thin ﬁlms. The surface proper- ties of an antiferromagnet, and the antiferromagnet –ferromagnet contact boundaries in particular, have been studied for a longtime, since they determine the practically important so-calledexchange bias e ﬀect, see Refs. 69and70. Even with an atomically smooth ﬁlm surface representing the ideal atomic plane, several scenarios can be imagined that lead to decompensation. Spins at the boundary may belong to di ﬀerent sublattices, as shown in Fig. 2(a) , or the same one, as seen on Fig. 2(b) .O n l yi nt h e ﬁrst case, the static magnetization of an ideal atomically smooth surface of the antiferromagnet is equal to zero. On the contrary, in the second case the boundary is magnetized, and such a boun-dary is uncompensated. For nano ﬁlms with an uncompensated boundary and a thickness of about 10 nm, which corresponds to20–30 atomic planes, there is a di ﬀerence in the behavior of ﬁlms with an even or odd number of atomic planes; in the latter case, there is a decompensation of the sample ’s total spin to the scale of several percent. In real systems with a boundary roughness on anatomic scale, there can exist an intermediate case of a partiallycompensated boundary. In this scenario, decompensation occurs alongside possible macroscopic inhomogeneities such as spin disclinations, see Refs. 71and 72, which were observed in thin chromium ﬁlms. 73Low Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 45,000000 (2019); doi: 10.1063/1.5121265 45,000000-938 Published under license by AIP Publishing.As far as we know, the e ﬀects of decompensation or the appearance of macroscopic defects, such as spin disclinations, have not been discussed with respect to the operation of antiferromag-netic spintronic devices. 2.2. Spin dynamics (Landau-Lifshitz equation) The spin dynamics of ferromagnets are described by the well-known Landau –Lifshitz equation; 74see also monograph,75 which is usually written for magnetization @M @t¼/C0γ[M/C2Heff]þRþT,Heff¼/C0δW[M] δM, (1) where γ=gμB/ℏ,γ≈2.8 MHz/Oe at g= 2. The e ﬀective magnetic ﬁeldHeﬀis deﬁned as the variational derivative of the energy (more precisely, the nonequilibrium thermodynamic potential) of the fer-romagnet W=W[M], written in the form of the magnetization density functional M=M(r,t). The ﬁrst term describes the non- dissipative dynamics of M, and the remaining terms determine the processes that do not conserve energy. Here, two terms are distin- guished: Rdescribes the dissipative processes, and Tdetermines the torque caused by the action of the spin-polarized current (spintransfer torque). It is important to note that at R= 0 and T= 0, Eq. (1)con- serves the magnetization length, ∂(M 2)/∂t=2 (M/C2∂M/∂t) = 0. This condition is a key component of the standard phenomenologicaltheory of magnetism. Usually, the terms RandTare chosen so that the condition M 2=M2 s= const is also maintained for non- conservative dynamics. Two alternative forms of writing Rcorre- spond to this condition, one is the original proposed by Landauand Lifshitz RLL, and the other is the form of Gilbert RG, RLL¼λHeff/C01 M2 s(Heff/C1M)M/C20/C21 , RG¼αG1 MsM/C2@M @t/C20/C21 :(2) It can be shown that these two forms are completely equiva- lent and are reduced to each other when re-identifying the con- stants, see monograph76for details. Let us discuss the importance of condition \| M\| = const and the possibility of going beyond it. In real magnets, when \| M\| devi- ates from the equilibrium (at a given temperature T) magnetization value M0(T), the occurring relaxation will be rather fast, see Ref.75. The use of Eq. (1)implies that such longitudinal relaxation is weakly coupled with the transverse dynamics of spins. In recentyears, the longitudinal evolution of magnetization has been studiedusing ultrafast heating of magnets by femtosecond laser pulses, which has led to the observation of unexpected and rather unusual eﬀects. For a ferrimagnetic alloy of rare-earth and transition metals GdFeCo, a fast (about a picosecond) “switching ”of the direction of the particle ’s total magnetic moment was observed after exposure to a femtosecond pulse. 56–58The description of this e ﬀect required the development of a consistent theory describing the longitudinal evolution of spins in various magnets. It is interesting to note thatthe basis of this theory dates back to the 80 s articles of V. G.Baryakhtar, who, on the basis of Onsager ’s formalism, constructed a general view of the relaxation terms Rhaving an exchange and relativistic nature. 77–81The Landau –Lifshitz equation with relaxa- tion terms proposed by V. G. Baryakhtar is now commonly calledthe LLBar equation. 82,83In particular, these studies showed that the relaxation term of exchange origin must violate the condition M2=M2 s, accepted in the phenomenological theory of magnetism for almost a century. The Landau –Lifshitz equation with the Bloch relaxation term (LLB) is also widely used, see Refs. 84–86, which also does not conserve the magnetization modulus. Soon after thepublication of Refs. 77–81, it was shown that taking into account the non-conservation of the modulus \| M\| is important for describ- ing the relaxation of nonlinear excitations (various solitons) inmagnets. 87–90The use of this equation for ferrimagnets made it possible to theoretically describe the abovementioned “switching ” eﬀects,91,92and also to point out the possibility of interesting e ﬀects of inhomogeneous longitudinal evolution of the magnet spin density.83,93–96However, a detailed description of the longitudinal spin dynamics is beyond the scope of this work, and the latestresults can be found in monograph. 76 Let us return to Eq. (1)and discuss the form of term T, which determines the e ﬀect the spin current has on the ferromagnet mag- netization (spin torque). If an electric current with a partial polari-zation of electron spins ﬂows along a certain direction ^p,^p 2=1 , taking into account the condition \| M\| = const, it is natural to accept the following expression for T(see Refs. 7–9for example): T¼σj Ms[M/C2(M/C2^p)], (3) FIG. 2. Different variants of the contact boundary between an antiferromagnet with two sublattices (magnetic atoms belonging to different sublattices are indi-cated by red and blue circles in the ﬁgure) with non-antiferromagnetic material, the atoms of which are indicated by green circles.Low Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 45,000000 (2019); doi: 10.1063/1.5121265 45,000000-939 Published under license by AIP Publishing.where jis the electric current density, the formulas for σin various spin torque regimes are given in Appendix 1 . It is convenient to use the parameter τ=σj, which has a frequency dimension.7Note that in principle, it is also possible to torque the longitudinal com-ponent. Its inclusion is important to describing the longitudinalspin dynamics; however, to our knowledge, this question has not been discussed. It is useful to note that the non-conservative terms RandT can be written in the standard way through the dissipative functionof the ferromagnet Q=Q G+QSTT RG¼γMsm/C2δQG δ(dM=dt)/C18/C19 , T¼γMsm/C2δQSTT δ(dM=dt)/C18/C19 ,(4) where m=M/Msand the contributions to Qare determined by the formulas QG¼αG 2γMsð dr@M @t/C18/C192 , QSTT¼τ γMsð dr@M @t(M/C2p)/C18/C19 ,(5) at the same time, the rate of the ferromagnet ’s energy change is determined by the expression dW dt¼/C02QG/C0QSTT: (6) The value QG> 0 has a de ﬁnite sign, while the sign of the spin current contribution QSTTdepends on the relative orientation of the spin current polarization pand the magnetization M. The latter contribution can play the role of both positive and negative friction, which determines the possibility of the “anti-damping ”eﬀects. Note that Eq. (6)diﬀers from the usual relation dW/dt=–2Q, which is valid only for mechanical systems with a dissipative func-tion that is quadratic with respect to generalized velocities. 97 3. SPIN DYNAMICS OF FERRIMAGNETS BASED ON THEGENERALIZED SIGMA-MODEL The approach based on the Landau –Lifshitz Eq. (1)is valid not only for “pure”ferromagnets, but also for describing the low- frequency dynamics (frequencies lower than the corresponding“exchange ”value, which will be speci ﬁed below) for all magnets with considerable magnetization in the ground state. In particular,if the spin lengths s 1=\|s1\| and s2=\|s2\| are noticeably di ﬀerent, s1-s2∼s1,2, then this equation is applicable to ferrimagnets.98,99 On the other hand, the use a relation such as Eq. (1)for the anti- ferromagnet vector Lis strictly forbidden by symmetry: the conser- vative part of this equation is invariant with respect to time inversion, but not invariant with respect to any odd element of the antiferromagnet symmetry group.24It is well known that, provided the magnetization is small, the dynamics of an antiferromagnet can be described using the closed equation for the unit antiferromagnetism vector l. In this approach, the antiferromagnet magnetization vector is a subordinate variableand is determined by the vector land its time derivative ∂l/∂t. The dynamic equations of motion for the ﬁeld of the unit vector lare commonly referred to as the sigma-model equations; their use sig- niﬁcantly simpli ﬁes the analysis of both the linear and nonlinear dynamic e ﬀects in antiferromagnets (see monographs and reviews Refs. 50,51,59, and 100). It turns out that near the spin compensa- tion point, the ferrimagnet dynamics are also described by some version of the sigma-model. Several alternative approaches can be used to derive this equation (see Refs. 49,53,64,66, and 67.W e use the simplest and most intuitive method, which is based on theuse of a system of two Landau –Lifshitz Eq. (1)for two spin groups (for brevity we limit ourselves to sublattices, although this approach is applicable to both crystalline and amorphous magnets) with M 1, M2magnetizations, or spin densities s1,s2,s1,2=-M1,2/g1,2μB, and hereinafter the spin is measured in Planck constant units. In the case of nonequivalent spins, working through spin den- sities is more consistent,91since it allows for a direct transition to quantum mechanics. It is enough to compare the schematic repre- sentation of the quantum equation of motion for the spin operator i/C22h@^s @t¼[^s,^H], where ^His the Hamiltonian of the system, and [^s,^H] is the operator commutator with the corresponding descrip- tion of the Landau –Lifshitz Eq. (1)in terms of the classical quan- tity, spin density s,/C22h@s/@t¼[s/C2δW=δs]. This method is also technically convenient, since it makes it possible to avoid takinginto account the di ﬀerent g-factors of di ﬀerent spins during the derivation of equations. Relations such as Eq. (1)for magnetization M 1,2can be easily rewritten in terms of spin densities s1,2, M1,2=−g1,2μBs1,2. Next, we need to choose the formulation for the phenomenological energy W=W[s1,s2], which depends on a large number of parameters that determine both the individual sublatticeproperties and the interaction between them. There are a lot more of these parameters than for an antiferromagnet, in the case of which certain simpli ﬁcations arise by virtue of sublattice equiva- lence. However, the formulation of the energy can be greatly sim-pliﬁed in the case that interests us, near the angular momentum (spin) compensation point s 1=s2, hereinafter s1,2=\|s1,2\|. In this case, instead of spin densities, it is convenient to use the same combinations that applied for an antiferromagnet, m¼s1þs2 stot,l¼s1/C0s2 stot,stot¼s1þs2: (7) The vectors mandlare connected by two simple relations: m/C1l¼(s1/C0s2)=stot,m2þl2¼1þ[(s1/C0s2)=stot]2: (8) In the linear approximation with respect to the small parame- ter (s1-s2)/stot/C281 , we can assume that m2+l2= 1 for a ferrimagnet, as well as for an antiferromagnet. We can expect the magnitude of vector mto be small near the compensation point. Because of this, we can limit ourselves to the main approximation of the vector mcomponents whenLow Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 45,000000 (2019); doi: 10.1063/1.5121265 45,000000-940 Published under license by AIP Publishing.formulating the energy expression. This greatly simpli ﬁes the energy structure that must be used to adequately describe the dynamics of ferrite. In actuality, the homogeneous exchange energyof a ferrimagnet can contain three invariants: l 2,m2andm×l. However, by virtue of condition (8),m×lreduces to a constant, andl2is expressed in terms of m2. Therefore, only one energy invariant can remain, for which it is convenient to choose m2. When formulating the energy of inhomogeneous exchange and rel-ativistic interactions, we can restrict ourselves to invariants contain-ing the vector land consider it to be a unit vector, 1 2= l. The inclusion of terms that are bilinear with respect to components m andl, such as mzlzfor a uniaxial ferrimagnet, can lead to some sin- gularities in soliton dynamics,101but does not change the nature of the eﬀects discussed in this review and is not considered further. As a result, the energy density can be written in a simple and uni-versal way: w¼Eex 2m2þA 2(∇l)2þwa(l)/C0MH 0/C01 2MH m: (9) Here, the ﬁrst two terms determine the exchange energy, Eex and Aare the homogeneous and inhomogeneous exchange con- stants, respectively, wa(l) is the anisotropy energy, and the last two terms describe the contribution of the external magnetic ﬁeldH0 and magnetostatic energy. These, and only these, terms depend on the magnetization M, and not on the spin densities. Below, it will be shown how vector Mcan be easily written using vector l, see Eq.(13), and therefore we can also assume that this part of the energy depends only on the vector l. The equations describing the phenomenological spin density theory for ferrimagnet sublattices s1ands2can easily be rewritten through the vectors landm; their structure is exactly the same as for an antiferromagnet /C0/C22hstot@m @t¼m/C2δW δm/C20/C21 þl/C2δW δl/C20/C21 , /C0/C22hstot@l @t¼l/C2δW δm/C20/C21 þm/C2δW δl/C20/C21 :(10) Here, W=W[l,m] is the energy functional (non-equilibrium thermodynamic potential), the quantities hm,l=−δW/δ(m,l) play the role of the e ﬀective ﬁelds for a ferrimagnet because they are written in energy density dimensions. Note that in these equationsthe signs at ∂l/∂tand∂m/∂tdiﬀer from those chosen for magneti- zation Min the Landau –Lifshitz Eq. (1); as explained above, the sublattice magnetizations M 1andM2are antiparallel to the spins of the electronic sublattices s1ands2. When writing Eq. (10), the non- conservative contributions of various origins are omitted for thesake of simplicity; they will be restored in the ﬁnal equation for the vector l. If the lengths of the spin densities s 1=\|s1\|a n d s2=\|s2\| and are noticeably di ﬀerent, s1-s2∼s1,2[for a more accurate criterion, see Eq.(12)], the linear approximation solution to Eq. (10) deﬁnes two branches of magnons, one of which is low-frequency and almost coincides with that obtained from the simple Landau –Lifshitzequation for a ferromagnet (1). The magnons of the second branch have an activation that is about the same as the exchange integral between sublattices Eex; their frequencies are in the infrared range. This limiting case is described in detail in monographs.98,99However, here there is another limiting case of “antiferromagnetic ”behavior that is interesting, in which both frequencies are comparable and small vis-à-vis the exchange frequency, but signi ﬁcantly exceed the “ferromagnetic ”frequency of relativistic origin. To analyze this case, consider the energy formulation (9),a n d note that all terms in the equation for ∂l/∂tare bilinear with respect to the components of vectors landm. It is clear that in the presence of an exchange term, the inclusion of small relativistic terms with the same structure is an excess of accuracy. If we leave only the exchanget e r m ,t h i se q u a t i o ni ss i m p l i ﬁed to the form /C22hstot@l @t¼/C0Eex(l/C2M), which allows us to write Eex[Ml2/C0l(l/C2M)]¼/C0/C22hstotl/C2@l @t/C0/C1 :Thus, same as for an antiferromagnet, the vector m,p r o p o r t i o n a lt ot h e total spin s=s1+s2,i sas l a v ev a r i a b l ea n di sw r i t t e no n l yt h r o u g h vector land its time derivative. Given the conditions of Eq. (8),t h e total density of spins s=s1+s2is written as s¼(s1/C0s2)lþ1 ωex(s1þs2)@l @t/C2l/C18/C19 : (11) This is where the characteristic exchange frequency ωex=Eex/(s1+s1)/C22his introduced. Note that one can also introduce the ferrimagnet exchange ﬁeld, say, using the formula 2 μBHex=/C22hωex. This value is convenient for estimates but does not have the same uni- versal meaning as for antiferromagnets. In particular, the g-factor values can vary for ferrimagnet sub lattices, and it is not clear which v a l u es h o u l db eu s e di nt h i sd e ﬁnition. Equation (11) for the total spin contains two terms. The ﬁrst is typical of the “ferromagnetic ”behavior of a ferrimagnet; it states that the total spin is parallel to the vector l. When only this term is taken into account, it turns out that the backs of the sublatticesremain collinear (antiparallel) even in the dynamics. The secondterm is typical of antiferromagnets and describes the noncollinear- ity of sandl. Equation (11) allows us to indicate the limits of applicability for the standard “ferromagnetic ”approach. Indeed, what is interest- ing is the case when the characteristic frequency /C22ω/differencej(l/C2 @l @t)jis small compared to the exchange frequency ωex. It follows that the spin dynamics of a ferrimagnet cannot be reduced to purely ferro- magnetic behavior when the inequality ( s1–s2)/(s1+s2)/C20/C22ω=ωex/C281 is fulﬁlled. Usually /C22ω/differenceﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃωrωexpwhere ωr<<ωexhas a relativistic origin. As a result, we get that a speci ﬁc“antiferromagnetic ”behav- ior takes place only when a rather stringent condition is met:49 v¼s1/C0s2 s1þs2/C20/C22ω ωex/differenceﬃﬃﬃﬃﬃﬃﬃωr ωexr /C281: (12) Depending on the type of magnet and the nature of the motion of the spinsﬃﬃﬃﬃﬃﬃ ωr ωexq /difference3/C210/C02/C010/C03, this characteristic value of νis very small, but can vary over a fairly wide range. Below, this ratio will be speci ﬁed for concrete problems.Low Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 45,000000 (2019); doi: 10.1063/1.5121265 45,000000-941 Published under license by AIP Publishing.The magnetization of a ferrimagnet M=–μB(g1s1+g2s2) can be rewritten as M¼/C01 2μB(s1þs2)[(g1/C0g2)lþ(g1/C0g2)m]: (13) In the “antiferromagnetic ”limit, near the spin compensation point, the quantity \| m\|∼max( ν,/C22ω/ωex) << 1. For magnets such as GdFeCo, the value g1–g2∼0.2 exceeds the expected values of νand /C22ω/ωex. Therefore, in Eq. (13), the ﬁrst contribution is dominant, and the energy in the external magnetic ﬁeldHand the energy of the magnetic dipole interaction can be described by the expression wH¼/C0μefflHþ1 2Hm/C18/C19 , μeff¼/C01 2μB(s1þs2)(g1/C0g2),(14) where the magnetostatic ﬁeld is determined using the standard magnetostatics equations, see Ref. 75, in which M=μeﬀl(recall that μeﬀ< 0). It can be seen that a su ﬃciently weak magnetic ﬁeld of the “relativistic ”scale (about μBHr∼/C22hωr) has a noticeable e ﬀect on the behavior of a ferrimagnet that does not have an insigni ﬁcant value ofg1–g2. For antiferromagnets, the characteristic ﬁelds are much larger. For example, the ﬁeld of the spin –ﬂop transition has an exchange-relativistic order of magnitude. Therefore, hereinafter werestrict ourselves to the case of su ﬃciently weak ﬁelds, \| H\|≤H r. For such ﬁelds, the speci ﬁc dynamic e ﬀects that are caused by the magnetic ﬁeld and known to occur for antiferromagnets do not appear. Therefore, the derivation of the dynamic terms for the sigma-model can be carried out under the condition H= 0, while the contribution of the magnetic ﬁeld is taken into account in the static energy wH. Using the explicit form of the energy (9), the smallness of the external magnetic ﬁeld, and Eq. (11), the desired equation of the generalized sigma-model can be written as containing only thevector l /C0v@l @t¼1 ωexl/C2@2l @t2/C18/C19 /C0c2∇2l/C20/C21 þl/C2@ωr @l/C18/C19 þαGl/C2@l @t/C18/C19 þτ(l/C2(l/C2p)): (15) Here, the decompensation parameter ν<< 1 is determined by Eq.(12),cis the characteristic velocity, which coincides with the spin wave velocity at s1=s2, c¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Aωex /C22h(s1þs2):r (16) As is the case for a pure antiferromagnet, the velocity cis determined only by exchange interactions and signi ﬁcantly exceeds the characteristic velocities for ferromagnets (one example of the exchange enhancement of dynamic parameters). For convenience,the function Ωr(l) is introduced, ωr(l)¼1 /C22h(s1þs2)wr(l), (17) having frequency dimensions. This function determines the energy density of relativistic interactions wr=wr(l), which includes the anisotropy energy, as well as the contributions of the external mag-netic ﬁeld and magnetic dipole interaction energy in the form of Eq.(14), the presence of which is associated with the incomplete compensation of the sublattice magnetic moments. The last two terms in Eq. (15) determine non-conservative dynamics, namely the Gilbert-type damping and the spin transfer torque with polarization p. The dimensionless Gilbert constant α G, and the constant τ=σj, which has a frequency dimension, see Eq. (3), have the meaning of e ﬀective constants describing the total contribu- tion of both sublattices. More general relaxation terms, such as theexchange relaxation terms introduced by V.G. Baryakhtar are impor- tant for describing the longitudinal evolution of spins during spin “switching ”in GdFeCo-type ferrimagnets under ultrafast heating, 91,92 but their consideration is beyond the scope of this review. Given exact spin compensation ( s1=s2,o rν=0 ) , E q . (15) coin- cides with the equation of the Lorentz-invariant sigma-model for a selected velocity c(16). An external magnetic ﬁeld disrupts the Lorentz invariance due to the “antiferromagnetic ”contribution of the ﬁeld, which is bilinear with respect to the components of vectors H and∂l/∂t.W i t has u ﬃciently weak ﬁeld, at H/C28ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃHexHap,i tc a nb e ignored, see the review in Ref. 59for details. The presence of a formal Lorentz invariance greatly simpli ﬁes the analysis soliton dynamics; see some examples in Refs. 102 and 103 and a recent review in Ref. 59. It is important that, given a considerable g1–g2∼g1,2,s u c h Lorentz-invariant exchange-accelerated dynamics of the vector lat s1→s2is also conserved for the ferrimagnet, but it is accompanied by considerable changes in the ferrimagnet magnetization M. The equation of the sigma-model (15) in the non-dissipative limit can be obtained by varying the Lagrangian L[l],l×δL/δl=0 . The ferrimagnet Lagrangian L[l]=T+G–Wincludes the kinetic energy T(a term that is quadratic in the time derivative ( ∂l/∂t)2)a n d the potential energy W, as well as the gyroscopic term G. Here, the expressions for TandWare the same as for an antiferromagnet, T¼/C22h(s1þs2) 2ωexð dr@l @t/C18/C192 ,W¼ð drA 2(∇l)2þwr(l)/C26/C27 :(18) As for the gyroscopic term, it has the same structure as that of a ferromagnet, and is written through a singular vector function,the vector potential of the Dirac monopole ﬁeldA=A(l) with a single magnetic charge, rot lA=l, see Refs. 104and105 G¼/C0/C22h(s1/C0s2)ð drA@l @t/C18/C19 ¼/C0/C22hv(s1þs2)ð drA@l @t/C18/C19 :(19) Recall that the vector potential Ais deﬁned only up to a certain calibration, while the ﬁctitious magnetic ﬁeld rot lA, which is part of the equations of motion, is gauge-invariant. For aLow Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 45,000000 (2019); doi: 10.1063/1.5121265 45,000000-942 Published under license by AIP Publishing.monopole ﬁeld, one can choose the expression A=(n/C2l=(1þn/C2l), where nis a constant unit vector, n2=1 . The vector potential of the Dirac monopole ﬁeld has a singularity; for this expression, the singularity (Dirac string) is located on thehalf-line l=−n. 104,105 Knowing the Lagrangian is useful when applying the collective variable approach. The use of the Lagrangian makes it possible to construct the energy-momentum tensor of the vector lﬁeld and write out the ferrimagnet ’s basic integrals of motion. The ferrimag- net energy is equal to the sum of the “kinetic energy ”and the “potential energy ”of vector l,E=T+W,TandWare determined by Eq. (18). However, integrals of motion such as the ﬁeld momen- tum of the vector lﬁeld, or the angular momentum, are not invari- ant with respect to the above gauge transformations. In particular,the formula for the momentum of the magnetization ﬁeldPcan be written as P i¼/C0/C22h(s1þs2) ωexð@l @t/C1@l @xi/C18/C19 drþP(0) i, P(0) i¼/C0/C22h(s1/C0s2)ð A/C1@l @xi/C18/C19 dr,(20) where the term P(0) iis clearly not gauge-invariant. Thus, the deter- mination of the ﬁeld momentum of the vector lﬁeld, and therefore the momentum of a ferrimagnet ’s excited states, runs into certain problems. However, in the case of a ferrimagnet, the form of the vector potential Ais the same as that of a ferromagnet, and the momentum problem for these two types of magnets is equivalent.To solve this problem in practically important cases (analysis ofdomain walls, vortices, and skyrmions), constructive methods have been developed with respect to ferromagnets, see Refs. 106–114, for example; these methods transfer almost automatically to thedescription of ferrimagnets. To analyze nonlinear spin dynamics, it is convenient to use angular variables for the unit vector l, l 1¼sinθcosw,l2¼sinθsinw,l3¼cosθ, (21) where l1,2,3is the projection of vector lonto some orthogonal axes 1, 2 and 3. It is convenient to choose the polar axis 3 such that itcoincides with the direction of the ferrimagnet ’s easy axis, so that the ground state corresponds to θ=0 ,π. In angular variables, the expressions for the kinetic and potential energy, see Eq. (18), take the form T¼/C22h(s1þs2) 2ωexð dr@θ @t/C18/C192 þsin2θ@w @t/C18/C192"# , (22) W¼ð drA 2@θ @xi/C18/C192 þsin2θ@w @xi/C18/C192"# þwr(θ,w)() , (23) and the gyroscopic term in the Lagrangian L=T+G–Wis clearly a function of the gauge. It is often the case that n=–e3is chosen, sothat the gyroscopic term has the same form as what was adopted in many studies on the spin dynamics of ferromagnets, G¼/C0/C22h(s1/C0s2)ð dr(1/C0cosθ)@w @t: (24) Naturally, the gyroscopic term is proportional to the uncom- pensated backbone and vanishes at s1=s2. The equations for variables θandwcan be written as /C0νsinθ@w @t¼1 ωex@2θ @t2/C0c2∇2θ/C18/C19 /C01 ωexsinθcosθ@w @t/C18/C192 /C0c2(∇w)2"# þ@Ωr @θþαG@θ @t/C0τsin2θ(p2cosw-p1sinw), (25) ν@θ @tsinθ¼1 ωex@ @tsin2θ@w @t/C18/C19 /C0c2∇(sin2θ∇w)/C20/C21 þ@Ωr @wþαG@w @tsin2θ-τp3sin2θ þτsinθcosθ(p1coswþp2sinw)¼0: (26) Here, the Gilbert form is chosen for the dissipative term, and the contribution of the spin torque is calculated for an arbitrarycurrent polarization p=(p 1,p2,p3). 4. NONLINEAR HOMOGENEOUS SPIN OSCILLATIONS Let’s begin by analyzing the simplest example of spin dynam- ics, in the form of uniform spin oscillations for a purely uniaxialferrimagnet. For such a magnet, the anisotropy energy dependsonly on the projection of vector lonto the chosen axis of the magnet ( zaxis), w a=wa(l2 z), or in angular variables wa=wa(θ), and the angular variable wenters the equations only through its derivatives. The external magnetic ﬁeld is not taken into account. It is enough to restrict ourselves to the simplest form of anisotropyenergy with one anisotropy constant K, and for a magnet with an “easy axis ”anisotropy it is convenient to write it as w a(θ)¼K 2(l2 xþl2 y)¼K 2sin2θ: (27) In other words, for the function Ωa(θ)=(ωa/2)sin2θintroduced above, the characteristic frequency ωa,/C22hωa=K/(s1+s2)c o r r e s p o n d s to the anisotropy energy, see Eq. (17). To start, consider the natural spin vibrations of the system without accounting for dissipation pro- cesses. It is easy to verify that at wa=wa(θ), Eqs. (21) and (22) allow for a simple solution in the form of w=ωt,θ=θ0=c o n s t , which describes the precession of vector laround the zaxis with a frequency ω. For this nonlinear dynamic mode θ0≠0,π, and the fre- quency ωand precession amplitude θ0(the angle between land theLow Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 45,000000 (2019); doi: 10.1063/1.5121265 45,000000-943 Published under license by AIP Publishing.selected axis) are related by νω¼cosθ0ω2 ωex/C0ωa/C18/C19 : (28) Let’s discuss the physical meaning of this expression in various limiting cases. First of all, note that with exact compensa-tion, when ν= 0, the frequency value does not depend on the pre- cession amplitude, which is typical for an antiferromagnet withanisotropy energy in the form of Eq. (27) (see Ref. 59for details). Thus, for the antiferromagnetic case of ν= 0, this means that dynamics are possible either with the exact equality ω 2=ωaωexor at cos θ0= 0. It turns out that in this “antiferromagnetic ”case the dynamics of vector lfor any ω2≠ωaωexis aﬂat rotation, θ0=π/2. This property of the antiferromagnet ’s spin dynamics is well known59and poses signi ﬁcant problems for the creation of antifer- romagnetic spintronic auto-oscillators, see Refs. 18and43–45and Appendix 1 . However, the signi ﬁcant dependence the frequency has on amplitude, i.e. the conical precession with the angle valueθ 0≠π/2, which depends on the frequency, is restored for an arbi- trarily small, but ﬁnite,ν. Note that, for a considerable νvalue, Eq. (28) reﬂects the well- known fact that a ferrimagnet has two modes with signi ﬁcantly diﬀerent frequencies. It is appropriate to refer to them as a ferro- magnetic mode, with a low (purely relativistic) frequency ωFM=−(ωa/ν)cosθ0, and an exchange mode, the frequency of which is νωex/cosθ0and is determined by the exchange interaction. Diﬀerent signs of the frequencies mean that the precession of vector lfor these two modes occurs in opposite directions, the magnon energies for both modes being positive. The directions of the precession of vector land the magnetiza- tion are opposite. The frequency ωFMis determined only by relativis- tic interactions, but increases when approaching the compensationpoint (formally, at ν→0i td i v e r g e sa s1 / ν). The frequency of the second mode is of the order of ω ex, but it decreases when approach- ing the compensation point. It is important to note the di ﬀerent behavior of the dependence on amplitude, the frequency ωFM decreases with increasing amplitude, and the frequency of the exchange mode increases. It is understood that at ν∼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ωa=ωexp the frequencies of these modes become comparable, and at the same time both frequencies become exchange-relativistic, of the order ofﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃωaωexp. It is convenient to write the mode frequencies through dimensionless variables in the following form: ω ω0¼/C22ν 2c o s θ0+ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ /C22ν 2c o s θ0/C18/C19 þ1s ,/C22ν¼νﬃﬃﬃﬃﬃﬃﬃωex ωar , (29) here the frequency ω0=ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃωaωexpis the frequency of antiferromag- netic resonance, and naturally the quantity /C22νarises, such that /C22ν/difference1 in the characteristic region of ferromagnetic behavior ν∼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ωa=ωexp /C281, see Eq. (12). A similar behavior of the magnetic resonance frequencies has been observed for many ferrimagnets, see monographs.98,99In a recent article,115this behavior was investigated by a purely optical method, by exciting thin (20 nm thick) ﬁlms of an amorphousGd22Fe74.6Co3.4ferrimagnet with a femtosecond laser pulse and optically detecting the signal (all optical pump probe) ( Fig. 3 ). One can also construct a more general solution describing a nonlinear traveling spin wave. It corresponds to the precession ofthe vector lwith constant amplitude and a phase that depends on the coordinate r,θ=θ 0=const, w=kr−ωt, here khas the meaning of a wave vector. The dependence of the nonlinear wave frequency ω0(k),k=\|k\|o nkand amplitude θ0is determined by the formulas νωexω¼cosθ0[ω2/C0ω2 0(k)],ω0(k)¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ω2 0þc2k2q , (30) where ω0(k) coincides with the frequency in the “antiferromag- netic”limit at ν=0. To summarize, we can say that the spin dynamics of ferrimag- nets are characterized by the presence of a fast (with exchange- relativistic frequency) conical spin precession. These properties offerrimagnets make it possible to create an e ﬀective terahertz range generator with spin current excitation. 116To discuss the details important to the operation of such devices, the properties of the system of Eqs. (25) and (26) are considered in more detail, with allowance for non-conservative terms. First and foremost, note that Eq. (25) allows for an exact pre- cession solution θ=θ0=const, w=ωteven if the dissipation is taken into account, in the event that the spin current polarization is directed along the magnet ’s axis of symmetry, p=±ez. In this case, px=0 ,py= 0, and the non-conservative terms in this equation vanish. As a result, Eq. (25) determines the relationship between the solution parameters in the same form as for a conservativesystem sinθ 0[vωωexþcosθ0(ωaωex/C0ω2)]¼0: (31) Here, the speci ﬁc form of the anisotropy energy Ωa=(ωa/2) sin2θis taken into account, but the sign of the anisotropy constant is not speci ﬁed, i.e. this analysis is equally applicable for easy axis and easy plane cases of anisotropy. In Eq. (26), the non-conservative terms are assembled into a compact expression and give [ αG(∂w/∂t)–τ]sin2θ0= 0. It follows from this formula that there are two possibilities: either θ0=0 ,π, and the magnet is in one of two static states, with l=ezorl=–ez, or there is a stationary dynamic state with θ0≠0,π, in which the precession frequency is determined by the intensity of the spin torque, ω¼τ=αG¼(σ=αG)j, (32) i.e. the frequency is proportional to the spin current density. A joint analysis of these conditions determines the di ﬀerent states of a ferrimagnet, both static and dynamic, in the presence of a spin current. We begin by analyzing the static states θ0=0 ,π, which corre- spond to l=ezandl=–ez. For easy axis anisotropy ( ωa> 0), these states simply determine one of the magnet ’s equivalent ground states. However, the precession solution is also applicable to theLow Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 45,000000 (2019); doi: 10.1063/1.5121265 45,000000-944 Published under license by AIP Publishing.easy-plane ferrimagnet, for which ωa< 0. In this case, the ground state corresponds to an arbitrary orientation of vector l⊥ezin the easy plane ( xy), i.e. the values θ0=π/2 and w¼w0= const. Note that in the case of an easy plane magnet, the precession motionrepresents a speci ﬁc, essentially non-linear, spin dynamics regime that does not allow for a simple transition to the linear case. In par- ticular, the linear theory spin wave spectrum cannot be obtained by a limiting transition from Eq. (30). One of its magnon branches is gapless, and the second has a ﬁnite gap with a frequency ω gap=ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ jωajωexþν2ω2 exp . Let us consider the forced dynamics for an easy plane ferri- magnet. It can be shown that for any arbitrarily small τ/αG, the ground state with θ0=π/2 is unstable, and a precession arises with a frequency ωdeﬁned by Eq. (32). This result is similar to that obtained previously for purely uniaxial antiferromagnets.20,17In the presence of weak anisotropy in the basal plane, there is a threshold current value.44However, for an antiferromagnet, only purely ﬂat motion with θ0=π/2,w=ωt=(σj/αG)tis possible, while for a ferrimagnet, vector lmust exit from the basal plane. The exit angle is determined by Eq. (31), which is conveniently written in terms of dimensionless variables for an easy-plane ferrimagnet cosθ0¼/C22ν/C22ω 1þ/C22ω2,/C22ν¼νﬃﬃﬃﬃﬃﬃﬃﬃωex jωajr ,/C22ω¼ω ω0,ω0¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ jωajωexp :(33) It is easy to see that for 0 < /C22ν< 2 (the quantity /C22νcan have any sign, but for the sake of certainty, we consider the positive values of /C22ν), this equation has the solution −1 < cos θ0< 1, i.e. precession exists for all values of current j. A numerical solution of the equations of motion showed that for any initial state with the vector lin theeasy plane, after turning on the current, the system goes into a dynamic state with stationary precession that corresponds to the current.116Thus, both of these dynamic states are stable, see Fig. 4 . It should be noted that static solutions with l=±ezor cosθ0= ±1formally exist for all values of /C22νand frequency (current), but they realize the maximum anisotropy energy. These states are marked by thin dashed lines in Fig. 4 . The analysis showed that these states are unstable at /C22ν<2. If the value is /C22ν.2, then the behavior of the magnet is more complicated. In this case, precessional dynamics also exist and arestable at frequencies ω<ω crit 1andω>ωcrit 1, in fact, at j<jcrit 1and j>jcrit 2, where the critical current values are determined by the formulas 2(σ=α)jcrit 2¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ /C22ν2/C04p þ/C22ν,2 (σ=α)jcrit 1¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ /C22ν2/C04p /C0/C22ν: (34) Ifjcrit 1,j,jcrit 2, then the values \|cos θ0\|>1 correspond to the formal solution of Eq. (33).I tt u r n so u tt h a ti nt h i sf r e q u e n c yr a n g e , one of these states that happens to be a continuous continuation of the dependences cos θ0(ω)a t\| c o s θ0\| < 1, acquires stability.116These states are marked by solid lines in Fig. 4 . Thus, the spin current at jcrit 1,j,jcrit 2“pushes ”the magnet into one of these static states with l=±ez,c o sθ0= ±1, which corresponds to the maximum possi- ble value of the anisotropy energy. The optimal value from the point of view of a useful signal value is the amplitude θ0=π/4, at which the EMF assumes the maximum possible value, see Appendix 1 . It is easy to show that this value can be realized at /C22ν>ﬃﬃﬃ 2p , and that it corresponds to two fre- quency values,ﬃﬃﬃ 2p/C22ωopt 1,2¼(/C0/C22ν+ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ /C22ν2/C02p . As such, the system ’s behavior, and in particular the characteristic frequency values, areextremely sensitive to the decompensation parameter. Recall thatthe normalized parameter /C22νis coupled with the real (extremely small) spin decompensation ν¼(s 1–s2)/(s1+s2)v i ae x p r e s s i o n FIG. 3. The temperature dependence of the magnetic resonance frequencies for the Gd 22Fe74.6Co3.4ferrimagnet; ωFMR andωex(solid and empty symbols) correspond to the upper and lower spin vibration modes. The values of TMand TA, which are the points of magnetization and angular momentum (spin) com- pensation, respectively, and the value of the Curie temperature Tc, are also given. The ﬁgure is taken from Ref. 115. FIG. 4. The azimuthal angle θfor steady-state motion as a function of the pre- cession frequency ω(actually, it is the intensity of the spin torque, σj=αGω) for various values of the effective spin decompensation parameter /C22ν(shown in the Figure) for an easy plane ferrimagnet.116Low Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 45,000000 (2019); doi: 10.1063/1.5121265 45,000000-945 Published under license by AIP Publishing.ν¼/C22vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ωa=ωexp /C28/C22ν. Since all typical values of /C22νare about equal to one, the corresponding di ﬀerences in the sublattice spins are suﬃciently small and do not exceed 10−2. The frequency values in this part of the text and on Fig. 4 are given in characteristic fre- quency units ω0¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ jωajωexp . This frequency is exchange-relativistic, but its value for materials like GdFeCo, which have a relatively weak exchange interaction between transition and rare-earth element ions, may not be very high about 100 –200 GHz, see Appendix 2 . However, characteristic frequencies, such as /C22ωopt 2, at which the useful variable signal is optimal, can have values (3 –5)ω0and reach values of the order of THz. A ferrimagnet with an easy axis type anisotropy exhibits a more complex behavior, mainly because a spin current with a di ﬀerent sign of polarization ( p=±ez) acts di ﬀerently on its two possible ground states, l=±ez. Here, in addition to the e ﬀects of exciting spin precession with a considerable amplitude θ0,θ0≠π/2, switching between the states l=ezandl=–ezis also possible for certain polari- zation directions.116 5. PRECESSION SOLITONS (MAGNON DROPLETS) IN UNIAXIAL FERRIMAGNETS The purely uniaxial ( zis the chosen axis) model of the ferri- magnet, in which Ωrdepends only on θ, allows for the analysis of a wide class of soliton solutions. As for any uniaxial magnet, in this case ∂Ωr/∂w= 0, and the equation for w(25) takes the form of a continuity equation, which de ﬁnes the conservation of the total spin Stot z¼Ðs zdr, see Eq. (11). The general type of solitons that are not related to the exact integrability of the equation, corre- sponds to two-parameter states (moving magnon droplets) with aﬁxed value of S tot zand soliton momentum P. These solitons are described by solutions that depend on two parameters, the solitonvelocity v, and the spin precession frequency ωin the reference frame moving with velocity vtogether with the soliton, so the solu- tion has the form θ¼θ(r/C0vt),w¼/C0ωtþψ(r/C0vt), (35) see original studies, 49,64,102,103,117and reviews and monographs59,118–121 for more details. For convenience, the frequency sign is chosen so that in the ferromagnetic limit the frequency is positive, see Eq. (28). Various types of precession solitons in ferromagnets,122–125 which are referred to as “magnon droplets ”throughout literature, are discussed as active elements of nanogenerators excited by aspin-polarized current. Such nanogenerators have serious advan-tages over systems with uniformly magnetized particles. 126–129 Therefore, it is useful to discuss the properties of precession soli- tons for the case of ferrimagnets (antiferromagnetic solitons havealready been considered in a recent review). 59 In contrast to an antiferromagnet, the Lorentz invariance is absent for a ferrimagnet, and the scenario is the same as for a ferro- magnet: moving solutions can only be constructed in the one- dimensional (1D) case.49For a soliton moving along the xaxis, a two-parameter 1D solution can be found in the form θ(ξ),w¼/C0ωtþψ(ξ), (36)where the variable ξ=x−vtis introduced. Using this substitution, it is easy to obtain an explicit expression for dψ/dξ, (c2/C0υ2)dψ dξ¼υωþvωex 2 cos2(θ=2)/C20/C21 , (37) and then write a second-order equation using ordinary derivatives forθ(ξ).49This equation has a ﬁrst integral, which is conveniently represented in the form l2 0 2dθ dξ/C18/C192 þU(θ)¼E¼0, U(θ)¼A(1/C0cosθ)/C01 2B sin2θþΔtg2θ 2/C18/C19 :(38) Here, the value of the integral Εis chosen based on the condi- tion that, the magnet is in the ground state when it is far from thesoliton, θ=0 ,dθ/dξ=, and the exchange length l 0is introduced l0¼c ω0¼ﬃﬃﬃﬃ A Kr , (39) which is a characteristic parameter of a magnet and determines, for example, the thickness of a static 180-degree domain wall. Theremaining parameters are determined by the formulas A¼/C22ν~ω (1/C0~υ2)2,B¼1 (1/C0~υ2)/C0~ω2 (1/C0~υ2)2,Δ¼/C22ν 2(1/C0~υ2)2, (40) where for brevity the notation ~υ¼υ c,~ω¼ω=ω0is used, and so is /C22ν¼νﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ωex=ωap from the previous section (recall that /C22ν/difference1 the decompensation value is small, ν/differenceﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ωa=ωexp /C281) The general solution to Eq. (38) can be written in explicit form tg2θ 2/C18/C19 ¼κ2l2 0 21ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ A2þ2BΔp ch2(κξ)þ(AþΔ), (41) where the value κ,κ2l2 0= 4(B –A–Δ/2) determines the width of the soliton ’s localization region δξ,δξ=1 /κ. The region of admissible parameter values is determined by the condition κ2> 0, and the equation κ2= 0 determines the boundary of this region. It is easy to imagine this condition as ωþνωex 2/C16/C172 þυ2 c2ω2 0þν2ω2 ex 4/C18/C19 /C20ω2 0þν2ω2 ex 4/C18/C19 , (42) i.e. for any ν≠0, the region of admissible soliton parameters on theυ,ωplane lies inside the ellipse with the center shifted down from the origin (for precision, we assume that ν>0 )b y νωex/2, see Fig. 5 . The maximum value of the soliton velocity, as in the case of an antiferromagnet, is equal to the velocity c, which is achieved atLow Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 45,000000 (2019); doi: 10.1063/1.5121265 45,000000-946 Published under license by AIP Publishing.ω=−νωex/2. In the limiting case ν→0, this ellipse, as is the case for an antiferromagnet, becomes symmetric with respect to the velocity axis. The ferromagnetic limiting case is obtained by the limiting transition /C22ν2¼ν2ω2 ex=ω2 0/C291, see below. Ferrimagnets exhibit a signi ﬁcantly greater variety of soliton states than ferromagnets or antiferromagnets. Theoretically, all ofthese states can be derived by considering the limiting cases of Eq.(41), but it is easier to use a qualitative analysis. The mechani- cal analogy can be used to qualitatively consider the solution type,which would mean comparing the distribution in the soliton θ(ξ) against the motion of a ﬁctitious material point with the coordinate θand velocity dθ/dξ, in the potential U(θ). In this case, the integral of motion (38) E=T+U(θ) represents the mechanical energy, seeFig. 6 . Atθ→0, the potential ’s asymptotic behavior is determined by the value κ 2,U(θ)=−κ2θ2/2.The soliton must correspond to a sol- ution with θ→0,dθ/dξ→0a tξ=±∞, i.e. the separatrix solution of Eq. (38) with the value Ε= 0, its asymptotics θ(ξ)∝exp(−κ\|ξ\|) as ξ→±∞. This solution describes the following motion: the repre- sentative point leaves (having an in ﬁnitely small speed at ξ=−∞) the equilibrium at θ= 0 and moves right to the stopping point θ0, which is determined by condition U(θ0) = 0. It is easy to see that when Δ≠0 [for Δ> 0 see Eq. (40)] this point is located at θ0<π. Having reached this value, the representative point turns back totheθ= 0. The same behavior takes place at Δ= 0, but Α>0 , i t i s typical for solitons in ferromagnets. However, such solutions do not limit the variety of solitons. The singular points of the solution to Eq. (41) are,ﬁrstly, points lying at the boundary of the region of existence of the κ 2=0 solution, and secondly, such parameter values for which Α=0 o r Δ= 0. It is the latter case that is characteristic of the transition to a “pure”antiferromagnet with ν= 0, for which the parameters Α=0andΔ= 0. In this limit, the stopping point corresponds to the value θ0=π, and the soliton is a 180-degree domain wall with internal precession, which can move with velocity v, see details in Refs. 59,103, and 130. Such a wall is a topological soliton with a π0-topological charge. This solution is fundamentally di ﬀerent from the localized solution of Eq. (41). However, in the case being con- sidered here, involving any arbitrarily small ν≠0 leads to qualita- tive changes in the form of the solution. Ifν≠0 and the soliton is stationary, but there is a precession with a frequency ω, then Δ= 0, but Α∝νω≠0. In this case, the character of the vector ldistribution depends on the sign of the fre- quency, more precisely, the sign of the product νω.A tνω> 0, the turning point lies at θ0<πand there is a localized soliton, while at νω< 0 the motion of the representative point stops only at θ=2π, seeFig. 6 . In the latter case, the soliton is a 360-degree domain wall of the vector l. The same directions of the vector lcorrespond to this wall as ξ→+∞andξ→−∞, but it has a non-trivial topology (π1-topological charge). A solution describing the 180-degree domain wall of vector l(a topological soliton with a π0-topological charge) at nonzero ν=0 arises only when both the speed and frequency are equal to zero. This is an important result: for a purely uniaxial model of a ferrimagnet with an arbitrarily small ν≠0, the motion of the domain wall is impossible. The dynamics of the domain walls in the presence of anisotropy in the basalplane will be considered in the next section. Let us consider the behavior of a soliton as its parameters υ andωapproach the region of existence boundary for localized solu- tions to Eq. (42). For small values of κ, when expanding the poten- tial with respect to θ, the following terms must be taken into account, with U(θ)∝−κ 2θ2+( A+ Δ)θ4.I fΑ+Δ> 0 (the points of the top half of the ellipse in Eq. (42) correspond to this condition), then the value of θat the stopping point is small, θ2 0∝κ2/(Α+Δ). FIG. 5. The soliton region of existence κ2> 0 for various values of parameter /C22ν; two solid lines (full ellipses) show the boundaries of typical ferrimagnet regions with /C22ν¼1 and /C22ν¼2 (the value is indicated near the curve); two dashed lines show the antiferromagnetic ( /C22ν¼0) and ferromagnetic ( /C22ν¼10 selected) limits. FIG. 6. The shape of the “potential ”U(θ) in the integral of motion (38) in the soliton region of existence ( κ2> 0) for various characteristic cases, see text for more details.Low Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 45,000000 (2019); doi: 10.1063/1.5121265 45,000000-947 Published under license by AIP Publishing.In this case, at κ2→0, the amplitude of the soliton tends to zero, θ∝κ, and the size of the localization region δξdiverges as 1 / κ, and at κ2= 0 the soliton completely delocalizes and disappears. In the lower half of the ellipse, where Α+Δ< 0, the behavior is funda- mentally di ﬀerent: the value θ0isﬁnite even at κ2→0. In this region, the soliton amplitude is ﬁnite at κ2→0, but the magnetiza- tion’s dependence on the coordinate ξbecomes algebraic: tg2θ 2/C18/C19 ¼jAþΔj (AþΔ)2(ξ=l0)2þΔ=2, (43) i.e. the so-called algebraic soliton arises. Far from the compensation point, when the inequality ν/C29ﬃﬃﬃﬃﬃ ωa ωexq ,1/C29ν, is valid, the soliton solution in Eq. (41) transi- tions to an analogous solution for a ferromagnet. In this case, the boundary of soliton states is transformed as follows: the center of the ellipse is in ﬁnitely removed downward from the origin, so that the top part of the ellipse goes into a parabola, ω ωFMþυ2 υ2 FM/C201,ωFM¼ωa ν,υFM¼2 νﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Aωa /C22h(s1þs2)r , (44) and the lower part of the ellipse does not appear at all (it corre- sponds to frequencies of about ωex, and the characteristic size of the soliton becomes about the order of the constant lattice).Therefore, in such a ferromagnetic limit, the soliton frequency islimited only from above, in particular, algebraic solitons such as those described in Eq. (43) do not exist. Note that in this case, the values of the characteristic “ferromagnetic ”frequency ω FMand velocity υFMare the same as in a ferromagnet: the frequency is determined by the anisotropy energy, and the velocity contains thefrequency ω a[instead of ωex, which is the case for the characteristic “antiferromagnetic ”velocity c, see Eq. (16)]. Both of these quanti- ties contain, however, the factor 1/ νand increase as they approach the compensation point. Non-one-dimensional soliton states can be similarly con- structed in a uniaxial ferrimagnet, in particular, non-topological two-dimensional (2D) or three-dimensional (3D) magnon droplets of the form w=ωt,θ=θ(r), where r2=x2+y2orr2=x2+y2+z2in the 2D or 3D cases, respectively. The function θ=θ(r) is described by equation l2 0d2θ dr2þD/C01 rdθ dr/C18/C19 þνω ωasinθ-1/C0ω2 ω2 0/C18/C19 sinθcosθ¼0, (45) where Dis soliton dimension. In many respects, these solitons are similar to the corresponding solitons in ferromagnets. In particular,they all exist only at a positive value νω>0, and their size increases indeﬁnitely at ω→0. 65As is the case for a ferromagnet, it is not possible to construct moving solutions. Since νω=ωa=ω/C22ν=ω0for a typical ferrimagnet, near the com- pensation point ( ν/differenceﬃﬃﬃﬃﬃ ωa ωexq or/C22ν/difference1), the characteristic frequency value for these solitons is about ω0. This frequency is, like other characteristic frequencies, exchange-ampli ﬁed. Refs. 126 and 127 made proposals for nanogenerators based on the excitation ofsolitons (magnon droplets with sizes of about l0) in ferromagnets. For such systems, a small generation line width is realized. If it is possible to make the same devices using a ferrimagnet close to thespin compensation point as an active element, the generationfrequency will increase signi ﬁcantly (inﬃﬃﬃﬃﬃ ωex ωaq /difference30/C0100 times), see frequency estimates in Appendix 2 . 6. THE DOMAIN WALL DYNAMICS IN BIAXIAL FERRIMAGNETS 6.1. General considerations and model formulation As noted above, for a purely uniaxial model of a ferrimagnet (easy axis - zaxis) with an arbitrarily small ν≠0, the motion of a 180-degree domain wall is impossible. Indeed, such a wall separates the regions of the magnet with the values lz= +1 and lz=–1, and when it moves with velocity υ, the quantityÐ lzdξ,d(Ð lzdξ)/dt=2υ inevitably changes. At s1≠s2(ν≠0), the quantity lzis directly related to sz,stot z¼(s1–s2)Ð lzdξ, see Eq. (11), such that dstot z/dt=2 (s1–s2)υ. On the other hand, for any form of a purely uniaxial anisotropy energy wa=wa(l2 z) the total value of the z-spin projec- tion stot zÐszdξis conserved: the total spin commutes both with the exchange Hamiltonian and with the Hamiltonian describing the uniaxial anisotropy. The motion of the wall for any arbitrarily weak spin uncompensation s1≠s2(ν≠0) is possible only if some terms that do not conserve stot zare taken into account. Note that in and of itself, the valueÐlzdξfor two-sublattice magnets is not conserved, even in a pure exchange approximation, see Refs. 78,80, and 92for more detail. Thus, for pure antiferromagnets, or precisely at the compensation point ( s1=s2), there is no such restriction, and the 180-degree domain wall can move even in a purely uniaxial antifer-romagnet or weak ferromagnet; its velocity is limited only by theLorentz contraction and coincides with c, see Refs. 50,51, and 131. The non-conservation of s tot zcan be due to the presence of crys- talline anisotropy in the basal plane of the magnet, and (or) themagnetic dipole interaction. For a 180-degree ﬂat wall, the latter is described by a density 2 π(Me ξ)2,eξis the unit vector along the direc- tion of wall movement, and we assume that eξis perpendicular to the easy axis ez.Note that this source of wall motion was considered by Landau and Lifshitz in the classical study,74wherein the motion of a domain wall was ﬁrst considered, as well as by Walker,132who studied the motion of the wall at considerable speeds and found its limiting velocity. If the direction of soliton motion along one of the crystalline axes of a biaxial magnet is selected (say, the x-axis), then this dipole energy is equivalent to the uniaxial anisotropy energywith an axis in the basal plane of the magnet. We choose the relativistic interaction energy in the form w a1 2K(l2 xþl2 y)þ1 2Kpl2 x, (46) where K> 0 is the uniaxial anisotropy constant (here zis the easy axis) and Kp=Kan pþ4πM2 s.0 describes the e ﬀective anisotropy in the basal plane, here Kan pis the crystalline anisotropy energy. In angular variables, this energy can be conveniently written as wa=(K/2)sin2θ(1 +ρsin2w), hereinafter the parameter ρ=Kp/Kis not assumed to be small.Low Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 45,000000 (2019); doi: 10.1063/1.5121265 45,000000-948 Published under license by AIP Publishing.We begin by analyzing the dynamics of the domain wall for a biaxial ferrimagnet with an anisotropy energy of the form of Eq. (46), without taking into account dissipative processes. If dissipation isweak enough, i.e. the relaxation constant α<< 1, then its inclusion does not change the structure of the domain wall found at α=0 . Forced wall motion with allowance for dissipation and some external force moving the wall, can be studied on the basis of e ﬀective equa- tions of motion for the collective variable, the wall coordinate. 6.2. The structure and limiting velocity of the domain wall A moving 180-degree domain wall is described by solutions like simple waves, for which l=l(ξ),ξ=x–υt. Moreover, Eqs. (25) and(26) forθand w, without taking into account dissipa- tive terms, assume the form of a system of ordinary di ﬀerential equations: A(1/C0υ2=c2)θ00-A(1/C0υ2=c2)(w0)2sinθcosθ /C0K(1þρsin2w)sinθcosθþ/C22h(s1-s2)υw0sinθ¼0, (47) A(1/C0υ2=c2)(w0sin2θ)0/C0ρKsin2θsinwcosw /C0υ/C22h(s1-s2)θ0sinθ¼0, (48) where the prime denotes the derivative with respect to ξ. It is easy to see that the system of Eqs. (47) and (48) has an exact solution that describes a moving 180-degree domain wall. For a ferromag- net, it was constructed by Walker.132This solution (it is commonly called the Walker solution) corresponds dw=dξ= 0, i.e. w¼w(υ)¼const, and the vector lin the wall rotates in a ﬁxed plane. Indeed, assuming that w0= 0, Eq. (47) gives a simple relation [l0(υ)]2θ00= sin θcosθ, where l0(υ) = const, for ξ(θ). This equation can be integrated once and written l0(υ)θ0¼+sinθ,l0(υ)¼l0ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1/C0υ2=c2 1þρsin2ws , (49) the value l0(υ) is the thickness of the domain wall moving with the velocity υ, and l0=ﬃﬃﬃﬃﬃﬃﬃﬃﬃ A=Kp is the exchange length. Further, substi- tuting Eq. (49) into Eq. (48), it is possible to ﬁnd the ratio of the azimuthal angle wand the wall velocity υin the form υﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1/C0υ2=c2p ¼ﬃﬃﬃﬃﬃﬃﬃ AKp /C22h(s1/C0s2)ρsinwcoswﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þρsin2wp ¼Φ(w): (50) These formulas make it possible to construct a solution that determines the structure of the wall cosθ¼thx/C0υt l0(υ)/C18/C19 : (51) Let us discuss the ratio of the parameters that de ﬁne the solu- tion. First of all, we note that the right-hand side of Eq. (50)contains a monotonically increasing velocity function υ, and the left-hand side is a bounded function wthat vanishes at w=0 ,π/2, π, and so on. The ﬁxed walls correspond to these values of the angle w. The maximum value of the right side of the equation is denoted by υw, max [ Φ(w)];υw¼ﬃﬃﬃﬃﬃﬃﬃ AKp /C22h(s1/C0s2)ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þρp /C01/C16/C17 : (52) This value determines the value of the limiting wall velocity υc, υc¼υWcﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ υ2 Wþc2p : (53) Let us discuss the value of this limiting velocity υc. Note that ifυW<<c, then the limiting velocity is close to υW,υc≈υW.I fw e consider a ferromagnet, i.e. replace /C22h(s1–s2) with the spin density /C22hs0, then the value υWcoincides with the Walker limiting velocity for a domain wall in a ferromagnet.132This velocity, in contrast to the“antiferromagnetic ”purely exchange velocity c∝ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃAωexp,i s proportional υw∝ﬃﬃﬃﬃﬃﬃﬃﬃAωap, and therefore the ratio υw/ccontains the small parameterﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ωa=ωexp (which is another example of the anti- ferromagnets ’dynamic parameters being exchange enhanced). On the other hand, for a ferrimagnet, the quantity υW∝1/vformally diverges as ν→0. Thus, in the characteristic “antiferromagnetic ” region ν∼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ωa=ωexp or/C22ν∼1a tρ∼1, the values υw/differencec. However, υwis proportional to the parameter (ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1þρp/C01) and goes to zero for a purely uniaxial magnet with ρ=0. For small ρandν, there is a nonanalytical dependence of υwon these parameters, υw∝ρ/ν. Thus, for any real ferrimagnet with an arbitrarily small butnonzero value of the decompensation parameter ν, the limiting velocity υ cvanishes at ρ= 0. However, in the formal limit ν= 0, i.e. exactly at the compensation point and arbitrarily small ρ≠0(ﬁnite values of the parameter ρarise when taking into account the mag- netic dipole interaction, which does not go to zero at the spin com-pensation point at g 1≠g2), the limiting velocity is equal to the minimum phase velocity of spin waves c, which is determined only by the exchange interaction parameters. This result is characteristic only of compensated magnets. The energy of the wall (hereinafter, values given are per unit area of the wall) is determined by the expression E(υ)¼E0ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þρsin2wp ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1/C0υ2=c2p ,E0¼2ﬃﬃﬃﬃﬃﬃﬃ AKp : (54) In the terminology established for ferromagnet domain walls, those with w¼0 and w¼π=2 will be referred to as Bloch and Néel, respectively. The rotation of vector lin the easy plane yz corresponds to the Bloch wall, its energy EBbeing equal to E0¼2ﬃﬃﬃﬃﬃﬃﬃ AKp , and its thickness lBcoinciding with l0=ﬃﬃﬃﬃﬃﬃﬃﬃﬃ A=Kp .F o r the Néel wall, vector lrotates in the less favorable xzplane, and its energy is of course higher than E0, and equal to EN=E0ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1þρp, whereas its thickness lN=l0/ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1þρp. For any arbitrarily small ν≠0, the value of wfor the Bloch wall increases with increasing wallLow Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 45,000000 (2019); doi: 10.1063/1.5121265 45,000000-949 Published under license by AIP Publishing.velocity, and decreases for the Néel wall. The possible values of wall velocities υcannot exceed the limit value υc(53).A t υ¼υcboth domain walls are identical. If ν= 0, then the value of the angle w does not depend on the wall velocity, and for any υ,cthere are only domain walls with w= 0, and w=π/2. 6.3. Forced motion of the domain wall The solution obtained above describes the motion of a domain wall “due to inertia ”, i.e. it excludes driving forces and dis- sipative processes. For practice, it is important to know the velocity of the domain wall ’s forced motion due to external force. This type of force arises when a magnetic ﬁeld directed along the easy axis of a magnet is applied. In this case, the energy densities of the magnet“to the right ”and“to the left ”of the wall di ﬀer by F=2M sHz, and the value of Fhas the meaning of the force acting on a unit of the wall area (magnetic pressure). In recent years, methods based on the use of spin-polarized current have been used more and moreoften, see Ref. 52. However, for weak dissipation, it is possible to study the motion of the wall without specifying the nature and source of this force. In this case, it is believed that both the dissipa- tion and the external ﬁeld are quite weak, so that the wall structure is deﬁned by the expressions obtained for a given wall velocity υ, without taking into account the dissipation and external force. Inthis case, the wall position is determined by its coordinate X=X(t), and the wall velocity υ¼dX=dt. For stationary motion, the analy- sis reduces to taking into account the energy balance: the wallvelocity is found from the condition that the drag force F diss(υ) bal- ances the force of magnetic pressure, Fdiss(υ)þF¼0: For the wall to move at a constant speed, the drag force is determined by the dissipative function of the magnet Q, which describes the dissipation rate of the domain wall energyF diss(υ)¼/C02Q=υ. For a dissipative function in the Gilbert form (26) and a Walker solution, a simple expression is obtained: Q¼υ2αG/C22h(s1þs2) 2l0(υ)¼υ2αG/C22h(s1þs2) 2l0ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þρsin2w 1/C0υ2=c2,s (55) where l0(υ)is the thickness of the moving domain wall, which is determined by Eq. (49). At small values of the force, the velocity is small and linearly depends on the magnitude of the force, υ¼μF,μ¼l0(0) αG/C22h(s1þs2), (56) where μhas the meaning of wall mobility, l0(0) is the thickness of a ﬁxed certain type of wall (Bloch or Néel). However, with an increase in the external force, the dependence of the wall velocityon the force magnitude is rather complicated, more complicated than in the limiting cases of a ferromagnet or an antiferromagnet. In particular, the velocity has a nonanalytic behavior at smallvalues of parameters ρandν. Let us discuss this dependence. First of all, we note that for a “pure”antiferromagnet ( ν= 0), the limiting velocity is υ c¼c, and the quantity Fdiss(υ)¼/C02Q=υ increases inde ﬁnitely as υ→υc. This means that the wall velocitymonotonously tends to cwith increasing F, υ(F)¼μFcﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ (μF)2þc2p : This behavior was experimentally obtained when studying the motion of domain walls in orthoferrites, see Refs. 50and51. This “Lorentz-invariant ”dependence was observed up to very strong ﬁelds and was violated only when the uniform state in the “disad- vantageous ”region of the magnet, in which the magnetization M was antiparallel to the ﬁeldH, became completely unstable. In this case, an “explosive ”instability of this phase was experimentally observed, which simulated an “over-limit ”wall motion at a veloc- ity that signi ﬁcantly exceeded the limit.50,51,133At the same time, a quasi-stationary motion of the magnetization inhomogeneityarose with a velocity substantially greater than υ c:Note that Schlöman demonstrated that the Walker limiting velocity is less than the minimum spin wave velocity υ(þ), and the expression forυ(þ)can be obtained from Eq. (52) and (53) by substituting (ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1þρp/C01) with (ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1þρpþ1).134This same relation υc/C20υ(þ) is valid for all magnets.135However, the over-limit motion of the wall with a velocity exceeding υ(þ)should be accompanied by Cherenkov magnon radiation.136But the above assumption that both dissipation and the external ﬁeld are weak is deliberately violated for over-limit motion, and therefore this question is not discussed further. In the case of a ferrimagnet ( ﬁnite value of ν), the situation is fundamentally di ﬀerent: υc,c,and the magnitude of the drag force Fdiss(υ) is bounded from above, Fdiss(υ)/C20Fmax. For the maximum value of the force, it is easy to obtain Fmax=ραG/2ν. Here, there is once again a nonanalytical dependence on the parameter ρ/ν. The steady motion velocity is written as υ¼μFcﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ (μF)2þ(c2=2)(2þρ+ρﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1/C0F2=F2 maxp )q , (57) where μis the mobility of the Bloch domain wall (the mobility of the Néel wall is less, and it is equal to μ/ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1þρp), the signs “–“and “+”in Eq. (57) correspond to the Bloch and Neel walls. This dependence is shown in Fig. 7 for various values of the compensa- tion parameter. Note that the velocity value υcis achieved when the force is F=Fc<Fmaxis less than Fmax, and the velocity at F=Fmaxis less than υc. However, the di ﬀerence between these values, υ(Fmax)and υc, and FcandFmax, is small even at large values ofρ, see Fig. 7 . Note a general property of domain walls in magnets with ν ≠:0 one type of velocity corresponds to two types of walls with diﬀerent energies. This is manifested, in particular, in the fact that for a given value of the force F<Fmaxthere are two di ﬀerent values of the wall velocity, see Eq. (54). The question of the stability of one of these walls is fundamentally important. Moreover, for thetop branch of υ(F)a t F c<F<Fmaxthe velocity decreases with increasing force and negative di ﬀerential mobility is realized, dυ(F) dF,0. This condition is usually associated with instability. The same property also holds for domain walls in a ferromagnet, but inLow Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 45,000000 (2019); doi: 10.1063/1.5121265 45,000000-950 Published under license by AIP Publishing.their case it is known that both branches of the dependence corre- spond to stable motion. This fact is explained by the fact that, atυ≠0, the Bloch and Néel walls in a ferromagnet correspond to diﬀerent values of the momentum P,s e eR e f . 119. The dependence E(P) for the domain walls in a ferromagnet is such that for υ≠0 only one value of energy and velocity corresponds to each momentum. For an antiferromagnet ( νis strictly equal to zero), the situa- tion is fundamentally di ﬀerent: there are also two types of walls, but a wall with higher energy is absolutely unstable, see Ref. 131. Since the question of wall stability has not been investigated for aferrimagnet with a small but ﬁnite value of ν≠-, we will try to explain these seemingly contradictory properties of the walls forqualitative reasons. This can be done by analyzing the dependence of the wall energy on its momentum P. Studying the dependence of wall dynamics using momentum Pis also useful for another reason. It was noted above that the stationary motion of the domain boundary is possible only atF≤F max. This result (the presence of a critical force value) is char- acteristic of uncompensated magnets; in particular, it is also valid for a domain wall in a “pure”ferromagnet, but formally there is no such restriction for compensated magnets. A natural questionarises: how will the wall move if a force F>F maxis applied to a magnet with a domain wall? Recall that for small αG, the value of Fmaxproportional to αGis also small.For a ferromagnet, this problem is solved in the classic study by Walker and Schryer, who showed that an unsteady wall motion arises in a constant magnetic ﬁeldHz>Hmaxthat exceeds the critical ﬁeld, including oscillations with a frequency of ω∼γHz137 (the so-called Walker –Schryer supercritical mode). This type of motion was repeatedly observed in experiments on the wall move- ment of magnetic domains in materials with cylindrical domains(bubble domains). 138For ferrimagnets, the analysis of such a motion is rather complicated, but using the momentum of the wall makes it more clear and visual. Using the dependence E(P) is not only a more convenient way to analyze forced motion, especially in the non-stationary case, it is also more consistent from the point of view of mechanics. Indeed, when applying the collective variable approach, the coordinate ofthe domain wall Xis used as the generalized coordinate. The wall energy in a ferromagnet or ferrimagnet with ν≠0 is not a simple function of wall velocity υ¼ dX dt; it is neither equal to E0+mυ2=2, where mis the e ﬀective wall mass, nor E0/ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1/C0υ2=c2p , which would be the case in classical or relativistic mechanics. Constructing the equations of motion for X(or, equivalently, con- structing the Lagrange function) is a non-trivial task (for more details on Lagrangian formalism for a ferromagnet wall see Refs. 119 and 120). On the other hand, wall energy recorded through its momentum represents the Hamilton function for the collective coordinate X. Knowing the Hamilton function allows one to construct consistent mechanics for the motion of the wall. The momentum of the domain wall P, like any other magnetic soliton, can naturally be determined from the density of the ﬁeld momentum of the magnetization ﬁeld,117see also Refs. 118and119. As noted in Sec. 2, a rigorous determination of the energy- momentum tensor for magnets with ν≠0i sd i ﬃcult due to the presence of a gyroscopic term that contains a nonanalytic function, the vector potential of the Dirac monopole ﬁeld, the shape of which depends on the selected gauge. Naturally, this problem alsocarries over to determining the momentum of a domain wall. However, for this particular case the situation is clear. The momen- tum P 0can be written asÐ1 –1AlðÞ@l @x/C18/C19 dxwhere Ais the dummy vector potential introduced in Sec. 3, and P0is the integralÐ A(l)dl along the trajectory on the sphere l2= 1, which describes the wall. This value is gauge-variant. However, the di ﬀerence between the momenta of the two di ﬀerent walls is de ﬁned by the closed-circuit integralÞA(l)dl, which, by virtue of the Stokes theorem, is equal to the ﬂow of the ﬁctitious ﬁeldB=r o t 1A(l) through the sphere region bounded by these two trajectories. Thus, the di ﬀerence in momenta for the two walls turns out to be gauge-invariant,107,111–113and it is determined by the area of the sphere segment lying between the trajectories that describe the walls (for the considered case with l(−∞)=–ez,l(∞)=ezthese trajectories exit through the south pole of the sphere and enter through thenorth). For the Walker solution, these trajectories are meridional lines with w1,2¼const, and therefore, this area is equal to 2( w1/C0w2). Assuming that for a ﬁxed Bloch wall with w=0 the momentum is zero, we obtain that the momentum of the wall is proportional to the angle w,w¼w(υ). Let us demonstrate this using the example FIG. 7. A domain wall ’s rate of forced stationary motion for various values of the decompensation parameter ν, as a function of the driving force magnitude (schematically). A suf ﬁciently large value of ρ= 0 was chosen to demonstrate the difference between the Bloch and Néel walls. The normalization of the force F0was chosen such that the mobility of the walls with respect to the normalized force F/F0is the same in all cases, while the value of the decompensation parameter νdetermines the relationship between F0andFmax:a tFmax=F0 there is almost a “ferromagnetic ”behavior, curves with Fmax=3F0and Fmax=5F0correspond to νvalues that are 3 and 5 times smaller The dashed line corresponds to the “antiferromagnetic ”limitν= 0 for the Bloch wall (at ν=0 the Néel wall is unstable). For curves with Fmax=F0andFmax=3F0the horizon- tal dotted lines show the values υ(Fmax),υc,and the vertical dotted lines show the force Fc<Fmax, which corresponds to the maximum velocity υc.Low Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 45,000000 (2019); doi: 10.1063/1.5121265 45,000000-951 Published under license by AIP Publishing.of well-known dependences for a simple ferromagnet. The wall energy EFM=E0ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þρsin2wp is a periodic momentum function that has the simple form: EFM(P)¼E0ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þρsin2(πP=P0,FM)p , (58) where the period per atomic chain of the ferromagnet is P0, FM=2π/C22hS=a,Sthe spin of the atom, and ais the interatomic distance. The value P0,FMis quite large (comparable to the size of the Brillouin zone PB=2π/C22h=a). For a ﬂat wall, the period is P0, FM=2π/C22hs, the momentum is calculated per unit wall area, and sis the spin density. Note that a periodic dependence with a period that is twice as large as the Walker value arises when describing the dynamics of a kink in the one-dimensional Ising model, in thepresence of an external magnetic ﬁeld perpendicular to the easy axis. 139The same dependence takes place for one-dimensional magnetic solitons, see Refs. 119and 120for details. The periodic dependence of energy on momentum, caused by the geometricproperties of the kinetic part of the Lagrangian, is a fairly generalproperty of magnetic solitons, see Refs. 107,111, and 112. In this regard, it is useful to make a general comment. The periodic dependence of energy on momentum is widely known for the motion of a quantum particle in a periodic potential;in this case, it is a consequence of Bloch ’s quantum theorem. Same as for an electron in a crystal, the periodicity in momentum shouldlead to the fact that the response to a constant force (electric ﬁeld for an electron) is a particle ’s oscillating motion (the so-called Bloch oscillations). This non-trivial e ﬀect is usually associated with quantum mechanics, but is actually not associated with quantumeﬀects. An indispensable condition for its implementation is that the Hamilton have a periodic dependence on momentum. In our case, periodicity with respect to momentum is present in classical theory and has nothing to do with any quantum e ﬀects. A detailed analysis of this problem, containing a comparison of the quantum and classi-cal approaches, is given in the review by A.M. Kosevich. 140 For a domain wall in a ferromagnet, the energy EFM(P)i sa simple periodic function of momentum, such that only one energyvalue corresponds to each value of the wall momentum, seeEq.(58). In particular, the momenta of the ﬁxed Bloch and Néel walls with w¼0 and w¼π 2diﬀer by the value P0,FM/2. The use of this fact makes it possible to clearly explain Walker and Schryer ’s above result137as a classical analogue of Bloch oscillations. Let us discuss the nature of the dynamics in detail. We begin with a simpler case of wall motion under the action of an externalforce Fwithout accounting for attenuation. In this case, the Hamilton equation dP/dt=Fcan be integrated for any dependence the force has on time. In the case of a constant force, P=Ft, i.e. w¼πFt=P0,FM, which determines the oscillating dependence of the wall velocity. If we take into account the wall friction, the equationis more complicated dP/dt=F+F diss(P), here Fdiss(P) is the friction force Fdiss(υ), expressed in terms of momentum. For a ferromagnet, theFdiss(P) function is proportional to the wall energy EFM(P),Fdiss ∝υE(P), is periodic and bounded above. It is clear that for an exter- nal force exceeding the maximum friction force max[ Fdiss(P)], the wall momentum increases inde ﬁnitely with time and oscillatory motion takes place.Using the same approach to analyze the momentum of a ferri- magnet and taking into account Eqs. (19) and(20), as well as the speciﬁc formulas (49)–(52) describing the structure of the domain wall, it is possible to write the momentum of the domain wall (perunit wall area) in the form P¼2/C22h(s 1/C0s2)wþυ c2E(υ), (59) where the ﬁrst term is the “ferromagnetic ”contribution P0, the second is typical for relativistic mechanics, and E(υ) is the energy of a domain wall moving with velocity υ. Note that for ν≠0 this energy is determined by formula (54), and its dependence on speed does not reduce to a relativistic factor 1/ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1/C0υ2=c2p , but is also related to the function w(υ):The purely relativistic dependence is restored only at ν= 0, in this case wis independent of velocity and takes only two values: w¼0o rw¼π/2. It is clear that when the angle changes by the value Δw=πN, where Nis an integer, the energy and velocity of the wall will not change, and the momentum will increase by 2 /C22hπ(s1/C0s2)N: Therefore, the energy of the domain wall in a ferrimagnet with ν≠0 is formally a periodic function of momentum, E(P+P0)=E(P). By virtue of the Hamilton equations, the wall velocity υ;dX dt¼dE(P) dP. T h ev a l u eo ft h ep e r i o d P0is determined by the formula P0¼2/C22hπ(s1/C0s2), (60) the period P0is small in proportion to the decompensation parameter s1-s2,P0=νP0,FM. Correspondingly, the gyroscopic term ’sc o n t r i b u - tion to the momentum is small: for small /C22ν, the value of the limiting velocity is close to c, and the magnitude of the second term can sig- niﬁcantly exceed P0. Unlike the case of ferromagnet, the analytical dependence E(P) cannot be found, and therefore we restrict ourselves to a qualitative and numerical analysis. In this case, it is convenient to use the dimensionless variables ~P¼cP E0,~E¼E E0,~υ¼υ cin these variables ~P¼/C22νwþ~υ~EandeP0¼/C22νπ(recall that /C22ν¼νﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ωex=ωap and /C22νis about equal to one in the characteristic region of an almost com- pensated ferrimagnet, for which ν/differenceﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ωa=ωex/C281):p Let us start with cases of small /C22νvalues. At ν→0, we have the case of an antiferromagnet for which the energies of the two walls areE(P)=ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ E2 0þc2P2p where E0=EBorE0=ENfor the Bloch or Néel wall, respectively (See Fig. 8 ). Here, the limiting velocity is equal to c, and the energy ~E¼1=ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1/C0~υ2p and momentum are not formally limited (the limit is only associated with the condition that the wall thickness l0(υ)¼l0ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1/C0υ2=c2p should be signi ﬁcantly greater than the interatomic distance, a, in the continuum descrip- tion). Two energy values correspond to each momentum value:E N(P)>EB(P). This could explain the well-known fact that the Néel wall, the energy of which is greater than the energy of the Bloch wall, is absolutely unstable50,131 At small but ﬁnite values of /C22νin the form of dependence E(P) there are two qualitative changes: ﬁrst, the maximum value of energy Emaxand momentum Pmax(P=Pmaxcorresponds to the maximum velocity of the wall, υ¼υc) become ﬁnite; second, the Neel ﬁxed wall corresponds to a non-zero momentum equalLow Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 45,000000 (2019); doi: 10.1063/1.5121265 45,000000-952 Published under license by AIP Publishing.to ±P0/2. Recall that the energy is also a periodic function of momentum, with a period of P0. It is clear that at small /C22ν→0, Pmax>P0/2. This relation remains valid for su ﬃciently small /C22ν,/C22νc. In the case of ν<νc, the dependence E(P) is qualitatively the same as that of a ferromagnet, and only one energy value corre- sponds to each momentum value. It can be said that – Pmax<P<Pmaxcorresponds to the Bloch wall, and –P0/2 <P<–Pmax and Pmax<P<P0/2 to the Néel wall, within the same period. It should be expected that both of these walls, as is the case for a ferro-magnet, are stable (the question of wall stability in ferrimagnets has not yet been investigated). The quantitative di ﬀerence in the dynam- ics of the ferrimagnet domain wall from the case of a ferromagnet isthat at ν<ν c,b u t ν≈νc, the E(P) dependences has fairly long regions where υ¼dE(P) dP/differenceconst :This value corresponds to the limiting value of the wall velocity, υ¼υc,a n di sq u i t el a r g e .S i n c e the Walker –Schryer supercritical mode (the oscillating dependence of the wall velocity on external force) is dP/dt=F=c o n s t , t h e w a l l moves with speed υ/C25υcfor a su ﬃciently long period of time. The dynamics of the domain wall in the Gd 23Fe67.4Co9.6ferri- magnet were experimentally studied in Ref. 53. The wall velocity under the action of a magnetic ﬁeld of up to 1 kOe was measured for a microstrip sample with a thickness of 20 nm, a width of 5 μm, and a length of 65 μm. The authors noted a sharp increase in the wall velocity to υ∼1.5 km/s when approaching the spin compensa- tion point, and explained this increase by way of the dynamic eﬀects’“exchange acceleration ”at this point. This value is smallerthan the “antiferromagnetic ”speed c, which reaches 4 km/s in our sample, according to our estimates, see Appendix 2 . In this article, the“over-limit ”motion mode is also considered analytically at υ/C28cand numerically. An increase in the e ﬀective mobility /C22μis shown, which determines the average speed of the over-limitmotion /C22υ,/C22υ¼/C22μH. When approaching the spin compensation point, the values of /C22μreached 20 km/(s /C2T). These results show the possibility of implementing ultrafast spintronics using domainwall dynamics and based on the use of ferrimagnetic nano ﬁlms with almost compensated spins. To conclude this section, it is useful to make one remark regarding the possibility of studying the motion of domain walls in more general ferrimagnet models, such as when more complexanisotropy w a(θ,w) is taken into account, or in the presence of an external magnetic ﬁeld perpendicular to the magnet ’s easy axis. In any case, the structure of the moving wall is described by a system of two second-order equations like Eqs. (47) and(48) for the vari- ables θ=θ(ξ),w=w(ξ). For the simplest biaxial anisotropy wa(θ,w) of the form (46) considered above, this system reduces to a pre-cisely integrable ﬁnite-dimensional (with two degrees of freedom) dynamical system, which determines the existence of the exact Walker solution. 142–144For an arbitrary form of wa(θ,w) this system is not integrable, and it is not always possible to ﬁnd its analytical solution, since it requires analyzing the dynamical system in a four-dimensional phase space. When going beyond the scope of integra- ble problems, the value of the ferromagnet ’s limiting velocity υ FM critcan increase signi ﬁcantly.145–151The equations describing the wall structure of ferromagnets and ferrimagnets di ﬀer only by the factor (1 /C0υ2 c2)for terms with second derivatives; therefore, they can be reduced to each other by a simple renormalization of the FIG. 9. The dependences E(P) for two types of wall, at a decompensation parameter that is almost critical ( /C22ν¼0:7 on the Figure) and at /C22ν¼1./C22νc. The/C22νvalues are indicated near the curves, and ρ¼0:5:Here, in contrast to the case ν<νcinFig. 8 , one period of this dependence (shown in the ﬁgure) includes all possible momentum values (taking into account the sign) andenergy. FIG. 8. The dependences E(P) (inEBunits) for two types of walls with a value ofρ= 0.5 for low decompensation ( /C22ν¼0:2, solid lines) and the purely antiferro- magnetic case ( ν= 0). Here, the long-dashed line is the data for the advanta- geous Bloch wall, and the short-dashed line is the data for the disadvantageousNeel wall. Momentum normalization by ~P¼2/C22h(s 1þs2)ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ωa=ωexp is chosen such that in ~Punits, the period is equal to P0=π/C22ν~P. For a wall with /C22ν= 0.2, only one segment of the periodic dependence that includes all possible momen- tum (taking the sign into account) and energy values is presented. For clarity,small sections corresponding to the periods that follow are also given. Verticallines indicate the boundaries of the “magnetic Brillouin zone ”P=±P 0/2.Low Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 45,000000 (2019); doi: 10.1063/1.5121265 45,000000-953 Published under license by AIP Publishing.constants and the variable ξ. Therefore, a series of results obtained for ferromagnets can be transferred to non-integrable models of ferrimagnets close to spin compensation. In particular, the valueof the limiting wall velocity can be found by replacing υ W!υFM crit in Eq. (53). 7. FEATURES OF TOPOLOGICAL SOLITONS, SKYRMIONS, AND VORTICES IN FERRIMAGNETS 7.1. Static structure and gyroscopic dynamics The study of topologically non-trivial magnetic states is one of the priorities of modern physics of magnetism. The presence of a nonzero topological charge leads to the additional stability of such states with respect to various extraneous in ﬂuences, such as thermal noise, which is useful for systems that record information.Magnetic topological solitons are interesting in that they can bepresent in magnetic particles of micron and submicron sizes, and even form the ground state of such nanomagnets. In recent years, there has been interest in two-dimensional non-homogeneousstates with non-trivial topology, magnetic vortices, and localizedtopological solitons, which are now referred to as skyrmions. Thelatter are named in honor of T. Skyrme, who proposed using the stable topological solitons of a nonlinear meson ﬁeld to describe baryons. 152,153Note that Skyrme himself never considered two- dimensional solitons, but studied either three-dimensional prob-lems and solitons with a π 3-topological charge152,153or simple one-dimensional ( x,t) models based on the Klein –Gordon sinus- oidal equation.154Despite this, the term “skyrmion ”isﬁrmly entrenched in the physics of magnetism and will be used below. We begin by analyzing the static properties of solitons, and then discuss their dynamics. Note that the structure of stationary sol- itons for a magnet, which can be described in terms of a unit vector, does not depend on the type of magnet —a ferromagnet, antiferro- magnet, or ferrimagnet. Certain dynamic features of these solitonsare also similar. Therefore, at the beginning of this section, thesequestions will be discussed rather brie ﬂy, and a more detailed analy- sis can be found in a recent Ref. 59. On the other hand, the options for creating skyrmions and vortices, as well as ways they can be usedin spintronic devices, di ﬀer signi ﬁcantly. The question of stability plays a major role in the physics of two-dimensional magnetic soli-tons, and the problems that arise for vortices and skyrmions are fun- damentally di ﬀerent. It is convenient to consider all these questions separately, which will be done in the concluding parts of this section. For the model of a purely uniaxial magnet, stationary soliton states can be written in the form θ 0(r),w¼qχþw0, (61) where q= ±1, ±2, …is an integer that determines the topological properties of solitons, and w0is an arbitrary constant. If the mag- netic dipole interaction is not considered, the function θ0(r)i s determined by the solution of the di ﬀerential equation:59,118–121 d2θ0 dr2þ1 rdθ0 dr/C0q2 r2sinθ0cosθ0/C01 A@wa @θ0¼0, (62)where wa(θ) is the anisotropy energy. The condition that there be no singularity at the center of the soliton gives that at r→0, θ=Crqorθ=–π−Crq. The second condition corresponds to the fact that the antiferromagnet is in the ground state far from thesoliton, i.e. as r→∞, the variable θ→π/2 for vortices in an easy plane magnet or θ→0,πfor skyrmions in an easy axis magnet. The size of the angle deviation region θ 0(r) from the equilibrium value, i.e., the natural size of the soliton ’s inhomogeneity (skyrmion radius or the size of the vortex core) is l0=ﬃﬃﬃﬃﬃﬃﬃﬃﬃ A=Kp , in other words, about tens of nm. For vortices and skyrmions, the value of the π2-topological charge is essential, as it corresponds to the mapping of the magnet plane ( x,y) onto the sphere 12= l. Such a mapping is characterized by a topological invariant Q: Q¼1 4πð εαβsinθ@θ @xα@w @xβdxdy , (63) εαβis the absolute antisymmetric tensor. For a skyrmion, the value ofQtakes only integer values, Q= 0, ±1, ±2, …The value of Qfor the vortex is half-integer, Q=–qp/2,where the integer p=± 1 deﬁnes the sign lz= ±1 in the center of the vortex core and is called the vortex polarization. The states of a vortex with Q= ±1/2 di ﬀer topologically and cannot be translated into each other by a continu-ous deformation. Generally speaking, the π 1charge is the main topological charge of vortices, and in our case this is the vorticity q, but this quantity plays a smaller role for magnetic vortices. In par- ticular, for vortices in soft magnetic particles, only the value q = 1is realized, see below. We proceed to consider the motion of topological solitons, vortices, and skyrmions. It is not possible to construct an exact sol- ution describing a non-one-dimensional soliton in a ferromagnet moving at a considerable velocity. We use the collective variableapproach, which is based on the assumption that l=l (0)(r-Rs) and ∂li/∂t=–(v/C2∇)l(0) i, where Rs=Rs(t) and v=dRs/dtare the coordi- nate and velocity of the soliton, and l(0)is the solution that describes the motionless soliton. In an antiferromagnet there is aformal Lorentz invariance that must also manifest for a ferrimagnetat an exact spin compensation ν= 0. In this case, for any type of soliton R scoordinate there should be relativistic dynamics. If we restrict ourselves to the case of low velocities, υ<<c, for ν= 0 the eﬀective equation of motion for Rshas a Newtonian form, md2Rsdt2¼F(t), where Fis the force acting on the soliton, and mis the e ﬀective soliton mass, m¼E0=c2,E0is the resting soliton energy. Here Fcontains both the potential contribution, Fpot=–∇U(Rs), wherein the potential energy Uis determined by the inhomogeneity of the magnet and/or magnetic ﬁeld parameters, and the dissipative contribution, which at a low soliton velocityhas the form of a viscous friction force, F d=–vη, where ηis the viscosity coe ﬃcient. For a ferrimagnet with a small but ﬁnite spin decompensation (νis small but nonzero), the soliton equation of motion can be constructed using the Hamilton formalism, taking into account thedeﬁnition of the soliton momentum, see Eq. (20). At low velocities, theﬁrst term in this formula takes the standard Newtonian form, P=m v+P(0). The external force serves as the measure of theLow Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 45,000000 (2019); doi: 10.1063/1.5121265 45,000000-954 Published under license by AIP Publishing.change to momentum, i.e. dP/dt=F. Further, the equation of motion can be rewritten as mdv/dt=F+FG, where the gyroscopic force (gyroforce) FG=–dP(0)/dt. It is easy to show that, unlike P(0), the quantity dP(0)/dtcontains only ∇l×Aand is gauge-invariant [see Ref. 59for the general form A(l)]. In the case of a ferrimagnet ∇l×A=1 , for the two-dimensional distribution of spins l=l(x,y), the quantity FG=(v×ez)Gwhere the gyroscopic constant Gis expressed in terms of the topological invariant Q(63), mdv dt¼(v/C2ez)GþF,G¼4π/C22h(s1/C0s2)Q: (64) Note that using a singular vector potential A(l) is a tall order from the perspective of rigorous mathematics. However, the pres-ence of a gyroforce in the form of Eq. (64) for topologically non- trivial states —not only skyrmions or vortices, but also for previ- ously studied cylindrical domains and Bloch lines in ferromagnets—has been reliably established experimentally. 138A theoretical analysis of the gyroscopic dynamics of solitons due to theirtopological π 2charge was carried out on the basis of a number of di ﬀerent theoretical approaches,106,109,110,113,114including a direct analysis of the Landau –Lifshitz equation,106and con ﬁrmed the existence of a gyroforce of the form Eq. (64) for ferromagnets. For vortices, this equation, written without the inertial term(m ¼0, i.e., ( ez/C2v)G=F), is called the Thiele equation and has been tested in many experiments on the dynamics of magnetic vor- tices in magnetically soft ferromagnet particles, see reviews Refs. 21 and22, as well as strongly coupled vortex pairs.155The structure of the gyroscopic terms, particularly that of the vector potential, is thesame for the Landau –Lifshitz equation and for ferrimagnets. Therefore, the transfer of gyroforce results to ferrimagnets is quite clear, and the validity of Eq. (64) is not in doubt. We note that the inertial term for solitons in ferromagnets described by theLandau –Lifshitz equation was also discussed by many authors, but the results are contradictory to this day, and will not be dis- cussed. However, in the sigma-model description of a ferrimagnet,containing (in contrast to the Landau –Lifshitz equation) the second derivative of the vector l,d 2l/dt2, the appearance of the inertial term and the mass of the soliton mare quite clear. Equation (64), which generalizes the Thiele equation with allow- ance for the inertial term and the possible smallness of the gyro-scopic e ﬀects at a small, but ﬁnite, spin compensation, can be appropriately referred to as the generalized Thiele equation. Thisequation was used in a theoretical analysis of the dynamics of a magnetic vortex in a ferrimagnet in Ref. 55. After this brief discussion of the general problems in ferrimag- netic topological solitons, let us turn to an analysis of the speci ﬁc properties of vortices and skyrmions. 7.2. Vortices in small ferrimagnetic particles Magnetic vortices, which have been studied for more than twenty years, can realize the ground state of an almost circularnanoparticle made of a magnetically soft ferromagnet such as per-malloy. This is the regard in which vortex stability is discussed. In this case, the vortex distribution of the form of Eq. (61) with q= 1 and with the values w 0¼π 2andw0=−π/2 ensures themagnetic- ﬂux-closure inside the particle. In this case, the only source of the demagnetizing ﬁeld are the surface magnetic poles (nonzero value Mz=Mscosθ), which gives Hm=–ez4πMscosθfor a suﬃciently thin particle. This ﬁeld is concentrated in the small region of the vortex core, and the demagnetizing energy of thevortex state is lower than the homogeneous state. In other words, the vortex is stable due to a decrease in the magnetostatic energy and is an alternative to the usual domain structure known for bulkmagnets. For magnetically soft particles, the anisotropy is portrayed by the energy w m=2πM2 scos2θ, and the size of the vortex core is determined by the value lm=ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ A=4πM2 sp (in literature about mag- netic vortices it is often denoted as l0). For the widely used magnet- ically soft permalloy Ni 80Fe20theﬁeld is 4 πMs∼10 kOe and the value of lmis about 5 nm. As is the case for a usual domain structure, the vortex state is realized only for particles with relatively large dimensions. A single- domain quasi-homogeneous state is more advantageous for smallerparticles. Estimates show that for a circular particle made of softmagnetic material, the product of the particle radius Rand its thickness Lis characteristic: the vortex state is the ground state if RL>3 0l 2 m.156To guarantee the manifestation of a vortex, the single- domain state must be unstable in the zero ﬁeld, which is realized under the more stringent condition RL>4 5l2 m.157For this reason, most of the vortex experiments were performed for particles with thickness greater than 10 nm. However, for applications in spintronics, it is important to stabilize the magnetic vortex in theparticle with the smallest thickness, preferably no more than 5 nm. For ferrimagnets considerable easy plane anisotropy can be present at the spin compensation point, which determines the size of the vortex core l 0(about several nm). However, their magnetiza- tion is small (4 πMs∼1 kOe for GdFeCo), and the value of lmis large, much larger than for permalloy. Typical lmvalues for ferri- magnets with spin compensation are 30 –40 nm. It is useful to note that for antiferromagnets with weak ferro- magnetism, such as hematite α-Fe2O3, iron borate FeBO 3, or ortho- ferrites, the value of Msvalue is very small, even lower than for ferrimagnets; for example, for iron borate it is 4 πMs= 120 Oe. A considerable easy plane anisotropy exists in these materials, and thedimension of the vortex core l 0is about several nm, but lmis about several hundred nm (for iron borate lm= 220 nm). However, unlike the vortices in ferromagnetic particles, the weak ferromagneticmoment M weak=HD(ez/C2l)/Hex, where HDis the Dzyaloshinskii ﬁeld, does not leave the plane of the particle. Therefore, for a parti- cle in the form of any rotating object with an axis parallel to the hard axis, the demagnetizing ﬁeldHmis equal to zero even in the core region.158Therefore, despite the smallness of the magnetic moment, the vortex state in such antiferromagnets can be advanta-geous for a particle that is su ﬃciently small,ﬃﬃﬃﬃﬃﬃ RLp .0:4μm. For ferrimagnets, the magnetization distribution is the same as vector l, and the distribution of ﬁeldH mrepeats that which occurs for a ferromagnet. Therefore, the above criteria are alsoapplicable, and the typical particle sizes for which a vortex state isfavorable are determined by the ratio of l mvalues, as they are several times larger than for standard ferromagnets such as permal- loy. For vortices in the CoTb ferrimagnet, typical particle sizes areestimated as R∼1μm and L∼100 nm. 55These sizes are too largeLow Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 45,000000 (2019); doi: 10.1063/1.5121265 45,000000-955 Published under license by AIP Publishing.for applications in spintronics. One way to reduce them is to use the Oersted ﬁeld, which is created by electric current, is always present in a system such as a magnetic nanocontact, and allows forthe stabilization of vortices even at particle thicknesses less than5 nm. 159–162Another possibility for reducing the size of a particle with a vortex is associated with the use of hybrid nanostructures, in which a ferromagnetic ﬁlm or nanoparticle are impacted by dipole scattering ﬁelds created by another magnetically hard layer with a certain geometry.23It is important to note here that the vortex state energy gain in all these cases is proportional to the ﬁrst degree of the small parameter, the magnetization Ms, while the standard energy of the demagnetizing ﬁelds is proportional to M2 s.T h e r e f o r e ,t h ei n d i - cated mechanisms of vortex stabilization by external ﬁelds are more eﬀective for ferrimagnets than for standard ferromagnets with large values of Ms. Thus, we can hope that the problem of realizing the vortex state for su ﬃciently small ferrimagnet particles is solvable. The interest in magnetic vortices is largely related to their dynamic properties. A ferromagnet vortex is characterized by gyro-tropic dynamics, in which the vortex core moves along a circular tra-jectory with a large radius, with the motion frequency ranging fromhundreds of MHz to 1 –2G H z 159–164(see also Reviews 21and22). This mode of motion can be excited by a spin-polarized current.165–167Vortex generators have record-breaking characteris- tics, an extremely narrow generation line, and a relatively high signalpower. 159–162Their disadvantages include a low frequency value. It can be hoped that using ferrimagnets close to the spin com- pensation point will signi ﬁcantly increase the operating frequency of the spin torque generator. For ferrimagnetic vortices, the dynam-ics are determined by the generalized Thiele Eq. (64), which for the case of a vortex in a circular particle, taking into account the spin-polarized current and dissipation, can be written as m d2Rs d2tþGez/C2dRs dt/C18/C19 ¼Fm/C0ηdRs dtþτs0/C22hL(ez/C2Rs), (65) where Fm=−κ(Rs)Rsis the restoring force, Rs=\|Rs\|d eﬁned by the magnetostatic energy of the interaction between the vortex and theparticle edge. 164,168 κ(Rs)¼κ0 1/C0(Rs=2R)2,κ0¼2πM2 s20L2 9R, (66) Land Rare the thickness and radius of the disk, and mis the eﬀective mass of the vortex. The last two terms determine the non- conservative forces, viscous friction, and the force caused by the action of the spin current. These forces can be written in the same way as those of a ferromagnetic vortex.165Further on it is easier to show that under the condition of balance between the non- conser-vative forces ηω=τs 0/C22hL, it is possible for steady-state vortex motion along a circular orbit with radius Rsto be realized, with the fre- quency of motion ωdetermined by the formula ω¼G 2m+ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ G 2m/C18/C192 þ[ω0(Rs)]2s ,ω0(Rs)¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ κ(Rs) m*s :(67)Estimates show that despite the small magnetization of amor- phous ferrimagnets near the compensation point, this frequency value is about tens of GHz, and signi ﬁcantly exceeds the gyroscopic vortex frequency for ferromagnets. For solitary vortices its value isless than GHz, for a closely coupled pair it can reach 3 –4 GHz. 7.3. Skyrmions —stability and dynamics Let us now consider the properties of ferrimagnetic skyrmions and the possibility of realizing such states in ferrimagnets. Skyrmions are interesting in that they can have rather small sizes (up to several nm) and a strongly localized structure. 169–174Unlike vortices, they are practically insensitive to the shape of the sample,can exist in continuous ﬁlms or magnetic nanoribbons, and can move freely enough along the sample. In fact, skyrmions resemble well-known cylindrical magnetic domains (magnetic bubbles), on the basis of which a magnetic memory is created that does notcontain mechanical moving elements. 138The motion of skyrmions is easily controlled by electric currents .175–177These properties allow us to hope that new types of new devices can be created on the basis of skyrmions for storing and processing information with extremely high density,177–179which have all the advantages of systems on cylindrical domains, but with a characteristic size of theorder of tens of nm (dimensions the cylindrical domain is di ﬃcult to make smaller than one micron). For skyrmions, as well as for other non-one-dimensional static topological solitons with ﬁnite energy, the stability problem is very important. The general assertion known as the Hobart –Derrick theorem, 180,181states that non-one-dimensional stationary localized soliton solutions for a model such as Eq. (18), in which the energy includes quadratic terms in the order parameter gradients (vector l components) and the anisotropy energy, are unstable. Note thatthis theorem is not applicable to solitons with in ﬁnite energy, such as hedgehogs (Bloch points) 89or the vortices considered above, as well as to some discrete models in which static two-dimensional topological solitons can exist.182A soliton can also stabilize by way of internal dynamics and magnetization precession.124,125However, for static stable skyrmions to exist, it is necessary to go beyond theframework of the standard continuum model with an energy func- tional of the form of Eq. (18). Let us brie ﬂy explain the conditions that the energy of a magnet must have in order for two-dimensional stable solitonstates (skyrmions) to exist therein. A good starting point is the well- known Belavin –Polyakov solution, 183obtained for an isotropic magnet with an energy of the form of E=(A/2)Ð(∇l)2dr, having the form tg(θ 2)¼(R r)jQj ,w¼Qχþw0, (68) where Ris the soliton radius, and Qis its topological charge. Below only the case Q= 1 is considered, for solitons with Q> 1(see Refs. 184and185). The energy of this soliton does not depend on the radius R,EBP=4πAQ(per unit length). If we take into account the anisotropy energy, then the character of the dependence θ(r)a t distances r>l0becomes exponential, and the anisotropy contrib- utes to the energy in the form of ΔEa≃8πKS2R2ln(l0/R).125As aLow Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 45,000000 (2019); doi: 10.1063/1.5121265 45,000000-956 Published under license by AIP Publishing.result, the soliton energy becomes a function of the radius E(R), and this function has no minimum at R≠0, which determines the collapse of the soliton. In fact, the size of the soliton in such amodel will decrease until it reaches a value close to the interatomicdistance a, when topological arguments based on the assumption of a smooth analytical dependence l(r) cease to apply. 182 The question is, how can a skyrmion be stabilized? Models have been proposed, in which the energy contains the followingdegrees of magnetization gradients, such as a 2A0(∇2l)2with A0>0 , which give an energy correction in the form of A0(a/R)2that stabi- lizes the skyrmion.186–188In fact, Skyrme himself took such terms into account to obtain stable three-dimensional solitons. However, there is a de ﬁnite advantage to the two-dimensional problem: mini- mizing the energy by accounting for this type of term gives asmall but macroscopic value of the soliton radius, R 0=ﬃﬃﬃﬃﬃﬃ al0 0p l0 0¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ A0=Kp , i.e. at l0 0/differencel0the value is a /C28R0/C28l0:The possibility of stabilizing a small-radius skyrmion in a magnetic ﬁlm on a non- magnetic metal or graphene substrate via the long-range Ruderman–Kittel –Kasuya –Yosida interaction through the electrons of the substrate was theorized in Ref. 189. However, such skyrmion stabi- lization scenarios have not yet been experimentally implemented. Quite a long time ago, another way of stabilizing skyrmions was proposed, by way of a contribution from the so-called Dzyaloshinskii – Moriah interaction, which is linear with respect to magnetizationgradients, written as aDM×(∇×M)[ f o ra n t i f e r r o m a g n e t si ti s aDl×(∇×l)],. 187,190,191Such terms are possible for crystalline m a g n e t sw i t h o u ta ni n v e r s i o nc e n t e r( f o re x a m p l e ,F e G eo rM n S imagnets with a B20 lattice, or the c uprate plane of the YBaCuO anti- ferromagnetic phase. 192) Their contribution to the energy of the Belavin –Polyakov soliton is aDRQ , i.e. a skyrmion with a certain sign of the topological charge Qcan stabilize for any sign of D.T h i s is the stabilization m echanism that has been experimentally imple- mented: hexagonal skyrmion lattices have been experimentallydetected in crystals with a B20 structure. 193,194T h es a m ei n t e r a c t i o n can occur in thin magnetic ﬁlms on a heavy metal substrate with strong spin-orbit interaction,195in which case the interaction energy is described by the expression aDs[Mz(∇/C2M)–(M/C2∇)Mz], where thezaxis determines the direction of the normal to the ﬁlm surface. Skyrmions stabilized by the surface Dzyaloshinskii –Moria interaction have been observed in ultrathin Co/Pt, Ir/Co/Pt ﬁlms and many others at room temperature.196–200 All skyrmion stabilization mechanisms considered above are purely static, and are possible not only for ferromagnets, but alsofor antiferromagnets and ferrimagnets, including near the spin compensation point. The advantages of using ferrimagnets near the spin compensation point are quite clear. First, they have “relativis- tic”skyrmion dynamics with an exchange rate (in any case, at small decompensation values, s 1→s2), which increases the theoreti- cal performance limit of a memory system. Secondly, the gyroforce decreases (and goes to zero at the compensation point s1=s2), which has been experimentally observed in Ref. 201. By virtue of this, the skyrmion moves along the applied force. In particular,there should be no e ﬀect of the skyrmion being “pushed out ”to the strip border, in order for it to move through a magnetic nanostrip. Note, however, that the dynamics of skyrmions in ferrimag- nets have not been studied, and it is di ﬃcult to say at what value ofdecompensation the antiferromagnetic Lorentz-invariant dynamics transition to ferromagnetic. It should be noted skyrmion dynamics in various magnets has not been adequately studied. The onlyexception is the case of a pure antiferromagnet, where there aresimple Lorentz-invariant laws. For ferromagnets, there is disagree-ment in the research community about the (seemingly simple at ﬁrst glance) dynamic characterization of the skyrmion e ﬀective mass. This mass was calculated long ago, and di ﬀerent approaches yielded similar results. 109,202,203In Ref. 204, the authors noted that the motion of a magnetic skyrmion observed by time-resolvedx-ray holography can only be described if the skyrmion mass is taken into account. This mass was claimed to be large, more than can be expected from a simple estimate of the total mass of thedomain wall bounding the skyrmion. On the other hand, it hasrecently been asserted that a skyrmion has no inertial properties inan ideal (defect-free) ferromagnet. 205A discussion of this problem can be found in the recent Ref. 206. For a ferrimagnet, the presence of aﬁnite skyrmion mass, by virtue of the fact that terms with second derivatives with respect to time are present in the equationsof motion, is not in doubt. However, an estimate of the “ferromag- netic”contribution at s 1≠s2(ν≠0), and the value of νat which these contributions become comparable, is of great interest. 8. CONCLUSION The study of the “antiferromagnetic ”spin dynamics of ferrimagnets with spin compensation, and especially ultrafast spin-tronics for such materials, is rapidly developing. In the current sit-uation, it is di ﬃcult to make predictions about how this development will go, what e ﬀects will turn out to be the most important, and what materials will be preferable as far as applica-tion. On the other hand, a vast series of results have been obtainedin the framework of developing “ordinary ”ferromagnetic spin- tronics and in the newer ﬁeld of antiferromagnetic spintronics. These results are important for understanding the spintronics of ferrimagnets, but they are di ﬃcult to describe in a short review. However, the author hopes that a systematic presentation of thevarious aspects of ferrimagnetic spin dynamics will allow an inter- ested reader to understand its speci ﬁcity and see its practical usefulness. In conclusion, I would like to express my deepest gratitude to V. G. Baryakhtar, Craig E. Zaspel, A. K. Kolezhuk and A. L.Sukstansky for many years of cooperation in the ﬁeld of dynamics of solitons and magnetic vortices. I am grateful to the authors of Ref. 115who kindly agreed to reproduce their experimental data in this review. This work was partially supported by programNo. 1/17-H of the National Academy of Sciences of Ukraine andthe Target Training Department of the Taras Shevchenko National University of Kyiv at the National Academy of Sciences of Ukraine (project “Elements of superfast neural systems based on antiferro- magnetic spintronic nanostructures ”). APPENDIX 1: SPIN CURRENT AND SPIN-TORQUE AUTO-OSCILLATOR CIRCUITS The simplest (conceptually-speaking) method of creating a spin current is based on the fact that the conduction electrons of a ferromagnetic metal are “magnetized ”due to the presence ofLow Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 45,000000 (2019); doi: 10.1063/1.5121265 45,000000-957 Published under license by AIP Publishing.magnetic ordering. Thus, in a layered magnetic nanostructure (Fig. 10 ), an electron ﬂux passing through a ferromagnetic layer with a ﬁxed magnetization (this layer is called a polarizer) becomes spin polarized (spin polarization e ﬃciency is introduced ε=(s↑–s↓)/(s↑+s↓) with its value being determined by the prop- erties of the polarizer). Further, this spin is transferred through athin (1 –2 nm) layer of normal metal into a layer of a magnetically soft ferromagnet (the so-called free layer) and implements the spintorque of the free layer magnetization. For this spin-torque generation scheme, an estimate of the parameter τ SC=σSCj included in the dynamic Eqs. (1),(3)or(15) is quite simple τSC¼εgμB 2eLj, (A1) where jis the electric current density (measurements are often done according to the total current I=jS, where Sis the contact area), 01 < ε≤is the spin polarization e ﬃciency, e> 0 is the elec- tron charge, and Lis the thickness of the free layer (see Ref. 7for details). The physical meaning of coe ﬃcient σis quite transparent: this quantity contains the ratio of the electron magnetic momentgμ B/2≈μBto its charge e. Naturally, such a scheme can be applied only to conductive magnets. However, the spin current is not necessarily related to the translational motion of electrons. For example, in the exchange approximation, the total spin of the magnet is conserved and theequation of motion of the spin density takes the form of the conti-nuity equation ∂s i/∂t+∂Πi,α/∂xα. The quantity, Πi,αdetermines the transfer of the ith component of the spin, i.e. the spin current. Πi,α is a bivector, and the Greek and Latin symbols the indices in the coordinate and spin spaces (see Refs. 77–81). At present, it is con- sidered promising to use the so-called spin Hall e ﬀect, which can be used for the spin torque of both magnetic dielectrics and metals.207,208This e ﬀect was predicted many years ago209,210and consists of the fact that when an electric current Jcﬂows through anormal metal, there is a spin ﬂux perpendicular to the current Jc (Fig. 11 ). Note that this e ﬀect is not related to the presence of a magnetic ﬁeld and is determined by the spin-orbit interaction — speciﬁcally by the relationship between the spin direction and the momentum (velocity) of the electron. As a result, an accumulationof spins with opposite direction can occur on the opposite surfaces of the sample parallel to the current. The nature of the e ﬀect, especially the mutual directions of the electric current J c, spin current Js, and spin polarization p, can be understood by analogy with the classical Hall e ﬀect, which takes place in an external magnetic ﬁeldH(for magnets, the same role can be played by magnetization M) (see Fig. 11 ).The direction of currents [electric Jcand Hall JH) and the ﬁeldH(or magnetization M] are shown in Fig. 11(a) . Similarly, for the vector Jc, the direction of the spin ﬂuxJs, and the spin current polarization pconstitute the three orthogonal vectors [see Fig. 11(b) ]. The spin Hall e ﬀect can be used for the spin torque of a layer of magnetic material in a two-layer “normal metal –magnet ” system (see Fig. 12 ). The intensity of the e ﬀect is determined by the spin-orbit interaction. Therefore, heavy metals such as platinumare chosen as the current carrier, for which this interaction is strong. The formula for the characteristic constant τ=τ SHE, which determines the torque e ﬃciency, has the form35 τSHE¼jgrθSHeλρ 2π/C22hs0LmagntgLHM 2λ: (A2) The expression for τis not as transparent as (A1); it contains the characteristics of both layers, as well as the value of the FIG. 12. A diagram of spin torque excitation of an active magnetic element (on the Figure FL stands for free layer) due the spin Hall effect, as an electriccurrent ﬂows through a heavy metal layer (HM in the Figure). The vertical lilac arrow indicates the direction of the spin current, and the notations for the direc- tion of electron ﬂow in the metal, the polarization of the spin current, and the magnetization of the active element, are the same as in Fig. 10 . FIG. 10. A diagram of a layered magnetic nanostructure. The letters denote: P - polarizer, FL - free layer, and between them a layer of non-magnetic metal is shown. The long blue arrow indicates the direction of electron motion, the thinner dark red arrows indicate the direction of magnetization in the polarizerand the magnetization precession in the free layer, the short red arrow with theletter pindicates the polarization direction of the spin current, the short black arrow indicates the direction of the chosen free layer axis. FIG. 11. A comparison of the usual Hall effect (a) and the spin Hall effect (b).Low Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 45,000000 (2019); doi: 10.1063/1.5121265 45,000000-958 Published under license by AIP Publishing.so-called spin-mixing conductance gr, which characterizes the boundary between the magnet and metal (see Ref. 35). The value of the spin –Hall angle determines the properties of the metal; for platinum, θSH∼0.1 rad., see Ref. 211,s0is the spin density of the magnet, ρandλare the electrical resistivity and spin di ﬀusion length for the metal, Lmagn andLHMare the layer thicknesses of the magnet and heavy metal, respectively. A useful high-frequency signal within the framework of this design can be obtained using the inverse spin Hall e ﬀect, which consists of the fact that the magnetization oscillationscreate a spin current J (ISHE ) s ,w h i c h ﬂows back from the metal to the magnet. A simple formula is obtained for this current, in the case of an antiferromagnet or ferrimagnet near spincompensation: J ISHE s¼/C22hgr 2π(l/C2@l @t)¼ezω/C22hgr 2πsinθcosθ: This spin current creates a variable EMF in the metal, i.e. a useful signal. For the case of uniform rotation of the vector l around the ezaxis with a frequency ω, the above-mentioned prop- erty of the system is obtained: the variable signal depends on the precession angle and goes to zero when the vector lis purely planar. APPENDIX 2: REAL FERRIMAGNET PARAMETERS Classical antiferromagnets, such as orthoferrites, transition metal oxides NiO, MnO, CoO, hematite α-Fe2O3, or iron borate FeBO 3, have been studied for many decades (see Ref. 24). The reso- nance properties and ﬁeld-induced spin- ﬂop phase transitions have been studied in reference to them, and important parameters such as the exchange or anisotropy ﬁelds, and inhomogeneous exchange constants, have been determined. The properties of amorphous fer-rimagnets, which are interesting in terms of this review, especiallythose that are important for describing ultrafast dynamics, have been studied in less detail. It is useful to provide the available data for at least some of them in order to be able to estimate the extentof their dynamic parameters. The anisotropy ﬁelds, or inhomogeneous exchange constants, can be determined by standard “ferromagnetic ”methods. In particular, it is di ﬃcult to expect that the inhomogeneous exchange constant will change strongly when passing from thevalue of decompensation ν∼0.1 to the value ν∼0.01. However, a particular problem is presente d by the homogeneous exchange constant E ex, and the associated characteristic exchange frequency ωex=Eex/(s1+s1)/C22h,s e eE q . (9). Their values do not manifest themselves in any way in the “ferromagnetic ”region of parame- ters, at ν/C29ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ωa=ωexp ∼10−2. The amorphous ferrimagnet GdFeCo attracts much of the researchers ’attention due to the discovery of ultrafast (faster than a picosecond) magnetization switching under the action of femtosec- ond laser pulses.25This was the material used in Ref. 53to study the motion of domain walls. Let us provide the data that are knownfor this material, and for similar ferrimagnets. Combining the mean ﬁeld method with the numerical atomic spin simulation makes it possible to calculate the completephase diagram of a magnet. 212Such calculations were performed for the Gd 25(FeCo) 75alloy, i.e. for a composition at which the compensation point is close to room temperature (next, as is thecase in this article, we neglect the presence of a small amount ofcobalt and discuss only the interaction between the gadoliniumand iron). Comparing the calculation results with the experimen- tal data made it possible to determine the values of all exchange integrals. In particular, the energy of the exchange interaction ofGd atoms and transition elements was determined, which isresponsible for uniform exchange. This energy (per atom) can berepresented ε Gd-FecSGdcSFe/2, where cSGdandcSFeare unit vectors that determine the direction of the sublattice spins, and each exchange connection is taken into account only once. Using cSFe=m+l, cSGd=m–land the condition that m2+l2=c o n s t ,s e eE q s . (7)and(8), it is found that the energy density of homogeneous exchange canbe written as w ex=εGd–Fem2(nGd+nFe), where nGd=sGd/SGdand nFe=sFe/SFeare the atomic densities of gadolinium and iron, which can be written using the corresponding spin densitiess Gd,sFeand the spin values of these elements SGd= 7/2 and SFe=1 .212As such, the homogeneous exchange constant in Eq. (9) is determined by the formula Eex=εGd–Fe(nGd+nFe). Assuming that εGd–Fe= 4.8/C210–21J/atom,56,212,213using the sublattice magnetizations near the compensation point MGd= 100 G and MFe= 1100 G (in the SI system M(SI) Gd¼105A/m and M(SI) Fe¼1:1/C2105A/m), and g-factors gFe= 2.2 and gGd=2 , w e get Eex= 3.6/C2109erg/cm3 (in SI units Eex= 3.6/C2108J/m3). Accordingly, the exchange frequency ωex= 3.1/C210131/s, or ωex/2π= 5 THz. The corresponding exchange ﬁeld can be de ﬁned asHex=(/C22h/2μB)ωex, for which we get Hex= 1.75 MOe (175 T). The inhomogeneous exchange constant has been determined f o rm a n yf e r r i m a g n e t sw i t hd i ﬀerent compositions; for (GdTb) (FeCo) alloys with about 25% content of rare-earth ions, itsvalue is A=5 . 2/C210 7erg/cm ( A=5 . 2/C210–12J/m) and weakly depends on the ratio of Gd and Tb.214Using these parameters, it is possible to estimate the value of the characteristic velocity c= 3.7 km /s. Note that the values of the exchange frequency and velocity are somewhat lower than for standard antiferromagnets with acomparable value of the Néel temperature. For example, for ortho-ferrites, the values are H OF ex= 6 MOe and c= 20 km/s. The fact is that for antiferromagnets both magnetic ordering and the homoge- neous exchange constant are determined by the same exchangeinteraction between the nearest neighbors. The exchange interac-tion rare-earth and transition element spins in GdFeCo-type ferri- magnets is weaker than the interaction of transition elements with each other. It is also important that the concentration of rare-earthions is low. Despite this, the exchange enhancement of ferrimag-netic dynamic parameters is quite substantial. In particular, theexpected value of the limiting velocity of domain walls exceeds what can be expected for ferromagnets. The values of the eigenfre- quencies of spin vibrations are also high: even assuming aminimum value of the anisotropy ﬁeld of about H a∼4πMs, where Ms=MFe–MGd∼100 G, we get a small value ωa=( 2μB//C22h) Ha∼3.5 GHz, but the value ω0=ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃωaωexp∼300 GHz. At the same time, the homogeneous spin vibration frequencies excited by the spin-polarized current near the spin compensation point can reachterahertz values, see Sec. 4.Low Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 45,000000 (2019); doi: 10.1063/1.5121265 45,000000-959 Published under license by AIP Publishing.REFERENCES 1R. Skomski, J. Phys .: Condens. Matter 15, R841 (2003). 2J. Stöhr and H. C. Siegmann, Magnetism From Fundamentals to Nanoscale Dynamics (Springer, Berlin, 2006). 3F. Saurenbach, U. Walz, L. Hinchey, P. Grunberg, and W. Zinn, J. Appl. Phys. 63, 3473 (1988). 4M. N. Baibich, J. M. Broto, A. Fert, F. N. Van Dau, F. Petro ﬀ, P. Eitenne, G. Creuzet, A. Friederich, and J. Chazelas, Phys. 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1.5008572.pdf	Coupled magnetic and elastic dynamics generated by a shear wave propagating in ferromagnetic heterostructure A. V. Azovtsev , and N. A. Pertsev Citation: Appl. Phys. Lett. 111, 222403 (2017); View online: https://doi.org/10.1063/1.5008572 View Table of Contents: http://aip.scitation.org/toc/apl/111/22 Published by the American Institute of PhysicsCoupled magnetic and elastic dynamics generated by a shear wave propagating in ferromagnetic heterostructure A. V . Azovtsev and N. A. Pertsev Ioffe Institute, St. Petersburg 194021, Russia (Received 7 October 2017; accepted 10 November 2017; published online 1 December 2017) Using advanced micromagnetic simulations, we describe the coupled elastic and magnetic dynamics induced in ferromagnet/normal metal bilayers by shear waves generated by the attachedpiezoelectric transducer. Our approach is based on the numerical solution of a system of differential equations, which comprises the Landau-Lifshitz-Gilbert equation and the elastodynamic equation of motion, both allowing for the magnetoelastic coupling between spins and lattice strains. Thesimulations have been performed for heterostructures involving a Fe 81Ga19layer with the thickness ranging from 100 to 892 nm and a few-micrometer-thick ﬁlm of a normal metal (Au). We ﬁnd that the traveling shear wave induces inhomogeneous magnetic dynamics in the ferromagnetic layer,which generally has an intermediate character between coherent magnetization precession and the pure spin wave. Owing to the magnetoelastic feedback, the magnetization precession generates two additional elastic waves (shear and longitudinal), which propagate into the normal metal. Despitesuch complex elastic dynamics and reﬂections of elastic waves at the Fe 81Ga19jAu interface, periodic magnetization precession with the excitation frequency settles in the steady-state regime. The results obtained for the magnetization dynamics at the Fe 81Ga19jAu interface are used to evalu- ate the spin current pumped into the Au layer and the accompanying charge current caused by the inverse spin Hall effect. The calculations show that the dc component of the charge current is high enough to be detected experimentally even at small strains /C2410/C04generated by the piezoelectric transducer. Published by AIP Publishing. https://doi.org/10.1063/1.5008572 The magnetoelastic interaction between spins and strains renders possible excitation and control of the mag- netic dynamics by mechanical stimuli such as elastic wavesand strain pulses. This strain-mediated approach is promising for the development of advanced spintronic devices with greatly reduced power consumption because strains can be generated by voltages applied to a piezoelectric material mechanically coupled to a ferromagnet. Hence, high ohmicenergy losses associated with conventional excitation techni- ques exploiting magnetic ﬁelds or spin-polarized currents can be avoided, which makes ferromagnetic-piezoelectric hybrids suitable for the low-power computing applications. Such hybrids were widely considered as appropriate buildingblocks for electric-write nonvolatile magnetic memories 1–5 and electrically tunable microwave devices6–8and could be proposed for designing the next generation of logic devicesbased on spin waves. 9 Owing to the interest in strain-mediated spintronics, the magnetoelastic phenomena in ferromagnetic heterostructures have been intensively studied experimentally during the past decade. In particular, it was shown that bulk sound wavesinjected into a ferromagnetic ﬁlm pump spin currents into an attached paramagnetic metal as a consequence of the energy transfer from the sound waves to spin waves. 10Several experimental studies of the magnetic dynamics induced by surface acoustic waves11–17and acoustic pulses generated by femtosecond laser pulses18,19have also been reported. Furthermore, the strain-induced generation of traveling spin waves in a ferromagnetic ﬁlm coupled to a piezoelectric sub-strate subjected to an ac voltage has been recently demon- strated experimentally. 20To better understand the observed magnetoelastic phe- nomena and formulate guidelines for the optimization of “straintronic” devices, it is imperative to develop a detailedtheoretical description of elastically driven magnetic dynam- ics in ferromagnetic heterostructures. The rigorous treatment of dynamic magnetoelastic problems requires the solution of a set of coupled differential equations for the magnetization and displacement. 21,22The linearization of these equations under the assumption of small perturbations of the initial state makes it possible to obtain approximate analytical solutions for some simple conﬁgurations.23,24Micromagnetic simula- tions represent a much more powerful approach, but usually they are performed without solving the elastodynamic equa-tion involving the coupling term, 25–28which leads to an approximate description of the magnetic dynamics. Only a few models attempting to allow for the backaction of magne-tization changes on the displacement ﬁeld have been devel- oped. 29–31However, none of these models provide a rigorous description of dynamic magnetoelastic problems because Liang et al. solved weak forms of partial differential equa- tions instead of their strong forms,29,30whereas Peng et al. omitted the inertial term in the elastodynamic equation.31 In this letter, we study elastically driven magnetic dynamics with the aid of advanced micromagnetic simula-tions, which fully take into account the magnetoelastic cou- pling between spins and lattice strains and properly evaluate dipolar interactions between oscillating spins. The hetero- structure considered in our study has the form of a (001)-ori- ented single-crystalline or highly textured polycrystallineferromagnetic layer mechanically coupled to a piezoelectric transducer and covered by a ﬁlm of a normal metal (Fig. 1). 0003-6951/2017/111(22)/222403/5/$30.00 Published by AIP Publishing. 111, 222403-1APPLIED PHYSICS LETTERS 111, 222403 (2017) We assume that the transducer creates a periodic mechanical displacement at the interface and simulate thus induced elas-tic and magnetic dynamics in the ferromagnet/normal metalbilayer. Using the results obtained for the magnetization pre- cession in the ferromagnetic layer, we calculate the time- dependent spin current pumped into the normal metal andevaluate the accompanying charge current. The micromagnetic simulations were performed using home-made software enabling us to solve numerically a sys-tem of differential equations, which comprises the Landau-Lifshitz-Gilbert (LLG) equation for the local magnetizationM(t) and the elastodynamic equation of motion for the dis- placement u(t), both solved in their strong forms. Since the LLG equation implies that the magnetization magnitudejMj¼M sis ﬁxed, which is a good approximation well below the Curie temperature, it can be cast into the form dm dt¼/C0c ð1þa2Þm/C2Heffþam/C2ðm/C2HeffÞ ½/C138 ; (1) where m¼M/Msis the unit vector deﬁning the magnetiza- tion direction, cis the electron’s gyromagnetic ratio, ais the dimensionless Gilbert damping parameter, and Heffis the effective magnetic ﬁeld acting on M. The effective ﬁeld was calculated with the account of all relevant contributions as described in our previous paper.27Characteristic features of our approach include accurate calculation of dipolar interac-tions between differently oriented oscillating spins and theaddition of a magnetoelastic contribution H meltoHeff. For cubic ferromagnets considered in this work, the componentsofH melare given by the relations Hmel x¼/C0 ð 1=MsÞ½2B1exxmx þB2ðexymyþexzmzÞ/C138,Hmel y¼/C0ð 1=MsÞ½2B1eyymyþB2ðeyxmx þeyzmzÞ/C138, and Hmel z¼/C0ð 1=MsÞ½2B1ezzmzþB2ðezxmxþezymyÞ/C138, where B1and B2are the magnetoelastic coupling constants andeij¼ð1=2Þðui;jþuj;iÞare the lattice strains ( i,j¼x,y,z, and indices after a comma here and below denote differentia- tion with respect to the corresponding coordinates). The elasto-dynamic equation of motion of a cubic ferromagnet reads (nosummation over repeated indices i¼x,y,z,j6¼i,a n d k6¼i,j) q F@2ui @t2¼cF 11ui;iiþcF 44ui;jjþui;kk ðÞ þcF 12þcF 44/C0/C1 uj;ijþuk;ik ðÞ þB1ðm2 iÞ;i þB2ðmimjÞ;jþðmimkÞ;k/C2/C3; (2)where qFis the density and cF 11,cF 12, and cF 44are the elastic stiffnesses at constant magnetization in the Voigt notation. To describe the elastodynamics of a cubic normal metal with the density qNand elastic constants cN ij, we used the standard equation of motion, which differs from Eq. (2)by the absence of magnetoelastic terms. It should be noted that all material constants, including the parameters K1andK2of the cubic magnetocrystalline anisotropy,27are deﬁned in the crystallographic reference frame ( x1,x2,x3) with the axes oriented along the cubic [100], [010], and [001] directionsparallel to the x,y, and zaxes shown in Fig. 1. The partial differential Eqs. (1)and(2)were supple- mented by the following set of boundary conditions: At the ferromagnet/transducer (F jTr) interface, the displacement u F was assumed to satisfy the relations uF x¼uF z¼0 and uF y ¼umaxsinxt, where the amplitude umax and frequency x¼2p/C23of the displacement uF yalong the yaxis (see Fig. 1) represent the input parameters of our simulations. For the ferromagnet/normal metal (F jN) interface, we employed the usual mechanical conditions relating displacements (uF¼uN) and stresses ( rF xj¼rN xj,j¼x,y,z) in the adjacent materials. The opposite boundary of the N layer was consid- ered mechanically free ( rN xj¼0). The free magnetic bound- ary condition @M=@x¼0 was used for the magnetization at both surfaces of the F layer. Since the LLG equation is known to be “stiff,” the numerical integration was realized with the aid of the pro-jective Euler scheme, where the condition jmj¼1i ss a t i s - ﬁed automatically. A ﬁxed integration step dt¼5f sa n d computational cells with the dimensions 2 /C22/C22n m 3 smaller than the exchange length kex/C254 nm were employed in the calculations.27Note that, although the considered problem is one-dimensional ( manduvary along on the x axis only), an ensemble of three-dimensional computation cells is needed to evaluate exchange and dipolar interac- tions between the spins. The simulations were performed for Fe 81Ga19jAu bilayers with the total thickness of 3 lm and the Fe 81Ga19 thickness tFranging from 100 to 892 nm. To stabilize the single-domain state in the Fe 81Ga19layer and to create mag- netization precession via the effective-ﬁeld component Hmel y¼/C0 B2exymx=Ms,w ei n t r o d u c e da ne x t e r n a lm a g n e t i c ﬁeld with the components Hx¼8 kOe and Hz¼0.5 kOe, at which the equilibrium magnetization orientation deviates from the in-plane zdirection by an angle h/C2528/C14. The follow- ing values of Fe 81Ga19and Au parameters were used in the simulations: saturation magnetization Ms¼1321 emu cm/C03,32 damping parameter a¼0.017,33exchange constant A¼1.8 /C210/C06erg cm/C01,34K1¼1.75/C2105erg cm/C03,K2¼0,35B1 ¼/C00.9/C2108erg cm/C03,B2¼/C00.8/C2108erg cm/C03,36cF 11¼1.62 /C21012dyne cm/C02,cF 12¼1.24/C21012dynecm/C02,cF 44¼1.26/C21012 dynecm/C02,qF¼7.8gcm/C03,37cN 11¼1.924 /C21012dynecm/C02, cN 12¼1.63/C21012dynecm/C02,cN 44¼0.42/C21012dynecm/C02,a n d qN¼19.3gcm/C03.38The maximal displacement umaxinduced at the F jTr interface was set to 0.02nm, which provides initial strains exy(/C23) with the amplitude /C2410/C04in the Fe 81Ga19layer at the microwave excitation frequencies /C23/C245–10GHz. The simulations showed that periodic in-plane displace- ment at the F jTr interface induces a shear wave propagating FIG. 1. Ferromagnet/normal metal bilayer mechanically coupled to a piezo- electric transducer, which generates an elastic wave propagating across the ferromagnetic (F) layer of thickness tFand the normal-metal (N) layer of thickness tN. The magnetization direction min the unstrained F layer devi- ates from the in-plane zaxis by an angle hdue to the applied magnetic ﬁeld Hwith the nonzero perpendicular-to-plane component.222403-2 A. V . Azovtsev and N. A. Pertsev Appl. Phys. Lett. 111, 222403 (2017)across the F jN bilayer, which is qualitatively similar to the generation of transverse microwave phonons in permalloy ﬁlms by a quartz transducer.39Figure 2(a) demonstrates the representative spatial distribution of the shear strain exyðx;tÞ in the traveling wave before its reﬂection from the free sur- face of the N layer. Since the phase velocity vt¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ c44=qp of transverse waves in Fe 81Ga19(vF tﬃ4/C2103ms/C01) is signiﬁ- cantly higher than in Au ( vN tﬃ1.475 /C2103ms/C01), the wave- length ktreduces about three times in the Au layer. Remarkably, after multiple partial reﬂections of the wave at the F jN interface (transmittance /C250.95), almost sinusoidal strain distribution with increased amplitude emax xy/C255/C210/C04 sets in the F layer with the thickness tF¼kF t. Owing to the magnetoelastic torque Tmel¼m/C2Hmel created by the strain exy, the shear wave induces magnetization precession in the F layer. To obtain maximal precession amplitude, the simulations were performed for mechanical excitations with frequencies /C23around the resonance frequency /C23res¼8.75 GHz of the coherent precession in the unstrained Fe81Ga19layer, which was determined by separate simula- tions. Figure 3(Multimedia view) shows spatial distributions of the direction cosines mi(x,t)i nF e 81Ga19layers with differ- ent thicknesses tF. As demonstrated by Fig. 3(a), the preces- sion is almost uniform in the layer with the thickness tF¼100 nm much smaller than the wavelength kF t¼446 nm of the driving shear wave. In contrast, the magnetic dynamics becomes highly inhomogeneous when tFequals or exceeds kF t, acquiring the form of a spin wave (SW) [see Figs. 3(b) and 3(c)]. Although the SW proﬁle is non-sinusoidal, the SW wavelength is governed by that of the shear wave, and the SW frequency is equal to the excitation frequency. The simulations also revealed the excitation of two addi- tional elastic waves in the F jN bilayer, which have the form of a shear wave exzðx;tÞand a longitudinal one exxðx;tÞ[see Fig.2(b)]. These waves have much smaller amplitudes than the driving shear wave (maximal strains /C2410/C06instead of /C2410/C04), being absent when the magnetoelastic terms in Eq. (2)are set to zero. Therefore, the magnetization precession in the F layer lies at the origin of the revealed phenomenon, which demonstrates the backaction of magnetic dynamics on lattice strains. After reaching the Fe 81Ga19jAu interface, the weak secondary waves penetrate into the Au layer, where the longitudinal wave propagates with signiﬁcantly higher veloc- ityvN l¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ cN 11=qNp ﬃ3.16/C2103ms/C01than the transverse one [Fig. 2(b)]. The magnetically induced elastic dynamics changes the effective ﬁeld Heffinvolved in Eq. (1)and so should affect the magnetization precession. This effect was evaluated by comparing the results of accurate simulations with those of approximate calculations disregarding the feedback created by the magnetoelastic terms in Eq. (2). The comparison showed that the feedback-induced relative changes in the perturbations Dmiof direction cosines during the magnetiza- tion precession are smaller than 2% at all studied conditions. This is due to the weakness of secondary elastic waves and the absence of substantial inﬂuence of magnetic dynamics on the driving shear wave, which has the unperturbed wave- length kF t¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ cF 44=qFp /C23and only slightly different amplitude (by no more than 1%) at the considered excitationfrequencies. Thus, the magnetoelastic feedback does not sig- niﬁcantly modify the magnetization dynamics in our case. Figure 4(a) demonstrates the temporal evolution of the magnetization precession at the Fe 81Ga19jAu interface for the heterostructure comprising the 446-nm-thick Fe 81Ga19 layer. Despite rather complex elastic dynamics, a steady-state magnetization precession with the constant frequencyand amplitude sets in after a short transition period of /C241n s (note that the reﬂectance of the driving shear wave at theFjN interface is only about 0.05). The end of the magnetiza- tion vector moves along an elliptical trajectory, and the max-imal angular deviation from the equilibrium direction isabout 3 /C14. The steady-state regime lasts until the shear wave reﬂected from the free boundary of the N layer r eaches the F one (time period of 2.7–4 ns), after which the magnetic dynam-ics becomes irregular due to the interference of the reﬂectedwave with the one constantly generated by the transducer. The results obtained for the magnetization precession at the F jN interface were used to evaluate the spin current pumped into Au by the dynamically strained Fe 81Ga19layer. The spin-current density Jsat the interface was calculated from the approximate relation Jsﬃð/C22h=4pÞRe½gr "#/C138m/C2dm= dtappropriate for thick F layers, where gr "#is the complex reﬂection spin mixing conductance per unit area of the F jN contact.40Fig. 4(b) demonstrates time dependences of the normalized projections of Json the coordinate axes. The ac FIG. 2. Elastic waves in the Fe 81Ga19jAu bilayer at the excitation frequency /C23¼9 GHz. (a) Spatial distribution of the strain exyðx;tÞin the driving shear wave at the time t¼1.73 ns. (b) Magnetically induced secondary elastic waves. The plots show the strains exx(x,t) and exz(x,t)a t t¼0.85 ns. The Fe81Ga19thickness equals 446 nm, and the coordinate xis given in units of the computational cell size amounting to 2 nm.222403-3 A. V . Azovtsev and N. A. Pertsev Appl. Phys. Lett. 111, 222403 (2017)component of the spin current clearly dominates, but the ﬁl- tering of high-frequency oscillations shows that signiﬁcantdc spin current hJ siis generated as well. The nonzero com- ponents hJs xiandhJs ziof the latter are plotted as a function of the Fe 81Ga19thickness in Fig. 5. According to our numerical estimates, the slowing down of the magnetization precession,which is caused by the spin pumping into the normal metaland can be accounted for by an increase in the Gilbert damp-ing parameter for computational cells adjacent to the inter-face, 41is expected to be rather small in Fe 81Ga19layers with thicknesses tF/C29kex.Owing to the inverse spin Hall effect, the pumped spin current generates a charge current in the N layer, which hasthe density J c¼aSHð2e=/C22hÞðes/C2JsÞ, where aSHis the spin Hall angle, eis the elementary positive charge, and esis the unit vector in the spin-current direction.42Taking aSH /C250.0035 for pure Au and using the theoretical estimate ðe2=hÞRe½gr "#/C138/C254:66/C21014X/C01m/C02obtained for the reﬂec- tion spin mixing conductance of the Fe jAu interface,40,43we evaluated densities of the spin and charge currents ﬂowing inthe Au layer near the Au jFe 81Ga19interface. As the distance from the interface increases, the injected spin current reducesdue to spin relaxation and diffusion. 44Allowing for the decay FIG. 3. Elastically driven magnetization dynamics in Fe 81Ga19layers with the thicknesses of 100 nm (a), 446 nm (b), and 892 nm (c). The plots showspatial distributions of the changes Dm iin the direction cosines of oscillating magnetization at /C23¼9 GHz in comparison with the time-dependent strain exy in the driving shear wave. Static images correspond to the time tﬃ2.5 ns, at which the shear wave has already experienced several reﬂections from the FjN interface, but the wave reﬂected from the free boundary of the N layer has not reached the F layer. Dynamical temporal evolutions are shown for the time period of 2 ns. Multimedia views: https://doi.org/10.1063/1.5008572.1 ; https://doi.org/10.1063/1.5008572.2 ;https://doi.org/10.1063/1.5008572.3 FIG. 4. (a) Magnetization precession at the Fe 81Ga19jAu interface. The plots show temporal evolutions of the direction cosines mifor the 446-nm-thick Fe81Ga19layer. (b) Time dependence of the spin current pumped by the Fe81Ga19layer into the adjacent normal metal. The components Js iof the spin-current density at the interface are normalized by the quantity ð/C22h=4pÞRe½gr "#/C138. The excitation frequency equals 9 GHz. FIG. 5. Inﬂuence of the Fe 81Ga19thickness on the dc spin current pumped into the Au layer and the total dc charge current ﬂowing in this layer at /C23¼9 GHz. The plots show the nonzero components hJs xiandhJs ziof the dc spin current at the Au jFe81Ga19interface normalized by ð/C22h=4pÞRe½gr "#/C138and the charge current hIc yicalculated for the Au layer of thickness tN/C29nsdand width wN¼10lm.222403-4 A. V . Azovtsev and N. A. Pertsev Appl. Phys. Lett. 111, 222403 (2017)of the dc spin current hJs ziin Au having the spin diffusion length nsd¼35 nm,42we calculated the total dc charge current hIciin the Au layer with the thickness tN/C29nsdand width wN¼10lm. It was found that hIci, which ﬂows along the y axis, strongly depends on the thickness of the Fe 81Ga19layer (see Fig. 5). Remarkably, the dc charge current generated at the excitation frequency /C23¼9 GHz amounts to 3–15 nA, which can be readily measured by modern picoammeters. In summary, we presented a rigorous micromagnetic approach to the solution of magnetoelastic problems, whichis distinguished by full account of the interplay betweenspins and strains in magnetic materials. The simulations per-formed for ferromagnet/normal metal bilayers traversed by ashear wave demonstrated the coupling of elastic and mag-netic dynamics and efﬁcient spin pumping into the normalmetal. 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1.1610797.pdf	Current-induced precessional magnetization reversal H. W. Schumacher, C. Chappert, R. C. Sousa, and P. P. Freitas Citation: Applied Physics Letters 83, 2205 (2003); doi: 10.1063/1.1610797 View online: http://dx.doi.org/10.1063/1.1610797 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/83/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Field and current-induced magnetization reversal studied through spatially resolved point-contacts J. Appl. Phys. 107, 103909 (2010); 10.1063/1.3407539 Thermomigration-induced magnetic degradation of current perpendicular to the plane giant magnetoresistance spin-valve read sensors operating at high current density J. Appl. Phys. 106, 113908 (2009); 10.1063/1.3260250 Current-induced magnetization reversal mechanisms of pseudospin valves with perpendicular anisotropy J. Appl. Phys. 102, 093902 (2007); 10.1063/1.2803720 Current-driven magnetization reversal in exchange-biased spin-valve nanopillars J. Appl. Phys. 97, 114321 (2005); 10.1063/1.1927707 Micromagnetic simulations of current-induced microwave excitations J. Appl. Phys. 97, 10C708 (2005); 10.1063/1.1852434 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, 01 Dec 2014 03:51:14Current-induced precessional magnetization reversal H. W. Schumachera)and C. Chappert Institut d’Electronique Fondamentale, UMR 8622, CNRS, Universite ´Paris Sud, Ba ˆtiment 220, F-91405 Orsay, France R. C. Sousa and P. P. Freitas Instituto de Engenharia de Sistemas e Computadores, Rua Alves Redol, 9, P-1000 Lisboa, Portugal ~Received 20 June 2003; accepted 21 July 2003 ! We report magnetization reversal in microscopic current-in-plane spin valves by ultrashort current pulses through the device. Current densities of the order of 1011A/m2with pulse durations as short as 120 ps reliably and reversibly switch the cell’s free-layer magnetization. Variations of the pulseparameters reveal the full signature of precessional switching, which is triggered by the transversemagnetic ﬁeld generated by the device current. This current switching mode allows for the designof a two-terminal nonvolatile magnetic memory cell combining ultrafast access times and highmagnetoresistive readout. © 2003 American Institute of Physics. @DOI: 10.1063/1.1610797 # In magnetic memory or storage applications, the content of a magnetic bit is generally written by means of a magneticﬁeld. In the case of a magnetic random access memory, 1such ﬁeld is, e.g., generated by an integrated ﬁeld line array. An-other way to induce magnetization reversal in a magnetic cellis the use of high electrical current densities through thedevice. This ﬁeld has recently attracted a lot of interest bothfor its possible applications and for the discovery of interest-ing physical phenomena. 2–11Various physical effects have been observed and discussed to induce such current inducedmagnetization reversal ranging from the spin torque exertedby a spin polarized current 2–7to current induced domain wall motion8,9and reversal by the Oersted ﬁelds generated by the device current.10,11Here, we report a further mechanism in the current induced switching of a spin valve ~SV!cell, the current induced precessional magnetization reversal. Using the giant magnetoresistance ~GMR !of current-in- plane spin valves we study the response of the magnetizationof the SV’s free layer to ultrashort current pulses. Currentpulses with densities of the order of 10 11A/m2and durations as short as 120 ps are found to induce reliable and full re-versal of the free-layer magnetization. Furthermore, theswitching reveals the full characteristics of precessionalswitching 12–18induced by the transverse Oersted ﬁeld of the current pulse, namely, reversible toggling of the magnetiza-tion by consecutive identical pulses and periodic transitionsfrom switching to nonswitching under variation of pulse pa-rameters. This switching mode allows for the design of two-terminal nonvolatile magnetic memory cells combining ul-trafast access times and high magnetoresistive readout. The experiments are carried out on exchange biased spin valves consisting from bottom to topof Ta 65 Å/Ni 81Fe1940Å/Mn 78Ir2280Å/Co 88Fe1243 Å/ Cu 24 Å/Co 88Fe1220Å/Ni 81Fe1930 Å/Ta 8 Å. The 50-Å- thick magnetic free layer is composed of the upper Co 88Fe12 and Ni 81Fe19layers and is situated directly under the Ta cap layer on top of the SV stack @cf. Fig. 1 ~a!#. A SV cell is shown in Fig. 1 ~b!. The lateral dimensions are 1.2533.6mm2with a total thickness of 31 nm. The pinned mag- netic layer is aligned along the long SV dimension by ex-change bias.The electrical contacts ~C!overlap with the ends of the SV cell and are connected to high bandwidth coplanarwaveguides. 19They allow injection of ultrashort current pulsesIPusing a commercial pulse generator and the mea- surement of the SV’s current-in-plane GMR before and afterpulse application. The transient pulses are monitored using a50 GHz sampling oscilloscope.The pulse duration T Pcan be adjusted between 120 ps and 10 ns ~at half maximum !with rise times down to 45 ps ~from 10% to 90% !. A magnetic ﬁeld line situated on the back of the chip allows us to reset the magnetization into a predeﬁned saturated state after cur-rent pulse testing. Furthermore, external ﬁelds along the easyaxis are applied via an external coil. Figure 2 ~b!shows a GMR loop of the device in Fig. 1 ~b! as a function of the external easy axis ﬁeld H easy. Due to overlap of contacts and SV the GMR signal mainly probesthe magnetization within the center of the SV. Flux closuredomains underneath the contacts thus do not contribute to theloop.The loop is shifted to an offset ﬁeld of H off528 Oe due a!Author to whom correspondence should be addressed; electronic mail: schumach@ief.u-psud.fr FIG. 1. ~a!Sketch of the spin valve ~SV!used in the experiments. The free layer is situated at the top of the SV stack. Other SV layers are not indicatedfor clarity. The current pulse I Palong the long device axis generates an Oersted ﬁeld HOeresulting in a transverse ﬁeld pulse HPin the center of the free layer. ~b!Electron micrograph of a spin valve cell. The ends of the 1.25mm33.6mm wide cell are covered by the electrical contacts ~C!.APPLIED PHYSICS LETTERS VOLUME 83, NUMBER 11 15 SEPTEMBER 2003 2205 0003-6951/2003/83(11)/2205/3/$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: 137.149.200.5 On: Mon, 01 Dec 2014 03:51:14to coupling of the pinned and the free layer.20In the follow- ing measurements this offset ﬁeld will be always compen-sated by the external ﬁeld ( H off5Heasy). Figure 2 ~a!shows a current pulse measured after trans- mission through the same SV. The pulse has a rise time of 65ps and a fall time of 125 ps. The pulse duration is T pulse 5120 ps with a current amplitude of 19.5 mA. The corre- sponding peak device current density is of the order of 5310 11A/m2. To test the SV response to such pulse, ﬁrst the free-layer magnetization is reset into the low resistance ~par- allel!state ~corresponding to 0% GMR !and the GMR is measured. Then, the current pulse is applied followed by asecond GMR measurement after pulse application. Multiplerepetition of such measure/reset cycles allows us to test thereliability of the switching properties of the current pulses.The such-derived response of the free-layer magnetization to the pulse shown in Fig. 2 ~a!is given in Fig. 2 ~c!. The GMR is plotted versus the index of the consecutive current/resetpulses. Every current pulse switches the magnetization fromthe low to the high resistance state. The GMR change of 2%corresponds to full free-layer reversal with every pulse @cf. Fig. 2 ~b!#. The 120 ps pulse of 5 310 11A/m2current density thus induces a full and reliable switching of the SV’s free-layer magnetization. A further important feature of this cur-rent induced SV switching is found in Fig. 2 ~d!. Here, a second spin valve with similar lateral dimensions but havinga higher GMR ratio was used. In contradistinction to theformer experiment, here, the external reset pulses were omit-ted. Figure 2 ~d!shows the GMR response to the consecutive application of identical current pulses of constant polarity. The pulse duration is 140 ps with current density of ;5.5 310 11A/m2. Despite of the lack of the reset pulses, alsohere, the GMR changes by 64.5% with every applied cur- rent pulse and consecutively switches from high to low re-sistance and vice versa. Each current pulse thus reversiblytogglesthe magnetization between the two easy directions. Such reversible switching is characteristic of precessionalmagnetization reversal as recently observed. 15,17 For the precessional switching of magnetic cells,15–18a fast rising ﬁeld pulse is applied along the in-plane magnetichard axis and thus perpendicular to the equilibrium directionof the magnetization M. As shown in Fig. 1 ~a!, injection of an in-plane current pulse I Palso generates an Oersted ﬁeld HOeinside the device as indicated by the dotted circular arrow.As the free layer is situated on the upper surface of thespin valve stack, this results in an in-plane transverse ﬁeldpulseH Pin the center of the free layer. For the device shown in Fig. 1 ~b!the in-plane transverse ﬁeld can be estimated to be of the order of 5 Oe/mA corresponding to ;100 Oe ﬁeld for the above-mentioned pulse. Note, that the in-plane ﬁeld isnot constant over the whole width of the device. However, inthe center region H ponly weakly changes resulting in a de- crease of only 10% within 90% of the total width of thedevice. To unambiguously show the precessional nature of the current induced switching, the switching properties weremonitored over a wide range of pulse parameters. Preces-sional switching by transverse pulses should then reveal pe-riodic transitions from switching to nonswitching with vary-ing pulse parameters. 13,17Figure 3 ~a!shows a gray-scale map of the switching reliability ^uDGMR u&, plotted versus pulse duration and nominal current pulse amplitude. The pulse am-plitude is varied in steps of 1 dB. ^uDGMR u&is the absolute GMR change per current pulse normalized to full reversal,and averaged over a series of pulses. ^uDGMR u&’1~white ! indicates stable switching, and 0 ~black !no switching. The switching was again tested from the low to the high resis- FIG. 2. Current induced precessional magnetization reversal. ~a!Current pulse inducing switching after transmission through the device shown inFig. 1 ~b!.T P5120ps,IP519.5 mA. ~b!Quasistatic easy axis hysteresis loop of the same device. The GMR change in the loop center is ;2% ~arrow !.~c!Current induced reversal testing by consecutive application of the current pulse shown in ~a!and a reset pulse. The easy axis loop offset is compensated. The 120 ps pulse of 5 31011A/m2current density induces the full switching of the free-layer magnetization of the device. ~d!Reversible current induced switching in a further SV. The free-layer magnetization istoggled with every consecutive current pulse of 140 ps duration and 5.5 310 11A/m2current density. FIG. 3. Higher order current induced precessional switching. ~a!Measured switching reliability, ~b!simulated data. Gray-scale map of the switching reliability ^uGMR u&as a function of pulse duration and current amplitude ~a! or transverse Oersted ﬁeld ~b!. White: ^uGMR u&51~reliable switching !; black ^uGMR u&50~no switching !. Switching order nis indicated in ~b!.2206 Appl. Phys. Lett., Vol. 83, No. 11, 15 September 2003 Schumacher 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: 137.149.200.5 On: Mon, 01 Dec 2014 03:51:14tance state by resetting the sample prior to current pulse ap- plication @cf. text to Fig. 2 ~c!#. The arrow in Fig. 3 ~a!marks the full reversal induced by the 19.5 mApulse shown in Fig.2~a!. Note, that the current values given on the right are valid for long pulse durations. Below 300 ps pulse duration thetransient current decreases linearly down to about 60% dueto instrumental limitations. Figure 3 ~b!shows the results of a simulation of the current induced switching over the corre-sponding parameter range. Again, white indicates switchingand black no switching. The simulation was performed bysolving the Landau–Lifshitz–Gilbert equation 21dM/dt5 2g(M3Heff)1(a/Ms)(M3dM/dt), in the single spin ~or single domain !approximation. Here, gis the gyromagnetic ratio,Heffthe effective ﬁeld, athe Gilbert damping param- eter, and MSthe saturation magnetization. The free-layer magnetization is modeled with saturation magnetization of4 pMS513500 Oe and demagnetizing factors NX/4p 50.0005 ~easy axis !,NY/4p50.0055 ~in-plane hard axis !, andNZ/4p50.994 ~out of plane !suitable for the measured total anisotropy of the device. Only the in-plane componentH pof the transverse Oersted ﬁeld HOegenerated by the cur- rent pulses was taken into consideration. Again, the ﬁeldswere derived from the measured transient current pulses. As seen in Fig. 3, the measured switching behavior is very well reproduced by the model of switching by the in-plane transverse Oersted ﬁeld. The periodic transitions fromswitching to nonswitching with increasing pulse duration areclearly observed. Three light ~switching !regions separated by dark ~nonswitching !regions are found. The ﬁrst region @markedn50i n ~b!#corresponds to so-called zero-order switching. 17Here,Mperforms approximately a half- precessional turn about the transverse Oersted ﬁeld duringpulse application. The pulse duration approximately equals ahalf of a precession period for the given ﬁeld ( T pulse ’1 2Tprec) resulting in a reversal of Mafter pulse decay. In the adjacent dark region ~no switch !Mperforms about a full precessional turn during pulse application ( Tpulse’Tprec) re- sulting in relaxation towards the initial direction of Mupon pulse termination ~no switch !. For further increase of Tpulse, switching occurs whenever Tpulse’(n11 2)Tprec, withnbeing an integer deﬁning the order of the switching process. Thereversal by the 120 ps, 19.5 mA pulse @cf. Figs. 2 ~a!and 2~c!#is marked by the arrow in the upper panel. Note that only in this region, i.e., for zero-order ( n50) switching, full and stable reversal ~corresponding to a white color !is found. For longer pulses and weaker currents only less reliableand/or switching of smaller domains of the sample is ob-tained ~gray!. A reason for this might be the above- mentioned inhomogeneity of the Oersted ﬁeld towards theborders of the device. This inhomogeneity also results in avariation of the precession frequencies over the width of thedevice favoring the breaking up of the free-layer magnetiza-tion into domains, especially for higher switching orders. In conclusion, we have observed current induced preces- sional switching of the magnetization in microscopic spinvalves. Switching by current densities around 5 310 11A/m2and ultrashort pulse durations down to 120 ps was observed. The switching showed the full characteristicsof precessional switching induced by the transverse Oerstedﬁeld created by the current pulse. The maximum GMR ratioof our cells was of the order of 5%. However, much highervalues of 20% and more should be obtainable in optimizedspin valves. 22Then, current induced precessional switching could be used to switch a nonvolatile, two-terminal memorycell combining the key features of ultrafast access times andhigh magnetoresistive readout. The authors would like to thank J. Miltat and M. Bauer for valuable discussions. The authors further acknowledgeﬁnancial support by the European Union by Marie CurieGrant No. HPMFCT-2000-00540, the Training and Mobilityof Researchers Program under Contract No. ERBFMRX-CT97-0147, and a NEDO contract ‘‘Nanopatterned Mag-nets.’’ 1S. Tehrani, B. Engel, J. M. Slaughter, E. Chen, M. DeHerrera, M. Durlam, P. 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1.3562884.pdf	Thermally activated transitions in a system of two single domain ferromagnetic particles Dorin Cimpoesu, , Alexandru Stancu, , Ivo Klik, , Ching-Ray Chang, , and Leonard Spinu Citation: Journal of Applied Physics 109, 07D339 (2011); doi: 10.1063/1.3562884 View online: http://dx.doi.org/10.1063/1.3562884 View Table of Contents: http://aip.scitation.org/toc/jap/109/7 Published by the American Institute of PhysicsThermally activated transitions in a system of two single domain ferromagnetic particles Dorin Cimpoesu,1,a)Alexandru Stancu,1,b)Ivo Klik,2,c)Ching-Ray Chang,2,d) and Leonard Spinu3 1Department of Physics, Al. I. Cuza University, Iasi 700506, Romania 2Department of Physics, National Taiwan University, Taipei, Taiwan 107 3Advanced Materials Research Institute (AMRI) and Department of Physics, University of New Orleans, New Orleans, Louisiana 70148, USA (Presented 17 November 2010; received 8 September 2010; accepted 14 December 2010; published online 8 April 2011) Numerical simulations based on the stochastic Langevin equation are applied here to a system of two uniaxial single domain ferromagnetic particles with antiferromagnetic dipolar coupling. The hysteresis loops of a strongly coupled systems exhibit fully demagnetized, intermediate metastable conﬁgurations which separate the two fully saturated states. At small magnetostatic couplings, onthe other hand, and at sufﬁciently weak damping, the intermediate metastable conﬁguration becomes only partially demagnetized. This state cannot be associated with any single local minimum of the free energy function. VC2011 American Institute of Physics . [doi:10.1063/1.3562884 ] INTRODUCTION In perpendicular magnetic recording, one can use larger particles without increasing their anisotropy too much. How- ever, interaction ﬁelds in this arrangement become quitehigh, and in the presence of thermal agitation also extremely complex. 1–3We study here a pair of thermally activated par- ticles with dipolar coupling of antiferromagnetic type. Sucha study may provide a benchmark for more complex investi- gations and our result, indeed, does provide a new perspec- tive on the effect of interactions in particulate media. The thermally activated dynamics of a pair of parallel uniaxial particles were studied by Lyberatos and Chantrell 4 who applied Langevin equation simulations5,6to the special case of low energy barrier, high damping, when the line con- necting the particles makes a bond angle b¼0 with the easy axes. The authors found in particular that the reversal bysymmetric fanning, predicted by Chen et al ., 7takes place only at strong coupling and high energy barrier. In this paper, we apply Langevin equation simulations to the special case of antiferromagnetically (AFM) coupled identical uniaxial particles with bond angle b¼p=2. We identify two qualitatively different reversal modes: a processwhich takes place only at low damping and low AFM cou- pling, and a process which takes place at high damping and strong coupling as well. We use the results of Chen et al. 7 who described the energy surface of two identical magneto- statically coupled uniaxial particles acted on by an external ﬁeld parallel to the particles easy axes. For brevity we intro-duce here merely the coupling strength q¼M 2 sV=ð2Kr3Þ where Kis the anisotropy constant, Vthe activation volume,Msthe saturation magnetization, and rthe distance between the particles. We also introduce the standard nucleation ﬁeld Hn¼2K=Ms, and the reduced ﬁeld h¼H=Hn. The bond angle b¼p=2 everywhere. The analysis of Chen et al.7allowed Klik et al.8,9to write down a three level master equatio n for the occupation probabil- ities of the three distinct metastable conﬁgurations "", "# þ #" ,a n d ##of the system. At zero or sufﬁciently low applied ﬁeld these metastable states represent four local minimaon the free energy surface of the two-particle system. At the other extreme is the case of a very large applied ﬁeld which allows the existence of only a single, fully magnetized state.The thermally activated transi tion rates between the metastable states are determined by barrier heights separating them. The prefactor of the thermal relaxation rate was taken to be con-stant, and all back-reversal processes 10were excluded. The rate (master) equation yields the tim e dependent occupation proba- bilities of the three metastable states. At sufﬁciently smallﬁelds 9it may schematically be represented by the diagram ""$"# þ #"$## : In the limit q!0 the three level master equation goes over to the free particle limit. In order to describe the ther mally activated dynamics of the two-particle system, we emp loy a Langevin stochastic equa- tion of motion based on the standard Gilbert equation11aug- mented by a stochastic thermal ﬁeld which is assumed to be a Gaussian random.12It is assumed that the ﬂuctuating ﬁelds act- ing on the different magnetic moments are independent. We interpret this equation in the sense of Stratonovich,13and numerically integrate it using an implicit midpoint time-integra- tion technique.14No temperature dependence of the anisotropy constant and saturation magnetization are taken into account. The applied ﬁeld driving the loop is htðÞ¼ hmaxcos 2pft, where fis the sweep frequency of the ﬁeld. In our simulationsa)Electronic mail: cdorin@uaic.ro. b)Electronic mail: alstancu@uaic.ro. c)Electronic mail: iklik@phys.ntu.edu.tw. d)Electronic mail: crchang@phys.ntu.edu.tw. 0021-8979/2011/109(7)/07D339/3/$30.00 VC2011 American Institute of Physics 109, 07D339-1JOURNAL OF APPLIED PHYSICS 109, 07D339 (2011)f¼1MHz. Starting from the initial positive saturated state, we let the system evolve during a time interval Dt¼2fNðÞ/C01, Nbeing the number of desired points on a curve. The average of the magnetization over the mentioned temporal window is then computed. This involves many thousands of steps in the numerical integration, since we set the time-step to be around1 psec. On a given time interval, a set of N r¼106stochastic realizations of the stochastic process has been performed, and their statistics have been calculated. Then the evolution of thesystem on the next time interval is computed, and so on, until the system reaches negative saturation. For each temporal window the mean normalized magnetic moment along theapplied ﬁeld is m z¼1 NrXNr i¼1mðiÞ 1zþmðiÞ 2z/C16/C17 ; where mðiÞ jzis the average (over the temporal window) projec- tion along the applied ﬁeld of the magnetization of the j-th particle ( j¼1;2) in the i-th stochastic realization. The probabilities (occupation numbers) nlthat the sys- tem ﬁnds in the l/C0th state ðl¼1;2;3Þare computed as n1¼1 NrXNr i¼1mðiÞ 1zþmðiÞ 2z 2 ! ;ifmðiÞ 1z/C210 and mðiÞ 2z/C210 0; otherwise;8 >>>>< >>>>: n2¼1 NrXNr i¼1jmðiÞ 1zjþjmðiÞ 2zj 2 ! ;ifmðiÞ 1zmðiÞ 2z<0 0; otherwise;8 >>< >>: n3¼1 NrXNr i¼1jmðiÞ 1zjþjmðiÞ 2zj 2 ! ;ifmðiÞ 1z<0 and mðiÞ 2z<0 0; otherwise:8 >>< >>: Here n1is the occupation probability of the saturated state "",n2is the occupation probability of the two demagnetized conﬁgurations "# þ #" ,n3and is the occupation probability of the inversely saturated state ##. In Fig. 1we show the reduced mean magnetization mz together with the occupation probability n2. The reduced inverse temperature here is q¼KV=kBT¼42, where kBis Boltzmann’s constant, and Tis temperature, and the reduced coupling strength q¼0:4. The ﬁgure is in accord with the predictions of the master equation formalism:9The initial saturated ""state reverses under the action of the ﬁeld and goes over into the demagnetized conﬁguration "# þ #" . This conﬁguration is metastable, and exists with occupation prob- ability n2¼1 over a ﬁnite interval of the reversing ﬁeld h. On further ﬁeld reversal the demagnetized state becomes unstable and gradually switches into the saturated state ##.I t is interesting to note that the transition ""!"# þ #" is very fast and virtually independent of the damping constant a, while the decay of the demagnetized state is slower, and strongly dependent on a. The fastest decay takes place here ata¼0:5, and the decay slows down with both increasingand decreasing ain accordance with the thermal decay theory.15 The hysteresis loop of Fig. 1is standard and presents no particular surprises. The situation changes, however, if the coupling strength is slightly decreased to q¼0:2, as shown in Fig. 2. In this case the domain of attraction of the demag- netized conﬁguration is reduced, and the magnetization loops exhibit a partially demagnetized metastable state, whichagain exists over a ﬁnite interval of the external ﬁeld. At large dissipative strength the magnitude of the nonzero meta- stable magnetization is quite small, but it increases appreci-ably at small values of a. We interpret this ﬁgure in terms of a reversal process in which the saturated state ""decays into the demagnetized state and is either trapped there or contin-ues immediately into the reverse saturated state ##. This pic- ture is supported by an analysis of the random sojourn time which the system spends in the demagnetized state "# þ #" . Forq¼0:4 Fig. 3shows the broad distribution of sojourn times associated with a thermally activated Markovian pro- cess. At q¼0:2, on the other hand, the sojourn times distri- bution has two peaks as is apparent form the plot of Fig. 4. In addition to the broad peak of overbarrier transitions there exists also a sharp, very narrow peak at very short sojourntimes. This peak corresponds to a process during which the system is not trapped in the demagnetized state at all, but FIG. 1. (Color online) The reduced mean magnetization mzalong the driv- ing ﬁeld (top), and the occupation probability n2of the demagnetized state "# þ #" (bottom) vs the external biasing ﬁeld htðÞ¼ hmaxcos 2pft. The reduced inverse temperature q¼42, the driving ﬁeld frequency f¼1MHz, the reduced coupling strength q¼0:4, and the dissipation constant a¼1;0:5;0:1;0:05;0:01, and 0 :005 as labeled. Inset (up): exploded view of mzdata for ﬁrst four values of a; the curves follow in the same order also in the bottom panel.07D339-2 Cimpoesu et al. J. Appl. Phys. 109, 07D339 (2011)proceeds directly, on a high energy trajectory, to the fully reversed conﬁguration. The trapping probability is high at large a, where strong energy dissipation prevails, but small at small a, where the system may be thermally activated to ahigh energy at switching, and then continue to execute almost deterministic motion over a signiﬁcant period of time. The partially demagnetized metastable states shown in Fig.2cannot be deduced from the free energy surface of the system, but are determined solely by the details of the dy- namics. With increasing coupling strength these states gradu- ally go over to fully demagnetized states, and they vanishaltogether in the limit q!0 where no intermediate metasta- ble state exists. ACKNOWLEDGMENTS This work was partially supported by Romanian PNII 12-093 HIFI, PNII-RP3 Grant No. 9/1.07.2009, and by NSF under Grant No. ECCS-0902086. We also acknowledge Tai- wan Grant No. NSC 98-2112-M-002-012-MY. A.S. alsoacknowledges the ﬁnancial help from National Taiwan University. 1J. L. Dormann, D. Fiorani, and E. Tronc, Adv. Chem. Phys. XCVIII , 283 (1997). 2J. M. Shaw, S. E. Russek, T. Thomson, M. J. Donahue, B. D. Terris, O.Hellwig, E. Dobisz, and M. L. Schneider, Phys. Rev. B 78, 024414 (2008). 3V. L. Safonov and H. N. Bertram, Phys. Rev. B 65, 172417 (2002). 4A. Lyberatos and R. W. Chantrell, J. Appl. Phys. 73, 6501 (1993). 5S. V. Titov, H. Kachkachi, Y. P. Kalmykov, and W. T. Coffey, Phys. Rev. B72, 134425 (2005). 6J. L. Garcia-Palacios and F. J. Lazaro, Phys. Rev. B 58, 14937 (1998). 7W. Chen, S. Zhang, and H. N. Bertram, J. Appl. Phys. 71, 5579 (1992). 8I. Klik and C. R. Chang, Phys. Rev. B 52, 3540 (1995). 9I. Klik, C. R. Chang, and J. S. Yang, J. Appl. Phys. 76, 6588 (1994). 10I. Klik and Y. D. Yao, J. Magn. Magn. Mat. 282, 131 (2004). 11T. L. Gilbert, Phys. Rev. 100, 1243 (1955);T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004). 12W. F. Brown, Jr., Phys. Rev. 130, 1677 (1963). 13D. V. Berkov and N. L. Gorn, J. Phys.: Condens. Matter 14, L281 (2002). 14M. d’Aquino, C. Serpico, G. Coppola, I. D. Mayergoyz, and G. Bertotti, J. Appl. Phys. 99, 08B905 (2006). 15I. Klik and Y. D. Yao, J. Magn. Magn. Mat. 290, 464 (2005). FIG. 2. (Color online) Same as Fig. 1, but with coupling strength q¼0:2. FIG. 3. (Color online) The sojourn times distribution in the demagnetized state"# þ #" . The reduced inverse temperature q¼42, the reduced cou- pling strength q¼0:4, and the dissipation constant a¼1;0:5;0:1;0:05; 0:01, and 0 :005 as labeled. FIG. 4. (Color online) Same as Fig. 3but with coupling strength q¼0:2.07D339-3 Cimpoesu et al. J. Appl. Phys. 109, 07D339 (2011)
1.4838655.pdf	Zero field high frequency oscillations in dual free layer spin torque oscillators P. M. Braganca, K. Pi, R. Zakai, J. R. Childress, and B. A. Gurney Citation: Applied Physics Letters 103, 232407 (2013); doi: 10.1063/1.4838655 View online: http://dx.doi.org/10.1063/1.4838655 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/103/23?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Dependence of the colored frequency noise in spin torque oscillators on current and magnetic field Appl. Phys. Lett. 104, 092405 (2014); 10.1063/1.4867257 Spin-torque nano-oscillator based on a synthetic antiferromagnet free layer and perpendicular to plane polarizer J. Appl. Phys. 113, 113908 (2013); 10.1063/1.4795160 Injection locking at zero field in two free layer spin-valves Appl. Phys. Lett. 102, 102413 (2013); 10.1063/1.4795597 Frequency shift keying in vortex-based spin torque oscillators J. Appl. Phys. 109, 083940 (2011); 10.1063/1.3581099 Spin transfer oscillators emitting microwave in zero applied magnetic field J. Appl. Phys. 101, 063916 (2007); 10.1063/1.2713373 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: 130.113.111.210 On: Fri, 19 Dec 2014 08:02:27Zero field high frequency oscillations in dual free layer spin torque oscillators P . M. Braganca,a)K. Pi,b)R. Zakai, J. R. Childress, and B. A. Gurney HGST, 3404 Yerba Buena Rd., San Jose, California 95135, USA (Received 25 September 2013; accepted 15 November 2013; published online 3 December 2013) We observe microwave oscillations in relatively simple spin valve spin torque oscillators consisting of two in-plane free layers without spin polarizing layers. These devices exhibit two distinct modes which can reach frequencies >25 GHz in the absence of an applied magnetic ﬁeld. Macrospin simulations identify these two modes as optical and acoustic modes excited by thecoupling of the two layers through dipole ﬁeld and spin torque effects. These results demonstrate the potential of this system as a large output power, ultrahigh frequency signal generator that can operate without magnetic ﬁeld. VC2013 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4838655 ] Generating high frequency electrical signals at frequen- cies approaching 50 GHz and above has strong interest due to applications in the ﬁelds of communications and radar.1In particular, structures using the spin angular momentum trans-fer effect 2,3known as spin torque oscillators (STOs) have been investigated4–8in recent years as potential high fre- quency sources due to a combination of compact device size(<100 nm), frequencies up to tens of GHz, and frequency tun- ability through the application of magnetic ﬁelds and/or elec- tric currents. Here, magnetization oscillations induced by spintransfer effects in one ferromagnetic (FM) layer of a spin valve device generate an RF voltage due to the magnetoresistance (MR) between the ferromagnetic layers. Previous studies 1,5,9,10 have demonstrated STO frequencies between 20 and 50 GHZ can be achieved using applied magnetic ﬁelds on the order of 1 T. However, such large ﬁelds are impractical for potentialapplications, especially mobile ones such as cell phones and radios. An alternative direction for STO research has been to excite STOs at very low or zero applied magnetic ﬁelds usinga variety of methods, such as perpendicular anisotropy, 11–14 spin torque excitation of vortex core oscillations,15–17oscilla- tions in synthetic antiferromagnet (SAF) systems,18–21or “wavy torques.”22While these approaches have been success- ful in generating auto-oscilla tions in STOs, they have been limited to frequencies below 13 GHz, as higher frequenciesare more difﬁcult to generate without substantial external ﬁeld. Thus, an alternative method of combining low ﬁeld STO excitations with high ( >20 GHz) oscillation frequencies would be of signiﬁcant interest. In this letter, we discuss high frequency signals meas- ured at small applied magnetic ﬁelds in STOs with a rela-tively simple design consisting of two free ferromagnetic layers separated by a relatively thin (4 nm) Cu spacer. Here, unlike the Fe/Cr/Fe system in Ref. 20, there is no signiﬁcant interlayer exchange (RKKY) coupling between free layers, with the dominant interactions between the free layers con- sisting of dipolar ﬁeld coupling and spin torque. We observetwo distinct modes depending on applied magnetic ﬁeldamplitude, which can reach frequencies greater than 25 GHz. Macrospin modeling of this system identiﬁes these two modes as acoustic and optical modes of the system where ei- ther spin torque or dipolar coupling dominate. Our devices were fabricated from sputter-deposited mul- tilayer stacks consisting of 2 nm CoFe (FM1)/4 nm Cu/4 nm CoFe (FM2). We chose to pattern these ﬁlms into 30 /C2 60 nm 2elliptical nanopillars to induce a small amount of shape anisotropy for the free layers, which was done using ebeam lithography and ion milling as described elsewhere.23 Top and bottom leads were patterned into a coplanar wave- guide conﬁguration with overlap of top and bottom leads minimized to limit capacitive losses. Figure 1shows a repre- sentative dV/dI vs. applied magnetic ﬁeld transfer curve for one such device at ﬁelds oriented both close to the easy axis and along the hard axis of the STO. For ﬁelds along the hardaxis as shown in Fig. 1(b), we see a gradual reduction in dV/dI with increasing applied magnetic ﬁeld H a, characteris- tic of scissoring of the two FM layers towards Ha.24 In Fig. 1(a), we orient the applied ﬁeld at a small angle (/C2430/C14) away from the long axis of the ellipse, as it results in the sharpest switching seen in these devices, nominally due FIG. 1. Easy axis (a) and hard axis (b) transfer curves for dual free layer spin torque oscillator devices. Insets show orientation of applied magnetic ﬁeld with respect to long axis of devices.a)Email: patrick.braganca@hgst.com b)Present address: Headway Technologies, Inc., 628 South Hillview DriveMilpitas, California 95035, USA. 0003-6951/2013/103(23)/232407/4/$30.00 VC2013 AIP Publishing LLC 103, 232407-1APPLIED PHYSICS LETTERS 103, 232407 (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: 130.113.111.210 On: Fri, 19 Dec 2014 08:02:27to device irregularities and slight canting of the ellipse during lithography and patterning. Due to the lack of a pinned refer- ence layer in these samples, determining values for the dipo- lar and coercive ﬁelds of the respective layers from the easyaxis transfer curve is non-trivial. To estimate these ﬁelds, we have made the assumption that the strength of the dipolar ﬁeld is proportional to the thickness of the ferromagneticlayer, such that the thicker layer FM2 exerts twice the dipole ﬁeld on FM1 as vice versa. We also make the simplifying assumption that the coercive ﬁelds of FM1 and FM2 are approximately the same despite the nominal volume differen- ces, which does not qualitatively affect the modeling resultspresented below. Using these assumptions, we estimate the dipole ﬁelds to be H dip,FM 1¼300 Oe, Hdip,FM 2¼600 Oe, and coercive ﬁelds Hc,FM 1¼Hc,FM 2¼150 Oe. Five other samples measured displayed similar easy axis transfer curves with dipolar and coercive ﬁeld values within 6100–150 Oe of these values. Figure 2shows the high frequency response of two rep- resentative devices biased with DC current, /C00.8 mA for the device measured in Fig. 2(a) and/C00.7 mA for the device in Fig.2(b), where negative current is deﬁned as electrons ﬂow- ing from FM2 to FM1. The individual spectra in these phase diagrams were obtained by sweeping Hafrom þ500 to /C0500 Oe with a ﬁeld angle 30/C14off the long axis of the devi- ces, just as was done for the quasi-static measurement. For all 6 devices measured in this experiment, including the twoshown in Fig. 2, we observe two distinct modes at frequen- cies greater than 20 GHz, with one (mode 1) excited in the approximate ﬁeld range /C0300 Oe /C20H a/C20300 Oe and the sec- ond (mode 2) occurring for \| Ha\|/C21300 Oe. The oscillations disappear very quickly beyond 6500 Oe, where the layers become oriented parallel in the quasi-static case. Here, themode around zero ﬁeld surprisingly has a higher frequency than the mode at larger ﬁelds, as illustrated in Fig. 2(c) where we have plotted cuts through the phase diagram inFig. 2(a) at zero ﬁeld and /C0400 Oe. This particular device shows a spectral peak at 33.3 GHz with a full width at half maximum Df¼180 MHz and integrated power P¼72.4 pW at zero ﬁeld and a peak at 26.6 GHz with Df¼140 MHz and P¼128.5 pW for the higher ﬁeld mode. Similar linewidths and powers were found for these two modes in all the sam- ples measured. We note here that the measured signalpowers are lower than the actual output of our STOs due to the fairly high capacitance of 1.1 pF between the top leads and the substrate, leading to substantial capacitive loss. The implications of this loss will be discussed below. To understand the origins of the high frequency modes seen in these STOs, we used ﬁnite temperature macrospin sim- ulations to model a dual free layer spin valve in a time interval of 200 ns. These simulations solve the Landau-Lifshitz-Gilbert (LLG) equation at ﬁnite temperature T ¼300 K with a Slonczewski spin-torque term 2,25included to account for spin torque interactions between the two layers. The ferromagneticlayers have saturation magnetization M s¼1680 emu/cm3, Gilbert damping parameter a¼0.014,26thicknesses tFM1¼2n m a n d tFM2¼4 nm, a value of K¼2 for the torque asymmetry parameter27and spin polarization P¼0.37. The elements are assumed to have a 30 /C260 nm2elliptical cross section resulting in a small shape anisotropy ﬁeld Hc¼150 Oe and dipole ﬁelds Hdip,FM 1¼300 Oe, Hdip,FM 2¼600 Oe, extracted from the easy-axis transfer curves as discussed above. Here, it is assumed the dipole ﬁeld magnitude staysconstant and points in a direction opposite to the magnetiza- tion of the generating layer at any given moment of time. Thermal effects were modeled on both ferromagnetic layersusing a randomly ﬂuctuating Langevin ﬁeld H thwith x,y,a n d zcomponents drawn from a Gaussian distribution of zero mean and standard deviationﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2akBT=cMsVDtp ,28,29where kB is Boltzmann’s constant, cis the gyromagnetic ratio, Vis the volume of the free layer, and Dt¼2 ps is the time step used in solving the LLG equation. FIG. 2. High frequency phase diagram for STO device 1 (a) and device 2 (b) depicting the two ﬁeld regimes corre- sponding to optical (mode 1) and acoustic (mode 2) oscillation modes for the two ferromagnets. (c) Power spectral density for the oscillator in(a) at 0 and /C0400 Oe, showing well deﬁned spectral peaks with large output powers at different ultrahigh frequen- cies. The integrated powers of the spec- tral peaks are 72.4 pW at Ha ¼0O e and 128.5 pW at Ha ¼/C0400 Oe, and we observe a hop of /C246.5 GHz between modes.232407-2 Braganca et al. Appl. Phys. Lett. 103, 232407 (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: 130.113.111.210 On: Fri, 19 Dec 2014 08:02:27Simulations were run with a current bias of /C00.8 mA for a range of Habetween 500 and /C0500 Oe oriented at 30/C14off the long axis of the ferromagnets, similar to the conditions for the device in Fig. 2(a). The simulations show two distinct modes, similar to the experimental results, with a higher fre- quency mode 1 at /C2428 GHz around zero ﬁeld which hops to a lower frequency mode 2 at /C2418 GHz for \| Ha\|/C21/C24250 Oe. To understand the origin of these distinct modes, we have plotted the trajectories of the two free layers for modes 1 (Ha¼/C050 Oe) and 2 ( Ha¼/C0400 Oe) in Figs. 3(a) and3(b), respectively. At low Ha, the mode excited (Fig. 3(a)) involves the two layers oscillating in a scissor-like motion. This isbetter illustrated in Fig. 3(c), where we plot the y component of magnetization for both layers over a 500 ps time span. We clearly see that the layers oscillate approximately in-phase,reaching the positive and negative maxima of the orbit along the y-axis at the same time, such that that they alternate between an anti-parallel orientation and one in which theyare oriented /C24120 /C14with respect to one another. In the ﬁeld regime associated with mode 2, we see a very different pre- cessional orbit (Fig. 3(b)), where the two layers have ﬂopped orientation such that the thicker FM2 layer points in the direction of Ha. We ﬁnd by plotting the y component of mag- netization (Fig. 3(d)) that the layers now oscillate roughly out of phase in an attempt to keep a 180/C14orientation betweenthem although a MR signal is seen as their precessional amplitudes are quite different due to the differences in criti- cal currents, leading to a more complex precessional motion of both layers with multiple harmonic components.30To fur- ther correlate the simulations to our experimental results, we have fast Fourier transformed the simulated normalized mag- netoresistance,31RðtÞ¼ð 1/C0^m1/C1^m2Þ=½ðK2þ1ÞþðK2/C01Þ ð^m1/C1^m2Þ/C138, for both modes seen. The results are shown in Figs. 3(e) and3(f) for/C050 Oe and /C0400 Oe, respectively, and show spectral peaks at frequencies similar to those seen experimentally. There are lower frequency modes seen for the/C0400 Oe case not seen experimentally, which may be suppressed by sample nonuniformities and/or micromagnetic effects not considered in the modeling. From this analysis, we have determined that the spin tor- que coupling and dipole ﬁeld interactions between the two ferromagnets results in optical (in-phase scissoring) and acoustic (out of phase) normal modes. At small Ha, the effec- tive ﬁeld (applied and dipole ﬁelds) acting on the thinner fer- romagnetic layer is larger than on the thicker ferromagnetic layer such that the critical currents of the two layers becomesimilar despite the differences in volume. Thus, spin torque effects dominate the dynamics of both layers, resulting in a scissor motion which generates a larger oscillation fre-quency since the MR signal frequency will be double the FIG. 3. Macrospin simulations of opti- cal (a) and acoustic (b) modes for dual free layer STO devices. Simulations show optical mode is a scissoring modewhere the layers oscillate in phase reach- ing maximum y value concurrently (c) while acoustic mode is an out of phase mode (d) where ferromagnets try to remain oriented at 180 /C14with respect to each other during their precessional motion. Fast Fourier transform of simu-lated normalized magnetoresistance at (e)/C050 Oe and (f) /C0400 Oe shows fre- quency components similar to experi- mental results.232407-3 Braganca et al. Appl. Phys. Lett. 103, 232407 (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: 130.113.111.210 On: Fri, 19 Dec 2014 08:02:27ferromagnetic resonance (FMR) frequency of a single layer. AsHaincreases for either ﬁeld polarity, the critical current Ic of the thicker FM2 layer increases, since FM2 tends to align itself with applied ﬁeld. In this case, spin torque mainlydrives the thinner ferromagnet and the thicker ferromagnet moves due to the dipole and applied ﬁelds acting upon it, which keeps the two ferromagnetic layers antiparallel withrespect to one another for a large portion of the dynamic motion. These phenomena explain the origins of the two modes seen experimentally. The signal amplitude for small H acalculated from the macrospin simulations corresponds to a MR amplitude ofroughly 40% of the total magnetoresistance DR. Assuming DR¼1Xas determined from the easy axis transfer curve shown in Fig. 1(a), we ﬁnd the estimated integrated signal power should be approximately 581 pW. We have calculated capacitive loss due to a 1.1 pF capacitance between the top leads and substrate to be approximately /C07.5 dB at 33 GHz, leading to a estimated measured integrated power of 103 pW, within reasonable agreement with the measured values of 72.4 pW for the device in Fig. 2(a) and 103.3 pW for the device shown in Fig. 2(b). If the capacitance between top leads and the substrate was decreased by a factor of 10, say by increasing the thickness of thermal oxide grown on a Sisubstrate before depositing the STO stack, the capacitive loss would decrease to /C01.4 dB at a frequency of 33 GHz, leading to an increase in power of approximately a factor of4. This analysis conﬁrms that not only does this STO design result in ultrahigh oscillation frequencies, but also in large output powers due to the oscillations incorporating a largefraction of DR. The more complex structure of the acoustic mode oscillations at higher ﬁelds makes calculations for measured power more complicated as the measured power isdistributed between more than one mode. In conclusion, we have studied simple dual free layer spin valves which exhibit two distinct oscillatory modesdepending on the strength of the applied magnetic ﬁelds. These oscillations correspond to modes brought on by sepa- rate regimes of spin torque dominated (low ﬁeld) and dipoleﬁeld dominated (larger ﬁeld) coupling between the two fer- romagnetic layers. Simulations have shown that small changes in FM layer anisotropy and dipolar coupling on theorder of 100–200 Oe can result in variations of 3–6 GHz in oscillation frequency, which explains the device-to-device variation we have seen. We have analyzed capacitive lossesfor our particular substrates and identiﬁed that a more opti- mized substrate could result in much larger output powers even for high frequencies /C2430 GHz, which would be extremely advantageous for applications in communications and radar. The output power could be further increased for high frequencies by using a low resistance tunnel barrier asopposed to a metallic spacer since the parallel interlayer cou- pling usually induced by ultrathin MgO barriers will be more than compensated by the strong dipolar coupling betweenthe ferromagnetic layers, resulting in much higher DRand output power without signiﬁcantly affecting the excitation of the modes described above.The authors would like to thank J. C. Sankey for provid- ing us with the macrospin simulation used in this work. We would also like to thank H.-W. Tseng for useful conversa- tions and discussion of this work. 1S. Bonetti, P. Muduli, F. Mancoff, and J. Akerman, Appl. Phys. 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1.4915093.pdf	Fabrication of MnAl thin films with perpendicular anisotropy on Si substrates Efrem Y . Huanga)and Mark H. Kryder Data Storage Systems Center and Department of Electrical and Computer Engineering, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, Pennsylvania 15213, USA (Presented 7 November 2014; received 21 September 2014; accepted 6 November 2014; published online 16 March 2015) For the ﬁrst time, perpendicularly magnetized L10-ordered MnAl thin ﬁlms were demonstrated using a MgO seed layer on Si substrates, which is critical to making spintronic devices. Fabricationconditions were selected by systematically varying sputtering parameters (ﬁlm thickness, DC sputtering power, in situ substrate temperature, and post-annealing temperature) and investigating structural and magnetic properties. Strong perpendicular magnetic anisotropy with coercivity H cof 8 kOe, Kuof over 6.5 /C2106erg/cm3, saturation magnetization Msof 300 emu/cm3, and out-of-plane squareness Mr/Msof 0.8 were achieved. These MnAl ﬁlm properties were obtained via DC magne- tron sputtering at 530/C14C, followed by 350/C14C annealing under a 4 kOe magnetic ﬁeld oriented perpendicular to the ﬁlm plane. VC2015 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4915093 ] I. INTRODUCTION Perpendicularly magnetized ferromagnetic thin ﬁlms have applications in permanent magnets, hard disk drives (HDDs), nonvolatile memory tec hnologies, spintronic devi- ces, etc. With a predicted high perpendicular magnetic ani- sotropy (PMA) of 1.5 /C2107erg/cm3, magnetization of 2.37 lB/Mn, and high magnetic energy product of 12.6 MGOe,1as well as being rare-earth and precious metal- free, L10-ordered s-phase MnAl has rightfully attracted much attention over the years.2–13For spin-transfer-torque random access memory (STT-RAM), s-MnAl ﬁlms with high anisotropy Kuof over 107erg/cm3and low Gilbert damping constant aof 0.0067are particularly attractive for simultaneously achieving high thermal stability ( KuV/KbT >60) and low critical switching current density (Jc<5/C2105A/cm2),14,15both necessary for the realization of multi-level cell (MLC) STT-RAM devices.16 Bulk s-MnAl has the L10-ordered CuAu-type structure with alternating Mn and Al monolayers in the c-axis direc- tion and is formed martensitically as a metastable phase;17,18however, in thin ﬁlms s- M n A li ss t a b l ea ta m b i - ent temperature. The formation of s-MnAl in thin ﬁlms is highly sensitive to deposition conditions and composition, with Hosoda et al. reporting an optimal target composition of Mn 48Al52.7Nevertheless, while there has been consider- able work developing s-MnAl ﬁlms on substrates, such as GaAs (001),2–6MgO (001),7–9and glass,10,11there have been no reports of high-PMA s-MnAl ﬁlms on Si sub- strates.13In this study, we report on the fabrication and characterization of sputter-deposited s-MnAl ﬁlms above MgO (001) seed layers on Si substrates. Structural andmagnetic properties were optimized by varying MnAl ﬁlm thickness, DC sputtering power, in situ substrate tempera- ture, and post-annealing temperature.II. EXPERIMENTAL METHOD The substrates used in this study were natively oxidized 1 in. (100) Si wafers. All ﬁlms we re deposited in a high-vacuum Leybold-Heraeus Z-400 sputte ring system at base pressures below 3 /C210/C07Torr. Film stacks followed the structure Si sub- strate/MgO (20 nm)/MnAl (10–50 nm)/Ta (5 nm). First, a 20 nm MgO seed layer was RF sputtered (0.015 nm/s ﬁlm growth rate, 10 mTorr Ar gas pre ssure) onto a Si substrate at room temperature. The substrate was then heated in situ to vari- ous temperatures (23–570/C14C), which helped enhance the MgO (001) texture. Next, a 10–50 nm MnAl ﬁlm was DC magnetronsputtered from a vacuum hot-pressed Mn 48Al52target onto the MgO while it was held at that sam e temperature at a deposition rate of 0.31–0.78 nm/s with an Ar gas pressure of 4 mTorr. Thesubstrate was subsequently allowed to cool to room tempera- ture, and a 5 nm Ta capping layer was DC magnetron sputtered at a deposition rate of 0.083 nm/s using Ar gas pressure of 4mTorr. Lastly, the sample wa s annealed in a Micro Magnetics SpinTherm-1000 magnetic thermal annealing system with a base pressure under 5 /C210 /C07T o r ra n daﬁ x e d4k O eﬁ e l dp e r - pendicular to the ﬁlm plane at various temperatures (250–350/C14C). Calibrations for in situ substrate temperatures were performed using a Type K chr omel–alumel thermocouple. Texture, microstructure, and magnetic properties of the ﬁlm stacks were investigated using x -ray diffraction (XRD), trans- mission electron microscope (TEM), alternating gradient ﬁeld magnetometer (AGFM), and phys ical property measurement system (PPMS). Thickness-d ependent order parameters Swere calculated for the MnAl ﬁlms from the integrated peak intensity ratios I001/I002extracted from out-of-plane h/2hXRD pat- terns.19–21Magnetic anisotropy constants were determined according to Ku¼HkMs/2, where Hk¼Hsþ4pMsis the anisot- ropy ﬁeld, Hsis the hard-axis (in-plane) saturation ﬁeld, and Ms is the saturation magnetization. III. RESULTS The effects of in situ sputtering temperature ( Ts) on for- mation of s-MnAl can be seen from the h/2hXRD patternsa)Author to whom correspondence should be addressed. Electronic mail: efrem@cmu.edu. 0021-8979/2015/117(17)/17E314/4/$30.00 VC2015 AIP Publishing LLC 117, 17E314-1JOURNAL OF APPLIED PHYSICS 117, 17E314 (2015) shown in Fig. 1. The 30 nm MnAl ﬁlms were sputtered using DC power of 40 W and annealed at Ta¼350/C14C. The MnAl (001) and (002) peaks were measured to be around 24.8/C14and 50.9/C14, respectively. The peaks at 33.0/C14, 38.2/C14, and 61.7/C14 belong to Si, due to the alignment of substrates during the scan. At sputtering temperatures below 350/C14C, no signiﬁcant s-MnAl was observed in the ﬁlms. As Tsincreased above 350/C14C,s-MnAl began to form, reaching a maximum order parameter Sof 0.98 at Ts¼410/C14C. This high degree of ordering ( S>0.94, rocking curve FWHM angle /C245/C14)w a s maintained for Tsup to 530/C14C, beyond which rapid degrada- tion of the s-phase took place, with Ts¼570/C14C resulting in very little s-MnAl. Instead, the nonmagnetic e-phase became dominant. The c-axis lattice constants we re calculated from the out-of-plane 2 h/C2424.8/C14s-MnAl (001) peaks as 3.58–3.59 A ˚for all ﬁlms with signiﬁcant s-MnAl. In-plane XRD scans revealed a-axis lattice constants of 3.92–3.95 A ˚for s-MnAl and 4.19–4.21 A ˚for MgO. Unlike previous studies,4,7these values are very close to the reported value of c¼3.57 A ˚and a¼3.92 A ˚for bulk s-MnAl.17The epitaxial growth relation- ship between MgO and s-MnAl is shown in Fig. 2: MgO [100] (001)//MnAl [100] (001). The dependence of coercivity ( Hc), squareness ( Mr/Ms, where Mris the out-of-plane remanent magnetization with no applied ﬁeld), saturation magnetization ( Ms), and anisot- ropy constant ( Ku)o n Tswere measured and are shown in Fig. 3. The 30 nm MnAl ﬁlms were sputtered using DC power of 40 W and annealed at Ta¼350/C14C. From these data,Ts¼530/C14C appeared to produce MnAl ﬁlms with the highest PMA. This substrate temperature was therefore used for fur-ther studies. The thickness dependence of the magnetic properties and microstructure of MnAl ﬁlms was also examined. Fromthe out-of-plane h/2hXRD patterns shown in Fig. 4, one can see that ﬁlms under 10 nm produced poor L1 0-ordering, and increasing ﬁlm thickness to 50 nm produced more nonmag-netic e-phase rather than s-phase. The ﬁlm thickness was consequently selected to be 30 nm. The effects of DC sputtering power on magnetic proper- ties were studied and are shown in Fig. 5. Films produced using DC sputtering power of 30–40 W demonstrated thehighest perpendicular coercivity and squareness. Additionally,ﬁlms deposited with DC power less than 30 W contained sig-niﬁcant e-phase, suggesting that low sputtering powers do not impart sufﬁcient energy upon the Mn and Al atoms forthem to order properly. Ultimately, 40 W, which depositedMnAl ﬁlm at a rate of 0.63 nm/s, was chosen for furtherstudy as it produced MnAl ﬁlms with high H cand moderately high Ms. The impact of magnetic annealing temperature ( Ta) was investigated and the results are plotted in Fig. 6. Error bars reﬂect possible range of values adjusting for thickness ofinterdiffusion layer between MnAl and Ta cap (maximum of5 nm measured for T a¼350/C14C, shown in TEM image below). Out-of-plane magnetic properties were improvedwith increasing T a, although Msdecreased as Tawas increased to 300/C14C. Saturation magnetization Mswas FIG. 1. Out-of-plane h/2hXRD patterns of 30 nm MnAl ﬁlms deposited at various in situ substrate temperatures, 40 W DC power, and Ta¼350/C14C. FIG. 2. Schematic of epitaxial growth relationship at MgO/ s-MnAl interface. FIG. 3. Sputtering temperature de- pendence of (a) out-of-plane coercivity Hcand squareness Mr/Msand (b) satu- ration magnetization Msand anisot- ropy constant Kuof 30 nm MnAl ﬁlms deposited using 40 W DC power and annealed at Ta¼350/C14C.17E314-2 E. Y . Huang and M. H. Kryder J. Appl. Phys. 117, 17E314 (2015)partially recovered at Ta¼350/C14C, which was selected as the ﬁnal annealing temperature. The ﬁnal set of deposition conditions was thus deter- mined to be 40 W DC sputtering power (which produced adeposition rate of 0.63 nm/s), 30 nm MnAl ﬁlm thickness, in situ sputtering temperature of T s¼530/C14C, and magnetic annealing temperature of Ta¼350/C14C. Using these parame- ters, a MnAl ﬁlm was fabricated and characterized. As plot-ted in Fig. 7(a), the ﬁlm demonstrated high PMA with H cof 8k O e , Kuof 6.5 /C2106erg/cm3,Msof 300 emu/cm3,a n d out-of-plane squareness Mr/Msof 0.8, with a measured ﬁlm composition of Mn 54.0Al46.0 and order parameter of S¼0.94. The TEM cross-sectional image in Fig. 7(b) shows signiﬁcant clumping of MnAl above the MgO (001)seed layer, suggesting the need for underlayers with reduced lattice mismatch and higher surface binding energy to improve wetting of the deposition surface. 22The TEMimage also indicates a 5 nm region of diffusion between the MnAl and Ta cap, supporting the idea of a magnetically“dead” layer as proposed by Cui et al . 8This 5 nm region was subtracted from the effective ﬁlm thickness when cal- culating the order parameter, saturation magnetization, andanisotropy constant. It was observed throughout the investigation that improvements in out-of-plane coercivity and squareness were invariably accompanied by similar increases in in- plane coercivity and squareness. Unlike Cui’s work on GaAs(001) substrates, 6no XRD peak was ever observed near 2h¼47/C14, which was proposed as corresponding to a partially in-plane MnAl (110) orientation. Instead, interface or diffu- sion effects at higher temperatures may be the main contribu-tors to in-plane magnetic behavior in our ﬁlms. This study showed that there are narrow regions of dep- osition conditions for producing s-MnAl ﬁlms with high PMA, and this fact is qualitatively in agreement with recent studies done by other groups: the successful formation of s- MnAl thin ﬁlms is highly sensitive to deposition and post- annealing conditions. However, whereas Hosoda et al. 7and Nieet al.4found the optimized sputtering temperatures to be Ts¼200/C14C and 350/C14C on MgO (001) and GaAs (001) sub- strates, respectively, we observed no s-phase MnAl in ﬁlms deposited at those temperatures on Si (100) substrates. Instead, much higher temperatures were required, whichlikely contributed to the high ﬁlm roughness. Therefore, we believe further work is necessary to develop underlayers with increased surface binding energy that would improvewetting by the MnAl, thereby enabling the use of lower dep- osition temperatures and promoting smooth, continuous growth of L1 0-ordered s-MnAl thin ﬁlms with improved PMA on Si substrates. FIG. 4. Out-of-plane h/2hXRD patterns of MnAl ﬁlms with various thick- nesses deposited using 40 W DC power, Ts¼530/C14C, and Ta¼350/C14C. FIG. 5. DC sputtering power depend- ence of (a) out-of-plane coercivity Hc and squareness Mr/Msand (b) saturation magnetization Msand anisotropy con- stant Kuof 30 nm MnAl ﬁlms deposited atTs¼530/C14C, and Ta¼350/C14C. FIG. 6. Magnetic annealing tempera- ture dependence of (a) out-of-plane coercivity Hcand squareness Mr/Ms and (b) saturation magnetization Ms and anisotropy constant Ku(error bars indicate possible range of values adjusting for thickness of MnAl/Ta interdiffusion layer). 10 nm MnAl ﬁlms were deposited using 40 W DC power and Ts¼530/C14C.17E314-3 E. Y . Huang and M. H. Kryder J. Appl. Phys. 117, 17E314 (2015)IV. CONCLUSION We investigated the structural and magnetic properties ofL10-ordered s-MnAl thin ﬁlms sputtered on Si substrates by systematically varying MnAl ﬁlm thickness, DC sputter-ing power, in situ substrate temperature, and post-annealing temperature. The 30 nm MnAl ﬁlm fabricated using 40 W DC sputtering power, T s¼530/C14C, and Ta¼350/C14C exhibited a high degree of ordering ( S¼0.94) and large PMA with out-of-plane Hcof 8 kOe, Kuof 6.5 /C2106erg/cm3,Msof 300 emu/cm3, and Mr/Msof 0.8. For the ﬁrst time, the excel- lent magnetic properties of s-MnAl thin ﬁlms were thus demonstrated on Si substrates, opening the possibility ofMnAl-based thin ﬁlms being used for perpendicular mag- netic tunnel junctions (pMTJs), particularly for STT-RAM applications.ACKNOWLEDGMENTS The authors would like to thank the Data Storage Systems Center at Carnegie Mellon University for support of this work. They would also like to thank Hoan Ho, VigneshSundar, the CMU Nanofabrication Facility technical staff(Chris Bowman, Carsen Kline, and James Rosvanis), and theCMU MSE Department staff (Tom Nuhfer, Jason Wolf, andAdam Wise) for the insightful discussions and invaluableassistance provided throughout this project. 1J. H. Park, Y. K. Hong, S. Bae, J. J. 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Phys. 52, 063003 (2013). 10C. Y. Duan, X. P. Qiu, B. Ma, Z. Z. Zhang, and Q. Y. Jin, Mater. Sci. Eng., B 162, 185 (2009). 11M. A. Angadi and V. Thanigaimani, J. Mater. Sci. Lett. 11, 1213 (1992). 12G. A. Fischer and M. L. Rudee, J. Magn. Magn. Mater. 213, 335 (2000). 13L. J. Zhu, S. H. Nie, and J. H. Zhao, Chin. Phys. B 22, 118505 (2013). 14J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). 15T. Taniguchi and H. Imamura, Phys. Rev. B 78, 224421 (2008). 16M. H. Kryder and C. S. Kim, IEEE Trans. Magn. 45, 3406 (2009). 17A. J. J. Koch, P. Hokkeling, M. G. v. d. Steeg, and K. J. de Vos, J. Appl. Phys. 31, S75 (1960). 18A. J. McAlister and J. L. Murray, J. Phase Equilib. 8, 438 (1987). 19A. Cebollada, D. Weller, J. Sticht, G. R. Harp, R. F. C. Farrow, R. F. Marks, R. Savoy, and J. C. Scott, Phys. Rev. B 50, 3419 (1994). 20S. D. Granz and M. H. Kryder, J. Magn. Magn. Mater. 324, 287 (2012). 21B. D. Cullity, Elements of X-Ray Diffraction (Addison-Wesley, Massachusetts, 1956). 22J. Venables, Introduction to Surface and Thin Film Processes (Cambridge University Press, Cambridge, New York, UK, 2000). FIG. 7. (a) Out-of-plane/in-plane magnetic hysteresis loops and (b) 50 000 magniﬁcation TEM cross-sectional image of the Si substrate/MgO (20 nm)/ MnAl (30)/Ta (5) ﬁlm stack deposited using 40 W DC power, Ts¼530/C14C, andTa¼350/C14C.17E314-4 E. Y . Huang and M. H. Kryder J. Appl. Phys. 117, 17E314 (2015)Journal of Applied Physics is copyrighted by AIP Publishing LLC (AIP). Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. For more information, see http://publishing.aip.org/authors/rights- and- permissions.
1.3594661.pdf	Frequency dependence of spin pumping in Pt/Y3Fe5O12 film Kazuya Harii, Toshu An, Yosuke Kajiwara, Kazuya Ando, Hiroyasu Nakayama, Tatsuro Yoshino, and Eiji Saitoh Citation: Journal of Applied Physics 109, 116105 (2011); doi: 10.1063/1.3594661 View online: http://dx.doi.org/10.1063/1.3594661 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 Joule heating-induced coexisted spin Seebeck effect and spin Hall magnetoresistance in the platinum/Y3Fe5O12 structure Appl. Phys. Lett. 105, 182403 (2014); 10.1063/1.4901101 Spin current generation from sputtered Y3Fe5O12 films J. Appl. Phys. 116, 153902 (2014); 10.1063/1.4898161 Interface-dependent magnetotransport properties for thin Pt films on ferrimagnetic Y3Fe5O12 Appl. Phys. Lett. 104, 242406 (2014); 10.1063/1.4883898 Acoustic spin pumping: Direct generation of spin currents from sound waves in Pt/Y3Fe5O12 hybrid structures J. Appl. Phys. 111, 053903 (2012); 10.1063/1.3688332 Surface-acoustic-wave-driven spin pumping in Y3Fe5O12/Pt hybrid structure Appl. Phys. Lett. 99, 212501 (2011); 10.1063/1.3662032 [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: Fri, 12 Dec 2014 16:14:05Frequency dependence of spin pumping in Pt/Y 3Fe5O12film Kazuya Harii,1,2,3, a)Toshu An,2,3Y osuke Kajiwara,2Kazuya Ando,2,3Hiroyasu Nakayama,2,3 Tatsuro Y oshino,2and Eiji Saitoh2,3,4,5 1Department of Applied Physics and Physico-Informatics, Keio University, Yokohama 223-8522, Japan 2Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 3CREST, Japan Science and Technology Agency, Sanbancho, Tokyo 102-0075, Japan 4The Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan 5PRESTO, Japan Science and Technology Agency, Sanbancho, Tokyo 102-0075, Japan (Received 2 October 2010; accepted 11 April 2011; published online 15 June 2011) The frequency dependence of magnetization precession in spin pumping has been investigated using the inverse spin-Hall effect in a Pt/Y 3Fe5O12bilayer ﬁlm. We found that the magnitude of a spin current generated by the spin pumping depends weakly on the applied microwave frequency.This weak dependence, which is attributed to the compensation between the frequency change in the spin-pumping cycle and the dynamic magnetic susceptibility, is favorable for making a spin-current-driven microwave demodulator. This behavior is consistent with a model calculationbased on the Landau-Lifshitz-Gilbert equation combined with the spin mixing. VC2011 American Institute of Physics . [doi: 10.1063/1.3594661 ] Spin pumping, the generation of spin currents from a mag- netization precession excited b y a microwave, has attracted much attention recently.1–15In a ferromagnetic/paramagnetic ﬁlm, the spin pumping driven by a magnetization precessioninjects a spin current into the par amagnetic layer. This injected spin current is converted into a dc electric voltage using the inverse spin-Hall effect (ISHE) in the paramagnetic layer. Thiseffect can be used as amplitude demodulation in an amplitude- modulation signal transmission using microwave demodula- tion, because the spin pumping generates a rectiﬁed electricvoltage that is proportional to the microwave amplitude, 5only when a microwave frequency ful ﬁlls the ferromagnetic reso- nance condition. Here, in this a mplitude demodulation method, the frequency tuning is achieved by controlling the strength of an external magnetic ﬁeld. The spin pumping originates from dynamical coupling between magnetization and conduction-electron spins, and thus higher-frequency operation is basically favorable. Because a spin current is emitted to an attached paramag-netic metal in each cycle of the magnetization precession, the spin current generated by the spin pumping is expected to be proportional to the precession frequency f FMR: js/fFMR.1Here we show that, in an operation of the spin pumping in a magnetic ﬁlm, the generated spin current varies rather moderately with fFMR. Figures 1(a)and1(b)show a schematic illustration of the sample used in this study. The sample is a paramagnetic Pt/ ferrimagnetic insulator Y 3Fe5O12(111) (YIG) bilayer ﬁlm comprising a 10-nm-thick Pt layer and a 2.4- lm-thick single- crystal YIG layer. The surface of the YIG layer is of a 1.3 mm/C23.5 mm rectangular shape. The YIG ﬁlm was grown on a Gd 3Ga5O12(111) single-crystal substrate via liquid phase epitaxy. The Pt layer was fabricated on the YIG layer via ion-beam sputtering. Two electrodes are attached to theends of the Pt layer. The sample system is placed on a microstrip-microwave guide. During the measurement, a 20-mW-excitation microwave with a frequency fgenerated by a vector-network analyzer was introduced to the micro-strip waveguide, and an external magnetic ﬁeld Halong the ﬁlm plane was applied perpendicular to the direction across the electrodes (see Fig. 1(b)). The magnetization precession excited by the applied microwave injects a dc pure spin FIG. 1. (Color online) (a) A schematic illustration of the spin pumping and inverse spin-Hall effect in the Pt/Y 3Fe5O12ﬁlm.MðtÞis the magnetization in the Y 3Fe5O12layer. jsdenotes the spatial direction of the generated spin current. rdenotes the spin polarization carried by the spin current. (b) A schematic illustration of the Pt/Y 3Fe5O12ﬁlm used in the present study. His an external magnetic ﬁeld. his a rf-magnetic ﬁeld. (c) The microwave fre- quency ( f) dependence of S11for the Pt/Y 3Fe5O12ﬁlm. The peak labeled “FMR” is a uniform mode (ferromagnetic resonance mode), the peaks in the area labeled “MSBVW” are magnetostatic backward volume modes, and the peaks in the area labeled “MSSW” are magnetostatic surface modes. (d) Microwave frequency ( f) dependence of the electromotive force for the Pt/Y 3Fe5O12ﬁlm. The solid circles and the open circles represent the experi- mental data when the external magnetic ﬁeld is applied perpendicular (solid circles) and parallel (open circles) to the direction across the electrodes. Thesolid curve shows a ﬁtting result using a sum of ﬁve Lorentz functions for the solid circles.a)Author to whom correspondence should be addressed. Electronic mail:kharii@imr.tohoku.ac.jp. 0021-8979/2011/109(11)/116105/3/$30.00 VC2011 American Institute of Physics 109, 116105-1JOURNAL OF APPLIED PHYSICS 109, 116105 (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: 141.212.109.170 On: Fri, 12 Dec 2014 16:14:05current with a spin polarization rparallel to the magnetiza- tion-precession axis into the paramagnetic layer under the spin-wave resonance condition.15This injected spin current can be detected electrically by means of the strong ISHE inthe Pt layer. 4,9,15Because YIG is an insulator, this ISHE mea- surement is not interfered with by effects in the ferromagnetic layer, e.g., the anomalous-Hall effect. Here, the relationbetween the electric ﬁeld E ISHEinduced by the ISHE, the spa- tial direction of the spin current js,a n d ris given by16,17 EISHE/r/C2js: (1) The spin-wave resonance signal of the Pt/YIG ﬁlm can be detected by measuring the S11parameter using the vector- network analyzer. We measured the spin-wave resonance signal and the electric voltage between the electrodes attached to the Pt layer. All of the measurements were per-formed at room temperature. Figure 1(c)shows the spin-wave resonance spectrum, or S 11(dB), for the Pt/YIG ﬁlm as a function of the microwave frequency f. In this spectrum, multiple resonance signals appear. These multiple signals are attributed to the spin-wave mode in the YIG layer. Here, we deﬁne the ferromagneticresonance (FMR) frequency for the most prominent peak (FMR mode) as f FMR. The peaks for f>fFMRare magneto- static surface modes (MSSW), and the peaks for f<fFMR are magnetostatic backward volume modes (MSBVW).18 Figure 1(d) shows the dc electric voltage signals for the Pt/YIG ﬁlm when the external magnetic ﬁeld is applied per-pendicular ( h¼0) and parallel ( h¼90 /C14) to the direction across the electrodes. At h¼0, voltage signals appear at the spin-wave resonance and FMR ﬁelds, indicating that theelectromotive force is induced by the ISHE in the Pt layer affected by the spin-wave resonance in the YIG layer. 15We conﬁrmed that the electromotive force disappears ath¼90 /C14, a situation consistent with Eq. (1). The spectral shape of the electromotive force is well reproduced by usinga sum of Lorentz functions as shown in Fig. 1(d) (solid line), consistent with the prediction of the spin pumping.6,15 Figures 2(a)and2(b) show the fdependence of the spin- wave resonance spectra and the electric voltage signals. In thefrequency range under 3 GHz, the FMR peak in the S 11spec- trum is strongly suppressed by the Suhl instability.18,19Here, by changing H,fFMRis varied systematically, consistent with Kittel’s formula:20fFMR¼ðl0ceff=2pÞﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ HFMRðHFMRþMsÞp , as shown in Fig. 2(c),w h e r e ceff¼1:78/C21011(T/C1s)/C01is the effective gyromagnetic ratio and l0Ms¼0:172 T is the saturation magnetization for the YIG ﬁlm estimated from the resonance frequency fFMR.21The microwave-absorption power PabatfFMRis proportional to the incident microwave power Pin, as shown in Fig. 2(d), indicating that PabforPin /C2020 mW is lower than the saturation of the FMR absorption when fFMR¼3:51 GHz. Here, Pabis estimated as the S11 spectrum for the resonance in the YIG layer from which the spectrum without resonance ( His changed) is subtracted. In Fig. 3, the ~x/C172pfFMR=ðceffl0MsÞdependence of VISHE=h2is shown (solid circles). VISHE is the voltage at fFMR.his the rf-ﬁeld amplitude at fFMRestimated by the rela- tionh2¼Pab=ðvpfFMRl0v00 FMRÞ. Here, vis a volume in which the irradiated microwave is absorbed. Because vcannot be estimated accurately, we assume that vis the whole volume of the YIG layer, vYIG.v00 FMR is the imaginary part of the complex magnetic susceptibility under FMR conditions: v00 FMR¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ 1þ4~x2p þ1 2a~xﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ 1þ4~x2p ; (2) where ais the Gilbert damping coefﬁcient for the YIG/Pt ﬁlm. In the frequency range without the Suhl instability, VISHE=h2was found to decrease slightly with increasing ~x, as shown in Fig. 3. Given that VISHEis proportional to js/C22/C22/C22jjsj as shown in Eq. (1), this result indicates that the spin current FIG. 2. (Color online) (a) Microwave frequency ( f) dependence of the spin-wave resonance spectra for the Pt/Y 3Fe5O12ﬁlm. The transition of the line colors corresponds to an increase of the external magnetic ﬁeld. (b) Microwave frequency ( f) dependence of the electric voltage signals for the Pt/Y 3Fe5O12ﬁlm. (c) Field ( H) dependence of the FMR frequency ( fFMR) for the Pt/Y 3Fe5O12ﬁlm. The solid circles represent the experimental data. The solid line shows a ﬁtting result using Kittel’s formula with the effective gyromagnetic ratio for the Y 3Fe5O12layer. (d) The microwave-absorption power ( Pab) due to FMR plotted against the power of incident microwave ( Pin) for the Pt/Y 3Fe5O12ﬁlm. The solid circles represent the experimental data, and the solid line shows a linear ﬁt- ting result.116105-2 Harii et al. J. Appl. Phys. 109, 116105 (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: 141.212.109.170 On: Fri, 12 Dec 2014 16:14:05induced by the spin pumping decreases with increasing fFMR, rather than the intuitively expected state of js/fFMR. This behavior is explained by the compensation between the magnetization-precession frequency and the spin current generated by a cycle of the precession j1 s(Ref. 1); they both depend on the frequency, but in different manners: js ¼fFMR/C1j1 s, where j1 s/C17/C22h 2g"# r1 M2 sð1=fFMR 01 2pMðtÞ/C2dMðtÞ dt/C28/C29 zdt: (3) Here, g"# ris the real part of the mixing conductance and hMðtÞ/C2dMðtÞ=dtizis the zcomponent of MðtÞ/C2dMðtÞ=dt. The zaxis is directed along the magnetization-precession axis. Using Eq. (3)and the Landau-Lifshitz-Gilbert equation, we ﬁnd js=h2to be js=h2¼/C22hg"# rl0ceff 4pa2Msﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ 1þ4~x2p þ1 2ð1þ4~x2Þ: (4) The decrease in VISHE=h2is reproduced by Eq. (4)as shown in Fig. 3(solid line). Here, hSHEis the spin-Hall angle in Pt, or the efﬁciency of the conversion of the spin current to an electric current.22We use a¼4:34/C210/C04, estimated by the S11spectrum, and g"# rhSHE/C251:73/C21016m/C02, which is larger than in our previous study.15A possible reason for this difference is that the approximation of v¼vYIGis over-esti- mated. In this analysis, we assumed that g"# rhSHEis constant for the whole frequency range. The slight decrease in VISHE=h2is explained by a decrease in j1 s. Because j1 sis proportional to the area of the magnetization-precession trajectory7S,Sdecreases with increasing fFMR due to the decrease in the magnetization- precession angle with the increase in the external magneticﬁeld HFMR that is necessary for achieving ferromagnetic resonance. In summary, we measured the frequency dependence of magnetization-precession in spin pumping in a Pt/Y 3Fe5O12 ﬁlm using the inverse spin-Hall effect. We found that in thisﬁlm, the spin-current density decreases slightly with increasingprecession frequency, which is well reproduced by a model calculation based on the Landa u-Lifshitz-Gilbert equation combined with a standard model of spin pumping. This resultis favorable for making a microwave demodulator detection device based on spin pumping and the inverse spin-Hall effect. The authors thank S. Takahashi, Y. Fujikawa, and H. Kurebayashi for valuable discussions. This work was sup- ported by a Grant-in-Aid for Scientiﬁc Research in PriorityArea “Creation and control of spin current” (19048028) from MEXT, Japan, a Grant-in-Aid for Scientiﬁc Research (A 21244058) from MEXT, Japan, Global COE for the Mate-rials Integration International Center of Education and Research from MEXT, Japan, a Grant for Industrial Technol- ogy Research from NEDO, Japan, and Fundamental ResearchGrants from CREST-JST, PRESTO-JST, and TRF, Japan. 1Y. Tserkovnyak, A. Brataas, and G. 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1.384637.pdf	Ultrasonic measurement of larynx vibratory pattern Sandra L. Hamlet height and vocal fold Department of Hearing and Speech Sciences, University of Maryland, College Park, Maryland 20742 (Received 14 November 1978; accepted for publication 14 March 1980) This paper describes the modification and extension of an ultrasonic through-transmission technique for examining vocal fold activity. The purpose was to achieve improved lateral resolution in the cranio-caudal dimension. A transducer was specially designed to produce an eliptical beam, so that a measurable signal might be received through the extreme upper and lower edges of the vibrating vocal folds. An interpretation of amplitude variation within the ultrasonic waveform is offered, with illustrative data shown for a single subject. PACS numbers: 43.70.Bk, 43.80.Sh, 43.35.Yb INTRODUCTION Medical ultrasound has been used by a number of investigators to examine vocal fold activity during phonatiOno Techniques which have proved successful include single transducer pulse echo (Mensch, 1964; Kitamura, 1967; Asano, 1968; Hertz, Lindstrom, and Sonesson, 1970; Hamlet, 1972a; Holmer, Kitzing and LindstrSm, 1973), a combination of pulse echo and pulse through transmission termed "ultrasonoglotto- graphy" (Kitamura et al., 1969; Miura, 1969; Kaneko et al., 1974; Kaneko et al., 1976), and the transmission of continuous wave ultrasound through the larynx (Bor- done-Sacerdote and Sacerdote, 1965; Hamlet and Reid, 1972; Holmer and Rundqvist, 1975). Further develop- ment of the lastJmethod is described here. A continuous wave (cw) through transmission system re- quires two transducers matched in frequency, to be located on opposite sides of the larynx. When an ultra- sonic beam is passing through the vibrating vocal folds reception of the signal is regularly interrupted when- ever they are open during the vibratory eyeleo The received signal thus appears as a series of pulses. By rectifying and alemodulating the received signal ultra- sonic data can be readily recorded on tapeø Such a signal (see Fig. 1), contains information about a num- ber of features of vocal fold vibration and laryngeal activity, some of it quite straightforward to recover, and some requiring interpretation relative to other simultaneously recorded dataø It is likely that ultra- sonic signals received through the vibrating vocal folds contain more information on biophysical features of phonation than we presently know how to extract or interpret. The vocal fundamental frequency (Holmer and Rund- qvist, 1975) and the duty cycle of vibration (Hamlet, 1973) have been measured ultrasonically. By comparing the temporal relations between ultrasonic and micro- phone signals, the cw through transmission technique has also been used to describe the times of vocal tract excitation within vocal fry vibratory cycles (Hamlet, 1971). For these types of information, ultrasound is not the only possible source of data. Electroglotto- graphy might have served as well. Electroglottographic signals are similar in appear- ance to those obtained with ultrasound when rectified and demodulated. However, the two techniques differ in one major respect. An ultrasonic signal is much more sensitive to transducer placemen[ on the neck. Varigtions in amplitude of received signals, their pulse shape, and degree of modulation can be observed (Ham- let, 1972a; Holmer and Rundqvist, 1975). For some applications, ultrasonic signal dependence on exact transducer position is a drawback. On the other hand, this very sensitivity is a characteristic that may poten- tially be exploited to advantage. Two attempts have been made to put to advantage the positional sensitivity encountered in using the cw ultra- sonic technique. In the first attempt, signal changes were sought corresponding to transmission indepen- dently through the top and bottom of the vocal folds. The pair of transducers was swept vertically down the neck through the level of the vocal folds. With the microphone signal as a reference, temporal relation- ships of the uppermost and lowermost signal received through the vocal folds were compared to see if evi- dence of the "vertical phasing" of vocal fold vibration could be seen. This was largely unsuccessful. A dif- ference in timing of the opening and closure of the up- per and lower edges of the vocal folds could not be discriminated, except in a male speaker with a deep bass voice phonating at a fundamental frequency of 87 Hz (Hamlet, 1972b). The transducers used were 6 mm in diameter with a frequency of 5 MHz (unfocused). The effective beam width at the laryngeal midline was esti- mated to be about 2.5 min. Two factors account for the inability to discriminate the activity of the upper and lower portions of the vocal folds independently, using these transducers. Intensity of the ultrasonic beam drops off with divergence from the axis, and only the attenuated edge portion of the beam can be used to pass through the uppermost or lowermost part of the vocal folds. Additionally, since the cross section of the beam was circular, a smaller area of tissue was being transmitted through at the top and bottom of the vocal folds than if the entire diameter of the beam were available. Success in tracking the vibra- tions of the upper and lower portions of the vocal folds has been reported by other investigators (Kaneko et al., 121 J. Acoust. Soc. Am. 68(1), July 1980 0001-4966/80/070121-06500.80 ¸ 1980 Acoustical Society of America 121 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 130.63.180.147 On: Sun, 23 Nov 2014 01:07:39ACCELEROMETER SIGNAL DEMODULATED ULTRASONIC SIGNAL ZERO CROSSING 5 ms FIG. 1. Tracing of oscillogram showing accelerometer and ultrasonic signals. Points of vocal fold closure (short arrow) and opening (long arrow) are indicated, as well as the zero crossing of the accelerometer signal used as a timing refer- enc e. 1974) using two transducers differently angulated. A pulsed ultrasonic technique with time motion display was used. A second attempt to take advantage of positional sen- sitivity involved having a subject speak an entire sen- tence while the paired transducers were held stationary against the neck. As the larynx moved up and down with articulatory and intonational features of the sentence, the received signal disappeared and reappeared. This was recorded on tape. The transducers were then moved to another position, slightly upwards or down- wards, and the sentence repeated. By sampling at a number of vertical transducer locations it was possible to infer from the composite picture the vertical motions of the larynx. The validity of this procedure rests on two assumptions: that repeated utterances of the same sentence would be similar in production, and that the amplitudes of the received signals could be used as a guide in determining the vertical midpoint of the vocal folds. This procedure was used to determine larynx positional changes as part of initial speech compensa- tion for a dental prosthesis (Hamlet and Stone, 1976). Since the changes were fairly substantial (3-10 mm) it was possible to demonstrate the direction, and to esti- mate their extent. Greater accuracy in determining larynx changes was desirable, however. Temporal discrimination is maximized with a con- tinuous wave technique, and depth information is sac- rificed completely. Possibilities for manipulating spatial resolution are thus limited to lateral (across the beam) resolution. Depending on the question of in- terest, either better or poorer lateral resolution may be sought. For an application such as pitch tracking with an ultrasonic signal as input, the aim is to have continuous information about vocal fold activityø A beam that is broad in the vertical dimension would be an advantage, since the vocal folds would remain within the ultrasonic field in spite of laryngeal movement during speech. Holmer and Ruudqvist (1975) have used a 30-x 5-mm rectangular transducer for this purpose. Unfortunately, in their paper there was no description of the expected ultrasonic field produced by that transducer. In seeking finer lateral resolution, the general approach would likely be to attempt focusing or use of higher frequencies or the near field. An al- ternative approach is described here, which rests more heavily on the interrelationship of beam characteristics, and anatomical features and expected physiological func- tioning of the structure to be examined. The vocal folds are longer in the anterior-posterior (longitudinal) dimension than they are thick (vertical dimension). During phonation at low pitches, such as in the range usually employed for speech, the vibratory pattern shows complexity in the vertical dimension (phasing) which is of theoretical interest. Positional adjustments of the larynx during running speech are primarily shifts in the vertical position of the larynx. For these reasons, fine lateral resolution is more im- portant to have in one dimension, the vertical, than in the other. One of the problems encountered in using circular beam transducers and attempting to obtain signals through only the upper or lower edge of the vocal folds, was that the area of the beam passing through the edge of the vocal folds was small. A broader beam in the longitudinal dimension should permit a greater area of tissue to be transmitted through, with the expected re- suit that a measurable signal might be received through the upper or lower edge of the vocal folds. Thus, in sacrificing fine resolution in the longitudinal dimension, the potential for obtaining better effective resolution in the vertical dimension is enhanced. A sound source which produces a field broader in one dimension than another is a rectangular plate. In the far field the radiation pattern is eliptical, with the long axis of the sound beam oriented perpendicular to the long axis of the rectangular source. Such a sound field does not diverge as rapidly as that from a circular pis- ton (Morse and Ingard, 1968). It seemed that an elip- tical ultrasonic beam produced by a rectangular piezo- electric element would offer the potential advantages sought. In addition, because of the more gradual diver- gence of the beam with axial distance from the source, the beam characteristics would be expected to remain similar when such a transducer was used on individuals having different neck diameters. I. TRANSDUCER DESIGN AND TESTING For a transmitting transducer, an ultrasonic beam was sought with dimensions at the midline of the larynx of 8 mm x 2 mm, ideally. The midline of the larynx lies usually between 20 and 25 mm from the side of the neck. The 8 mm dimension was chosen because it would likely be somewhat shorter than the vibrhting length of the vocal folds for adults, and thus wduld permit essentially total reflection of sound back from the glottal rim when the vocal folds were open. This, in turn, would result in a received signal with sharp definition between open- ness and closure. Computer solutions for the radiation pattern from a 122 J. Acoust. Soc. Am., Vol. 68, No. 1, July 1980 Sandra L. Hamlet: Ultrasonic measurement of larynx height 122 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 130.63.180.147 On: Sun, 23 Nov 2014 01:07:39rectangular plate were obtained, for a range of dimen- sions and frequencies. These computations were done by the Fresnel method described by Neubauer (1965a, 1965b). Because of the small wavelengths at ultrasonic frequencies, the problem of choosing rectangular ele- mentdimensions involved not only finding dimensions which would theoretically produce the desired sound field, but which could also be practically built as a transducer for medical use. A rectangular element, 6 mm x 2 mm, with a fre- quency of 1.8 MHz, was chosen as best meeting both criteria. Figure 2 shows plots of the radiated sound field along both the narrow and broad dimensions of the beam. These values were obtained using 1540 m/s for the sound velocity. Design criteria for the receiving transducer were slightly different than for the transmitter. Given the expected narrowness of the beam in one dimension, it was anticipated that the orientation of the transmitter and receiver would be critical, and a source of potential difficulty in use on a subject. Although it would be pos- sible to orient the transducers properly before applica- tion to the subject, the larynx may be tipped somewhat from the true horizonal, and differently so in different individuals, or at different vocal pitches. The desired beam orientation would be with the long dimension of the eliptieal field parallel to the length of the vocal folds. So that the transmitting transducer could be rotated slightly to orient the field relative to the plane of the vocal folds without adjusting the receiver, a square element was chosen for the receiver (6 mm x 6 ram, with a frequency matched to the transmitter). Transducers with these characteristics were specially ordered from KB-Aerotech, Inco, in Lewistown, PA. The piezoelectric elements were mounted at the end of plastic cylinders 12 mm in diameter, for ease of appli- cation to the subject. The transducers were unfoeused, damped and inductively tuned, and were tested to meet electrical current leakage specifications for human safety. Beam profiles were obtained for each trans- dueer, by the manufacturer, by plotting pulse amplitude 1.2 1.2 1.0 1.0 ' 0.8 ' 0.8 ß 'e.- O. fi "e.- O. fi 0.,4 o.,4. 0.2 O2 0 2 4 6 8 I0 0 2 4 6 8 I0 DISTANCE Y (ram) DISTANCE X (ram) ' y X Z FIG. 2. Calculated sound field produced by a 6-ram x 2-ram rectangular piston, at an axial distance of 24 min. Frequency is 1.8 MHz. Units on the ordinates follow the convention in Neubauer (1965a, 1965b), where is the uniform piston velocity amplitude, and k is the wave- number (2 received as the transducer was moved past a series of stainless steel rods, running perpendicular both to the axis of the transducer and to its direction of travel, and submerged to various depths in a water-filled testing tank. Their criterion of lateral resolution potential is the width of the beam profile at the half amplitude level at a specific distance from the transducer. By this cri- terion, the beam of the transmitter was 3.5 mm x 8 mm at an axial distance of 30 ram, agreeing fairly well with the calculated field illustrated in Fig. 2. The beam width of the reeiver was 5 mm x 5 mm at the same axial distanceø Further testing was done to aid in determining signal criteria for establishing the vertical depth of contact during closure of the vibrating vocal folds. A small testing tank was constructed (15 em on each side), having holes through two parallel sides for mounting the transdueerSo Because ew ultrasound was to be used, the inside surfaces were coated with 6.25-mm-thiek mater- ial designed to absorb ultrasound and reduce echoes by 30 dB (Corsaro and Klunder, 1979). This material is now commercially available from Consumer Usage Laboratories, Inc., Rockville, Marylandø When the transducers were mounted in the testing tank, their piezoelectric faces pointed directly at each other. The distance between them could be varied-- testing was done in the 4-5 em rangeø The tank was filled with water to simulate ultrasonic transmission through body soft tissues. The testing tank also had a provision for placement, raising and lowering of samples midway between the transducers. To simulate conditions of ultrasonic transmission through the larynx, parallel metal plates were used to represent the air spaces above and below the glottis. It was confirmed that no signal was re- eeived when one of these plates was placed to block the ultrasonic beam. A horizontal slot between the plates simulated the closed vocal folds, through which ultra- sound could be received. The width of this slot could be varied--widths from 2 to 12 mm were tested. The metal plate arrangement was lowered through the region of the ultrasonic beam in 0.5-ram steps, and the amplitude of the received signal recorded. It was found that for slots 5 mm wide and larger, the maximum received signal was as great as that for an unobstructed field. The position of the plates at which a signal with half this amplitude was received defined the edges of the slot. In other words, it was possible to determine the width of the slot by the vertical distance over which signals within 6 dB of the maximum were reeeivedo For slots less than 5 mm wide, the maximum received sig- nal was less than for an unobstructed field, and a drop in amplitude to 0.7 of the maximum received for a particular width gave a better estimate of the edge of the slot. Calibration of the transmitting transducer was done at the National Bureau of Standards to determine its acoustic power output at expected driving voltages. A modulated radiation pressure method was used (Green- span, Breekenridge, and Tsehiegg, 1978). In use on a 123 J. Acoust. Soc. Am., Vol. 68, No. 1, July 1980 Sandra L. Hamlet: Ultrasonic measurement of larynx height 123 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 130.63.180.147 On: Sun, 23 Nov 2014 01:07:39subject the transducer is driven by a 3 V (p-p) signal at its resonance frequency. The calculated total power output would be 0.63 mW, based on this calibration. II. DATA RECORDING AND ANALYSIS PROCEDURES For recording data, a subject was seated in a dental chair with head stabilized. The head rest portion of the dental chair (an older mode!), was modified so that the transducers could be mounted upon it, adjusted against the subject's throat, and moved manually up and down. The trajectory of the transducers was an arc with a minimum radius of 20 cm. Since the total vertical dis- tance transducers need to be moved in order to traverse the thickness of the vocal folds is less than 2 cm, this motion resulted in only minimal anterior-posterior changes in the ultrasonic field. The gross positioning of the ransducers for a given sample of sustained phonation was determined empir- ically by the appearance of a signal during phonation which had a discontinuous carrier, characteristic of reception through the vocal folds (Hamlet and Reid, 1972)o Recived signals were monitored on an oscillo- scope throughout a recording procedure. When a sig- nal through the vocal folds had been obtained, the sub- ject was requested to sustain phonation while the trans- mitter was rotated slightly. An orientation was sought at which the received signal was maximum. In subjects tested thus far, this orientation was with the broad axis of the eliptical field at or very near the true horizontal, for a vocal fundamental frequency near the subject's habitual speaking pitch. The vertical position of the ultrasonic transducers was tracked with a strain gauge displacement trans- ducer and associated bridge amplifier, which has been used previously for studying jaw motions during speech (Abbs and Gilbert, 1973)o Multichannel FM recordings were made of four types of signals: microphone, ultra- sonic, strain gauge monitor, and pretracheal acceler- ometer. The importance of the last mentioned signal will be discussed below. An attempt was first made to determine larynx height and depth of vocal fold contact from these data. The amplitude of the ultrasonic signal received during a cough or swallowing was taken as a reference for each subject, since these are the largest that can be obtained through the larynx. When the largest signal received during phonation was smaller than this (which is the case for subjects thus far tested), it was assumed that the depth of tissue in contact during phonation was less than 5 mm. Thus, the upper or lower edge of the re- gion of closure should correspond to that transducer location at which the received signal was reduced by 3 dB. Larynx height was taken as midway behveen the ultrasonically defined upper and lower edges--a location where there was usually also a clear maximum in re- ceived signal amplitude. Figure 3 shows sample data from a male subject. The task was to sustain phonation using the vowel /i/. A separate breath was taken for each frequency produced, and only those samples with similar sound pressure , (range of 3.25 dB) were compared. As can be seen, 9- z 6 c 5 4 IOO 150 200 250 50 FUNDAMENTAL FREQ. (Hz) x 25- >-'" 20- n- E 15- _-r 5- O IOO ß ß ß ß I () I I 150 20 250 300 FUNDAMENTAL FREQ. (Hz) FIG. 3. Vertical range of vocal fold contact and larynx height at various fundamental frequencies. Data are from one male subject. larynx height changed nearly 25 mm over about an oc- tave change in vocal fundamental. Larynx height chan- ges with variation in fundamental frequency have been measured previously also, and are consistent with the data shown here for subjects with untrained voices (see, for example, Shipp 1975). The attempted measurement of depth of contact yielded some surprises. Although the maximum amplitude of received signals was such that this depth was assumed to be less than 5 mm, the resulting depth of contact determination (using a-3-dB amplitude criterion) often- times was greater than 5 mm (see Fig. 3). This was also the case for female subjects at the lower fre- quencies in their ranges. In interpreting these data, it should first be remem- bered that the criterion of depth determination was based upon a very simple geometric situation, and that the actual sound field in the larynx may be considerably different. Also, with this methodology, particular points on the vocal folds are not being tracked, but rather activiby is being viewed through an ultrasonic window. The vibratory complexity in the vertical di- mension for low vocal pitches may be such that no more than 5 mm of tissue was in contact at any time, but the location of that contact moved throughout the vibratory cycle. A composite view, as obtained here, would identify the lowermost and uppermost edges of the entire range of positions in contact, for which there was a transmission pathway through the larynx. Thus, this measure is being referred to as the range of contact, rather than the depth of contact. An attempt was also made to discriminate closure and opening times for the upper and lower portions of the vocal folds. For such a task it was critical to have a timing standard that remains synchronized with cycle- by-cycle fundamental [requency perturbations. Pre- 124 J. Acoust. Soc. Am., Vol. 68, No. 1, July 1980 Sandra L. Hamlet: Ultrasonic measurement of larynx height 124 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 130.63.180.147 On: Sun, 23 Nov 2014 01:07:39viously, the speech waveform was used for this purpose (Hamlet, 1972b). Here, a pretracheal accelerometer signal was employed, because its waveform is simple, and certain of its features, such as a rapid signal rise near the time of vocal fold closure, are vowel indepen- dent. Measurements were made of the timing of vocal fold closure and opening relative to the zero crossing of the accelerometer signal (see Fig. 1). These mea- surements were made at sampling points corresponding to 0.5-mm increments of change in ultrasonic trans- ducer position. Figure 4 shows the results of such measurements for a male subject phonating at a vocal fundamental of 100 Hz. Zero ms is the time of the accelerometer signal zero crossing, and the vertical line to the right indi- cates the duration of one vibratory cycle. The ordinate represents the range of transducer positions, with zero being the vocal fold midpoint (corresponding to the point used for larynx height). Positive transducer pos- itions refer to those locations above the vocal fold mid- point, and negative to those below the midpoint. To the right, a vertical bar indicates the measured range of contact. The horizontal lines represent the duration of closure measured at the various transducer locations. Above are tracings of the ultrasonic signal waveform as seen at locations A, B, and C, respectively. Zero ms in these waveform tracings also refers to the zero crossing of the accelerometer signal. The first thing to note is that within the range of trans- ducer positions defining the range of contact, there is very little differentiation of the timing of closure and opening. Both above and below this range differentiation does appear, however. At these extreme transducer 0 4.19 0 0.86 4.19 0 0.86 ms ms ms 12 8 4 0 --4- --8- -12 0 oC B øA 4 8 IO ms FIG. 4. Ultrasonic signal waveforms received through the lower, middle, and upper portions of the vocal folds. Below is a summary of the duration of closure as measured at vari- ous transducer locations. 0 5 4 4 o o 4 -4 i i i ! i i 0 , 4 6 ms 0 2 4 ms FIG. 5. Durations of closure measured at various transducer locations, and the waveform obtained through the midpoint of the vocal folds, for the two lowest pitched samples illustrated in Fig. 3. positions only the very edge of the ultrasonic beam, or even only the side lobes, would be transmitted. Even so, as can be seen from the waveform tracings, the signal-to-noise ratio is still good enough to make con- fident measurements. This provides evidence that a rectangular element provides better effective lateral resolution than a circular transducer element, for this particular application. The second major feature to note in Fig. 4 is that the timing of closure and opening of the upper and lower portions is represented in the waveform received through the midpoint of the vocal folds (waveform B). The low amplitude portion at the onset of closure (up to 0.86 ms) represents closure of the bottom part of the vocal folds. This feature is not represented in wave- form C, which is received through only the upper por- tion of the folds. After 0.86 ms in waveform B, there follows a large amplitude middle portion, which would re- present the completion of closure and vertical deforma- tion, because of tissue incompressibility and relative lack of longitudinal tension at a low vocal fundamental (Titze, 1976). The low amplitude portion of waveform B prior to vocal fold opening (after 4.19 ms), represents the opening of the bottom portion of the vocal folds, with tissue contact only of the superior margins. This fea- ture of waveform B is not represented in waveform A, which is received through the lower edge of the folds. This particular sample was chosen for its clarity as an illustration. Waveforms more representative of typical ultrasonic signals received during speech are shown in Fig. 5. These examples are for the two lowest pitches repre- sented in Fig. 3. The formats of Figs. 4 and 5 are similar. Indications of discrete closure of either the bottom or top portions of the vocal folds are indicated by "hips" on the waveshape. These features disappear fro'm the waveform at higher vocal fundamentals in the speaking range, and there is usually no obtainable sig- nal through the vocal folds above about 400 Hz. 125 J. Acoust. Soc. Am., Vol. 68, No. 1, July 1980 Sandra L. Hamlet' Ultrasonic measurement of larynx height 125 ' Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 130.63.180.147 On: Sun, 23 Nov 2014 01:07:39Physiological data have been illustrated to show the feasibility and potential of this ultrasonic technique, not to suggest normarive data. Since diagnostic ultrasound provides a medically safe method for studingthepara- meters of phonation discussed in this paper, and the use of a rectangular transducer element extends the applic- ability of the through transmission technique, this method should prove useful in both normative and clin- ical investigations of phonation. ACKNOWLEDGMENTS The author wishes to thank Frank Breckenridge and Charles Tsehiegg, National Bureau of Standards, Wash- ington D.C., for performing the transducer power out- put calibration. This research was supported by NIH grant DE-03 631. Abbs, J. H., and Gilbert, B. N. (1973). "A strain gage trans- duction system for lip and jaw motion in two dimensions," J. Speech Hear. Res. 16, 248-256. Asano, H. (1968). "Application of the ultrasonic pulse-method on the larynx" (in Japanese), J. Oto-laryngol. Japan 71, 895- 916. Bordone-Sacerdote, C., and Sacerdote, G. (1965). "Investiga- tions on the movemen of the glottis by ultrasounds," Proc. 5th Int. Congr. Acoust. Corsaro, R. D., and Klunder, J. D. (1979). "Tank coatings for ultrasonic echo reduction," J. Acoust. Soc. Am. Suppl. 1 65, S46. Greenspan, M., Breckenridge, F.R., and Tschiegg, C. E. (1978). "Ultrasonic transducer power output by modulated radiation pressure," J. Acoust. Soc. Am. 63, 1031-1038o Hamlet, S. L. (1971). "Location of slope discontinuities in glottal pulse shapes during vocal fry," J. Acoust. Soc. Am. 50, 1561-1562. Hamlet, S. (1972a). 'Vocal fold articulatory activity during whispered sibilants," Arch. Otolaryngol. 95, 211-213. Hamlet, S. L. (1972b). "Interpretation of ultrasonic signals in terms of phase difference of vocal fold vibration." J. Acoust. Soc. Am. 51, 90. Hamlet, S. L., and Reid, J. M. (1972). "Transmission of ul- trasound through the larynx as a means of determining vocal fold activity," IEEE Trans. Biotaed. Eng. 19, 34-37. Hamlet, So L. (1973). 'Vocal compensation: An ultrasonic study of vocal fold vibration in normal and nasal vowels," Cleft Palate J. 10, 267-285. Hamlet, S. Lo, and Stone, M. (1976). "Compensatory vowel characteristics resulting from the presense of different types of experimental dental prostheses," J. Phonetics 4, 199-218. Hertz, C., Lindstrm, B., and Sonesson, B. (1970). 'Ultra- sonic recording of the vibrating vocal folds," Acta Oto- laryngol. 69, 223-230. Holmer, N-G., Kitzing, P., and Lindstrm, K. (1973). "Echoglottography, ultrasonic recording of vocal fold vibra- tions in preparation of human larynges," Acta Otolaryngol. 75, 454-463. Holmer, N-G., and Rundqvist, H. E. (1975). 'Ultrasonic reg- istration of the fundamental frequency of a voice during nor- mal speech," J. Acoust. Soc. Am. 58, 1073-1077. Kaneko, T., Kobayashi, N., Asano, H., Miura, T., Naito, J., Hayasaki, K., and Kitamura, T. (1974). "Ultrasonoglottog- raphy," (in French), Ann. Oto-laryngol. )aris 91, 403-410. Kaneko, T., Kobayashi, N., Tachibana, M., Naito, J., Haya- saki, K., Uchida, K., Yoshioka, T., and Susuke, H. (1976). 'Ultrasonoglottography: Glottal width and the vibration of the vocal cords" (in French), Rev. Laryngol. 97, 363-369. Kitamura, T. (1967). 'Ultrasonoglottography: A preliminary report," Japanese Med. Ultrason. 5, 40-41. Kitamura, T., Kaneko, T., Asano, H., and Miura, T. (1969). "Ultrasonic diagnosis in oto-rhino-laryngology," Eye, Ear, Nose Throat. Mon. 48, 121-131. Mensch, Bo (1964). "Analysis of isolated vocal cord move- ments by ultrasonic exploration" (in French), C. R. Soc. Biol. 158, 2295-2296. Miura, T. (1969). 'qVlode of vocal cord vibration--study of ul- trasonoglottography" (in Japanese), J. Oto-laryng. Japan. 72, 895-1002. Morse, P.M., and Ingard, K. U. (1968). Theoretical Acous- tics (McGraw-Hill, New York), p. 393. Neubauer, W. G. (1965a). "Application of the Fresnel method to the calculation of the radiated acoustic field of rectangular pistons," NRL Rpt. 6286. Neubauer, W. G. (1965b). 'Radiated field of a rectangular pis- ton," J. Acoust. Soc. Am. 38, 671-672. Shipp, T. (1975). '*Vertical laryngeal position during continu- ous and discrete vocal frequency change," J. Speech Hear. Res. 18, 707-718. Titze, I. R. (1976). 'qDn the mechanics of vocal-fold vibra- tion," J. Acousto Soc. Am. 60, 1366-1380. 126 J. Acoust. Soc. Am., Vol. 68, No. 1, July 1980 Sandra L. Hamlet: Ultrasonic measurement of larynx height 126 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 130.63.180.147 On: Sun, 23 Nov 2014 01:07:39
5.0001730.pdf	AIP Conference Proceedings 2220 , 110037 (2020); https://doi.org/10.1063/5.0001730 2220 , 110037 © 2020 Author(s).Frequency tunability via spin hall angle through spin hall spin torque nano oscillator Cite as: AIP Conference Proceedings 2220 , 110037 (2020); https://doi.org/10.1063/5.0001730 Published Online: 05 May 2020 H. Bhoomeeswaran , R. Bakyalakshmi , T. Vivek , and P. Sabareesan Frequency Tunability via Spin Hall Angle through Spin Hall Spin Torque Nano Oscillator Centre for Nonlinear Science & Engineering, School of Electrical and Electronics Engineering, SASTRA Deemed University, Thanjavur, Tamil Nadu - 613401, India. a) Corresponding author: sendtosabari@gmail.com Abstract. The present work deals with the theoretical modeling of Spin Hall Spin Torque Nano Oscillato r (SHSTNO) using four different Ferromagnetic Materials (FM) (Py, Co, CoFeB and Ni) with Current In Plane (CIP) geometry. The d evice comprised of bilayer (Pt/X ) (i.e.)a top Ferromagnetic Free Layer (FFL) which is notated as (X) along with the heavy metal ( Pt). In this work, we tried to tune the frequency of the device by using different materials in FFLand by altering the Spin Hall Angle (SHA), which is notated as (θ). The sustained oscillation in the FFL is studied by the governing Landau - Lifshitz - Gilbert - Slonczewski (LLGS) equation. The device works under the principle called Spin Hall Effect (SHE) that originates from spin - orbit scattering paves for the deflection of conduction electrons with opposite signs oriented in the opposite direction. When the spi n reaches FFL due to the phenomena called Spin Transfer Torque (STT), persistent oscillation occurs, resulting in the emission of frequency in the microwave regime. The SHA (θ) is highly tunable up to 0.8 from 0.1. The author mentions that one can vary (θ) to the maximum and use the low Saturation Magnetization (SM) based material in FFL for maximum frequency tunability. The results presented in the article can find the application as spin - wave emitters for magnonic applications where the spin waves may use for transmission and processing information. INTRODUCTION SH ST NO, an embryonic field in spin - based electronics , [1,2] which is a bilayer device that earns a separate place for it because of its smaller size as compared to other electronic and the spintron ic oscillators. As far as the electronic oscillator is concerned, there are specific limitation s in terms of device output frequency, size, and complication in integration with on - chip. On the other hand, Spin Torque Nano Oscillator [STNO] which is a typical spintronic oscillator of trilayer structure emits high frequency as compared to the electronic oscillators. Researchers axiom that in the future, while comparing with all other oscillators SH ST NO tops the chart in term s of emitting the frequency. The advantage of bilayer than trilayer is smaller, the device easy to fabricate and commercialize and also, the current required to operate the device is low. For these reasons, SHNO became one of the active research areas off late [3,4]. Researchers already started to work on SHNO, but the main problem they are facing is the tunability of frequency [5]. Zahedinejad et al. [6] tuned the frequency of about 9 GHz by applying field 0.76 T with out of plane configuration. Divinskiy et al. [7] tuned the frequency to 9.5 GHz with in - plane angle and by 2000 Oe. Giordino et al. [8] tuned the frequency of about 9.98 GHz for the same Pt/Py device. Durrenfeld et al. [9] tuned the frequency of about 17.57 GHz in their work using a 20 nm Pt/P y SHNO device using out of plane configuration. Also, he tried to synchronize the Pt/CoFeB SHNO device and obtained the maximum frequency of about 22 GHz by varying the external magnetic field [10]. Instigated by the above ideas and the technological inter est, t he authors have modeled the four types of SH ST NO devices separately [Pt/Co , Pt/Py, Pt/Fe and Pt/CoFeB] and tried to tune the frequency by varying the SHA (θ) which will be discussed in the forthcoming sections. This device is an ideal candidate for t he nanoscale oscillators. The paper is organized as follows: In section 2, device architecture and the governing equation are discussed. Section 3, encompasses the frequency emission by the device. Finally, concluding remarks are made in Section 4. H Bhoomeeswaran, R Bakyalakshmi, T Vivek and P Sabareesan a) 3rd International Conference on Condensed Matter and Applied Physics (ICC-2019) AIP Conf. Proc. 2220, 110037-1–110037-5; https://doi.org/10.1063/5.0001730 Published by AIP Publishing. 978-0-7354-1976-6/$30.00110037-1 FIGU RE 1. S chematic representation of SHSTNO device. DEVICE AND ITS DYNAMICAL EQUATION SHSTNO [1] is a bilayer device with Current In Plane (CIP) geometry, as shown in Figure.1 comprised of heavy metal along with a FM material. In Figure.1, top FFL, materials s uch as Co, Py, Co FeB, and Ni are taken . This device works under the principle called SHE ,[3,4] which originates from spin - orbit scattering paves for the deflection of conduction electrons with opposite signs oriented in the opposite direction. The thicknes s of the Pt is taken as 5 nm and all other FM in FFL as 3 nm. It is well known that the Hall Effect (HE) in nonmagnetic metals or semiconductors refers to the build - up of a so - called Hall voltage across an electrical conductor, transverse both to the elect ric current in the conductor and to an external magnetic field perpendicular to the current. Edwin Hall found that the SHE in FM results ten more times greater than HE in nonmagnetic materials. The SHE is then a limiting case of this phenomenon in material s without any spontaneous magnetization but with substantial spin - orbit coupling. As there is no spin imbalance in a nonmagnetic material, the asymmetric spin - dependent scattering does not result in any Hall voltage. What it does result in, however, it is a net spin current transverse to the charge current and, in a steady - state, the build - up of spin accumulation zones at the edges of the conductor, which can be directly observed by optical means and which is used in spin - current driven devices. Thus, so th e generated spin current injected into reaches the free layer, STT will occur. It is due to the transformation of conduction electrons from one layer to other made the FFL magnetization to precess. Eventually, it ends up in generating the frequency, which will be in the order of the microwave regime [5]. The governing LLGS equation, including SHE and STT effect, may be written as: = − ⨉ − ⨉ ⨉ − [ ⨉ ( ⨉ ) ] + [ ( ⨉ ) ] The first term of the LLGS equation is the precession term that conserves magnetic energy and regulates the precessional frequency of magnetization dynamics. The second term is the damping term that depletes the energy during magnetization dynamics. The third term is the spin transfer term p roportional to a j1 either amplifies or attenuates the precessional motion so that purely rely on the current flow direction. The final term a j2 is field - like torque exists only when the thickness of the free layer (d) is much smaller than the decay of tran sverse spin current. The effective magnetic field ( H ) that acts on the free layer comprised of anisotropy as well as demagnetization field and it may be written as H = H + H (i.e.) H = K M - 4π M where and are the unit vectors along x and z directions. The num erical parameters are: γ = 2.21× 10 - 7 ms/A, is gyro magnetic ratio, α = 0.01 is the Gilbert damping factor, a j 1 = [(2µJ θ/(1+α 2 )ed M s 3 )g( . )] and a j 2 = [(2µJα θ/(1+α 2 )ed M s 2 )g( . )] are the STT coeffi cients. g( . )= [ - 4(1+S 3 )(3+cosθ)/4S 3/2 ] - 1 . Here, S = 0.5 is the polarization factor , µ is the permeability of free space, e is the electron’s charge, d is the thickness of the free layer, J is the current density, M s is the FFL saturation magnetization , K is the magneto - crystalline anisotropy coefficient and the magnetization of the FFL is . M s and K for Co, CoFeB, Fe, Py are taken as 1.449×10 6 A/m, 1.2×10 6 A/m, 1.712×10 6 A/m, 0.795×10 6 A/m and 500 ×10 3 J /m 3 , 28.6 ×10 3 J /m 3 , 48 ×10 3 J /m 3 , 2 ×10 3 J /m 3 respectively. θ is the SHA and it vary from 0.1 to 0.8. 110037-2RESULT AND DISCUSSIONS FIGURE 2. (a) . A plot of SHA ( θ ) vs . Frequency (GHz). (b) . A p lot of SHA ( θ ) vs . Quality Factor for the SHSTNO devices. The LLGS equation governs the FFL magnetization dynamics of the SHSTNO device, and it is numerically computed by using the embedded Runge - Kutta Fourth order procedure. The applied current density for all the SHSTNO device is of about 9.4 ⨉ 10 10 A/m 2 with zero applied magnetic field. The numerical computation of LLGS equation provides the output of magnetization as a function of time. The obtained output osc illations are transformed into f requency from time domain to frequency domain using the Fast Fo urier Transform (FFT) method [11]. After t aking the F ourier T ransform , we shall get the f requency (GHz) vs . A mplitude (arbitrary unit) for all the devices are achieved. In this article, we vary the SHA ( θ), from 0.1 to 0.8. But in more general case, it may vary up to 0.99 from 0.01. But keeping the experimental feasibility in mind, we varied up to 0.8 from 0.1 for all the four devic es and studied the response of f requency. Figure.2 (a) is plotted for t he various SHA (θ) against the f requency of the all four SHSTNO devices, and (b) is plotted for SHA (θ) against the Quality Factor (QF) of the all four SHSTNO devices. The thickness (d) of FFL is taken as 3 nm for all the four SHSTNO devices. From Figure. 2 (a), it is apparent that the f requency is directly prop ortional to the SHA (θ). In other words, f requency increases if we increase the SHA (θ) and all the devices behave in a similar patte rn. Moreover, it is found that f requency is highly tunable in all the SHA (θ) for all the four devices. In particular, Py b as ed SHSTNO device emits maximum f requency of about 71.75 GHz than other devices with maximum SHA θ = 0.8 whereas, for the same θ , CoFeB, Co and Fe emit 67.8, 32.2 and 48.6 GHz for d = 3 nm. Also, the QF of for all the four devices is calculated separately. Interestingly, the maximum QF of about 257, is obtained for Py based SHSTNO device with θ = 0.8 and d = 3 nm whereas for the same θ, CoFeB, Co and Fe emits 170, 125 and 128. At θ = 0.8, not only the maximum QF but also the max imum f requency is obtained. From Figure.2 ( b), it is evident not only the f requency but also the QF is directly proportional to the SHA (θ), (i.e.) QF increases if we increase the SHA (θ) and all the devices behave i n the similar pattern. 110037-3 FIGURE 3 . A plot of Frequency (GHz) vs . Amplitude (arbitrary unit) for Py based SHSTNO at θ = 0.8 . The inset plot of Figure.3 (Left) denotes the M x vs . time (ns) for Py based SHSTNO device at θ = 0.8 , and the inset plot of Figure.3 (Right) denotes the corresponding trajectory. Large value of SHA (θ), with less FF L thickness paves for high f requency, b ecause it has high energy than lower SHA (θ) leads to faster STT emits high f requency. Also, if FFL thickness is low, then more effic ient magnetization precession takes place. It is noted that lower S M based FM materials emit high f requency in FFL. In our case, Py has lower SM than other FM materials emits high f requency as well as high QF than other SHSTNO devices. The author emphasize s that for achieving maximum frequency tunability and QF, one might have to choose the low SM based FM material in FFL with th e minimal thickness and tandemly one has to fix the SHA (θ) to the maximum. The reason for maximum frequency is not only of lower SM material but also it depends on the heavy metal that we have taken. Here, Pt is the heavy metal and the advantage of it over other materials such as Tungsten ( W ) , Ta ntalum (Ta) are Pt is capable of interacting the alloys with other metals to form FM or very near FM. Also, Pt provides smooth layer to promote the growth of subsequent films. The sputtered Pt atoms have high atomic mass expect to have more energy than FM. So bonding of FM is weaker than Pt, consisting the penetration of Pt in FM layers. For these reasons , Pt is taken as heavy metal in SHSTNO device. In the observed results, alloys such as Co 20 Fe 60 B 20 and Ni 80 Fe 20 emits higher frequency than metals such as Co and Fe, because Pt interacts strongly with alloys than metals . As compared to both t he alloys, Ni 80 Fe 20 [ 80% Ni+ 20% Fe=100%] e mits higher frequency than Co 20 Fe 60 B 20 [ 20% Co+ 60% Fe=80%, (B is diamagnetic)] b ecause the former has more influence on FM than later paves for high frequency emission . F igure.3 is plotted for the f requency (GHz) vs . A mplitude (arbitrary unit) for Py based SHSTNO device (because it emits maximum f requency) at θ = 0.8. The inset plot of Figure.3 (Left) denotes the M x vs . Time (ns) for Py based SHSTNO device at θ = 0.8, and the inset plot of Figure.3 (Right) denotes the corresponding trajectory . QF is the dimensionless parameter and it is calculated individually for all the devices using the fundamental formulae [QF = Center Frequency (Peak)/Bandwidth (∆F)] and it is tabulated in Table.1 . In short, the calculated method f or QF will be briefly disclosed on the authors another article [4]. From the results it is evident that the frequency is linearly proportional to QF, because (Peak) is directly proportional to QF ( from the above expression ) . It means more the frequency mor e the QF (narrower and shar per the P eak is) . 110037-4T ABLE 1. Table is tabulated for Frequency (GHz) and QF for all the four devices against the SHA ( θ) . In the table, Frequency (GHz) is represented as F and the Quality Factor is represented as QF. CoFeB Fe Py Co S. No F Q F F QF F QF F QF 1 0.1 6 30 3.2 18 2 61 4 .5 23 2 0.2 12.8 65 6.5 34 10 89 9 .2 46 3 0.3 20.9 84 9.9 51 19 103 14.9 62 4 0.4 29.6 102 14 64 28 146 21.7 87 5 0.5 38.7 129 18.3 74 39 178 27.6 99 6 0.6 48.1 137 22.8 85 49 228 34.4 108 7 0.7 57.9 166 27.5 102 61 244 41.4 116 8 0.8 67.8 170 32.3 125 71.75 257 48.6 128 CO NCLUSION T he magnetization precession dynamics for all the four SHSTNO devices [Pt/Co, Pt/Py, Pt/Fe and Pt/CoFeB] are investigated. The precession dynamics for FFL are studied by solving the governing LLGS equation, numerically by the macro magnetic framework. Tuning of frequency is achie ved by varying the SHA, (θ) and the author sparks that for more frequency tunability one has to fix the (θ) to the maximum and tandemly consider the low SM in FFL. Here, Py which has low SM among all FM emits frequency up to 72 GHz for θ = 0.8. Because, if (θ) is more and less SM in the device, the STT mechanism is stronger, engenders high output frequency. It is to be noted that, not only maximum frequency but also the maximum Quality Factor of about 257 is achieved in Py based device with maximum SHA (θ) as 0.8. Moreover, the results are taken for constant current density throughout the work of about 9.4 ⨉ 10 10 A/m2wi th zero applied magnetic field. The author emphasizes that these can find the application as spin-wave emitters for magnonic applications. A CKNOWLEDGEMENTS P. Sabareesan & H. Bhoomeeswaran acknowledges the Department of Science and Technology (DST), Government of India, for the award of SERB – Young Scientist Project (SB/FTP/PS – 061/2013). RE FERENCES 1. J. C. Slonczewski, J Magn. Magn. Mater. 159, L1-L7 (1996). 2. L. Berger, Phys. Rev. B 54, 9353 (1996). 3. W. H. Rippard ,et al., Phys. Rev. Lett. 92,027201 (2004) . 4. H. Bhoomeeswaran, T. Vivek and P. Sabareesan, AIP Conf. Proc. 1942 , 130030 (2018). 5. H. Bhoomeeswaran and P. Sabareesan, Appl. Phys. A 125, 513 (2019). 6. C. Zheng, et al., Chin. Phys. B. 28, 037503 (2019). 7. H. Kubota, et al., Appl. Phys. Express 6, 103003 (2013). 8. D. Houssameddine, et al., Nat. Mater 6, 447 (2007). 9. I. Firastrau, et al., J. Appl. Phys 113, 113908 (2013). 10. S. M. Mohseni, et al., Phys. Status Solidi 5, 432 (2011). 11. H. Bhoomeeswaran and P. Sabareesan, IEEE Trans. Magn. 54, 4 (2018). 110037-5
1.3243687.pdf	Modulation of propagation characteristics of spin waves induced by perpendicular electric currents X. J. Xing, Y . P . Yu, and S. W. Lia/H20850 State Key Laboratory of Optoelectronic Materials and Technologies, School of Physics and Engineering, Sun Yat-sen University, Guangzhou 510275, People’s Republic of China /H20849Received 8 July 2009; accepted 13 September 2009; published online 9 October 2009 /H20850 We have theoretically and numerically investigated the effect of perpendicular currents on the propagation characteristics of spin waves on a wire composing the free layer of a spin valve. Ourtheory shows that the single Slonczewski’s spin-transfer torque can cause the spin-wave Dopplereffect and modify the spin-wave attenuation and that the ﬁeldlike torque makes negligiblecontribution due to its relatively small magnitude. Micromagnetic simulations conﬁrm thesetheoretical predictions and further reveal that spin waves at suprathreshold currents are instable andbecome chaotic with increased time. Finally, selective tuning of the spin-wave attenuation isdemonstrated by using local, individual currents. © 2009 American Institute of Physics . /H20851doi:10.1063/1.3243687 /H20852 With the rapid development of microfabrication tech- niques, spin waves /H20849SWs /H20850in conﬁned magnetic structures 1–3 have attracted signiﬁcant interest because they deﬁne the basic timescale of spin dynamics.2,3Recently, several device concepts based on processing of SW signals on/H20849sub/H20850micrometer-scale waveguides have been proposed, e.g., SW logic devices. 4–6Some of them have been experimen- tally demonstrated.5,6However, even for the frequently used Permalloy materials with almost the lowest damping param-eter /H20849/H110110.01 /H20850, the dissipation of SWs is fatal. This greatly restricts further progress of these devices toward practical application. Most recently, Seo et al. 7offered a way of re- ducing the damping of a magnetic wire system by means ofthe Zhang and Li’s 8nonadiabatic spin torque, which was mostly studied as external force to drive domain-wall motionin one-dimensional magnetic wires. 8,9In a recent work, Piz- ziniet al.10reported an exceedingly high domain-wall veloc- ity in zigzag spin-valve stripes. Later, Khvalkovskiy et al.11 numerically modeled a wire-shaped spin-valve system and attributed the high domain-wall velocity to the Slonczewski’sand ﬁeldlike torques arising from the perpendicular currentacross the spin valve. Generally speaking, the Slonczewski’s spin torque can perform as an antidamping torque. 12–15Fine treatment can relate the Slonczewski’s and the accompanying ﬁeldlike torques to two effective ﬁelds,11,13each of which serves as the source of both precessional and antidamping motions.13 Accordingly, a perpendicular current is able to modify propa-gation characteristics of SWs externally excited at an end ofa wire and traveling along the wire. Until now, no reportshave yet been made to address the issue despite the well-known antidamping effect of the Slonczewski’s torque. 12–15 In the letter, we studied how the Slonczewski’s spin torque affects SW propagation on a magnetic wire which isthe free layer of a spin valve. The theoretical results showthat the Slonczewski’s torque itself can lead to the Dopplereffect and inﬂuence the SW attenuation. The simulation re-sults verify these theoretical ﬁndings and additionally showthat the SWs can be locally tuned by using independent currents. The model system is a wire-shaped spin-valve structure 10,11composed of a magnetic free layer, a nonmag- netic spacer, and a magnetic ﬁxed layer. The free layer has anin-plane magnetization along the wire length. The magneti-zation of the ﬁxed layer is antiparallel to that of the freelayer. The nonmagnetic spacer is assumed to be thick enoughto avoid any interlayer interaction between the magneticlayers. 10A sinusoidal excitation ﬁeld with the magnitude of 100 Oe is applied to a local region ranging from 200 to 208nm to the left edge of the free layer to steadily excite thelowest-mode SW with frequency f/H20849 /H9275f=2/H9266f/H20850. The current proﬁle is shaped through the delicately deﬁned top electrodes and is assumed to be uniform in the perpendicular direction.To describe the SW dynamics, the Landau-Lifshitz-Gilbert/H20849LLG /H20850equation including the spin-torque terms 11,14is em- ployed, m/H6109˙=−/H9253m/H6109/H11003H/H6109eff+/H9251m/H6109/H11003m/H6109˙−/H9253aJm/H6109/H11003/H20849m/H6109/H11003mˆp/H20850−/H9253bJm/H6109 /H11003mˆp. /H208491/H20850 For deﬁnition of the notations, see supplemental material /H20849Ref. 16/H20850. In micromagnetic simulations,17we focus only on the dynamics in the free layer. The free layer is 10 /H9262m /H20849or more /H20850in length, 120 nm in width, and 6 nm in thickness and was discretized into meshes with a size of 4 /H110034 /H110036n m3. At time t=0, the sinusoidal ﬁeld and the dc current with zero rise time were switched on. The ﬁeldliketorque was taken into account, but the Oersted ﬁeld andthermal ﬂuctuations were not considered. Typical permalloymaterial parameters were used. 18The saturation magnetiza- tion Ms=8.6/H11003105A/m, the exchange stiffness A=1.3 /H1100310−11J/m, and /H9251=0.01. The magnetocrystalline aniso- tropy is neglected and the spin polarization of the current istaken to be P=0.7. For the small-amplitude SWs satisfying the linear ap- proximation, the variable magnetization can be expanded ina series of plane waves of magnetization. 3Approximately, the SWs attenuate exponentially with the increased propaga-tion distance.7Consequently, mcan be expanded in the form,a/H20850Author to whom correspondence should be addressed. Electronic mail: stslsw@mail.sysu.edu.cn.APPLIED PHYSICS LETTERS 95, 142508 /H208492009 /H20850 0003-6951/2009/95 /H2084914/H20850/142508/3/$25.00 © 2009 American Institute of Physics 95, 142508-1m=eˆx+m0e−x/Le−i/H20849kx−wt/H20850, where m0is the SW amplitude ful- ﬁlling m0/H112701/H20849i.e.,m0x/H11270m0y,m0z/H20850andLis the SW attenuation length. After detailed calculations /H20851supplemental material /H20849Ref. 16/H20850/H20852, the main theoretical ﬁndings are arrived at /H9275f2−4/H20849/H92530/H9251kD /L/H20850/H9275f =/H925302aJ2+/H925302/H20849Hk+Hd+Dk2−bJ/H20850/H20849Hk+Dk2−bJ/H20850 +/H925302/H20851/H20849D2/L4/H20850−/H20849D/L2/H20850/H208492Hk+Hd+6Dk2−2bJ/H20850/H20852 /H20849 2/H20850 and L=2/H92530Dk /H9251/H9275f−aJ/H9275f Hk+/H20849Hd/2/H20850+Dk2−bJ. /H208493/H20850 Equation /H208492/H20850is the dispersion relation accounting for the current-induced torques and the SW attenuation. The SWDoppler effect7,19–21caused by the current can be clearly observed when the L-related terms are discarded. Here, the values of these ﬁelds, Hk,Hd,Dk2,aJ, and bJ, should be estimated to illustrate respective contributions of aJandbJ to the Doppler shift. aJ=/H6036JPg /H20849/H9258/H20850//H92620deM s/H20851see supplemental material /H20849Ref. 16/H20850for the deﬁnition of notations /H20852and bJ=/H9264aJwith/H9264having a typical value of about 0.1.22 Using totally the same values of P,d,A, and Msas set in micromagnetic simulations, we obtain Hk/H110116.2/H11003104A/m, Hd/H110117.4/H11003105A/m, Dk2/H110112.4/H11003105A/m, aJ/H110113.6 /H11003104A/m, and bJ/H110110.36/H11003104A/m, if kandJtake the characteristic values of 0.1 nm−1and 1 /H110031012A/m2, re- spectively. These estimations reveal that the Doppler shiftmainly comes from the Slonczewski’s torque and that thecontribution of the ﬁeldlike torque is trivial since b Jis hun- dreds of times smaller than Hk+Dk2andHk+Hd+Dk2. Figure 1shows SW dispersion relations for a series of current densities. It is observed that the Doppler shift, /H9004k,i s always negative regardless of the direction of the appliedcurrent. That is because the Slonczewski’s torque enters Eq. /H208492/H20850in the form of a J2rather than aJ. In addition, the Doppler shift is not evident unless the current density exceeds /H110111 /H110031012A/m2/H20849see inset in Fig. 1/H20850. But, unfortunately, that is not visible for suprathreshold currents due to chaotic dynam-ics. Therefore, the dispersion curves for positive currents areplotted for only the currents well below the threshold value.Equation /H208493/H20850presents that the SW attenuation length can be modiﬁed by the current-induced torques. The contributionof the ﬁeldlike torque b Jcan also be ignored as in Eq. /H208492/H20850 according to similar arguments. The Slonczewski’s torque aJ can enhance or reduce the SW attenuation depending on the current direction. When aJ/H110210 the attenuation length is de- creased. When aJ/H110220 the attenuation length is increased; once aJattains a threshold value, the attenuation length goes inﬁnite suggesting equiamplitude propagation of SWs; fur-thermore, as a Jexceeds the threshold value, the attenuation length becomes negative corresponding to the ampliﬁcationof SWs. Figure 2/H20849a/H20850plots the attenuation length of SWs versus the current density. Evidently, the threshold current values/H20849where the attenuation length changes sign /H20850vary with the frequency of SWs. Top inset of Fig. 2/H20849a/H20850presents a combi- nation of the simulation and theoretical results. It is observedthat the theoretical results are well reproduced by the simu-lation results except for a discrepancy in the magnitude ofthe attenuation length. The good agreement between simula-tions and theory is also evidenced by the fact that they pre- dict almost identical threshold current values /H20851bottom inset of Fig. 2/H20849a/H20850/H20852. Figures 2/H20849b/H20850–2/H20849e/H20850display the spatial distribu- tion of the normalized M zmagnetization along the wire. Fig- ure2/H20849b/H20850shows the SW at zero current, Figs. 2/H20849c/H20850and2/H20849d/H20850 show current-modulated SWs with decreased and increasedattenuation lengths, respectively, and Fig. 2/H20849e/H20850plots the am- pliﬁed SW at a suprathreshold current. Now, we can point that it is more efﬁcient for the Slon- czewski’s torque to control the attenuation of SWs than tocause the Doppler effect of SWs. Regarding the impact onthe attenuation of SWs, the efﬁciency of the perpendicularcurrent is at least tens of times higher than that of the in-plane current 7because of the large ratio of the magnitudes of1015202530 0.00 0.04 0.08 0.12 0.1 60.055 0.0600123456-0.016-0.0080.000J/CID1/CID1x1012A/m2/CID2/CID2 -6.0 -5.4 -4.8 -4.2 -3.6 -3.0 -2.4 -1.8 ±1.2 ±0.6 0.0 k/CID1/CID1nm-1/CID2/CID2f( GHz)f (GHz) 26 20 14 /CID1/CID1k/CID1/CID1nm-1/CID2/CID2\|J\|/CID1/CID1x1012A/m2/CID2/CID2 FIG. 1. /H20849Color online /H20850Dispersion relations /H20849without L-related contribution involved /H20850of the SW at current densities Jranging from −6 /H110031012to 1.2 /H110031012A/m2. Top inset: The Doppler shift as a function of the current density at three selected frequencies. Below 1 /H110031012A/m2, the Doppler shift is small. Bottom inset: Zoom-in view of a portion of the dispersionrelation curves clearly exhibiting the shifts in wave vector k. 0369 036-0.010.000.01-0.010.000.01(d) x/CID1/CID1/CID1/CID1m/CID2/CID2 x/CID1/CID1/CID1/CID1m/CID2/CID2(c) (e)Mz/Ms(b)-5 -4 -3 -2 -1 0 1 2 3 4 5-40-2002040 -1 0 1123 15 30 451234f( G H z ) 10 14 20 30 40 50L/CID1/CID1/CID1/CID1m/CID2/CID2 J/CID1/CID1x1011A/m2/CID2/CID2(a) simulation theory simulationstheoryJthreshold /CID1/CID1x1011A/m2/CID2/CID2 f (GHz) FIG. 2. /H20849Color online /H20850/H20849a/H20850The attenuation length Lof the SWs at given frequencies vs current density J. Top inset: Comparison of simulation and theoretic L/H20849J/H20850relations for three selected frequencies. Bottom inset: Com- parison of simulation and theoretic results of the threshold current vs fre-quency. /H20849b/H20850-/H20849e/H20850Spatial distribution of M z/Mscomponent for SWs at various current densities: /H20849b/H20850zero, /H20849c/H20850−1/H110031011A/m2,/H20849d/H208501/H110031011A/m2, and /H20849e/H20850 1.5/H110031011A/m2. The frequency of the SW is 14 GHz and the moment is 10 ns.142508-2 Xing, Yu, and Li Appl. Phys. Lett. 95, 142508 /H208492009 /H20850the Slonczewski’s torque and the Zhang and Li’s nonadia- batic torque.11 Micromagnetic simulations indicate that SWs at suprath- reshold currents are instable and become fully chaotic withtime /H20851compare Figs. 3/H20849a/H20850and3/H20849b/H20850/H20852. The instability and cha- otic dynamics might be due to the high-order terms in theexpanded LLG equation /H20851supplemental Eq. /H208496/H20850/H20849Ref. 16/H20850/H20852. Tentatively, it is found that the SWs at suprathreshold cur-rents can be stabilized to some extent by using a bias ﬁeld of/H110111000 Oe directed along the free-layer magnetization. Fig- ures 3/H20849c/H20850and3/H20849d/H20850plot SWs on the ﬁeld-biased wire at su- prathreshold currents, which, different from the unbiasedones, is still stable at 20 ns without presenting chaotic be-haviors. Signiﬁcantly, the attenuation of SWs can be locally con- trolled by perpendicular currents. To modify the attenuationof SWs, it is not required to apply the current to the fullrange of the length of the spin valve, and where there arerequirements to tune the attenuation of SWs there the currentshould be utilized. Figures 4/H20849a/H20850and4/H20849b/H20850show SWs locally tuned with perpendicular currents. In each case, currents areapplied to three independent regions deﬁned by the separatedtop electrodes /H20851schematics /H20849I/H20850and /H20849II/H20850/H20852. It is evident that the propagating SWs are selectively modulated by the currents.In Fig. 4/H20849b/H20850, the SW travels away from the source to right at small amplitude in the ﬁrst region, the signal is subsequentlyampliﬁed by a larger current for probing, and ﬁnally theampliﬁed SW is switched off to reduce reﬂections around the boundary. These ﬁndings might ﬁnd application in signalprocessing in SW devices. 4–6For example, a positive sub- threshold current can be used to expand the transmissiondistance of SWs, a suprathreshold current can be used tolocally amplify SWs for read-out, and a negative current canperform as a gate to switch the propagation of SWs. In conclusion, for the wire-shaped spin valve covered with delicately deﬁned top electrodes, the perpendicular cur-rents can cause the Doppler effect and control the attenuationof SWs propagating on the free layer. These effects originatemainly from the Slonczewski’s spin torque. The SW be-comes instable when it is subjected to suprathrehsold cur-rents. The attenuation of SWs can be selectively modulatedby using currents across separated top electrodes. These ﬁnd-ings could potentially be applied in SW devices. The National Natural Science Foundation of China /H20849Grants No. 60977021 and No. U0734004 /H20850funded this work. 1F. Montoncello and F. Nizzoli, in Magnetic Properties of Laterally Con- ﬁned Nanometric Structures , edited by G. Gubbiotti /H20849Transworld Research Network, Kerala, 2006 /H20850, and references therein. 2S. O. Demokritov and B. Hillebrands, in Spin Dynamics in Conﬁned Mag- netic Structures I , edited by B. Hillebrands and K. Ounadjela /H20849Springer, Berlin, 2002 /H20850. 3S. O. Demokritov, B. Hillebrands, and A. N. Slavin, Phys. Rep. 348,4 4 1 /H208492001 /H20850. 4R. Hertel, W. Wulfhekel, and J. Kirschner, Phys. Rev. Lett. 93, 257202 /H208492004 /H20850. 5T. Schneider, A. A. Serga, B. Leven, B. Hillebrands, R. L. Stamps, and M. P. Kostylev, Appl. Phys. Lett. 92, 022505 /H208492008 /H20850. 6M. P. Kostylev, A. A. Serga, T. Schneider, B. Leven, and B. Hillebrands, Appl. Phys. Lett. 87, 153501 /H208492005 /H20850. 7S.-M. Seo, K.-J. Lee, H. Yang, and T. Ono, Phys. Rev. Lett. 102, 147202 /H208492009 /H20850. 8S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 /H208492004 /H20850. 9L. Heyne, M. Kläui, D. Backes, T. A. Moore, S. Krzyk, U. Rüdiger, L. J. Heyderman, A. F. Rodríguez, F. Nolting, T. O. Mentes, M. Á. Niño, A.Locatelli, K. Kirsch, and R. Mattheis, Phys. Rev. Lett. 100, 066603 /H208492008 /H20850. 10S. Pizzini, V . Uhlí ř, J. V ogel, N. Rougemaille, S. Laribi, V . Cros, E. Jiménez, J. Camarero, C. Tieg, E. Bonet, M. Bonﬁm, R. Mattana, C.Deranlot, F. Petroff, C. Ulysse, G. Faini, and A. Fert, Appl. Phys. Express 2, 023003 /H208492009 /H20850. 11A. V . Khvalkovskiy, K. A. Zvezdin, Y . V . Gorbunov, V . Cros, J. Grollier, A. Fert, and A. K. Zvezdin, Phys. Rev. Lett. 102, 067206 /H208492009 /H20850. 12S. Petit, N. de Mestier, C. Baraduc, C. Thirion, Y . Liu, M. Li, P. Wang, and B. Dieny, Phys. Rev. B 78, 184420 /H208492008 /H20850. 13D. Cimpoesu, H. Pham, A. Stancu, and L. Spinu, J. Appl. Phys. 104, 113918 /H208492008 /H20850. 14D. C. Ralph and M. D. Stiles, J. Magn. Magn. Mater. 320,1 1 9 0 /H208492008 /H20850. 15P. M. Braganca, O. Ozatay, A. G. F. Garcia, O. J. Lee, D. C. Ralph, and R. A. Buhrman, Phys. Rev. B 77, 144423 /H208492008 /H20850. 16See EPAPS supplementary material at http://dx.doi.org/10.1063/ 1.3243687 for the detailed procedure used to deduce the theoretical results and the schematic showing the current distribution. 17M. J. Donahue and D. G. Porter, http://math.nist.gov/oommf/ . 18In simulations, g/H20849/H9258/H20850/H20849Ref. 16 /H20850was set to be 1/2 for direct comparison with the theoretical results. 19Y . B. Bazaliy, B. A. Jones, and S.-C. Zhang, Phys. Rev. B 57, R3213 /H208491998 /H20850. 20J. Fernández-Rossier, M. Braun, A. S. Nuñez, and A. H. MacDonald, Phys. Rev. B 69, 174412 /H208492004 /H20850. 21V . Vlaminck and M. Bailleul, Science 322, 410 /H208492008 /H20850. 22M. A. Zimmler, B. Özyilmaz, W. Chen, A. D. Kent, J. Z. Sun, M. J. Rooks, and R. H. Koch, Phys. Rev. B 70, 184438 /H208492004 /H20850.-0.04-0.020.000.020.04 0369 036-0.02-0.010.000.010.02Mz/Ms(a) (b) (c) (d) x/CID11/CID80m/CID12/CID12 x/CID11/CID80m/CID12 FIG. 3. Spatial variation of Mz/Mscomponent at 5 ns /H20851/H20849a/H20850and /H20849c/H20850/H20852and 20 ns /H20851/H20849b/H20850and /H20849d/H20850/H20852. The frequency of the SW is 14 GHz and the current density is 1.5/H110031011A/m2. SWs in /H20849c/H20850and /H20849d/H20850are under a bias ﬁeld of 1000 Oe along the magnetization direction of the free layer. Application of the bias ﬁeldlowers the instability, delaying the occurrence of chaotic dynamics. 0369-0.12-0.060.000.060.12 0369-0.012-0.0060.0000.0060.012 33 ns after J1action x/CID1/CID1/CID1/CID1m/CID2/CID2 x/CID1/CID1/CID1/CID1m/CID2/CID2(b)Mz/Ms(a) 30 ns after J action xz Schematic (I) Schematic (II)mfree mPHbiasJe- Hac mfree mPHbiasJ1 e- HacJ3 J2 FIG. 4. /H20849Color online /H20850Selective tuning /H20849Ref. 16/H20850of SW attenuation. Hbias =1000 Oe and f=14 GHz. /H20849a/H20850J=1.5/H110031011A/m2; in each current range the SW keeps equiamplitude propagation /H20851Schematic /H20849I/H20850/H20852./H20849b/H20850J1=1.5 /H110031011,J2=3/H110031011, and J3=−3/H110031011A/m2; thirty nanoseconds after the ﬁrst current is switched on, the second and third currents are launched/H20851Schematic /H20849II/H20850/H20852. The ﬁrst current sends SW signal to a sufﬁciently far dis- tance, the second current ampliﬁes the signal for read-out, and the thirdcurrent makes the SW attenuate rapidly to avoid reﬂections at the rightboundary.142508-3 Xing, Yu, and Li Appl. Phys. Lett. 95, 142508 /H208492009 /H20850
1.580115.pdf	Solid/liquid/gaseous phase transitions in plasma crystals Hubertus M. Thomas and Gregor E. Morfill Citation: Journal of Vacuum Science & Technology A 14, 501 (1996); doi: 10.1116/1.580115 View online: http://dx.doi.org/10.1116/1.580115 View Table of Contents: http://scitation.aip.org/content/avs/journal/jvsta/14/2?ver=pdfcov Published by the AVS: Science & Technology of Materials, Interfaces, and Processing Articles you may be interested in Order-parameter-aided temperature-accelerated sampling for the exploration of crystal polymorphism and solid- liquid phase transitions J. Chem. Phys. 140, 214109 (2014); 10.1063/1.4878665 Solid-liquid phase transitions in single crystal Cu under shock and release conditions J. Appl. Phys. 115, 143503 (2014); 10.1063/1.4871230 Lindemann measures for the solid-liquid phase transition J. Chem. Phys. 126, 204508 (2007); 10.1063/1.2737054 Solid-liquid phase transitions in CdTe crystals under pulsed laser irradiation Appl. Phys. Lett. 83, 3704 (2003); 10.1063/1.1625777 Infrared Spectra of Some Alkyl Silanes and Siloxanes in Gaseous, Liquid, and Solid Phases J. Chem. Phys. 20, 905 (1952); 10.1063/1.1700592 Redistribution subject to AVS license or copyright; see http://scitation.aip.org/termsconditions. Download to IP: 130.113.111.210 On: Mon, 22 Dec 2014 02:38:57Solid/liquid/gaseous phase transitions in plasma crystals Hubertus M. Thomasa)and Gregor E. Morﬁll Max-Planck-Institut fu ¨r Extraterrestrische Physik, 85740 Garching, Germany ~Received 16 October 1995; accepted 14 December 1995 ! We present further observations of the solid/liquid and liquid/gaseous phase transition in plasma crystals. Plasma crystal is the term used to describe the recently discovered ‘‘state’’that a colloidalplasma may assume under certain conditions—a state which has properties resembling those ofmetals. During the melting transition from solid to liquid the system passes through an intermediate‘‘ﬂow and ﬂoe’’stage that has not been observed in other model crystals before. It may well be thatthis intermediate stage is a general feature of the solid/liquid phase transition in crystals. In this caseit is clearly important. The fact that this stage could be detected for the ﬁrst time is a consequenceof the unique properties of plasma crystals: global charge neutrality, very fast response and littledamping, easy experimental control, detailed imaging, and ﬁne time resolution of the dynamics ofindividual particles ~‘‘atoms’’ !.©1996 American Vacuum Society. I. INTRODUCTION The detailed study of the processes accompanying the melting of a crystal is of great interest for solid state physics.Many effects associated with this transition, annealing, forexample, are not understood in detail and their investigationis very complicated in real atomic or molecular systems. Dy-namical analyses on a molecular or atomic level are all butimpossible experimentally; structure analysis can be per-formed by, e.g., refraction experiments which provide only atime and space average of the structure and calculations inFourier space. The recently discovered plasma crystals, 1–3which consti- tute a new type of crystalline system, have some uniqueproperties: detailed imaging and ﬁne time resolution of thedynamics of individual ‘‘atoms.’’These properties allow theinvestigation of processes such as the solid/liquid/gaseousphase transitions, self-organization, coherent and incoherent dynamical structures, etc., and may help us to understand thephysics behind them. For instance, the possibility of intro-ducing lattice defects to study their dynamics and thereforetheir energetics during a soft heating and cooling experimentmay solve some questions concerning the processes that leadto annealing. The plasma crystal is the solid phase of the so-called ‘‘dusty plasmas,’’ which are systems consisting of chargeddust particles ~of micron size !embedded in a neutral plasma ~ions and electrons !that may contain neutral gas as well.The particles interact via their Coulomb forces and the dust cloudmay form a gaseous, liquid, or solid phase depending on thekinetic energy of the particles and their charge. 4It can be viewed as a macroscopic model for a crystal and comple-ments the ion crystals 5–7and electron crystals8,9on the atomic scale and colloidal crystals in aqueous solutions10–12 on the macroscopic scale. Comparing the plasma crystalswith the well known systems mentioned above, we clearlyregister similarities with the colloidal crystals in aqueous so- lutions. The plasma crystals differ from these, however,through the diffuse medium between the particles and theirsizes, which are in the micron range. The medium is a lowdensity plasma; as a result the plasma crystals do not sufferfrom the disadvantages of ﬂuid systems, namely, the opticalthickness, the strong overdamping, the slow adjustmenttimes, and the difﬁculties in controlling the system. For ex-ample, equilibration times are typically a million timesfaster. For two-dimensional colloidal crystals in aqueous solu- tions a melting theory that can also be used for two-dimensional ~2D!plasma crystals, was developed. This theory is developed in Refs. 13–17 and was ﬁrst applied toplasma crystals by Quinn et al. 18It refers to correlation func- tions that are calculated from the positions of the particles.These are then used to determine the ‘‘phase’’ or ‘‘state’’ ofthe system. In most cases the pair g~r!and bond-orientational g 6(r) correlation functions11,19are used for structure analysis and for comparison with the so-called KTHNY meltingtheory. 5This theory identiﬁes a so-called ‘‘hexatic’’ phase, intermediate to the crystalline and liquid phase. Here we list commonly accepted criteria11involving g~r! andg6(r) for identifying the various phases. In the crystal- line phase, g(r)}r2h(T)andg6(r)5const, where h(T)<1/3 and is weakly temperature dependent. In the hexatic phase, g(r)}exp(2r/j) andg6(r)}r2hwith 0,h<1/4. In the liquid phase, g(r)}exp(2r/j),g6(r) }exp(2r/j6) and j5j6. Here jandj6are scale lengths for translational and orientational orders, respectively. In the liq-uid phase jis smaller than in the hexatic. In the gas phase g(r)'1 after an increase from 0 and g6(r) oscillates around zero so that the ﬁts no longer make sense. These criteria arebased on the KTHNY theory; other empirical criteria, suchas the numbers of nearest neighbors, are sometimes used aswell. 20 II. EXPERIMENTAL SETUP The plasma crystal experiment1is performed in a radio frequency discharge chamber especially designed for basica!Permanent address: DLR-Institut fu ¨r Raumsimulation, Linder Ho ¨he, 51140 Ko¨ln, Germany. Electronic address: Hubertus.Thomas@europa.rs.kp.dlr.de 501 501 J. Vac. Sci. Technol. A 14(2), Mar/Apr 1996 0734-2101/96/14(2)/501/5/$10.00 ©1996 American Vacuum Society Redistribution subject to AVS license or copyright; see http://scitation.aip.org/termsconditions. Download to IP: 130.113.111.210 On: Mon, 22 Dec 2014 02:38:57plasma research, the so-called GEC RF Reference Cell. The GEC RF Reference Cell is described in Ref. 21; our modiﬁ-cations for the plasma crystal experiment were: ~i!the upper electrode system was removed and re- placed by a ring electrode without insulator; ~ii!a window was installed on the top chamber ﬂange; ~iii!a dust dispenser ~a movable sieve, mounted over the hole in the upper ring electrode !was built for par- ticle injection. These changes allow a vertical view into the plasma perpendicular to the electrodes. The rf ﬁeld between thetwo electrodes partially ionizes the neutral gas ~krypton ! at a pressure between 0.1 and 0.5 mbar ~ionization fraction of 10 27–1026). Monodispersive melamine/formaldehyde spheres of ~6.960.2!mm diameter are injected into the plasma, where they become charged and are levitated by aconstant electrostatic ﬁeld in the sheath of the lower elec-trode. For suitable plasma parameters the particles orderthemselves in a regular lattice and form a disk shaped cloudof more than 100 lattice distances in the horizontal directionand a few ~with a maximum of 18 !lattices in the vertical direction. Observations were made by illuminating a planewith a sheet of laser light.The two-dimensional structure canbe observed with a charge coupled device ~CCD!camera with a macrolens positioned over the upper ring electrodeand can be stored on a VCR. The CCD camera as well as theillumination system is adjustable in the vertical direction sothat one can obtain three-dimensional pictures or one canfollow particles as they move about. The latter is of interestwhen the plasma parameters are changed. III. PLASMA CRYSTAL EXPERIMENTS In the experiment described here, we study the phase tran- sition of a plasma crystal to its liquid and gas phase. Theparticles that constitute the crystal are observed individuallyand their motion is followed in two dimensions. The phasetransition is initiated by a decrease of the neutral gas pres-sure. This, in turn, leads to an adjustment of the plasmaparameters ~see, e.g., Ref. 22 !which determine the values of Gand k.Gandkare dimensionless parameters normally used to describe the thermodynamics of colloidal systemsinstead of temperature and density. 23,24The so-called Cou- lomb coupling parameter Gis deﬁned as the ratio of the Coulomb energy between two neighboring particles to theirkinetic energy and kis the ratio of the lattice distance Dto the Debye screening length. In this way it is easy to controlthe plasma crystal through its melting phase transition. Figure 1 ~a!shows the trajectories of the particles in the crystalline phase at a neutral gas pressure of 0.42 mbar. Astatic analysis of the particle positions according to theKTHNY 2D melting theory was performed and clearly indi-cates the crystalline structure. The g(r) andg 6(r) functions are shown in Figs. 2 and 3. The ﬁt to g(r) leads to a power law slope proportional to r2h, with h50.059. This is much lower than the above mentioned value of 1/3 as a criterionfor the solid phase.The Debye–Waller factor, responsible forthe broadening of the peaks in g(r), isb50.013. The ﬁt tog 6(r) in Figure 3 ~1signs!results in ﬁt parameters of j65306 for the exponential ﬁt and h50.01 for the power law ﬁt. Another structural analysis for hexagonal crystals isto count the fraction of particles having six bonds ~sixfold coordination !. This fraction is .90% for the data shown in Fig. 1 ~a!, also indicating the crystalline phase. Apart from the static structure analysis, the plasma crystal experimentalso allows dynamical analyses both of single particles andthe whole cloud. It is evident that the hexagonal structure fornearly all of the 392 particles ~mean particle number, aver- aged over all frames !in the frame is stationary.Afew single particles near dislocations in the lattice exhibit minorchanges in their positions during the observation time of 1 s.An arrow shows the direction of the motion if the distancetraveled is larger than D/10 in a second. In the marked win- dow larger scale motion can be seen. This motion is initiatedby the migration of a single particle to another lattice plane.This particle oscillates vertically in and out of the ﬁeld ofview before it disappears. Then reordering of the neighbor-ing particles takes place. In Fig. 1 ~b!the trajectories of the particles are shown for a lower pressure ~0.38 mbar !. It is obvious that the hexagonal structure is not as well established as in Fig. 1 ~a!. The bond- orientational correlation function gives us a scale length of j6519 lattice constants and h50.12, comparable with the values pertaining to the so-called hexatic phase ~intermediate phase, which has quasi-long-range orientational order, butshort-range translational order !. The percentage of particles with sixfold symmetry is 83%. The mobility of the particles has increased and some local changes in the structure ap-peared. In the marked window a position change coupledwith an out-of-plane motion can be observed. A particlemoves towards a neighboring particle which disappears inthe vertical direction. At the same time a particle appearsclose to the starting position of the moving one. Then reor-dering takes place. The changes in the structure due to a further decrease of the pressure are shown in Figs. 1 ~c!and 1 ~d!, both at a pres- sure of 0.36 mbar. Figure 1 ~d!shows the sequence following directly that shown in Fig. 1 ~c!. From Fig. 1 ~c!it is obvious that the local motion of the particles has increased althoughthe ﬁt to the bond-orientational correlation function ~scale length of 23 and 26 lattice constants, respectively !, and the sixfold symmetry ~over 80% for both !are comparable to those of Fig. 1 ~b!. More out-of-plane motion occurred, lead- ing to a reordering and ﬂow of the particles in the observedplane. In the marked window ~dotted lines !a new particle is seen to appear leading to the displacement of the originalparticles around it. In the second window ~dashed lines !a particle is seen to disappear. The particles to the right movetowards the new formed dislocation to restore order. Theseparticles rearrange to ﬁll up the vacant spot. This motioninﬂuences particles as far as about six lattice distances. Areverse ﬂow occurred in the lattice line above the latter. In Fig. 1 ~d!we recorde d2st oshow not only local motion502 H. M. Thomas and G. E. Morﬁll: Solid/liquid/gaseous phase transitions in plasma crystals 502 J. Vac. Sci. Technol. A, Vol. 14, No. 2, Mar/Apr 1996 Redistribution subject to AVS license or copyright; see http://scitation.aip.org/termsconditions. Download to IP: 130.113.111.210 On: Mon, 22 Dec 2014 02:38:57and oscillations but directed larger ﬂows. The particles from the dotted window in Fig. 1 ~c!exhibit an ‘‘eddylike’’ ﬂow pattern now. It can be seen that many particles participate inordered macroscopic ﬂows while many others seem to staystill and are unaffected by the ﬂow. This transition stage between solid and liquid is reminiscent of an ocean with iceﬂoes, and is described as the ‘‘ﬂow and ﬂoe’’ stage. 25 Figure 1 ~e!depicts a liquid phase.At this neutral gas pres- FIG. 1. Trajectories of particles as observed over successive video frames ~the number of frames and the mean lattice constant are indicated in the upper left-hand corner; the time between successive frames is 0.02 s !at different neutral gas pressures ~indicated in the upper right-hand corner !corresponding to different phases of the plasma crystal. ~a!The crystalline phase, ~b!–~d!a transition phase ~possibly hexatic !,~e!ﬂuid, and ~f!gas~the following sides !. The direction of the trajectories is marked with an arrow if the particles stay in the ﬁeld of view in all frames and the traveled distance from beginning to end is larger than D/10 in 1 s.503 H. M. Thomas and G. E. Morﬁll: Solid/liquid/gaseous phase transitions in plasma crystals 503 JVST A - Vacuum, Surfaces, and Films Redistribution subject to AVS license or copyright; see http://scitation.aip.org/termsconditions. Download to IP: 130.113.111.210 On: Mon, 22 Dec 2014 02:38:57sure of 0.29 mbar we found a scale length of one lattice constant. At some locations randomlike motion can now beobserved. The last phase that we observed has to be regarded as a gas phase @Fig. 1 ~f!#.At a pressure of 0.22 mbar the particles move randomly through the ﬁeld of view and out of it. TheCoulomb interaction between the particles is so weak at thisstage, that particles only interact when they come close to-gether. Their kinetic temperature, calculated from the dis-tances traveled in a given time, has strongly increased fromabout room temperature in the crystalline phase to 2.4 eV inthe liquid phase at 0.29 mbar and 4.4 eV at the gas phase at0.22 mbar. The last two values are determined by ﬁtting aMaxwellian ﬁt to the measured velocity distribution func-tion. The temperature would increase further if the neutral gas pressure were decreased even more, but the particlescannot be followed then because their motion is too rapid forour CCD camera. The particles show large segments of theirtrajectories ~of a few lattice constants !during the exposure time of 0.02 s.This strong increase of the kinetic temperatureof the particles is not yet understood in detail ~a comparable result can be found in Ref. 26 !. One effect is the reduced damping of the particles through friction with the neutrals atlower pressures. Another effect may be a higher drag on theparticles by the ions which pass through the sheath and maytherefore collide with the particles on their way to the lowerelectrode. The mean free path of the ions increases with de-creasing neutral gas pressure and becomes comparable withand eventually exceeds the sheath thickness. Hence the ac-celeration of the ions in the sheath potential is more effectiveand no longer hindered by scattering on the neutrals. The melting can be followed also by analyzing the pair correlation between the particles in the different phases. Thethree-dimensional plot ~Fig. 4 !shows changes in the transla- tional order of the system with time, t, wheret50 corre- sponds to the beginning of the melting ~pressure reduction ! sequence. At t50~crystalline phase !, the peaks of g(r) can clearly be identiﬁed as the peaks of the ideal hexagonal lat-tice. With passing time ~decreasing pressure !the heights of the peaks decrease, their number is reduced due to the over-lapping of neighboring peaks and due to the disappearance ofthe correlation peaks at larger distances. This developmentcontinues until even the ﬁrst peak disappears, and g(r) takes on the constant value of unity at all distances r/D. IV. CONCLUSION We have presented new and detailed observations of the solid/liquid/gaseous phase transitions in plasma crystals. Dueto the special properties of plasma crystals it was possible toobserve the dynamics of these transitions in unprecedenteddetail. A new intermediate phase, the so-called ﬂow and ﬂoe FIG. 2. Normalized pair correlation function vs normalized distance. Experi- mental and least-squares ﬁt pair correlation function are shown as well asthe dfunctions for a perfect crystal ~the latter are shown as peaks with correct positions and relative heights using an arbitrary vertical scale !. The ﬁt yielded the mean interparticle spacing D5273mm,h50.059, and the Debye–Waller factor b50.01. FIG. 3. Bond-orientational correlation function g6vs normalized distance for the different phases ~pressures !from Fig. 1 and exponential and power law ﬁts. The ﬁt parameters to the power law ~dashed lines, h!and exponential ﬁts~solid lines, j6) for the calculated bond-orientational correlation func- tions are shown on the right side. The signs correspond to 1~0.42 mbar !,* ~0.38 mbar !,d~0.36 mbar !,L~0.32 mbar !,n~0.29 mbar !,h~0.22 mbar !, and3~0.20 mbar !. FIG. 4. Three-dimensional plot of the normalized pair correlation function vs normalized distance and time.The neutral gas pressure is proportional to thetime; 0 s corresponds to 0.42 mbar and the beginning of the depressuriza- tion. The decrease of the order of the system due to melting of the plasmacrystal can be seen here clearly.504 H. M. Thomas and G. E. Morﬁll: Solid/liquid/gaseous phase transitions in plasma crystals 504 J. Vac. Sci. Technol. A, Vol. 14, No. 2, Mar/Apr 1996 Redistribution subject to AVS license or copyright; see http://scitation.aip.org/termsconditions. Download to IP: 130.113.111.210 On: Mon, 22 Dec 2014 02:38:57stage, was found between the solid and the liquid phases. If this stage is not peculiar to the plasma crystal but rather agenerally occurring stage in the crystalline melting process,its discovery could be of major signiﬁcance. Evidence fromthe limited experimental work done so far points in the di-rection that the ﬂows may be spatially associated with latticedefects ~in the lattice plane under observation or in the near- est neighbors !. This will have to be conﬁrmed by further experimental work. It is then reasonable to link this stage tothe annealing process and investigate this too. It may well bethat the heterogeneous ﬂow and ﬂoe structure allows, and atthe same time limits, the degree of annealing possible. Theunique properties of the plasma crystals should allow us toinvestigate this in detail. ACKNOWLEDGMENTS The authors would like to thank Lorenz Ratke for helpful discussions. This work is supported by DARA. 1H. 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Morﬁll: Solid/liquid/gaseous phase transitions in plasma crystals 505 JVST A - Vacuum, Surfaces, and Films Redistribution subject to AVS license or copyright; see http://scitation.aip.org/termsconditions. Download to IP: 130.113.111.210 On: Mon, 22 Dec 2014 02:38:57
1.5144841.pdf	Appl. Phys. Lett. 116, 102401 (2020); https://doi.org/10.1063/1.5144841 116, 102401 © 2020 Author(s).Complex switching behavior of magnetostatically coupled single-domain nanomagnets probed by micro-Hall magnetometry Cite as: Appl. Phys. Lett. 116, 102401 (2020); https://doi.org/10.1063/1.5144841 Submitted: 13 January 2020 . Accepted: 23 February 2020 . Published Online: 09 March 2020 N. Keswani , Y. Nakajima , N. Chauhan , T. Ukai , H. Chakraborti , K. D. Gupta , T. Hanajiri , S. Kumar , Y. Ohno , H. Ohno , and P. Das COLLECTIONS This paper was selected as an Editor’s Pick Complex switching behavior of magnetostatically coupled single-domain nanomagnets probed by micro-Hall magnetometry Cite as: Appl. Phys. Lett. 116, 102401 (2020); doi: 10.1063/1.5144841 Submitted: 13 January 2020 .Accepted: 23 February 2020 . Published Online: 9 March 2020 N.Keswani,1 Y.Nakajima,2N.Chauhan,2T.Ukai,2H.Chakraborti,3K. D. Gupta,3T.Hanajiri,2 S.Kumar,2 Y.Ohno,4,a)H.Ohno,4and P. Das1,b) AFFILIATIONS 1Department of Physics, Indian Institute of Technology, Delhi, New Delhi 110016, India 2Bio-Nano Electronics Research Centre, Toyo University, Saitama 3508585, Japan 3Department of Physics, Indian Institute of Technology, Bombay, Mumbai 400076, India 4Research Institute of Electrical Communication, Tohoku University, Sendai 980-8577, Japan a)Present address: Faculty of Pure and Applied Physics, University of Tsukuba, Tsukuba, Ibaraki, 305-8573, Japan. b)Author to whom correspondence should be addressed: pintu@physics.iitd.ac.in ABSTRACT We report here the results of two-dimensional electron gas based micro-Hall magnetometry measurements and micromagnetic simulations of dipolar coupled nanomagnets of Ni 80Fe20arranged in a double square ring-like geometry. We observe that although magnetic force microscopy images exhibit single domain like magnetic states for the nanostructures, their reversal processes may undergo complex behavior. The details of such reversal behavior are observed as speciﬁc features in micro-Hall magnetometry data, which are comparable with themicromagnetic simulation data. Published under license by AIP Publishing. https://doi.org/10.1063/1.5144841 Patterned magnetic nanostructures open up possibilities to study geometry dependent interesting magnetic behavior that may be useful for modern spintronic devices. 1,2Therefore, controlled fabrication of nanostructures by lithography techniques resulted in a ﬂurry of research activities recently.3A nanomagnet of strong shape anisotropy can mimic a single macrospin, which can act as a binary switch due to its two stable states.4–6Although such nanomagnets contain a large number of spins interacting via strong exchange interaction, their net macrospin-like behavior can be used as a building block of potential single spin-like logic circuitry operating at room temperature.Energetically, using such nanomagnets interacting predominantly via dipolar interactions is advantageous compared to solid state based logic circuits. 7Several developments toward designing such practical devices exploiting the role of individual nanomagnetic states have taken place.6However, recent developments in creating more complex structures make it interesting to utilize the collective behavior of the nanomagnets for such practical applications.8–11It has been demon- strated that magnetostatically coupled nanomagnetic system based computational logic devices have further advantage of nonvolatility in addition to their low-power requirement.3,10,12–18The magnetization switching behavior of such nanomagnets in a magnetostaticallycoupled environment may be complicated by the mutual interactions. Therefore, it is clear that in order to realize potential applications of such nanomagnetic structures, engineering and control of the mag- netic states of such structures are essential. This requires an in-depth understanding of the switching behavior of the nanomagnets in adipolar coupled environment. Moreover, recent developments in the use of such shape anisotropic nanomagnets mimicking Ising spin-like behavior have opened avenues to create arrays in different geometries and explore the underlying physics. 19–24While switching behaviors of nanomagnets of simple geometries have been adequately reportedin the literature, 25,26little is known when such nanomagnets are placed in complex arrangements within a dipolar coupled environment.27In general, extracting the detailed behavior from aver- age magnetic measurements of large arrays is not straightforward. Typically, for global measurements of magnetization, large numbers(typically in the range of several thousands to a few millions) of such nanomagnets are used. 28,29Minute information regarding magnetiza- tion changes during the reversal process in these nanomagnets gets eliminated from the average data obtained from global measurements. Therefore, studies on individual building blocks of such complexstructures are desired. Appl. Phys. Lett. 116, 102401 (2020); doi: 10.1063/1.5144841 116, 102401-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplIn this work, we investigated the collective switching behavior of nine dipolar coupled nanomagnets of strong shape anisotropy arranged in two square ring-like geometry. Such rings are building blocks of dif-ferent engineered systems, 12,30and therefore, understanding their switching behavior in this geometry may be helpful to elucidate their behavior in more complex structures. In our earlier work of micromag-netic simulations, we observed simultaneous magnetization reversals of nanomagnets, which are in the next nearest neighbor positions, sugges- ting an indirect coupling of the nanomagnets in the given system. 31In order to understand the role of the next nearest neighbor nanomag- netic elements in the ring structure, two nanomagnets of the same dimensions as the others were patterned at the next nearest neighborpositions without changing the symmetry as shown in Fig. 1(a) . Although our magnetic force microscopy (MFM) measurements show that all these nanomagnets are in a single domain state, detailed magne-tization reversals studied by employing the highly sensitive two- dimensional electron gas (2-DEG) based micro-Hall magnetometry technique reveal several interesting features. Micromagnetic simula-tions show that these features are due to changes in the magnetic state of individual nanomagnets in the dipolar coupled environment. For our studies, an array of Hall devices was fabricated from a molecular beam epitaxy (MBE)-grown modulation-doped GaAs/AlGaAs heterostructure. 32T h e2 - D E Go ft h eh e t e r o s t r u c t u r el i e s approximately 230 nm below the surface. Hall bars of 2 /C22lm2are patterned using electron-beam lithography (EBL) followed by wet-chemical etching. Ohmic contacts with the 2-DEG were ensured by fol- lowing metallization steps involving Ni, Au, and Ge layers. 33The detailed fabrication steps are discussed in our earlier work.34Shape anisotropic nanomagnets of Ni 80Fe20of dimensions 300 /C2100/C225 nm3 are deﬁned on the active area of Hall bars using a second EBL step in combination with the lift-off process (see Fig. 1 ). The center-to-center distance between each nanoisland is 450 nm. A Ti layer of thickness 5 nm was used to increase the adhesion of the Permalloy on the GaAssurface. A capping layer of Al of thickness 5 nm was deposited on Ni 80Fe20to prevent oxidation of the magnetic layer. Entire deposition was carried out using electron beam induced deposition without break-ing the vacuum. The measurements were carried out using an oxford instruments’ Heliox cryostat. The sheet carrier density ( n)a n dm o b i l i t y (l) of the 2-DEG were determined to be 3.59 /C210 11/cm2and 3.7 /C2105 cm2/V s, respectively, at T¼300 mK. For the magnetic measurements, the external magnetic ﬁeld was applied in plane as shown in Fig. 1(b) . In this measurement geometry, the measured Hall voltage ( VH)i sp r o - portional to the average z-component of the magnetic stray ﬁeld ( hBzi) emanating from the magnetic nanostructures. This was conﬁrmed fromthe Hall voltage measurements on an empty Hall cross (not shown). The measurements were carried out using the standard Lock-In technique. Figure 2(a) shows the topography of the nanostructures as obtained using an atomic force microscope (AFM). Irregularities inthe shapes of the nanoislands are observed which arises from the weakadhesion of Ni 80Fe20on the GaAs surface as well as the lift-off process. However, the corresponding magnetic force microscopy (MFM) data in the remanent state exhibit clear bright and dark patches in eachnanomagnet, showing that all the nanomagnets are magnetically inthe single domain state [see Fig. 2(b) ]. As mentioned above, such sin- gle domain state can be considered as a classical analog of an Ising-like macrospin and has been exploited in artiﬁcial spin ice (ASI) as well as logic devices. Moreover, the data show that the three horizontal as wellas six vertically placed nanomagnets are ferromagnetically aligned.Such ferromagnetically aligned nanomagnets in two different sublatti-ces form onion type loops in the two ring-like arrangements, which is evident from Fig. 2(b) . 35The MFM data further indicate that the two nanomagnets at next nearest neighbor positions do not interact signiﬁ-cantly with the elements involved in the ring structure. This is evi-denced by the observation of ferromagnetic alignment of thecorresponding vertical nanomagnets. FIG. 1. (a) SEM image of nanomagnets grown on a GaAs/AlGaAs Hall sensor. (b) Schematic diagram of the measurement setup. FIG. 2. (a) AFM image of the magnetic nanoislands at room temperature. (b) Corresponding MFM image showing the magnetic contrasts of single domainislands of Ni 80Fe20. Dotted loops with arrows are a guide to the eye for the magneti- zation directions of the nanoislands. Magnetization of the square rings forming onion type loops are shown by four other arrows. (c) Micro-Hall magnetometry data showing hysteresis in Hall voltage of 2-DEG vs external magnetic ﬁeld. The fea-tures in the hysteresis loops are identiﬁed by numbers and arrows. Upsweep anddownsweep features at corresponding ﬁelds are identiﬁed by the same numbers.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 116, 102401 (2020); doi: 10.1063/1.5144841 116, 102401-2 Published under license by AIP PublishingNext, in order to investigate the detailed switching behavior of these nine dipolarly coupled nanomagnets, measurements of hBziof the nanomagnetic system were carried out using the high-sensitivemicro-Hall magnetometry method. The measurements were carried out at T¼1.6 K. At this temperature, the mean free path of carriers is estimated to be /C243:3lm. Thus, the electronic transport in the 2-DEG at the experimental temperature occurs in the ballistic regime. Due to the strong shape anisotropy ( K s/C246:5/C2104Jm/C03) of these nanomag- nets, the nanomagnets are athermal. The saturation ﬁeld ( Bext)f o r these nanomagnets is /C24200 mT. The measurements were carried out for an in-plane external ﬁeld of 6300 mT applied along the [10] direc- tion, which is the easy axis for the horizontally placed nanomagnetic islands [hard axis for the vertical islands, see the inset of Fig. 2(c) ]. As the ﬁeld is swept within 6300 mT, a hysteresis is observed in the Hall voltage of the 2-DEG with several reproducible features for both the down and up sweep of the ﬁeld, as shown in Fig. 2(c) . These changes in the Hall voltage reﬂect the changes in the magnetic state of the dipo- larly coupled nanomagnets. The data exhibit distinct steps as well as broad peaks at speciﬁc ﬁeld values while sweeping the ﬁeld along both the directions. Speciﬁcally, we observe the major features as a sharp drop of signal at /C24682 mT(1), three peaks at /C24615 mT(2), /C24750 mT(3), and /C247110 mT(4), respectively and two sharp jumps at/C247125 mT(5) and /C247150 mT(6), respectively. Here, the ﬁrst sign is for the downsweep and the second for the upsweep ﬁeld, respec-tively. The numbers in brackets are to identify the features as also indi- cated by the arrows in Fig. 2(c) . Interestingly, two features as described above are observed in the ﬁrst quadrant of the V H-Bloop. While the sharp jumps may indicate magnetization switchings of individual nanoislands,27the origin of other reproducible features such as peaks appearing in both ﬁeld sweep directions are not immedi- ately clear. These results may indicate speciﬁc changes occurring within the coupled nanostructures induced by an external ﬁeld whichis not directly accessible by MFM measurements. We note here that the high sensitivity of the 2-DEG based Hall sensors have been used to detect nucleation and annihilation of magnetic vortices in individual nanostructures, interaction of domain walls with Peierls potential, etc. 36–39In order to understand the observed features which are likely to be the results of complex switching processes involving the multiple nanostructures, we carried out micromagnetic simulations of the entire system in the presence of an in-plane ﬁeld applied along the[10] direction. Our ground state simulations were performed using the ﬁnite difference based Object Oriented Micro Magnetic Framework (OOMMF). 40Typical experimentally reported values of saturation magnetization Ms¼8.6/C2105A/m, the exchange stiffness constant A¼13 pJ/m, and a damping constant of 0.5 for Ni 80Fe20are used for the calculations.41The magnetocrystalline anisotropy is neglected in the computation. For the simulations, we used the exact experimental structure of the nanomagnets as obtained by AFM. For the nanomagnetic system under consideration, any deviation from the single domain Ising-like state is reﬂected in the magnetization My along the [01] direction. The inset of Fig. 3 shows the simulated results of hysteresis in Myfor the nanostructures. For comparison, the simula- tion data as well as the experimental data for only the downsweep ﬁeldare plotted in the main ﬁgure. As can be seen from the plot, the simu- lation results capture several experimentally measured features remarkably well. Particularly, we observe clear features due to the magnetic activity before remanence, the peak at about 751 mT, andseveral other sharp jumps in the ﬁeld range where features in the experimental data are observed. The results allow us to investigate indetail the exact micromagnetic state of individual nanomagnets andthe changes of these states in the dipolarly coupled environment whichis induced by the external ﬁeld. At the ﬁrst quadrant of the M yvsBloop (i.e., before remanence), the simulation results exhibit features at 30 mT, 27 mT, and 18 mT,respectively. As the ﬁeld is downsweeped from the saturation, weobserve that the detailed micromagnetic behavior of some of the nano- structures changes differently which is most likely due to the irregular- ities in the structures and the different local dipolar ﬁelds which thenanostructures experience. These micromagnetic analysis are showninFig. 4 .F o r B<B sat, the magnetization of the nanomagnets 4, 5, and 9[ a si d e n t i ﬁ e di n Figs. 2 and4(a)] starts to rotate toward their easy axis whereas 6 and 8 start to form vortices (see the discussions below).The magnet 7 behaves differently which, at about 27 mT, suddenlyswitches stabilizing a horse-shoe type loop at the lower ring whereas the upper ring shows an onion-type loop as shown in Fig. 4(c) .A t about 18 mT, both nanomagnets 6 and 8 forms vortices where featuresare observed both in M yas well as experimentally measured VH[see Fig. 4(d) ]. Between /C2418 mT and /C050 mT, the vortex cores for mag- nets 6 and 8 appears to shift from the lower end to the upper end. At about /C051 mT, the island 1 switches thus forming a microvortex state at the upper ring. Therefore, the upper ring has a microvortex and thelower ring has a horse-shoe state as seen in Fig. 4(f) . At this ﬁeld, a peak is observed in the experiments and a corresponding reverse step is observed for M y.A t/C093 mT, magnetization of the island 3 switches thus converting the lower ring to the two head-to-head and two tail-to-tail type loops [see Fig. 4(g) ]. At/C24/C0 96 mT, island 2 switches thereby forming a horse-shoe type state for both the rings as shown in Fig. 4(h) . These changes in Mappear to result in a peak in the average hB ziwhich is observed between /C097 mT and /C0110 mT in the mea- sured Hall voltage. With the switching of nanomagnet 2, the magnet- izations of all the horizontal islands align along the applied ﬁeld direction. As the ﬁeld reaches /C24/C0 129 mT, the island 4 undergoes a sudden change in magnetization orientation showing a switching likebehavior, thus a horse-shoe state in upper ring changes to an onion state [see Fig. 4(i) ]. In general, such a sudden change in magnetization FIG. 3. Magnetization Myas obtained from the simulation as well as Hall voltage VHdue to the average z-component of stray ﬁeld hBzimeasured in the Hall voltage of 2-DEG. The inset shows the complete hysteresis loop obtained from the simulation.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 116, 102401 (2020); doi: 10.1063/1.5144841 116, 102401-3 Published under license by AIP Publishingfor this nanomagnet is unexpected as the applied ﬁeld direction is along the hard axis. The corresponding sharp step in VHvsBdata is observed at this ﬁeld. At /C24/C0 132 mT, the magnetization of island 8 suddenly changes, exhibiting a switching-like behavior. Next, the island 5 switches at /C0138 mT [see Fig. 4(k) ]. This reversal converts the horse-shoe state to the onion state in the lower ring,which forms onion states in both the rings. Finally, at –207 mT, themagnetization of the vertical nanoisland 9 switches, thereby satu-rating the magnetization as shown in Fig. 4(l) . It is interesting to note that with the applied external ﬁeld direction in this case, whichis along the hard direction for the vertically placed nanomagnets,gradual rotation of magnetization and not sudden switching inindividual nanomagnets are expected for magnets 4, 5, 8, and 9. However, clearly, these nanomagnets show a sudden switching-like behavior demonstrating that the dipolar interaction may lead tonontrivial micromagnetic states in nanomagnetic systems whichare otherwise in a single domain state as also observed from theMFM data as shown in Fig. 2(b) . To determine if there is any role of dipolar interaction in stabilizing magnetic vortex states in nano-magnets 6 and 8, the micromagnetic states of these nanostructuresin the isolated state are investigated. It is observed that for theapplied ﬁeld along the hard direction, the nanomagnets indeedexhibit magnetic vortex states without any dipolar interaction (not shown). However, the nanomagnet 4 shows a single domain state at remanence. This is observed in the dipolar coupled environment aswell. This shows that the vortices as observed in the two nanomag-nets may be due to the speciﬁc geometry of the nanomagnets. Theexperimental and simulation results clearly show that although thenanomagnets show a single domain like behavior, the real struc-tures may undergo complex switching processes which do not con-form to Stoner–Wohlfarth like behavior for ideal single domainmagnetic states. In conclusion, we have investigated the magnetization reversal behavior of dipolarly coupled nanomagnets forming two coupled ring-like structures. The remanent state as observed using MFM shows thatall the nanomagnets are in a single domain state. However, detailed(continuous) ﬁeld dependent high-sensitive measurements of averagestray ﬁelds emanating from the nanomagnets show features which canbe identiﬁed due to speciﬁc micromagnetic states in the nanomagnets.This is explained with the micromagnetic simulation results, whichmatch reasonably well with the experimental data. The results suggestthat local irregularities affect the exact micromagnetic state, thereby converting two islands of single domain dimensions to a magnetically vortex state. Our results also demonstrate the remarkable ability of the2-DEG based micro-Hall magnetometry method used in the ballistictransport regime in detecting changes in stray ﬁelds due to localmicromagnetic changes. We gratefully acknowledge Yasuhiko Fujii and Masahide Tokuda for technical assistance in the fabrication of Hall devices.N.K. is thankful to the University Grants Commission (UGC),Government of India, for providing the research fellowship for thiswork. P.D. acknowledges the partial ﬁnancial support throughcollaborative research and education under IIT Delhi-BNERC,Toyo University’s joint Bio-Nano Mission program. Part of thiswork was carried out at the Nano Research Facility (NRF) and theHigh Performance Computing (HPC) Centre of IIT Delhi. FIG. 4. Micromagnetic states of the interacting nanostructures at different magnetic ﬁeld values corresponding to the features observed in the hysteresis loop shown in Fig. 3 . The numbers in (a) are to identify the nanostructures. The dotted arrows refer to themagnetization switchings of the nanomagnets at the corresponding external ﬁelds.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 116, 102401 (2020); doi: 10.1063/1.5144841 116, 102401-4 Published under license by AIP PublishingREFERENCES 1A. Hoffmann and S. D. Bader, Phys. Rev. Appl. 4, 047001 (2015). 2T. P. Dao, M. M }uller, Z. Luo, M. Baumgartner, A. Hrabec, L. J. Heyderman, and P. 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1.3058680.pdf	Reducing the critical current for spin-transfer switching of perpendicularly magnetized nanomagnets S. Mangin, Y. Henry, D. Ravelosona, J. A. Katine, and Eric E. Fullerton Citation: Appl. Phys. Lett. 94, 012502 (2009); doi: 10.1063/1.3058680 View online: http://dx.doi.org/10.1063/1.3058680 View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v94/i1 Published by the AIP Publishing LLC. 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 09 Jul 2013 to 128.42.202.150. 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_permissionsReducing the critical current for spin-transfer switching of perpendicularly magnetized nanomagnets S. Mangin,1,a/H20850Y . Henry,2D. Ravelosona,3J. A. Katine,4and Eric E. Fullerton5 1LPM, Nancy-Université, UMR CNRS 7556, F-54506 Vandoeuvre Cedex, France 2IPCMS, UMR CNRS 7504, Université Louis Pasteur, F-67034 Strasbourg Cedex 2, France 3IEF , UMR CNRS 8622, Université Paris Sud, F-91405 Orsay Cedex, France 4San Jose Research Center, Hitachi-GST, San Jose, California 95135, USA 5CMRR, University of California, San Diego, La Jolla, California 92093-0401, USA /H20849Received 23 September 2008; accepted 22 November 2008; published online 6 January 2009 /H20850 We describe nanopillar spin valves with perpendicular anisotropy designed to reduce the critical current needed for spin transfer magnetization reversal while maintaining thermal stability. Byadjusting the perpendicular anisotropy and volume of the free element consisting of a /H20851Co/Ni /H20852 multilayer, we observe that the critical current scales with the height of the anisotropy energy barrierand we achieve critical currents as low as 120 /H9262A in quasistatic room-temperature measurements of a 45 nm diameter device. The ﬁeld-current phase diagram of such a device is presented. © 2009 American Institute of Physics ./H20851DOI: 10.1063/1.3058680 /H20852 The ability of a spin-polarized current to reverse the magnetization orientation of nanomagnets1–4should enable a range of magnetic devices such as high performance mag-netic memories. However, several advances are needed torealize practical devices. 5One key for memory applications is the reduction in the current required to reverse the magne-tization of the free layer while maintaining thermal stability.There has been a range of approaches to lower the criticalcurrents 6–10and it has recently been demonstrated that samples exhibiting perpendicular magnetic anisotropy/H20849PMA /H20850provide a pathway to low critical currents and large thermal stability.10Current switching for various PMA me- tallic systems has been demonstrated10–14and has recently been incorporated with tunnel barriers.15 In the macrospin approximation, the critical current for spin-transfer reversal of a PMA free layer at zero tempera-ture is given by I C0=−/H208732e /H6036/H20874/H9251MSV g/H20849/H9258/H20850pHeff, /H208491/H20850 where MSandVare the saturation magnetization and vol- ume,/H9251is Gilbert’s damping constant, and pis the spin po- larization of the current. The factor g/H20849/H9258/H20850depends on the rela- tive angle /H9258of the reference- and free-layer magnetization vectors.2,16Heff=/H20849HK/H11036−4/H9266MS+Happ+Hdip/H20850is the effective ﬁeld acting on the free layer, which contains contributions from the perpendicular applied ﬁeld Happ, the dipolar ﬁeld from the reference layer Hdip, and the uniaxial PMA ﬁeld HK/H11036. The factor −4 /H9266MSarises from the demagnetizing ﬁeld of the thin ﬁlm geometry. The thermal stability of the free element is determined by the height of the energy barrier UK=/H20851MSV/H20849HK/H11036 −4/H9266MS/H20850/H20852/2 between the two stable magnetization conﬁgu- rations. Thus the critical current is directly proportional to the energy barrier in the absence of external ﬁeld H=Hdip +Happ=0,5,10IC0=−/H208732e /H6036/H208742/H9251 g/H20849/H9258/H20850pUK. /H208492/H20850 Within the assumptions of Eq. /H208492/H20850, low critical currents can then be achieved by reducing the /H9251/pratio and minimiz- ingUK, while maintaining thermal stability. In this letter, we describe measurements of current-induced reversal of Co/Nifree layers with PMA that are thermally stable and reverse incurrents as low as 120 /H9262A/H208497/H11003106A/cm2/H20850in quasistatic room-temperature measurements. The low currents result from lowering UKby lowering both the PMA and volume of the free layer. a/H20850Electronic mail: stephane.mangin@lpm.u-nancy.fr.-2 -1 0 130.731.0-10 0 1030.731.0 -400 0 40030.731.0Hdip=0 . 8k O edV/dI(Ω) Happ(kOe)dV/dI(Ω) Happ(kOe)(a) (b)dV/dI(Ω) IDC(µA)(c) FIG. 1. /H20849Color online /H20850Differential resistance as a function of perpendicular- to-plane applied ﬁeld Happat zero current /H20851/H20849a/H20850and /H20849b/H20850/H20852and dc bias current Idc for zero net external ﬁeld, H=Happ+Hdip=0 /H20849c/H20850for a 45 nm diameter nano- pillar measured at room temperature.APPLIED PHYSICS LETTERS 94, 012502 /H208492009 /H20850 0003-6951/2009/94 /H208491/H20850/012502/3/$23.00 © 2009 American Institute of Physics 94, 012502-1 Downloaded 09 Jul 2013 to 128.42.202.150. 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 samples studied are similar to those described in Ref. 10. The magnetic structure consists of aP t /H208493n m /H20850//H20851Co/H208490.25 nm /Pt/H208490.52 nm /H20850/H20850/H20852/H110035/Co/H208490.2 nm /H20850/ /H11003/H20851Ni/H208490.6 nm /H20850/Co/H208490.1 nm /H20850/H20852/H110032/Co/H208490.1 nm /H20850 reference layer and a Co /H208490.2 nm /H20850//H20851Ni/H208490.6 nm /H20850/Co/H208490.1 nm /H20850/H20852/H110035 free layer separated b ya4n mC u layer. Compared to Ref. 10, the free layer contains one additional /H20851Ni/Co /H20852repeat and, more importantly, is notterminated b ya3n mP t layer. Removing this Pt layer could alter the effective spin polarization /H20849p/H20850 /H20849Refs. 17and18/H20850and the damping /H20849/H9251/H20850in the free layer.19 However, the primary effect is to suppress the strong contri- bution of the Co–Pt interfacial anisotropy lowering the PMAof the free layer. We focus on the differential resistance R ac=dV /dImea- sured on a circular 45 nm diameter nanopillar /H20849Figs. 1and2/H20850. The low and high resistance states of the device correspondto the parallel /H20849P/H20850and antiparallel /H20849AP/H20850alignments of the reference and free layers. From the major loop of Fig. 1/H20849a/H20850, the reversal of these magnetizations occurs for applied ﬁeldsof/H1101112 and /H110111 kOe, respectively. The minor loop performed with the hard layer along the positive ﬁeld direction /H20849Fig.1/H20850 reveals a free layer coercive ﬁeld of 0.42 kOe and a dipolarﬁeld /H20849H dip/H20850originating from the hard layer of 0.80 kOe. In the remainder the data will be presented as a function of the net external ﬁeld H=Happ+Hdipacting on the free layer. Racas a function of the dc bias current, for a net ﬁeld H=0 /H20849Happ=−0.80 kOe /H20850, is shown in Fig. 1/H20849c/H20850. Cycling the current switches the spin valve from the P state to AP state and back with critical currents of +120 and −110 /H9262A, re- spectively. These correspond to a current density of /H110117/H11003106A/cm2. Both the current and current density are sig- niﬁcantly lower than previously reported values for PMAmetallic devices.10–14The corresponding H=0 values for Co/Ni free layers of higher coercivity /H208492.6 kOe /H20850in Ref. 10 are 1.45 mA /H208493.9/H11003107A/cm2/H20850. A comparison of the key parameters for the two types of Co/Ni free layers is given in Table I. To gain a better view of the spin transfer phenomena, we measured a complete /H20849ﬁeld H, current Idc/H20850phase diagram at room temperature. From the resistances along the decreasing /H20849RacP→AP/H20850and increasing current /H20849RacAP→P/H20850branches, we plot the sum RacP→AP+RacAP→P/H20849Fig.2/H20850. The sum strongly enhances reversible processes but also reveals the region of the /H20849H,Idc/H20850 parameter space where hysteresis /H20849bistability /H20850occurs. The hysteretic reversal between the P and AP states is observed inthe ﬁeld range −1 kOe /H11021H/H11021+0.7 kOe. For ﬁelds outside this range, reversible transitions occur, which is different toRef. 10where reversible transitions are only observed in negative ﬁelds. The bottom-left and upper-right boundaries of the region of bistability /H20849Fig. 2/H20850deﬁne the evolutions with Hof the critical currents I CP→AP/H20849H/H20850andICAP→P/H20849H/H20850, respectively. Along the major part of these boundaries, the critical currents vary linearly with H/H20849Fig.3/H20850, as expected from Eq. /H208491/H20850. The cor- responding slopes are dICP→AP/dH=−2.9/H1100310−4A/kOe and dICAP→P/dH=−2.2/H1100310−4A/kOe /H20849Table I/H20850. These slopes cor- respond to the prefactor in Eq. /H208491/H20850, though slightly modiﬁed by ﬁnite temperature effects. They can be compared to thosereported in Ref. 10in Table Ito estimate the role of the parameters that the prefactor in achieving the lower criticalcurrents reported here. We ﬁnd that the factor of 2 differ-ences in the slopes can be accounted for by the factorof 2 differences in the free layer volumes. Therefore, weconclude that there is no dramatic difference in the−/H208492e//H6036/H20850 /H9251MS/g/H20849/H9258/H20850pratios of the Co/Ni free layers with and without Pt termination, and the reduction in ICis thus as- cribed to changes in UK. To further compare the sample studied presently to the one in Ref. 10, it proves useful to compare the corresponding IC/VH Cratios /H20849Table I/H20850. For a uniaxial macrospin system with the external ﬁeld along the easy axis, the zero tempera-ture coercive ﬁeld is H C0=2UK/VM S. Then, assuming that Eq. /H208492/H20850holds, IC0scales linearly with HC0and the free-layer volume V/H20849and MS/H20850. At ﬁnite temperature, the critical current and the coercive ﬁeld are affected by thermalﬂuctuations8,20,21according toTABLE I. Properties for nanopillar 1 /H20849studied in the present letter /H20850and nanopillar 2 from Ref. 10:Vis the free layer volume, HCits coercive ﬁeld, andICis the critical current for reversal of the free layer. Nanopillar 1 Nanopillar 2 V/H2084910−18cm3/H20850 5.9 11.2 HC/H20849kOe /H20850 0.42 2.65 IC/H20849/H9262A/H20850 120 1450 IC/VHC/H208491013A/kOe /cm3/H20850 4.8 4.9 dICP→AP/dH /H2084910−4A/kOe /H20850 /H110022.9 /H110026.4 dICAP→P/dH /H2084910−4A/kOe /H20850 /H110022.2 /H110024.0-0.40 . 00 . 4-101IDC(mA)P AP H(kO e) FIG. 3. /H20849Color online /H20850Field variation in the switching currents at 300 K /H20849open symbols /H20850and 20 K /H20849solid symbols /H20850. Squares and circles correspond to AP→P transitions /H20849ICAP→P/H20850and P→AP reversals /H20849ICP→AP/H20850, respectively.-2 -1 0 1 2-1000-50005001000 APP H (kOe)IDC(µA)-3.50 0.10 3.50 FIG. 2. /H20849Color /H20850Sum of the differential resistances RacP→AP+RacAP→Pmeasured while sweeping the dc bias current Idcfrom +1 to /H110021m A /H20849RacP→AP/H20850and from /H110021t o+ 1 m A /H20849RacAP→P/H20850. To eliminate Joule and Peltier effects /H20851see Fig. 1/H20849c/H20850/H20852 the parabolic background signal was subtracted.012502-2 Mangin et al. Appl. Phys. Lett. 94, 012502 /H208492009 /H20850 Downloaded 09 Jul 2013 to 128.42.202.150. 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_permissionsIC/H20849T/H20850=IC0/H208751−kBT UKln/H20849tPf0/H20850/H20876 /H208493/H20850 and HC/H20849T/H20850=HC0/H208771−/H20875kBT UKln/H20873tPf0 ln 2/H20874/H20876n/H20878, where tpis the experimental time scale, f0is the attempt frequency for crossing the anisotropy barrier by thermal ac-tivation /H20849/H1101110 9s−1/H20850, and nis an exponent that depends on the reversal mechanism. It follows that, for a given sample, taking the ratio of the critical current to the coercive ﬁeldremoves much of the temperature effects so that in a reason-able approximation I C/H20849T/H20850/HC/H20849T/H20850/H11015IC0/HC0provided that MS remains close enough to its zero temperature value. The low anisotropy Co/Ni free element studied here has a volume and a room temperature coercive ﬁeld reduced byfactors of 1.9 and 6.3, respectively, as compared to the highanisotropy one of Ref. 10. Thus, one expects the critical current to be 12 times lower as observed experimentally/H20849Table I/H20850and is reﬂected in the almost identical values of I C/VH Cfor the two devices. From the measurements of ad- ditional devices of different shapes and sizes, we could con-clude that the ratio I C/Vis nearly constant and the deviations are consistent with anisotropy distributions often observed inPMA nanostructures.22 The room-temperature value of K can be estimated from Eq. /H208493/H20850using the room-temperature coercivity and assuming HC0=2 K /M or from the low temperature coercivity. From these approaches we estimate UK=/H2084945/H1100610/H20850kBTat room temperature. Thus the PMA of the Co/Ni free layer is at the minimum value required to ensure long term thermal stabil-ity. With the above values and Eq. /H208493/H20850we estimate I C0 /H110112IC.23Any further reduction in ICwill only be made pos- sible by a control of /H9251/g/H20849/H9258/H20850pand/or the design of more complex architecture or system design. With varying temperature, we ﬁnd the critical current does not scale with the coercive ﬁeld as may be expected. At20 K, the free layer coercivity increases fourfold, whereasthe average critical current increases by a factor of 8 com- pared to that measured at 300 K. I CP→APandICAP→Pstill vary linearly with ﬁeld /H20849Fig. 3/H20850but the slopes increase to dICP→AP/dH=−4.2/H1100310−4A/kOe and dICAP→P/dH=−6.2 /H1100310−4A/kOe, roughly twice the room-temperature values. This increase in slope is not expected from thermal effectsand reﬂects the temperature dependence of intrinsic proper-ties of the materials. While a small increase in M Sis ex- pected upon decreasing T, the most likely explanation for the increased slopes is an increase in /H9251. It is known that effects such as sidewall oxidation of the pillar can have dramaticeffects on the low temperature damping parameter.8A dou-bling of /H9251at low temperature would explain both the change in slope of IC/H20849H/H20850and the relatively large currents needed to reverse the magnetization. In summary, we have fabricated 45 nm diameter spin valve devices based on Co/Ni multilayer elements, whichexhibit perpendicular anisotropy, are thermally stable, andrequire current as low as 120 /H9262A for spin transfer switching in zero magnetic ﬁeld. The critical current is found to scalewith the anisotropy energy barrier U K, as expected for a uniaxial system. Since thermal stability limits further signiﬁ-cant reductions in U K, further decreases of ICfor such a system must come from control of /H9251//H20849g/H20849/H9258/H20850p/H20850or system design. 1L. Berger, Phys. Rev. B 54, 9353 /H208491996 /H20850. 2J. C. Slonczewski, J. Magn. Magn. Mater. 159,L 1 /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. 4E. B. Myers, D. C. Ralph, J. A. Katine, R. N. Louie, and R. A. Buhrman, Science 285, 867 /H208491999 /H20850. 5J. A. Katine and E. E. Fullerton, J. Magn. Magn. Mater. 320, 1217 /H208492008 /H20850. 6D. Chiba, Y. Sato, T. Kita, F. Matsukura, and H. Ohno, Phys. Rev. Lett. 93, 216602 /H208492004 /H20850. 7P. M. Braganca, I. N. Krivorotov, O. Ozatay, A. G. F. Garcia, N. C. Emley, J. C. Sankey, D. C. Ralph, and R. A. Buhrman, Appl. Phys. Lett. 87, 112507 /H208492005 /H20850. 8O. Ozatay, P. G. Gowtham, K. W. Tan, J. C. Read, K. A. Mkhoyan, M. G. Thomas, G. D. Fuchs, P. M. Braganca, E. M. Ryan, K. V. Thadani, J.Silcox, D. C. Ralph, and R. A. Buhrman, Nature Mater. 7,5 6 7 /H208492008 /H20850. 9Z. Diao, Z. Li, S. Wang, Y. Ding, A. Panchula, E. Chen, L. -C. Wang, and Y. Huai, J. Phys.: Condens. Matter 19, 165209 /H208492007 /H20850. 10S. Mangin, D. Ravelosona, J. A. Katine, M. J. Carey, B. D. Terris, and E. E. Fullerton, Nature Mater. 5, 210 /H208492006 /H20850. 11T. Seki, S. Mitani, K. Yakushiji, and K. Takanashi, Appl. Phys. Lett. 89, 172504 /H208492006 /H20850. 12H. Meng and J. P. Wang, Appl. Phys. Lett. 88, 172506 /H208492006 /H20850. 13D. Ravelosona, S. Mangin, Y. Lemaho, J. A. Katine, B. D. Terris, and E. E. Fullerton, Phys. Rev. Lett. 96, 186604 /H208492006 /H20850. 14T. Seki, S. Mitani, and K. Takanashi, Phys. Rev. B 77, 214414 /H208492008 /H20850. 15M. Nakayama, T. Kai, N. Shimomura, M. Amano, E. Kitagawa, T. Na- gase, M. Yoshikawa, T. Kishi, S. Ikegawa, and H. Yoda, J. Appl. Phys. 103, 07A710 /H208492008 /H20850. 16M. D. Stiles and J. Miltat, Spin Dynamics in Conﬁned Magnetic Structures III, Springer Series Topics in Applied Physics /H20849Springer, Berlin, Heidel- berg, 2006 /H20850, Vol. 101. 17S. Urazhdin, N. O. Birge, W. P. Pratt, Jr., and J. Bass, Appl. Phys. Lett. 84, 1516 /H208492004 /H20850. 18Y. Jiang, T. Nozaki, S. Abe, T. Ochiai, A. Hirohata, N. Tezuka, and K. Inomata, Nature Mater. 3, 361 /H208492004 /H20850. 19S. Mizukami, Y. Ando, and T. Miyazaki, J. Magn. Magn. Mater. 226, 1640 /H208492001 /H20850. 20M. P. Sharrock, J. Appl. Phys. 76, 6413 /H208491994 /H20850. 21R. H. Koch, J. A. Katine, and J. Z. Sun, Phys. Rev. Lett. 92, 088302 /H208492004 /H20850. 22T. Thomson, G. Hu, and B. D. Terris, Phys. Rev. Lett. 96, 257204 /H208492006 /H20850. 23The critical current at zero temperature IC0can be deduced from the criti- cal current ICobtained for various current pulse width /H20849Ref. 9/H20850.012502-3 Mangin et al. Appl. Phys. Lett. 94, 012502 /H208492009 /H20850 Downloaded 09 Jul 2013 to 128.42.202.150. 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_permissions
1.2143983.pdf	Diagnostic Application of Pulse Echo Ultrasound to the Abdomen Joseph H. Holmes Citation: The Journal of the Acoustical Society of America 42, 1167 (1967); View online: https://doi.org/10.1121/1.2143983 View Table of Contents: http://asa.scitation.org/toc/jas/42/5 Published by the Acoustical Society of America74TH MEETING ß ACOUSTICAL SOCIETY OF AMERICA TUESDAY, 14 NOVEMBER 1967 MONTE CARLO HOTEL, 2:30 P.M. Monte Carlo Session. Ultrasonic Visualization II: Medical Specialities (Joint session AIUM/ASA) DENNIS WHITE AND WILLIAM M CKINNEY, Joint Session Chairmen Invited Paper (25 minutes) MC1. Ultrasonic Encephalography. CHARLES GROSSMAN (nonmember), Department of Neurology, University of Pittsburgh.--A review of clinical applications of diagnostic ultrasound, in conjunction with EEG, will be presented. MC2. Ultrasonic Stereotaxis. DOUGLAS GORDON, Richmond, Surrey, England.mUltrasonic visualization of human tissues has aimed at obtaining information simultaneously from a number of different depths from the surface. In consequence, only weakly focused transducers have been employed. In stereotaxic surgery when the highest possible accuracy is essential in three dimensions, it is preferable to sacrifice all other considerations to reducing the size of the focus to the minimum and to move the transducer bodily so that the focus scans a single plane of the body which can be in any of the three anatomical axes. As echoes are obtained from what is virtually a point source, the effect of the angulation of the surface is much less important than when plane waves are used. Using conventional gating circuits, it is possible to select only those echo pulses that occur exactly at the focus. These are recorded automatically on electrosensitive paper, either directly or with enlarged scale, by use of a pantograph. The problems involved in obtaining short well-damped pulses for diagnosis and high acoustic power continuously for destructive purposes from the same transducer have been overcome by employing high harmonics for diagnosis and a spherically ground ceramic bowl with no added damping. In actual lab- oratory experiments on cats, it has been possible to localize the major blood vessels to 0.1 mm three dimensionally and to distinguish between the surface of the hemisphere and the roof of the lateral ventricle only 5 mm below. MC3. Ultrasonography in Ophthalmology. GILBERT BAuM (nonmember), Albert Einstein College of Medicine, Bronx, New York 10461.---This paper is a historical review of the use of ultrasound in ophthalmology. The?paper will review its earliest therapeutic applications, the use of A and B Mode for diagnostic purposes, the localization of intraocular and intraocal foreign bodies by both the A and B Mode techniques, ultrasonic measurement of eyesize, and the use of intense focused ultrasound for the production of focal chorioretinal lesions. The present state of these applications will also be discussed. M C4. Ultrasound Cardiograrn in Clinical and Physiological Studies. CLAUDE JOYNER (nonmember), RICHARD PYLE (nonmember), AND JOHN GRUBER (nonmember), Edward B. Robinette, Foundation and Cardiovascular Clinical Research Center Hospital of the University of Pennsylvania, Philadelphia, Pennsylvania.mUltrasound cardiograms have been obtained from over 3000 subjects over the past 6 yr. The accuracy of this method for the assessment of mitral valve disease, as described in earlier reports, has been confirmed in this large patient study. This method of external, safe study has been found to equal cardiac catherization in accuracy when judged by findings at operation. Equally reliable evaluation of tri- cuspid valve disease has been obtained, and actually found superior to catherization in preoperative assessment. Valve substance, pliability, and mobility can be determined from the ultrasound records. The prediction of valve characteristics determining whether replacement of a valve with a prosthesis is needed, has been quite accurate. This preoperative infor- mation is not obtained from catherization studies. The be- havior of the mitral valve, accounting for the opening snap in mitral stenosis, and the Austin-Flint murmur of aortic regurgitation, have been defined by simultaneous record of direct video display of the mitral valve ultrasound with the phonocardiogram. The use of the mitral ultrasound as a reference for analysis of sound records will be presented. The ultrasound method of determining pericardial fluid has been confirmed as valid but subject to record error and misinter- pretation. The ultrasound cardiogram is 100% accurate in prediction of mitral valve sizes and valve leaflet character. The accuracy in determination of pericardial fluid is less reliable. MC5. Sonographic Interpretation--Pitfalls and Break- throughs. L.JOS I. VON MlCSKY (nonmember), Department of Obstetrics and Gynecology, College of Physicians and Surgeons, Columbia University and Bioacoustical Laboratory, Woman's Hospital, St. Luke's Hospital Center, New York, New York.m Contrary to the impression conveyed by recent publications, the problems arising in diagnositc ultrasonics are not primarily of a technical nature. The difficulty now lies in deciphering the vast amount of information recorded on the sonograms and in elucidating the tissue's structural characteristics responsible for the echo pattern obtained. Since relatively little is known and understood of the physical mechanisms attending the propagation of high-frequency acoustic waves in biological materials, the first step toward intelligent interpretation is to explore new theoretical and experimental avenues for gaining insight into the sources of tissue echoes. The use of morphologic interpretive criteria alone has been rendered grossly inadequate by the well-known fact that a number of different conditions can produce sonographic patterns that appear nearly identical on visual inspection. The various methods of data processing practiced in our laboratory are discussed in detail, emphasizing a general tendency twoard some form of quantification. New techniques involving color-translating isodensitracing, ultra- sonic holography, quantitative evaluation of compound scan sonograms, and automatic pattern recognition are mentioned. MC6. Diagnostic Application of Pulse Echo Ultrasound to the Abdomen. Jos.ea H. HOLMES (nonmember), University of Colorado Medical Center, Denver, Colorado 8022rz.--Present studies indicate that ultrasonic pulse echo compound scanning techniques are of value in demonstrating intra-abdominal pathology and thus may prove to be a valuable diagnostic aid in a wide variety of diseases. Equipment used employs either contact scanning or water-path scanning. The liver, kidney, and spleen all transmit sound well and thus can be The Journal of the Acoustical Society of America 1167 74TH MEETING ß ACOUSTICAL SOCIETY OF AMERICA readily outlined to determine size and position. In the liver, which transmits sound well, abnormal echo patterns are ob- served with such lesions as cirrhosis, tumor, abscess, chloecy- stitis, congestion, and hepatitis. Renal cysts and tumor give abnormal echo patterns. The ultrasonic technique has had greatest diagnostic application on examining abdominal masses. Fluid filled structures like the stomach and the bladder are outlined readily by ultrasound, and it is possible to delineate structural distortion, pressure from adjacent structures, and to estimate the amount of fluid present. The pancreatic area, abdominal aorta, and vena cava can be visualized in some patients with appropriate use of time-varied gain and depth control. Ultrasonic controls that must be used to obtain proper visualization will be discussed. TUESDAY, 14 NOVEMBER 1967 SILVER CHIMES EAST, 8:00 P.M. Workshop Session. Underwater Acoustics Workshop: Ships and Sonar DONALD Ross, Chairman Invited Papers (20 minutes) Underwater acoustics is closely tied to the ships that take it to sea. On the one hand, the sonar system is affected by the ship (surface and submarine), and the equipment design and performance reflects the ship characteristics. On the other hand, the installation of the sonar system aboard the ship influences the ship's performance and its design. It is the purpose of this workshop to discuss these sonar system ship inter- actions. WU1. Effects of Sonar Domes on Ship Performance in Smooth Water. G. STUNTZ, Naval Ship Research and Development Center, Carderock, Maryland. WU2. Motion Performance Characteristics of ASW Ships. RAY WEIMTEI (nonmember), Naval Ship Research and Development Center, Carderock, Maryland. WUS. Hydromechanics Problems of Variable Depth Sonar. REsE FOLB (nonmember), Naval Ship Research and Development Center, Carderock, Maryland. WU4. Effect of Ship Motions on the Design of Sonar Equipment. G. W. Bar>smw, Naval Under- sea Warfare Center, San Diego, California. WUS. Influence of Acoustics on the Evolution of the Submarine. WALTER L. USN Underwater Sound Laboratory, New London, Connecticut. WEDNESDAY 15 NOVEMBER 1967 SILVER CHIMES EAST, 9:00 A.M. Session K. Ultrasonic Visualization III' Image Formation, Conversion, and Display WESLEY L. NYBORG, Chairman Invited Papers (25 minutes) K1. Acousto Optics and Theft Application to Ultrasonic Visualization. H. W. JONES (nonmember), University of Wales, Swansea, England.--This contribution reviews the present position of acousto- optics in relation to ultrasonic visualization. The basic principles relating to matching, reflection: and refraction are briefly reviewed. Detailed consideration is given to the problem of matching at the front face of an image convertor. The attenuation in liquids and solids is considered and related to the diffraction effects that arise in lenses and mirrors. The results of calculation of the relative importance of diffraction and geometric aberrations in lenses and mirrors accounting for absorption and mode conversion effects are given. The use of zone plates and other artifices to improve performance is discussed. Finally, the problems of ultrasonic illumination are considered in relation to reflection and transmission modes of operation. K2. Investigation on Electronic Image Conversion in Pulsed Operation. R. POrtoMAN, G. tLIG, ANY E. SCaXTZER, LaboratoriumJr Ultraschall, Aachen, Germany.--The two possible methods of the electronic ultrasonic image conversion--the CW or impulse-reflex technique--are confronted and compared. For short distances, an unequivocal superiority of the impulse-reflex-technique results. The low coverage at high frequencies proves to be hindering, which mainly depends on the damping in the transmission channel and by this on the frequency, but furthermore, also on the shape of the object that is to be imaged. At equal number of image elements, the diameter of the plate must in- 1168 Volume 42 Number 5 1967
1.3373833.pdf	A frequency-controlled magnetic vortex memory B. Pigeau, G. de Loubens, O. Klein, A. Riegler, F. Lochner, G. Schmidt, L. W. Molenkamp, V. S. Tiberkevich, and A. N. Slavin Citation: Applied Physics Letters 96, 132506 (2010); doi: 10.1063/1.3373833 View online: http://dx.doi.org/10.1063/1.3373833 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/96/13?ver=pdfcov Published by the AIP Publishing Articles you may be interested in All-electrical operation of magnetic vortex core memory cell Appl. Phys. Lett. 99, 262505 (2011); 10.1063/1.3673303 Irreversibility of field-induced magnetostructural transition in NiCoMnSb shape memory alloy revealed by magnetization, transport and heat capacity studies Appl. Phys. Lett. 96, 112503 (2010); 10.1063/1.3365181 Nanosecond magnetic switching of ferromagnet-antiferromagnet bilayers in thermally assisted magnetic random access memory J. Appl. Phys. 106, 014505 (2009); 10.1063/1.3158231 Transition between onion states and vortex states in exchange-coupled Ni – Fe Mn – Ir asymmetric ring dots J. Appl. Phys. 99, 08G303 (2006); 10.1063/1.2164435 Double-barrier magnetic tunnel junctions with GeSbTe thermal barriers for improved thermally assisted magnetoresistive random access memory cells J. Appl. Phys. 99, 08N901 (2006); 10.1063/1.2162813 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: 188.82.181.120 On: Wed, 02 Apr 2014 21:39:19A frequency-controlled magnetic vortex memory B. Pigeau,1G. de Loubens,1,a/H20850O. Klein,1A. Riegler,2F . Lochner,2G. Schmidt,2,b/H20850 L. W. Molenkamp,2V. S. Tiberkevich,3and A. N. Slavin3 1Service de Physique de l’État Condensé (CNRS URA 2464), CEA Saclay, 91191 Gif-sur-Yvette, France 2Physikalisches Institut (EP3), Universität Würzburg, 97074 Würzburg, Germany 3Department of Physics, Oakland University, Michigan 48309, USA /H20849Received 27 January 2010; accepted 28 February 2010; published online 1 April 2010 /H20850 Using the ultralow damping NiMnSb half-Heusler alloy patterned into vortex-state magnetic nanodots, we demonstrate a concept of nonvolatile memory controlled by the frequency. Aperpendicular bias magnetic ﬁeld is used to split the frequency of the vortex core gyrotropic rotationinto two distinct frequencies, depending on the sign of the vortex core polarity p=/H110061 inside the dot. A magnetic resonance force microscope and microwave pulses applied at one of these two resonantfrequencies allow for local and deterministic addressing of binary information /H20849core polarity /H20850. ©2010 American Institute of Physics ./H20851doi:10.1063/1.3373833 /H20852 One of the most important goals of the modern informa- tion technology is the development of fast high-density non-volatile random access memories /H20849RAM /H20850that are energy ef- ﬁcient and can be produced using modern planar micro- andnanofabrication methods. Magnetic nano-objects offer aconvenient way to store binary information through theirbistable properties, but the development of practical mag-netic RAM requires to ﬁnd a performant mechanism to re-verse the magnetization inside individual cells. 1One of the ways is to take advantage of the high dynamical susceptibil-ity of magnetic nano-objects made of low dissipation mate-rials at their ferromagnetic resonance frequency. In a vortex-state magnetic nanodot, 2the static magneti- zation is curling in the dot plane, except in the dot centerwhere it is forming an out-of-plane vortex core 3of typical size of the exchange length lex/H112295–10 nm. The core can be directed either perpendicularly up or down relative to the dotplane, this bistability being characterized by the core polarity p=/H110061. Recent experiments demonstrated that the core po- larity can be reversed in zero magnetic ﬁeld through the ex-citation of the gyrotropic rotation of the vortex core about itsequilibrium position. 4,5AtH=0, the frequency of this gyro- tropic mode, f0, is identical for both core polarities but the sense of this rotation depends on p. Thus, for a given core polarity, the circular polarization of the microwave ﬁeld– right or left depending on the sign of p–discriminates the occurrence of the resonant microwave absorption by thevortex-state magnetic dot.6When the radius rof the core orbit increases, a distortion of the core proﬁle characterizedby the appearance of a tail having the magnetization direc-tion opposite to that of the original core polarity occurs. 5,7,8 The magnitude of this tail depends solely on the linear ve- locity V=2/H9266f0rof the vortex core.7When the latter reaches the critical value Vc=/H208491/3/H20850/H9275MlexatH=0 /H20851where /H9275M =/H9253/H92620M0,/H92620is the permeability of the vacuum, M0 the saturation magnetization of the magnetic material, and/H9253its gyromagnetic ratio /H20852, the core polarity suddenly– within few tens of picoseconds9–reverses.Still, reliable control of an individual cell in a large array based on resonant switching, which takes full advantage ofthe frequency selectivity of magnetic resonance, has to berealized. In this paper, we demonstrate a frequency-controlled memory with resonance reading and writingschemes using vortex-state NiMnSb nanodots placed in aperpendicular magnetic bias ﬁeld H/HS110050. In our experimental realization, local addressing of the core polarity is achievedby means of a magnetic resonance force microscope /H20849Fig.1/H20850. The key role of the static magnetic ﬁeld Haligned along the axis of the vortex core– Hbeing the sum of an homoge- neous bias and of a small additional local component–is tointroduce a controlled splitting of the frequency of the gyro-tropic mode depending on the core polarity. 10Therefore, the polarity state of individual magnetic dots can be selectivelyaddressed by controlling the frequency of a linearly polar-ized microwave pulse excitation. When the core polarity isequal to p=+1 /H20849i.e., the core is parallel to H/H20850, the rotational frequency f +is larger than the frequency f−corresponding to the core polarity p=−1. This frequency splitting is directly proportional to H10,11/H20851see Fig. 2/H20849a/H20850/H20852, f+/H20849H/H20850−f−/H20849H/H20850=2f0/H20849H/Hs/H20850, /H208491/H20850 where Hsis the magnetic ﬁeld required to saturate the dot along its normal and f0is the frequency of the gyrotropic a/H20850Author to whom correspondence should be addressed. Electronic mail: gregoire.deloubens@cea.fr. b/H20850Present address: Institut für Physik, Martin-Luther-Universität, Halle Wit- tenberg, 06099 Halle, Germany. FIG. 1. /H20849Color online /H20850Prototype of a frequency-controlled magnetic memory realized by means of a magnetic resonant force microscope. Thememory elements are vortex-state NiMnSb disks of diameter 1 /H9262m and thickness 44 nm separated by 10 /H9262m.APPLIED PHYSICS LETTERS 96, 132506 /H208492010 /H20850 0003-6951/2010/96 /H2084913/H20850/132506/3/$30.00 © 2010 American Institute of Physics 96, 132506-1 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: 188.82.181.120 On: Wed, 02 Apr 2014 21:39:19mode at H=0, which can be approximated by the analytical expression:2f0=/H2084910 /9/H20850/H20849/H9275M/2/H9266/H20850/H9252, where /H9252=t/R,tis the thickness and Rthe radius of the dot. To design a practical memory cell it is necessary to choose the static magnetic ﬁeld Hin such a way that the ﬁeld-induced gyrotropic frequency splitting Eq. /H208491/H20850exceeds the linewidth /H9004fof the gyrotropic mode which can be approximately expressed as /H9004f/H11229/H9251vf0, where /H9251v=/H9251/H208511 +ln/H20849R/Rc/H20850/2/H20852is the damping parameter for the gyrotropic mode,12/H9251is the dimensionless Gilbert damping constant of the dot magnetic material and Rc/H11011lexis the vortex core ra- dius. Thus, the minimum perpendicular bias ﬁeld is given by Hmin/H11229/H20849/H9251v/2/H20850Hs /H208492/H20850 /H20851see Fig. 2/H20849a/H20850/H20852. It follows from Eq. /H208492/H20850, that to reduce Hmin,i t is necessary to choose the dot magnetic material with lowdamping and to increase the aspect ratio /H9252of the dot, as this leads to the decrease in the saturation ﬁeld Hs. The design of our experimental frequency-controlled magnetic vortex memory is presented in Fig. 1. The memory elements are circular magnetic dots made of an epi-taxial, ultralow damping half-Heusler alloy with highCurie temperature, NiMnSb /H20849001/H20850/H20849 /H9251=0.002, /H92620M0 =690 mT, TC=730 K /H20850.13,14Their aspect ratio /H9252/H112290.1/H20849t =44 nm, R=500 nm /H20850is relatively large, and they are sepa- rated from each other by 10 /H9262m. The detailed magnetic characterization of such NiMnSb dots was performed in Ref.10and yields the saturation ﬁeld /H92620Hs=800 mT. An elec- tromagnet is used to produce a tunable perpendicular mag-netic ﬁeld homogeneous on all the dots, and oriented perpen-dicular to the plane. This static ﬁeld creates the abovementioned splitting of the gyrotropic frequencies for oppo-site core polarities. The dots are placed at the extremity of animpedance-matched gold microwave strip-line which pro-vides an in-plane linearly polarized microwave magneticﬁeld h. Since hcontains both right and left circular compo- nents, it couples to the gyrotropic rotation of the vortex corefor both core polarizations p=/H110061. This microwave ﬁeld with variable frequency fis used to resonantly excite gyro- tropic rotation of the vortex core in a magnetic dot. If his weak, the amplitude of this gyrotropic rotation is relativelysmall but sufﬁcient to read the polarity of the rotating core /H20849without destroying it /H20850using the technique of magnetic reso- nance force microscope /H20849MRFM /H20850, which is illustrated in Fig. 1and described in detail in Ref. 15.I fhis sufﬁciently large and has the frequency corresponding to the gyrotropic reso-nance frequency for a given core polarity /H20849e.g., f +forp= +1/H20850, the velocity of the vortex core rotation induced by this ﬁeld reaches the critical value, and the core polarity is re-versed /H20849written /H20850. Achieving a dense memory requires to address the vor- tex core polarity state of a selected magnetic nanodot insidean array. In our experimental memory prototype of Fig. 1,w e meet this challenge using MRFM. In the framework of thistechnique the magnetic probe glued to a soft cantilever isscanned horizontally over the different magnetic dots. Thisprobe is a 800 nm diameter sphere made of amorphous Fe/H20849with 3% Si /H20850, and its role is twofold. First of all, it works as a sensitive local probe capable of detecting the change of thevertical component of magnetization caused by the gyrotro-pic rotation of the vortex core in a single magnetic dot. Thus,it is possible to read the polarity of the vortex core in aselected dot 10/H20851Fig.2/H20849b/H20850/H20852. Second, the dipolar stray ﬁeld of the magnetic probe creates an additional local bias ﬁeld ofabout 20 mT /H20849i.e., roughly twice as large as /H92620Hmin /H1122913 mT /H20850, which allows one to single out the particular magnetic dot situated immediately under the probe in theinformation writing process. The presence of the local biasﬁeld created by the probe shifts the gyrotropic frequency inthis dot by about /H9004f, thus allowing one to choose the fre- quency of the microwave writing signal in such a way, thatthe reversal of core polarity is done in only the selected dot,without affecting the information stored in the neighboringdots. In our experiments the total static magnetic ﬁeld was chosen to be /H92620H=65 mT /H112295/H92620Hmin, which gave the fol- lowing gyrotropic frequencies of the experimental magneticdot: f +=254 MHz, f−=217 MHz, and f0=236 MHz /H20849see Fig.2/H20850. The frequency linewidth of the gyrotropic rotation in the experimental dot was of the order of /H9004f/H112298 MHz. The process of reading of the binary information stored in a magnetic dot is illustrated by Fig. 2/H20849b/H20850. The amplitude of the cantilever oscillations is measured while the frequency ofthe weak reading microwave signal is varied in the intervalcontaining f +andf−. The results of these measurements are shown for the cases when the vortex core polarity was set at p=+1 or p=−1 at the beginning of the microwave frequency sweep.16It is clear from Fig. 2/H20849b/H20850that the core polarity can be detected not only from the resonance signal frequency,which is different for opposite core polarities, but also fromthe sign of the MRFM signal, 10which is positive for p=+1 and negative for p=−1. The writing process in a dot with initial core polarity equal to p=+1 is illustrated by Fig. 3. Figure 3/H20849a/H20850shows the frequency fwof the strong writing pulses of width /H9270w =50 ns and power Pw/H11229100/H9262W/H20849corresponding to a micro- wave magnetic ﬁeld of /H92620h=0.3 mT /H20850, while Fig. 3/H20849b/H20850shows the frequency frof the weak reading signal of power Pr FIG. 2. /H20849Color online /H20850/H20849a/H20850Frequency splitting induced by a perpendicular magnetic ﬁeld between the gyrotropic modes corresponding to the polarities p=/H110061. The shaded area illustrates the broadening of the gyrotropic mode. /H20849b/H20850MRFM absorption signals at /H92620H=65 mT for p=/H110061.132506-2 Pigeau et al. Appl. Phys. Lett. 96, 132506 /H208492010 /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: 188.82.181.120 On: Wed, 02 Apr 2014 21:39:19/H1122910/H9262W, which is supplied continuously and is interrupted every second in order to apply a strong writing pulse. Thefrequency of the weak reading signal can be kept close toeither f +orf−, and the amplitude of cantilever oscillations measured by MRFM provides the reading of the core polar-ity, as presented in Fig. 3/H20849b/H20850for the f +case. The application of the ﬁrst writing pulse to a particular selected dot having the initial polarity p=+1 results in the excitation of the vortex core rotation at f+=254 MHz with an amplitude sufﬁcient to bring the vortex core to the thresh-old speed corresponding to the core polarity reversal.7Once inverted, the ﬁnal state p=−1 is out of resonance with the writing pulse /H20849asf−/H11021f+−/H9004f/H20850, so that the polarity cannot be switched back to p=+1. It is clear from Fig. 3that the writ- ing pulses of the carrier frequency f+/H20849that we shall call /H9016+-pulses /H20850change the vortex core polarity from p=+1 to p =−1, while the writing pulses of the carrier frequency f− /H20849/H9016−-pulses /H20850change the core polarity from p=−1 to p=+1. For the chosen parameters of the writing pulses the polarityreversal is deterministic: the reversal efﬁciency has beentested several hundred times without any failure, implying asuccess rate better than 99%. We also note that the applica-tion of the /H9016 +-pulse to the magnetic dot with the polarity p=−1 /H20849and application of a /H9016−-pulse to the dot with p=+1/H20850 does not have any effect on the vortex core polarity in thedot. Moving the MRFM probe to the neighboring dots duringthe reading sequence allows one to check that the core po-larity in adjacent dots /H20849situated 10 /H9262m away /H20850is unaffected by the core reversal process in the selected dot. Thus, it hasbeen demonstrated that the frequency-selective deterministicmanipulation of the binary information has been achievedlocally. Although the experimental device shown in Fig. 1can be used as a prototype for the development of a frequency-controlled magnetic memory, a series of improvements canbe imagined to make a more practical solid-state variant/H20849Fig.4/H20850. First, it would be useful to increase the dot aspect ratio to /H9252=t/R=1 in order to reduce the dot saturation ﬁeld Hs, and, therefore, the minimum perpendicular bias magnetic ﬁeld to /H92620Hmin/H112295m T /H20851see Eq. /H208492/H20850/H20852. In this case, a static bias ﬁeld of only 20 mT, that could be produced by a per-manent magnet placed underneath the substrate, should be sufﬁcient to ensure reliable operation of the memory. Sec-ond, the dots of the practical variant should be arranged in aregular square array, where addressing of a particular dot isachieved by local combination of the static and microwaveﬁelds at the intersection of a word and a bit lines. The wordline could be made in the form of a pair of wires runningparallel to each row of dots at a 100 nm separation distance.A bias current I w=5 mA would be sufﬁcient to create an additional perpendicular ﬁeld of 10 mT at the addressed row,causing an additional shift of the resonance frequency byabout a full linewidth. The bit line could be made as animpedance matched wire running above each column of dots,producing the in-plane linearly polarized microwave ﬁeld h. Third, it would be useful to replace the MRFM detection ofFig.1, which contains mechanically moving parts, by local electrical detectors of the absorbed power for the readingprocess. Finally, the proposed design offers the possibility tocreate a multiregister memory by stacking dots of differentaspect ratios /H9252on top of each other, as they will have differ- ent resonance frequencies of the vortex core rotation. This research was partially supported by the French Grant Voice ANR-09-NANO-006-01, EU Grants DynaMaxunder Grant No. FP6-IST-033749 and Master NMP-FP7-212257, Contract No. W56HZV-09-P-L564 from the U.S.Army TARDEC, RDECOM, and NSF under Grant No.ECCS-0653901. 1H. W. Schumacher, C. Chappert, R. C. Sousa et al. ,Phys. Rev. Lett. 90, 017204 /H208492003 /H20850. 2K. Y. Guslienko, J. Nanosci. Nanotechnol. 8,2 7 4 5 /H208492008 /H20850. 3T. Shinjo, T. Okuno, R. Hassdorf et al. ,Phys. Rev. Lett. Science 289,9 3 0 /H208492000 /H20850. 4B. Van Waeyenberge, A. Puzic, H. Stoll et al. ,Phys. Rev. Lett. Nature /H20849London /H20850444, 461 /H208492006 /H20850. 5K. Yamada, S. Kasai, Y. Nakatani et al. ,Phys. Rev. Lett. Nature Mater. 6, 270/H208492007 /H20850. 6M. Curcic, B. Van Waeyenberge, A. Vansteenkiste et al. ,Phys. Rev. Lett- .Phys. Rev. Lett. 101, 197204 /H208492008 /H20850. 7K. Y. Guslienko, K.-S. Lee, and S.-K. Kim, Phys. Rev. Lett. 100, 027203 /H208492008 /H20850. 8A. Vansteenkiste, K. W. Chou, M. Weigand et al. ,Nat. Phys. 5, 332 /H208492009 /H20850. 9R. Hertel and C. M. Schneider, Phys. Rev. Lett. 97, 177202 /H208492006 /H20850. 10G. de Loubens, A. Riegler, B. Pigeau et al. ,Phys. Rev. Lett. 102, 177602 /H208492009 /H20850. 11B. A. Ivanov and G. M. Wysin, Phys. Rev. B 65, 134434 /H208492002 /H20850. 12K. Y. Guslienko, Appl. Phys. Lett. 89, 022510 /H208492006 /H20850. 13B. Heinrich, G. Woltersdorf, R. Urban et al. ,J. Appl. Phys. 95,7 4 6 2 /H208492004 /H20850. 14P. Bach, A. S. Bader, C. Rüster et al. ,Appl. Phys. Lett. 83,5 2 1 /H208492003 /H20850. 15O. Klein, G. de Loubens, V. V. Naletov et al. ,Phys. Rev. B 78, 144410 /H208492008 /H20850. 16By saturating all the dots in a large positive or, respectively, negative static magnetic ﬁeld. FIG. 3. /H20849Color online /H20850Local frequency control of the binary information demonstrated at /H92620H=65 mT. /H20849a/H20850The writing is performed every second by applying a single microwave pulse /H20849/H9270w=50 ns, Pw=−11 dBm /H20850whose car- rier frequency is tuned at either f+orf−./H20849b/H20850The reading /H20849Pr=−19 dBm /H20850is performed continuously between the writing pulses by MRFM using a cy-clic absorption sequence at the cantilever frequency /H20849f c=10 kHz /H20850. FIG. 4. /H20849Color online /H20850Proposed solid state design of the frequency- controlled magnetic memory.132506-3 Pigeau et al. Appl. Phys. Lett. 96, 132506 /H208492010 /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: 188.82.181.120 On: Wed, 02 Apr 2014 21:39:19
1.4870865.pdf	Structural and dynamical magnetic response of co-sputtered Co 2FeAl heusler alloy thin films grown at different substrate temperatures Anjali Y adav and Sujeet Chaudhary Thin Film Laboratory, Department of Physics, Indian Institute of Technology Delhi, New Delhi 110016, India (Received 3 January 2014; accepted 28 March 2014; published online 7 April 2014) The interdependence between the dynamical magnetic response and the microstructural properties such as crystallinity, lateral crystallite size, structural ordering of the co-sputtered polycrystalline Co2FeAl thin ﬁlms on Si (100) are studied by varying the growth temperature from room temperature (RT) to 600/C14C. Frequency (7–11 GHz) dependent in-plane ferromagnetic resonance (FMR) studies were carried out by using co-planar waveguide to estimate Gilbert damping constant (a) and effective saturation magnetization ( 4pMeff). The improvement in crystallinity, larger crystallite and particle sizes of the ﬁlms are critical in obtaining ﬁlms with lower aand higher 4pMeff. Increase in the lattice constant with substrate temperature indicates the improvement in the structural ordering at higher temperatures. Minimum value of ais found to be 0.005 60.0003 for the ﬁlm deposited at 500/C14C, which is comparable to the values reported for epitaxial Co 2FeAl ﬁlms. The value of 4pMeffis found to increase from 1.32 to 1.51 T with the increase in deposition temperature from RT to 500/C14C. The study also shows that the root mean square ( rms) roughness linearly affects the FMR in-homogenous line broadening and the anisotropy ﬁeld. VC2014 AIP Publishing LLC . [http://dx.doi.org/10.1063/1.4870865 ] INTRODUCTION In the recent years, Co-based full heusler alloys have been extensively used in spintronic devices. The outstanding properties, namely, very high spin polarization at room tem- perature (RT) and high Curie temperature make them moresuitable for magnetic tunnel junctions (MTJ’s), spin transfer switching devices, spin torque nano-oscillators, and also as efﬁcient spin injectors into semiconductors. 1–5The ferromag- netic materials which have low damping constant are highly desirable in MTJ’s as spin injecting electrode to further reduce the critical current density of magnetization switching.The superior thermal stability of these alloys is an additional advantage. Among these full heusler alloys, Co 2FeAl (CFA) has gained special attention because, in addition to high spinpolarization and high curie temperature (827 /C14C),1it also pos- sesses very low damping constant ( a)and shows perpendicu- lar magnetic anisotropy in CFA/MgO structures.6,7Wang et al. reported a 330%–340% tunnel magnetoresistance in Co2FeAl/MgO/CoFe MTJ’s.8So far, ferromagnetic reso- nance (FMR) studies of this heusler alloy have been reportedby Qiao et al. on GaAs (001) by molecular beam epitaxy, Mizukami et al. and Belmeguenai et al. on MgO (001) sub- strates, and Ortiz et al. reported FMR studies on MgO and Cr buffered MgO (001) substrate, by sputtering. 6,9–11All these studies have be carried out by using post deposition annealing process. The large diversity reported in avalues varying from 0.001 to 0.04 (Refs. 6,9–13) is thus found to depend on growth technique, choice of substrate and also on the method used to evaluate Gilbert damping constant. The former isattributed to the fact that the structural and magnetic proper- ties of heusler alloys are strongly affected by manufacturing process. A systematic investigation into the change in mag-netization dynamics that can be wrought by substrate temper- ature (T s) used for the growth of Co 2FeAl thin ﬁlm has notbeen undertaken in detail to date. The understanding of mag- netization dynamics of free Co 2FeAl ferromagnetic layer and its correlation with microstructure and root mean square (rms) roughness in such device structures is critical in realiz- ing high speed spin transfer torque (STT) based devices. In the present paper, we investigated the effect of growth tem- perature on the dynamical magnetic response using FMR inthe pulsed dc-magnetron sputtered Co 2FeAl thin ﬁlms. In par- ticular, the interdependence between Gilbert damping con- stant, effective magnetization of co-sputtered Co 2FeAl thin ﬁlms deposited on Si substrate with the rmsroughness and improvement in crystalline quality of the samples have been established. Our results are understood in terms of varyinggrowth kinematics prevailing during growth of Co 2FeAl thin ﬁlm as conditions vary by changing the T s. EXPERIMENTAL DETAILS The Co 2FeAl heusler alloy thin ﬁlms were prepared by co-sputtering of Co, Fe, and Al targets (200diameter) using pulsed dc-magnetron sputtering. The stoichiometry of theCFA thin ﬁlms was optimized ﬁrst by varying the dc-power applied to different targets. The elemental composition of the thin ﬁlm was conﬁrmed by using energy dispersive X-rayspectroscopy (EDX). Prior to deposition, the Si( 100)s u b - strates were cleaned ﬁrst with acetone and then in propanol using ultra-sonic cleaning bath. The native SiO 2present on the Si substrates was removed by dipping in 5% HF solution for 2 min. After this, the substrates were blown dried and loaded in the vacuum chamber for deposition. CFA thin ﬁlmof thickness 140 nm were grown on Si(100) substrates at dif- ferent T s; RT, 400/C14C, 500/C14C, and 600/C14C at a growth rate of 7 nm/min. These ﬁlms will be referred henceforth as SRT,S400, S500, and S600, respectively. The substrate tempera- ture was calibrated by measuring the temperature using the 0021-8979/2014/115(13)/133916/6/$30.00 VC2014 AIP Publishing LLC 115, 133916-1JOURNAL OF APPLIED PHYSICS 115, 133916 (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: 220.225.230.107 On: Thu, 10 Apr 2014 04:19:56Chromel-Alumel thermocouple glued directly on the substrate by Ag-paint, and also by reading the temperature as read by the thermocouple mounted within the heater, simultaneously.It was found that the temperature on the top of Si substrate was lower than the temperature indicated by heater’s display panel. Thus, the heater temperature was appropriately set toget the desired substrate temperature on the top of Si sub- strate. The base pressure of the chamber was better than 2/C210 /C06Torr and the working pressure was 4 /C210/C03Torr. Prior to heating, the base pressure of the chamber was better than 2.0 /C210/C06Torr. With substrate-heating, the chamber was further pumped for 2–3 h for obtaining the lowest basepressures, which ranged from 2.0 to 2.8 /C210 /C06Torr. The crystallographic structure of the samples was characterized by using a PANalytical X’pert PRO X-ray diffractometerequipped with Cu K asource. The XRD measurements were done in grazing angle X-ray diffraction (GAXRD) mode for structural analysis of thin ﬁlms. Atomic force microscopy(AFM) studies were performed in tapping mode by using Brucker’s Dimension Icon atomic force microscope. The can- tilever with nominal spring constant of 42 Nm /C01and resonant frequency in the range 230–410 kHz was used for imaging. The relative atomic compositions were studied using Oxford instruments (Model-Swift ED3000) EDX. The elementaldepth proﬁles of the ﬁlms were measured by using an ION-TOF make TOF-SIMS V (Secondary ion mass spec- trometry) equipped with two ion sources. In dynamic mode,the ﬁlm of an area of 300 /C2300lm 2was sputtered for depth proﬁling using 1 keV oxygen ions (I /C24250 nA) and the pulsed 25 keV Bi 1þion beam (I /C241.1 pA) raster scan over the ﬁlm (area /C24100/C2100lm2) for analysis. FMR measurements were performed by using 50 Xcharacteristic impedance co-planar waveguide (CPW) connected to a vector networkanalyzer (VNA). The effective magnetization ( 4pM eff)a n d Gilbert damping constant ( a) were evaluated by performing FMR in presence of in-plane applied magnetic ﬁeld. All thesemeasurements were done at room temperature. RESULTS AND DISCUSSION Structural properties Full heusler alloys (X 2YZ) consists of four interpenetrat- ing fcc sublattices. The half-metallicity in heusler alloy is known to be fragile against atomic disorders. The so called L21structure is the most ordered structure and ensures 100% spin polarization. In L21structure of CFA, when the Fe and Al atoms occupy their sites randomly, it transforms into B2 structure. However, in the presence of atomic disorderbetween all three atoms Co, Fe, and Al in a unit cell, the struc- ture is labeled as A2structure. In many cases, the structural disorder can be identiﬁed by XRD analysis. A full heusleralloy’s XRD pattern can be divided into odd superlattice dif- fraction, even superlattice diffraction and fundamental diffrac- tion. The odd superlattice diffraction (h, k, and l ¼odd numbers) can only be observed when the L2 1structure is formed. The even superlattice diffraction (h þkþl¼4nþ2) appears in both L2 1andB2structures. The fundamental dif- fraction becomes independent of atomic ordering when hþkþl¼4n.14Figure 1shows the GAXRD pattern of thesamples deposited at different values of Ts. It is observed that CFA grows in polycrystalline form in all the cases but with different crystalline properties. The observed XRD patternshows three reﬂections with h, k, and l values (220), (400), and (422) suggesting the presence of A2phase. The (200) and (400) reﬂections remain absent, even after increasing the sub-strate temperature up to 600 /C14C. It shows that the coherence length of B2andL21type long range ordering still does not exceed the sensitivity limit of XRD. The diffractogram ofSRT sample shows broader (220) peak, indicating the pres- ence of smaller crystallite size. The intensity of the (220) peak i n c r e a s e sw i t hi n c r e a s ei nT sfrom RT to 500/C14C and decreases at T s¼600/C14C. Fig. 1(b) shows the zoomed view near the (220) peak. It is observed that the full width at half maxima (FWHM) sharply decreases from 0.62/C14to 0.37/C14on increasing Tsfrom RT to 400/C14C, and further decreases to 0.33/C14at 500/C14C. However, at Ts¼600/C14C, FWHM is increased to 0.40/C14. The observed variation in both the parameters (FWHM and peak intensity) of the diffraction peaks strongly reveals the understandable improvement in the crystallinity of thin ﬁlms with increase in Tstill 500/C14C. This evolution in peak in- tensity could be understood in terms of ad-atom mobility, sur- face energy at the interfaces (substrate and growing ﬁlm), and thermal diffusivity.15The decrease in intensity at Ts¼600/C14C might be possible due to either the mis-orientation of planes within the grains or to the predominance in the atomic disor- der in unit cell. The lateral crystallite size ( tc)w e r ee s t i m a t e d using (220) reﬂection by employing Debye Scherrer formula. The t cwas found to increase from 18 nm for the SRT to 30 nm for S400 and then to 33 nm for S500 sample as shownin Fig. 2(a). The self-surface diffusion of ad-atoms plays a crucial role in the growth of grains. The increase in substrate FIG. 1. (a) Glancing angle X-ray diffraction pattern for the SRT, S400, S500, and S600 samples. The (220), (400), and (422) reﬂection are from the Co2FeAl thin ﬁlm. (b) Selected area of the diffraction pattern near (220) reﬂection of the Co 2FeAl samples. FIG. 2. (a) Evolution of crystallite size and lattice constant as a function of substrate temperature of Co 2FeAl samples. (b) rms roughness and average particle size as a function of substrate temperature evaluated by AFM.133916-2 A. Y adav and S. Chaudhary J. Appl. Phys. 115, 133916 (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: 220.225.230.107 On: Thu, 10 Apr 2014 04:19:56temperature enhances the surface diffusion of ad-atoms into equilibrium sites and promoting island-coalescence resulting in increase in the grain size.16However, in case of S600 sam- ple, the lateral crystallite size decrease to 28 nm which might be possible due to enhanced surface diffusion along the planes within the grain, and also in part to the higher ad-atoms ﬂuxreceived by the planes with higher surface energy. Consequently, at higher growth temperature of 600 /C14C, a higher growth rate in the transverse direction might havecaused the reduction in lateral grain size. 17 The peak-position of (220) reﬂection is found to shift towards lower 2hvalues for the samples SRT to S500. It implies an increase in d-value of CFA (220), and therefore, an increase in lattice constant with Ts, are shown in Fig. 2(a). The cubic lattice constant aoincreases with Tsup to 500/C14C and then reduces in S600 ﬁlm. The closest value of lattice constant with bulk CFA (lattice structure L21,ao¼0.5730 nm) was found to be 0.5703 60.0005 nm for the sample grown at 500/C14C, which is slightly smaller than the reported value.18 This could be attributed to the change in nature of the phasefrom nano-crystalline to crystalline. Another possible reasoncould be the decrease of micro-strain in different ﬁlms, which may be correlated with improvement in stoichiometry of the ﬁlms with increase in T s. The increase in Tsincreases the ad- atom’s mobility and their energy which makes these atoms more reactive and helps to get settle at the appropriate sites on the growing surface; this in turn lowers the concentration ofintrinsic point defects and improves the crystallinity as well as stoichiometry. 16 AFM studies The AFM images of SRT, S400, S500, and S600 sam- ples are shown in Fig. 3with identical vertical length scale of 31 nm. It is observed that there is a marked difference in the surface morphology with Ts. The appropriate parameter in characterizing the surface morphology of thin ﬁlms is thermsroughness ( qrms),19which expresses the standard devia- tion of the Z-values for the sample area, as given by equation qrms¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ XN n¼1ðZn/C0/C22ZÞ2 Nvuut; where, Nis the number of points in a given area and /C22Zis the mean value of surface height relative to central plane, and Zn is the vertical height of the surface at a given point n. Figure 2(b) shows that the qrmsof CFA ﬁlms increased from 0.6 nm to 8 nm as the substrate temperature is increased from RT to600 /C14C. However, the average particle size increases from 55 nm to 110 nm up to 500/C14C and then decrease to 95 nm at 600/C14C, which is closely correlated to the crystallite size determined by XRD. The high growth temperature can stim- ulate the migration of grain boundaries and aid the coales- cence of more and more grains during the high temperaturegrowth process. 20Also at high T s, more energy is available to the atoms so that they can diffuse and occupy the appro- priate site in the crystal lattice and form grains with lowersurface energy leading to larger grain size at higher T s.21The resulting increase in grain size enhances the surface rough- ness. The AFM images reveal that CFA thin ﬁlms exhibit 2Dlike growth at room temperature and as T sincreases, it shows 3D columnar type growth, explaining the observed changes inqrmsof the ﬁlms. EDX and TOF-SIMS analyses The residual gas composition, as inferred from the resid- ual gas analyzer, of the base vacuum was found to be con- sisting of O 2,N2, hydrocarbons, and H 2O vapors with their respective partial pressures as /C246/C210/C07Torr (for O 2), /C249/C210/C07Torr (for N 2),/C2410/C09Torr (for hydrocarbons), and/C247/C210/C07Torr (for H 2O vapors). Thus, the possibility of contamination of ﬁlm as well as substrate by oxygen and FIG. 3. Typical three dimensional 1/C21lm2AFM images of (a) SRT, (b) S400, (c) S500, and (d) S600 samples recorded in tapping mode. The Z-height is 31 nm in all the images.133916-3 A. Y adav and S. Chaudhary J. Appl. Phys. 115, 133916 (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: 220.225.230.107 On: Thu, 10 Apr 2014 04:19:56water vapors could be detrimental to the ﬁlm quality, partic- ularly at high growth temperatures. At high growth tempera- ture, there could be an inter-diffusion of atomic species fromﬁlm (Co, Fe, and Al) and substrate (Si) since the diffusion is a thermally activated process. This could lead to composition variation, particularly near the ﬁlm/substrate interface. Wehave in the present case delved further into these possibilities by performing EDX and SIMS measurements on these ﬁlms. Table Ipresents the atomic ratios (Co/Al and Fe/Al) as obtained from the relative atomic percentage of the Fe, Co, and Al by neglecting the contributions from carbon, oxygen, and Si recorded in their EDX spectra. Although, this is notabsolute, but the relative effect of increasing T sis readily evident. It is observed that the Co/Al ratio approaches from a Co rich value of 2.38 to the ideal value of 2.0 as T sis increased from RT to 400/C14C, and then to 500/C14C, followed by a degradation to a lowest value of 1.90 at T s¼600/C14Ci n S600. On the other hand, the Fe/Al ratio shows relativelyless and monotonic decrease with T s(from 0.94 in SRT to 0.79 in S600). Thus, compared to Fe, the relative proportion of Co is signiﬁcantly affected with change in T s. The SIMS depth proﬁles recorded at RT for the SRT, S400, S500, and S600 ﬁlms are presented in Figs. 4(a)–4(d). It can be observed that all the ﬁlms exhibited a ﬂat elementalproﬁle indicating composition uniformity over the ﬁlm thick- ness. Small enhancement in the elemental counts observed near the ﬁlm-substrate interface, which is predominantlyseen for cobalt, is attributed to the known increase in the atomic yield in metallic specimens due to the presence of oxygen. 22In the present case, the source of oxygen is in part due to inevitable formation of SiO 2before ﬁlm growth and also due to the fact that 1 keV oxygen ion beam is used for depth proﬁling during the SIMS measurement. In addition, aclose inspection near the trailing edges of the Co, Fe, and Al near the ﬁlm-substrate indicates an increasing tendency of inter-diffusion at ﬁlm/substrate interface as T sis increased. This is highlighted by a thick arrow (in Figs. 4(a)–4(d)) whose vertical shift could be taken as an indicator about the extent of inter-diffusion at the ﬁlm/substrate interface.However, the most remarkable and signiﬁcant effect of T sis clearly evident in ﬁlm sputtered at highest T sof 600/C14C, in that the trailing edges of Co, Fe and Al are distinctly wellseparated from the growing edge of the Si proﬁle, in striking contrast to the S500, S400, or SRT ﬁlms. In view of high partial pressures of oxygen and water vapors, the depth pro-ﬁle of S600 clearly suggests signiﬁcant oxidation of Si-substrate owing to higher T sof 600/C14C. On one hand, this SiO 2formed at the ﬁlm-substrate interface has somehow suppressed the elemental inter-diffusion at the ﬁlm/substrate interface (akin to a barrier layer), but it has signiﬁcantly affected the ﬁlm microstructure and its overall surfaceroughness, consistent with the XRD and AFM ﬁndings. Ferromagnetic resonance analysis FMR measurements were done by sweeping the external magnetic ﬁeld from 2000 Oe to zero ﬁeld values at ﬁxed microwave frequency while measuring the transmission sig-nalS 21. Figure 5(a) shows the ﬁeld dependent S21signal of SRT sample obtained at different frequencies. Generally, the distortions of the FMR line shapes originate from the mixingof the absorptive and dispersive components. The absorptive and dispersive parts indicated by LandD, respectively, are ﬁtted well with the following equation: 23 S21¼LDH2 ðHext/C0HrÞ2þDH2þDDHðHext/C0HrÞ ðHext/C0HrÞ2þDH2þC: (1) Here, S21is a linear combination of Lorentzian and disper- sive line shape, LandDare amplitudes of the absorptive and dispersive parts, Hextis the external dc-magnetic ﬁeld, H risTABLE I. EDX analysis of Co 2FeAl ﬁlms samples. Elemental composition (at. %) Atomic ratios Film sample Co Fe Al Co/Al Fe/Al SRT 55.1 21.8 23.0 2.38 0.94 S400 55.6 20.3 24.0 2.31 0.84S500 51.9 22.0 26.0 1.99 0.84S600 51.5 21.5 27.0 1.90 0.79 FIG. 4. SIMS depth proﬁles recorded on the ﬁlm sputtered at different T s, (a) RT, (b) 400/C14C, (c) 500/C14C, and (d) 600/C14C. The vertical shift of the thick arrow with change in T sindicates the extent of elemental inter-diffusion at the ﬁlm/substrate interface. FIG. 5. (a) Frequency dependent FMR signal recorded for the SRT sample.(b) Symmetric and anti-symmetric contributions to the asymmetric Lorentz line shape ﬁt for SRT sample recorded at 7.5 GHz frequency. A small con- stant background is found and added to the anti-symmetric contribution.133916-4 A. Y adav and S. Chaudhary J. Appl. Phys. 115, 133916 (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: 220.225.230.107 On: Thu, 10 Apr 2014 04:19:56the resonance ﬁeld, DHis the FWHM, and Cis a constant. Figure 5(b) shows the best ﬁt of the FMR signal exciting at 7.5 GHz using Eq. (1)with Hr¼513 Oe and DH¼42.6 Oe. For clarity, the contribution of absorptive (symmetric) and dispersive (asymmetric) contributions to the FMR signal is shown in Fig. 5(b)separately using the ﬁtted values. To determine the Meffanda, the FMR measurements are carried out between 7 and 11 GHz range in steps of 0.5 GHz. It is found that the Hrincreases with increase in frequency, as is expected for a ferromagnetic thin ﬁlm. The dependence ofHron applied microwave frequency can be well under- stood by the Kittel’s formula24for the ﬁlms f¼c0ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðHrþHkÞðHrþHkþ4pMef fÞq ; (2) where, c0¼glB/his the gyromagnetic ratio, Hkis the mag- netic anisotropy ﬁeld, and 4 pMeffis the effective magnetiza- tion ﬁeld. The 4pMeffandHkare evaluated by ﬁtting the Hr vs.fplot using Eq. (2). The solid lines in Fig. 6(a) present the ﬁtted results for all the samples. The 4pMeffvalue so obtained is found to vary signiﬁcantly with change in Ts. Whereas, Hkincreases from 5.8 Oe to 80 Oe with the increase in Tsfrom RT to 600/C14C, the 4pMeffincreases from 1.32 T to 1.51 T when Tsvaries from RT to 500/C14C, and at Ts¼600/C14C,4pMeffreduce to 1.50 T. These 4pMeffvalues are quite well comparable with those reported for the annealed epitaxial CFA heusler alloy ﬁlms grown on MgOsubstrates capped with Ta and Cr. 10The growth temperature dependence of the 4pMeffcan be understood as originating from the increase in structural ordering25in the samples as indicated by their XRD patterns. To determine a, the fre- quency dependent linewidth ðDHðfÞÞis ﬁtted with the fol- lowing equation:26 DHðfÞ¼DHin/C0homoþ4pa cf; (3) where, DHin/C0homo is known as in-homogeneous broadening and is equal to the intercept at zero frequency. The slope ofDH(f) is proportional to a, which gives information about the intrinsic contribution to FMR linewidth. Figure 6(b) shows the linewidth (DH)as a function of frequency and their corre- sponding ﬁtted curve (solid lines) for the samples. The ﬁtted values of aare found to decrease from 0.0072 60.0004 to 0.0050 60.0003 with increase in T sfrom RT to 500/C14C,however, at Ts¼600/C14C, the avalue increases to 0.0076 60.0006. The lowest value of aobserved at Ts¼500/C14Ci s found to be comparable to the published data.11DH0values are found to increase from 2.6 60.3 Oe to 112.8 610.0 Oe with increase in Ts. Zero frequency intercept ( DH0) of the ﬁt- ted curve is typically associated with the extrinsic contributionto the linewidth which arises due to inhomogeneties in the sample. From Fig. 6(b), it is observed that at a constant fre- quency the linewidth increases with increase in T s. The possi- ble causes leading to the linewidth broadening with increase inTscould be anisotropy-broadening and/or two-magnon grain boundary scattering,27,28which are affected by ﬁlm microstructure, defects and/or surface roughness present in the sample. The dependence of avalue on T sis shown in Fig. 7(a). The decrease in avalue with T scorrelates very well with the XRD and SIMS results. In that the Tsdriven increase (decrease) in the crystallinity, lattice constant, crystallite size, and the particle size, which is estimated from the AFM stud-ies, is consistent with the decrease (increase) found in a(Fig. 7(a)) and also in 4pM eff(Fig. 7(b)). The later also indicates the enhancement in the magnetic properties of the thin ﬁlmsasT sis increased to 500/C14C. The maximum 4pMeffobserved in ﬁlm sputtered at T s¼500/C14C is consistent with the ideal value of 2.0 of Co/Al atomic ratio as revealed by SIMS analy-ses. This, in fact, is an indirect evidence of the reduction in atomic disorder resulting from increase in T sfrom RT to 500/C14C. On the other hand, the observed increase in DH0and Hk(asTsis increased from RT to 600/C14C) is found to be well correlated with the surface roughness. Hence, the qrmsthus appears to play a signiﬁcant role in the inhomogeneous line-width broadening. The signiﬁcant difference in the nucleation and growth kinetics in case of ﬁlm sputtered at highest T sof 600/C14C, which has eventually resulted in signiﬁcant increase in surface rough- ness and decrease in lattice const ant, lateral crystallite size and particle size is due to oxidation of the substrate, which are re-sponsible for the observed increase in damping constant as T s is increased from 500/C14C to 600/C14C. It is therefore concluded that the substrate temperature of 500/C14C is found to be opti- mum vis- /C18a-vis the least value of a¼0.005060.0003 in these pulsed dc-sputtered ﬁlm. It is to be noted that this value of ain our polycrystalline ﬁlm is larger only by a factor of /C242c o m - p a r e dt ot h a tr e p o r t e db yM i z u k a m i et al.6for epitaxial CFA ﬁlms. The possible cause for this difference in the avalues is attributed to the structural inhomogeneities present in our poly-crystalline ﬁlm as discussed in the previous para. The struc- tural inhomogeneity associat ed with the grain boundaries FIG. 6. (a) Resonance ﬁeld as a function of frequency and (b) linewidth as a function of frequency for the SRT, S400, S500, and S600 samples. Measured data shown by symbols and ﬁtted curve shown by solid line. FIG. 7. (a) Fitted values of the damping constant and inhomogeneous broad- ening ( DHo), and (b) effective magnetization and anisotropy ﬁeld as a func- tion of growth temperature for Co 2FeAl thin ﬁlms.133916-5 A. Y adav and S. Chaudhary J. Appl. Phys. 115, 133916 (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: 220.225.230.107 On: Thu, 10 Apr 2014 04:19:56causes the non-uniformity in the internal magnetizing ﬁeld due to the prevailing defect structure. However, our optimally least avalue of 0.0050 in the polycrystalline CFA heusler alloy ﬁlm grown at Ts¼500/C14C is better than the avalue of 0.008 reported for epitaxial Co 2FeSi heusler alloy ﬁlms29and is quite comparable to those exhibited by epitaxial Co 2MnSi heusler alloy ﬁlms.30This might be possible, as discussed above, due to the different growth kinetics present during the ﬁlm-growth at higher T sin the present case as opposed to annealing after the deposition. This is understa ndable since the optimum value of substrate temperature during deposition provides enough energy to the ad-atoms for surface mobility and helps them tosettle at the appropriate sites. CONCLUSIONS In summary, we have brought out a correlation between the crystalline structure, surface roughness, and dynamicmagnetic properties of Co 2FeAl heusler alloy thin ﬁlm by studying the effects of varying the substrate temperature. The study shows that the growth temperature plays a signiﬁ-cant role in attaining a high effective magnetization ﬁeld and a low damping constant. The effective magnetization ﬁeld signiﬁcantly increases to 1.50 T with increase in growth tem-perature from RT to 500 /C14C, and gets more or less saturated thereafter. The least value of a¼0.005060.0003 is evi- denced at the optimally appropriate growth temperature of500 /C14C for the polycrystalline Co 2FeAl. It is concluded that the magnetization dynamics of Co 2FeAl thin ﬁlms can be tai- lored by changing the granular microstructure. ACKNOWLEDGMENTS One of the authors (A.Y.) would like to thank MHRD, Government of India for the research fellowship. Weacknowledge the thankful discussions with Dr. R. K. Kotnala and Dr. N. Karar of CSIR-NPL, New Delhi. 1S. Trudel, O. Gaier, J. Hamrle, and B. Hillebrands, J. Phys. D: Appl. Phys. 43, 193001 (2010). 2R. Okura, Y. Sakuraba, T. Seki, K. Izumi, M. Mizuguchi, and K. Takanashi, Appl. Phys. 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McCleskey, and Q. X. Jia, Thin Solid Films 101, 492 (2005). 21Z. B. Fanga, Z. J. Yana, Y. S. Tana, X. Q. Liua, and Y. Y. Wanga, Appl. Surf. Sci. 241, 303 (2005). 22K. Elst and W. Vandervorst, J. Appl. Phys. 73, 4649 (1993). 23N. Mecking, Y. S. Gui, and C. M. Hu, Phys. Rev. B 76, 224430 (2007). 24C. Kittel, Phys. Rev. 73, 155 (1948). 25Y. V. Kudryavtseva, V. A. Oksenenkoa, V. A. Kulagina, J. Dubowikb, and Y. P. Lee, J. Magn. Magn. Mater. 310, 2271 (2007). 26C. E. Patton, J. Appl. Phys. 39, 3060 (1968). 27N. Mo, J. Hohlfeld, M. Islam, C. S. Brown, E. Girt, P. Krivosik, W. Tong, A. Rebei, and C. E. Patton, Appl. Phys. Lett. 92, 022506 (2008). 28B. Heinrich, J. F. Cochran, and R. Hasegawa, J. Appl. Phys. 57, 3690 (1985). 29M. Oogane, R. Yilgin, M. Shinano, S. Yakata, Y. Sakuraba, Y. Ando, and T. Miyazaki, J. Appl. Phys. 101, 09J501 (2007). 30R. Yilgin, M. Oogane, Y. Ando, and T. Miyazaki, J. Magn. Magn. Mater. 310, 2322 (2007).133916-6 A. Y adav and S. Chaudhary J. Appl. 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1.2362760.pdf	Dynamic magnetization processes in magnetostrictive amorphous wires A. P. Chen, A. Zhukov, J. González, L. Domínguez, and J. M. Blanco Citation: Journal of Applied Physics 100, 083907 (2006); doi: 10.1063/1.2362760 View online: http://dx.doi.org/10.1063/1.2362760 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/100/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Evolution of Barkhausen jumps in annealed amorphous wire having vanishing values of magnetostriction J. Appl. Phys. 100, 043914 (2006); 10.1063/1.2335385 Energetic model of ferromagnetic hysteresis: Isotropic magnetization J. Appl. Phys. 96, 2753 (2004); 10.1063/1.1771479 Interacting amorphous ferromagnetic wires: A complex system J. Appl. Phys. 85, 2768 (1999); 10.1063/1.369592 Influence of bending stresses on the magnetization process in Fe-rich amorphous wires (abstract) J. Appl. Phys. 81, 4035 (1997); 10.1063/1.364928 Density of magnetic poles in amorphous ferromagnetic wires: An example of weak chaos J. Appl. Phys. 81, 5725 (1997); 10.1063/1.364649 [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: Wed, 17 Dec 2014 19:16:25Dynamic magnetization processes in magnetostrictive amorphous wires A. P . Chen,a/H20850A. Zhukov, and J. González Department of Material Physics, Chemistry Faculty, P .O. Box 1072, 20080 San Sebastián, Spain L. Domínguez and J. M. Blanco Department of Applied Physics I, EUITI, UPV/EHU, Plaza Europa I, 20080 San Sebastián, Spain /H20849Received 3 November 2005; accepted 13 August 2006; published online 25 October 2006 /H20850 We have performed the theoretical studies on the longitudinal dynamic magnetization process of magnetostrictive amorphous wires characterized by a large single Barkhausen jump /H20849magnetic bistability /H20850based on our previous experimental measurements on these wires. The domain structures of these wire samples consist of a single domain inner core with axial magnetization surrounded bythe outer domain shell with the magnetization oriented perpendicular /H20849/H9261 s/H110220/H20850or circular /H20849/H9261s/H110210/H20850to the wire axis. In the present work we use the resultant magnetization vector M/H6023tilting /H9258angle to z axis to describe the sample’s domain structures. In terms of solving the Landau-Lifshitz-Gilbert equation followed by M/H6023the analytical solution of the dimensionless axial component of the magnetization mz=MZ/Mshas been obtained, and mz/H20851t/H20849H0,fe/H20850,/H9270,/H9253/H20852is a function of the ﬁeld amplitude H0, ﬁeld frequency fe, and the samples’ material parameters such as the damping constant /H9270and the gyromagnetic ratio /H9253. The function mz/H20851t/H20849H0,fe/H20850,/H9270,/H9253/H20852allows us to study the dynamic properties of the magnetization process of a wire sample. It has been found that the switching time ts, the switching ﬁeld Hsw, and the dynamic coercive ﬁeld Hdcdepend on a magnetic ﬁeld and material parameters. We found that the parameter /H9251=/H9253/H9270//H208491+/H92702/H20850related to the rate of M/H6023, rotating the direction of the effective ﬁeld, plays an important role in the magnetization process. By ﬁtting the experimental data to the theoretical magnetization curve the value of the damping constant /H9270of the magnetostrictive amorphous wires can be estimated. © 2006 American Institute of Physics . /H20851DOI: 10.1063/1.2362760 /H20852 I. INTRODUCTION In the past several years, there were many attempts to obtain the general theoretical shape of the hysteresis loop ormagnetization curve for ferromagnetic materials. 1–8Gener- ally there are two theoretical methods, namely, the Preisach-Nèel model 1which allows to reproduce quite well the shape of hysteresis loops, but their physical meaning seems to beunclear, and the other is based on the micromagnetics theoryof Brown 4and Aharoni,5however, this second method yields nonlinear equations which are difﬁcult to solve.6,8 In addition, it is well recognized that the magnetization process is closely related to the domain structure of the ma-terial, the orientation and amplitude of the external magneticﬁeld, as well as the type of the magnetization process. Con-sequently, many theoretical models have been proposed todescribe the magnetization process of different magneticsamples according to different conditions. For example, re-garding the magnetic hysteresis phenomenon being a resultof the domain wall motion when impeded as it is pinnedduring the magnetization process, the mean ﬁeld approxima-tion has been successfully used 6to deduce a simple differen- tial equation of the state for a ferromagnetic material. It wasshown that the solution of this equation could demonstratethe features associated with the initial magnetization curve,major and minor hysteresis loops, in agreement with the ex-perimental results. Another example concerning thin ferro-magnetic ﬁlms, a realistic shape of hysteresis loop describing the incoherent rotations of the magnetization process, wasobtained in Ref. 8by introducing an additional internal ﬁeld in the Stoner-Wohlfarth model. On the other hand, the magnetic bistability effect exhib- ited by various magnetic amorphous materials /H20849wires, rib- bons, and microwires /H20850has been a topic of growing interest during the last few years 9–17because of their very prominent technological applications as sensing elements. The charac-teristic feature of such magnetic bistability is the appearanceof a rectangular hysteresis loop at low applied magnetic ﬁeldand, consequently, the axial magnetization process takesplace by a single large Barkhausen jump /H20849LBJ /H20850. This phe- nomenon was satisfactorily interpreted in terms of nucleationof reversed domain inside the internal single domain with the consequent domain wall /H20849DW /H20850propagation. 11–18Perfectly rectangular shape of the hysteresis loop has been related to adepinning of such DW at certain magnetic ﬁeld associatedwith a very high propagation velocity. 9,13 Drastic change of the magnetization at applied magnetic ﬁelds just above the switching ﬁeld gives rise to sharp volt-age pulses appearing in a secondary pickup coil during thelarge Barkhausen jump, which promises to be very useful fordifferent technological applications. 19,20Furthermore, the do- main structure of magnetostrictive amorphous wires consistsmainly of a single domain inner core with the magnetizationparallel to the wire axis that is surrounded by the outer do-main shell with the magnetization oriented perpendicular/H20849/H9261 s/H110220/H20850or circular /H20849/H9261s/H110210/H20850to the wire axis. As a conse-a/H20850Electronic mail: aipingcz@hotmail.comJOURNAL OF APPLIED PHYSICS 100, 083907 /H208492006 /H20850 0021-8979/2006/100 /H208498/H20850/083907/4/$23.00 © 2006 American Institute of Physics 100, 083907-1 [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: 137.149.200.5 On: Wed, 17 Dec 2014 19:16:25quence, the magnetostrictive amorphous wires offer us a nearly ideal material to study the magnetization processes.Applying magnetic ﬁeld H exwith the frequency feand am- plitude H0along the wire axis z, the domain walls move rapidly within the core, and according to the Landau-Lifshitz-Gilbert equation the resultant magnetization vector M/H6023of the wire would rotate. Therefore, the aim of this paper is to attempt to obtain a quantitative relation between M zand quasi-dc external axial magnetic ﬁeld Hexduring the magnetization processes of magnetostrictive amorphous wires by solving the Landau-Lifshitz-Gilbert equation and to study the dynamic propertiesof the hysteresis loop of these amorphous wires. II. THEORY In the theory the resultant magnetization vector of the magnetostrictive amorphous wires is represented by the vec- torM/H6023, which follows the Landau-Lifshitz-Gilbert /H20849LLG /H20850 equation, /H11509M/H6023 /H11509t=/H9253/H20851M/H6023/H11003H/H6023eff/H20852−/H9270 Ms/H20875M/H6023/H11003/H11509M/H6023 /H11509t/H20876, /H208491/H20850 where /H9253is the gyromagnetic ratio, /H9270is a phenomenological damping constant, Msis saturation magnetization, and H/H6023effis the effective magnetic ﬁeld. In the experimental measure-ment, the amorphous wires were subjected to an axialquasi-dc ﬁeld, H ex/H20849t/H20850, having a triangular wave form related to the frequency feand amplitude H0as21 Hex/H20849t/H20850=4feH0t. /H208492/H20850 In this case, if the effective magnetic ﬁeld has an axial com- ponent only it can be written as Hefz=Hex−Hn, /H208493/H20850 where Hnis the nucleation ﬁeld, which can be assumed to have the following form: Hn/H20849t/H20850=4feH0t. /H208494/H20850 It means that if t/H11021t, the domain wall only undergoes re- versible displacement around its equilibrium position. Whent/H11022t , the domain wall moves irreversibly and the velocity of the domain wall motion can be measured experimentally;13 meanwhile, the resultant magnetic vector M/H6023of the wire un- der the action of Hex/H20849t/H20850will rotate giving rise to the hyster- esis loop. In terms of the cylindrical coordinate /H20849r,/H9278,z/H20850/H20849see Fig. 1/H20850, with zparallel to the wire axis, in the case of /H9261s/H110220 the components of the magnetization M/H6023can be written as Mr=Mssin/H9258, M/H9278=0 , /H208495/H20850 Mz=Mscos/H9258, and in the case of /H9261s/H110210 there areMr=0 , M/H9278=Mssin/H9258, /H208496/H20850 Mz=Mscos/H9258, where /H9258is the angle between the magnetization vector M/H6023 and the zaxis. In the case of /H9261s/H110220, the components of the LLG equa- tion /H208491/H20850can be given as dM r dt=/H9270cos/H9258dM/H9278 dt, dM/H9278 dt=−/H9253MsHefzsin/H9258+/H9270/H20873dM Z dtsin/H9258−dM r dtcos/H9258/H20874, /H208497/H20850 dM Z dt=−/H9270sin/H9258dM/H9278 dt. Following Eq. /H208497/H20850, the time derivative of the dimensionless axial component of the magnetization, mz=MZ/Ms=cos/H9258, can be expressed as dcos/H9258 dt=/H9253/H9270 /H208491+/H92702/H20850/H20849Hefzsin2/H9258/H20850=/H9251sin2/H9258Hefz, /H208498a/H20850 where /H9251=/H9253/H9270 /H208491+/H92702/H20850. /H208498b/H20850 Here it is noted that /H9251is one of the important parameters in the magnetization process since it represents the rate of M/H6023 rotating in the direction of the effective ﬁeld. In the case of /H9261s/H110210, introducing /H208496/H20850into /H208491/H20850it can be veriﬁed that the axial component motion equation is alsoexpressed by /H208498/H20850. According to /H208498/H20850,m zhas two stable states, /H9258=0 and /H9258 =/H9266. Considering the expressions /H208492/H20850–/H208494/H20850and integrating /H208498/H20850 we can obtain the time dependence of mzas FIG. 1. The components of the magnetization Min the cylindrical coordi- nates /H20849r,/H9278,z/H20850:/H20849a/H20850/H9261s/H110220 and /H20849b/H20850/H9261s/H110210.083907-2 Chen et al. J. Appl. Phys. 100, 083907 /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: 137.149.200.5 On: Wed, 17 Dec 2014 19:16:25mz/H20849t/H20850=th/H20851Arth /H20849m0/H20850±2feH0/H9251/H20849t−t/H208502/H20852, /H208499/H20850 where m0denotes the mz/H20849t/H20850att=t; the sign /H11001corresponds to the external magnetic ﬁeld Hex/H20849t/H20850applied along positive /H11001 or negative /H11002zaxial direction. On the other hand, following relation /H208492/H20850we can write t−t=/H20849Hex−Hn/H20850/4feH0. /H2084910/H20850 Introducing /H2084910/H20850into /H208499/H20850the magnetization curve can be ob- tained as mz/H20849t/H20850=th/H20851Arth /H20849m0/H20850±/H9252/H20849Hex−Hn/H208502/H20852, /H2084911/H20850 where /H9252=/H20849/H9251/8/H20850/H208491/H0fe/H20850. It can be seen that mzis a function of the applied ﬁeld amplitude H0and ﬁeld frequency feas well as the material parameters. The time dependence of themagnetization m z/H20849t/H20850described by expression /H208499/H20850and the ﬁeld dependence mz/H20849Hex/H20850described by /H2084911/H20850allow us to study ana- lytically the dynamic properties of the magnetization process of these amorphous wires. III. DISCUSSION A. Switching time ts The switching time tsdeﬁned where mz/H20849t/H20850changes its orientation to the opposite can be deduced from Eq. /H208499/H20850as ts=1.6 /H20849feH0/H9251/H208501/2. /H2084912/H20850 Expression /H2084912/H20850gives the reasonable result that tsbecomes shorter as fe,H0, and/H9251increase. On the other hand, the parameter /H9251expressed by /H208498b/H20850 mainly depends on the damping constant /H9270and it has the maximum value of /H9251=/H9253/2, where /H9270=1. Figure 2shows the time dependence of mzdetermined numerically from /H208499/H20850for m0=0.99, feH0=720 A/ms, and /H9270=0.01, 0.1, 1, respectively. It can be seen that the switching time tsas well as the relax- ation time tr, where mz/H20849t/H20850becomes zero, decrease as the magnitude of the damping constant /H9270increases. Generally /H9270 is the material parameter associated with the paths of the tipof the magnetization precession around the effective mag- netic ﬁeld. When /H9270/H110221, the path of M/H6023follows the energy gradient, which is equivalent to the static situation accordingto Ref. 22. B. Switching ﬁeld Hsw Following expression /H2084911/H20850at the same H0and fe, the switching ﬁeld Hswis proportional to the nucleation ﬁeld Hn and the material parameters /H9253and/H9270, Hsw/H11008Hn+1 /H92511/2. /H2084913/H20850 Figure 3shows the ﬁeld dependence of mzwith Hnas pa- rameter for /H9270=0.1 and /H20849/H9253/H0fe/H20850=8. It can be seen that the magnitude of the switching ﬁeld Hswis proportional to the nucleation ﬁeld Hn. The nucleation ﬁeld Hncan be obtained by minimizing the total Gibbs free energy /H9278tgiven by the following expression:22 Hn=2Ku /H92620MS+/H20849N/H11036−N /H20648/H20850MS, /H2084914/H20850 where Ku=/H208493/2 /H20850/H9261s/H9268is the stress induced magnetic aniso- tropy constant and N /H20648andN/H11036are the demagnetizing factors for magnetization parallel and perpendicular to the zaxis. For the magnetostrictive conventional amorphous wires,Fe 72.5Si12.5B15and Co 72.5Si12.5B15, the stress induced aniso- tropy constant Ku/H20849J/m3/H20850are 2200 and 240,9respectively. The different values of demagnetizing factors N /H20648andN/H11036thus can be calculated from the corresponding ratio of the wire innercore and the outer shell /H20849N /H20648is proportional to the diameter of inner core dinbeing D//H208812 for/H9261s/H110220 and D//H208813 for/H9261s/H110210, respectively, where Dis the wire diameter /H20850. From this differ- ence one can explain why at the same ﬁeld amplitude H0and frequency fe=60Hzthere is a difference in values of the switching ﬁeld Hsw, 12 and 6.4 /H20849A/m /H20850for/H9261S/H110220 and /H9261S /H110210, respectively.9 FIG. 2. The time dependence of zcomponent mzwith/H9270=0.01,0.1,1 as the parameters for m0=0.99 and feH0=720 A/ms, respectively. FIG. 3. The ﬁeld dependence of mzwith the nucleation ﬁeld Hnas param- eter for /H9270=0.1 and /H20849/H9253/H0fe/H20850=8.083907-3 Chen et al. J. Appl. Phys. 100, 083907 /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: 137.149.200.5 On: Wed, 17 Dec 2014 19:16:25C. Dynamic coercive ﬁeld Hdc For many technological applications of amorphous wires at ac ﬁelds the amplitude H0and frequency fedependences of the coercivity of amorphous wires have attracted mostattention. 21,23–25Assuming mz=0 in expression /H2084911/H20850we can acquire the expression of the dynamic coercive ﬁeld Hdc: Hdc=Hn+C/H20849feH0/H208501/2, /H2084915/H20850 where C=/H20851−Arth /H20849m0/H20850/H208498//H9251/H20850/H208521/2. The similar dependence of /H20849feH0/H208501/2was pointed out in Refs. 21and24in the case of small frequency in the amorphous wires and in Ref. 23for the amorphous ribbons. D. The damping parameter /H9270 As mentioned above the parameter /H9251represents the rate ofM/H6023rotating in the direction of H/H6023effand plays an important role in the magnetization process. Figure 4shows the ﬁeld dependence of mzwith /H9251=104, 4.4 /H11003104, 5.4 /H11003104 /H20849s A/m /H20850−1as the parameter for Hn=1,H0=15 /H20849A/m /H20850, and fe=50 s−1. It clearly demonstrates that increasing /H9251de- creases the dynamic coercive ﬁeld Hdc; meanwhile, the large Barkhausen jump becomes more evident. The value /H9251can be determined by /H2084911/H20850as /H9251=8H0fe/H20849atanh mz−atanh m0/H20850 /H20849Hex−Hn/H208502. /H2084916/H20850 Fitting the experimental measurement data of the wire sample we can estimate the value of /H9251. By introducing into /H208498b/H20850the damping constant /H9270can be calculated. It is about 0.32 for the conventional amorphous wire Fe 72.5Si12.5B15at the ﬁeld amplitude H0=10.5 A/m and fe=60 s−1and 0.005for Co 72.5Si12.5B15wire at the H0=100 A/m and fe=60 s−1. Additionally expression /H2084916/H20850combined with /H208498b/H20850may develop one method to estimate the value /H9270of the magneto- strictive amorphous wire. IV. CONCLUSION The dynamic magnetization process in bistable amor- phous wire has been interpreted in terms of the solution of the LLG equation for the magnetization vector M/H6023under the action of the effective magnetic ﬁeld H/H6023eff. It has been dem- onstrated that the switching time ts, the switching ﬁeld Hsw, and the dynamic coercive ﬁeld Hdcdepend on both extrinsic /H20849H0andfeof the external magnetic ﬁeld Hex/H20850and intrinsic /H20849/H9270and/H9253of material /H20850parameters. It also has shown that the parameter /H9251related to the rate of M/H6023rotating onto the effec- tive ﬁeld plays an important role in the magnetization pro-cess. By ﬁtting the experimental data to the theoretical mag-netization curve the value of the damping constant /H9270can be estimated. 1F. Preisach, Z. Phys. 94, 277 /H208491935 /H20850. 2R. M. Del Vecchio, IEEE Trans. Magn. MAG-16 ,8 0 9 /H208491980 /H20850. 3M. A. Rahman, M. Poloujadoff, R. D. Jackson, J. Perrard, and S. D. Gowda, IEEE Trans. Magn. MAG-17 , 3253 /H208491981 /H20850. 4W. F. Brown, J. Appl. Phys. 30,6 2 /H208491959 /H20850. 5A. Aharoni, J. Appl. Phys. 30,7 0 /H208491959 /H20850. 6D. C. Jiles and D. L. Atherton, J. Appl. Phys. 55,6 /H208491984 /H20850. 7F. Ossart and T. A. Phung, J. Appl. Phys. 67,9 /H208491990 /H20850. 8J. Nowak, J. Appl. Phys. 72,4 /H208491992 /H20850. 9K. Mohri, F. B. Humphrey, K. Kawashima, K. Kimura, and M. Muzutani, IEEE Trans. Magn. 26, 1789 /H208491990 /H20850. 10A. M. Severino, C. Gómez-Polo, D.-X. Chen, and M. Vázquez, J. Magn. Magn. Mater. 103,1 1 7 /H208491992 /H20850. 11A. P. Zhukov, M. Vázquez, J. Ve1ázquez, H. Chiriac, and V . Larin, J. Magn. Magn. Mater. 151, 132 /H208491995 /H20850. 12B. K. Ponomarev and A. P. Zhukov, Sov. Phys. Solid State 26, 1795 /H208491984 /H20850. 13A. Zhukov, Appl. Phys. Lett. 78, 3106 /H208492001 /H20850. 14J. González, J. Appl. Phys. 79, 376 /H208491996 /H20850. 15V . Zhukova, A. Zhukov, J. M. Blanco, J. González, and B. K. Ponomarev, J. Magn. Magn. Mater. 249, 131 /H208492002 /H20850. 16M. Vázquez and A. Zhukov, J. Magn. Magn. Mater. 160,2 2 3 /H208491996 /H20850. 17R. Varga, K. L. García, M. Vázquez, A. Zhukov, and P. V ojtanik, Phys. Rev. B 70, 024402 /H208492004 /H20850. 18J. Yarnasaki, M. Takajo, and F. B. Humprhrey, IEEE Trans. Magn. 29, 2545 /H208491993 /H20850. 19V . Larín, A. Torcunov, S. Baranov, M. Vázquez, A. Zhukov, and A. Hemando, Spain Patent No. P 9601993 /H20849September 20, 1996 /H20850. 20A. Zhukov, J. González, J. M. Blanco, P. Aragoneses, and L. Domínguez, Sens. Actuators, A 81, 129 /H208492000 /H20850. 21A. Zhukov, M. Vázquez, J. Ve1ázquez, C. García, R. Valenzuela, and B. Ponomarev, Mater. Sci. Eng., A 226–228 , 753 /H208491997 /H20850. 22H. Kronmüller and M. Fähnle, Micromagnetism and the Microstructure of Ferromagnetic Solids /H20849Cambridge University Press, Cambridge, 2003 /H20850, p. 96. 23A. Zhukov, Mater. Des. 14, 299 /H208491993 /H20850. 24P. Aragoneses, J. M. Blanco, L. Domínguez, A. Zhukov, J. González, and K. Kuladowski, J. Phys. D 31, 494 /H208491998 /H20850. 25K. Hoselitz, J. Test. Eval. 20, 201 /H208491980 /H20850. FIG. 4. The ﬁeld dependence of mzwith /H9251=104, 4.4/H11003104, 5.4 /H11003104/H20849sA/m /H20850−1as the parameter for Hn=1,H0=15 A/m, and fe=50 s−1.083907-4 Chen et al. J. Appl. Phys. 100, 083907 /H208492006 /H20850 [This article is copyrighted as indicated in the article. 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1.4917334.pdf	Intrinsic Gilbert damping constant in epitaxial Co 2Fe0.4Mn0.6Si Heusler alloys films Augustin L. Kwilu,1,a)Mikihiko Oogane,2Hiroshi Naganuma,2Masashi Sahashi,1 and Yasuo Ando2 1Department of Electronic Engineering, Tohoku University, 6-6-05, Aza-Aoba, Aramaki, Aoba-ku, Sendai 980-8579, Japan 2Department of Applied Physics, Tohoku University, 6-6-05, Aza-Aoba, Aramaki, Aoba-ku, Sendai 980-8579, Japan (Presented 7 November 2014; received 22 September 2014; accepted 9 December 2014; published online 14 April 2015) The (001)-oriented and (110)-oriented epitaxial grown Co 2Fe0.4Mn 0.6Si ﬁlms were fabricated by magnetron sputtering technique in order to investigate the annealing temperature dependence ofthe intrinsic Gilbert damping constant ( a). The stuck ﬁlms, deposited on MgO and Al 2O3a-plane substrates, respectively, were annealed at various temperatures ranging from 400/C14C to 550/C14C. The X-ray diffraction analysis was conducted to conﬁrm that all the ﬁlms were epitaxially grown. Inaddition, the ferromagnetic resonance measurements as well as the vibrating sample magnetometer were carried out to determine their magnetic properties. A small aof 0.004 was recorded for the sample with 001-oriented Co 2Fe0.4Mn 0.6Si (CFMS (001)) and 110-oriented CFMS (CFMS (110)) annealed at 450/C14C.VC2015 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4917334 ] I. INTRODUCTION Heusler alloys1are potential components for spintronics devices based on their half-metallicity.2Their high and efﬁ- cient generated input spin current makes them reliable candi- dates in spin transfer torque magnetic random access memory, in Hard disk drive (HDD) read head, and for very sensitivemagnetic sensor devices. 3Moreover, recently, their use as free or reference layers in the spin valve architectures was proposed to overcome the challenging shortage of outputpower observed in Spin Torque Nano-Oscillator, in an in- plane precession mode, by increasing the giant magnetoresist- ance (GMR) effect. 4The magnetoresistance can be increased, for instance, throughout the current perpendicular to the plane GMR (CPP-GMR) devices with Co 2Fe0.4Mn0.6Si (CFMS) electrodes.5The X-ray diffraction (XRD) measurements6–8 performed on Heusler alloys established that chemical order- ing of their components is bestowing theirs ferromagnetic properties. The constant ais one of the parameters, which characterizes Heusler alloys and based on which spintronics devices are designed. The constant aof Co 2-Heusler was reported previously in the case of Co 2MnAl and Co 2MnSi (CMS).9,10Although the attention was primarily focused on Co2MnAl and CMS characterization, CFMS attracted our attention for its very high Curie temperature above that ofCMS and other Heusler. Prior to our work on CFMS, the rela- tive concentration was calculated based on a systematic study ofafor a variety of Co 2FexMn1/C0xSi with respect to Fe con- centration x.11The Fe concentration xdependence of a showed that CFMS with x¼0.4 has the smallest damping constant. This work, however, is focused on the dependenceon the annealing temperature of ain CFMS. In fact, the annealing temperature affects both the crystallinity and thechemical ordering in CFMS. We fabricated the ﬁlms whose CFMS layers were, respectively, annealed at various tempera- tures. We determined the internal magnetic properties of the CFMS layer of each ﬁlm by Ferromagnetic resonance (FMR)measurements. The constant aof CFMS layer is estimated from the ﬁtting of the experimental data related to the out-of-plane angular dependence of resonance ﬁeld ( H R) and that of the peak-to-peak linewidth ( HPP) of FMR spectra of the ﬁlms. The ﬁtting equations are derived from the Landau-Lifshitz-Gilbert (LLG) equation of motion. II. EXPERIMENTAL METHOD Cr(40)/Co 2Fe0.4Mn 0.6Si(30)/Ta(3) and Ta(3)/Cr(40)/ Co2Fe0.4Mn 0.6Si(30)/Ta(3) ﬁlms were, respectively, depos- ited on MgO(001) (for (001)-orientation) and Al 2O3a-plane (for (110)-orientation) substrates in order to investigate thecrystal orientation effect on a. The ﬁlms were epitaxially grown by dc magnetron sputtering technique under a basepressure of 4 /C210 /C07Pa and under argon pressure of 0.1 Pa. Prior to the deposition substrates were in-situ ﬂushed at 700/C14C and then cooled down during 1 h. In the ﬁlms’ com- position, both Cr and Ta buffer layers decrease the surfaceroughness and improve the crystallinity in CFMS ﬁlms. 12A 3-nm-thick Ta layer was deposited as capping layer to pre- vent the oxidation of CFMS. Furthermore, the CFMS layers were annealed at various temperatures between 400/C14C and 550/C14C. The magnetic properties of the samples are deter- mined through the saturation magnetization with a vibratingsample magnetometer (VSM). The FMR measurements wereconducted at room temperature using an X-band microwavesource whose frequency is 9.4 GHz. The X-band’s micro-wave power, modulation ﬁeld, and modulation frequency are1 mW, 0.1 mT, and 100 kHz, respectively. a)Electronic mail: kwilu6@ecei.tohoku.ac.jp. 0021-8979/2015/117(17)/17D140/4/$30.00 VC2015 AIP Publishing LLC 117, 17D140-1JOURNAL OF APPLIED PHYSICS 117, 17D140 (2015) III. EXPERIMENTAL RESULTS AND DISCUSSION Figures 1(a)and1(b)show the XRD ( h=2hs c a n )o f( 0 0 1 ) and (110)-oriented ﬁlms, each with a CFMS layer annealed at particular temperature between 400/C14C and 550/C14C. Figure 1(a)shows (002) and (004) CFMS characteristic superlattice peaks of the B2ordered structure,13which indicate that each sample contains a (001)-orientation and B2ordered structure. In Figure 1(b), the peaks (110) and (220) indicate that all sam- ples contain a (110)-oriented CFMS layer. In addition, we car- ried out the XRD /-scan on the (220) reﬂection for both the (100)-oriented CFMS and (110)-oriented CFMS samples, respectively. The results of this /-scan pole ﬁgure analysis for all the samples (400/C14Ct o5 5 0/C14C) in a speciﬁc orientation are nearly similar. Hence, for each crystal orientation, we dis- played only the typical /-scan on the (220) reﬂection for the sample annealed at 450/C14C. Figure 1(c)shows a /-scan (220) reﬂection from the sample (100)-oriented CFMS annealed at 450/C14C. The four-folder symmetry peaks in /-scan (220) con- ﬁrm the epitaxial growth of CFMS on MgO substrate. Similarly, in Figure 1(d), the four-folder symmetry peaks in /-scan (220) conﬁrm the epitaxial growth of CFMS on Al 2O3 substrate. However, XRD /-scan on the (111) reﬂection, which characterizes the L21ordering, was very weak for both (100)-oriented CFMS and (110)-oriented CFMS samples. All the ﬁlms were characterized by the B2order parameter S B2, which is approximately around 0.6. The magnetic properties of the samples are illustrated in Figure 2,w h i c hi n d i c a t e st h e dependence on the annealing temperature of MSin (001)- and (110)-oriented thin-ﬁlms. In both (001) and (110)-oriented thin-ﬁlms, M Sincreases with respect to the increase of the temperature from 400/C14Ct o4 5 0/C14C. For the higher tempera- tures ( >450/C14C), M Sdecreases. This decreasing is attributed to the interdiffusion between the CFMS components and the buffer layer (Cr) at high temperatures.14,15 As part of the FMR experiments, we submitted the sam- ples to an effective magnetic ﬁeld H whose direction is given by the angle theta with the normal of the surface of thesample, as illustrated by the coordinate system in Figure 3. The direction of the magnetization Mof the ﬁlm is also measured from the normal to the sample by h. To conduct the out-of-plane analysis of the FMR spectra, we used theLLG equation of motion 1 cdM dt¼/C0 M/C2H ðÞ þa cjMjM/C2dM dt/C18/C19 ; (1) where the effective magnetic ﬁeld H, acting on M, takes into account the internal microwave ﬁeld and where the gyro-magnetic ratio cis given in terms of the Bohr magnetron constant l Band Lande-factor g by c¼glB=h. The reso- nance conditions of the ferromagnetic resonance, based on alinear approximation, 16,17is determined from Eq. (1)by the following relations: x c/C18/C192 ¼H1/C2H2; (2) H1¼HRcosðhH/C0hÞ/C04pMeffcos 2h; (3) H2¼HRcosðhH/C0hÞ/C04pMeffcos2h; (4) FIG. 1. (a) and (b) XRD ( h=2hscan) patterns for the (001) and (110)-oriented ﬁlms annealed at various temperatures from 400/C14C to 550/C14C .( c )a n d( d )P o l e ﬁgures for (220) reﬂections with CFMS (001) and (110)-oriented ﬁlms annealed at 450/C14C. FIG. 2. Annealing temperature dependence of the saturation magnetization(M S); the open circles are the data points for (001)-oriented CFMS and the square represents the data points for (110)-oriented CFMS.17D140-2 Kwilu et al. J. Appl. Phys. 117, 17D140 (2015)where HRis the resonance ﬁeld, 4 pMeffis the effective demagnetization ﬁeld, and xis the microwave frequency. Experimentally, we determine the empirical angular depend- ence measure of the linewidth DHPPof ferromagnetic spectra. Besides, the value of DHPPis expressed as a sum of three other different linewidths,14namely, the linewidth attributed to the intrinsic damping ( DHa PPÞ,which attributed to the dis- persion of magnitude of M(DH4pMeff PPÞand which is due to the demagnetization ﬁeld ( DHhH PPÞ. These three linewidths are evi- dently correlated to the resonance ﬁeld in terms of differential equation of HRwith respect to ( hH), (4pMeff), and ( x=c), respectively. The value of acan be obtained in two steps: The calculated function HRðhHÞis ﬁtted to the experimental data HRversus hHwhile adjusting the value of Lande-factor g and that of 4 pMeff. Next, the best ﬁtting parameter values obtained for hHand 4pMeffare inputted in the theoretical expression of DHa PPto ﬁt the experimental data DHa PPversus hHwhile adjusting the value of Dð4pMeffÞ,DhHanda.F i g u r e 4shows particularly the ﬁtting of experimental data for the sample with CFMS at 400/C14C. Figure 4(a)shows the out-of- plane angular dependence of HR.F i g u r e 4(b)shows the ﬁtting of experimental data of DHPP. In this case, the estimated ais 0.007, Dð4pMeffÞis approximated to 0.009, and 4 pMeffis around 1.29 with a Lande-factor g of around 1.81. Figure 5 shows the dependence on the annealing temperature of afor both (001) and (110)-oriented ﬁlms annealed at various tem-peratures from 400 /C14Ct o5 5 0/C14C. For 400/C14C and 450/C14C, the intrinsic damping constants are the same for CFMS (001) and CFMS (110). At 400/C14C, a high avalue of 0.007 is obtained in both cases. The damping reaches the smallest value of around0.004 at 450 /C14C for CFMS (001) and CFMS (110). Above 450/C14C, the values of aare higher than 0.004 and increase with the temperature. The exceeding values of afor the tem- perature higher than 450/C14C, which can be regarded here as the optimal temperature, and the gap between ain (001) and (011)-oriented CFMS can be attributed to the diffusion ofCFMS components through the ﬁlms at the high annealing temperature ( >450 /C14C). From this diffusion results, the disappearance of the half-metallicity of CFMS at the interfaceCFMS/Cu. Therefore, afor the (011)-oriented CFMS samples is much more affected at the interface than that of(001)-oriented CFMS. 18,19Theoretically, the value of ais attributed to the spin-orbit interaction. A small spin-orbit interaction generates a small damping constant.20Moreover, from our measurements on a, the effect of the crystal orienta- tion is not signiﬁcant. In fact, the values of aare the same for (001) and (011)-oriented CFMS between 400/C14C and 450/C14C, where the samples display a small a. FIG. 3. The co-coordinate system illustrating the angel handhHof the nor- mal to the plan of the sample with magnetization vector and the effective magnetic ﬁeld, respectively. FIG. 4. (a) and (b) The out-of-plane angular dependence of the resonanceﬁeld and of peak-to-peak linewidth of the FMR spectra for the ﬁlm annealed at 450 /C14C. The full circles stand for experimental data points and the solid lines are ﬁtting results. FIG. 5. Dependence on the temperature of the intrinsic Gilbert damping con- stant. The open down-oriented triangles are the data points for (001)-ori- ented CFMS and the open up-oriented triangles represent the data points for (110)-oriented CFMS.17D140-3 Kwilu et al. J. Appl. Phys. 117, 17D140 (2015)IV. SUMMARY We fabricated (001)-oriented and (110)-oriented epitax- ial grown ﬁlms with the layer CFMS annealed at various temperatures from 400/C14C to 550/C14C by magnetron sputter- ing. The ﬁlms were characterized by a B2ordering structure. The VSM measurements show an enhancement of saturation MSup its maximum at 450/C14C. We have studied the depend- ence on the annealing temperature of ain CFMS through the out-of-plan angular dependence of the resonance ﬁeld and linewidth of FMR spectra. The smallest intrinsic dampingconstant aof 0.004 was recorded for the sample annealed at 450 /C14C. These results infer that the optimal condition, i.e., the annealing temperature of CFMS layer at which there is a high half-metallicity is 450/C14C. The effect of crystal orienta- tion on the damping constant is not signiﬁcant for (001)-ori-ented ﬁlms and (110)-oriented ﬁlms. ACKNOWLEDGMENTS This work was conjointly supported by Japanese Ministry of education (Monbukagakusho, MEXT) as well asthe Strategic Japanese–German Cooperative program (ASPIMATT) of the Japan Science Technology Agency, the Funding Program for World-Leading Innovative R&D onScience and Technology (FIRST program) of Japan Society for Promotion of Science (JSPS), and grand-in-aid for scientiﬁc research S (No. 24226001). 1F. Heusler, Verh. Dtsch. Phys. Ges. 5, 219 (1903). 2R. A. de Groot et al. ,Phys. Rev. Lett. 50, 2024 (1983).3M. Oogane and T. Miyazaki, Magnetic random access memory, Research Signpost, 37/661(2), Fort P.O., Trivandrum-695 023, Kerala,India; H. Jin and T. Miyazaki, Springer Series in Mater. Sci. 158, 433 (2012). 4H. B. Huang, X. Q. Ma, Z. H. Liu, C. P. Zhao, and L. Q. Chen, AIP Adv. 3, 032132 (2013). 5J. Sato, M. Oogane, H. Naganuma, and Y. Ando, Appl. Phys. Express 4, 113005 (2011). 6P. J. Webster, J. Phys. Chem. Solids 32, 1221 (1971). 7H. H. Potter, Proc. Phys. 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Yilgin, S. Kazan, N. Akdogan, M. Obaida, H. Inam,and K. Westerholt, J. Supercond. Novel Magn. 25(8), 2605–2609 (2012). 19H. Chudo, K. Ando, K. Saito, S. Okayasu, R. Haruki, Y. Sakuraba, H.Yasuoka, K. Takanashi, and E. Saitoh, J. Appl. Phys. 109, 073915 (2011). 20V. Kambersky, “On the Landau-Lifshitz relaxation in ferromagnetic met- als,” Can. J. Phys. 48, 2906–2911 (1970).17D140-4 Kwilu et al. J. Appl. Phys. 117, 17D140 (2015)Journal of Applied Physics is copyrighted by AIP Publishing LLC (AIP). Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. For more information, see http://publishing.aip.org/authors/rights- and- permissions.
1.5020168.pdf	Perspective: Stochastic magnetic devices for cognitive computing Kaushik Roy , Abhronil Sengupta , and Yong Shim Citation: Journal of Applied Physics 123, 210901 (2018); doi: 10.1063/1.5020168 View online: https://doi.org/10.1063/1.5020168 View Table of Contents: http://aip.scitation.org/toc/jap/123/21 Published by the American Institute of PhysicsPerspective: Stochastic magnetic devices for cognitive computing Kaushik Roy,a)Abhronil Sengupta, and Y ong Shim School of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana 47907, USA (Received 20 December 2017; accepted 14 March 2018; published online 5 June 2018) Stochastic switching of nanomagnets can potentially enable probabilistic cognitive hardware consisting of noisy neural and synaptic components. Furthermore, computational paradigmsinspired from the Ising computing model require stochasticity for achieving near-optimality in sol- utions to various types of combinatorial optimization problems such as the Graph Coloring Problem or the Travelling Salesman Problem. Achieving optimal solutions in such problems arecomputationally exhaustive and requires natural annealing to arrive at the near-optimal solutions. Stochastic switching of devices also ﬁnds use in applications involving Deep Belief Networks and Bayesian Inference. In this article, we provide a multi-disciplinary perspective across the stack ofdevices, circuits, and algorithms to illustrate how the stochastic switching dynamics of spintronic devices in the presence of thermal noise can provide a direct mapping to the computational units of such probabilistic intelligent systems. Published by AIP Publishing. https://doi.org/10.1063/1.5020168 I. INTRODUCTION Neural network models, inspired by the computational primitives and interconnectivity in the biological brain, arepresently outperforming humans at various cognitive tasks. 1,2 However, their hardware implementation using CMOS tech- nologies suffers from huge computational resource require- ments. This has resulted in the exploration of several post- CMOS technologies such as spintronics,3,4chalcogenides,5,6 and others that can provide orders of magnitude energy improvement in comparison to CMOS implementations. This is due to the inherent mapping of the underlying operational physics of these devices to the computational units of neural algorithms coupled with the possibility of “in-memory” com- puting due to the non-volatility of such resistive technologies. Most of these “neuro-mimetic” algorithms are based on deterministic computational units—driven by the fact that the underlying CMOS hardware used to implement such algo- rithms is deterministic in nature. However, stochasticity observed in the switching of various post-CMOS technologies has opened up new possibilities of envisioning probabilistic neural hardware enabled by stochastic devices. Interestingly,it is believed that the brain is also characterized by noisy sto- chastic neurons and synapses that perform probabilistic com- putation. 7Hence, exploration of such stochastic neuromorphic platforms might open up new avenues at mimicking the bio- logical brain. The potential advantages of such a computing framework from the hardware implementation perspective are manifold. As will be explained in Sec. III, they allow neural/ synaptic state compression (in turn, leading to scaled device implementations) due to the additional time-domain encoding of information probabilistically. In other words, traditionally used multi-bit deterministic neural/synaptic units can now bereplaced by single-bit units (enabled by stochastic magnetic devices) where the single-bit device state is updated probabil- istically over time. This can be achieved because the loss ininformation due to bit compression can be encoded in the probabilistic transitions of the single-bit unit observed over a period of time. Simultaneously, they allow for sub-thresholdoperation of devices (in order to exploit the stochastic switch-ing regime, these devices have to be operated below the criti-cal current requirement for deterministic switching), therebyleading to energy consumption reductions. The concept of leveraging the underlying stochastic device physics of spin devices started with their usage astrue random number generators. Essentially, the magnet canbe biased to switch with equal probability to either of its twostable states. 8–11Stochastic logic implementations based on such spin random number generators have also been pro-posed. 12Probabilistic synaptic learning based on the concept of switching a magnetic synapse with a ﬁxed probability wasexplored in Ref. 13. The ﬁrst work on proposing the concept of a magnet behaving as a “stochastic bit” (exploiting the entire range ofthe analog probabilistic switching regime of a nanomagnet)—behaving as a conditional random number generator producinga probabilistic output pulse stream with the probability beingconditioned on the magnitude of the input stimulus can befound in Ref. 14for neural inference applications. Thereafter, this was followed by a plethora of work exploring several neu-romorphic and other unconventional computing paradigmsenabled by such magnetic “stochastic bits.” 15–21In this per- spective article, we review different stochastic spiking neuralcomputing paradigms that can be potentially enabled by the stochastic device physics of spintronic devices. We provide motivation for the implementation of Restricted Boltzmannmachines and Deep Belief Networks based on such stochasticinference units. We extend the discussion to another variant ofBoltzmann machines (in particular, Ising computing models)that can be used to solve different combinatorial optimizationproblems (by serving as a natural annealer). We also considerthe implementation of Bayesian inference networks where thestochastic spin devices can directly mimic the inference units.A detailed device-circuit-system level perspective is provided a)Electronic mail: kaushik@purdue.edu 0021-8979/2018/123(21)/210901/11/$30.00 Published by AIP Publishing. 123, 210901-1JOURNAL OF APPLIED PHYSICS 123, 210901 (2018) for the various proposals mentioned above, based on our ear- lier work and some of the other recent work in this ﬁeld, fol-lowed by a discussion of our outlook on future possibilities in this ﬁeld. Note that we are limiting our discussion to various unconventional non-von Neumann computing paradigms inthis text which can be enabled by stochastic spintronic devices.The inherent stochasticity of spin devices can also potentiallyﬁnd use as on-chip temperature sensors 22a n di nl o g i ci m p l e - mentation.23,24However, note that the delay incurred in proba- bilistic logic implementation using such stochastic magnetswould be signiﬁcantly higher than a corresponding determinis-tic CMOS logic implementation since the average output of the logic has to be observed over a large enough time window to infer the output with maximum probability. II. SPINTRONIC DEVICES FOR STOCHASTIC COMPUTING In order to provide a direct mapping to the computa- tional primitives of neuromorphic and other post-Boolean unconventional computational paradigms, a nanoelectronic device is required that is characterized by a multi-bit staterepresentation which can be tuned in response to the magni-tude of an external stimulus. The state is usually represented by the conductance of the device. Recent experiments on elongated ferromagnet-heavy metal heterostructures with astabilized chiral transitory magnetization proﬁle (referred toas the “domain wall”) that exhibits current induced domain wall motion 25–27have revealed the possibility of designing such multi-bit “neuro-mimetic” devices.28–32The three- terminal device structure is shown in Fig. 1(a)wherein a tun- nel junction is formed between a magnetic “pinned” layer (magnetization proﬁle uniformly pinned in a particular direc- tion) and magnetic “free” layer (magnetization proﬁle with astabilized domain wall region). The magnetic stack lies ontop of a heavy metal underlayer where input current ﬂowing through that layer between terminals T2 and T3 results in the movement of the domain wall due to spin-Hall effect 33 induced transverse spin current injection on the ferromagnetlying on top. Depending on the location of the domain wall, the proportion of “free” layer magnetization parallel to the “pinned” layer magnetization proﬁle can be varied, therebyresulting in the variation of the Magnetic Tunnel Junction(MTJ) device conductance between terminals T1 and T3.Such a programmable multi-bit state representation in spin devices can be used to emulate the computational primitives of neurons and synapses in neuromorphic computing frame- works. Although such devices are still in the preliminary stages of development, recent experiments have demon-strated multi-level programmable resistive states in FeB- MgO magnetic stacks. 34 However, with progressive scaling of such magnetic bilayer structures, it is expected that these devices might lose their multi-bit state representation properties and therefore may only exhibit binary states. Hence, a rethinking of tradi- tional neural algorithms (enabled by deterministic multi-bitcomputing units) is required in order to enable them for binary synapse/neural units. Interestingly, spintronic devices are characterized by stochasticity during the switching pro- cess which can be harnessed to compensate for the loss of information due to binary state representation. In other words, the inherent stochasticity of spintronic devices can be used to compress the multi-bit state representation of neural and synaptic units to binary state representations. The stochastic spin device that will be primarily consid- ered in this article is shown in Fig. 1(b). The device is similar to the domain wall motion based device structure [shown in Fig. 1(a)], except that the ferromagnet lying on top of the heavy metal layer is a scaled mono-domain magnet with in- plane magnetic anisotropy that can be switched by the in- plane transverse spin current generated by current ﬂowing through the underlying heavy metal (spin-orbit torque 35). The thermal noise can be theoretically modeled as an addi- tional thermal ﬁeld,36,37Hthermal ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ a 1þa22kBT cl0MsVDtq G0;1(where G0,1is a Gaussian distribution with zero mean and unit stan- dard deviation, kBis the Boltzmann constant, Tis the temper- ature, and Dtis the simulation time-step) in the Landau- Lifshitz-Gilbert (LLG) magnetization dynamic equation38 given by d^m dt¼/C0cð^m/C2HeffÞþa^m/C2d^m dt/C18/C19 þ1 qNsð^m/C2Is/C2^mÞ; (1) where ^mis the unit vector of “free” layer magnetization, c¼2lBl0 /C22his the gyromagnetic ratio for the electron, ais Gilbert’s damping ratio, Heffis the effective magnetic ﬁeld, FIG. 1. (a) Multilevel resistive states can be encoded in the device structure shown above by programming the position of the domain wall due to the pass age of a charge current of appropriate magnitude between terminals T2 and T3. The device state can be “read” between terminals T1 and T3 using the tunneling junction. (b) The magnitude of “write” current ﬂowing through the heavy metal can switch a mono-domain magnet in a similar device structure probabili sti- cally depending on the magnitude of the “write” current.210901-2 Roy, Sengupta, and Shim J. Appl. Phys. 123, 210901 (2018)Ns¼MsV lBis the number of spins in the free layer of volume V (Msis the saturation magnetization and lBis the Bohr mag- neton), and Isis the input spin current. Figure 2(a)depicts the temporal proﬁle of the stochastic magnetization dynamics of an elliptic magnet with dimen- sionsp 4/C2100/C240/C21:2n m3in response to a current pulse of magnitude 80 lA and duration 0.5 ns ﬂowing through the heavy metal underlayer. Figure 2(b) depicts the variation of the switching probability of the mono-domain magnet in response to the magnitude of the external current stimulus for different values of the duration of the current pulse atroom temperature. The simulation parameters correspond to experimental measurements performed in CoFe- bWbilayer structures. 39Note that we are using stochastic spin-orbit tor- que switching based devices in this discussion for energy- efﬁcient magnetization reversal. Due to multiple repeated scattering of injected spins at the magnet-heavy metal inter-face (and therefore transfer of multiple units of spin angularmomentum to the magnet lying on top), spin-orbit torque based magnetization reversal requires much less switchingcurrent in comparison to standard spin-transfer torque based magnetization reversal. Additionally, decoupled “write” and “read” current paths in spin-orbit torque based device struc- tures assist in independent optimizations of the “write” and “read” peripheral circuitries. However, the concepts intro-duced in this article can be easily extended to include other innovations in the material stack (for instance, using magne- toelectric oxide based devices 40). Proof-of-concept experiments demonstrating stochastic magnetization switching in ferromagnet-heavy metal bilayerstructures have been also demonstrated. 19Figure 3(a)depicts a 1.2 lm wide Hall-bar structure consisting of a Ta(10 nm)/ CoFeB(1.3 nm)/MgO(1.5 nm)/Ta(5 nm) (from bottom to top)material stack with perpendicular magnetic anisotropy. Input charge current ﬂows between IþandI– terminals, while the ﬁnal stable magnetization state is determined by the anoma-lous Hall effect resistance between terminals Vþand V–. Note that the switching is performed in the presence of an external in-plane magnetic ﬁeld since the perpendicularanisotropy magnet cannot be solely switched by in-plane spins generated by current ﬂowing through the heavy metal underlayer. Figure 3(b)represents the experimental measure- ments for the switching probability of the magnetic stack with a variation in the magnitude of the current pulse being used for switching (with the pulse width being ﬁxed at10 ms). Note that the non-linear variation of the switching probability of the magnet with the magnitude of the current pulse ﬂowing through the heavy metal underlayer resemblestheoretical simulations depicted in Fig. 2. Such proof-of-con- cept experiments can be easily extended to device structures depicted in Fig. 1(b), where a Tunnel Junction is used as the read-out mechanism (exhibiting 2–3 times larger resistance variation in comparison to Hall-bar structures) for compati- bility with peripheral CMOS circuitry. The barrier height of the magnet (deﬁned as the product of the magnetic anisotropy and the magnet volume) deter- mines the current range that can be used for stochastic mag- net switching. As the magnet volume is scaled down, themagnitude of the current range useful for stochastic switch- ing reduces, thereby increasing the energy efﬁciency of theFIG. 2. (a) Temporal stochastic LLG dynamic simulation of the magnetiza- tion proﬁle of a nanomagnetic elliptic disk of volumep 4/C2100/C240 /C21:2n m3with a saturation magnetization of Ms¼1000 kA/m and a damp- ing factor; a¼0.0122 in response to a current pulse of magnitude 80 lA and duration 0.5 ns. mX,mY, and mZare the X, Y, and Z components of magneti- zation, respectively, where mYis the magnetization component being switched. (b) Variation of the switching probability of the magnet with the magnitude of the “write” current ﬂowing through the heavy metal layer for different values of the pulse duration. FIG. 3. (a) Hall-bar structure consistingof a Ta (10 nm)/CoFeB (1.3 nm)/MgO (1.5 nm)/Ta (5 nm) (from bottom to top) material stack. 19Input current ﬂows between terminals IþandI–, while the magnetization state is detected by a change in the anomalous Hall-effect resistance measured between terminalsVþandV–. (b) Experimental measure- ments of the switching probability of t h eH a l l - b a rw i t hav a r i a t i o ni nt h e amplitude of the current pulse ﬂowing through the heavy metal underlayer for a ﬁxed pulse width of 10 ms. 19210901-3 Roy, Sengupta, and Shim J. Appl. Phys. 123, 210901 (2018)device. However, in highly scaled devices having barrier height /C241kBT, the magnet undergoes random telegraphic switching in the nanosecond time scale. Figure 4(a) depicts the magnetization dynamics of a 1 kBTmagnet under no bias current ﬂowing through the heavy-metal (HM). The average magnetization over a long enough time window is approxi- mately 0. On the other hand, the dwell time in either one of the stable states can be modulated in the presence of an external bias current [Fig. 4(b)]. Note that such superpara- magnetic MTJs operating in the telegraphic regime has been referred to as “p-bits” by authors in Refs. 23and 24. Experiments have demonstrated telegraphic switching inMTJ stacks, 21,42,43with a barrier height as low as /C2411kBT.44 Scaling magnets to even lower barrier heights ( <5kBT) might be difﬁcult from the fabrication perspective. The potential advantage of utilizing random telegraphic switching as the stochastic computing (SC) element lies in its energy efﬁcient operation. While /C2471lA current is required for 0.5 ns to switch a 20 kBTbarrier height magnet with 50% probability [Fig. 2(b)],15thereby leading to an I2Rtenergy consumption of /C241fJ, zero bias current is required to achieve 50% switching probability in a /C241kBTdevice. Note that, in practical device implementation, 50% switching probability may not be achieved exactly at zero bias current due to the presence of device imperfections, stray ﬁelds, and magnetic coupling between elements. Also, the device being highly sensitive to noise and variations requires appropriate periph- eral circuits for proper functionality. These design tradeoffs will be explained in detail in Sec. III. We will consider the entire gamut of stochastic spin devices from random tele- graphic noise induced stochasticity in highly scaled devices to non-telegraphic thermal noise induced stochasticity in mag- nets with a higher barrier height and their corresponding implications for peripheral circuit design corresponding tovarious stochastic computing paradigms, namely, Stochastic Neural Networks, Ising computing, and Bayesian inference. III. STOCHASTIC SPIKING NEURAL NETWORKS Spiking Neural Network (SNN) based neuromorphic computing paradigms consider neural communicationthrough synaptic junctions in the form of spikes, therebyenabling event-driven hardware operation. The two maincomputational units under consideration in SNNs are theneuron and the synapse. The computing elements or the neu-rons process incoming spikes transmitted from the fan-inneurons weighted by the synaptic weights and propagate out-put spikes to the fan-out neurons. Different abstractions of the functionalities of spiking neurons are used in the litera- ture and can be roughly divided into two categories—deter-ministic (for instance, Leaky-Integrate-Fire models 45) and stochastic.46Interestingly, the stochastic switching character- istics of spin devices [depicted in Fig. 2(b)] enables us to directly map such characteristics to a stochastically ﬁringspiking neuron that performs rate encoding in response to themagnitude of the external input. While CMOS hardware canbe used to implement deterministic/stochastic spiking neuronmodels, they do not offer a direct mapping and require multi-ple transistors for a single neuron implementation. 47For instance, emulation of stochastic spiking neuron functionality would require a random number generator circuit where theprobability can be tuned in accordance with an external input.In addition to the area overhead, CMOS implementations willalso be limited by the required power consumption in contrastto spintronic devices which are magneto-metallic devices andrequire low currents for probabilistic switching. Other resis-tive technologies emulating stochastic neurons have also beenproposed in the literature, 48,49but they usually do not offer a direct mapping to stochastic neuron functionalities, are char-acterized by much higher operating current and voltage levels,and require power-hungry peripheral design for interfacing with synaptic crossbar arrays. 14 Let us ﬁrst consider the hardware mapping of the neuron functionality to an equivalent spintronic implementation. Inother words, the probability of spiking of the neuron at a par-ticular time-step is a non-linear function of the instantaneousmagnitude of the input being provided to the neuron. This, inturn, implies that the average rate of the output spike train ofthe neuron would be a function of the rate of the input spike train. The operation of the device is explained in Fig. 5. During the “write” phase (WR activated), the resultant inputcurrent to the neuron at a particular time-step ﬂows throughthe underlying heavy metal and switches the MTJ probabilis-tically depending on the magnitude of the bias current. Aftera “relax” phase, the “read” phase (RD activated) is used todetermine the ﬁnal state of the MTJ at the correspondingtime-step. Reading the neuron MTJ state is performed byinterfacing it with a reference MTJ, as shown in Fig. 5, which is always ﬁxed to the anti-parallel state. The neuronMTJ’s state determines the state of the output inverter being driven by the resistive divider circuit. In case the MTJ switches or “spikes,” the MTJ is reset back to the initial statefor the next time-step of operation. Note that the LLG FIG. 4. Simulation study of the random telegraphic switching of a superpar- amagnet of barrier height 1 kBT(a) under no bias and (b) under a bias current of 1.5 lA.41210901-4 Roy, Sengupta, and Shim J. Appl. Phys. 123, 210901 (2018)magnetization dynamics (discussed in Sec. II) bear resem- blance to a Leaky-Integrate-Fire spiking neuron model.14 Such a magnetization characteristic along with the thermal noise and the sequential “write”-“read” mode of neuron operation results in the abstraction of the neuron computing mechanism as that of a stochastically spiking neuron.14 Unsupervised learning using such stochastic MTJ neuronshas been explored in Ref. 14. Such devices can directly mimic the stochastic inference units in Restricted Boltzmann Machines and Deep Belief Networks. 14Supervised learning platforms exhibiting high classiﬁcation accuracies /C2497% in typical digit recognition workloads and requiring /C2420/C2less energy consumption in comparison to an equivalent 45 nm CMOS implementation have been also investigated.15 The analysis on the scaling effects of stochastic spin devi- ces for neuromorphic computing has been performed in Ref. 41. As mentioned previously, scaling magnetic device dimen- sions result in reduced energy consumption for stochastic opera- tion. However, as the scaling tends to the “super-paramagnetic”regime, the magnet undergoes volatile telegraphic switching. Such a volatile device operation entails the “asynchronous” mode of network operation since parallel “read” and “write” operations are now required for the MTJ (unlike the synchro- nous clocked “write” and “read” cycles used to operate the MTJ for non-superparamagnetic MTJs). The “read” and “write” ports of the neuron MTJ are activated simultaneouslydue to the low data retention time of the magnet. The system is not driven in a synchronous fashion by any clock signal, and spikes generated by the neuron output inverters drive the next set of fan-out neurons in an asynchronous fashion. Note that asynchronous parallel “read” and “write” operations are also not suited for high barrier height magnets in the non- telegraphic regime (10–20 k BT) from the delay perspective since telegraphic switching would occur in the /C24ls–ms time- scale in this scenario. As the barrier height is scaled, the reten-tion failure probability of the magnet during a speciﬁed “read” cycle will increase. Analysis performed in Ref. 41 reveals that the barrier height of the magnet should be greater than 4.6 k BTto ensure that the retention failure probability is less than 1% during a “read” time cycle of 1 ns (required time for worst-case corner simulations of the “read” circuit in the 45 nm technology node). Hence, magnets with barrier heights less than 5 kBTare more suited for the asynchronous scheme of operation mentioned above. The lower power consumption in superparamagnets as neural inference elements is achieved at the expense ofreduced error resiliency. Since the “write” and “read” opera- tions occur in parallel for magnets switching in the tele- graphic regime, the “read” current can signiﬁcantly bias the probabilistic switching of the device. Magnetic ﬁelds gener- ated by nearby electric currents may also serve to bias the device stochasticity. The situation is worsened by the fact that the “write” and “read” currents are in the same range due to the signiﬁcantly lower “write” current requirement for stochastic switching in such scaled devices. Hence, the “read” circuit for the neuron MTJ (Fig. 5) needs to be highly optimized such that the read current is maintained at the min- imal value. Note that this is not a design issue in higher barrier height magnets since “read” and “write” cycles are de-coupled in time. Furthermore, the gradient or the rate of change of switching characteristics of such magnets in response to input current magnitude is extremely high. For instance, the stochastic switching characteristics undergo a full swing from 0 to 1 approximately in the range of 61lA for a 1 kBTmagnet.41In other words, the stochastic switching characteristics are highly sensitive to variations in the magni- tude of the external bias input current which, in turn, results in reduced classiﬁcation accuracy or a similar performance metric of any pattern recognition system with variations in the supply voltage, synaptic conductances, or CMOS periph- erals.41For instance, variation analysis performed in Ref. 41 for a standard digit recognition problem on a two-layer con- volutional neural network architecture enabled by asynchro- nous operation of 1 kBTbarrier height magnets reveals /C245% accuracy degradation for the 20% variation in the synaptic resistive elements, /C246% accuracy degradation for the 25 mV variation in crossbar supply voltage, and 3% accuracy decre- ment for worst-case corner simulation with 2 rvariations in the CMOS read circuit. In contrast, the synchronous imple- mentation with higher barrier height magnets is resilient to variations in the crossbar supply voltage and read circuit, while a small degradation of /C243% classiﬁcation accuracy is observed for variations in the synaptic elements of the resis- tive crossbar array. Note that such sensitive operation in response to noise and other non-idealities is not speciﬁc to a 1kBTmagnet but is valid for superparamagnets operating in the telegraphic switching regime (barrier height in the range of 1–5 kBT). In addition to neural functionalities, the stochastic switching of nanomagnets can also be utilized to implement probabilistic learning in binary synapses.13,16The unsuper- vised (learning without any information of the labels of FIG. 5. A stochastically switching nano- magnet is interfaced with other periph- erals to realize the functionality of a stochastically spiking neuron. During the “write” phase (WR activated), a currentpulse with varying magnitudes probabil- istically switches the neuronal device. After a relaxation phase ( t RELAX ), the magnetic state is sensed using the refer- ence MTJ ( MTJ REF) and the inverter, which generates an output HIGH signal (spike) in case the magnet switched.210901-5 Roy, Sengupta, and Shim J. Appl. Phys. 123, 210901 (2018)incoming data patterns) Spike-Timing Dependent Plasticity (STDP) Hebbian learning rule51dictates that the synaptic strength should increase (decrease) if the post-neuron spikes after (before) the pre-neuron. This learning rule helps to pro- mote temporal correlation between the pre-neuron and the post-neuron, i.e., in case the post-neuron ﬁred after the pre- neuron, then the synaptic weight joining the two neurons increments to assist and temporally correlate the two neurons even more. In other words, the causal agent (pre-neuron) gets more strongly connected to the effect (post-neuron). The opposite scenario of temporal de-correlation occurs for post-neuron spiking before the pre-neuron. While this learn- ing rule is based on deterministic multi-bit synapses, recent work has considered implementation of an alternative sto- chastic version of this algorithm for binary synapses in MTJ synaptic crossbar arrays.13Probabilistic versions of more bio-realistic STDP characteristics [based on measurements for rat hippocampal glutamatergic synapses51and depicted in Fig. 6(a)] have been also explored in the literature for MTJ synapses16 pðw0!1Þ¼Aþexp/C0Dt sþ/C18/C19 ;Dt>0; pðw1!0Þ¼A/C0expDt s/C0/C18/C19 ;Dt<0:(2) Here, Aþ,A–,sþ, and s–are constants and Dt¼tpost–tpre, where tpreandtpostare the time-instants of pre- and post- synaptic ﬁrings, respectively. We will refer to the case of Dt>0(Dt<0) as the positive (negative) time window for learning. p(w0!1) represents the probability of the binary synapse to switch from a low state to a high state and vice versa. Such a probabilistic update can be implemented in a stochastic spin synapse as shown in Fig. 6(a). The two peripheral access transistors are used to decouple the “write” and “read” current paths. POST serves as the “write” control signal and is activated whenever the post-neuron ﬁres ( t2). At the commencement of the positive timing window ( t1), the PRE line is driven by a linearly increasing voltage that biases the MSTDP transistor in saturation, whenever the POST signal is activated. Due to the increase in the gate volt- age of the MSTDP transistor with time once the pre-neuron spikes, the magnitude of programming current ﬂowing through the spin device (when the POST signal is activated) reduces as the delay between the pre-neuron ﬁring and post- neuron ﬁring increases. The biasing region of the MSTDP transistor is determined to ensure that the current ﬂowing through the heavy metal varies in such a manner that the switching probability of the MTJ varies exponentially with the spike timing difference. This pertains to the implementa- tion of the positive time window for STDP learning. More details on the circuit-level implementation can be obtained in Ref. 16. Figure 6(b) represents a possible arrangement of the spin synapses in an array fashion joining pre-neurons A and B to post-neurons C and D. Analysis performed in Ref. 50indicates that an “All- Spin” Stochastic SNN (where the neurons spike stochasti- cally in response to input spike currents, while the synapses are also programmed probabilistically according to theSTDP learning rule) can achieve /C2470% recognition accuracy on a typical Modiﬁed National Institute of Standards andTechnology database pattern recognition problem for 200 neurons. Note that the accuracy can be increased by increas- ing the number of neurons in the system. Figure 7(a)repre- sents the learnt synaptic weights of the 200-neuron network FIG. 6. (a) STDP implementation in the spin synapse is shown. At the receipt of a spike from the pre-neuron at time-instant t1, the gate voltage of theMSTDP transistor starts increasing linearly. When the post-neuron ﬁres at time-instant t2, an appropriate amount of programming current ﬂows through the heavy metal underlayer depending on the delay between the pre- neuron and post-neuron spikes. (b) Possible arrangement of spintronic syn- apses in an array fashion for pre-neurons A and B connecting to post- neurons C and D.210901-6 Roy, Sengupta, and Shim J. Appl. Phys. 123, 210901 (2018)at the end of 2000 training epochs. The average energy con- sumption of the network of 200 excitatory neurons during the training period was 1.16 fJ per time-step for the “write,”“read,” and “reset” operations of the neuron. 50The spin- tronic synapse, on the other hand, consumed a maximum of36 fJ per spiking event for realizing the stochastic-STDP learning algorithm. 50In contrast, CMOS implementations have been reported to consume energy on the order of pJ.50 The “All-Spin” Stochastic SNN architecture (the program-ming transistors for STDP implementation are shown in the inset) is depicted in Fig. 7(b). It is worth noting here that such networks, in principle, are “Binary Networks” beingcharacterized by binary neuron and binary synaptic units. IV. STOCHASTIC ISING SPIN MODEL While stochastic spin devices can be used to construct Restricted Boltzmann machines and Deep Belief Networks (a particular category of the generative neural network model thatconsists of stochastic neural inference units 53)b a s e do nt h ep r o - posal discussed in Sec. III, let us shift our attention to a probabil- istic version of the Ising computing model that can be enabled by such device stochasticity. In fact, Boltzmann machines areconsidered to be a variant of Ising spin models. 54The model comprises single-bit random variables whose probability ofbeing in a certain state is manipulated by the magnitude of the input stimulus. The input magnitude is determined by the inter- action with other variables which is deﬁned through the inter-connections and associated interconnection coefﬁcients. One of the problems that can be solved efﬁciently through the Ising spin model is a combinatorial optimization problem. 55,56The goal of the combinatorial optimization problem is to assign a set of binary values to the variablessuch that a cost function for the given problem becomes maximum (or minimum). However, solving such problems based on a general purpose processor turns out to be inefﬁ-cient due to the complexity of the problems (the complexityincreases exponentially as the number of related variablesincreases). Instead, unconventional computing models, such as Ising spin model, are able to ﬁnd an optimal solution (ornear-optimal solutions) within a reasonable time by mapping the problem to the process of ground state search of some metrics, for instance, Ising Hamiltonian (system energy) of the Ising model. The Ising spin model is a mathematical model to describe the behavior of magnetic spins and coupling between them. 56 Due to its combinatorial interpretation and inherent ability toconverge towards a lower energy state, the Ising model hasbeen researched extensively. 52,57In the conventional Ising spin model, the behavior of the spins [shown in Fig. 8(a)]i s governed by the Hamiltonian below H¼/C0X i<jJijsisj/C0XN i¼1hisi; (3) where s iis the spin state of the i-th spin (can have either up or down state), J ijis the interconnection coefﬁcient with neighboring spins, and h imodels the external ﬁeld. Here, the states of the spins are updated through the interaction with its neighbors in such a way that the spins eventually con- verge to a set of states, which causes the given Hamiltonian to achieve the minimum possible energy value. Therefore, starting from random spin states with high energy, the sys- tem evolves towards a minimum energy state through the coupling mechanism, where the coupling strength is assigned based on the type of problem to be solved. Finally, the FIG. 7. (a) System-level simulation demonstrating the learnt synaptic weights of the 200-neuron (depicted for 14 /C214 neurons) network at the end of 2000 training epochs for an “All-Spin” Stochastic SNN.50(b) “All-Spin” Stochastic SNN implementation where stochastic MTJ synapses drive magneto-metallic stochastic neurons (programming transistors are shown in the inset). FIG. 8. (a) Conventional Ising spin model with spin states (s i), interconnec- tion weights (J ij), and external magnetic ﬁeld (h i). (b) Hamiltonian (energy of the system) with respect to the spin states. The energy proﬁle has a global minimum energy state and multiple local minimum energy states. The annealing process prevents the system being stuck into a local minima.20,52210901-7 Roy, Sengupta, and Shim J. Appl. Phys. 123, 210901 (2018)solution is obtained by examining the ﬁnal state of the spins at the minimum Hamiltonian level. The two main functions under consideration in the Ising spin model are the “Annealing” and the “Majority Vote.” The majority vote function is used to determine the next state of each spin. In the conventional Ising spin model, four near-est spins are voted at either up or down state that will be the next state of the center spin based on the majority rule. One of the drawbacks of such a heuristic optimization solver isthat the system easily gets stuck at local minima [Fig. 8(b)]. To address this issue, various types of annealing processes are commonly used to introduce noise (random bit ﬂip) tothe system. 58,59The implementation of such annealing func- tions based on conventional CMOS technology is known to be very expensive in terms of the silicon area and power con-sumption. 52Interestingly, the stochastic switching of nano- magnetic devices can act as a natural annealer in such scenarios. Nanomagnetic implementations for the Ising spinmodel have been proposed in Ref. 18(superparamagnets in the telegraphic regime) and in Ref. 20(for higher barrier height magnets in the non-telegraphic regime). The relativeadvantages/disadvantages for using highly scaled magnet switching in the telegraphic regime have been discussed in Sec.IIIand also hold true for this case. Let us ﬁrst consider the implementation of the majority vote function based on the stochastic spin-device under con- sideration. The main idea of the majority vote lies in the deci-sion of the next spin state depending on the majority rule with votes from the neighbors. Note that the switching probability of the spin device increases as the magnitude of input currentincreases. Hence, the majority voting rule can be imple- mented by passing an input current pulse through the HM underlayer, whose magnitude is proportional to the number ofvoters from the neighboring spins [implying high (low) switching probability with more (less) voters]. The implemen- tation of this functionality is rather simple, and it requiresmultiple current sources and switches that would be turned on/off through the individual votes from the neighbors. In the case of the annealing function, the stochastic switching of thenanomagnet can be exploited as a natural randomizer. Since there is always a chance of ﬂipping the spin state in an unwanted direction, this introduces a natural noise to the sys-tem without any additional hardware. Moreover, the random bit ﬂips can occur from multiple spins at the same time, thereby helping the system to avoid local minima. The detailed hardware implementation of the stochastic Ising model on the proposed device-circuit conﬁguration is shown in Fig. 9. The proposed device-circuit conﬁguration has been used to solve Maximum-cut (Max-cut) and Graph Coloring problems in Ref. 20. The maximum-cut problem could be formulated as deﬁning two mutually exclusive subsetsof spins by connecting edges of two regions so that the sum- mation of the weights along the edges becomes the maximum. 60The goal of Graph Coloring, a famous non- deterministic polynomial-time-complete problem,61is to check whether it is possible to color n-vertices with k-colors in such a way that two adjacent vertices have different colors. Figure10represents the results of the Max-cut problem through changes in the Hamiltonian over time along with visualizedspin states at speciﬁc iterations. For this simulation study, /C243900 spins are used, and the interconnection coefﬁcients are programmed such that the spin states show the digit numbers 0 to 4 without noise at the lowest possible energy state. Different variants of combinatorial optimization prob- lems, for instance, the Traveling Salesman Problem (TSP), can also be solved based on the proposed device-circuit primitives for the stochastic spin device. 40The TSP problem tries to ﬁnd the shortest path that a traveling salesman can take by visiting each city in a problem just once and then returning to the ﬁrst city. The basic mechanism required to form a hardware optimization solver for speciﬁc problems, for instance, interpreting the Hamiltonian62of the problem to build a system and evolving the system to ﬁnd an optimal solution with the lowest energy, is similar to the discussionpresented earlier. Two practical TSP problems with a size of 15-city and 26-city have been solved based on the proposed hardware implementation and are compared to the solutions from a software TSP solver. The suggested solution from the hardware Ising solver is compared to the software solverbased on the Lin-Kernighan 63heuristic algorithm and shows 100% exact solution for the 15-city problem. V. BAYESIAN INFERENCE In addition to the core hardware primitive for the Ising spin model, which comprises random variables with FIG. 9. The proposed device-circuit conﬁguration for the single Ising spin model. FIG. 10. Application of the spin-based Ising model to the maximum-cutproblem: System-level simulation depicting the (a) system energy transition over the iterations. (b) Visualized spin state at particular stages. 20210901-8 Roy, Sengupta, and Shim J. Appl. Phys. 123, 210901 (2018)symmetric interconnections, the spin-device with stochastic switching characteristics can also be used to construct a proba- bilistic inference engine. Probabilistic inference from real-timeinput data is regarded as one of the potential pathways toward cognitive intelligence. In addition, recent studies have revealed that the inferencing and decision making occurring in the bio-logical brain resemble Bayesian inference. 64Probabilistic information processing through Bayesian inference can be modeled by a probabilistic directed acyclic graph, termed asthe Bayesian Network (BN), which will be explained later. A hardware belief network design based on high barrier height magnets, initially biased towards the magnetic hard axis, wasproposed in Ref. 17. However, the proposed device operation is based on dipolar coupling between magnetic layers which might be potentially difﬁcult from the fabrication perspective.The proposal in Ref. 19, based on the stochastic spin-orbit tor- que based device discussed in this text, demonstrates a device- circuit-system level implementation of the Bayesian Network(BN) with directed connection. Implementation of BN based on Muller C elements using stochastic magnetic devices has also been explored by authors in Ref. 21. The BN is a graphical model to represent conditional independencies between the variables. The BN consists of a node, a random variable, and a link which represent directdependencies between the variables. These dependencies are quantiﬁed through the conditional probabilities that model probabilistic impacts from the parent nodes to a particularnode. Based on this basic network, the inference operation is performed to estimate the probability of the hidden causes from the given observed situation by following Bayes’ rule. There are multiple proposals regarding the implementa- tion of the Bayesian inference engine based on conventional CMOS. 65–68However, the computational complexity required for the series of ﬂoating point calculations makes these pro- posals infeasible considering the hardware footprint and energy consumption. Instead, the use of Stochastic Computing (SC)69 b a s e do naP o i s s o ns p i k eg e n e r a t o r ,e n a b l e db ys t o c h a s t i cs p i n devices (considering the spin device switching to exhibit Poisson statistics), can lead to a concise and an energy efﬁcienthardware implementation. SC, proposed in the 1960s, 69repre- sents a particular number as the probability of a deterministic event. The unique representation of a certain quantity as aprobabilistic switching event (Poisson spike train) makes the corresponding arithmetic building blocks (such as multiplica- tion and addition) required for the entire system designextremely simple and consequently used for many applications ranging from SC computers 70to neural networks.71Based on SC, the probability of a certain event can be estimated bycounting the number of spikes from a variable over a long enough time window. The probabilistic information from a causal variable is transferred to the nearby variables throughthe directed interconnection of the BN in the form of pulse stream. Figure 11(a) shows an implementation of the random variable of the BN using a stochastic spin-device (with CMOS peripherals) along with its interconnection to a neigh- boring variable. The pulse stream that encodes probabilityinformation from the ﬁrst variable is presented to the second variable via simple CMOS logic (denoted as Interface) andacts as the Write (WR) command for the interfaced unit. Based on the basic BN, complex inference operation to esti- mate the probability of hidden causes is also possible byintroducing additional arithmetic building blocks such as multiplication and division [Fig. 11(b) ]. Such computations become quite simple and hardware-friendly in the domain ofSC. For instance, the multiplication (union function between the variables) can be performed through a simple logic AND gate. For the detailed implementation including divisionoperation between two Poisson pulses, readers are directed to Ref. 19. To validate the feasibility of the proposed system, BN hardware with 4 variables along with additional peripherals for the inference operation has been demonstrated in Ref. 19. Figure 12represents the timing waveforms for different vari- ables and inference operations [P ðSjWÞand P ðRjWÞ]. The probabilities of events are estimated by counting the number FIG. 11. (a) Information transfer between the variables through CMOS interfaces for the BN. (b) The inference operation is performed through the division and multiplication operations between two Poisson pulses based onthe given network. FIG. 12. The probability of occurrence of various events from the BN with 4variables [in Fig. 11(b) ] along with the inference results [for instance, PðSjWÞrepresents the probability of event “S” on the given situation of event “W”] is estimated by counting the number of pulses from each vari- able or from an additional building block during 100 epochs. The simulation results, represented in the format A/B, denote that Ais the analytical (exact) solution, while Bis the estimated probability by counting the pulses from each output.210901-9 Roy, Sengupta, and Shim J. Appl. Phys. 123, 210901 (2018)of spikes and are subsequently compared to the exact ana- lytic solution. These results show that the estimated probabil- ity is approximately equivalent to the exact solution within100 epochs. The accuracy of the solution can be improved by increasing the number of monitored samples over a longer duration of time. VI. CONCLUSIONS In conclusion, computing paradigms connecting the sto- chastic device physics of spin devices to algorithm frame- works operating probabilistically over time can potentially lead to an alternative compact representation of cognitivehardware. While achieving such stochastic behavior inher- ently is difﬁcult from single CMOS transistors, spintronic devices with their stochastic switching at non-zero tempera- tures can potentially serve as the building block—“stochastic bits”—of such probabilistic hardware platforms. Proof-of-concept experiments at a single-device level for stochastic cognitive computing have been already dem- onstrated. We believe that the next breakthrough in this ﬁeld would lies in the actual hardware fabrication of such a sys- tem of stochastic magnetic devices interconnected through peripherals to realize probabilistic cognitive frameworks. It remains an open question whether such magnets can bescaled to barrier heights of the order of 1 k BTand still per- form reliably in the presence of different forms of noise and non-idealities. However, such superparamagnetic devices can enable a new genre of low-power asynchronous cogni- tive computing platforms. Furthermore, augmenting such device concepts with alternate device physics for switching (like the magnetoelectric effect or topological insulatorinduced switching) or read-out is an open area of exploration to increase the power, energy, and area efﬁciency of these devices in comparison to CMOS hardware. 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1.3068620.pdf	Nonexponential magnetization thermal decay of a single-domain particle: Numerical computations using the dynamic Fokker–Planck equation Kezhao Zhanga/H20850 Hitachi Global Storage Technologies, 5600 Cottle Road, San Jose, California 95193, USA /H20849Presented 11 November 2008; received 15 September 2008; accepted 7 November 2008; published online 13 February 2009 /H20850 Nonexponential thermal decay of magnetization in a single-domain particle has been studied by numerically solving the Fokker–Planck equation as an initial value problem as well as an eigenvalueproblem. The probability of not switching and switching time distribution is calculated for a widerange of applied ﬁelds and K uV/kBTvalues. In the low and intermediate energy barrier region, the switching time distribution is nonexponential. The switching time distribution is nonmonotonic andhas one peak. For time less than the peak time, the distribution can be well ﬁtted with inverseGauss distribution; for time longer than the peak time, the distribution is exponential. The timeconstant of the exponential decay is equal to the inverse of the smallest eigenvalue of theFokker–Planck equation. Furthermore, the switching time at the peak location of the distributionis a logarithmic function of the smallest eigenvalue. © 2009 American Institute of Physics . /H20851DOI: 10.1063/1.3068620 /H20852 I. INTRODUCTION In the Néel–Brown1model for thermally assisted switch- ing of single-domain magnetic particles, the probability forthe magnetization not to switch decays exponentially withtime. However, nonexponential behavior of a single-domainparticle has been observed in numerical solutions to the sto-chastic Landau–Lifschitz equation /H20849LLE /H20850. 2–4Instead of using the stochastic LLE approach, which requires large number ofMonte Carlo runs to obtain adequate statistics, we numeri-cally solved the Fokker–Planck equation 1,5for the probabil- ity of magnetization orientation as an initial value dynamicproblem as well as the traditional eigenvalue problem. II. NUMERICAL METHOD Brown derived the Fokker–Planck equation for the prob- ability density of the magnetization orientations Wfrom Gil- bert’s equation with a random noise added to the total effec-tive magnetic ﬁeld. The Fokker–Planck equation can bewritten in a compact vector form as /H11509W /H11509t=ar0·/H20849/H11612E/H11003/H11612W/H20850+b/H11612·/H20849W/H11612E/H20850+/H9260/H116122W, /H208491/H20850 where r0=M /Ms.Mis the magnetization vector and Msis the saturation magnetization. Eis the magnetic energy den- sity.a=−/H9253//H20851/H208491+/H92512/H20850Ms/H20852,b=/H9251a, and/H9260=bkBT/V./H9253is the gy- romagnetic constant and /H9251is the damping constant in Gil- bert’s equation. Vis the volume of the particle. Tis the temperature. kBis the Boltzmann constant. The term with a in Eq. /H208491/H20850is related to the precession. The bterm involves the alignment of the magnetization with the external ﬁeld.The /H9260term is the diffusion term due to thermal ﬂuctuation, which tends to make the distribution more uniform.The Fokker–Planck equation is spatially deﬁned on the surface of a unit sphere. In order to solve the ﬁrst passagetime or switching time problem, an absorbing boundary isimposed such that W=0 in a cap region around one of the poles on the sphere surface. 2 To solve the equation numerically, the surface of the unit sphere is triangulated into curved “triangular” patches.Within each patch, the probability density Wis quadratically interpolated from its values at the three vertices and threemidpoints on the edges using the isoparametric mapping ba-sis functions /H9278,2,6 W/H20849u,v/H20850=/H20858 i=16 Wi/H9278i/H20849u,v/H20850, where /H9278i/H20849u,v/H20850are quadratic functions that map a standard triangle on the Euclidean plane to a curved triangular patch on the unit sphere. As a result, the curved surface of the unitsphere is parametrized with the two independent local vari-ables uand vso that differential and integral calculus can be carried out on the sphere surface.7 The Galerkin method6is then used to discretize the par- tial differential equation /H20851Eq. /H208491/H20850/H20852into a set of ordinary dif- ferential equations by multiplying both sides of the equationwith the basis functions /H9278iand then integrating over the sphere surface. The resulting ODEs can be written in a ma-trix form BdW dt=AW, /H208492/H20850 where W=/H20849W1,..., Wn/H20850Tis the list of probability density at the nodal points of the triangular mesh with nbeing the number of nodes in the mesh. Both AandBare sparse square matrices. In addition, Bis symmetric and positive deﬁnite.a/H20850Electronic mail: kezhao.zhang@hitachigst.com.JOURNAL OF APPLIED PHYSICS 105, 07D307 /H208492009 /H20850 0021-8979/2009/105 /H208497/H20850/07D307/3/$25.00 © 2009 American Institute of Physics 105 , 07D307-1Equation /H208492/H20850can be solved as an eigenvalue problem by writing W=/H9274exp /H20849−/H9261t/H20850. The eigenvalues /H9261and the eigen- functions /H9274satisfy −/H9261B/H9274=A/H9274. /H208493/H20850 A variant of Lanczos method8is used to calculate the eigen- values. Although the eigenvalue problem calculates the relax- ation rates, it does not provide the detailed dynamics of therelaxation process, which is obtained by solving Eq. /H208492/H20850as an initial value problem with the backward-differential-formulamultistep method. 9 The probability of not switching or the probability that the magnetization has not reached the absorbing boundary attime tis P n/H20849t/H20850=/H20885 S/H11032W/H20849t/H20850dS, /H208494/H20850 where S/H11032is the unit sphere surface excluding the region with the absorbing boundary, where W=0. The probability density function of the switching time is − dPn/H20849t/H20850/dt. III. RESULTS AND DISCUSSION Both the probability of the magnetization not to switch as a function of time and the switching time distribution havebeen calculated for a single-domain particle with uniaxialanisotropy. Initially the magnetization is saturated along theeasy axis and M=M s. The external ﬁeld is applied in the opposite direction. The surface of the unit sphere is dividedinto 636 curved triangular patches with 1274 nodes. For thismesh, we choose the absorbing boundary such that W/H20849t/H20850=0 forM/H11349−0.9324 M s. Thermally assisted magnetization decay is exponential only when the energy barrier is very large, either when theapplied ﬁeld is small or when K uV/kBTis large. In the low and intermediate energy barrier region, the relaxation behav-iors are nonexponential. For example, as shown in Fig. 1, with K uV/kBT=20 and applied ﬁelds /H20849normalized by Hk/H20850 ranging from 0.8 to 0.99, the linear-log plots of Pn/H20849t/H20850are curved at the starting region of the decay /H20849see the inset ofFig. 1/H20850, indicating nonexponential behaviors. At a longer time scale, the curves are straight lines, indicating exponen-tial decays. The probability density functions corresponding to P n/H20849t/H20850 in Fig. 1are shown in Fig. 2. Unlike the exponential distri- bution whose probability density function is a monotonicallydecreasing function, the probability density function ofswitching time distribution increases monotonically forswitching time t/H11021 /H9270peakand decreases monotonically for t /H11022/H9270peak, where /H9270peakis the peak location of the distribution. The peak location /H9270peakdepends on the applied ﬁeld and KuV/kBT, as shown in Fig. 3./H9270peak, normalized by the mean switching time /H20849MST /H20850, decreases sharply when the applied ﬁeld is small and KuV/kBTis large /H20849e.g., h=0.8 and KuV/kBT/H1102250/H20850. A smaller value means a smaller portion of the thermal decay process is nonexponential and the overallmagnetization decay is more exponential-like. For switching time t/H11021 /H9270peak, the switching time distribu- tion can be empirically ﬁtted very well with inverse Gaussiandistribution. 10An example is shown in Fig. 4, where the probability of not switching and the least square ﬁt with thecomplementary cumulative distribution function of inverseGaussian distribution are plotted for t/H11021 /H9270peak; ﬁtting with an exponential function for t/H11022/H9270peakis shown in the inset of the same ﬁgure. As can be seen, in both cases the ﬁtting is ex-cellent. Since P n/H20849t/H20850decays exponentially for t/H11022/H9270peak, the decay time constant is obtained from curve ﬁtting and compared with the smallest eigenvalue of Eq. /H208493/H20850. For a wide range ofh/Equal0.99 h/Equal0.9 h/Equal0.8 20 40 60 80 1001. 0.5 0.2 0.1 0.05 0.02 0.01 Time /LParen1Hk/Minus1Γ/Minus1/RParen1Pn 24681012140.50.60.70.80.91. FIG. 1. Probability of not switching Pn/H20849t/H20850forKuV/kBT=20 and various applied ﬁelds. Normalized applied ﬁeld h=Happlied /Hk, where Hk=2Ku/Ms. A short time scale is shown in the inset.h/Equal0.99 0.9 0.8 Τpeak 10 20 30 40 50 60 700.000.020.040.060.08 Switching Time /LParen1Hk/Minus1Γ/Minus1/RParen1Probability Density FIG. 2. Distributions of switching time: KuV/kBT=20. h/Equal0.99 h/Equal0.95 h/Equal0.9 h/Equal0.8 1 2 5 10 20 50 1000.20.30.40.50.60.7 KuV kBTΤpeak MST FIG. 3. /H9270peak, normalized by the MST, depends on the applied ﬁeld hand KuV/kBT.07D307-2 Kezhao Zhang J. Appl. Phys. 105 , 07D307 /H208492009 /H20850applied ﬁelds and KuV/kBTvalues, the time constant of the exponential decay for t/H11022/H9270peakis equal to the inverse of the smallest eigenvalue /H20849Fig. 5/H20850. Finally, for all applied ﬁelds ranging from 80% to 99% of the coercivity and KuV/kBTvalues ranging from 0.5 to 100,/H9270peakis a logarithmic function of the smallest eigenvalue /H20849Fig. 6/H20850. Usually the nonexponential behaviors of thermally as- sisted magnetization relaxations are attributed to distribu-tions in magnetic materials of certain physical propertiessuch as grain size and energy barrier. Results in this studyshow that nonexponential behaviors are an intrinsic propertyof the dynamics of a single-domain magnetic particle de-scribed by the stochastic Gilbert’s equation and the Fokker–Planck equation. The exponential behavior is an approxima- tion in the limiting case with large energy barrier. We alsoshow that the eigenvalue of the Fokker–Planck equation issufﬁcient to quantify such exponential time constant. The exact mechanism for the nonexponential behaviors is still unclear but is likely related to the energy barrier. Inthe Néel–Brown model, the energy barrier is high so that theequilibrium within the distribution near the energy minimawill be established much faster than the equilibrium betweenthe minima. As a result, the switching process is dominatedby the exponential behavior of the switching over the energybarrier. When the energy barrier is low, as investigated in thisstudy, the time scale of reaching equilibrium near the energyminimum becomes more comparable to that of switchingover energy barrier, resulting in the initial very small decay-ing of the probability of not switching. Finally the logarithmic dependence of the peak location of switching time distribution /H9270peakon the smallest eigen- value for wide range of applied ﬁelds and KuV/kBTvalues hints at some possible underlying principle of the stochasticdynamics yet to be discovered, which calls for further stud-ies. 1W. F. Brown, Jr., Phys. Rev. 130, 1677 /H208491963 /H20850. 2K. Zhang, Ph.D. thesis, University of California, San Diego, 1998. 3E. D. Boerner and H. N. Bertram, IEEE Trans. Magn. 34, 1678 /H208491998 /H20850. 4G. Grinstein and R. H. Koch, Phys. Rev. B 71, 184427 /H208492005 /H20850. 5W. F. Brown, Jr., IEEE Trans. Magn. 15, 1196 /H208491979 /H20850. 6K. H. Huebner, D. L. Dewhirst, D. E. Smith, and T. G. Byrom, The Finite Element Method for Engineers , 4th ed. /H20849Wiley-Interscience, New York, 2001 /H20850. 7D. J. Struik, Lectures on Classical Differential Geometry , 2nd ed. /H20849Dover, New York, 1988 /H20850. 8G. H. Golub and C. F. Van Loan, Newblock Matrix Computations , 2nd ed. /H20849The Johns Hopkins University Press, Baltimore, 1989 /H20850. 9S. D. Cohen and A. C. Hindmarsh, LLNL Report No. UCRL-MA-118618, 1994. 10R. Chhikara, The Inverse Gaussian Distribution: Theory, Methodology, and Applications /H20849CRC, New York, 1988 /H20850.0 2 4 6 80.750.80.850.90.951. Time /LParen1Hk/Minus1Γ/Minus1/RParen1Pn 0 10 20 30 40 50 600.0010.010.11 FIG. 4. Calculated probability of not switching data /H20849circles /H20850can be ﬁtted with inverse Gaussian distribution /H20849solid line /H20850fort/H11021/H9270peakand exponential distribution /H20849inset /H20850fort/H11022/H9270peak:h=0.9 and KuV/kBT=10. 100 104106100104106 Τd1/Slash1Λ1 FIG. 5. The inverse of the smallest eigenvalue 1 //H92611vs the decay time constant /H9270dobtained from the ﬁtting of an exponential to numerically cal- culated Pn/H20849t/H20850fort/H11022/H9270peak. The dashed line is the ﬁt to the calculated data /H20849shown in dots /H20850. Slope is 0.996 /H110060.001; intercept is −0.0256 /H110060.003. r2 =0.999 95. h=0.99,0.95,0.9,0.8,0.6,0.4,0.2,0. KuV/kBT =0.5,1,5,10,20,50,100. The unit of /H9270disHk−1/H9253−1.0.1 0.05 0.01 0.005010203040 Λ1Τpeak FIG. 6. Relation between the smallest eigenvalue /H92611and/H9270peak. The dashed line is the least square ﬁt to the data with function /H9270peak=alog10/H92611+b, where a=−19.95 /H110060.57, b=−6.26 /H110060.58, and r2=0.979. Note the logarith- mic scale on the horizontal axis. h=0.99,0.95,0.9,0.8. KuV/kBT =0.5,1,5,10,20,50,100. The unit of /H9270peakisHK−1/H9253−1.07D307-3 Kezhao Zhang J. Appl. Phys. 105 , 07D307 /H208492009 /H20850
1.4986324.pdf	Inverse spin Hall effects in Nd doped SrTiO 3 Qiuru Wang , Wenxu Zhang , Bin Peng , and Wanli Zhang Citation: AIP Advances 7, 125218 (2017); doi: 10.1063/1.4986324 View online: https://doi.org/10.1063/1.4986324 View Table of Contents: http://aip.scitation.org/toc/adv/7/12 Published by the American Institute of Physics Articles you may be interested in Spin galvanic effect at the conducting SrTiO 3 surfaces Applied Physics Letters 109, 262402 (2016); 10.1063/1.4973479 Research on c-HfO 2 (0 0 1)/ -Al2O3 (1 -1 0 2) interface in CTM devices based on first principle theory AIP Advances 7, 125001 (2017); 10.1063/1.5001904 Magnetic and electrical transport properties of perovskite manganites Pr 0.6Sr0.4MxMn1-xO3 (M = Fe, Co, Ni) AIP Advances 7, 125002 (2017); 10.1063/1.5008978 Investigation on high-efficiency Ga 0.51In0.49P/In0.01Ga0.99As/Ge triple-junction solar cells for space applications AIP Advances 7, 125217 (2017); 10.1063/1.5006865 Edge-spin-derived magnetism in few-layer MoS 2 nanomeshes AIP Advances 7, 125019 (2017); 10.1063/1.4989477 Laser pulse shape dependence of poly-Si crystallization AIP Advances 7, 125102 (2017); 10.1063/1.4998221AIP ADV ANCES 7, 125218 (2017) Inverse spin Hall eﬀects in Nd doped SrTiO 3 Qiuru Wang, Wenxu Zhang,aBin Peng, and Wanli Zhang State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science and Technology of China, Chengdu 610054, People’s Republic of China (Received 4 June 2017; accepted 6 December 2017; published online 14 December 2017) Conversion of spin to charge current was observed in SrTiO 3doped with Nd (Nd:STO), which exhibited a metallic behavior even with low concentration doping. The obvious variation of DC voltages for Py/Nd:STO, obtained by inverting the spin diffusion direction, demonstrated that the detected signals contained the contribution from the inverse spin Hall effect (ISHE) induced by the spin dependent scattering from Nd impurities with strong spin-orbit interaction. The DC voltages of the ISHE for Nd:STO were measured at different microwave frequency and power, which revealed that spin currents were successfully injected into doped STO layer by spin pumping. The linear relation between the ISHE resistivity and the resistivity induced by impu- rities implied that the skew scattering was the dominant contribution in this case, and the spin Hall angle was estimated to be (0.17 0.05)%. This work demonstrated that extrinsic spin dependent scattering in oxides can be used in spintronics besides that in heavy elements doped metals. © 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/10.1063/1.4986324 I. INTRODUCTION The perovskite-type 3 d0oxide SrTiO 3(STO), as an insulator with a wide band gap of about 3.25 eV ,1has attracted much attention for its potential physical properties, such as superconduc- tivity,2quantum paraelectricity3and ferroelectricity.4These excellent physical performances can be introduced by the doping of a small amount of carriers through generating oxygen vacancies5 or adding dopants such as Cr, La, Nb and Nd.6–9In recent decades, due to the increasing interest in spintronics, the spin Hall effect (SHE) in doped materials has been intensively investigated,10,11 which is an important method for converting charge currents into spin currents. In addition to the intrinsic SHE originating from the intrinsic spin-orbit interaction (SOI) in the band structure, the SHE in the doped system is enhanced by the SOI effect in impurity, called the extrinsic SHE. The magnitude of the extrinsic SHE relies on the distinction of the SOI between the host and the impurity. For example, Fert et al. found the SHE with large values of spin Hall angle in copper doped with 5dheavy metals.12Here the intrinsic SHE in Cu is negligibly weak and the SHE signals mainly arise from scattering by impurities presenting strong SOI. Gradhand et al. proved the giant SHE in heavy metal Au induced by skew scattering at C and N impurities.13The reciprocal effect of SHE is called the inverse spin Hall effect (ISHE), by which spin currents are converted into charge currents as a result of the SOI in nonmagnetic materials as schematically shown in Fig. 1(a). It also provides a way to engineer the magnitude of the ISHE. As reported in the literature, the SOI effect in graphene increased linearly with the impurity coverage.14There are two types of mechanisms to take account for this extrinsic effect, namely the skew scattering15and the side jump.16The former arises from asymmetric scattering from impurities due to the spin-orbit coupling, and the latter can be viewed as a consequence of the anomalous velocity. Normally the ISHE from extrinsic mechanism is observed in metals or semiconductors doped with heavy elements like Ir, Nb, etc. The doped oxides are much aAuthor to whom correspondence should be addressed. Electronic mail: xwzhang@uestc.edu.cn 2158-3226/2017/7(12)/125218/6 7, 125218-1 ©Author(s) 2017 125218-2 Wang et al. AIP Advances 7, 125218 (2017) FIG. 1. (a) A schematic illustration of Py/Nd:STO for the measurement of DC voltages. The magnetic ﬁeld dependence of the DC voltages of (b) Py/Nd:STO and (c) Py/STO before ( Vbe) and after ( Vaf) ﬂipping at f= 2.8 GHz, P= 32 mW. The inset in b shows the VISHE before ( VISHE-be ) and after ( VISHE-af ) ﬂipping, respectively. less studied, although the controllability of carriers is well demonstrated. The inﬂuence to the SOI and SHE is unexplored. In this work, we present the ISHE measurements on STO substrates lightly doped with non- magnetic impurities Nd (Nd:STO). The spin current is injected from the ferromagnetic permalloy (Py) ﬁlm into the adjacent doped STO by spin pumping at the ferromagnetic resonance (FMR), and subsequently transformed into a charge current via the SOI effect induced by the spin skew scattering on the Nd impurities, which can be detected by the shorted microstrip transmission line technique.17 II. RESULTS AND DISCUSSION Fig. 1(a) shows a schematic illustration of Py on Nd:STO used in this experiment. The com- mercially available STO single crystals are cut into size of 0.2 510 mm3, and the doping concentrations of Ndis 0.05 wt%. The ferromagnetic Py ﬁlms with thickness about 20 nm are deposited on these substrates by magnetron sputtering. During the measurements, the sample is placed at the center of microstrip ﬁxture, where the external magnetic ﬁeld is applied perpendicular to the direction across the electrodes in the ﬁlm plane. In the Py/Nd:STO samples, the nonmagnetic layers no longer suppress the further diffusion of the spins due to the introduction of free elec- trons from impurities, which enables the spin currents to be effectively injected by spin pumping. Thus, the conversion of spin to charge current can be detected via the ISHE. As predicted by Fert, it can be explained by resonant scattering from impurity states split by the SOI, namely the spin Hall angleSH= 3dsin(22 1) sin1/(5) (withdbeing the impurity spin-orbit constant, being the resonance width, 1and2being the mean phase shift at the Fermi level).12The effect is expected to be larger when the electrons are more localized, where is smaller, as in the case of doped insulators compared with that in metals. Since the voltages measured are contributed from the spin rectiﬁcation effect (SRE), anomalous Hall effect (AHE) and ISHE and each of them may have symmetric and antisymmetric Lorentzian contributions, it is difﬁcult to separate them. In the present report we utilize the two-step measurement with sample ﬂipping to obtain the ISHE sig- nals (V ISHE) in Nd:STO,18–20while the remaining voltage signals are considered to originate from the SRE and AHE, which are simply denoted by V SRE. In order to keep the sample at the same position in the microstrip ﬁxture, an insulating STO chip with same thickness as the sam- ple is covered on the surface of Py/Nd:STO. The sandwich structure ensures the Py ﬁlm is at the same positions before and after being ﬂipped. As shown in Fig. 1(b), the DC voltages detected in doped STO are obviously different before ( Vbe) and after ( Vaf) ﬂipping, which implies that these signals are attributed not only to the contribution from the SRE in Py ﬁlms, but also to the contri- bution from the ISHE in doped layers. This signiﬁcant change arises from the reversal of the sign ofVISHE due to the inverted spin injection by the sample ﬂipping as shown in the inset of Fig. 1(b), and the signals of samples before and after ﬂipping can be respectively described as the addition (Vbe=VSRE+VISHE) and subtraction ( Vaf=VSRE-VISHE) of the DC voltages of two effects. Thus, theVISHE is separated from the VSREthrough the subtraction of experimental data ( Vbe- Vaf). As depicted in the inset, the line shape of VISHE is symmetric, which is typical as also shown in previous studies.21,22125218-3 Wang et al. AIP Advances 7, 125218 (2017) In contrast, there is no spin current injected into the interface of Py/STO because of the insulating STO suppressing the further diffusion of the spins. Thus, the DC voltage of this sample detected in this case is attributed to the contribution from the SRE in Py ﬁlm, the line shape of which is a combination of the symmetric and asymmetric Lorentzian components as plotted by the red squares in Fig. 1(c). The SRE, rectifying the microwave current at the FMR, is associated with the precessing magnetization and microwave electric ﬁeld,17but independent on the spin diffusion direction.23When the undoped samples are inverted at the steady external magnetic and microwave electric ﬁeld, the voltage signal of Py/STO is not distinctly different from that of the sample before ﬂipping as depicted by the blue circles in Fig. 1(c). We also notice that the voltage amplitude measured with the un-doped sample is about 3 times larger than the doped one. This is due to the shunting effect of the conducting substrate and the conductance difference of the Py layer. Fig. 2 shows the external magnetic ﬁeld dependence of VISHE for Py/Nd:STO at different microwave power P, in which the VISHE increases linearly with P. Furthermore, the values of VISHE at resonance ﬁeld Hrfor both doped samples are proportional to Pas shown in the inset, which is consistent with the model of the DC spin pumping. The spin current density generated by the spin pumping can be expressed as js=G"# 2~hrf2 82266666644Ms +q (4Ms )2+ 4!2 (4Ms )2+ 4!2377777752e ~, (1) where G"#, ,},hrf,, 4Msand!denote the spin mixing conductance, gyromagnetic ratio, Dirac constant, amplitude of the microwave magnetic ﬁeld, Gilbert damping coefﬁcient, effective saturation magnetization and microwave angular frequency, respectively. It indicates that the DC spin current is proportional to the time averaged Gilbert damping term onto the external magnetic ﬁeld direction, which depends linearly on the square of the magnetization-precession amplitude.24,25 In terms of the relation between VISHE andjs,VISHE =(SHwN)js(withSH,w,Nand being the spin Hall angle, electric resistivity, length of the sample and the spin-polarization vector, respectively),26the electric voltage due to the ISHE is also proportional to the square of the amplitude of magnetization-precession, namely the microwave power P.27In our case, the measured linear dependence is consistent with this simpliﬁed consideration, which demonstrates the voltages with symmetric Lorentz shape extracted by sample ﬂipping are due entirely to the ISHE induced by the spin pumping. Fig. 3 shows the variation of VISHEat different microwave frequency ranging from 2.4 to 4.8 GHz, in the step size of 0.4 GHz with ﬁxed P= 32 mW. As depicted in Fig. 3(a), the peak position of voltage curve for doped STO, namely the resonance ﬁeld Hr, increases with the FMR frequency f. It is well known that the relation between Hrandfﬁts the Kittel equation f=( =2)pHr(Hr+ 4Ms), where Msis the saturation magnetization of Py. Based on the ﬁtting as plotted by dotted lines in Fig. 3(a), the effective saturation magnetization 4Msfor for Py/STO and Py/Nd:STO are determined to be 11.5 kOe and 10.9 kOe, respectively. The difference of this parameter between pristine and doped FIG. 2. The magnetic ﬁeld dependence of VISHE at various microwave power P. The inset shows Pdependence of VISHE at the resonance ﬁeld Hr.125218-4 Wang et al. AIP Advances 7, 125218 (2017) FIG. 3. Ferromagnetic dependence of (a) resonance ﬁeld and (b) FMR linewidth for Py/STO and Py/Nd:STO, respectively. samples is very small (about 5%), which may arise from the difference in surface roughness of substrates. In addition, it can be clearly seen that voltage curves of the ISHE broaden with increasing the FMR frequency. As shown in Fig.3(b), the FMR linewidth H of doped sample is larger than that of the pristine STO, which experimentally provides the evidence of spin injection induced by the spin pumping28The Gilbert damping constant can be extracted from the function H=H0 + 2f/ , whereH0is the inhomogeneous contribution to the linewidth.29By ﬁtting the data, we obtain the damping parameters Py/STO = 0.010 and Py/Nd:STO = 0.016. When the spin current is allowed to leak into a normal conducting material, the magnetization vector loses torque during this process. The effect can be included into the phenomenological damping parameter in the Gilbert equation.30,31According to the enhanced damping contribution =Py/Nd:STO Py/STO induced by the effective spin absorption in doped STO, the spin mixing conductance G"#can be determined by G"#=4MsdF gB, (2) where dF,g, andBare the thickness of Py ﬁlm, Land ´e factor, and Bohr magneton, respectively. Using Eq. (2), the G"#at the interface is estimated as 5.97 1019m-2for Py/Nd:STO, which is in the same order of the values for the Py/Bi (1.06 1019m-2)32and Py/n-Ge/Pd samples (2.15 1019m-2),33 demonstrating that the Py/Nd:STO also enables the efﬁcient spin pumping. Intrinsic STO is an insulator with bandgap larger than 3.0 eV . Fig. 4(a) shows the transport properties of doped STO. The increase in the resistivity with increasing temperature is a typical metallic behavior caused by the introduction of impurities, consistent with the single crystal work of Tufte et al.34and Robey et al.35The mobility is 5 cm2V-1s-1at room temperature and increases to 90 cm2V-1s-1at 80 K. The carrier density extracted from this data is of order 1019cm-3and independence on temperature, which can be explained by small donor binding energy of STO due to its large dielectric constant.36In this case, an impurity band is formed in doped STO through the overlapping electronic pictures of individual impurity states. When the impurity band overlaps with the conduction band, it results in the transformation from insulator to semiconductor or metal. FIG. 4. (a) Temperature dependence of resistivity , mobility and carrier density nin Nd:STO. (b) Evaluated ISHE resistivity ISHE as a function of the resistivity induced by impurities.125218-5 Wang et al. AIP Advances 7, 125218 (2017) As in the case of the extrinsic SHE, the skew scattering and side jump are two major contributions to the ISHE in doped samples. In order to ﬁgure out the mechanism in Nd:STO, we measured the ISHE resistivity ISHE at different temperatures by ISHE =wRISHEIC/(xIS), where wis the width of sample,RISHE is the amplitude of the ISHE resistance, ICis the charge current induced by the ISHE, ISis the effective spin current injected into the Nd:STO and xis a correction factor.37It is established that the mechanisms in the extrinsic ISHE can be determined based on the relationship between the ISHE and the resistivity induced by Nd impurities with strong SOI, i.e. ISHE/ for the skew scattering and ISHE/2for the side jump.38As shown in Fig. 4(b), the ISHE is approximately proportional to the , which provides an indication for the dominant contribution from skew scattering by impurities. The spin Hall angle SH, deﬁned as the ratio of ISHE and , is estimated as (0.17 0.05)%, which is smaller than that for alloys ( SH= 0.6% for Ag doped with Ir, and SH= 2.1% for CuIr),39but larger than that for semiconductors, such as SH= 0.01% for p-Si26andSH= 0.02% for n-GaAs.40The enhancement may come from the extrinsic spin dependent scattering, where the intrinsic contribution is tiny in STO. III. CONCLUSIONS In conclusion, we have shown that doping with Nd leads the STO to the transformation from insulator to metal. By the method of the two-step measurement with sample ﬂipping, the ISHE, giving rise a conversion of spin to charge current caused by the spin skew scattering from impurities, is observed in Nd:STO and separated from the SRE in terms of their different relationship with the direction of spin diffusion. 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1.1496761.pdf	Electromagnetic secondary instabilities in electron temperature gradient turbulence C. Holland and P. H. Diamond Citation: Physics of Plasmas (1994-present) 9, 3857 (2002); doi: 10.1063/1.1496761 View online: http://dx.doi.org/10.1063/1.1496761 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/9/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Coupling of ion temperature gradient and trapped electron modes in the presence of impurities in tokamak plasmas Phys. Plasmas 21, 052101 (2014); 10.1063/1.4875342 Hot-electron generation by “cavitating” Langmuir turbulence in the nonlinear stage of the two-plasmon–decay instability Phys. Plasmas 19, 102708 (2012); 10.1063/1.4764075 Gyrokinetic secondary instability theory for electron and ion temperature gradient driven turbulence Phys. Plasmas 14, 112308 (2007); 10.1063/1.2812703 Instability of convective cells in ion temperature gradient turbulence Phys. Plasmas 9, 1565 (2002); 10.1063/1.1467927 Nonlocal theory and turbulence of the sheath-driven electron temperature gradient instability Phys. Plasmas 8, 750 (2001); 10.1063/1.1343513 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: 129.120.242.61 On: Sun, 30 Nov 2014 18:01:15Electromagnetic secondary instabilities in electron temperature gradient turbulence C. Hollanda)and P. H. Diamond University of California, San Diego, La Jolla, California 92093-0319 ~Received 10 April 2002; accepted 30 May 2002 ! The electron temperature gradient mode has been proposed as a primary mechanism for electron transport. The possibilities of magnetic secondary instabilities ~‘‘zonal’’ magnetic ﬁelds and magnetic ‘‘streamers’’ !are investigated as novel potential mechanisms for electron transport regulation and enhancement, respectively. In particular, zonal magnetic ﬁeld growth and transportregulationisinvestigatedasanalternativetoelectrostaticzonalﬂows.Growthratesandimplicationsfor electron thermal transport are discussed for both electrostatic and magnetic saturationmechanisms. The possibility of magnetic streamers ~mesoscale radial magnetic ﬁelds !, and their potential impact on electron thermal transport, is also considered. © 2002 American Institute of Physics. @DOI: 10.1063/1.1496761 # I. INTRODUCTION Akey issue in magnetic conﬁnement fusion is the under- standing of microturbulence which is believed to driveanomalously high levels of transport. Although this problemhas been intensively studied in the context of ion-temperature gradient ~ITG!turbulence ~likely the primary cause of ion particle and heat transport !a similar understand- ing of electron transport has not been achieved. There areseveral outstanding issues in the area of electron transport.The foremost issue is the need to identify the underlyinginstability process causing said transport. Several pieces ofexperimental evidence point towards the electron tempera-ture gradient ~ETG!mode. Work by Stallard et al. 1suggests that electron transport coefﬁcents are weakly affected or un-affected by the shear ﬂows believed to regulate the ITGmodes, suggesting an electron transport mechanism whichhas a smaller characteristic scale and larger growth rate than the ion turbulence. The ETG mode satisﬁes both of thesecriteria. Their observations also suggest that the measuredelectron temperature gradient is close to the marginallystable value of the linear ETG mode. More recently, Ryteret al.have published evidence that electron tranport and tem- perature proﬁles are determined by a critical gradient length. 2 However, it should be noted that there are other modes ~no- tably short-wavelength collisionless trapped electron modes3 and ‘‘drift-islands’’4,5!which may also be able to explain some of these results. Indeed, it has not been conclusivelyshown that only one mode is responsible for electron trans-port. Also particularly challenging is to understand electrontransport mechanisms which can function in the presence oftransport barriers or other conditions which quench particleand thermal ion transport. In this paper, we restrict ourselvesto the ETG mode. Traditionally, one calculates the magnitude of turbulent transport based on mixing-length or quasilinear estimates ofthe turbulent ﬂux. As such calculations require a determina- tion of the turbulent spectrum, the question of nonlinear satu-ration mechanisms naturally arises.Analytic 6,7and computa- tional work8has demonstrated that ITG turbulence saturates via a nonlinear transfer of energy to shear ﬂow modes,termed zonal ﬂows, which are toroidally and poloidally sym-metric. The zonal ﬂows have a predominantly poloidal ﬂow component ~certainly no radial ﬂow !, preventing them from tapping the free energy of the system to drive transport, andare damped due to ion–ion collisions. The combined systemof zonal ﬂows and turbulence can be described by apredator–prey-type model, in which total wave energy isconserved. Because of the close analogy with the ITG mode,we investigate the ETG mode 9–11for similar dynamics. It should be noted that such ﬂows have been observed in simu-lations of ETG turbulence, 12in the special case of a magnetic ﬁeld with a local minimum, and with lDe.re. Due to in- tuitive expectations that electromagnetic effects are more im-portant in the ETG case than ITG, we also investigate thepossibility of zonal magnetic ﬁeld generation as a possiblesaturation mechanism. Zonal magnetic ﬁelds are mesoscalepoloidal magnetic ﬁelds with k y5kz50, which would satu- rate the turbulence via random magnetic shearing instead of‘‘ﬂow’’shear ~seeAppendix B for more details !. The genera- tion of such ﬁelds in the context of explaining the low tohigh conﬁnement ~L–H!transition has been studied by Guz- daret al. 13Zonal ﬁelds are also discussed in Gruzinov et al.14and Diamond et al.7In this paper, we are interested in studying their general effectiveness as saturation mecha-nisms for ETG turbulence. We also consider zonal ﬂow/ﬁeldgeneration in the context of a random phase approximation~RPA!modulational instability, appropriate for fully devel- opedwave-turbulence, rather than the four-mode coherent/ parametric approach taken by Guzdar and co-workers. A more recent developement in the study of ITG and ETG modes has been the discovery of ‘‘streamers,’’ 7,11,15,16 which are radially extended convective cells. In particular, ithas been argued that streamers represent a mechanism for a!Electronic mail: cholland@physics.ucsd.eduPHYSICS OF PLASMAS VOLUME 9, NUMBER 9 SEPTEMBER 2002 3857 1070-664X/2002/9(9)/3857/10/$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: 129.120.242.61 On: Sun, 30 Nov 2014 18:01:15describing the bursty or ‘‘intermittent’’ transport often ob- served in simulation and experiment, and provide a route toenhancing transport well beyond gyro-Bohm levels. In simu-lations of ETG turbulence, electrostatic streamers have beenobserved in certain parameter regimes. 11Jenkoet al.argue that these streamers are a necessity for enhancing electro-static ETG transport to experimentally relevant levels. Fol-lowing the zonal ﬂow/ﬁeld analogy, we also investigate thepossibility of magnetic streamers . Magnetic streamers are radialmesoscale magnetic ﬁelds, produced by secondary in- stability, with the potential for greatly increased thermaltransport. They are extended cells in B xandBy, providing a radial magnetic connection mechanism. They also representan intriguing extension of a traditional convention of ETGturbulence, which is to heuristically invoke inverse cascadeprocesses as a mechanism for increasing the correlationlength of the turbulence to the electron skin depth, with aresultant increase in turbulent ﬂux. The structure of the paper is as follows: In Sec. II, we present the analytic model used, and discuss the basic phys-ics and linear dispersion relations. In Sec. III, zonal modesare investigated, whereas streamer physics is studied in Sec.IV. A summary and discussion of the results is given in Sec.V. II. MODEL EQUATIONS The full description of the electromagnetic ETG mode in general geometry, including nonlinear effects, requires a for-midable set of equations. In this work, we use the modelpresented in Horton et al., 10in a local limit; a similar set of equations is used in Ref. 16. Equations for electron vorticityand pressure, as well as Ohm’s Law, are used to describe theevolution of the electrostatic potential, electron pressure, andparallel magnetic potential. This model assumes that there isno magnetic shear or parallel magnetic ﬂuctuations, but doesinclude the diamagnetic heat ﬂux. It also assumes the ions tobe fully adiabatic, since k ’ri>1. The perpendicular mag- netic ﬁeld ﬂuctuations are then driven purely by the currentarising from the ﬂuctuating electron parallel velocity, allow- ing us to write vi5„’2Ai, using the normalizations deﬁned below. The dominant nonlinearities are assumed to be vE3B ’f5$f,f%andB˜’’f52(b/2)$Ai,f%, again using the normalizations deﬁned below. The model equations are ~2t1„’2!]f ]t1~12e1~11h!„’2!]f ]y1e]p ]y1„’2]Ai ]z 52$f,„’2f%1b 2$Ai,„’2Ai%, ~1! SS2b 21„’2D] ]t2b 2~11h!] ]yDAi2] ]z~f2p! 52$f,„’2Ai%2b 2$Ai,f2p%, ~2!]p ]t1~11h2Ge~12t!!]f ]y12Ge]p ]y1G„’2]Ai ]z 52$f,p%1Gb 2$Ai,„’2Ai%. ~3! The Poisson brackets are deﬁned as $f,g%5zˆ"(f 3g). The various quantities are normalized as f 5(Ln/re)ueuf˜/Te,Ai5(2LnvT/brec)ueuA˜i/Te,p 5(Ln/re)p˜/p0,Lf52dlnf/dx,h5Ln/LTe,e5Ln/LB,t 5Te/Ti,b58pp0/B02,x,y!x,y/re,z!z/LN,t !Lnt/vTe, and G55/3. In particular, we have normalized the ﬁeld quantities to the mixing length level ~i.e.,f,Ai, p.re/Lnaccording to mixing length estimates, or are ;1 with this normalization !. Note that damping terms, particu- larly thermal conduction, are neglected here ~restricting the validity of the equations to regimes of weak collisionality,appropriate for the core region of tokamaks !. Simulations by Labit and Ottaviani 17suggests that their effects are weak. Physically one can interpret the nonlinear terms as: elec- trostatic and magnetic Reynolds stresses driving the vorticity,current and magnetic ﬁeld advection in Ohm’s Law, and con-vection of pressure along with the magnetic Reynolds stressdriving the pressure. It is also useful to bear in mind thatEqs. ~1!and~2!suggest that the relevant basic length scale for fisrei5re/t1/2, whileAiwill scale with the collision- less skin depth c/vpe5b21/2~in the normalized units used here!. Basic dynamics and linear physics :Anumber of authors have investigated and documented the linear physics of ETGmodes ~see, e.g., Refs. 9, 10 !. Therefore, we provide only a brief overview here. First, it is useful to consider theenergetics of the mode. Deﬁning E f51 2d3x(tufu2 1u„’fu2),EA51 2d3x(b/2u„’Aiu21u„’2Aiu2), and Ep 5G/2d3xupu2, it is easy to show that ] ]t~Ef1EA1Ep!5S11h G1etDEd3xpS2]f ]yD ~4! using the identity d3xf$f,g%50. The energy of the system grows as the electrostatic turbulent ﬂux QturbES5d3xpvx 5d3xp(2(]f/]y)) extracts energy from the mean gradi- ents. It is interesting to note that in this model, magneticﬂuctuations redistribute energy between the ﬁelds, but thatthe electromagnetic ﬂux Q EM5viBxp/B0does not contribute to the growth of total ﬂuctuation energy. Let us now consider the linear dispersion relation. Fou- rier analyzing in time and space, we can combine Eqs. ~1!– ~3!together to ﬁnd the linear dispersion relation, v2~v1v31ekyv4!2ki2k’2~G~v11eky!1v32v4!50,~5! where we have deﬁned v15~t1k’2!v1~12e2~11h!k’2!, ~6! v25b 2~v2~11h!ky!1k’2v, ~7! v35v22Geky, ~8! v4511h2Ge~12t!ky. ~9!3858 Phys. Plasmas, Vol. 9, No. 9, September 2002 C. Holland and P. H. Diamond 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: 129.120.242.61 On: Sun, 30 Nov 2014 18:01:15In the limit that kiis unimportant, and neglecting the diamagnetic heat ﬂux, one can determine the dispersion re-lation to be SSb 21k’2Dv2b 2~11h!kyD~~t1k’2!v2 1kyv1e~11h!ky2!.0. ~10! The solutions are a marginally stable drift oscillation arising from the parallel dynamics, and the electrostatic curvature-driven ETG mode, which is the mode of interest. The solu- tion in this limit is v05vr01ig052@ky/2(t1k’2)# 1iukyuA(e/t1k’2)(h2hc)1/2,hc.1/(4et)21. It is impor- tant to note that as the model used is only valid for k’re <1, a more detailed derivation of quantities such as hcto include full ﬁnite Larmor radius ~FLR!effects is not neces- sarily meaningful, and potentially misleading. We treat theﬁnite bandkieffects perturbatively to ﬁnd their contribu- tions to the dispersion relation. One ﬁnds dvr5k’2ki2ky 2~t1k’2!ad2bc ~avr02bky!21a2g02, ~11! dg521 g0k’2ki2 2~t1k’2!~avr02bky!~cvr02dky!1acg02 ~avr02bky!21a2g02, ~12! a5(b/2)1k’2,b5(b/2)(1 1h),c511G(t1k’2), andd 5h112G. It is easy to see that kieffects are stabilizing, and introduce a frequency shift whose sign is parameter de-pendent. The growth rate from Eq. ~5!is plotted in Fig. 1 for typical parameters ~ t51,h53,e50.1, and b50.04!. The stabilizing effects of kias well as FLR stabilization at high k’are clearly seen. III. ZONAL MODE EQUATIONS A. Zonal mode generation We ﬁrst consider zonal modes, because of greater famil- iarity with their nonlinear dynamics. Conceptually, weassume that there is a spectrum of non-axisymmetric, ‘‘fast’’/small scale ~small but ﬁnite k i,k’;re21!modes rep- resenting the turbulence. Then based on experience with thegeneric drift waves and the ITG mode 6,7we average the base equations over fast time and space scales ~denoted by tildes ! and investigate the possibility of a modulational instability for a ‘‘slow’’( k!re21) mode with poloidal and toroidal sym- metry ( ]/]y5]/]z50). For the slow mode, averaging yields ~2t1„x2!]f¯ ]t52$f˜,„’2f˜%1b 2$A˜i,„’2A˜i% 1Cf~f¯!, ~13! S2b 21„x2D]A¯i ]t52$f˜,„’2A˜i%1b 2$A˜i,f˜2p˜% 1CA~A¯i!, ~14!]p¯ ]t52$f˜,p˜%1Gb 2$A˜i,„’2A˜i%2Cp~p¯!. ~15! Note that the polodial symmetry ( ]/]y50) of the slow mode makes it completely insensitive to the lineardrive terms of the base equations, and reﬂects that such modes arenecessarily nonlinearly generated. The C fterm represents a generalization of the Rosenbluth–Hinton collisional dampingterm for electrostatic zonal ﬂows, 18withCAandCprepresent- ing collisional parallel resistivity and diffusion for A¯iandp¯, respectively. Physically, the zonal modes are damped by aneffective friction between the kinetic electron response to themode and trapped electrons. It should also be noted that inthe spirit of analogy between ITG and ETG physics, onemight expect C f}nee, andCA}nei, relative to niiin the ITG case. As nee.nei@nii, it is likely that collisional damping of zonal modes may be even more important in ETG thanITG turbulence. We now assume that we can describe the underlying ﬂuctuations via a quasilinear approach, such that A˜i,k.kzv32v4 v2v32Gki2k’2f˜k5RAf˜k, ~16! p˜k.v2v42Gki2k’2 v2v32Gki2k’2f˜k5Rpf˜k. ~17! We can reexpress the nonlinear terms as functions of uf˜ku2, using these quasilinear responses, and known properties ofthe Poisson brackets ~see Appendix A !,a s FIG. 1. Linear growth rate for t51,h53,e50.1, and b50.04.3859 Phys. Plasmas, Vol. 9, No. 9, September 2002 Electromagnetic secondary instabilitie 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: 129.120.242.61 On: Sun, 30 Nov 2014 18:01:15~t2„x2!]f¯ ]t52„x2Ed3kS12b 2uRAu2Dkxkyduf˜ku2 2Cf~f¯!, ~18! Sb 22„x2D]A¯i ]t52i„xEd3kSk’21b 2~12Rp!2i„xkxD 3kyRAduf˜ku22CA~A¯!, ~19! ]p¯ ]t5i„xEd3kSRp1i„xkxGb 2uRAu2Dkyduf˜ku2 2Cp~p¯!. ~20! To close these equations, we exploit the scale separation be- tween the underlying turbulence and the slow mode by usingthe wave-kinetic equation to calculate the response of theturbulence to the zonal modes. Such an approach exploits thefact that large-scale modulations of the small-scale ﬁeldsconserve the action or quanta number ~N k5Ek/vk, where Ek is the energy of mode k!of the small-scale ﬁelds. This ap- proach is valid due to the time-scale separation between theslow and fast modes. Generically, there will be an adiabaticinvariant of the form N k5Nk~uf˜ku2,uA˜i,ku2,up˜ku2!. ~21! Standard substitution of the quasilinear relations allow us to writeNk5Nk(uf˜ku2), which in turn allows us to express the modulated nonlinear drive terms as functions of Nkvia duf˜ku25(duf˜ku2/dNk)dNk5LkdNk. The adiabatic invariant Nkcouples the turbulence to the slow mode via the wave- kinetic equation, ]Nk ]t1]vnl ]k]Nk ]x2]vnl ]x]Nk ]k5gnlNk2DvNk2,~22! where vnlandgnlare the frequency and growth rate of the underlying turbulence in the presence of the slowly-varyingﬁelds, and the ﬁrst term of the right-hand side representslinear growth of the turbulence, while the second term cor-responds to higher-order interactions. To ﬁnd vnl, one can modify the linear mode equations to reﬂect that the primaryeffect of the slowly-varying mode on the small scales is con- vection of fast modes by the slow, via ]t!]t1$f¯,%,ki !kz2(b/2)$A¯i,%; we also note that inclusion of a slow varying pressure will create an effective pressure gradient heff5h2„xp¯. One can then write vnlas the sum of the origi- nal linear dispersion relation and an effective Doppler shiftfrom the slowly varying ﬁelds ( vgi5]vk/]kz), vnl.v~k’,ki!1k’V¯E3B 5vklin1vgidki1k’V¯E3B 5vklin2k’S„Sf¯2b 2vgiA¯iD3zˆD, ~23! where we have taken dvk/dh.0, as it enters only through FLR effects. Equation ~23!underscores that the small-scale turbulence will be sheared by boththe electrostatic and elec-tromagnetic potentials, i.e., it feels both ﬂow and magnetic shear ~seeAppendix B for a more complete discussion !.I ti s also important to note that the growth rate is modiﬁed by thepresence of the slow modes.The modiﬁed growth rate can beexpressed as gnl.gk1b 2]gk ]kzk~„A¯i3zˆ!2]gk ]h„xp¯. ~24! The wave-kinetic equation then takes the form, ]N ]t1vgN1ky„x2Sf¯2b 2vgiA¯iD]N ]kx .Sgk2b 2]gk ]kzky„xA¯i2]gk ]h„xp¯DN2DvN2. ~25! Expressing the action density as the sum of a mean back- ground and a coherent response ( Nk5Nk1dN), one can lin- earize the wave-kinetic equation to ﬁnd an expression for dN, dNq52q2kySf¯q2b 2vgiA¯qDR~qvgx!]Nk ]kx 2iqSkyb 2]gk ]kzA¯q1]gk ]hp¯qDR~qvgx!Nk. ~26! gqandqare the growth rate and wave number of the large scale perturbation, and R(qvgx)51/(Dvk1(gq1ıqvgx)), where Dvknow encompasses both the linear growth rate and nonlinear frequency shift of the underlying turbulence, and gqis the growth rate of the zonal modes. One can now close the zonal mode equations via substitution of dNqinto Eqs. ~13!–~15!. However, it is useful to ﬁrst consider the various k-space symmetries of the nonlinear drive terms and dNq.I n particular, examination of the quasilinear responses indicatesthatR Ais odd in kz, whileRpis even.Additionally, vgiand ]gk/]kzare also odd in kz, while ]gk/]his even. Thus, upon substitution of dNqinto Eqs. ~13!–~15!, and integration overkz, one ﬁnds that the equation for zonal magnetic po- tential decouples from the electrostatic potential and pressureequations. Therefore, zonal magnetic ﬁeld dynamics are ef- fectively decoupled from electrostatic zonal ﬂow dynamics ! One can also use k xsymmetry to show that f¯qandp¯qare essentially independent, with p¯qacting as a passive scalar for the zonal mode case, as well as to simplify the zonal ﬁeldequation. The resulting equations of interest are ~t1q2!]f¯q ]t5q4Ed3kky2S12b 2uRAu2DR~qvgx! Lk 3S2kx]N¯k ]kxDf¯q2Cf~f¯q!, ~27!3860 Phys. Plasmas, Vol. 9, No. 9, September 2002 C. Holland and P. H. Diamond 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: 129.120.242.61 On: Sun, 30 Nov 2014 18:01:15Sb 21q2D]A¯q ]t5b 2q4Ed3k~2ky2RARevgz!R~qvgx! Lk 3S2kx]N¯k ]kxDA¯q1b 2q2Ed3k 3Sk’21b 2~12Rpre!DSky2RAim]g ]kzD 3R~qvgx! LkN¯kA¯q2CA~A¯q!. ~28! It is now straightforward to ﬁnd growth rates for the electro- static and magnetic modes, so that gqf5q4 t1q2Ed3kky2S12b 2uRAu2DR~qvgx! Lk 3S2kx]N¯k ]kxD2nf, ~29! gqA5bq4/2 b/21q2Ed3k~2ky2RARevgz!R~qvgx! Lk 3S2kx]N¯k ]kxD1bq2/2 b/21q2Ed3kSk’21b 2~12Rpre!D 3Sky2RAim]g ]kzDR~qvgx! LkN¯k2nAq2. ~30! We have explicitly rewritten CfandCAto demonstrate their physical meanings ~collisional friction and resistivity, respectively !. Examination of Eq. ~29!shows that the growth rate of the electrostatic zonal ﬂow is reduced relative to theITG case because of fully adiabatic ions, and the stabilizingeffects of the magnetic Reynolds stress.An interesting prob-lem is to elucidate the conditions for if/when the magnetic Reynolds stress will completely stabilize the growth of f¯q. Examination of Eq. ~16!forRAshows that uRAu2}kz2/ky2, one can then write ky2S12b 2uRAu2D5ky22b 2kz2f~k’2,b,t,e!. ~31! Equation ~31!shows that the competition between the elec- trostatic and magnetic Reynolds stresses can be cast as the difference between mean ky2andkz2of the turbulence, or in other words, that ( ky2)1/2must be greater than a critical wave numberkcforfto grow, where kc25b 2kz2f. ~32! If one estimates kz2.(e/qB)2ky2~whereqBis the safety fac- tor!, it would appear that in general the magnetic Reynolds stress would be, at best, weakly stabilizing @that is, one would expect b/2(e/qB)2f,1]. One must quantify the ‘‘proportionality function’’ f, and especially its bdepen- dence, to verify this suggestion. An understanding of zonal magnetic ﬁeld growth re- quires a more extensive analysis, as indicated by the relativecomplex of Eq. ~29!vs~30!. Such an analysis is most easilydone by ﬁrst considering some basic dependencies of the relevant quantities. In particular, one can estimate R Are }2kz/ky,RAim}2gkkz/ky2,]g/]kz}2kz/gk,vgz}sgn(ad 2bc)kz/ky. These expressions allow one to rewrite Eq. ~30! ~momentarily ignoring the damping nA!as gqA;q2b/2 b/21q2Ed3k~k’21L22!f1kz2 gkN¯k Lk 1sgn~ad2bc!q4b/2 b/21q2Ed3kf2kz2 gk1 Lk 3S2kx]Nk ]kxD. ~33! To interpret this result physically, it is useful to consider the original Ohm’s Law equation @Eq.~2!#. It is clear that the nonlinearites correspond to electrostatic convection of cur-rent and magnetic ﬁeld ﬂuctuations. Thus, the electrostatic turbulence ampliﬁes small-scale magnetic ﬂuctuations intolarger-scale magnetic ﬁelds! Zonal magnetic ﬁeld generation can be clearly viewed as a kind of small-scale dynamoaction. 19Observing that this derivation has assumed q<1, it is also interesting to note that the zonal ﬁeld growth is drivenprimarily by the term arising from modulation of the growthrate, rather than the frequency modulation term. In contrast,the modulation of the linear growth rate gives no contribu-tion to the electrostatic mode growth rate. One can also notethat like the linear ﬂuctuations, the zonal electrostatic poten-tial length scale is set by re, while the zonal ﬁeld length scale depends on the collisionless skin depth. Finally, itshould be noted that the zonal modes and turbulence form aconnected system, 6and that the zonal modes back-react on the turbulence even as the turbulence generates these modes.For a more complete discussion of this issue, the reader isagain referred to Appendix B. B. Estimations of transport As alluded to in the introduction, the key question for any investigation such as this is ‘‘What level of transport isthe mode expected to produce?’’ We address this questionhere. The turbulent radial heat ﬂux Q xis Qx5^p˜v˜x&5Kp˜S2]f ]yDL1^pviBx&5QxES1QxEM. ~34! Keeping only second order correlations, the electromagnetic heat ﬂux can be written as QxEM5p0^v˜iB˜x&1v¯i^p˜B˜x&1B¯x^p˜v˜i& 5b 2p0( kikyk’2uAku21b 2v¯i( kikyRpRAufku2 1B¯x( kk’2RARpufku250. ~35! Thek-space symmetries of each term ~kyfor the ﬁrst, kivia RAfor the second and third !reduce the electromagnetic heat3861 Phys. Plasmas, Vol. 9, No. 9, September 2002 Electromagnetic secondary instabilitie 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: 129.120.242.61 On: Sun, 30 Nov 2014 18:01:15ﬂux to zero. One might argue that the vanishing of the ﬁrst term is a function of using triply periodic boundary condi-tions. This term can be rewritten as ^v˜iB˜x&5b 2Edydz „’2A]A ]y 5b 2Edydz „xS]A ]x]A ]yD 1b 21 2Edydz „ySS]A ]yD2 2S]A ]xD2D 5b 2„xEdydzS]A ]x]A ]yD. ~36! Thus, the only remaining term upon averaging over ﬂux sur- faces is the radial gradient of the magnetic Reynolds stress,which will yield a transport much lower than that suggestedby static stochastic ﬁeld estimates. 20With some consider- ation, this result should not be surprising, as it is well knownthat ambipolarity limits the particle diffusion predicted bysuch estimates. It should also be noted that consideration ofthe energy equation @Eq.~4!#indicates that only electrostatic transport introduces energy into the system, while the mag-netic nonlinearites redistribute the energy between variousﬁelds. It is also instructive to consider potential transport aris- ing from parallel conduction. In a collisional regime ~e.g., near the edge, but not in the core !, a radial heat ﬂux of the formQ x5(B˜x/B0)Qi52kiuB˜x/B0u2dT0/dxmight be ex- pected, which would appear to have a potentially large mag-nitude. However, when one takes into account the fact thatthis expression is derived from Q i52n0ki„iT, andB"T .0, it becomes clear that collisional parallel transport along magnetic ﬂuctuations cannot lead to experimentally relevantlevels of electron heat transport ~particularly in the core re- gion!. We are then left with only the electrostatic heat ﬂux, Q xE3B5^p˜v˜x&.pe0vte( k~kyre!Im~Rp!Uef˜k TeU2 ~37! 54t2~11h!pe0vTeAe t~h2hc!1/2 3( kukyreuUef˜k TeU2 . ~38! UsingQ52n0xdTe/dx5pe0x/LT, and deﬁning xGB 5re2vTe/LT, one ﬁnds x xGB54t211h h2Ae t~h2hc!1/2 3( kukyreuSLn reD2Uef˜k TeU2 . ~39!We are now left with estimating the saturated intensity level of the turbulence, which is accomplished via use of a simplere-expression of the previously derived equations. Equations~29!–~30!are rewritten as ]f¯q ]t5SfEf¯q2nff¯q, ~40! ]A¯q ]t5SAEA¯q2nAq2A¯q. ~41! E5Nk/Lk5(Ln/re)2uef˜/Teu2is the intensity of the small- scale turbulence, and we have written gqf5SfEandgqA 5SAE. Note that the gradient length used for normalizing the base equations is Ln, but as the mode is driven by the temperature gradient, it is more appropriate to use LTwhen estimating mixing-length transport coefﬁcents. In steady-state, the turbulence intensity is set by the balance betweenthe nonlinear growth and linear damping of the zonal mode.The saturated intensity ~in normalized units !and correspond- ing thermal diffusivity ~in unnormalized units !for the case of electrostatic zonal ﬂow saturation are E f5nft1q2 q4Ae t~h2hc!1/2 k02kc, ~42! xf xGB54te11h h~h2hc!SnfLT vTeDt1~qre!2 ~qre!4k0 k02kc, ~43! wherek0is a mean kyof the turbulence, and kcwas deﬁned in Eq. ~32!. For zonal ﬁeld saturation, one ﬁnds EA5nAS11q2 bDk021 ~k021L22!f11q2f2Ah2hc e3tqB2,~44! xA xGB54t e11h h~h2hc!SnALT vTeD~11~q~c/vpe!!2! 3qB2re23 k0~~k021L22!f11q2f2!. ~45! Here,kz2is again estimated as ( e/qB)2k02, whereqBis the safety factor. Consideration of xAoffers an intriguing possibility. If q(c/vpe).1, Eq. ~45!suggests that xA}(h2hc)/b. Such a scaling would be very appealing, as it offers the possibilityof good agreement with experiment. In particular, oneachieves a b21-dependent scaling, without invoking in- creased correlation lengths of the turbulence, and consider-ing only electrostatic transport. This result offers not only aninteresting route to a neo-Alcator-type scaling, but may alsooffer some insight into the results presented in Ref. 21,which describes ETG turbulence creating a xedue only to electrostatic transport, but which exhibits b21scaling. Labit and Ottaviani also observe decreasing transport with increas-ing b.17Clearly, then, the saturation mechanism for zonal ﬁeld is crucial. We have taken here the simplest possibility,which is a purely collisional damping with no bdependence. If, however, zonal ﬁeld growth is limited by a ‘‘tertiary’’instability, 11,22,23one could easily imagine that the bscaling3862 Phys. Plasmas, Vol. 9, No. 9, September 2002 C. Holland and P. H. Diamond 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: 129.120.242.61 On: Sun, 30 Nov 2014 18:01:15ofxecould readily change. For a zonal magnetic ﬁeld, such an instability might take the form of something akin to amicrotearing mode, rather than the Kelvin–Helmholtz-typemodes described in Refs. 11, 22, and 23. Thus, stability ofzonal ﬁelds is an issue that demands further investigation.The requirement of zonal ﬁeld scale smaller than skin depthfor the b21scaling also highlights the importance of inves- tigating the scales of the secondary instabilities. In the con-text of bdependencies of xe, one should also consider the role of the k02kcterm for the electrostatic case, which rep- resents the competition between electrostatic and magneticReynolds stresses. It is clear that the magnetic Reynoldsstress is a stablizing factor for the electrostatic zonal ﬂow,and should be more effective for increasing b. Qualitatively, increased stabilization of the zonal ﬂow with bleads to a higher saturated intensity level, and thus a higher transportlevel. However, a more quantitiative investigation is needed.It is also interesting to note that both modes give different e scalings. Finally, it should be noted that the absolute magni-tudes of the predicited thermal diffusivities may be con-strained by their explicit dependence on collisionality, whichhas been assumed to be small in the analytic model usedhere. C. Discussion These simple considerations of transport suggest several interesting questions whose answers could shed more lighton the physics of electron transport. First, the physics ofcollisionality and zonal mode saturation remains a key issue.In ITG turbulence, the competition of ‘‘tertiary’’instabilities,back-reaction on the turbulence, and collisional ﬂow damp-ing as secondary instability saturation mechanisms is anopen issue. Investigation of analogous tertiary instabilitiesfor ETG secondary modes is an obvious question, and suchstudies are underway. In particular, whether such tertiary in-stabilities will be able to compete with nf,nA;neeis of particular interest. A more detailed investigation of nAand tertiary instabilites of the zonal ﬁeld is particularly warrantedin light of the potential bscalings for xeour analysis sug- gests. One could also make a more pessimistic observation,and note that if the relevant collisional time scale for ETG istruly nee, it is possible that the damping may kill the zonal modes outright unless the turbulence reaches a much higherintensity level, relative to the ITG case. The different effectsof bon electrostatic and magnetic modes are also interesting. An intriguing question to ask is if there is a critical bat which zonal ﬁelds become the dominant saturation mecha-nism, rather than zonal ﬂows. The limitation of negligiblemagnetic ﬂutter transport is counterintuitive to the ‘‘conven-tial wisdom’’ in ETG turbulence, which has often qualita-tively invoked ﬂutter transport as the dominant transportmechanism. However, negligible ﬂutter transport is in agree-ment with the simulation results of Jenko et al., 11,21as well as Labit and Ottaviani.17It would be interesting to determine what additional physics could be added to the model ~if any !, to break this constraint. Unless such physics is found, thislimitation would appear to invalidate many of the earliermodels. One would also like to quantify the effects of mag-netic shear and geometry on the transport, as well as the impact of nonadiabatic ions. Finally, we suggest that many ofthe predictions and questions raised in this section could beaddressed via existing codes, not the least of which would beto simply see if zonal ﬁelds are in fact generated in ETGsimulations. IV. MAGNETIC STREAMERS In both ETG and ITG simulations, radially extended electrostatic convective cells are observed. These cells,termed streamers, are found to greatly enhance the turbulenttransport. The possibility of zonal magnetic ﬁelds naturallyleads to question of magnetic streamers. By magneticstreamers, we mean radially extended convective cells in B x. They would be similar to magnetostatic convective cells, butare expected to have a ﬁnite real frequency. Based on previ-ous analytic studies of electrostatic streamers in ITG turbu-lence, we undertake a similar study here. The approach usedis similar to that of the zonal case, except now we look formodes with „ x!„y, and „z.0. Structurally, these equations will be similar to those of the zonal modes, and in particular,it is clear that the k zsymmetry of the ﬂuctuations will also decouple the magnetic streamer from the electrostic mode.The equation for magnetic streamer modes is Sb 22„y2D]A¯i ]t1b 2~11h!]A¯i ]y 5$f˜,„’2A˜i%2b 2$A˜i,f˜2p˜%1nA„y2A¯i. ~46! The equation is again closed via an appeal to wave-kinetics, with the adiabatic invariant response now dNq5S2q2kxvgi]Nk ]ky1iqkx]g ]kzNkDb 2R~qvgy!A¯q. ~47! Carrying out the analysis is a similar fashion to the zonal mode, one ﬁnds that the streamer has a real frequency Vq and growth rate gq, which are Vq5b 2~11h!q b/21q2, ~48! gq5q4b/2 b/21q2Ed3k~2kx2RARevgz!R~qvgx! Lk 3S2ky]N¯k ]kyD1q2b/2 b/21q2Ed3kSk’21b 2~12Rpre!D 3Skx2RAim]g ]kzDR~qvgx! LkN¯k2nAq2. ~49! It is useful to note that the structure of the streamer growth rate is quite similar to that of the zonal ﬁeld growth rate,suggesting that k x2kyasymmetries in the spectrum may be crucial for determining which has the larger growth rate. Having established the potential for magnetic streamer growth, it is important to assess their importance via inves-tigating the transport they are expected to produce. One is3863 Phys. Plasmas, Vol. 9, No. 9, September 2002 Electromagnetic secondary instabilitie 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: 129.120.242.61 On: Sun, 30 Nov 2014 18:01:15immediately confronted with the fact that in the model used, magnetic ﬂuctuations cannot induce a ﬂux ~see Sec. IIIB !. Several ways of extending the model which might allow sig-niﬁcant ﬂutter transport present themselves. The ﬁrst is toappeal to additional physics which could alter the phase shiftbetween viandBx. What such a mechanism would be ~per- haps a current contribution from nonadiabatic ions !, and whether the phase could be altered strongly enough to have ameaningful impact, are unclear. Alternatively, one mightsearch for a way to overcome the objections of Sec. IIIB totransport due to parallel conduction along ﬂuctuating ﬁeldlines. However, to make such a claim, one should have abetter understanding of the role of collisionality for large-scale modes. Perhaps the most appealing possibility is torelax the restriction on „ z, which would lead to linearcou- plings betwen the magnetic and electrostatic streamers. In-deed, simulations by Beyer et al. 15suggest that streamers are in fact composed of many different poloidal and toroidalmode numbers. In contrast to the zonal case, where the elec-trostatic and magnetic modes completely decouple, onewould have a single ‘‘electromagnetic’’ streamer with bothelectrostatic and magnetic components; these componentswould have independent nonlinear driving terms, butcoupled linear drives. Of particular interest would be to in-vestigate whether the linear stabilizing properties of the mag-netic component ~analogous to the line-bending stabilization effect of magnetic ﬂuctuations on the linear mode !or its nonlinear drive are dominant when it couples to the electro-static component. Such calculations are left for a future pub-lication. However, one observes that many of the same limi-tations and issues raised in the previous section appear againhere, highlighting the need to quantify the role of collisionaldamping and tertiary instabilities ~e.g., the physical mecha- nisms which determine streamer intensity !for streamers as well as zonal modes. V. CONCLUSIONS A thorough understanding of electron transport remains an open challenge to the magnetic fusion community. TheETG mode is often invoked as a potential mechanism forexplaining the anomalously high electron transport. In thispaper, we have investigated the possibility of secondary elec-trostatic and magnetic instabilities as potential saturation andtransport regulation mechanisms. In particular, we have in-vestigated in detail zonal magnetic ﬁelds as novel saturationmechanisms for the turbulence. Zonal magnetic ﬁelds aregenerated via electrostatic convection of magnetic ﬁeld andcurrent ﬂuctuations, in clear analogy with mean-ﬁeld dy-namo theory, and saturate the turbulence via random mag-netic shearing. It has been demonstrated that the underlyingk isymmetry of the ETG mode leads to a decoupling of the zonal magnetic ﬁeld from the ‘‘traditional’’ electrostaticzonal ﬂow. We have also extended the idea of magnetic sec-ondary instabilities to streamers. For streamers with q i.0, one again has decoupled electrostatic and magnetic stream-ers, where as these modes will linearly couple into a single‘‘electromagnetic’’streamer for ﬁnite q i. More detailed stud- ies of streamer physics in ETG are currently underway.Our investigations have raised as many questions as they have answered. The need for further study of magnetic Rey-nolds stress inhibition of zonal ﬂow growth has been dem-onstrated. The inability of magnetic ﬂutter to induce trans-port seems to invalidate many of the more qualitative modelsof electron thermal transport, but appears to be a direct con-sequence of the relation between current and magnetic ﬁeld.Most importantly, the need for a detailed understanding ofthe saturation mechanisms for ETG zonal modes and stream-ers, both electrostatic and magnetic, is a recurring conse-quence of our analysis. In particular, quantifying ‘‘tertiary’’instabilities and collisional damping for the various second-ary instabilities is key. It would also be useful to extend theanalysis to a nonlocal model, which would introduce mag-netic shear into the dynamics. Quantifying the role of mag-netic shear in tertiary instabilites, particularly for the zonalﬁeld and magnetic streamer, would be of particular interest.Another crucial issue for both ITG and ETG turbulence isthat of pattern selection, that is, whether zonal modes orstreamers are preferentially generated.At this time, the issueis unresolved, but will be addressed in future publications.Finally, we note that many of these questions should be trac-table to analysis by both existing simulations, and extensionsof existing analytic ITG investigations. ACKNOWLEDGMENTS The authors would like to thank E.-J. Kim, F. Jenko, W. Horton, B. Labit, and T. S. Hahm for valuable discussionsand critical readings of the manuscript. C.H. performed this research under an appointment to the Fusion Energy Sciences Fellowship Program, adminis-tered by Oak Ridge Institute for Science and Education un-der a contract between the U.S. Department of Energy andthe Oak Ridge Associated Universities. This research wassupported by Department of Energy Grant No. FG-03-88ER53275. APPENDIX A: PROPERTIES OF POISSON BRACKETS The Poisson brackets $f,g%5(f3g)zˆoffer a con- vient shorthand notation for writing many of the nonlinearterms encountered in plasma physics. In the course of modu-lation stability analysis, it is often helpful to rewrite the Pois-son brackets in terms of partial derivatives acting on both f andg. In particular, the following identities are often found to be of use: $f,g%5„yS]f ]xgD2„xS]f ]ygD, ~A1! $f,„’2f%5~„y22„x2!S]f ]x]f ]yD 2„x„ySS]f ]yD2 2S]f ]xD2D, ~A2!3864 Phys. Plasmas, Vol. 9, No. 9, September 2002 C. Holland and P. H. Diamond This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.120.242.61 On: Sun, 30 Nov 2014 18:01:15$f,„’2g%5„y2S]f ]x]g ]yD2„x2S]f ]y]g ]xD 2„x„ySS]f ]y]g ]yD2S]f ]x]g ]xDD 2H]f ]x,]g ]xJ2H]f ]y,]g ]yJ ~A3! 5„y2S]f ]x]g ]yD2„x2S]f ]y]g ]xD 2„x„ySS]f ]y]g ]yD2S]f ]x]g ]xDD 1„xS]2f ]x]y]g ]x1]2f ]y2]2f ]y2]g ]yD 2„yS]2f ]x2]g ]x1]2f ]x]y]g ]yD. ~A4! APPENDIXB:GENERALIZEDEFFECTOFRANDOM SHEAR AND GROWTH RATE MODULATIONON SMALL-SCALE TURBULENCE It has been noted previously6,7that the coupled system for electrostatic zonal ﬂows and drift waves form a closedsystem ~of a predator–prey form !which conserves total en- ergy. While the zonal ﬂows are generated by the drift waves,they also back-react on the turbulence via random shearingink-space. The back-reaction can easily be computed via quasilinear formalism, and one ﬁnds coupled equations of the form ( N ¯k5^Nk&), ]fqZF ]t5q2E k.k0d3kky2R~qvgx! LkS2kx]^Nk& ]kxD, ~B1! ]^Nk& ]t5] ]kx~q4ky2R~qvgx!ufqZFu2!]^Nk& ]kx 1g^Nk&2Dv^Nk&2. ~B2! It is easy to see that Eq. ~B2!describes how the random ﬂow shear leads to diffusion of the turbulence in kx, and will thus increase ^kx2&~i.e., ‘‘eddy shearing’’ !.We now wish to extend this result to include both the random magnetic shearing ef-fects of the zonal ﬁeld, as well as the effects of modulatingthe growth rate by the zonal magnetic ﬁeld and zonal pres-sure. The theory of random shearing by both zonal ﬁelds andﬂows is developed in Sec. IIIA.As usual, ray chaos, namely,the overlap of wave group and zonal phase velocity reso-nances, is necessary for the applicability of quasilineartheory. For a uniﬁed treatment of all effects, we rewrite theequation for ^Nk&as ]^Nk& ]t5] ]kxSq2kySf2q2b 2vgiA2qDdNqD 2iqSb 2]gk ]kzkyA2q2]gk ]hp2qDdNq1gk^Nk& 2Dv^Nk&2. ~B3!Equation ~26!fordNqcan then be substituted in, giving the generalized description for the back-reaction, ]^Nk& ]t5] ]kxSq4ky2R~qvgx!Ufq2b 2vgiAqU2D]^Nk& ]kx 1q2R~qvgx!Ub 2]gk ]kzkyAq2]gk ]hpqU2 3^Nk&1gk^Nk&2Dv^Nk&2~B4! )]^Nk& ]t5] ]kxDEM]^Nk& ]kx1gNL^Nk&1gk^Nk& 2Dv^Nk&2. ~B5! Thus, including zonal magnetic ﬁelds and pressure leads to a kxdiffusion coefﬁcient that reﬂects the electromagnetic char- acter of the shearing, as well as a quasilinear growth rate viamodulation of the linear growth rate. The new diffusion co-efﬁcent is an intuitive generalization of the electrostatic case,with f!f2(b/2)vgiA. For the case of zonal magnetic ﬁeld generation, it is useful to note that as both bandvgiare small quantities, it is possibile that random magnetic shear-ing may not be as effective a saturation mechanism as theﬂow shear. Clearly, case-by-case quantitative analysis is re-quired. It is also interesting to consider the nonlinear growth rate created via modulation of the linear growth rate, and in par-ticular, the effects of the zonal pressure. As ]gk/]h;(h 2hc)21/2, there is the possibility of the zonal pressure intro- ducing signiﬁcant energy into the turbulence. A nonlinearmodulation analysis, then, is required to treat the regime nearmarginality ( h;hc). It was found that for zonal modes, the pressure is essentially generated as a passive scalar by theelectrostatic mode, and uncorrelated with the zonal magneticﬁeld.Thus, in the electrostatic case, one might expect a com-petition between the random shearing of the zonal ﬂow as asaturation mechanism, and energy reintroduced into the tur-bulence via the zonal pressure. Again, this is an issue thatshould be addressed in a more quantitative fashion. Finally,we note that the introduction of zonal magnetic ﬁelds andpressures suggests interesting extensions of the predator–prey model developed in Diamond et al. 6 1B. W. Stallard, C. M. Greenﬁeld, G. M. Stabler et al., Phys. Plasmas 6, 1978 ~1999!. 2F. Ryter, F. Leuterer, G. Pereverzev et al., Phys. Rev. Lett. 86, 2325 ~2001!. 3F. Y. Gang, P. H. Diamond, and M. N. Rosenbluth, Phys. Fluids B 3,6 8 ~1991!; T. S. Hahm and W. M. Tang, ibid.3,9 8 9 ~1991!. 4B. B. Kadomtsev, Tokamak Plasma: A Complex Physical System ~Institute of Physics, Bristol, 1992 !. 5R. D. Sydora, Phys. Plasmas 8, 1929 ~2001!. 6P. H. Diamond, M. N. Rosenbluth, F. L. Hinton et al.,17th IAEA Fusion Energy Conference , Yokohama, Japan, 1998 ~InternationalAtomic Energy Agency, Vienna, 1998 !, IAEA-CN-69/TH3/1. 7P. H. Diamond, S. Champeaux, M. Malkov et al.,18th IAEA Fusion En- ergy Conference , Sorrento, Italy, 2000 ~International Atomic Energy Agency, Vienna, 2001 !, IAEA-CN-77/TH2/1. 8Z. Lin, T. S. Hahm, W. W. Lee et al., Science 281, 1835 ~1998!. 9Y. C. Lee, J. Q. Dong, P. N. Guzdar, and C. S. Liu, Phys. Fluids 30,1 3 3 1 ~1987!. 10W. Horton, B. G. Hong, and W. M. Tang, Phys. Fluids 31, 2971 ~1988!.3865 Phys. Plasmas, Vol. 9, No. 9, September 2002 Electromagnetic secondary instabilitie 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: 129.120.242.61 On: Sun, 30 Nov 2014 18:01:1511F. Jenko, W. Dorland, M. Kotschenreuther, and B. N. Rogers, Phys. Plas- mas7,1 9 0 4 ~2000!. 12Y. Idomura, M. Wakatani, and S. Tokuda, Phys. Plasmas 7,3 5 5 1 ~2000!. 13P. N. Guzdar, R. G. Kleva, A. Das, and P. K. Kaw, Phys. Rev. Lett. 87, 015001 ~2001!. 14I. Gruzinov, P. H. Diamond, and A. Smolyakov, Phys. Rev. Lett. ~submit- ted!. 15P. Beyer, S. Benkadda, X. Garbet, and P. H. Diamond, Phys. Rev. Lett. 85, 4892 ~2000!. 16R. Singh, P. K. Kaw, and J. Weiland, 18th IAEA Fusion Energy Confer- ence, Sorrento, Italy, 2000 ~International Atomic Energy Agency, Vienna, 2001!, IAEA-CN-77/TH2/4.17B. Labit and M. Ottaviani, Phys. Plasmas ~submitted !. 18M. N. Rosenbluth and F. L. Hinton, Phys. Rev. Lett. 80,7 2 4 ~1998!. 19S. Childress and A. Gilbert, Stretch, Twist, Fold: The Fast Dynamo ~Springer, Berlin, 1995 !. 20A. B. Rechester and M. N. Rosenbluth, Phys. Rev. Lett. 40,3 8~1978!. 21F. Jenko and W. Dorland, Plasma Phys. Controlled Fusion 43, A141 ~2001!. 22B. N. Rogers, W. Dorland, and M. Kotschenreuther, Phys. Rev. Lett. 85, 5336 ~2000!. 23W. Dorland, F. Jenko, M. Kotschenreuther, and B. N. Rogers, Phys. Rev. Lett.85, 5579 ~2000!.3866 Phys. Plasmas, Vol. 9, No. 9, September 2002 C. Holland and P. H. Diamond 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: 129.120.242.61 On: Sun, 30 Nov 2014 18:01:15
1.414495.pdf	Ahybrid ray –mode (wavefront –resonance) approach for analyzing acoustic radiation and scattering by submergedstructures I-Tai Lu Department of Electrical Engineering, Weber Research Institute, Polytechnic University, Route 110, Farmingdale, New York 11735 ~Received 18 April 1994; revised 10 July 1995; accepted 12 July 1995 ! Thispaperdiscussesahybridray–mode ~wavefront–resonance !approachtoanalyzewaveradiation andscatteringbyﬂuidloadedtargetswithinternalstructures.Theapproachconsistsofthefollowingthree methods: ~1!coupling of plate and shell modes at joints and junctions; ~2!spectral approaches ~such as ray asymptotics, collective rays, guided modes, resonances, ray modes, etc. !for separable and weakly nonseparable structures; ~3!a combination of methods ~1!and~2!for nonseparable structures. The general theory is applied to a prototype structure of revolution consisting of acylindrical pipe, hemispherical endcaps, a bulkhead, and a rib. First a conventional surface ray–normal mode approach is applied to the normal coordinate of each shell element, reducing thesubmerged structure into an equivalent multilayer–multiwave medium in the lateral domain. Amatrix Green’s-function formulation is then employed to systematically synthesize the acousticradiation or scattering returns in terms of angular spectra of surface modes of the structuralelements. This allows efﬁcient bookkeeping of various spectral objects such as ray, collective ray,modes, ray–mode, resonance, etc., to be maintained. © 1995 Acoustical Society of America. PACS numbers: 43.20.Tb, 43.20.Dk, 43.20.Gp, 43.40.Rj INTRODUCTION Wave scattering or radiation from underwater structures is of considerable interest. In the scattering case, acousticwaves incident from the water are partially reﬂected by theouter shell of the vessel. In addition, the incident waves ex-cite elastic vibrations of the internal structures of the vessel,which in turn reradiate into the water.Thus a related problemconsists of radiation into the water due to vibrating sourceswithin the vessel. Therefore, predicting the acoustic ﬁeld inthe scattering and radiation problem requires the ability tomodel and account for the internal structure of the vessel. There are many approaches to tackle this important type of problem. In the low-frequency regime, numerical methodsare obviously the best choice because of their ﬂexibility tomodel arbitrarily complex structures ~e.g., see Refs. 1–3 !. However, numerical approaches alone provide little physicalinsight, as well as becoming very inefﬁcient in the high-frequency regime. When applicable, analytical approachesare best, as they provide physical insight and numerical efﬁ-ciency through alternative representations ~e.g., see Refs. 4–6!. Unfortunately, they are appropriate only for separable canonical structures. In order to improve the modeling ﬂex-ibility, one can employ semianalytical approaches, or applyapproximations to the analytical approaches ~e.g., see Refs. 7–12 !. Some of these methods ~such as variational ap- proaches !work better in the low-frequency regime, while others ~such as asymptotic approximations !can usually be made in the high-frequency regime. To gain a wider appli-cable range, one may combine the numerical algorithms withanalytical solutions ~e.g., see Refs. 13–15 !. This usually re- quires substantial analytical and programming works. Ray-type approaches belong to the category of high-frequency asymptotic approximation. However, a three- dimensional ray approach is not always suitable, especiallyfor structures consisting of thin shell and plate elements be-cause of the existence of multiple length scales. Two recentpublications using three-dimensional rays are cited here inwhich one solves for a solid ~nonshell !structure, 16and the other employs a coarse approximation for thin shellproblems. 17To remedy this difﬁculty one has to realize the fact that wave motion on thin shell/plate elements can bedescribed more conveniently by surface rays of the shellmodes than by three-dimensional rays. Consider the simpli- ﬁed structure in Fig. 1 as an example. When bulk acousticwavelengths are much larger than the thicknesses of variousplate and shell segments that form the decks, bulkheads, andhull, one can approximate the vessel as a collection of platesand shells. ~Wave scattering from small structural details of the vessel are not considered for the moment. !Each plate or shell element supports ~quasi- !ﬂexural, longitudinal, and shear waves. These three fundamental wave species propa-gate independently in an element but couple to one anotherat the boundaries between elements, as well as at other dis-continuities. The analyzing strategy is to divide the vesselinto subregions so that the complexity of the original prob-lem can be reduced greatly. If the wavelengths of plate/shellmodes are smaller than the typical lateral dimensions ofthese plate/shell elements, ray shooting techniques can beused to efﬁciently describe the elastic vibrations of the struc-ture. In this approach, propagation of shell and/or platemodes along the lateral surface of the plate or shell is de-scribed by rays on two-dimensional surfaces. Starting at thesource, which is the result of acoustic waves incident fromthe water onto the outer shell in the scattering case, surfacerays of shell/plate modes are traced until they encounter a 114 114 J. Acoust. Soc. Am. 99(1), January 1996 0001-4966/96/99(1)/114/19/$6.00 © 1996 Acoustical Society of America Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.114.34.22 On: Sat, 22 Nov 2014 00:16:56junction or other scattering sites. The vibrations associated with reﬂected, transmitted, and scattered surface rays of theshell/plate modes can be computed using the junction orscattering properties obtained from an appropriate canonicalproblem. A number of recent papers are devoted to this kind of surface ray–shell mode development. Originally, theseworks were performed on separable structures, 18–28and then extended to include simple internal loading29–36and weakly nonseparable structures.37–39A general ray procedure for more complicated structures should consist of the followingthree relevant ingredients: ~1!plate and/or shell modes cou- pling at junctions and discontinuities, ~2!spectral techniques and alternative representations ~ray/mode !for solving radia- tion and scattering problems with separable structures andweakly nonseparable structures, and ~3!a combination of elements ~1!and~2!to construct a novel ray–mode approach for nonseparable structures where the problem complexity isreduced by using plate and/or shell modes. Extending thetime-harmonic results for transient excitations, one may sim-ply apply the FFT to the time-harmonic responses for syn-thesizing time-domain results. This is a progressing descrip-tion where the transient wave phenomena can be organizedin terms of wavefronts. These wavefronts travel from sourceto the scatterer, and then interact with the scatterer to gener-ate new wavefronts which travel to the sensor. Each wave-front carries the local information of the medium along thecorresponding ray trajectory. The main advantage of the ray-type approach is that it provides some physical insight of the energy ﬂow mecha-nism for the computed results.This insight can be used in theplanning of experiments, design, and the interpretation ofpractical measurements. Consider data inversion as an ex-ample. If some of the arriving wavefronts can be separatedfrom the measured data, one may be able to identify someproperties of the corresponding local scattering centers byusing the arrival times, amplitudes, and/or waveshapes ofthese waveforms ~e.g., see Ref. 40 !. However, when source- excited wave phenomena are resolved into ray constituents,wave species coupling at interfaces and boundaries generatea proliferation of wave ﬁelds ~in both time-harmonic and time-dependent domains !. This makes successive tracking ~even on the reduced two-dimensional structural surface !im- practical for all but a few such encounters. This is the mainobstacle for applying the ray-type approaches to general structures in practice. Moreover, since the wavefront descrip-tion is highly dependent on the arrangement of sources andreceivers, and is also sensitive to the details of the scatterer,it is difﬁcult to extract the global features ~independent of the excitation and receiving mechanisms !of the scatterer. An alternative approach is to model the wave prolifera- tion processes in terms of composite reﬂection/transmissionor transverse resonance which emphasizes propagationacross or along the junctions ~interfaces or joints !, respec- tively. In the oscillatory description of transient responses,the transverse resonance formulation is employed for all~three!coordinates. The wave phenomena are then synthe- sized by resonances of the structure as a whole ~e.g., see Refs. 41–47 !. These global resonances take the form of damped sinusoidal responses, characterized by complexresonant frequencies. While these frequencies are totally in-dependent of sources and sensors, the excitation amplitudesof these resonances are determined by the temporal and con-ﬁgurational spectra and the locations of sources. Some of theresonant frequencies ~usually, those with smaller damping terms !can be extracted out from the received signals by signal processing techniques and, therefore, can serve as thesignatures of the scatterer for target classiﬁcation. However,one loses the possibility of tracking individual wave ﬁelds inthis approach. The progressing and resonant approaches complement one another. In the time-harmonic domain, the spectral inter-vals sparsely ﬁlled by progressing ~ray!constituents will be densely ﬁlled by transverse resonance ~mode !constituents, and vice versa. In the time-dependent domain, the progress-ing~wavefront !description is effective in representing the early time arrivals because of time causality, but the oscilla-tory~global resonance !description is convenient for describ- ing the late arrivals when the medium has fully responded tothe excitation. With respect to data acquisition and interpre-tation, both arrival times ~of wavefronts !and resonant fre- quencies ~of oscillatory waves !are good physical observ- ables, in which the former contains information on the localscattering centers and the latter has information on the globalstructure. Therefore, it is desirable to combine the wavefrontand resonance descriptions of transient responses for achiev-ing numerical efﬁciency and for providing physical insight.This kind of hybrid formulation is essential especially forinversion problems such as identiﬁcation and classiﬁcation.A lot of work on hybrid wavefront–resonance approacheshas been done for canonical problems ~e.g., see Ref. 48 !, but not for nonseparable structures with internal loading ~to the best of our knowledge !. In this paper, our emphasis is on setting up a uniﬁed framework for the above-mentioned ray–mode approach for general nonseparable structures of revolution and on extend-ing the ray–mode approach in the time-harmonic case to thewavefront–resonance approach for transient response. By us-ing a ﬂexible arrangement, we retain some traveling ﬁeldsand account for the remaining ones collectively through theuse of modiﬁed collective reﬂection and transmission matri-ces, and/or through the use of some resonant modes. Further-more, we avoid detrimental effects of the proliferation of FIG. 1. Ray tracing in a submerged vessel composed of thin plates and shells. 115 115 J. Acoust. Soc. Am., Vol. 99, No. 1, January 1996 I-Tai Lu: Ray –mode radiation and scattering Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.114.34.22 On: Sat, 22 Nov 2014 00:16:56traveling-wave constituents by devising a wave object which is composed of a superposition of all relevant wave speciesin deﬁnite proportions. These proportions are maintained af-ter multiple complete reverberations, the only modiﬁcationbeing an adjustment of its overall amplitude. These ideashave been developed in our previous works 49,50dealing with a multiwave, multilayer media subject to simultaneous exci-tation and detection at several arbitrary locations.The formu-lation is structured so as to permit access to all interestingsubregions ~e.g., those containing a source or receiver, or causing a signiﬁcant change of ray phases !but to treat the remaining uninteresting regions in hidden form. This pro-vides conceptual and analytical economy by focusing atten-tion only on those ﬁeld variables that take part in wave pro-cesses which are considered important. Crucial in thisdevelopment is a proper ordering of these variables into twoarray wave vectors. A special objective of this ordering is tomake the formulation commensurate with the hybrid ray–mode scheme. This leads to a matrix Green’s function thatdiffers from those developed for the multilayer problem. 51–56 Various new spectral objects ~eigenreverberation, eigenray, and eigenmode !, theories ~ray/mode equivalent and eigenray/ eigenmode equivalent !, and computational algorithms ~col- lective ray and hybrid ray–mode method !have been derived from this formulation to furnish numerical advantages and toprovide physical insight. The computational aspects havealso been explored by numerical studies on wave propaga-tion in an elastic layer for both time-harmonic 57and transient58line source excitations. The matrix and alternative representations have also been extended to incorporatebeam-type source excitation, and to allow inhomogeneouslayers for both forward modeling and data inversion. 59,60The validity of the hybrid algorithm has been conﬁrmed, and pa-rameter regimes have been found wherein the hybrid ap-proach offers a competitive alternative to other options. In Sec. I, we consider the formulation of the problem in general separable curvilinear coordinates. This leads to ageneral representation of the Green’s function for the pres-sure ﬁeld in the surrounding ﬂuid as complex spectral inte-grals over products of individual one-dimensional character-istic Green’s functions. 61,62Alternative representations derived from this general form lead to spectral objects thatcan be interpreted in terms of basic wave phenomena includ-ing rays, modes, collective rays, ray–mode, wavefront–resonance, etc. The solutions are generally in the form oftranscendental functions. For ease of computation as well asphysical interpretation of the wave phenomena described bythese functions, it is desirable to explore asymptotic tech-niques emphasizing ‘‘high frequency.’’ Rather than dealingwith the transcendental functions associated with the variousseparable coordinate systems, we will discuss in Sec. II a systematic asymptotic procedure for approximating the dif-ferential equations directly. 61,62Then, one can obtain the WKB-type solution ~when the source or receiver is far away from turning points !, theAiry-type solution ~when it is near a turning point !, or other types of approximate solutions.61 This procedure can be applied to separable conﬁgurations without dealing with the corresponding transcendental func-tions. Most importantly, the systematic ray–mode approachis further extended to analyze nonseparable structures.As an example, a nonseparable structure of revolution is solved bya quasiseparable approach. In Sec. III, the previously devel-oped matrix Green’s function is extended to solve the re-duced nonuniform ~with joints and discontinuities !one- dimensional problem ~in the vdomain !of the nonseparable structure of revolution.Alternative representations emphasiz-ing vector rays, modes, eigenrays, collective rays, and hybridforms, etc., for the multiple wave species in the vdomain are brieﬂy summarized in Sec. IV.The combination of the hybridvector ray–lateral mode form in the lateral domains ~contain- ing different shell elements !with the conventional hybrid lateral ray–surface mode approach in the vertical coordinates~normal to the surfaces of their corresponding shell elements ! is totally novel. Summary and discussion are made in Sec. V.In theAppendix, the results are extended to weakly nonsepa-rable structures using an adiabatic approach. 63Numerical ex- amples of typical ray trajectories of structural modes, traveltimes of wavefronts, and complex frequencies of resonanceshave been computed using the approach outlined in this pa-per. However, due to the extensive scope of the problem,they will be published elsewhere. I. RAY–MODE (WAVEFRONT–RESONANCE) METHODS FOR SEPARABLE STRUCTURES This section presents an overview of various alternative ray–mode and wavefront–resonance methods in the time-harmonic and time-dependent domains for separable struc-tures. These methods are based on rigorous spectral tech-niques for separable problems. 61,62With no loss of generality, consider a scattering problem where both sourceand receiver are in the surrounding ﬂuid. Because of separa-bility, the time-dependent results can be obtained from thetime-harmonic solutions and the multidimensional resultscan be derived from the constituent multiple one-dimensional solutions. Since each one-dimensional solutionsustains various options, the combination of multiple one-dimensional solutions provides a variety of options for alter-native representations of the multidimensional results. We formulate the problem in terms of a time-harmonic Green’s-function problem in three dimensions, governed by ~¹21kf2!G~r,r8!52d~r,r8!, ~1! wherekfis the wave number of the surrounding ﬂuid of the structure, and r5(u,v,w) andr85(u8,v8,w8) are the three- dimensional orthogonal coordinates of the receiver andsource, respectively. By applying the Fourier transform to thetime-harmonic solutions, additional options of alternativerepresentations of the time-dependent results will be gener-ated due to the existence of different schemes for evaluatingthe Fourier transform. A. One-dimensional characteristic Green’s function The typical one-dimensional time-harmonic Green’s- function problem is summarized below. It yields representa-tion theorems in integral or eigenfunction form. The charac-teristic Green’s function is deﬁned as follows: 116 116 J. Acoust. Soc. Am., Vol. 99, No. 1, January 1996 I-Tai Lu: Ray –mode radiation and scattering Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.114.34.22 On: Sat, 22 Nov 2014 00:16:56Sd dxp~x!d dx1zq~x!1lxt~x!Dgx~x,x8;z;lx! 52d~x2x8! ~2! with boundary conditions Sp~x!d dx1a1,2Dgx~x,x8;z;lx!50,x5x1,2, ~3! where zis a ﬁxed parameter and lxis the characteristic spec- tral variable whose domain in the complex lxplane is re- stricted so as to ensure a unique solution for gx.The variable xstands for any of the individual ( u,v,w) coordinates that appear in the three-dimensional problem. Let gQxandgWxbe two independent solutions of ~2!which satisfy the boundary conditions ~3!atx1andx2, respectively. Then, gx5gQx~x,!gWx~x.!@2pW~gQx,gWx!#21, ~4! where the Wronskian is W~gQx,gWx!5gQxdgWx dx2dgQx dxgWx. ~5! In~4!,x,andx.represent the smaller and the greater, re- spectively, of x8andx. The corresponding eigenvalue prob- lem is Sd dxp~x!d dx1zq~x!1lxmt~x!Dfxm~x,z!50, ~6! with the same boundary condition as those on gxin~3!. Here, fxmis the eigenfunction and lxmthe eigenvalue. The completeness relation for the one-dimensional xdo- main can be stated in the form of a weighted delta functionrepresentation that involves a general spectral integration, d~x2x8! t~x8!521 2piR Cxgx~x,x8;z;lx!dlx. ~7! The contour Cxin~7!encircles all singularities ~poles at lxm and/or branch points !ofgxin the complex lxplane in the positive sense. The discrete and/or continuous eigenfunctionexpansion in ~7!is obtained therefrom by residue and/or branch cut evaluation. Formally, ~7!can be represented alter- natively in terms of an orthonormal eigenfunction expansion~with f¯xmdenoting the adjoint eigenfunction !: d~x2x8! t~x8!5( mgxm,gxm5fxm~x!f¯xm~x8!. ~8! In a discrete eigenfunction expansion, gxmis the residue of themth eigenvalue: gxm[limlx!lxm@2(lx2lxm)gx#. With regard to the continuous eigenfunction expansion, the discrete summation index in ~8!is replaced by a continuous ~integration !index and gxm5gx, and the integration contour is along the branch cuts. When there is a resonant denominator with pole singu- larities in gx, whose zeros at the eigenvalues lxmgenerate the spectral poles, it is possible to derive an alternative seriesbased on a power-series expansion of the resonant denomi-nators:g x5( jgx~j!. ~9! For example, if theWronskian Win~5!contains the resonant denominator, it can be expressed as W5H~12F!,H[gQxdgWx dx,F[121 HdgQx dxgWx, ~10! where the factor ~12F!accounts for the resonances. In other words,Fis the reverberation factor accounting for the phase and amplitude changes of a fundamental wave travelingthrough a round trip between the two boundaries x 1andx2. Then the element gx(j)in~9!can be written as gx~j!521 pHgQx~x,!gWx~x.!Fj. ~11! Substituting ~9!and~11!into~7!, one has a sum of spectral integrals. Each integral with an index jcan be viewed as the entirety of various generalized ray species with jreverbera- tion. Instead of using a ray expansion as in ~9!, it is some- times more desirable to have a partial ray sum plus a remain-der, g x5( j50J21 gx~j!1Rx~J!,Rx~J![gxFJ5gx~J! 12F, ~12! in which the remainder Rx(J)still contains the resonant de- nominator ~12F!, and hence all the spectral poles lxm. Note that the generalized ray term gx(j)is given in ~11!. Substitut- ing~12!into~7!, one has a partial sum of generalized rays and a remainder integral which can be used to generate ahybrid ray–mode representation in a multidimensionalGreen’s-function formulation. Thus the completeness rela-tion affords alternative options that emphasize the elemen-tary solutions with mode @standing wave as in ~8!#and ray @traveling wave as in ~9!#features. Rigorously constructed hybrid combinations using some of each are likewise pos-sible by using ~12!. For given ranges of physical parameters, the choice can be made so as to achieve favorable overallconvergence and a cogent description of propagation phe-nomenology. In the progressing description of a transient response, the Fourier transform of g x(j)in~11!gives the progressing wavefront traveling from source to receiver with jreverbera- tions between the two boundaries. The Fourier transform ofR x(J)in~12!gives the collective contribution from wavefronts traveling from source to receiver with Jor more reverbera- tions between the two boundaries. The residue evaluation ofthe Fourier transform ~temporal frequency integral !ofg xin ~4!orRx(J)in~12!yields the one-dimensional resonances gˆxm[lim vx!vxmi~vx2vxm!hs~v!exp@ivxmt#, h5gxorRx~J!, ~13! where vxmis themth resonant frequency and s~v!is the source spectrum. Thus a hybrid wavefront–resonant formu-lation can be obtained by applying the Fourier transform to~12!and deforming the integration contour of R x(J)into the 117 117 J. Acoust. Soc. Am., Vol. 99, No. 1, January 1996 I-Tai Lu: Ray –mode radiation and scattering Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.114.34.22 On: Sat, 22 Nov 2014 00:16:56steepest descent path.The sum of gx(j)integrals gives the ﬁrst jwavefronts, the enclosed residues of Rx(J)integral represent the late time resonances, and the asymptotics of the de-formed integration contour of R x(J)integral account for a re- mainder. B. Three-dimensional Green’s functions The three-dimensional Green’s function can be synthe- sized in terms of the one-dimensional characteristic solu-tions: G ~r,r8!51 ~2pi!2R CwR Cvgu~u,u8;lu!gv~v,v8;lv! 3gw~w,w8;lw!dlvdlw. ~14! The contour Cw(Cv) in the complex lw(lv) plane encloses in the positive sense all singularities of gw(gv) but no others. Additional singularities in the lworlvplanes arise from g(u,u8;lu). The functional dependence lu5lu(lv,lw)o r luonlvandlwis governed by the nature of the coordinate representation in the ( u,v,w) domains. Depending on the mode, ray, or ray–mode feature used to describe the wave motion in each coordinate, Eq. ~8!,~9!, or~12!, respectively, can be employed to expand the corre- sponding one-dimensional Green’s function gx,x5u,v,w, in~14!. This general procedure can provide many alternative representations for the three-dimensional Green’s function,and the details may be found in the literature. 61Here only three basic options are shown. At ﬁrst, we represent theGreen’s function Gin a general format as a sum of some basic wave objects G (jmn): G~r,r8!5( ~jmn!G~jmn!~r,r8!. ~15! If one is interested in a generalized ray approach, ~9!is em- ployed for expanding all of the three one-dimensionalGreen’s functions in ~14!, where the traveling-wave feature is emphasized in all three coordinates. Then, the basic waveobject in ~15!is a three-dimensional generalized ray: G ~jmn!~r,r8!51 ~2pi!2R CwR Cvgu~j!~u,u8;lu! 3gv~m!~v,v8;lv!gw~n!~w,w8;lw!dlvdlw. ~16a! If one uses the standing-wave feature to represent the wave motion in one coordinate ~say,u!, and uses the traveling- wave feature for the remaining two coordinates ~say,vand w!, then the basic wave object in ~15!becomes G~jmn!~r,r8!521 2pifuj~u!f¯uj~u8! 3R Cwgv~m!~v,v8,lvj!gw~n!~w,w8,lw!dlw. ~16b! Similarly, one can use the standing-wave feature to represent the wave motion in two coordinates ~say,uandv!, and usethe traveling-wave feature for the remaining coordinate. Then, the basic wave object in ~15!becomes G~jmn!~r,r8!5fuj~u!f¯uj~u8!fvm~v!f¯vm~v8! 3gw~n!~w,w8,lwjm!. ~16c! In the context of the speciﬁc conﬁgurations to be treated later on, we are particularly interested in the wave objects inthe form of ~16b!, which can be used for a ~structural mode !–~lateral ray !description of the ﬂows of acoustic en- ergy along ray trajectories in the surrounding ﬂuid and on thesurface of thin shells. However, ~15!with the element de- ﬁned in ~16b!is not an efﬁcient representation when the receiver or the source is in far zones. This is due to the factthat the surrounding ﬂuid is unbounded, requiring the use ofthe continuous eigenfunction expansion. To remedy thisproblem, we combine both the ray elements in ~16a!and the mode elements in ~16b!self-consistently to represent the to- tal ﬁeld.Applying the partial ray expansion ~12!to represent g u, whereudenotes the coordinate normal to the structure surface, and the ray expansion ~9!to represent both gvand gw, where (v,w) denote the remaining two lateral coordi- nates, one has G~r,r8!5( ~mn!( j51J21 G~jmn!~r,r8!1( ~mn!R~Jmn!~r,r8!. ~17! The element G(jmn)is a generalized ray integral deﬁned in ~16a!, and the remainder, R~Jmn!~r,r8!51 ~2pi!2R CwR CvRu~J!~u,u8;lu! 3gv~m!~v,v8;lv!gw~n!~w,w8;lw!dlvdlw, ~18! can be evaluated by deforming the integration contours to the steepest descent paths ~sdp!. The sdp results can be viewed as collective rays ~see Ref. 57 !, and the residue contributions from the spectral poles intercepted during the contour defor-mation give the ~structural mode !–~lateral ray !terms. The above-mentioned procedure for a cylindrical shell and aspherical shell has been published in Refs. 21 and 22 andRefs. 24 and 28, respectively. Here, our emphasis is on set-ting up a framework for the uniﬁed and systematic approachfor general separable structures to be discussed in the nextsection, which will be further extended for analyzing non-separable structures. In the progressing description of a transient response, the frequency Fourier transform of G (jmn)in~16a!gives the three-dimensional progressing wavefront traveling fromsource to receiver with j,m, andnreverberations between the two boundaries in the u, v, andwcoordinates, respec- tively. The frequency Fourier transform of G(jmn)in~16b! gives the one-dimensional mode ~transverse resonance !in theucoordinate, and two-dimensional progressing wave- front traveling from source to receiver with mandnrever- berations between the two boundaries in the vandwcoor- dinates, respectively. The frequency Fourier transform ofG (jmn)in~16c!gives the two-dimensional mode ~transverse 118 118 J. Acoust. Soc. Am., Vol. 99, No. 1, January 1996 I-Tai Lu: Ray –mode radiation and scattering Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.114.34.22 On: Sat, 22 Nov 2014 00:16:56resonance !in theuandvcoordinates, and the one- dimensional progressing wavefront traveling from source toreceiver with nreverberations between the two boundaries in thewcoordinates. The frequency Fourier transform of R (Jmn)in~18!gives the collective contribution from wave- fronts traveling from source to receiver with Jor more re- verberations between the two boundaries in the ucoordinate, and with mandnreverberations between the two boundaries in thevandwcoordinates, respectively. The residue evalu- ation of the frequency Fourier transform of R(Jmn)in~18! cannot yield the global ~three-dimensional !resonances be- cause of the traveling-wave representations in the vandw domains. To yield the global resonances, we use standing-wave forms in all three coordinates. This can be achieved byemploying the residue evaluation for the two spatial spectraltransforms and the frequency Fourier transform of Gin~14!. Thus, to obtain the hybrid wavefront–resonant formulationin the transient time domain, we have to apply an alternativepresentation other than the one in ~17!and~18!for the time- harmonic responses. We ﬁrst apply partial ray expansion in~12!to all three coordinates and yield G ~r,r8!5( n51N21 ( m51M21 ( j51J21 G~jmn!~r,r8!1R~JMN !~r,r8!. ~19! The element G(jmn)is a generalized ray integral deﬁned in ~16a!, and the new remainder, R~JMN !~r,r8!51 ~2pi!2R CwR CvRu~J!~u,u8;lu! 3Rv~M!~v,v8;lv!Rw~N!~w,w8;lw!dlvdlw, ~20! can be evaluated by deforming the integration contours to the steepest descent paths ~sdp!. Second, we apply the frequency Fourier transform to ~19!and deform the two spatial and one frequency spectral integration contours of R(JMN)into their corresponding steepest descent paths. The sum of G(jmn)in- tegrals gives the early arriving wavefronts, the enclosed resi-dues ofR (JMN)integral represent the late time global ~three- dimensional !resonances, and the asymptotics of integrations along the deformed contours of R(JMN)integrals account for the remainder. II. UNIFIED ASYMPTOTIC APPROACH The solutions of the homogeneous equation ~2!, which synthesize the characteristic Green’s function gxin~4!, are generally in the form of transcendental functions. For ease ofcomputation as well as physical interpretation of the wavephenomena described by these functions, it is desirable toexplore asymptotic techniques emphasizing high frequency.Here, we will discuss a systematic procedure which can beapplied to any separable conﬁgurations without dealing di-rectly with the corresponding transcendental functions.Then,this procedure is further extended to handle nonseparablestructures. Finally, as an example, a nonseparable structureof revolution is discussed in detail.A. One-dimensional characteristic function A uniﬁed asymptotic approach for all separable prob- lems can be done systematically by ﬁrst expressing Eq. ~2!in standard form via the transformation61,62 gx5fx/Ap~x!, ~21! which leads to Sd2 dx21gx2~x!Dfx521 Ap~x8!d~x2x8!, ~22! where the square of the equivalent wave number gxin thex coordinate is gx2~x!5zq~x!1lxt~x! p~x!21 Ap~x!d2 dx2Ap~x!. ~23! Similarly, the boundary conditions in ~3!are reduced to Sp~x!d dx1a1,221 2d dxp~x!Dfx50,x5x1,2.~24! For large values of the parameters zand~or!lxappearing in gx(x), one may employ asymptotic procedures to approxi- matefx. These depend critically on whether the coordinate variables ~x,x8!are located near the zeros ~if any !ofgx(x). Away from such zeros, fxcan be expressed by standard propagating or decaying WKB local plane-wave functions.In this case, the two independent solutions of ~22!are fQx~x!'expS2iE xt1x gx~t!dtD 1Gx1expSiE xt1x gx~t!dtD, ~25! fWx~x!'expS2iE xxt2gx~t!dtD 1Gx2expSiE xxt2gx~t!dtD, xt1<x<xt2, wherext1(xt2) represents the lower ~upper !turning point if gx(xt1)50 andx1<xt1<x@gx(xt2)50 andx<xt2<x2#. Otherwise xt1(xt2) represents the boundary point x1(x2). The reﬂection coefﬁcient Gx1(Gx2) is chosen to satisfy the turning point condition when the corresponding turning pointexists, or to satisfy the boundary condition ~24!atx 1(x2)i n the absence of a turning point. If xt1(xt2) reaches 2`~1`! @and therefore no reﬂection occurs at xt1(xt2)#, one can sim- ply set Gx1(Gx2) to zero, satisfying the radiation condition. Note that if urepresents the coordinate normal to the struc- ture surface, then Gx1contains the information on the struc- ture. By following ~4!and~5!and employing ~21!, the non- uniform asymptotic solution of ~2!can be written as 119 119 J. Acoust. Soc. Am., Vol. 99, No. 1, January 1996 I-Tai Lu: Ray –mode radiation and scattering Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.114.34.22 On: Sat, 22 Nov 2014 00:16:56gx~x,x8!'CxfQx~x,!fWx~x.!exp$iLx%$12Fx%21, Fx5Gx1Gx2exp$2iLx%,Lx5E xt1xt2gx~t!dt, ~26! Cx521 2igxAp~x!p~x8!. If both reﬂection coefﬁcients are not zero, the factor 1 2Fx accounts for the resonances.The ray expansion in ~9!and the partial ray expansion in ~12!are appropriate expressions. If both reﬂection coefﬁcients are zero, the reverberation factorF x50. Thus only one generalized ray exists and the expres- sions in ~9!and~12!are no longer relevant. Note that the reverberation factor Fxremains zero when only one reﬂec- tion coefﬁcient is zero. If the remaining nonzero reﬂectioncoefﬁcient does not contain a resonant denominator, there aretwo generalized rays @see~25!#and the ray expansion in ~9! is still applicable in a general sense. If the remaining nonzeroreﬂection coefﬁcient does contain a resonant denominator,there are one generalized ray and one collective ray. Bothoptions @ray expansion in ~9!and partial ray expansion in ~12!#are still applicable to the collective ray. These different possibilities deﬁne various ray species and are summarizedin Table I. A uniform asymptotic solution in the transitionregion near the turning point can be expressed in terms ofAiry functions ~see Refs. 61 and 62 !. When there are mul- tiple turning points, the detailed behavior of uniform ap-proximation depends on whether the turning points are iso-lated or clustered. Details of the general procedure may befound in the literature. 61B. Three-dimensional Green’s functions To simplify the notations, we use the superscript of (l) to represent the multiple superscripts (jmn), (Jmn), and (JMN)i n ~15!–~20!and thesubscripts x ain Table I. A typical generalized ray integral in ~16a!–~c!and a remainder integral in ~18!can always be expressed as FG~l! R~l!G5R CvR CwFA~l!exp~iw~l!! B~l! Dexp~ic~l!!Gdlvlw, ~27! where the integrand has been grouped so that A(l)andB(l) are slowly varying amplitudes, w(l)andc(l)represent rapidly varying phases, and the factor Daccounts for the resonant denominator. Here, A(l)andB(l)/Dare functions incorporat- ing dependence on the excitation, reﬂection coefﬁcients, etc.,while w(l)is of the form w~l!orc~l!5( x5u,v,wE Txgx~x!dx, ~28! whereTxis the projection of all segments of the ray trace on thexcoordinate, x5u,v,w. In the high-frequency regime, if the phase w(l)has a simple stationary point ( lvs,lws) deﬁned by ]w~l! ]lv50,]w~l! ]lw50a t lv5lvs,lw5lws ~29! and the function A(l)is regular near the stationary point (lvs,lws), then the ray integral G(l)in~27!can be approxi- mated asymptotically:61 G~l!'A~l!~lvs,lws!expFiw~l!~lvs,lws! 1ip 4( i512 sgn~di!G2p Audet~M!u, ~30! M[F]2w~l! ]lvs2]2w~l! ]lvs]lws ]2w~l! ]lvs]lws]2w~l! ]lws2G, wherediare the eigenvalues of the matrix M. If several stationary points exist, each contributes in the same manneras in ~30!if these saddle points are far from one another. Under the same condition, the remainder integral in ~27! can also be evaluated asymptotically: 49,57,58 R~l!'Rsdp~l!1( resRres~l!, ~31! where the steepest descent path contribution is very similar to that in ~30!of the ray integral Rsdp~l!'B~l!~lvs,lws! D~lvs,lws!expFc~l!~lvs,lws! 1ip 4( i512 sgn~di!G2p Audet~M!u, ~32!TABLE I. Ray species gx(j)andRx(J)in~12!. Gx150 gx(j)50i fjÞ0;Rx(J)50 Gx250 gx~0!5CxexpiUE xx8 gx~t!dtU Gx1Þ0 gx(j)50, ifjÞ0 Gx250 gx~0!5CxexpiUE xx8 gx~t!dtU Rx~J!5CxGx1expiHE xt1x 1E xt2x8Jgx~t!dt Gx1Þ0 Fgx~j! Rx~J!G5( a514 Fgx2~j! Rxa~J!G5( a514 CxFFj FJ 12FGDa Gx2Þ0Da55expiUE xx8 gx~t!dtU,a51 Gu1expiHE xt1x 1E xt1x8Jgx~t!dt,a52 Gu2expiHE xxt2 1E x8xt2Jgx~t!dt,a53 Gu1Gu2expiHLx2UE xx8 gx~t!dtUJ,a54 120 120 J. Acoust. Soc. Am., Vol. 99, No. 1, January 1996 I-Tai Lu: Ray –mode radiation and scattering Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.114.34.22 On: Sat, 22 Nov 2014 00:16:56M[F]2c~l! ]lvs2]2c~l! ]lvs]lws ]2c~l! ]lvs]lws]2c~l! ]lws2G, if the function B(l)/Dis regular near the simple stationary point ( lvs,lws) deﬁned in ~29!. The residue contributions from the spectral poles lvpintercepted during the contour deformation of the original contour Cvto its corresponding steepest descent path can be expressed as Rres~l!'2piB~l!~lvp,lws! ]D~lvp,lws!/]lvpexpFic~l!~lvp,lws! 1ip 4sgn~d!GA2p udu, ~33! d5]2c~l! ]lws2,D~lvp,lws!50,]c~l! ]lws50, where the second spectral integral with respect to lwis evaluated asymptotically by a stationary phase approxima-tion at the stationary point l ws. The pole lvpdetermines the wave numbers of the fundamental modes of the structure.Here, contributions from branch cuts are not shown. Uniformasymptotic solutions in the transition regions, when station-ary points, poles, or branch points are clustered, can be foundin the literature. 61 C. Surface ray–structural mode interpretation Although the Green’s function is represented formally in multiple sums in ~15!,~17!,o r~19!, usually only several terms are required for a practical calculation. For a structure~with a separable conﬁguration !submerged in an unbounded ﬂuid, only two saddle-point contributions and several resi-dues need to be computed for far-zone scattering returns.This is due to the following two reasons. First, the normalcoordinate ucan reach inﬁnity and G u2in~25!and~26!is zero~see Table I !. Therefore, the index Jin~17!or~19!is reduced to 1, and the jsum is deleted. Second, the terms related to nonzero morn, if they exist, do not have real saddle points deﬁned in ~29!or~32!, and their contributions are negligible. Each term in the form of ~30!or~32!can be compre- hended in terms of a rayinterpretation. The two saddle-point contributions account for the direct ray from source to re-ceiver and the specular reﬂection from the structure, respec-tively. The residue contributions in ~33!are related to the guided modes on the structure, which account for waves in the structure excited by the incident wave and reradiateacoustic energy back to the surrounding ﬂuid while propa-gating along the lateral surface of the structure. These resi-dues correspond to the poles of the reﬂection coefﬁcient G u1 at the structure surface. These poles determine the wave numbers ~denoted by km!of guided modes along the lateral dimensions ( u,v). Depending on the relation between the real parts of these wave numbers and the wave number kfin the surrounding ﬂuid, the wave mechanisms can be further classiﬁed intothree categories. When Re( km),kf, the waves are propagat- ing in the udirection ~normal to the structure surface !in the ﬂuid and are called leaky modes. A leaky mode is excitedwhen the projection of the wave number of the incident waveon the lateral surface is the same as the corresponding k m. By reciprocity, this mode sheds energy back into the ﬂuid viathe same phase matching condition, with the angle of radia-tion or excitation given by u5sin21Re(km)/kf. When Re(km).kf, the waves are evanescent in the udirection in the ﬂuid and are called trapped modes. In this case, uis complex, implying that the excitation and reradiation oftrapped modes are through the mechanism of evanescent tun-neling. If either one or both the source and the receiver arefar away from the structure, the trapped modes will radiatevery little and be weakly excited. When Re( k m)5kf, the waves are called creeping modes.Acreeping wave is excitedor shedding energy when the incident wave or radiatingwave, respectively, in the ﬂuid is tangent to the surface of thestructure, i.e., u590°. These waves have the velocity of the ﬂuid and carry no information about the structure. The leaky waves are of particular interest in our consid- eration here for the far-zone scattering from submergedstructures. Equation ~17!can then be rewritten in the follow- ing form: G'G dir1Gref1( mGm, ~34! where the direct ray Gdirand the reﬂected ray Gretare in the forms of ~30!and~32!, respectively. The sum in ~34!is over all non-negligible leaky mode contributions Gmwhich are in the form of ~33!. A schematic drawing of these spectral ob- jects is shown in Fig. 2. In the traditional convention, the time-harmonic re- sponse in ~34!is a hybrid ray–mode representation ~Gdirand Grefare rays and Gmis a mode !. But in the structure acous- tics practice, Gmis also called a ray. A more accurate name ofGmwould be surface ray–shell mode. To obtain the time- domain responses, we can apply the frequency Fourier trans-form to ~30!,~32!, and ~33!. For a narrow-band signal, the phases in ~30!,~32!, and ~33!can be considered as a linear function of frequency; then the corresponding travel times ofthese wave objects are w(l)(lvs,lws)/v,c(l)(lvs,lws)/v, andc(l)(lvp,lws)/v, respectively.To obtain the global reso- FIG. 2. The ray trajectories of a direct ray, reﬂected ray, a leaky mode, and a mode diffracted ﬁeld. 121 121 J. Acoust. Soc. Am., Vol. 99, No. 1, January 1996 I-Tai Lu: Ray –mode radiation and scattering Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.114.34.22 On: Sat, 22 Nov 2014 00:16:56nances, both spatial spectral integrals in ~27!and the tempo- ral spectral integral for the remainder R(l)will be evaluated by residue theorem. The global resonance can then be ap-proximated by Rˆ res~l!'~2pi!3B~l!~lvp,lwp! ]3D~lvp,lwp,vp!/]lvp]lwp]vp 3exp@ic~l!~lvp,lwp!2ivpt#, ~35!D~lvp,lwp,vp!50, where vpis the resonant frequency and lvpandlvpare the modal eigenvalues. The resonant condition D50i n~35!can also be approximated as Gx1Gx2exp~iLx!51,x5u,v,w. ~36! D. Extension to nonseparable structures The spectral integrals in ~14!,~16a!–~c!,~18!, and ~20! are valid only when the structures are separable. But theapproximate results in ~30!,~32!, and ~33!can be extended to nonseparable problems ~see the ray trajectories in Fig. 1 !. Thus the procedures discussed above serve as the frameworkfor a general ray–mode ~wavefront–resonance !approach for analyzing nonseparable structures. This kind of procedureshould include the following three key steps: ~1!preprocess- ing of canonical propagation and scattering problems such asplate/shell modes, excitation and detaching of leaky modes,junction coupling and diffraction, 29,64etc., ~2!computer modeling of a vessel by two-dimensional surfaces, and ~3! ray shooting or eigenray search. Most of the canonical topicsin the ﬁrst step have been addressed in this paper except forthe case with small obstacles. It poses no difﬁculty in theray–mode approach to include these obstacles in the struc-ture models as long as their scattering coefﬁcients areknown, which can be derived by various analytic, asymp-totic, numerical, or hybrid methods. 13–15 Upon completion of the preprocessing, the next step is to establish a computer data model to describe various two-dimensional surfaces for structure element representations~see Fig. 1 !. Their boundaries or interfaces represent trunca- tions or joints, respectively. Upon completion for the ﬁrstand second steps, one is ready to apply a typical ray shootingor searching technique to obtain various physical parameterssuch as scattered ﬁelds, energy ﬂow trajectories, travel times,resonances, etc.Atypical ray shooting procedure consists thefollowing four steps: ~a!emanation from the source and propagation in the ﬂuid; ~b!coupling into the structure through the phase matching mechanism; ~c!propagation in structure and coupling at joints; ~d!reradiation into the sur- rounding ﬂuid through phase matching. The ray traces insteps ~a!and~d!are governed by the stationary phase con- dition in ~29!, and those on structure surfaces in step ~c!are dictated by the stationary phase condition in ~33!. The phase matching scheme in steps ~b!and~d!is determined by the resonant condition in ~33!. The scattering direction by a line junction is determined by a Snell’s-type condition. 29Regard- ing the ray magnitude, the geometric spread in ﬂuid andstructure elements is governed by the square-root terms in~29!and~33!, respectively. The excitation and detaching co-efﬁcients are determined by the residue term B (l)/(]D/]lv). The coupling coefﬁcients at joints can be obtained by follow-ing the procedure in Ref. 29. One general approach forsearching eigenrays is to repeat the ray shooting procedureuntil the ray hits the receiver. An alternative approach con-sists of the following ﬁve steps: ~1!Divide the ray into seg- ments where each segment can be governed by wave equa-tions in a canonical geometry; ~2!represent each ray segment in terms of a spectral sum of fundamental wave constituents;~3!match the wave constituents of ray segments at bound- aries; ~4!sum up the phase terms of all ray segments as the total phase @see Eq. ~28!#;~5!the stationary phase condition gives the eigenray solution. This can be considered apiecewise-separable approach. E. Nonseparable structures of revolution In this section, we consider a simpler structure with ro- tational symmetry where the functional dependence of wavemotions on the azimuthal coordinate wis separable from those on the other two coordinates uand v. In Fig. 3, a simpliﬁed model for a large submerged vessel consists of aﬁnite cylindrical shell, a rib, a bulkhead, and two hemi-spherical endcaps. Since the structure has rotational symme-try about the zaxis, the angular spectral decomposition with respect to the fcoordinate is applicable to all structural el- ements, and the overall response can be synthesized in termsof the fspectra. Also, a guided mode representation ~where the modes in the endcaps, rib, pipe, and bulkhead are deﬁnedalong the rcoordinate, zcoordinate, rcoordinate, and z coordinate, respectively !satisﬁes the full elastic equations and boundary conditions at the surfaces of the plate or shellelements. Using a combination of guided modes and angularspectra, the problem is reduced to a ‘‘one-dimensional’’problem in the remaining coordinate, which is rfor the bulk- head and the rib, ufor the endcaps, and zfor the pipe. For convenience in later formulations and discussions, we willdenote the common fcoordinate of all structural elements byw, the normal coordinate to the surface of each plate element by u, and the remaining ~third!coordinate of each plate element by v. The solution in the remaining coordinate vis not truly a one-dimensional problem because the original structure is FIG. 3. The ﬁgure is a simpliﬁed model for a large submerged vessel. It consists of a ﬁnite cylindrical shell, a rib, a bulkhead, and two hemisphericalendcaps. 122 122 J. Acoust. Soc. Am., Vol. 99, No. 1, January 1996 I-Tai Lu: Ray –mode radiation and scattering Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.114.34.22 On: Sat, 22 Nov 2014 00:16:56nonseparable. However, we can project sources and receivers onto the surface of the hull for all possible excitation andobserving mechanisms. For example, sources in water willexcite leaky modes on the structure. In Fig. 4, sources inregion ‘‘1’’ ~‘‘2’’!, enclosed by the two solid ~dashed !lines and one-half of the right endcap, can excite a leaky modepropagating clockwise ~counterclockwise !on the right end- cap. The angles between the lines and the endcap are deter-mined by the phase matching condition of the correspondingleaky mode. Similarly, sources in regions ‘‘4’’ and ‘‘3’’ canexcite the corresponding leaky wave propagating clockwiseand counterclockwise, respectively, on the cylindrical pipe.By reciprocity, the receivers in regions 1 and 2 will observethe corresponding leaky wave propagating counterclockwiseand clockwise, respectively, on the right endcap, and the re-ceivers in regions 4 and 3 will observe the correspondingleaky wave propagating counterclockwise and clockwise, re-spectively, on the cylindrical pipe. The possible source orreceiver regions for the left endcap can be determined in thesame manner. Note that we need only the upper half of thestructure for each of the two graphs in Fig. 4 in the angularspectral domain because of the rotational symmetry of thestructure along the zaxis. The lower portion of the structure is shown to remind the readers of the original conﬁguration. By projecting sources and receivers onto the hull, the reduced problem in the vcoordinate becomes one dimen- sional. The essential feature of this formulation is that thesystem of equations becomes algebraic. Figure 5 shows thereduced conﬁguration in the vdomain without indicating the equivalent source and receiver locations. Points a and f de-note the two turning points of the guided modes in the two hemispherical endcaps, respectively. Point g denotes theturning points of the guided modes in the bulkhead. Note thatdifferent modes usually pertain to different turning points.Points b and e denote the junctions between the cylindricalpipe and the two endcaps. Point d ~c!denotes the junction between the cylindrical pipe with the bulkhead ~rib!. Point h denotes the truncation of the rib. These points act as scatter-ing~coupling !centers of guided modes in structural ele- ments. The lines between any pair of adjacent scattering cen-ters denote the propagation factors of guided modes. Sinceeach structural element supports multiple waves, each linedenotes a wave vector consisting of corresponding guidedmodes. We use two lines to connect each pair of scatteringcenters to depict the two wave propagation directions. The solution to this reduced one-dimensional problem in Fig. 5 can be synthesized by traveling waves ~e.g., see the signal ﬂow graph approach in Ref. 65 !to include all possible coupling phenomena ~reﬂection, transmission, and diffrac- tion!occurring at the joints and truncations. Generally speak- ing, the scattered ﬁeld can be written as G'GW11GW2, GW1'G dir1( eHGref1( mGmJ, ~37! GW2'( lG2l, whereGW1 denotes contributions from structure elements as if they were without truncation, joint, or internal loading,andGW2 denotes contributions from more than one struc- ture element where internal loading or ﬁniteness of the struc-ture plays an important role. The GW1 is very similar to that in~34!for the separable problems except the sum over e ~which is a depiction for summing over possible contribu- tions from all structure elements !. In Fig. 2, the direct ray, reﬂected ray, and leaky mode belong to GW1, and the modal diffraction belongs to GW2. A generic expression for GW2 waves can be obtained by ﬁrst using the residue theorem toevaluate the l vintegral in ~18!for each structure element and then combining these terms with proper coupling, dif-fraction, excitation, or detaching coefﬁcients ~denoted by T e!: FIG. 4. The reduced two-dimensional conﬁguration in ( u,v) coordinates. Sources in regions 1 and 2 can excite a leaky mode propagating clockwiseand counterclockwise, respectively, on the right hemispherical endcap.Sources in regions 4 and 3 can excite a leaky mode propagating clockwiseand counterclockwise, respectively, on the cylindrical pipe. FIG. 5. The signal ﬂow graph of the reduced one-dimensional conﬁguration in thevcoordinate. 123 123 J. Acoust. Soc. Am., Vol. 99, No. 1, January 1996 I-Tai Lu: Ray –mode radiation and scattering Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.114.34.22 On: Sat, 22 Nov 2014 00:16:56G2l'1 2piR Cw) e$Telim lv!lveRue~J!gve~m!%gw~n!dlw,~38! where the subscript erepresents various structure elements. The superscripts ( Jmn) have been deﬁned in ~18!and are related to the single subscript lofGlI. Except the coefﬁ- cientsTe, each term in ~38!can be obtained for analyzing its corresponding structure element, with or without ﬂuid load-ing, using the techniques derived before. Combining thephases and amplitudes, respectively, of these terms in theintegrand, the resulting spectral integral could be evaluatedby asymptotic approximation if the phase term of the inte-grand is rapidly varying. The stationary phase conditions@see~29!and~33!#and the transverse resonance condition @see~33!#determine the ray trajectories in the ﬂuid and on each of the structure elements. Therefore, the procedure isvery similar to the one previously discussed for solving sepa-rable structures. The key ingredient in this extension to ana-lyze nonseparable structures is the analysis of the couplingand diffraction coefﬁcients at junctions or discontinuities,which has been published in Ref. 29. So far the effects ofﬁniteness of the structure elements on the excitation or rera-diation of leaky modes have not been taken into consider-ation: These effects are proven to be of lower order in farzones and will be published elsewhere. 64 III. MATRIX GREEN’S-FUNCTION FORMULATION FOR NONSEPARABLESTRUCTURESOFREVOLUTION Since each structure element in Fig. 3 supports three propagating modes and these modes couple at joints andtruncations, traveling-wave ﬁelds proliferate. Thus a system-atic scheme is required to track successively all encountersof discontinuities for all GW2’s. In this section, a matrix Green’s-function formulation is employed to synthesize thewave motion on the structure due to simultaneous excitationsand reradiations on the hull. Noting that the structure hasrotational symmetry about the center axis, the angular spec-tral decomposition with respect to the wcoordinate is appli- cable to all structure elements. Thus the overall response canbe synthesized in terms of the angular spectra. Using theguided mode solutions for the ucoordinate, the reduced ‘‘one-dimensional’’problem in the vdomain has been sche- matized in Fig. 5. For convenience, the multiple reverbera-tions of modes in the rib ~between the truncation h and the joint c !and those in the bulkhead ~between the turning points g and the joint d !will be treated collectively @see Eqs. ~34a!– ~c!in Ref. 29 #. The structure in Fig. 3 or Fig. 5 is then reduced to a ﬁve-layer medium in the vdomainas shown in Fig. 6, where each layer supports three waves. The top andbottom boundaries represent the origins of the two endcapswhere mode coupling does not occur. The two middle inter-faces represent the junctions of the pipe with the rib and thebulkhead. Here, the multiple reverberations of modes in therib~between the truncation and the joint with the pipe !and in the bulkhead ~between its center and its joint with the pipe ! have been treated collectively. The other two interfaces rep-resent the joints of the pipe with the two endcaps. Let the subscript pdenote the pth layer and the sub- scriptsmpdenote the mth guided mode of the pth layer. Byspectral decomposition along the azimuthal coordinate wand the guided mode representation in the ‘‘normal’’ coordinateu, the one-dimensional wave solution in the third coordinate vof themth mode in the pth layer satisﬁes the following equation: Fd2 dvp21gv,pm2Ggv,pmb~vp!50,vpÞvpm8b, ~39! when the observation location vpis not at the excitation locationvpm8bwhereb5DorU, respectively, denoting the downgoing or upgoing wave. The equivalent wave number gv,pmis deﬁned in ~23!. Note that the wave number gu,pmin theucoordinate satisﬁes the resonant condition @D50i n ~35!or~36!#. The source location vpm8bis determined by the phase matching condition. The bridging across source levelis described by g v,pmb~vpm8b1!5gv,pmb~vpm8b2!6apmb,b5D,U,~40! whereapmbis the excitation strength, and the upper and lower signs go with DandU, respectively. Combine all the guided mode to deﬁne the wave vector in the pth layer as gIv,pb5@gv,p1b,gv,p2b,...,gv,pMb#,b5D,U. ~41! These wave vectors are denoted by the lines in the signal ﬂow graph in Fig. 6. Accordingly, the source vector is de-ﬁned as sI pb5@ap1b,ap2b,...,apMb#,b5D,U. ~42! FIG. 6. An equivalent ﬁve-layer medium of the structure in Fig. 3. Each layer supports multiwaves. The two array wave vectors @see~45!#propagate from an array coordinate vector VI8to an another array coordinate vector VI @see~44!and~46!#. 124 124 J. Acoust. Soc. Am., Vol. 99, No. 1, January 1996 I-Tai Lu: Ray –mode radiation and scattering Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.114.34.22 On: Sat, 22 Nov 2014 00:16:56Consider wave ﬁelds in ‘‘the ﬁve layers’’simultaneously and deﬁne an array wave vector as gIv~VI!5@gIv1~v1!,gIv2~v2!,gIv3~v3!,gIv4~v4!,gIv5~v5!#T, ~43! whereTdenotes the transpose operation. Here VIis the ver- tical array coordinate vector VI5@v1,v2,v3,v4,v5#T~44! andviis thevcoordinate in the Ith layer, I51,2,...,5. We will construct two array wave vectors from the ten upgoingand downgoing wave vectors in the ﬁve layers in Fig. 6.They are given by gI v~VI!5gIvD~VI!1gIvU~VI!, gIvD~VI!5@gIv1D~v1!,gIv2U~v2!,gIv3D~v3!,gIv4U~v4!,gIv5D~v5!#T, ~45!gIvU~VI!5@gIv1U~v1!,gIv2D~v2!,gIv3U~v3!,gIv4D~v4!,gIv5U~v5!#T. Note that we purposely alternate the superscripts of the vec- tor elements in ~45!, where the superscript ~UorD!of the array wave vectors in ~45!corresponds to the superscript of their ﬁrst vector element. To accommodate the arrangementof observables into two array wave vectors in ~45!, it is nec- essary to deﬁne various matrices and vectors adapted in thisdecomposition. From ~45!and Fig. 6, propagation from an arbitrary array coordinate vector VI 8to an another arbitrary array coordinate vector VIis described by gIvb~VI,VI8!5@E~VI,VI8!#gIvb~VI8!,b5U,D. ~46! The array propagation matrix [ E(VI,VI8)] is a diagonal block matrix: @E~VI,VI8!#5F@e1~v1,v18!# @e2~v2,v28!# @e3~v3,v38!# @e4~v4,v48!# @e5~v5,v58!#G. ~47! Thepth block element in ~47!, @ep~vp,vp8!# 5Fep1~vp,vp8! ep2~vp,vp8! epM~vp,vp8!G, ~48! is the propagation matrix in the pth layer. It is a MbyM diagonal matrix with its mth element representing the phase propagator of the mth mode: epm~vp,vp8!5E vp8vpgv,pmdvp. ~49! Regarding the boundary conditions of array wave vectors, we ﬁrst deﬁne the two array coordinate vectors at the inter-faces as VI U5@t12,t11,t32,t31,t52#T, ~50!VID5@t01,t22,t21,t42,t41#T. Second, we express the boundary conditions at top and bot- tom boundaries as gIv1D~t01!5@r0U#gIv1U~t01!,gIv5U~t52!5@r5D#gIv5U~t52! ~51! and express that at any interface between two adjacent layers asgIv,pU~tp1!5@rpD#gIv,pD~tp1!1@tpU#gIv,qU~tq2!, gIv,qD~tq2!5@tpD#gIv,pD~tp1!1@rpU#gIv,qU~tq2!, p51,2,3,4, q5p1. ~52! Here, [rpU(D)] and [tpU(D)] are the reﬂection and transmission, respectively, matrices at the pth interface due to an incident upgoing ~downgoing !wave vector. Hence, the boundary con- ditions of the array wave vectors at the interfaces become gIvD~VID!5@RDU~VID!#gIvU~VID!, ~53! gIvU~VIU!5@RUD~VIU!#gIvD~VIU!, with the array coupling matrices @RDU~VID!#5F@r0U# @r2D#@t2U# @t2D#@r2U# @r4D#@t4U# @t4D#@r4U#G~54! deﬁned at the array coordinate vector VID, and 125 125 J. Acoust. Soc. Am., Vol. 99, No. 1, January 1996 I-Tai Lu: Ray –mode radiation and scattering Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.114.34.22 On: Sat, 22 Nov 2014 00:16:56@RUD~VIU!#5F@r1D#@t1U# @t1D#@r1U# @r3D#@t3U# @t3D#@r3U# @r5D#G~55! deﬁned at the array coordinate vector VIU. Choosing an arbi- trary reference level VI, the boundary conditions become gIvD~VI!5@RDU~VI!#gIvU~VI!,gIvU~VI!5@RUD~VI!#gIvD~VI!. ~56! From ~46!and~53!, the coupling matrices at an arbitrary array coordinate vector VIare expressed as @RUD~VI!#5@E~VI,VIU!#@RUD~VIU!#@E~VIU,VI!#, ~57!@RDU~VI!#5@E~VI,VID!#@RDU~VID!#@E~VID,VI!#, @FU~VI!#5@RUD~VI!#@RDU~VI!#, ~58!@FD~VI!#5@RDU~VI!#@RUD~VI!#. This leads to the ~block tridiagonal !reverberation matrices. Thus the relation between the array wave vectors before and after propagating a complete excursion in the entirestructure is gIˆvb~VI!5@Fb~VI!#gIvb~VI!,b5U,D. ~59! When the self-consistent condition is met, gIˆvb~VI!5gIvb~VI!, ~60! we obtain the resonance condition of the structure along the third coordinate v, det$@I#2@FU#%5det$@I#2@FD#%50, ~61! where [I] is the identity matrix. Recall that the resonant condition in the u~‘‘normal’’ !coordinates of the structure elements has been satisﬁed by the guided mode representa-tion in this formulation. The resonant condition in the azi-muthal coordinate can be satisﬁed by setting the azimuthalspectral variable ~l w!1/2to be an integer n. Thus the equation in~61!determines the resonance condition of the entire structure if ( lw)1/25n. To obtain the solution for the array wave vectors we shift the source levels to their corresponding interfaces bydeﬁning the equivalent array source vectors as SI U~VIU!5@sIˆ1U~t12!,sIˆ2D~t11!,sIˆ3U~t32!, sIˆ4D~t31!,sIˆ5U~t52!]T, ~62!SID~VID!5@sIˆ1D~t01!,sIˆ2U~t22!,sIˆ3D~t21!, sIˆ4U~t42!,sIˆ5D~t41!]T. Their vector elements in the pth layer are sIˆpD~tp211!5@aˆp1D~tp211!,aˆp2D~tp211!,..., aˆpMD~tp211!], ~63!sIˆpU~tp2!5@aˆp1U~tp2!,aˆp2U~tp2!,...,aˆpMU~tp2!#,withp51,2,3,4,5. The mth elements in the above equivalent source vectors ~63!are the downgoing and upgoing sources of themth mode referred at the corresponding interface: aˆpmD5apm epmD~tp211,vpm8D!,aˆpmU5apm epmU~tp2,vpm8U!. ~64! The array wave vectors in ~45!can then be expressed as gIvb~VI!5gIvbc~VI!1gIvbb~VI!, $b5D,c5U%or$b5U,c5D%, ~65! where gIvbc~VI!5$@I#2@Fb~VI!#%21@Rbc~VI!#@Ec~VI,VIc!#SIc~VIc! ~66! is theb-type wave to c-type source SIc~cdenotes upgoing or downgoing when bdenotes downgoing or upgoing, respec- tively !, and gIvbb~VI!5$@I#2@Fb~VI!#%21@Eb~VI,VIb!#SIb~VIb! 2dIb~VI,VIb! ~67! is theb-type wave vector due to the b-type source SIb. The delta vector dIb~VI,VIb!5@Db~VI,VIb!#@Eb~VI,VIb!#SIb~VIb!,b5D,U ~68! removes the direct waves in those layers where they do not appear. Here the delta matrix @Db#is a diagonal matrix with itsIth diagonal element deﬁned as Diib5H1, 1 2, 0,vpm8D.vp~vpm8U,vp!, vpm8b5vp, else,b5UorD ~69! if the subscript iis corresponding to the mth mode in the pth layer. IV. SPECTRAL INTEGRAL AND ALTERNATIVE REPRESENTATIONS In Sec. III, the vdomain solution gIvin~45!is obtained in terms of the array wave vectors gIvU,Din~32!–~36!. The acoustic ﬁeld GIobserved at multiple receivers at (uj,vj,wj),j51,2,3,..., J, due to a point source at (u8,v8,w8) can then be expressed as a spectral integral GI5GIU1GID,GIb521 2piR( r,e@K#r$gIvb%edlw, b5U,D. ~70! Note that the direct wave and specular reﬂection contribu- tions for scattering problems, which can be taken into ac-count separately, are not included in ~70!. The matrix [ K] ris the propagating matrix accounting for the amplitude andphase changes in the uandwcoordinates from the surface of the hull to the receiver levels. The sum over rand over eis over all possible radiation and excitation mechanisms, re-spectively. These mechanisms include the phenomena ofleaky modes and the diffraction effects at joints ~see Fig. 2 ! We will concentrate on the excitation and radiation mecha- 126 126 J. Acoust. Soc. Am., Vol. 99, No. 1, January 1996 I-Tai Lu: Ray –mode radiation and scattering Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.114.34.22 On: Sat, 22 Nov 2014 00:16:56nisms of leaky modes where the consideration of the diffrac- tion effects at joints will be published elsewhere.64For con- venience, the sums over randeand the subscripts rande will be omitted. Each row of [ K] corresponds to a receiver, and the num- ber of rows of matrix [ K] is equal to the number of receivers ~i.e.,J!. Each element of a row vector of [ K] is related to a guided mode in a subregion of the structure.Thus the dimen-sion of each row vector of [ K] is equal to the dimension of gI v, which is PMwherePis the number of layers and Mis the number of guided modes in each layer. For example, P55 in Fig. 3, and M53 for thin shell structures because there are three fundamental modes in each structural ele-ment. Let the element K jiof [K] relate to the jth receiver at (uj,vj,wj) and let the other subscript ispecify the mth mode in the pth layer. If the receiver is located in the ‘‘illu- minated’’ region of the corresponding mode as illustrated inFig. 4, K ji5gw~wj,w8!expHiUE u`ujgu,pm~t!dtUJ. ~71! If the receiver is not located in the illuminated region, Kji50. Here, we neglect the diffraction effects due to ﬁnite- ness of guided modes, which is found to be negligible in thefar zones. 64In~71!,gwis the one-dimensional Green’s func- tion in the wcoordinate deﬁned in ~4!or~26!,u`is theu coordinate of the pth layer in Fig. 6 ~i.e., the outer surface of thepth structural element !, and gu,pmis the guin~23!with lvbeing speciﬁed to be the modal wave number of the mth mode in the pth layer: lv,pm. Note that u,u`, and gu,pmare related to the spherical shell if p51 or 5, and are related to the cylindrical shell if p52, 3, or 4. Regarding the source excitation strength apmbin~40!,w e follow the conventional procedure in ~33!and obtain apmb52CuCvB ]D/]lv,pmexpHUE v8vpm8b gv,pm~t!dtU 1UE u8u gu,pm~t!dtUJ, ~72! B/D[Gu1,D~lv,pm!50. Here,D50 is the resonance condition of the pth layer in the ucoordinate. lv,pmis the modal wave number of the mth mode in the pth layer. Gu1is the reﬂection coefﬁcient @see Eq.~25!#.The coefﬁcient Cu,vis deﬁned in ~26!.The equiva- lent source location of the mth mode in the pth layer,vpm8b,i s obtained by projecting the point source onto the hull surface~see Fig. 2 or 4 !based on the phase matching condition. The wave numbers gu,pmandgv,pmare deﬁned in ~23!. Equation ~70!can be extended to adapt to multiple sources by using the superposition principle.Acompact formcan be obtained by ﬁrst separating out the source coordinate~w 8!dependent terms in the one-dimensional Green’s func- tiongwfrom the matrix element Kji@see~71!#, and then putting them into the source vector SIU,DingIvb@see~40!, ~42!, and ~62!#. Thereafter, the ﬁeld vector GIand the sourcevectorSIU,Dare replaced by a ﬁeld matrix and a source ma- trix, respectively,where each of their columns corresponds toa point source. A. Eigenvector basis and wave basis The spectrum in ~70!of the leaky mode contributions is represented in the wave potential basis @see~43!#. Thus the reverberation matrices [ FU] and [FD]i n~58!account for phase and amplitude changes in all layers observed at thereference levels VIafter one complete reverberation in these layers. As discussed in Refs. 49, 50, 57, and 58, the wavecoupling implied can be removed by transforming from theoriginal wave potential basis to the eigenvector basis. It di-agonalizes the reverberation matrices [ F U,D] and then sca- larizes the entire ﬁeld formulation in the spectral domain.Therefore, alternative representations can be written either inthe conventional forms based on wave potential basis or inthe ‘‘scalar forms’’ based on eigendecomposition. The latterhave the advantage that the eigensolutions undergo multiplereverberation without change ~except for overall amplitude !. Thus the eigensolutions in multilayered media propagate likeordinary rays in a single wave layer, and the ray–modeequivalence in the eigenbasis becomes tractable. These prop-erties, as well as those pertaining to a variety of other alter-native representations, can be inferred from Refs. 49, 50, 57,and 58. For the present paper, we focus on the conventionalforms. B. Direct numerical integration The spectral integral in ~70!highlights the features as- sociated with the various waves that synthesize the equiva-lent source distributions on the hull.The ﬁeld solution can beobtained by direct numerical integration. Note that the rever-beration matrices [ F U,D] are block tridiagonal. Therefore, numerical techniques such as Gaussian elimination, blockJacobi, block Gauss–Seidel, etc., can be employed to evalu-ate the matrix inversion in ~66!and~67!with numerical sta- bility across evanescent layers due to the ordering adapted in~45!. This method becomes inefﬁcient at high frequencies and/or for large source–observer–scatterer separations dueto the strong oscillatory behavior of the integrand. C. Mode representation The behavior of the spectral integral in ~70!is domi- nated by its singularities. By closing the integration path atinﬁnity, one may derive a modal representation in terms ofthe discrete modes generated by residues at the poles. Themodal representation becomes inefﬁcient at high frequenciesand/or small source–observer separation where many modesare required. The integrand in ~70!has two sets of pole sin- gularities. One set is related to the vdomain solution and is determined by the resonant condition in ~61!.The other set is related to the wdomain solution. If the poles lvmof thev domain solution are enclosed, the modal solution becomes 127 127 J. Acoust. Soc. Am., Vol. 99, No. 1, January 1996 I-Tai Lu: Ray –mode radiation and scattering Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.114.34.22 On: Sat, 22 Nov 2014 00:16:56GI5( mGIm, GIm[2lim lw!lwm~lw2lwm!@K#$gIvD1gIvU%, ~73!det$@I#2@FU,D#%lwm50. The modes in ~73!are guided in the wdirection. If the poles of thewdomain solution are enclosed, the modal solution becomes GI5( n50` GIn,GIn[@K¯#n$gIvD1gIvU%,@K¯#n5@K¯ji,n#, K¯ji,n51 2pencos@n~w2w8!#expHiUE uujgu,pm~t!dtUJ, en51i fn50 and en52,nÞ0. ~74! The modes in ~74!are guided in the vdirection. D. Ray representation For convenience and clarity, the term ‘‘ray’’used in this paragraph and the rest of the paper represents the wave ob-jects obtained by applying ray techniques to the spectral in-tegral in ~70!which is derived by using the guided modes on the structural elements in the ucoordinate. Using the traveling-wave form of g wand expanding the denominator term$[I]2[FU,D]%21in~66!and~67!as $@I#2@FU,D#%215( j50` @FU,D#j, ~75! one generates a series of integrals GI5( j50` GIjU1GIjD,GIjb521 2piR@K#$gIv2jb%dlw, b5U,D, ~76! where gIv2jb5gIv2jbc1gIv2jbb, b5D,C5Uorb5U,C5D, ~77!gIv2jbc5@Fb#j@Rbc#@Ec#SIc,gIv2jbb5@Fb#j@Eb#SIb. By further expansion of all matrices and products, one ob- tains the conventional rays which express reverberations interms of a multitude of conventional ray ﬁelds undergoingcoupling and splitting at each interface. The present formu-lation in ~76!provides a ray generation scheme that groups these conventional ray ﬁelds according to their number ofreverberations, i.e., the number of ray segments, in variouslayers. These conventional ray ﬁelds can be evaluated bynumerical integration or by asymptotics ~saddle-point ap- proximation !. Due to ray proliferation, for large j, it may be preferable to evaluate ~76!~without further expansions of products, inversions, etc. !by asymptotics when valid, or by direct numerical integration along a rapid convergence pathin the complex l wplane.This is a new kind of collective ray, which incorporates all conventional rays belonging to GIintoa matrix formulation with an equivalent ray phase @see~28! in Ref. 57 #. In this connection, it may be noted that an alter- native composite treatment can be applied to multiples incertain layers by incorporating these rays into collective re-ﬂection and transmission coefﬁcients. The ability to loadconventional rays into collective formats depends on hownearly coincident the incident angles are of the correspond-ing conventional rays. Viewed in the spectral domain, nearlycoincident angles imply clustering of saddle points in thecorresponding ray integrals. Even with the help of collectiveschemes, the ray representation still becomes intractable andphysically obscure for large source–observer separationwhere many conventional and collective rays must be in-cluded. E. Hybrid ray–mode representation Our formulation accommodates the hybrid form in Refs. 49 and 50, which combines ray integrals, modes, and a re-mainder ﬁeld in unique proportion. It accounts in a compactmanner for the hierarchy of multiple reﬂected conventionalrays in terms of modes plus a remainder via a ray–modeequivalent. By ﬁnite series expansions $@I#2@FU,D#%215( j50L21 @FU,D#j1@FU,D#L 3$@I#2@FU,D#%21~78! we may write the total ﬁeld as a ﬁnite sum of ray integrals plus a remainder integral sILb: GI5( j50L21 $GIjU1GIjD%1sILU1sILD, ~79! msILb521 2piR@K#$rIv2Lb%dlw, where rIv2Lb5rIv2Lbc1rIv2Lbb, b5D,C5Uorb5U,C5D, rIv2Lbc5@Fb#L$@I#2@Fb#%21@Rbc#@Ec#SIc, ~80! rIv2Lbb5@Fb#j$@I#2@Fb#%21@Eb#SIb. In~79!, the collective ray integral GIjU,Dhas been deﬁned in ~76!. The integration contour of the remainder integral in ~79!can be deformed into a steepest decent path ~denoted by sdp; see Refs. 49, 50, 57, and 58 !, and hence the remainder integral becomes sILb[$sILb%sdp1( mGm, ~81! $sILb%sdp521 2piE sdp@K#$rIv2Lb%dlw, where the modes GImare deﬁned in ~73!and the sdp integral in~81!can usually be neglected with a proper selection of L. Equations ~79!–~81!furnish the hybrid formulation. The hy- 128 128 J. Acoust. Soc. Am., Vol. 99, No. 1, January 1996 I-Tai Lu: Ray –mode radiation and scattering Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.114.34.22 On: Sat, 22 Nov 2014 00:16:56brid format employed here is obtained from the vdomain solution and, therefore, the modes intercepted during thecontour deformation are in the form of ~73! F. Transient responses The time-dependent motion corresponding to a source excitation s~v!can be determined from the frequency- domain solution GIby applying the inverse transform G=5IFT@s~v!GI~v!#[1 2pE 2`1id`1id s~v!GI~v! 3exp~2ivt!dv,d.0,~82! where IFT denotes the inverse Fourier transform, and the integration path in the complex frequency plane extendsabove all singularities of the integrand to satisfy causality.The frequency dependence of GI, omitted in the notation of previous discussions, has been exhibited explicitly in ~82!. Various alternative representations of the time-harmonic ﬁeldGIdiscussed previously can be employed here. In the integral representation, substitute ~70!into~82!to obtain double ~l w andv!integrals. The time-dependent ray or remainder ﬁelds @substituting ~76!or~79!into~82!#also consist of double integrals, while the time-dependent modal ﬁeld @substituting ~73!or~74!into~82!#has only the frequency integral. Any of these integrals may be evaluated in three ways: ~a!direct numerical integration along the original contour, ~b!numeri- cal integration or asymptotic approximation when the origi-nal contour is deformed into a more rapid convergent path~some singularities may be intercepted during contour defor- mation !, and ~c!residue evaluation by closing the contour at inﬁnity. These various options enable us to represent thetransient response in terms of a great variety of alternativeexpressions. Detailed discussions have been made elsewhere~e.g., see Ref. 58 !, and we will concentrate on two basic options here, i.e., wavefronts and resonances. Wavefronts in time domain are usually related to rays in the frequency domain. The collective ray integrals in ~76!or the conventional ray integrals @obtained from the collective rays after expanding all matrix products in ~76!and~77!#are dispersive because guided modes on the structure elementsare involved in this formulation. Therefore, the well-knownCagniard method for nondispersive rays cannot be directlyapplied here. Some inversion algorithms using a weakly dis-persive assumption may be suitable here. Nevertheless, anasymptotic approximation for the l wintegral and the direct numerical integration approach for the vintegral ~along the original path or along a rapid convergent path !are usually the most convenient options to evaluate distinct wavefrontarrivals in our case. Resonances are modes of ﬁnitestructures satisfying self- consistent conditions in all coordinates. They can be ob-tained by evaluating the residues of the vintegral in the modal expressions @~82!together with ~73!or~74!#: G=5( m( n50G=mn, det $@I#2@FU,D#%lwm~vmn!50,G=mn[2ilim v!vmn~v2vmn!@K¯#n 3$gIvD1gIvU%exp~ivt!s~v!. ~83! The matrix [ K¯]nis deﬁned in ~74!. The explicit frequency dependence of the resonant equation in ~83!is shown where vmnis the resonant frequency. The hybrid wavefront– resonant approach can be obtained by applying the IFT in~82!to the hybrid ray–mode form in ~79!–~81!. V. SUMMARY AND DISCUSSION A general procedure for the ray–mode method in the time-harmonic domain and for the wavefront–resonancemethod in the time-dependent domain has been outlined foranalyzing wave scattering and radiation from submergedstructures. We start with the formulation of the problem ingeneral separable curvilinear coordinates and explore asymp-totic techniques emphasizing the ‘‘high-frequency’’ regime.Instead of dealing with the transcendental functions associ-ated with various separable coordinate systems, we employ asystematic asymptotic procedure which can be extended toanalyze nonseparable structures. As an example, a nonseparable structure of revolution is solved by a quasiseparable approach.Apreviously developedmatrix Green’s-function formulation 50for wave propagation in stratiﬁed media is extended to formulate the wave motionson a cylindrical shell with endcaps, internal rib, and bulk-head. In this formulation, ﬁeld variables are arranged in arrayvectors by following a special ordering technique to revealdominant wave processes and to load uninteresting ones in acollective form. This arrangement allows simultaneous exci-tation and detection at arbitrarily speciﬁed locations and pro-vides a physically appealing view pertaining to an array-typesource and receiver arrangement which is appropriate to de-scribe the various guided modes on the structure elements.Multiple coupling among these modes at junctions can thenbe built in systematically in terms of series of products ofspectral propagation and coupling coefﬁcients.To understandthe physical phenomena, we can single out any speciﬁcphysical process or treat a particular class of processes col-lectively. This formulation provides a uniﬁed and systematicapproach for deriving all alternative representations includ-ing mode, ray, spectral integral, ray–mode, collective ray,eigenray–eigenmode, wavefront–resonance, etc., in the non-uniform vcoordinate in both time-harmonic and time- dependent domains. This approach is totally novel and themerits and versatility of this approach may also provide onewith numerical efﬁciency. The extension of the matrix Green’s-function formula- tion to the ray structural acoustics is very appropriate be-cause the following three requirements of our problemtreated here can be met by the matrix formulation. At ﬁrst,the guided modes on the structure usually reradiate and/orare excited at various locations in each subregion, whichimplies that the reduced one-dimensional problem involvesarray-type sources and receivers even though the originalconﬁguration may contain only a source and receiver pair.Thus we need a formulation to deal with simultaneous exci- 129 129 J. Acoust. Soc. Am., Vol. 99, No. 1, January 1996 I-Tai Lu: Ray –mode radiation and scattering Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.114.34.22 On: Sat, 22 Nov 2014 00:16:56tations and observations efﬁciently. Second, the traveling waves on the structure surface proliferate rapidly and weneed a systematic way to keep track of the traveling waves orto treat them collectively. The third need is to obtain a solu-tion in hybrid formats for proper physical interpretation. Fortransient responses, the progressing description is effective inrepresenting the early time arrivals because of causality, butthe oscillatory description is convenient in describing the latearrivals where the structure has fully responded to the exci-tation. For time-harmonic responses, spectral intervals clus-tered with mode spectra can usually be represented by smallnumbers of ray spectra, and vice versa. Thus we need a for-mulation which can adapt to the hybrid ray–mode orwavefront–resonance formulation easily, and the matrix for-mulation in Ref. 50 is designed to meet the above-mentionedrequirements. The validity and accuracy of the presented ray–mode ~wavefront–resonance !approach depend on the validity of the following three subapproaches of the problem: ~a!modal solutions for each plate or shell component ~subregion !,~b! the high-frequency method ~such as rays !employed, and ~c! coupling at discontinuities ~e.g., junctions !. Since the modal representation is only convenient for plates or shells withuniform ~or, at most, weakly varying !material and geometri- cal properties, substructures not belonging to this categorywill be treated as scattering centers and will be analyzed bynumerical methods. By doing so, the error contributed fromthe modal approach will be minimized. Regarding the valid-ity of the ray approach, it is well known that the simple raysolution fails at various catastrophic regions such as caustics.Fortunately, there are ways ~such as uniform asymptotics, beam approach, Green’s-function approach, etc. !to remedy the difﬁculties. For small scatterers not suitable for ray-typeapproaches, numerical methods can be employed to obtaintheir scattering coefﬁcients. This information can then be fedinto the ray–mode program. The difﬁculty of ray prolifera-tion can also be overcome by using parallel processing orcollective approaches such as collective rays and modes. Re-garding the validity of coupling at discontinuities, the ap-proach in Ref. 29 only considers coupling through junctionsbetween substructures and neglects coupling through the sur-rounding ﬂuid. The coupling from the surrounding ﬂuid canbe taken into consideration by formulating a coupling systemof equation of the entire structure. We conclude that variouserrors can be controlled and quantiﬁed. ACKNOWLEDGMENT This work was supported by the Ofﬁce of Naval Re- search. APPENDIX: ADIABATIC EXTENSION TO WEAKLY NONSEPARABLE STRUCTURES When a general physical structure can be modeled as a weak deviation from a strictly separable canonical conﬁgu-ration, one may employ spectral scalings and adiabatic in-variants to account approximately for the weak nonseparabil-ity, generally manifested in the lateral dependence. Withinthe present format, this implies assumption of local separa-bility between the radius coordinate uand the lateral coordi- nates ( v,w). The constituent separate equations in the sur- rounding ﬂuid become, accordingly, @¹u21lu~v,w!#guu,u8;lu~v,w!52d~u2u8!, ~A1!@¹~v,w!21k22lu~v,w!#g~v,w!v,w,v8,w8;lu~v,w! 52d~v2v8!d~w2w8!, where the spectral separation parameter luand hence either one or both of the other two spectral parameters, lvandlw, are no longer constants but functions of the lateral coordi-nates ( v,w). The dependence of these parameters on ( v,w) is determined from an invariant. A general theory for three-dimensional adiabatic transform and for synthesizing three-dimensional Green’s functions has been published in Ref. 63.Its applications to model wave propagation on thin shellshave also been published in Ref. 37. Here we summarizeonly the results of a simple case where the two-dimensionalwave equation ~A1!in the lateral domain is separable ~e.g., through rotational symmetry of the structure !. Then the inte- gral representation of the three-dimensional Green’s functionbecomes G ~r,r8!51 ~2pi!2R CvR Cvguu,u8;lu~v! 3gvv,v8;lv~v!gw~w,w8;lw! 3Adlv~v!dlv~v8!dlw, ~A2! which can be treated in a manner similar to that discussed in Secs. I–IV. In ~A2!, we have assumed that lwis a constant and that luandlvare functions of ( v) but not of ( w). The symmetrized symbolic derivative in ~A2!is deﬁned as Adlv~v!dlv~v8!5F]Dq~v! ]q~v! ]Dq~v8! ]q~v8!G1/2 dq~v!, q~v!5Alv~v!, ~A3! or Adlv~v!dlv~v8!5F]Dq~v8! ]q~v8! ]Dq~v! ]q~v!G1/2 dq~v8!, q~v!5Alv~v!, ~A4! which will ensure the reciprocity of the overall Green’s func- tion. Here, ~A3!and~A4!are based on using the lateral co- ordinate of the receiver ( v) and the source ~v8!, respectively, as the reference coordinate. The function Drepresents the resonant denominator, and the condition Dq~v!5Dq~v8!5constant ~A5! is the invariant which scales the spectrum with the lateral coordinate ( v) so that it adapts locally to the changing struc- ture. 130 130 J. Acoust. Soc. Am., Vol. 99, No. 1, January 1996 I-Tai Lu: Ray –mode radiation and scattering Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.114.34.22 On: Sat, 22 Nov 2014 00:16:561L. 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1.3075850.pdf	Mechanism of microwave assisted magnetic switching Masukazu Igarashi,1,a/H20850Yoshio Suzuki,1Harukazu Miyamoto,1Youji Maruyama,2and Yoshihiro Shiroishi2 1Central Research Laboratory, Hitachi Ltd., 1-280 Higashi-Koigakubo, Kokubunji, Tokyo 185-8601, Japan 2Hitachi Global Storage Technology, Odawara, Kanagawa 256-8510, Japan /H20849Presented 13 November 2008; received 21 September 2008; accepted 15 December 2008; published online 19 March 2009 /H20850 The characteristic of microwave assisted switching for an isolated grain was investigated using the Landau–Lifshitz–Gilbert simulation. It was found that anticlockwise and clockwise polarized ﬁeldsassist magnetization to switch and to reswitch, respectively. Using larger linear polarized ﬁeld,sufﬁcient switching is not obtained. It was conﬁrmed the magnetic resonance effect on the reductionin the switching ﬁeld. It was also found that a large assist effect and a narrowing transition effectwere achieved in exchange coupled subgrains. © 2009 American Institute of Physics . /H20851DOI: 10.1063/1.3075850 /H20852 I. INTRODUCTION As recording bit density of hard-disk drives becomes higher, volume of a bit becomes smaller with larger mediaanisotropy energy to keep thermal stability. As a result, arequired magnetic ﬁeld is approaching a physical limit of thehead magnetic ﬁeld. Microwave assisted magnetic recording/H20849MAMR /H20850with a microwave generator of spin torque has recently been proposed as one of the promising candidates ofnext generation recording technology. 1It has been thought that the switching of magnetization in MAMR was assistedby magnetic resonance. 2–4However, the switching properties of MAMR might be different from those expected for con-ventional ferromagnetic resonance with sharp peak of ab-sorption because the switched area on /H9275-Hdiagram is widely distributed for MAMR. In this work, switching probability has been calculated with changing polarizations of the microwave to investigaterole of the resonance effect in MAMR because the ﬁeld di-rection of the microwave should be synchronized with theprecession of the magnetization when the resonance hap-pens. We have investigated the switching characteristic as-sisted by microwave for an isolated grain composed of sub-grains using the Landau–Lifshitz–Gilbert /H20849LLG /H20850simulation. II. CALCULATION MODEL A cylindrical grain with a diameter Dof 10 nm and a height tmagof 12 nm was used to simulate an isolated single grain. The grain was divided into two subgrains /H20849twin /H20850in the direction of the height /H20849z-axis /H20850. In each subgrain, the magne- tization rotates coherently and interacts with the neighbor bythe exchange coupling. The easy axis of each grain was inthe direction of the height. The value of the uniaxial perpen-dicular anisotropy energy K uwas assumed to be 13.5 Merg /cm3/H20849Hk=30 kOe /H20850. The time evolutions of the magnetizations in the subgrains were calculated by solving the LLG equation as shown in Eq. /H208491/H20850,5/H208491+/H92512/H20850dM/H6023 dt=−/H9253/H20849M/H6023/H11003H/H6023/H11032/H20850,H/H6023/H11032=H/H6023eff+/H9251M/H6023/H11003H/H6023eff M. /H208491/H20850 Here Heffwas the effective ﬁeld consisting of ﬁve terms /H20849Fig. 1/H20850: the applied ﬁeld Hext, the uniaxial anisotropy ﬁeld Ha, the demagnetizing ﬁeld Hd, the exchange ﬁeld from the neigh- boring subgrain Hexc /H20849the intersubgrain exchange surface en- ergy density wof 0.1 erg /cm2/H20850, and ac ﬁeld with the fre- quency facofHaccos /H208492/H9266fact/H20850, which was applied on the x-axis. The gyromagnetic ratio /H9253of 1.76 /H11003107/H20849Oe s /H20850−1and the Gilbert damping constant /H9251of 0.02 /H20849Refs. 6and7/H20850were used. There was no qualitative change though the values of /H9251 were increased up to 0.1. The switching time was investi-gated up to 3 ns with changing H ac,fac, and Hext. The angle between the external ﬁeld and the magnetization easy axis /H9258h was mainly set to be 10°. III. RESULTS AND DISCUSSION At ﬁrst we consider the magnetic resonance effect on the reduction in the switching ﬁeld. Generally linear polarizedﬁeld can be break down the anticlockwise component andthe clockwise component as expressed by Eq. /H208492/H20850, a/H20850Electronic mail: masukazu.igarashi.qu@hitachi.com. FIG. 1. Calculation model.JOURNAL OF APPLIED PHYSICS 105, 07B907 /H208492009 /H20850 0021-8979/2009/105 /H208497/H20850/07B907/3/$25.00 © 2009 American Institute of Physics 105 , 07B907-1Haccos /H208492/H9266fact/H20850ex/H6023=Hac 2/H20851/H20849cos /H208492/H9266fact/H20850ex/H6023+ sin /H208492/H9266fact/H20850ey/H6023/H20850 +/H20849cos /H208492/H9266fact/H20850ex/H6023− sin /H208492/H9266fact/H20850ey/H6023/H20850/H20852. /H208492/H20850 Based on the magnetic resonance, the clockwise polarized ﬁeld does not affect the resonance anymore. Figure 2shows the Hext-Hacdiagrams of the anticlock- wise, the clockwise, and the linear polarized ac ﬁelds at 50GHz. At the frequency the switching ﬁeld reached its mini-mal value with H acof 2 kOe. The brighter area means region switched within 3 ns. The gray area means transition region. The darker areameans region nonswitched. The hatched area means unstableregion, in which the direction of more than 30% grain’s mag-netizations is in the range between 20° and 160° from theeasy axis. When the anticlockwise polarized ﬁeld is used, theswitching ﬁeld decreases remarkably with increasing H ac.I t should be noted that switching needs no external ﬁeld at Hac over 4 kOe /H208490.13Hk/H20850. With the clockwise polarized ﬁeld ap- plied, no switched condition is observed even when Hextex- ceed the Stoner–Wohlfarth ﬁeld. It seems the magnetizationreswitches under the clockwise polarized ﬁeld. This is be-cause the precession direction for the switched magnetizationis the clockwise. With the linear polarized ﬁeld applied,roughly two times larger H acis needed under a constant value of Hext. Furthermore, with larger Hac, unstable condi- tion is observed. This is because magnetizations switch andreswitch frequently in that condition, that is, to saypseudohigh temperature condition. Thus it is conﬁrmed thatthe resonance effect is deeply related to the microwave as-sisted switching. Figure 3shows number of switched grains as a function ofH extfor isolate grains separated into two parts /H20849twin /H20850along with a single domain grains at 50 GHz with Hac’s of 1.0 and 2.5 kOe. The number was counted after 3 ns ﬁeld applica-tion. The switching ﬁelds decrease with increasing H ac. For twin the switching ﬁeld is 15% smaller and the switchingﬁeld dispersion is roughly 50% smaller than those for singledomain grain. Figure 4shows the time dependences of thez-components of the subgrain magnetizations M 1,M2and the ac energy absorptions of the subgrains I1,I2atHacof 5 kOe, facof 50 GHz, and Hextof 6 kOe. Initially, the subgrain magnetizations directed almost z-axis and inclined mutually by 1.5°. The ﬁrst switched subgrain magnetization is deﬁnedasM 1. The ac energy absorptions are calculated from the following equation /H20851Eq. /H208493/H20850/H20852as I1=/H20849H/H6023ac+H/H6023exc_1–2 /H20850·dM/H60231 dt, I2=/H20849H/H6023ac+H/H6023exc_2–1 /H20850·dM/H60232 dt. /H208493/H20850 Here Hexc_ i-jis the exchange ﬁeld of subgrain /H20855i/H20856from the neighboring subgrain /H20855j/H20856. It should be noted that the z-components of the subgrain magnetizations represent the FIG. 2. Hac-Hextdiagrams for the ac polarizations.HAC=2.5kOe Twin Single SingleTwinHAC=01.0kOe 05121024 051 0 Hext(kOe)Number of switched grains 15HAC=2.5kOe Twin Single SingleTwinHAC=01.0kOeHAC=2.5kOe Twin Single SingleTwinHAC=01.0kOe 05121024 051 0 Hext(kOe)Number of switched grains 1505121024 051 0 Hext(kOe)Number of switched grains 1505121024 051 0 Hext(kOe)Number of switched grains 15 FIG. 3. /H20849Color online /H20850Number of switched grains as a function of the external ﬁeld for isolate grains separated into two parts /H20849twin /H20850along with single domain grains /H20849Hk=30 kOe, /H9004Hk/Hk=5%,f=50 GHz, /H9258h=30° /H20850. FIG. 4. Time dependences of the z-components of the subgrain’s magneti- zations and the ac energy absorptions for /H20849a/H20850the ﬁrst switched subgrain and /H20849b/H20850the second subgrain.07B907-2 Igarashi et al. J. Appl. Phys. 105 , 07B907 /H208492009 /H20850absorption of energy from the static external magnetic ﬁeld. Both subgrain magnetizations show almost similar behaviorwith small vibration and do not switch up to t 1. From t1tot2, the difference in the phase of the vibration changes from 0 to /H9266. After t2,M1switched at ﬁrst, leading the switch of M2. The energy absorption of each subgrain is well correspondedto the subgrain magnetization. When a subgrain gets or losesthe energy, the z-components of magnetization decrease or increase, respectively. It should be noticed that M 1gets the energy and at the same time M2loses it around t2, leading the switch of M1. This means that the exchange of the energy between subgrains assists the switching of a subgrain. Figure 5shows the time dependences of the absolute value of the exchange ﬁeld Hexcbetween the subgrains. Hexc takes small value up to t1. After t1,Hexcincreases remarkably with increasing time. When the time is t2,Hexctakes a maxi- mum value of 2.0 kOe. This is consistent with the energyabsorption because the exchange of the energy between sub-grains is activated at t 2. There are several peaks of Hexcthat are observed at the time range between t2and t3. Once asubgrain switched, the other subgrain may be forced to switch by the exchange ﬁeld from the ﬁrst switched sub-grain. Thus a large effect of microwave assistance is achieved through exchange coupling between the subgrains in an iso-late grain. When a certain subgrain meets a resonance con-dition and its magnetization starts to rotate on, a large highfrequency exchange ﬁeld will affect resonance of the adja-cent subgrains and ﬁnally entire magnetic grain. IV. CONCLUSIONS The switching characteristic assisted by microwave for an isolation grain was investigated by the LLG simulation;following results have been obtained: /H208491/H20850Anticlockwise and clockwise polarized ﬁelds assist magnetization to switch and to reswitch, respectively.Using larger linear polarized ﬁeld, sufﬁcient switching isnot obtained. /H208492/H20850The magnetic resonance effect on the reduction in the switching ﬁeld is conﬁrmed. /H208493/H20850A large assist effect and a narrowing transition effect are achieved by exchange coupling between the subgrains. 1J. G. Zhu and X. Zhu, Microwave Assisted Magnetic Recording /H20849TMRC2007 /H20850/H20849unpublished /H20850, Paper No. B6. 2W. Scholz, S. Batra, Micromagnetic Modeling of Ferromagnetic Reso- nance Assisted Switching, /H20849MMM2007 /H20850/H20849unpublished /H20850, Paper No. CC-10. 3K. Rivkin, N. Tabat, and S. Foss-Schroeder, Appl. Phys. Lett. 92, 153104 /H208492008 /H20850. 4S. Okamoto, N. Kikuchi, and O. Kitakami, Appl. Phys. Lett. 93, 102506 /H208492008 /H20850. 5M. Igarashi, F. Akagi, and Y . Sugita, IEEE Trans. Magn. 37, 1386 /H208492001 /H20850. 6N. Inaba, S. Igarashi, F. Kirino, M. Fujita, K. Koike, and H. Kato, Phys. Status Solidi C 4, 4498 /H208492007 /H20850. 7M. Igarashi, T. Kambe, K. Yoshida, Y . Hosoe, and Y . Sugita, J. Appl. Phys. 85, 4720 /H208491999 /H20850. FIG. 5. Time dependences of the absolute value of Hexc.07B907-3 Igarashi et al. J. Appl. Phys. 105 , 07B907 /H208492009 /H20850
1.338824.pdf	Rareearth substitution in (BiYCa)3(FeSiGe)5O1 2 bubble films L. C. Luther, S. E. G. Slusky, C. D. Brandle, and M. P. Norelli Citation: Journal of Applied Physics 61, 325 (1987); doi: 10.1063/1.338824 View online: http://dx.doi.org/10.1063/1.338824 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/61/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Magnetic, dielectric, and magneto-dielectric properties of rare-earth-substituted Aurivillius phase Bi6Fe1.4Co0.6Ti3O18 J. Appl. Phys. 116, 154102 (2014); 10.1063/1.4898318 Effect of rare-earth Ho ion substitution on magnetic properties of Fe 3 O 4 magnetic fluids J. Appl. Phys. 99, 08M906 (2006); 10.1063/1.2163848 Magnetooptical properties of Bisubstituted epitaxial rareearth iron garnet thick films J. Appl. Phys. 61, 3256 (1987); 10.1063/1.338873 Reproducible growth and bubble properties of rareearthsubstituted YCaGeIG films AIP Conf. Proc. 29, 103 (1976); 10.1063/1.30536 THE TEMPERATURE DEPENDENCE OF BUBBLE PARAMETERS IN SOME RAREEARTH GARNET FILMS AIP Conf. Proc. 5, 120 (1972); 10.1063/1.3699406 [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: Sun, 23 Nov 2014 11:57:35Rare .. earth substitution in {BiYCah(feSiGe)S012 bubble films L. C. Luther, S. E. G. Slusky, C. D. Brandle, and M. P. Norem AT&TBell Laboratories, Murray Hill, New Jersey 07974 (Received 29 May 1986; accepted for publication 2 September 1986) The substitution ofY by Sm, Th, Gd, and Ho in (BiYCa)3 (FeSiGe) 5012 bubble garnet is shown to have large effects on the growth-induced anisotropy K ~. The presently accepted film composition intended for 6-or 8-JtID period bubble memory devices demands partial substitution ofY by Gd and Ho. However, comparing films grown under the same growth conditions it is observed that YGdHoBiIG films possess less K ~ than their Gd, Ho-free counterparts. Thus, to satisfy K ~ requirements, the supercooling during growth must be increased by 20 K to 80 or 85 K with undesirable effects on defect densities. A new film composition containing Sm, Th, and Gd has been formulated to satisfy aU known material property specifications for 6-or 8-Jtm period memory devices, It can be grown with only 45-50 K supercooling, INTRODUCTION For operation of bubble memory devices over an ex~ tended temperature range such as -55 to + 125 ·C, bis muth yttrium iron garnet is a promising film material.! However, it has been demonstrated for some time now that a simple composition with reduced moment such as (Y 1.9Bio.4 CIleJ.7 HFeSiGe) 5°12 is not suitable for extended temperature range bubble memory devices. It is necessary to match the temperature dependence of the bubble collapse field to that of the permanent bias magnet. This can be done to Ii point with Gd substitution for Y. Second, to ensure reliable bubble propagation, bubble mobility must be re duced. This can be done using Ho substitution.2 The Gd and Ho substitutions exact a price, however. Less growth-in duced anisotropy K ~ was realized after Gd and Ho substitu tion for otherwise constant composition parameters and growth conditions. (REBi)3Fes 012, where REis a rare earth or Y, differ wide ly with respect to growth-induced anisotropy. The purpose of this paper is to describe this K ~ loss to some extent and to propose alternate substitutions which have a positive effect on K ~. Our observations agree qualita tively with the findings of Fratello et al.3 who recently dem onstrated that the individual members of the series TABLE I. Melt compositions. Type I Type IIA Type lIB Oxide Wt. mol%' Wt. mol % Y203 2.75 2.72 2.72 SmOO3 0.35 0.41 Gd,03 0.66 0.39 0.45 Th20j 0.30 0.36 Ho,O] 0.73 CaO 2.14 2.40 2.40 Si02 6.45 5.48 5.48 GcO, 9.00 7.75 7.75 Fcz03 148.0 140.9 141.0 Mo03 138.5 9.0 114.4 9.0 137.6 PbO 2000 83.5 1650 83.5 1980 Biz03 375.0 7.5 309 7.5 371.0 "Mol % (llux only). EXPERIMENT Melts were prepared based on a flux consisting of PbO :83.5 mol %, Bi2 03 :7.5 mol %, and Mo03 :9.0 mol %. IOOr------------------------~--------,_, '" E 90 80 y 70 "-.,. ~ ... )-60 c. o ~ i 50 <I: o I.I,j u 15 40 z I ::!: I- ~ 30 o Q; ~ 20 10 2C 40 60 80 100 120 SUPER COOLING O() FIG. 1. Growth-induced anisotropy of substituted BiYIG as a function of supercooling. 325 J. AppL Phys. 61 (1). i January 1987 0021-8979/87/010325-03$02.40 @ 1986 American Institute of Physics 325 [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: Sun, 23 Nov 2014 11:57:35This flux has been shown superior to a simple PbO:Bi203 mixture with respect to Bi incorporation and K ~ genera tion.4-Representative melt compositions are given in Table I. Two melts were used to grow type-II material. One of them (lIB) contained ~ 15% more Sm, Tb, and Gd oxide than type IIA. Saturation temperatures were 883 ± 2 "C. Films were grown with supercooling ranging from 45 to 115 K. The following film properties were measured on all films: thickness, stripe width, collapse field, anisotropy field (sea of bubbles ), lattice constant, and Faraday rotation at 543 nm as described elsewhere.5 From these film properties a num ber of material properties were calculated: moment (4rrI\.fs)' strain-and growth-induced anisotropy (K~ and K ~ ), and Bi content (X Bi ). For several of the SmGdTb films, addi tional measurements included the temperature variation of collapse field and stripe width between -50 and + 150°C and the Gilbert parameter a, and the anisotropy field using ferromagnetic resonace at temperatures ranging from -60 to + 160°C. RESULTS The growth induced anisotropy K! of various films is shown in Fig. 1 as a function of supercooling. Films of com position (YBiCah (FeSiGe) s 012 yielded the anisotropy versus supercooling relation labeled YBi. Clearly Gd, Ho substitution leads to a loss in K;. On the other hand, the substitution of Sm, Tb, and Od into bismuth yttrium iron garnet has a beneficial effect on K!. Given the need for 45- 50 kerg/cm3 of K! for a 6-Jlm period device fabrication as indicated by the dashed lines on Fig. 1, the amount of super cooling (aTs) required with a GdHo melt is 80-85 K as shown in Fig. 1. For SmTbGd material this much K ~ can be obtained with only 45-50 K. Significant improvements in defect density and wafer yield are expected from such large reductions in t:.T,. Table II lists bubble and material parameters at room temperature for a number of representative films. No effort was made to bring the bubble parameters within a given spe cification rage. In fact, the moment is somewhat high for 6- or 8-.ttm period devices. It is, however, comparable for the two types of film. The temperature variation of the collapse TABLE n. Film and material properties. Film Type' 4126 4130 4151 4269 II 428~ II 4446 II 6.T$ b (K) 53 67 94 82 77 53 Th' GrR<I (I'm) (I'mimin) 2.g1 0.40 2.51 0.42 1.96 0.39 2.82 O.4S 2.34 0.46 2.12 0.42 SW' Ho' 41TMf! H/;.h (I'm) (G) (G) (Oe) 1,74- 453 682 1230 1.88 385 632 1790 1.53 404 680 1700 1.92 466 732 2420 1.72 436 710 2330 1.68 395 668 1870 "Type I Composition is Y 1.5 Gdo.2 HOc.2 Ca".7 Bio.4 Fe4.3 Sic .• Gen., 0,.2' Type II Composition is Y 1.6 SITIo.! Tho., Gd", 010.7 Bio .• Fe •.• Sio., GeO.3 0,2' bAT, = supercooling. eTh = thickness. dGrR = growth rate. ·SW = stripe width. 'Ho = collapse field. "47rM, = moment. hHk = anisotropy field. kao = lattice constant. 326 J. Appl. Phys., Vol. 61, NO.1, 1 January 1987 .. 3000 0 .. 0 2500 0 .. .. 2000 0 ., 0 .. >< 0 :x: 1500 .. c .. 0 1000 .. LL 4446 (ITS) 0 o LL4285 mAl 500 9100 -50 0 150 T{'C) FIG. 2. Anisotropy field of two films of Sm-, Tb-, Gd-substituted BiYIG (type II) as a function of temperature. field was adjusted to match the temperature variation of standard permanent bias magnets ( = -O.20%/deg) by fine tuning the Gd and the Tb and/or Ho contents. DISCUSSION Approximate composition estimates for the two types of films under consideration were obtained using Biolsi's6 dis tribution coefficient of rare earths with respect to Y. Type-I film is described as (Y1.5 Gdo.2 HOO.2 Cao.? BiOA ) Fe4.3 SioA GeO.3 012, while type-II film is given as (YI.6SmO.l Gda.2 Tbo.l RiOA Caa.6 )Fe4ASio.3 Gee.3 012, These composition estimates are probably correct within 0.1 atom per formula unit Cafu) for all components. However, for Bi the concentration (X Bi ) as determined from Faraday ad K"' K.~ I K! n~ xM; lIolHo 0 cr' r' (A) (kerg/crn3) (afu) (%/deg) (MHz/De) 12.38560.33.4 2.0 35.4 .33 12.3836 12.3966 12.4037 12.4019 12.3882 45.0 0 45.0 46.0 13.0 59.0 70.5 20.1 90.6 65.3 13.3 84.1 49.7 4.6 54.3 'Ku = uniaxial anisotropy. m K ~ = strain-induced Ku nK~ = growth-inducedKu "XB; = Bi content. 0.38 0.48 0.48 -0.210 C.43 --0.197 O.li C.25 -0.IS8 0.14 PH bl Ho = temperature coefficient of collapse field. (~(I/Ho)(dHr/dnIT~50'C)' qa = Gilbert damping parameter. ry = gyromagnetic constant. 2.68 Luther at al. 326 [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: Sun, 23 Nov 2014 11:57:35TABLE III. H k ratios at temperature extremes. LL4446 LL4285 8-pmBiGdHo 8-pm CaGe" 1.48 1.42 US 1.73 1.98 1.83 1.72 3.22 'CaGe refers to films of the general composition (YSmLuCah (FeGe) 5012' rotation measurements varies with supercooling as seen in Table II. The temperature dependence of the anisotropy field H k for representative films of composition IIA and IIB is shown in Fig. 2. Quantitative comparisons with other material compositions are made in Table III. Here the values of H k at -60°C and + 160°C are compared to H k at 25°C. The H k ratios are smaHest for type-I films and largest for CaGe (standard material) films. The low-temperature ratios for the type-II films are about halfway between those of the type-I and CaGe (standard material) films. The high-tem perature ratios for the type-II films are much closer to the high-temperature ratio for type-I film. These ratios are sig nificant because they serve as predictors for the high-and low-temperature performance of devices made on these ma terials. Type-I films, with their weak Hk -temperature de pendence (i.e., small ratios) have been shown to produce excellent device performance over a wide temperature range ( -5S to + 125 ·C). 1,7 Devices made on CaGe films have not operated well at low or high temperatures. No devices have been fabricated to date on type-II films. The ratios in Table III suggest that such devices would operate well at high temperatures but might show some problems at low temperatures. Tests of devices fabricated on type-II films are necessary to determine the accuracy of these predictions. It is also noteworthy that the small increase in rare earth content (about 15%) between type-II films 4285 and 4446 produces a noticeable increase in the ratios. This sug gests that the use of the Sm, Tb, Gd combination to increase K ~ would have to be very limited and closely controned to produce reproducible, good device material. Only device re sults will be able to show how much Sm, Tb, Od substitution can be tolerated before the operational temperature range of devices is restricted. The temperature dependence of the collapse field H biRo has been studied in detail for 8-,um period Gct, Ho bubble film and can be predicted from the empirical rela tion2 x +lY + 0.91 _ Gd Y'Ho %/deg, Xy +XGd +XHo 327 J. Appl. Phys., Vol. 61, No.1, 1 January 1987 For a 6-,um period (417M3 -600 0) the absolute value of HblHo, is larger by 0.02%/deg. For the type-I films with X Gd -0.2 and X Ho -0.2, the predicted temperature depen dence is -0.20%/deg. For the Sm, Th, Gd films (type II) no previous data were available. In the present study Gd20, was added until aH bHo value of -0.20 ± 0.01 %/deg was reached. The value of the Gilbert damping parameter a can be calculated from the FMR linewidth t:JI using8 MI = 2a(uly, where (;) is the cavity frequency and y is the gyromagnetic ratio. In type-I films, only Ho contributes linearly to AH, at the known rate of 7000 Oe per 3 atoms per formula unit Cafu),8 In the type-II films, Sm and Tb are present in equal concentrations and contribute at rates of 2000 and 12500 per 3 afu.8 Thus, it was anticipated and later confirmed that compared to Ho only one half the combined amount of Sm and Tb would be needed to yield the same value of a as type-! films. The damping requirements for device operation are not accurately known. A Gilbert damping coefficient of 0.11 has been shown to produce reliable bubble device operation. 2 CONCLUSIONS The growth-induced anisotropy of bismuth yttrium iron garnet bubble films is sensitive to rare-earth substitution. Both positive and negative effects are possible. Careful selec tion of rare earths and fine tuning their concentrations as guided by trade-off's in materi.al properties can probably re duce the amount of supercooling needed during growth and thereby lead to improvements in defect densities. However, device tests are necessary to ascertain whether such im provements would be at the cost of impaired low-tempera ture device performance. ACKNOWLEDGMENT This work was supported by the Tri-Servlce/NASA contract no. F3361S-81-C-1404. IS. E. G. SIusky, J. E. Ballintine, R. A. Lieberman, L. C, Luther, and T. J. Nelson, IEEE Trans. Magn. MAG· IS, 1286 (1982). zP. I. Bonyhard, F B. Hagedorn, Gov. Rept. AFWAL-TR-83-1l21 (1983), p. 13. -'V.!. Fratelio, S. E. G. Slusky, Co D. Brandle, and M. P. Norelli, J. Appl. Phys, 60, 2488 (1986). 4L. C. Luther and M. P. Norelli (unpublished). -'L, C. Luther, V. V. S. Rana, S. 1. Licht, and M. P. Norelli (unpublished}. ·W. A. Biolsi (unpublished) (For Gd, Ho, Th. k = 1.0, for Sm, k = 0.85, all with respect to Y. ) fL. G. Arbaugh Ir. and R. r. Fairholme, paper HE-6, Intermag. Conf., Phoenix, AZ, April 14-17, 1986. "w. H. von Aulock. ed. Handbook oJlificrowave Ferrite Materials (Aca demic, New York, 1965). Luther eta!. 327 [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: Sun, 23 Nov 2014 11:57:35
1.1855533.pdf	Numerical simulation of write-operation in a magnetic random access memory cell array with a magnetostatic interaction Y. Nozaki, H. Terada, and K. Matsuyama Citation: Journal of Applied Physics 97, 10P505 (2005); doi: 10.1063/1.1855533 View online: http://dx.doi.org/10.1063/1.1855533 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 Low write-current magnetic random access memory cell with anisotropy-varied free layers J. Appl. Phys. 104, 113901 (2008); 10.1063/1.3032894 Performance of write-line inserted magnetic tunneling junction for low-write-current magnetic random access memory cell J. Appl. Phys. 103, 07A711 (2008); 10.1063/1.2839288 Ultrafast direct writing scheme with unipolar field pulses for synthetic antiferromagnetic magnetic random access memory cells Appl. Phys. Lett. 87, 142503 (2005); 10.1063/1.2043236 Improved selectivity in magnetic random access memory arrays using hysterons J. Appl. Phys. 97, 10C502 (2005); 10.1063/1.1846971 Cell writing selection when using precessional switching in a magnetic random access memory J. Appl. Phys. 95, 1933 (2004); 10.1063/1.1641145 [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: 134.124.28.17 On: Tue, 11 Aug 2015 00:48:14Numerical simulation of write-operation in a magnetic random access memory cell array with a magnetostatic interaction Y. Nozaki, H. Terada, and K. Matsuyama Department of Electronics, Kyushu University, Hakozaki 6-10-1, Higashi-ku, Fukuoka 812-8581, Japan sPresented on 11 November 2004; published online 17 May 2005 d The margin for selective write-operation in a current coincident scheme has been numerically evaluated by considering a magnetostatic interaction in a magnetic random access memory cellarray.Foraconventionalmethod,themarginover20%cannotbeachievedasthecellsizeissmallerthan 0.2 30.4 mm2. This is mainly attributed to the degradation of ﬁeld localization created by a conductor current. The minimum cell size ensuring the practical margin can be decreased to 0.1630.24 mm2by using an opposing current ﬂowing through neighboring conductors. The margin is found to be remarkably decreased as a current pulse width becomes less than 0.4 ns because of agyromagnetic effect. © 2005 American Institute of Physics .fDOI: 10.1063/1.1855533 g As the bit integration density of magnetic random access memory 1sMRAM dhas been increasing towards a Gbit/cm2 order, the margin for selective write-operation and the ther- mal stability of the MRAM bits are expected to be sup-pressed due to bit-to-bit magnetostatic coupling. The cou-pling ﬁeld between the bits produces the distribution of theswitching ﬁeld of the bits, resulting in the suppression of theoperation margin for programming. Janesky et al.have ex- perimentally investigated the collective switching behaviorof the array of submicron patterned magnetic elements toevaluate the magnetostatic interaction ﬁeld between theelements. 2However, it is essential for MRAM applications to clarify the individual switching properties of the bits,which strongly depend on the magnetization directions of thesurrounding bits. The degradation of the ﬁeld distributioncreated by the conductor current should also be consideredfor denser cell conﬁgurations. In this paper, the inﬂuence ofthe bit-to-bit magnetostatic coupling on the switching prop-erties of the MRAM bit array has been investigated bymeans of numerical simulations using the Landau–Lifshitz–Gilbert sLLG dequation. The pulse width and the phase dif- ference between two orthogonal current pulses have alsobeen optimized to ensure the practical operation margin. The bit programming in a current coincident scheme was simulated for 9 39 matrix of Stoner-like bits with the rect- angular shape of F3F3tsF=60, 80, 100, and 150 nm d. Numerical integration of the LLG equation was performedusing the conventional fourth-order Runge–Kutta algorithm.The damping factor aused for the simulations was 0.008, which is appropriate for a typical soft magnetic material used for a free layer of a MRAM bit.3,4Here, the thickness of the bittwas deﬁned as satisfying the practical thermal stability sKuF2t=80kBTd. It has been considered that the maximum magnitude of the magnetic ﬁeld produced by a practical con- ductor current cannot exceed 100 Oe. When the bit iscooperatively switched by using both easy- and hard-axisﬁelds,H eandHh, with the same magnitude s,100 Oe d, the switching ﬁeld Hks=2Ku/Msdof the bit should be less than 277 Oe because the Stoner-like bits sat- isfy the relationship, He2/3+Hh2/3=Hk2/3. Assuming the satura-tion magnetization of the bit as 4 pMs=1.0 3104G, the maximum uniaxial anisotropy Kuavailable for the successful switching is evaluated to be 1.1 3105erg/cm3. In our simu- lations, these values were used as the Kuand the 4 pMsfor each bit. The cell integration period along the easy-axis pe was varied from 2 Fto 6F, whereas that along the hard-axis phwas ﬁxed as 2 F. The cross sections of the conductors creating the Heand theHhwereF3400 and F3300 nm, respectively, and the separation between these conductorsand the bit were 150 and 50 nm, respectively. For the case ofF=100 nm, the magnitude of the H eand theHhcreated by pulsed easy- and hard-axis switching currents, IeandIh, were 6.4 and 12 Oe/mA, respectively. The stray ﬁeld distributionin the array was calculated using the surface charge model.The magnitude of the stray ﬁeld H surrproduced by the sur- rounding bits was calculated at the center of the bit. In ourmodel, it was found that the magnitude of the H surrwas nearly proportional to 1/ r3, whererwas the distance from the bit. The magnitude of the Hsurrproduced by the fourth nearest neighbor bits along the easy-axis direction was lessthan 0.5 Oe, which was 1.4% of the H surrby the ﬁrst nearest neighbor bits and was much smaller than the Hkof the bit. Consequently, it was considered that the number of the ma-trix bits s939dused in our simulations was enough to evalu- ate the inﬂuence of the magnetostatic interaction among the bits on the switching properties. Figure 1 shows the p edependence of the current margin, which is deﬁned as 2 sIu−Ild/sIu+Ild, whereIuandIlare up- per and lower limits to switch only the selected bit, respec- tively. The rise time trs0%–100% dand the pulse width tw s50%–50% dof theIeare 20 and 40 ns, respectively. For the current pulse Ih,tr=10 and tw=70 ns. The phase of the Ie matches with that of the Ih. Under this condition, the quasi- static energy minimization is dominated in the magnetizationreversal mechanism because the pulse duration time is muchlonger than the typical relaxation time of a few nanoseconds.The lower limit I lcorresponds to the minimum current to switch the selected center bit with the magnetostatically moststable bit arrangement as schematically shown in Fig. 2 sad. The upper limit I uis given by the maximum current ampli-JOURNAL OF APPLIED PHYSICS 97, 10P505 s2005 d 0021-8979/2005/97 ~10!/10P505/3/$22.50 © 2005 American Institute of Physics 97, 10P505-1 [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: 134.124.28.17 On: Tue, 11 Aug 2015 00:48:14tude to switch the selected bit without disturbing the neigh- boring half-selected one with most magnetostatically un-stable conﬁguration illustrated in Fig. 2 sbd.As shown in Fig. 1, the margin for selective switching is reduced with decreas-ing both the Fand thep e. This is mainly attributed to the degradation of the localization of the easy- and hard-axisﬁelds created by the conductor currents. As the Fdecreases from 150 to 60 nm, the relative magnitude of the hard-axisﬁeld at the ﬁrst nearest neighbor bits is increased from 0.25to 0.57, and consequently the margin is completely vanishedatF=60 nm. To improve the ﬁeld localization, we propose another switching method, which allows the conductor currents toﬂow back through the neighboring two parallel lines as illus-trated in Fig. 3 sad. In this conﬁguration, the ﬁeld created by the opposing current ﬂow compensates the magnetic ﬁeldaffecting on the neighboring bits. The distribution of thehard-axis ﬁeld calculated for F=80 nm is shown in Fig. 3 sbd, where the results for the ﬂow back sFBdconﬁguration is shown as closed circles. For comparison, the ﬁeld distribu-tion calculated for a conventional single current ﬂow is alsoshown as open circles. As shown in Fig. 3 sbd, the relative magnitude of the hard-axis ﬁeld at the neighboring bits canbe reduced from 0.46 to 0.27 by using the FB method. Theoperation margin for the FB method sF=80nm dis shown in Fig. 1 as closed squares. It is found that the practical opera- tion margin sabove 20% dcan be achieved even for F =80 nm and p e=3F, corresponding to the bit density of 2.6 Gbit/cm2. Here, it should be noted that the efﬁciency of the ﬁeld generation with the FB method is decreased by halfcompared with that using the conventional method.For the improvement of the packing density of the MRAM, it is also important to clarify the inﬂuence of thebit-to-bit magnetostatic interaction on the energy barrier DE for the magnetization reversal. In our model, the total mag-netic energy of the bit is given by Es ud=Kusin2u −2KusHsurr/Hkdcossb−ud, where uandbare the angles of theMand theHsurrwith respect to the easy axis as illustrated in the inset of Fig. 4 sad. The solid line in Fig. 4 sadshows the calculated proﬁle of the DE, assuming the magnetostatically most unstable bit arrangement for F=80 nm and pe=3F.I n this case, the magnitude of Hsurr/Hkis 0.092 and bis 153°. As shown in this ﬁgure, it is found that the height of DEis suppressed due to the bit-to-bit magnetostatic interaction.Figure 4 sbdshows the cell size dependence of the reduced energy barrier DE/K uV. As thepedecreases from 6 Fto 2F, FIG. 1. Integration period pedependence of the current margin for the selective writing operation. The open circles, triangles, and squares indicatethe results calculated for F=150, 100, and 80 nm, respectively. The opera- tion margin for the ﬂow back method sF=80nm dis also shown as closed squares. FIG. 2. Schematic bit arrangements where the center bit is magnetostaticallysadmost stable and sbdmost unstable due to the bit-to-bit magnetostatic interaction. These arrangements are evaluated for the device conﬁgurationwithp e=2F. FIG. 3. sadSchematic diagram of the proposed ﬂow back conﬁguration. sbd Distributions of the hard-axis ﬁeld produced by the conductor current Ia. Open and closed circles indicate the ﬁeld produced by the current with thesingle sconventional dand the ﬂow back conﬁgurations, respectively. FIG. 4. sadSuppression of the energy barrier DEdue to the surrounding ﬁeld ofHsurr/Hk=0.092 at b=153°. This result is calculated for the cells for the case of F=80 nm and pe=3F.sbdReduced energy barrier DE/KuVas a function of the integration period pe.10P505-2 Nozaki, Terada, and Matsuyama J. Appl. Phys. 97, 10P505 ~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: 134.124.28.17 On: Tue, 11 Aug 2015 00:48:14the angle bgradually approaches 135°, where the lowering effect on the DEby an applied ﬁeld is generally the largest because of the astroid characteristics of the switching thresh-old in the Stoner-like particle. Furthermore, the magnitude oftheH surris also increased with decreasing the pe. The re- markable decrease of the DEwith decreasing the peis in- duced by these changes of the Hsurrand the bdependent upon the pe. The DEdecrease of over 30% from the intrinsic barrier height KuVis observed when the cell size becomes smaller than 0.16 30.24 mm2. Figure 5 shows the operation margin as a function of the phase difference tdefbetween the Ieand theIh. For both pulsed currents, the trand thetware ﬁxed to be 1 and 2 ns, respectively.The maximum margin of about 30% is achievedin the range of −1.0 łt defł+1.25 ns, where the maximum magnitude of the Ieis overlapped with that of the Ih. The margin for the positive tdefseems to be a little bit larger than that for the negative tdef. When the tdefis positive, the ﬁeld created by two orthogonal conductor currents is rotated from90° to 180° with respect to the initial magnetization direc-tion. The inverse rotation of the ﬁeld is achieved as t def,0. Fortdef=0, the magnetic ﬁeld is continuously applied at 135°, where the switching ﬁeld is minimized for a Storner-like particle. For the case of t def.0, the direction of the applied ﬁeld is varied simultaneously with the rotation ofmagnetization. Generally, the effective torque for the magne-tization rotation is produced by the ﬁeld component perpen-dicular to the magnetization. The t defdependence of the mar- gin may be induced by the efﬁciency of the magnetizationreversal.The pulse width dependence of the margin is shownin Fig. 6, where the ratio between rise time and pulse width,t r/tw, is ﬁxed to be 0.5. In our simulations, remarkable dif-ference between the results for tdef=0 and 0.375 twis not observed. The operation margin is markedly decreased as thet wbecomes less than 0.4 ns, where the switching and non- switching regimes alternatively appear due to a large-angleprecessional motion of magnetization. 5–8 In conclusion, we have simulated the selective write- operation in the array of MRAM bits magnetostaticallycoupled with each other. For the denser cell conﬁguration,the operation margin is suppressed not only due to the dis-persion of the switching ﬁeld caused by the bit-to-bit mag-netic interaction but also due to the degradation of the ﬁeldlocalization created by the conductor current. By using theproposed ﬂow back conﬁguration for the programming cur-rents to improve the ﬁeld localization, the current marginover 20% can be achieved for the cell size of 0.1630.24 mm2. For the pulse width lower than 0.4 ns, the mar- gin shows remarkable decrease due to the appearance of thegyromagnetic effect. As the cell size becomes less than0.1630.24 mm2, the reduction of the energy barrier associ- ated with the bit-to-bit magnetostatic interaction exceeds30% of the intrinsic uniaxial anisotropy of the bit. 1S. Tehrani et al., Proc. IEEE 91, 703 s2003 d. 2J. Janesky, N. D. Rizzo, L. Savtchenko, B. Engel, J. M. Slaughter, and S. Tehrani, IEEE Trans. Magn. 37, 2052 s2001 d. 3M. R. Freeman, W. Hiebert, and A. Stankiewicz, J. Appl. Phys. 83, 6217 s1998 d. 4C. E. Patton, Z. Frait, and C. H. Wilts, J. Appl. Phys. 46,5 0 0 2 s1975 d. 5M. Bauer, J. Fassbender, and B. Hillebrands, Phys. Rev. B 61, 3410 s2000 d. 6S. Kaka and S. E. Russek, Appl. Phys. Lett. 80, 2958 s2002 d. 7H. W. Schumacher, C. Chappert, P. Crozat, R. C. Sousa, P. P. Freitas, J. Miltat, J. Fassbender, and B. Hillebrands, Phys. Rev. Lett. 90, 017201 s2003 d. 8T. Devolder and C. Chappert, J. Appl. Phys. 95,1 9 3 3 s2004 d. FIG. 5. Operation margin as a function of the relative phase difference between the Ieand theIh. FIG. 6. Operation margin as a function of the duration time of the pulsed conductor current.10P505-3 Nozaki, Terada, and Matsuyama J. Appl. Phys. 97, 10P505 ~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: 134.124.28.17 On: Tue, 11 Aug 2015 00:48:14
1.4874135.pdf	Effect of Dzyaloshinskii–Moriya interaction on magnetic vortex Y. M. Luo, C. Zhou, C. Won, and Y. Z. Wu Citation: AIP Advances 4, 047136 (2014); doi: 10.1063/1.4874135 View online: http://dx.doi.org/10.1063/1.4874135 View Table of Contents: http://scitation.aip.org/content/aip/journal/adva/4/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Interface boundary conditions for dynamic magnetization and spin wave dynamics in a ferromagnetic layer with the interface Dzyaloshinskii-Moriya interaction J. Appl. Phys. 115, 233902 (2014); 10.1063/1.4883181 The effects of Dzyaloshinskii-Moriya interactions on the ferromagnetic resonance response in nanosized devices J. Appl. Phys. 115, 17C902 (2014); 10.1063/1.4870138 Influence of the Dzyaloshinskii-Moriya interaction on the spin-torque diode effect J. Appl. Phys. 115, 17C730 (2014); 10.1063/1.4867750 Chiral magnetization textures stabilized by the Dzyaloshinskii-Moriya interaction during spin-orbit torque switching Appl. Phys. Lett. 104, 092403 (2014); 10.1063/1.4867199 Magnetic vortex generated by Dzyaloshinskii–Moriya interaction J. Appl. Phys. 113, 133911 (2013); 10.1063/1.4799401 All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license. See: http://creativecommons.org/licenses/by/3.0/ Downloaded to IP: 192.231.202.205 On: Thu, 18 Dec 2014 17:50:41AIP ADV ANCES 4, 047136 (2014) Effect of Dzyaloshinskii–Moriya interaction on magnetic vortex Y . M. Luo,1C. Zhou,1C. Won,2and Y . Z. Wu1,a 1State Key Laboratory of Surface Physics, Department of Physics, and Advanced Materials Laboratory, Fudan University, Shanghai 200433, People’s Republic of China 2Department of Physics, Kyung Hee University, Seoul 130-701, Korea (Received 12 January 2014; accepted 18 April 2014; published online 28 April 2014) The effect of the Dzyaloshinskii–Moriya (DM) interaction on the vortex in magnetic microdisk was investigated by micro-magnetic simulation based on the Landau– Lifshitz–Gilbert equation. Our results show that the DM interaction modiﬁes the size of the vortex core, and also induces an out-of-plane magnetization component at theedge and inside the disk. The DM interaction can destabilizes one vortex handedness, generate a bias ﬁeld to the vortex core and couple the vortex polarity and chirality. This DM-interaction-induced coupling can therefore provide a new way to controlvortex polarity and chirality. C/circlecopyrt2014 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License. [http://dx.doi.org/10.1063/1.4874135 ] I. INTRODUCTION A novel antisymmetric exchange coupling1,2called the Dzyaloshinskii–Moriya (DM) interac- tion has recently attracted great interest. The DM interaction arises from spin-orbit scattering of electrons in an inversion asymmetric crystal ﬁeld, and it exists in systems with broken inversion symmetry, such as in speciﬁc metallic alloys with B20 structure3–7and at the surface or interface of magnetic multi-layers.8–10The existence of the DM interaction can induce chiral spin structures such as skrymion,3–10unconventional transport phenomena,11–13and exotic dynamic properties,14–16 many of which stimulated interest in fundamental magnetism studies and provided new possibilities for the development of future spintronic devices. Besides the DM-interaction-induced effects in bulk materials and thin ﬁlms, the practical con- sequences of the DM interaction in conﬁned structures such as magnetic nanodisks and nanostripes have begun to attract increasing interest.17–20The stable magnetic conﬁguration in sub-micrometer scale magnetic microdisk is a magnetic vortex, which can be characterized by an in-plane curlingmagnetization (chirality) and a nanometer-sized central region with an out-of-plane magnetization (polarity). 21,22The chirality can be clockwise ( C=−1) or counterclockwise ( C=1), and the po- larity can be up ( P=1) or down ( P=−1). The vortices can also be classiﬁed as left-handed vortex (CP=−1) and right-handed vortex ( CP=1).23,24In the classic model, the vortex chirality and polarization are not coupled and can be switched independently, but one recent experiment indicated that the DM interaction can break this symmetry.24A few theoretical investigations were carried out to explore the inﬂuence of the DM interaction on magnetic vortices.19,20Through Monte-Carlo simulation, Kwon et. al showed that even without the dipolar interaction the DM interaction could induce the vortex structure in a nanodisk with the radium less than 30nm.20Butenko et. al found that DM coupling can considerably change the size of vortices,19but ignored its effect on the magneti- zation at the disk edge which exists in the real magnetic vortex system. Usually the experimental studies on the magnetic vortex were performed in the magnetic disk with the diameter around themicrometer size. 21–24 aCorrespondence to: wuyizheng@fudan.edu.cn 2158-3226/2014/4(4)/047136/10 C/circlecopyrtAuthor(s) 2014 4, 047136-1 All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license. See: http://creativecommons.org/licenses/by/3.0/ Downloaded to IP: 192.231.202.205 On: Thu, 18 Dec 2014 17:50:41047136-2 Luo et al. AIP Advances 4, 047136 (2014) In order to have a measurable study on the effect of DM interaction on the magnetic vortex before the further experimental study, we performed a micro-magnetic simulation together with the existing demagnetization ﬁeld on a magnetic microdisk. Our results showed that the existence of the DM interaction not only shrinks or broadens the vortex core, but it also induces an out-of-planemagnetization component both at the edge and at the disk plane, which has a linear dependence on the DM interaction strength. Moreover, we found that the DM interaction can induce a bias ﬁeld on the vortex core, so that a clear bias effect can be observed through the vortex core switchingprocess. Thus the DM interaction can couple the vortex chirality and polarity, which provides a new possibility for manipulating the vortex chirality and polarity together. II. EQUATIONS AND METHODS In our simulation, the spin system is described by a 2-dimentional (2D) square lattice, with a local magnetic moment \|/vectorm\|=MSat each site. This model includes the ferromagnetic exchange interaction, the DM interaction, the magnetic dipole interaction, and Zeeman coupling, and the magnetocrystalline anisotropy was ignored. The Hamiltonian can be written as: E=− J/summationdisplay <i,j>/arrowrighttophalfmi·/arrowrighttophalfmj+D/summationdisplay <i,j>/arrowrighttophalfrij·(/arrowrighttophalfmi×/arrowrighttophalfmj)−μ0/arrowrighttophalfH·/summationdisplay i/arrowrighttophalfmi−1 2μ0/summationdisplay i/arrowrighttophalfmi·/arrowrighttophalfHd (1) where J,D,/arrowrighttophalfrij,μ0,/arrowrighttophalfHand/arrowrighttophalfHddenote the exchange constant, the DM constant, the distance vector between the spin sites iandj, magnetic permeability, the external magnetic ﬁeld and demagnetization ﬁeld, respectively./arrowrighttophalfHdis computed from the magnetization distribution through the magnetostatic equations.25,26In this study, we considered the DM interaction as those in the materials with B20 structure3–7that can induce the helical spiral stripes. The spiral period is determined by the ratio J/D,6and the helical direction is determined by the sign of D:n e g a t i v e Dproduces a left-handed helical structure, and positive Dproduces a right-handed helical structure.9 Generally, the spin conﬁguration can be simulated by numerically solving the Landau–Lifshitz– Gilbert (LLG) equation: d/arrowrighttophalfmi dt=−\|γ\|/arrowrighttophalfmi×/arrowrighttophalfHef f−α Ms/arrowrighttophalfmi×d/arrowrighttophalfmi dt(2) with the total effective ﬁeld/arrowrighttophalfHef f, the Gilbert gyromagnetic ratio γand the damping constant α. The total effective ﬁeld includes the exchange ﬁeld, the dipole ﬁeld, the DM ﬁeld, and the external ﬁeld, and can be written as: /arrowrighttophalfHef f=−1 μ0∂E ∂/arrowrighttophalfmi(3) where Eis the total energy of the system as expressed in Eq. (1). According to Eq. (1), the effective ﬁeld on the spin at the site iinduced by the DM interaction from the nearest neighboring spins at the sites jcan be expressed as: /arrowrighttophalfHi DM=−1 μ0∂D/summationtext j/arrowrighttophalfrij·(/arrowrighttophalfmi×/arrowrighttophalfmj) ∂/arrowrighttophalfmi=−D μ0Ms/summationdisplay j(/arrowrighttophalfmj×/arrowrighttophalfrij)( 4 ) In the continuous limit, the DM interaction can be written as27EDM=D/arrowrighttophalfm·(∇×/arrowrighttophalfm), thus its effective ﬁeld can be expressed as: /arrowrighttophalfHDM=−2D μ0Ms(∇×/arrowrighttophalfm)( 5 ) We realized the simulation by adding a DM interaction module into the standard micro-magnetic simulation software OOMMF.28The micro-magnetic simulation is usually considered to directly All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license. See: http://creativecommons.org/licenses/by/3.0/ Downloaded to IP: 192.231.202.205 On: Thu, 18 Dec 2014 17:50:41047136-3 Luo et al. AIP Advances 4, 047136 (2014) FIG. 1. Magnetization conﬁgurations in the vortices with different Dvalues. The color represents the out-of-plane magne- tization direction as indicated by the right color bar; the arrows denote the in-plane magnetization direction. compare with the real material system, so in the simulation we chose the typical parameters,29 such as Ms=8×105A/m, the exchange stiffness A=1.3×10−11J/mandα=0.01, but the DM interaction was regarded as a tuning parameter. In the simulation, the nanodisk diameter is 50 2 nm, and the thickness is 50 nm; in this case the stable magnetic conﬁguration is vortex. We only report the simulation results with the unit cell of 2 ×2×50nm3, thus the center spin can point to the direction exactly perpendicular to the ﬁlm plane. We also did the 2D calculation with a unit cellsize of 1 ×1×50nm 3on a diameter of 501nm, and obtained the same results. So the chosen unit cell of 2 ×2×50nm3is accurate enough for the current study. In order to make sure that the DM interaction will not signiﬁcantly change the Neumann boundary condition used in the OOMMFcode, 30we tested the calculation on 2 ×2 disk arrays with 100nm separation, and obtained the same results as shown in the single disk. Therefore, it is still valid to introduce the DM interaction module into the OOMMF code. In this paper, we only present the simulation results on the vortex with the counterclockwise chirality, and the similar results can be expected for the vortex with the clockwise chirality by reversing the sign of the DM interaction. III. RESULTS AND DISCUSSION To systematically study the effect of the DM interaction on a magnetic vortex, we ﬁrst simulated the stable vortex magnetic conﬁguration with up polarity and counterclockwise chirality for D=0, and then studied how the DM interaction inﬂuenced the magnetic conﬁguration by gradually varying theDvalue. Fig. 1shows the simulated magnetic conﬁguration with different Dvalues. There is a phase transition from a vortex state to a helical stripe state when the DM strength reaches a threshold Dcrit. This threshold depends on the disk parameters, and the Dcritvalue in this simulation is 1.76 mJ/m2. When \|D\|>Dcrit, the DM energy is strong enough to overcome the dipole energy in the system, so then the interplay between the DM energy and the exchange energy forms a helicalspin structure, which is close to the helical stripe phases in 2D thin ﬁlm. 6For\|D\|<Dcrit, the disk still keeps the vortex conﬁguration, but a positive Dcan widen the core, while a negative Dshrinks the core. This phenomenon is consistent with the previous study based on the analytical model.19 All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license. See: http://creativecommons.org/licenses/by/3.0/ Downloaded to IP: 192.231.202.205 On: Thu, 18 Dec 2014 17:50:41047136-4 Luo et al. AIP Advances 4, 047136 (2014) FIG. 2. (a) The size of the vortex core as a function of the Dvalue for the vortex with the counterclockwise chirality. The inset shows the typical Mzline proﬁle across the vortex core with two representative D values, rdenotes the distance to the center. The simulations were performed with the unit cell size of 2 nm and 1 nm respectively. The size of the vortex core isdeﬁned as the peak width at M Z=0.5 indicated by the red arrow. (b) and (c) show schematic drawings of the DM ﬁeld at the vortex core for (b) D>0a n d( c ) D<0. Yellow arrows represent the spin structures around the vortex core, and the red arrows denote the out-of-plane component direction of the DM ﬁeld. Moreover, if we continue to reduce the negative D value, the vortex core polarity can ﬁnally be switched by the DM interaction, as shown in Fig. 1(e). This fact means that there only exists the vortex with typical handedness while the DM interaction is sufﬁciently strong. In the simulation, thecritical value to switch the polarity is D switch=−1.1mJ/m2, thus only the single handedness vortex could be observed for \|D crit\|>\|D\|>\|Dswitch \|. If the negative D value is further reduced, the DM interaction will increase the size of the vortex core with the reversed polarity, until the vortex statebreaks into the helical strip phase (Fig. 1(f)) for the strong negative Dvalue. In Ref. 19, Butenko et al. mentioned that the radial stable solutions exist only below certain critical strength of the DM constant, which may be related to the threshold from the vortex state to the helical stripe state basedon the results in Fig. 1. Our simulation further show that the vortex core size depends on the DM interaction nearly linearly, as shown in Fig. 2(a), where the vortex core size is characterized by the width at half maximum of vortex core through the line proﬁle across the vortex core. The vortex core expands from 22 nm to 38 nm as Dincreases from 0 to 1.6 mJ/m 2, but shrinks continually to 8 nm as Ddrops to−1.1 mJ/m2, below which the core polarity ﬂips from up to down, forming a left-handed vortex. The simulation with smaller unit cell size of 1 ×1×50nm3shows similar result, and only small difference of core size can be found that for the core size less than 10nm, as shown in Fig. 2(a). The DM interaction can not only inﬂuence the vortex core size, but also inﬂuence the magneti- zation conﬁguration at the disk edge and inside the disk. Fig. 3(a)shows the line proﬁles across the vortex core with different DM interaction values. The effect of the DM interaction on the core sizecould be clearly identiﬁed through the line proﬁles. Generally in a system with D =0, the spins apart from the core lie in the ﬁlm plane due to the in-plane demagnetization ﬁeld. With the existence of the DM interaction, we found the magnetization at the disk edge could be titled away from the surfaceplane. The out-of-plane component M Zat the disk edge increases linearly with the DM interaction, and reversed its sign once D changes from positive to negative, as shown in Fig. 3(b). It should be noted that the edge MZcomponent decays rapidly within 30 nm away from the disk edge, thus the tilted edge magnetization induced by the DM interaction can only be observed experimentally by those modern magnetic imaging technologies with high spatial resolution, such as magnetic force microscope,22or spin polarized scanning tunneling microscope.21Moreover, we found that the DM All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license. See: http://creativecommons.org/licenses/by/3.0/ Downloaded to IP: 192.231.202.205 On: Thu, 18 Dec 2014 17:50:41047136-5 Luo et al. AIP Advances 4, 047136 (2014) FIG. 3. (a) MZline proﬁle across the vortex core with different D values for the vortex with C =1,rdenotes the distance to the center. The insert shows the magniﬁed MZproﬁle. The arrows indicate the negative dips induced by the stray ﬁeld from the out-of-plane magnetization in the neighboring core and edge. (b) The M Zat the disk edge and at r =150 nm as a function of the DM interaction constants. interaction could induce a weak out-of-plane magnetization component even in the disk plane. The inset in Fig. 3(a) shows the ampliﬁed magnetization proﬁles with different D values, which clearly proves the M Zcomponents depend on the sign of the DM interaction. Fig. 3(b) shows that M Zat r=150nmalso changes linearly with the D value, but M Zin the disk plane has the opposite sign with much smaller amplitude than at the disk edge. Recently, Rohart and Thiaville showed that, in ultrathin ﬁlm nanostructure with out-of-plane anisotropy, the interfacial DM interaction can also bend the magnetization at the edges towards to the in-plane direction at the edges.31This is different with our results that the in-plane magnetization was titled to the normal direction by the bulk-like DM interaction. Usually, the interfacial DM interaction All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license. See: http://creativecommons.org/licenses/by/3.0/ Downloaded to IP: 192.231.202.205 On: Thu, 18 Dec 2014 17:50:41047136-6 Luo et al. AIP Advances 4, 047136 (2014) energy between two neighboring spins can be described as EDM=/arrowrighttophalfDij·(/arrowrighttophalfSi×/arrowrighttophalfSj),8–10,31and/arrowrighttophalfDij is the DM vector which is perpendicular to the distance vector/arrowrighttophalfrij. Obviously, the interfacial DM interaction is different with the bulk DM interaction in Eq. (1). It would be interesting to further check the effect of the interfacial DM interaction on the vortex in the micro-size magnetic disk with the in-plane magnetization. The inﬂuence of the DM interaction on the vortex core can be attributed to the DM ﬁeld at the core. Eq. (4)points out that the DM ﬁeld direction at the vortex core is determined by its neighboring spin direction (the vortex chirality) and the sign of D. Since the vector/arrowrighttophalfrijis always in the ﬁlm plane, the in-plane magnetization of the neighboring spins would induce an effective perpendicular DM ﬁeld. For a vortex with the counterclockwise chirality, the perpendicular component of the DM ﬁeld at the core is upward when D>0(see Fig. 2(b)) and downward when D<0 (see Fig. 2(c)), thus the vortex core expands when its polarity is parallel to the DM ﬁeld, and shrinks when its polarity isopposite to the DM ﬁeld. The internal DM ﬁeld at the vortex core also lifts the energy degeneration between the left-handed and right-handed vortices, because vortex states have lower energy if the core polarity is parallel to the DM ﬁeld. In this way, a coupling effect between the vortex polarityand chirality can be expected. The tilting of the edge magnetization can also be understood by the DM ﬁeld at the edge. Eq.(4)indicates that the in-plane magnetization of neighboring spins can induce an out-of-plane DM ﬁeld. The spin at the disk edge has two neighboring spins along the circumferential direction, but the DM interaction from those two spins would be close to zero since their magnetization vectors are almost parallel to the distance vector /arrowrighttophalfrij. The spin at the disk edge also contains one neighboring spin along the radial direction inside the disk, thus the DM interaction from this inner neighboringspin will generate the perpendicular DM ﬁeld on the edge spin, and tilt the magnetization away from the surface plane. But this DM ﬁeld at the disk edge has the opposite sign as that at the core shown in Fig. 2, so the positive D value will induce a negative M Zat the edge, and the negative D value will induce a positive M Z. The strength of the DM ﬁeld should depend on the D value, thus the edge magnetization changes almost linearly with the D value. The out-of-plane component of the magnetization in the disk plate is also attributed to the DM interaction. For the spin inside the disk, the two neighboring spins along the circumferential direction only contribute zero DM ﬁeld, but the two neighboring spins along the radial direction induce the non-zero out-of-plane DM ﬁeld with the opposite sign. Due to the in-plane curling magnetization inthe vortex, the DM ﬁeld induced by the outer neighboring spin is slightly larger than that induced by the inner neighboring spin, so that the overall perpendicular DM ﬁeld has the opposite sign as that at the disk edge which is induced only by the inner spins, and the out-of-plane magnetization in the disk has the opposite dependence on the DM interaction, as shown in Fig. 3(b). Such DM ﬁeld in the disk plate induced by the in-plane curling magnetization can be calculated quantitatively from Eq. (5).I f assuming the magnetization outside the vortex core is lying in the ﬁlm plane and rotating around the disk center, the estimated out-of-plane DM ﬁeld is \|/arrowrighttophalfHDM\|=2D μ0Msrwith rrepresenting the distance to the disk center. This DM ﬁeld will induce a weak M Zagainst the in-plane demagnetization ﬁeld /arrowrighttophalfHd. Usually the demagnetization ﬁeld in the ﬁlm can be estimated as \|/arrowrighttophalfHd\|=Mswhich could be much larger than \|/arrowrighttophalfHDM\|, and thus the out-of-plane magnetization component can be estimated as Mz≈\|/arrowrighttophalfHDM\| \|/arrowrighttophalfHd\|≈2D μ0M2sr(6) So the Mzinduced by the DM interaction should be proportional to1 rwith a slope of2D μ0M2s.T h i s relation is not valid near the vortex core and the disk edge, where the titling angle is so large that the exchange interaction and the long-range dipolar interaction can’t be omitted. As an example, we found that the dipolar interaction from the perpendicular magnetization at the core and at the edgeinduced a stray ﬁeld on the neighboring spins with the opposite direction, and such stray ﬁeld can force the magnetization near the core and the edge to tilt and form a negative dip with the opposite out-of-plane magnetization component, as indicated by the arrows in Fig. 3(a). In order to reduce All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license. See: http://creativecommons.org/licenses/by/3.0/ Downloaded to IP: 192.231.202.205 On: Thu, 18 Dec 2014 17:50:41047136-7 Luo et al. AIP Advances 4, 047136 (2014) FIG. 4. MZline proﬁle of a disk with a diameter of 1502 nm and a D value of 0.6 mJ/m2,rdenotes the distance to the center. The left insert shows the magnetic conﬁguration of the disk, and the right insert shows the magniﬁed line proﬁle andthe ﬁtting curve from the 1 /rfunction. the inﬂuence of the stray ﬁeld from the magnetization at disk edge and disk center, we performed a simulation on a magnetic disk with a large diameter of 1502 nm and D=0.6mJ/m2.F i g . 4shows the calculated magnetization proﬁle, and MZat the core and at the edge are very close to those in the disks with the smaller diameters, and MZat the edge also shows a negative value. The MZ(r) with r in the range between 100 nm and 550 nm can be well ﬁtted with a1 rfunction, as shown in the inset of Fig. 4. The ﬁtted coefﬁcient is 1 .39nm, which is very close to the theoretical value of 2D μ0M2s=1.49nm.S oE q . (6)can effectively describe the out-of-plane magnetization component in the disk with a large enough diameter. Fig.2shows that the in-plane curling magnetization could induce a perpendicular DM ﬁeld on the vortex core, so that the DM ﬁeld would be expected to induce a bias ﬁeld while the core polarity is switched by the out-of-plane magnetic ﬁeld. In order to better illustrate the reversing process of the vortex core, we only present the hysteresis loop of the small selected area around the vortexcore, as shown by the green rectangle area (62 ×62nm 2)i nF i g . 5(a).F i g . 5(b) shows the typical hysteresis loop of the selected area on the disk with D=0.2mJ/m2. The applied magnetic ﬁeld is strong enough to saturate the magnetization along the normal direction, as shown by the insets inFig. 5(b). The magnetic conﬁgurations at the remanence show clear vortex states, however for the ﬁeld sweeping downward, the vortex always has up polarity and counterclockwise chirality, and for the ﬁeld sweeping upward from a negative saturation ﬁeld, the vortex has down polarity andclockwise chirality. It is clear that the vortex core polarization can be determined by the applied ﬁeld direction during the vortex creation process, and thus the simulation results indicate that the chiral direction always follows the core polarization in the vortex-formation process. So the DMinteraction couples polarity and circularity of the vortex in magnetic microdisk, and the right-handed vortex state is the energy favorable state for positive Dvalue. We also performed the simulation without the DM interaction, and did not observe the coupling effect between the polarity and the All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license. See: http://creativecommons.org/licenses/by/3.0/ Downloaded to IP: 192.231.202.205 On: Thu, 18 Dec 2014 17:50:41047136-8 Luo et al. AIP Advances 4, 047136 (2014) FIG. 5. (a) The selected green rectangular area (62 ×62nm2) around the vortex core. (b) The magnetic hysteresis loop of the selected area in the vortex with D=0.2mJ/m2. The insets show the magnetization conﬁgurations inside the selected area at saturation states and remanence states. (c) The minor loops of the selected area with different Dvalues. The insets show the magnetization conﬁgurations at 800 mT and −800 mT with the same chirality. circularity of the vortex. In the magnet vortex induced by the DM interaction in the nanodisk with the radius less than 30nm, Kwon et al.20also showed that the polarity and circularity of the skyrmion structure are coupled together. through the ﬁeld dependent simulation. As shown in Fig. 2, the DM interaction can provide a DM ﬁeld on the vortex core. This DM interaction not only modiﬁes the vortex core size, but also is the origin to couple the polarity and the circularity of the vortex. We further understood the handedness preference of the vortices induced by the DM interaction more clearly through the minor hysteresis loop, in which the magnetic ﬁeldis only strong enough to switch the vortex core polarization without changing the vortex circularity. In this case, the applied ﬁeld should be smaller than the saturation ﬁeld, i.e. less than 860 mT in the simulation. Fig. 5(c)show the typical minor loops of the selected area with different DM interaction constants. The insets of Fig. 5(c)show that only the core polarities can be switched without changing the counterclockwise chirality during the ﬁeld sweeping process. For the conventional vortex without DM interaction, the obtained loop is symmetrical, and the vortex core switching ﬁeld H − Cfor the polarization from up to down and the switching ﬁeld H+ Cfor the polarization from down to up have the same magnitude of 650 mT. However, for the vortex with the DM interaction, a clear offset of the switching ﬁelds can be observed. When D=0.2mJ/m2,H+ cis 550 mT, and H− cis−750 mT, so that this simulation conﬁrmed that there is a positive bias ﬁeld of 100 mT on the vortex core induced by the DM interaction, which is consistent with the physical picture in Fig. 2. It requires an extra ﬁeld to overcome the bias ﬁeld for the core polarization switching from up to down. We foundthat the bias ﬁeld can reverse its sign after the DM interaction becomes negative, and is proportional to the DM interaction. Thus the observed bias ﬁeld is a clear evidence to prove the existence of the effective DM ﬁeld induced by the DM interaction. However, we can only observe the biased vortex All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license. See: http://creativecommons.org/licenses/by/3.0/ Downloaded to IP: 192.231.202.205 On: Thu, 18 Dec 2014 17:50:41047136-9 Luo et al. AIP Advances 4, 047136 (2014) core loops for the DM interaction weaker than 0 .4mJ/m2, because for stronger DM interaction, the core cannot be switched when the applied ﬁeld is less than the saturation ﬁeld. Here it should be noted that the vortex core switching ﬁeld could be increased by choosing smaller unit cell size in the simulation,32but the bias ﬁeld induced by the DM interaction has little dependence on the unit cell size. In magnetic disks, it is difﬁcult to control the vortex circularity, and usually the asymmetry disks, such as edge-cut disks,33,34were applied to control the vortex chirality. Our simulation clearly demonstrated that the DM interaction can couple the polarity and chirality together, so that the vortex chirality in a circle disk can be controlled with its polarity by switching the external ﬁeld.Although the ordinary material used in the study on magnetic vortex, such as permalloy, is unlikely to contain large DM interaction, the DM strength in the B20 materials, such as Fe 0.5Co0.5Si, can be up to 0.48 mJ/m2,6,20and the discovered DM-interaction-induced effect is feasible to be realized experimentally in the magnetic nanodisk made by the B20 materials. IV. CONCLUSIONS We have studied the effect of the DM interaction on magnetic vortices by micro-magnetic simulation based on the LLG equation. The DM interaction in magnetic mcrodisks can inﬂuence the size of the vortex core, and destabilize one vortex handedness at intermediate DMI strength, and destabilize all vortex states into the helical stripe phase for strong DMI strength. The DM interaction can also induce an out-of-plane magnetization component at the edge and an opposite component atthe disk plane. We found that the effective DM ﬁeld could induce a bias ﬁeld on the vortex core while switching the core polarization with an out-of-plane magnetic ﬁeld, and further induce the coupling between the vortex circulation and polarity. Our calculations indicate that the DM interaction can bea new and efﬁcient way to control the vortex in magnetic microdisks. We acknowledge helpful discussions with Prof. Shufeng Zhang and Prof. Haifeng Ding. This work was supported by MOST (No. 2011CB921801, No. 2009CB929203), by NSFC of China(No. 10925416 and No. 11274074), by WHMFC (No. WHMFCKF2011008), and by the National Research Foundation of Korea Grant funded by the Korean Government (2012R1A1A2007524). 1T. Moriya, Phys. Rev. 120, 91 (1960). 2I. E. Dzialoshinskii, Sov. Phys. 5, 1259 (1957). 3A. Tonomura, X. Z. Yu, K. Yanagisawa, T. Matsuda, Y . Onose, N. Kanazawa, H. S. Park, and Y . Tokura, Nano. Lett. 12, 1673 (2012). 4X. Z. Yu, N. Kanazawa, Y . Onose, K. Kimoto, W. Z. Zhang, S. Ishiwata, Y . Matsui, and Y . Tokura, Nat. Mater. 10, 106 (2011). 5H. Wilhelm, M. Baenitz, M. Schmidt, U. K. R ¨oßler, A. A. Leonov, and A. N. Bogdanov, Phys. Rev. Lett. 107, 127203 (2011). 6X. Z. Yu, Y . Onose, N. Kanazawa, J. H. Park, J. H. Han, Y . Matsui, N. Nagaosa, and Y . Tokura, Nature 465, 901 (2010). 7S. M ¨uhlbauer, B. Binz, F. Jonietz, C. Pﬂeiderer, A. Rosch, A. Neubauer, R. Georgii, and P. B ¨oni,Science 323, 915 (2009). 8S. 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1.437609.pdf	The orthogonal gradient method. A simple method to solve the closedshell, openshell, and multiconfiguration SCF equations A. Golebiewski, Juergen Hinze, and E. Yurtsever Citation: The Journal of Chemical Physics 70, 1101 (1979); doi: 10.1063/1.437609 View online: http://dx.doi.org/10.1063/1.437609 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/70/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Gradient techniques for openshell restricted Hartree–Fock and multiconfiguration selfconsistentfield methods J. Chem. Phys. 71, 1525 (1979); 10.1063/1.438494 The Hellmann–Feynman theorem for openshell and multiconfiguration SCF wave functions J. Chem. Phys. 71, 1511 (1979); 10.1063/1.438423 An algorithm to solve open and closedshell and restricted MC–SCF equations J. Chem. Phys. 70, 3188 (1979); 10.1063/1.437906 Best Choice for the Coupling Operators in the OpenShell and Multiconfiguration SCF Methods J. Chem. Phys. 48, 1994 (1968); 10.1063/1.1669003 Open and ClosedShell SCF Method for Conjugated Systems J. Chem. Phys. 34, 1444 (1961); 10.1063/1.1731759 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: 131.193.242.67 On: Mon, 01 Dec 2014 06:02:24The orthogonal gradient method. A simple method to solve the closed-shell, open-shell, and multiconfiguration SCF equations A. Golebiewski a) Institute of Chemistry. Jagellonian University. Cracow. Poland Juergen Hinze and E. Yurtsever Department of Chemistry. University of Bielefeld. Bielefeld. Germany (Received 31 July 1978) A new, simple and efficient method of solving closed-shell, open-shell, and multiconfiguration SCF orbital equations is presented. It is based on the orthogonal gradient approach introduced originally in conjunction with the maximum overlap principle. Semiempirical test calculations show that the method converges rapidly. STATUS AND STATEMENT OF THE PROBLEM In the explicit calculation of electronic wave functions for atoms or molecules, the many-electron function 11< K is generally written as a sum of antisymmetrized prod uct functions (Slater determinants) <P1, constructed from orthonormal one electron functions, the spin orbitals l/!J. They in turn, consist of spatial functions, the or bitals cp", multiplied by either spin function O! and {3. In the special case of a closed shell or open shell system with 11< K = <P K, the expansion is restricted to one Slater determinant. Using the variational principle and mini mizing the expectation value of the total energy with re spect to the orbital form leads to the Hartree-Fock equation for the closed shell system or a set of equations in the more general case. With the orbitals expanded in terms of a finite set of basis functions Xq, the Hartree Fock operators (usually called now self-consistent field or SCF operators) are represented as a matrix or a set of matrices, composed of one-and two-electron inte grals over the basis functions chosen. The theory for the case of a single Slater determinant can be traced back to Hartree, 1 Fock,2 and Slater. 3 Analytic formulation was given for the closed shell case by Roothaan4 and Hall, 5 and for the open shell case by Lefebvre, 6 Roothaan, 7 Huzinaga, 8 and Birss and Fraga.9,10 Early multi configurational (MC-SCF) calcu lations were carried out by Hartreell and Jucys12 for atoms and by Das and Wahl13 for linear molecules. General MC-SCF equations were first derived by Mc Weeny14 and have been rederived in various forms by Adams, 15 Das and Wahl, 16,17 Gilbert, 18 Hinze, 19 Huzin aga, 20 Hirao and Nakatsuji, 21 Hinze and Roothaan, 22 Veillard and Clementi, 23 and Golebiewski. 24 In the closed shell case, or more generally, if a single Slater determinant is used to represent 11< K with an equal occupation of all shells of a given type, the wave func tion is invariant to a unitary transformation among the occupied orbitals. This freedom renders the possibil ity of simplifying the Hartree-Fock equation Fcpk alOne of the authors (A. G. ) acknowledges a grant from D. A. A. D. and partial support by the Institute of Low Tem peratures and Structural Research of the Polish Academy of Sciences, Wroclaw. =~CPIE'k by eliminating theoff-diagonaILagrangianmul tipliers E,k' The orbital equations are then pseudo eigenvalue equations which can be solved using conven tional matrix diagonalization techniques iteratively. Solution of this problem seldom presents any difficulty. In the more general open shell or MC -SC F case, where shells of the same type may have different occu pation numbers, different SCF operators (matrices) are obtained for different shells of the same type. This leads to several SCF equations coupled to each other via the off-diagonal Lagrangian multipliers. SpeCial coupling operator techniques have been developed to ab sorb the off-diagonal multipliers into the SCF operators and to generate a single operator for all orbitals, yield ing again the pseudo-eigenvalue problem. 7-10,21,22,25-27 Unfortunately, the routine application of these procedures is hampered by two serious difficulties: (a) the for malism needed is rather complex, and (b), more impor tantly, SCF convergence is slow and arduous at best, if it can be achieved at all. 28,29 There are interesting alternate procedures to derive the orbital equations variationally without introducing Lagrangian multipliers to satisfy the restrictive ortho normality conditions between the orbitals. 30,31 However, the resulting numerical problems to be solved are mathematically equivalent. Recently, a significant effort has been made to de velop alternate procedures to solve the open shell and/or MC-SCF problem. A tentative alternative is the direct minimization tech nique introduced by McWeeny.32 In this technique the first order denSity matrix is varied (or, sometimes, the orbitals) after implementing some constraints due to the orthonormality of orbitals. The change of the density matrix is then chosen along the negative energy gradi ent, or the negative conjugate gradient, to a degree governed by the matrix of second derivatives, the Hessian. Alternate methods differ mainly in the way the orthonormality constraints are introduced into the problem.33-37 In some, a symmetrical reorthonormal ization is performed after each iteration. 19,38 Sutcliffe has shown, however, that the Hessian matrix may be singular, and argues that this accounts for the slow con vergence.39,40 Polezzo's treatment34-36 is said to be free J. Chern. Phys. 70(03), 1 Feb. 1979 0021-9606n9/031101-06$Ol.00 © 1979 American Institute of Physics 1101 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: 131.193.242.67 On: Mon, 01 Dec 2014 06:02:241102 Golebiewski, Hinze, and Yurtsever: Orthogonal gradient method from this problem. Nevertheless, he required a large number of iteraticns (of the order of a hundred) even for something as simple as the closed shell case of H~. In the MC-SCF case the method becomes impractical be cause one has to deal with as many SCF matrices as there are different Hartree-Fock operators. 36 Some what unexpectedly rapid convergence (six to eight itera tions) is reported for a similar procedure by Kuprie vich. 37 Related is the 2 x 2 rotation method, according to which the "best" orbitals are obtained by a finite number of orthogonal transformations of two orbitals at a time, in an exact way, with no singularity problems. 41-43 The large number of rotations required makes the method not very suitable even for molecules of moderate size. Excellent convergence may be obtained using the gen eralized Brillouin theorem. 44-46,24 Conveniently the super-CI problem is first solved, 45 accounting for all single replacements of orbitals. The resulting wave function is then contracted by a proper redefinition of orbitals. A variant of this iterative scheme has been recently suggested by Ruttink and van Lenthe. 47 Draw backs are the relatively high complexity of the scheme and, particularly, the large size of the super-CI ma trix. In the last category of methods to be conSidered, ap proximate invariance of the SCF operators with respect to a variation of the orbitals is assumed. The orbitals are retransformed to ensure the Hermiticity of the ma trix of Lagrange multipliers. Suppose we conSider a system defined by n Hartree-Fock operators in a space spanned by N basis functions. In the method due to Colle et al. 29 every SCF step consists of (n/2) diago nalizations of NxN matrices. In the method due to Hinze and Yurtsever4B a Jacobi-like transformation is performed simultaneously for all the n SCF matrices. In both cases a good convergence is reported. However, dealing with n, or even (n/2) matrices of the size NxN is rather troublesome, particularly in the case of larger basis sets. In what follows an alternative solution of the latter problem is given, with better convergence and requiring less computer storage and time. THEORETICAL BACKGROUND Often techniques are developed in one particular field of science which are useful in other fields. This is just the case with a simple technique developed in conjunc tion with the maximum overlap prinCiple by Murrell49 and Golebiewski. 50 Let us define a set of m orthonormal functions in the form of a row vector, 8=(8182", €I",), (1) and another set of n orthonormal functions, cP = (cf>ICP2' •• cf>m'" cf>n) , (2) where n'2!m. The main goal of the maximum overlap prinCiple is to find a linear transformation of the latter functions, cP' = cP (UV) , (3) where U is the transformation matrix for first m func tions of cP' and V for the remaining n-m ones, such that A =2: (cf>~18,> (4) I is a maximum and the new functions (3) are still nor malized and orthogonal one to each other. Murrell49 has shown that condition (4) is equivalent to the requirement that the overlap matrix S'= (cf> ~ I 81 > ( cf>2181> (cf> ~ I 8",> 0 (cf>~18",) 0 be symmetric (or, more generally, Hermitian): S' = (S't . o o o (5 ) (6) Two Simple solutions of this condition have been given, by Gilbert51 and by Golebiewski. 50 In accordance with the latter one (cf>~cf>~'" CP~)=(cf>1cf>2'" cf>n)U, where U is an nxm rectangular matrix U = Sm(S;'S",tI/2 with the overlap matrix for the original functions. (7) (8) (9) An obvious generalization of this scheme is to consid er matrix elements of an arbitrary constant operator, instead of the overlap integrals, thus obtaining a trans formation which renders the representation of the oper ator Hermitian. MATHEMATICAL FORMULATION Assumptions, basic equations, and notation will fol low rather exactly the two papers of Hinze. 19,52 To facilitate the reading we recall here merely the equa tions and concepts necessary to understand the new method. We restrict the discussion to the nonrelativistic Born Oppenheimer Hamiltonian H= Vo+ Lh(i)+ Lg(i,j) , (10) I I>J where the meaning of the symbols is obvious. The total wave function W K is expanded into a finite set of Slater determinants, WK= L:4>rCIK , 1 4>1= (nl)-1/2det{l/Ilj(l)l/I'2(2) ••. l/I'n(n)}, (11) (12) constructed from a set of spin-orbitals, normalized, and orthogonal one to each other. Spin-orbitals con- J. Chern. Phys., Vol. 70, No.3, 1 February 1979 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: 131.193.242.67 On: Mon, 01 Dec 2014 06:02:24Golebiewski, Hinze, and Yurtsever: Orthogonal gradient method 1103 tained in >It K are called occupied. All other spin-orbit als are called empty. We restrict the discussion to the problem of optimum orbitals assuming thus that the CI problem has been solved already. Best spin-orbitals satisfy then the Fock-like MC-SCF equations, ace ace L.F(fJ)l/JJ= L. l/JJEJI, (13) J J where the summations run over occupied spin-orbitals and F(jj)=Yuh+ t rU•kl( l/J"lgll/J,) is the Fock-like operator. In Eq. (14) Yu=(>ItKla;a,I>ItK\ (14) (15) is the first-order reduced density matrix element and (16) is the second-order reduced density matrix element, both types of elements being defined with the help of Hinze'sl9 annihilation and creation operators, a, and a;. We certainly could keep the discussion as general as possible. In accordance with the common practice, however, we shall consider the restricted SCF and MC-SCF scheme in more detail. The spin-orbitals are usually written as spatial orbit als ¢ 1> multiplied by either spin function a and f3. In the restricted scheme, the same spatial orbitals are al ways used with either spin function. To account for this it is customary to introduce new reduced density matrix elements, defined for the spatial orbitals as (17) (18) where ~ stands for rf>,a, i for ¢,f3, etc. With this def inition most equations have formally the same appear ance as in the spin orbital space. As in Eqs. (17) and (18), we could distinguish the new quantities by prime. In order to simplify the notation, however, we shall drop the prime from now on. Thus, in analogy to Eq. (14), we introduce the Fock like operators m F(jj)=YlJh+~ru.",(¢"lgl¢/) , (19) and the common (at the start non-Hermitian) SCF oper ator m F=L.FW) Irf>,) (rf>,1 . U (20) We have assumed that the orbitals ¢t> rf>2' ••• , rf>m are occupied and all the others are empty. We note that for any orbital rf>r. occupied or empty, Fr, = (rf>r I FI rf>,) = f'Yfjh rJ+ :2::ru."I(¢r(rf>"lglrf>,)rf>,). (21) 'r J'" With this definition of the SCF operator we can rewrite the formula for the expectation value of the energy in compact form: E= Vo + t YlJhu + ~ ~ r,J•kl (¢, (rf>"lgl rf>,) CPJ) IJ 6n = Vo + ~ ~ ku + 4: hUYJI} • (22) Except in the case of atoms, we usually expand the orbitals into a finite set of basis functions, XI> ... , X", where n>m. We optimize the orbitals then within a definite subspace of the Hilbert space. Equivalently, we can span this subspace by the set of n orthonormal or bitals rf>h ... , rf>n' each orbital being a linear combina tion of the basis functions XI> .. , , Xn' We look then for "better" orbitals within this subspace, ¢;, ... , ¢~, which yield a lower total energy E, where, because of the orthonormality conditions, U·U= 1 , and 1 is an m Xm unit matrix. (23) (24) To solve this problem we need the nontrivial part of the Hartree-Fock matrix F in this subspace: Fl1 F12 F1m Fo= F21 F22 F2m (25) Fnl Fn2 Fnm All other elements, within this subspace, are zero. THE ORTHOGONAL GRADIENT METHOD Let us consider a variation of Eq. (22), taking into account the definition of F, Eq. (20). After straightfor ward although somewhat lengthy algebraic manipulations we find, for real functions and operators, that m ~E=2L.(~rf>/IFlcp/)+0(~2). (26) I This' is indeed the same expression one obtains when deriving the SCF equations with the Lagrangian multi pliers method. In the steepest descent method we put some con straints upon ~rf>/s due to the orthonormality require ments and look for changes which guarantee the largest decrease of energy. Equivalently, at least for small changes of orbitals, we may look for a set of new orthonormal orbitals ¢;, ... , rf>~ (23), such that m ~= L.(rf>~IFI¢,) (27) I is a minimum. In contrast to the steepest descent method, however, this formulation defines not only the direction of change (gradient), but also the magnitude of all changes. The minimum condition of Eq. (27) resembles the maximum condition of Eq. (4), with 8, replaced by rf>, and with the (approximately) invariant SCF operator F put in between. Continuing the comparison we find im mediately the desired solution U = Fo(FoFo)"1 12 • (28) In contrast to the maximum overlap principle, however, J. Chern. Phys., Vol. 70, No.3, 1 February 1979 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: 131.193.242.67 On: Mon, 01 Dec 2014 06:02:241104 Golebiewski, Hinze, and Yurtsever: Orthogonal gradient method negative roots of the inverse square root matrix must be taken, to ensure the minimum of E. The process is iterative in nature. After performing the orthogonal transformation, Eqs. (23) and (28), the matrix Fo must be recalculated. For this reason also the new empty orbitals are required, (¢~+1 ... ¢~) = (¢ l' .• ¢n)Y , where for Y, the n x (n -m) matrix, Y+U = 0 , Y+V = 1 . (29) (30) Apart from these two conditions, the empty orbitals can be defined in several ways. However, usually ¢ ~ -¢ k for k > m can be assumed. Then, (31) is likely to be the simplest solution, where Nk is the normalization factor. Alternatively, in accordance with Eqs. (30), (32) Thus Y consists alternatively of eigenvectors of UU+ of eigenvalue O. All remaining m eigenvectors correspond to the eigenvalue 1. The matrix UU+ is usually sparse so that the diagonalization does not take much time. We can also find a simple expression for the quantity ~ defined in Eq. (27). From Eqs. (23), (28), and (27) we obtain immediately ~=tr(FoFo)1/2 . (33) It is thus obvious why we had to take negative roots in the case of Eq. (28). In order to avoid divergence when still far from the converged limit a simple level shifting procedure can be designed, by adding arbitrary (in principle) constants, dj> to the diagonal elements of Fo. Taking the dt's suf ficiently large in absolute value we can always force U to be almost the unit matrix. A satisfactory physical explanation of the role of such a damping can be easily given. With the damping we require that m ~'=L {<¢~IFI¢I)+dl<¢~I¢I)} (34) I is a minimum; this means we look for the largest de crease of energy under the condition that the new orbit als resemble to some extent, governed by the weights dl' the original ones. In practice, we have not been forced to make use of this procedure, thus far. Also an alternate interpretation of the transformation, Eq. (28), can be given; it changes the Lagrange multi pliers matrix into a Hermitian one under the condition that all the m functions, /,= t jYlih +2: r/J,kl <¢klgl ¢I)} ¢J , (35) J t kl are insensitive to the transformation of occupied orbit als. These are the Fock functions defined by Hinze. 19 Indeed, now (36) and By assumption, F'=f>~<¢~I-tf,<¢~1 , I I and hence E~, = <¢~I F' I ¢~) -<¢~If,) ='t uJk<¢Jlf,)=(crF)k/' J (37) (38) (39) The matrix of Lagrangian multipliers should be Hermi tian. By this requirement (40) The proof that the matrix U given by Eq. (28) satisfies Eq. (40) is rather straightforward. SUBCASES The advantage of the orthogonal gradient method is its high internal simpliCity and its general character. Closed shell, open shell, and general MC-SCF cases are treated in the same way once the rectangular SCF matrix Fo (the nontrivial part of the F matrix) has been found. In the most general case the matrix element Frl, where1~i~mand1~r~n, isgivenbyEq. (21). In important subcases the expression for Frl simplifies significantly. Closed shell case The only nonvanishing reduced density matrix elements are YH, rH,JJ' and r/J,i/' In this case (41) where the shorthand notation (42) has been introduced. Single open shell case Let us consider the open shell case in accordance with the classic treatment of Roothaan. 7 Let ¢ jp ¢ 2, ••• , ¢P be the closed shell orbitals and ¢P+h ... , ¢m the open shell ones of the (m -p )-dimensional open shell. Now, for i=1, 2, ... , p and r=1, 2, ... , n Frl = 2(hrl + t {2(ri 1m -(rj Iji)} + f t {2(ri 1m -(rj Iji)}\ . J=P+\ '} On the other hand, for i =p + 1, P + 2, 2, ... , n Frl = 2f(hrl +"tt {2(ri Iii) -(rj Iii)} + / t {2a(ri Ijj) -b(rjlji)}\ J=p+l ~ (43) ... , m and r= 1, (44) where f is the fractional occupation of the open shell and a, b are coefficients tabulated by Roothaan. 7 Matrix J. Chern. Phys., Vol. 70, No.3, 1 February 1979 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: 131.193.242.67 On: Mon, 01 Dec 2014 06:02:24Golebiewski, Hinze, and Yurtsever: Orthogonal gradient method 1105 TABLE I. Convergence tests for methylene and ethylene within the MINDO/3 MC-SCF scheme. Values of (F ii -Fj/)maJC in a. u. Methylene Ethylene Iteration Jacobi-like rnethod48 This work Jacobi-like rnethod48 This work (0) 0.174849 0.174849 0.123324 0.123324 (1) 0.052723 0.041844 0.031055 0.021866 (2) 0.016799 0.009608 0.010372 0.004677 (3) 0.005575 0.002208 0.004042 0.001104 (4) 0.001903 0.000508 0.001554 0.000285 (5) 0.000663 0.000117 0.000594 0.000075 (6) 0.000234 0.000027 0.000228 0.000022 (7) 0.000084 . O. 000 006 0.000087 0.000007 (8) 0.000030 0.000001 0.000036 0.000003 (9) 0.000011 0.0000003 0.000015 0.000001 elements (43) can be identified with matrix elements of the closed shell operator, elements of type (44), with matrix elements of the open shell operator. Formulas (43) and (44) do not follow directly from Eq. (21); they involve some symmetrization processes as suggested by Roothaan.7 MC-SCF theory with double replacements A great deal of attention has been paid to the subcase of the general MC-SCF theory, where all Slater deter minants in Eq. (11) differ from each other by at least a double replacement. 23,24,36,53 In this case we get Frl = YI/hrl + t {r II,JJ(ri lij) + r lJ,Jl(rj Iii)} , (45) J with i running from 1 to m and r from 1 to n. The val ues of the reduced density matrix elements vary from case to case. TEST CALCULATIONS In order to test the convergence of the method a semi empirical MC-SCF scheme has been applied to methylene and ethylene, based on the Dewar MlNDO/3 model. 54 For comparison, a similar calculation has been carried out with the Jacobi -like method of Hinze and Yurtsever. 48 As seen from Table I the convergence is good and defin itely better than in the case of the Jacobi-like transfor mations. In a semiempirical SCF or MC-SCF calculation the most time consuming step is solving the SCF equations, and it is worth mentioning here that the orthogonal gradient method required less than one-half the time per iteration than the extended Jacobi procedure, in addition to needing fewer iterations. To be sure, this advantage will not be as great in ab initio computations, where the construction of the SCF matrices is the most time consuming step. However, even here the smaller number of iterations required and the less stringent storage requirements of the orthog onal gradient method, as well as its simplicity, make it appear rather attractive. We believe the orthogonal gradient method has the above stated advantages over all procedures to solve the MC-SCF (SCF) equations known to us. ACKNOWLEDGMENT This research was supported in part through a grant (Az. : Hi: 254/2) of the Deutsche Forschungsgemein schaft. One of us (A. G. ) is grateful for a DAAD stip end, allowing him a stay in Germany. ID. R. Hartree, Proc. Cambridge Phil. Soc. 24, 89 (1928). 2V. Fock, Z. Phys. 61, 126 (1930). 3J. C. Slater, Phys. Rev. 35, 210 (1930). 4C. c. J. Roothaan, Rev. Mod. Phys. 23, 69 (1951). 5G. G. Hall, Proc. R. Soc. (London) Ser. A 205, 541 (1951). GR. Lefebvre, J. Chirn. 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1.4971828.pdf	Low operational current spin Hall nano-oscillators based on NiFe/W bilayers Hamid Mazraati , Sunjae Chung , Afshin Houshang , Mykola Dvornik , Luca Piazza , Fatjon Qejvanaj , Sheng Jiang , Tuan Q. Le , Jonas Weissenrieder , and Johan Åkerman Citation: Appl. Phys. Lett. 109, 242402 (2016); doi: 10.1063/1.4971828 View online: http://dx.doi.org/10.1063/1.4971828 View Table of Contents: http://aip.scitation.org/toc/apl/109/24 Published by the American Institute of Physics Articles you may be interested in Route toward high-speed nano-magnonics provided by pure spin currents Appl. Phys. Lett. 109, 252401252401 (2016); 10.1063/1.4972244 Large influence of capping layers on tunnel magnetoresistance in magnetic tunnel junctions Appl. Phys. Lett. 109, 242403242403 (2016); 10.1063/1.4972030 Microwave emission power exceeding 10 µW in spin torque vortex oscillator Appl. Phys. Lett. 109, 252402252402 (2016); 10.1063/1.4972305 Tunneling electroresistance effect in ultrathin BiFeO3-based ferroelectric tunneling junctions Appl. Phys. Lett. 109, 242901242901 (2016); 10.1063/1.4971996 Low operational current spin Hall nano-oscillators based on NiFe/W bilayers Hamid Mazraati,1,2,a)Sunjae Chung,2,3,a)Afshin Houshang,1,3Mykola Dvornik,3 Luca Piazza,2Fatjon Qejvanaj,1,2Sheng Jiang,1,2Tuan Q. Le,2Jonas Weissenrieder,2 and Johan A˚kerman1,2,3 1NanOsc AB, Kista 164 40, Sweden 2Department of Materials and Nanophysics, School of Information and Communication Technology, KTH Royal Institute of Technology, Electrum 229, SE-16440 Kista, Sweden 3Department of Physics, University of Gothenburg, 412 96 Gothenburg, Sweden (Received 28 October 2016; accepted 24 November 2016; published online 15 December 2016) We demonstrate highly efﬁcient spin Hall nano-oscillators (SHNOs) based on NiFe/ b-W bilayers. Thanks to the very high spin Hall angle of b-W, we achieve more than a 60% reduction in the auto- oscillation threshold current compared to NiFe/Pt bilayers. The structural, electrical, and magnetic properties of the bilayers, as well as the microwave signal generation properties of the SHNOs, have been studied in detail. Our results provide a promising path for the realization of low-current SHNOmicrowave devices with highly efﬁcient spin-orbit torque from b-W.Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4971828 ] Spin Hall nano-oscillators (SHNOs) are promising micro- wave signal generators with a high degree of frequency tun-ability, both via the electrical current and the magnitude anddirection of an external magnetic ﬁeld. 1These SHNOs operate on the basis of spin-orbit torque (SOT),2which is produced when a spin current, created via the spin Hall effect3–8in a metal with high spin-orbit coupling (e.g., Pt,9W,10,11Ta12), is absorbed by an adjacent ferromagnetic layer. SOT can act as anegative spin-wave damping in the ferromagnet and above a certain threshold current, it can sustain a steady state auto- oscillation of the local magnetization. To date, most SHNOs have been based on NiFe/ Pt, 2,13–20YIG/Pt,21,22and CoFeB/Pt,23with SOT produced by a Pt layer. Many other heavy metals, such as W,10,11 IrCu,24,25and CuBi,26have however been investigated, revealing that the so-called spin Hall angle ( hSH), which describes the charge current to spin current conversion efﬁ-ciency, can exceed that of Pt. W is particularly interesting, as it has two different structural phases, called aandb,w i t h an order of magnitude difference in resistivity ( q a<qb), and a much larger hSHin the bphase.10,11,27However, b-W can only be obtained under restricted conditions, such aswell-controlled slow deposition rates, no heating, and thinlayer thicknesses. In this study, we demonstrate the fabrication and operation of nano-constriction based SHNOs employing b-W as the heavy metal. We ﬁnd a large negative value of h SH¼/C00.385 and a corresponding dramatic reduction of the SHNO threshold current by over 60% compared to Pt-based SHNOs.2,13–19,21–23 All thin ﬁlms were deposited at room temperature on c- plane sapphire substrates using magnetron sputtering in a 2.5mTorr Ar atmosphere, in an ultra-high vacuum (base pres- sure below 1 /C210 /C08mTorr) AJA Orion 8 sputtering system. First we carried out a detailed study of the thickness depen-dence of the W ﬁlm resistivity in ﬁlms deposited at a rate of0.07 A ˚/s. Figure 1shows that, as the thickness increases, theW resistivity decreases exponentially from a very high value (/C24400lXcm) at a thickness of 3 nm and ﬁnally drops to a near constant value of /C2430lXcm for thicknesses of 12 nm and above, which is close to the bulk W resistivity. This indi-cates that ﬁlms with a thickness below 10 nm are primarily ofb-phase, and a W thickness of 5 nm was therefore chosen for the fabrication of SHNO devices. Prior to the device processing, a bilayer consisting of 5n mN i 80Fe20(Py) and 5 nm b-phase W was deposited, with the thicknesses predetermined from X-ray reﬂectivity meas-urements on calibration ﬁlms. High-resolution transmission electron microscopy (HR-TEM) (Figure 1-inset) conﬁrms the thicknesses of the bilayer and also shows that the Py/ b-W interface is well-deﬁned without any signiﬁcant intermixing. The distance between lattice fringes in the W-layer was foundfrom a fast Fourier transformation to be approximately 2.50 A ˚, i.e., close to the (0 0 2) plane of b-W. 28This conﬁrms that the b-W crystal structure is indeed realized in the stack. To fabricate SHNOs, the bilayer was patterned into an array of 4 lm/C212lm rectangular mesas using photolithog- raphy and dry etching. Nano-constrictions with the width FIG. 1. Resistivity vs. thickness for W thin ﬁlms. Inset: Cross-section HR- TEM image of the stack and the result of fast Fourier transformation of the W-layer.a)H. Mazraati and S. Chung contributed equally to this work. 0003-6951/2016/109(24)/242402/4/$30.00 Published by AIP Publishing. 109, 242402-1APPLIED PHYSICS LETTERS 109, 242402 (2016) varying from 80 to 160 nm were subsequently fabricated in the center of these mesas by a combination of e-beam lithog- raphy and Argon ion milling. To determine hSHof tungsten, 6lm-wide bars were fabricated simultaneously next to the SHNO mesas, and characterized using spin-torque-induced ferromagnetic resonance (ST-FMR) measurements. Finally, a conventional ground–signal–ground (GSG) waveguide andelectrical contact pads for wide frequency range microwave measurement were fabricated using lift-off photolithography and Cu/Au sputtering on top of both the SHNO nano- constrictions and ST-FMR bars. A schematic of the device structure, an atomic force microscopy (AFM) image, and a scanning electron micros- copy (SEM) image of the fabricated 100 nm-wide nano-con- striction SHNO device are shown in Figure 2(a), where the direction of the applied in-plane ﬁeld and current are also deﬁned: the ﬁeld angle /¼0 /C14along þxandþ90/C14along þy, while a negative (positive) current means electrons ﬂowalong þ(/C0)y. The linear nano-constriction width dependence of the SHNO resistance shown in Figure 2(b) indicates that the device-to-device variation during the fabrication process was moderate. ST-FMR measurements based on homodyne detection were then carried out on 6 lm bar-shaped devices, using a 313 Hz-pulse-modulated microwave signal and a dc current injected simultaneously through rfand dcports, respectively, of a bias-tee. The dcvoltage response ( V dc) from the 313 Hz modulated microwave input is detected also through the dcport of the bias-tee using a standard lock-in ampliﬁer.21,29–32Figures 3(a) and3(b) show the resonance peaks extracted from the detected ST-FMR output spectra with the in-plane magnetic ﬁeld ( HIP) swept from 0 to2500 Oe at different constant frequencies fof the injected microwave signal. Microwave measurements were carried out using our custom-built setup in which a dccurrent was driven into the SHNO and the output microwave signal was picked up sepa- rately via the bias-tee. The output signal was ampliﬁed by 35 dB using a broadband (0.1–20 GHz) low-noise ampliﬁer(LNA) prior to an R&S FSQ26 spectral analyzer. Results of the microwave measurement in the applied in-plane and out- of-plane (OOP) ﬁelds are shown in Figures 4and5, respec- tively. All measurements were performed at room temperature. Figure 3(a) shows the ST-FMR results of a 6 lm-wide bar-shaped device in an in-plane magnetic ﬁeld ( H IP) applied along /H¼30/C14and swept from 0 to 2.5 kOe. All ST-FMR spectra have been ﬁtted to a function consisting of one sym-metric and one antisymmetric Lorentzian having the same resonance ﬁeld and linewidth 9 Vdc¼VSDH2þ4VAHIP/C0Hr ðÞ DH 4HIP/C0Hr ðÞ2þDH2þVoffset; (1) where Hris the resonance ﬁeld of the measurement spectra, DHis its linewidth (full-width half maximum, FWHM), VS andVAare the coefﬁcients of the symmetric and antisymmet- ric Lorentzian functions, respectively, and Voffset is a con- stant offset voltage. The color plot of the ST-FMR spectra is shown in Figure 3(a). The dependence of the resonance ﬁeld ( Hr) on the micro- wave frequency shows good agreement with the Kittel formula (solid red line),33and by considering the gyromagnetic ratio of c/2p¼28 GHz/T, the extracted effective magnetization from the ﬁt is l0Meff¼0.71 T. The inset in Figure 3(a) shows the FIG. 2. (a) Schematic of the device structure and the conﬁguration of the applied magnetic ﬁeld. Inset: AFM image of the active nano-constriction areawith zoomed-in SEM image of the nano-constriction. (b) Device resistance vs. constriction width. Inset: AMR measurement for constriction width ¼100 nm. FIG. 3. (a) ST-FMR measurement on a 6 lm-wide bar-shaped structure. Inset: ST-FMR linewidth vs. frequency. (b) ST-FMR linewidth vs. current in an in- plane magnetic ﬁeld along /¼30/C14(black dots) and 210/C14(red dots).242402-2 Mazraati et al. Appl. Phys. Lett. 109, 242402 (2016) linewidth, DH(white dots) as a function of frequency, together with a linear ﬁt33DH¼DH0þ4paf/c(red line), from which an inhomogeneous broadening ( DH0) of 1.8 Oe and a Gilbert damping ( a)o f8 . 3 2 /C210/C03can be extracted. To extract the spin Hall angle hSH, we then investigated the current-induced linewidth changes originating from theSOT of the W layer. Figure 3(b) shows how the linewidth depends linearly on the device current, with positive currentproviding a negative damping when the ﬁeld is along /¼ 30 /C14(black circle), and positive damping when the ﬁeld direc- tion is reversed to 210/C14(red circle). The spin Hall angle, deﬁned as the ratio of the spin and charge current densitiesh SH¼Js/Jc, is then extracted from the slopes in Figure 3(b) using the following equation:9,11,34 hSH¼dDH=dIDC 2pf csin/ HIPþ0:5Mef f ðÞ l0Mst/C22h 2eRPyþRW RPyAC; (2)where Msis the saturation magnetization, and tis the thick- ness of the magnetic layer, eis the electron charge and /C22his Planck’s constant, RNiFe andRWare the resistances of the NiFe and tungsten layers, and ACis the cross-sectional area of the measured device. We obtained hSH¼/C00.38560.009, which is comparable to previous studies.10,21 We now turn to the SHNOs. Figure 4(a)shows the mea- sured power spectral density (PSD) vs. SHNO drive current from a 140 nm wide nano-constriction in a ﬁeld of HIP¼400 Oe along /¼20/C14. The auto-oscillation frequency, linewidth, and integrated power are extracted by ﬁtting the peaks of all spectra with a Lorentzian function. Whereas theauto-oscillation frequency only depends weakly on the drive current, both the linewidth and the integrated power shows a rapid exponential improvement up until I DC¼/C00.9 mA, reaching a minimum linewidth of about 10 MHz, and a maxi- mum power of about 1 pW. Above this current level, the auto-oscillation frequency decreases, and both linewidth andpower deteriorate, possibly indicating the appearance of a different mode. 18,35,36 When the applied ﬁeld is tilted out-of-plane ( h¼80/C14; /¼20/C14) and increased to 8 kOe to align the Py magnetiza- tion (Figure 4(b)), the current dependence changes character to a much more non-monotonic frequency, with an initial redshift at low current, which changes to a clear blue shift at higher current. In contrast to the in-plane case, the integrated power shows a monotonic improvement with current, andthe linewidth improves to about 2 MHz, i.e., close to an order of magnitude better than in the in-plane case. As shown in Ref. 20this non-monotonic frequency behavior results from a gradual change in the location of the auto-oscillations. At low currents, the auto-oscillations start from the edges of the nano-constrictions. With increasing current, the auto-oscillation region then expands into the nano-constriction, and the point of maximum spin wave intensity moves inward. In this process, the mode center experiences a vary-ing ﬁeld landscape with the net effect being a frequency red- shift as the mode leaves the edges followed by a blue-shift as it further expands inward. The large spin Hall angle is expected to have a strong beneﬁcial impact on the threshold current ( I th) for micro- wave signal generation. To evaluate this, we measured morethan 20 SHNOs with different nano-constriction widths rang- ing from 80 to 160 nm. I this determined from linear ﬁts of the inverse microwave power vs. current,37,38as shown in the inset of Figure 5; all the measurements were carried out with HIP¼0.08 T and /¼20/C14. Figure 5shows that Ith depends linearly on the nano-constriction width, and the val- ues observed are more than 60% lower than those from pre- vious reports on Pt-based SHNOs.2,13,14,16–18,21,23 The effective threshold current ﬂowing through the W layer, shown on the right-hand y-axis of Figure 5, can be cal- culated by considering the ratio of the Py and W resistivities. This shows that a current of only about 75 lA in W can excite Py auto-oscillations in a 80 nm nano-constriction. From a linear ﬁt to all the measured devices, we can calcu- late the effective threshold current density in our W-basedSHNOs to be J th,eff’2/C2107A/cm2. We have demonstrated a reliable fabrication and opera- tion of W/Py based nano-constriction SHNOs. Our devices FIG. 5. Threshold current vs. constriction width in an 800 Oe in-plane mag- netic ﬁeld at /¼20/C14. Inset: threshold current extraction through a linear ﬁt of 1/power versus current at low current. FIG. 4. Power spectral density (PSD), extracted linewidth, and integratedpower of the auto-oscillations for (a) in-plane, and (b) out-of-plane magnetic ﬁelds.242402-3 Mazraati et al. Appl. Phys. Lett. 109, 242402 (2016) achieved a signiﬁcant reduction in the threshold current of more than 60% compared to Pt/Py devices. Our ﬁndings thus lay out a deﬁnite development path for low-current SHNOs using highly efﬁcient spin orbit torque from a W layer. This work was supported by the Swedish Foundation for Strategic Research (SSF), the Swedish Research Council (VR), and the Knut and Alice Wallenberg foundation (KAW).This work was also supported by the European Research Council (ERC) under the European Community’s Seventh Framework Programme (FP/2007–2013)/ERC Grant 307144“MUSTANG.” 1T. Chen, R. K. Dumas, A. Eklund, P. K. Muduli, A. Houshang, A. A. Awad, P. D €urrenfeld, B. G. Malm, A. Rusu, and J. A ˚kerman, Proc. IEEE 104, 1919 (2016). 2V. E. Demidov, S. Urazhdin, H. Ulrichs, V. Tiberkevich, A. 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(published online 2016). 21M. Collet, X. de Milly, O. d’Allivy Kelly, V. V. Naletov, R. Bernard, P. Bortolotti, J. Ben Youssef, V. E. Demidov, S. O. Demokritov, J. L. Prieto, M. Mu ~noz, V. Cros, A. Anane, G. de Loubens, and O. Klein, Nat. Commun. 7, 10377 (2016). 22A. Hamadeh, O. d’Allivy Kelly, C. Hahn, H. Meley, R. Bernard, A. H. Molpeceres, V. V. Naletov, M. Viret, A. Anane, V. Cros, S. O. Demokritov, J. L. Prieto, M. Mu ~noz, G. de Loubens, and O. Klein, Phys. Rev. Lett. 113, 197203 (2014). 23M. Ranjbar, P. Durrenfeld, M. Haidar, E. Iacocca, M. Balinskiy, T. Q. Le, M. Fazlali, A. Houshang, A. A. Awad, R. K. Dumas, and J. Akerman,IEEE Magn. Lett. 5, 3000504 (2014). 24Y. Niimi, M. Morota, D. H. Wei, C. Deranlot, M. Basletic, A. Hamzic, A. Fert, and Y. Otani, Phys. Rev. Lett. 106, 126601 (2011). 25M. Yamanouchi, L. Chen, J. Kim, M. Hayashi, H. Sato, S. Fukami, S. Ikeda, F. Matsukura, and H. Ohno, Appl. Phys. Lett. 102, 212408 (2013). 26Y. Niimi, Y. Kawanishi, D. H. Wei, C. 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1.4892168.pdf	Spin-Hall nano-oscillator: A micromagnetic study A. Giordano, M. Carpentieri, A. Laudani, G. Gubbiotti, B. Azzerboni, and G. Finocchio Citation: Applied Physics Letters 105, 042412 (2014); doi: 10.1063/1.4892168 View online: http://dx.doi.org/10.1063/1.4892168 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 Micromagnetic study of auto-oscillation modes in spin-Hall nano-oscillators Appl. Phys. Lett. 104, 042407 (2014); 10.1063/1.4863660 Non-stationary excitation of two localized spin-wave modes in a nano-contact spin torque oscillator J. Appl. Phys. 114, 153906 (2013); 10.1063/1.4825065 Macrospin and micromagnetic studies of tilted polarizer spin-torque nano-oscillators J. Appl. Phys. 112, 063903 (2012); 10.1063/1.4752265 Anisotropic spin-wave patterns generated by spin-torque nano-oscillators J. Appl. Phys. 109, 07C733 (2011); 10.1063/1.3566000 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: 129.101.79.200 On: Mon, 01 Sep 2014 16:04:05Spin-Hall nano-oscillator: A micromagnetic study A. Giordano,1M. Carpentieri,2A. Laudani,3G. Gubbiotti,4B. Azzerboni,1and G. Finocchio1 1Department of Electronic Engineering, Industrial Chemistry and Engineering, University of Messina, C.da di Dio, I-98166 Messina, Italy 2Department of Electrical and Information Engineering, Politecnico of Bari, via E. Orabona 4, I-70125 Bari, Italy 3Department of Engineering, University of Roma Tre, via V. Volterra 62, I-00146 Roma, Italy 4Istituto Ofﬁcina dei Materiali del CNR (CNR-IOM), Unit /C18a di Perugia c/o Dipartimento di Fisica e Geologia, Via A. Pascoli, 06123 Perugia, Italy (Received 16 April 2014; accepted 18 July 2014; published online 1 August 2014) This Letter studies the dynamical behavior of spin-Hall nanoscillators from a micromagnetic point of view. The model parameters have been identiﬁed by reproducing recent experimental data quan-titatively. Our results indicate that a strongly localized mode is observed for in-plane bias ﬁelds such as in the experiments, while predict the excitation of an asymmetric propagating mode for large enough out-of plane bias ﬁeld similarly to what observed in spin-torque nanocontact oscilla-tors. Our ﬁndings show that spin-Hall nanoscillators can ﬁnd application as spin-wave emitters for magnonic applications where spin waves are used for transmission and processing information on nanoscale. VC2014 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4892168 ] Experimental studies of bilayer composed by a heavy metal ﬁlm coupled with a thin ferromagnet have opened aroute on the development of a more efﬁcient category of spintronic devices where the magnetic state can be changed by the effects related to spin-orbit coupling, such asDzyaloshinskii-Moriya interaction, Rashba and spin-Hall effects. 1–7In particular, domain wall motion at very high ve- locity,3magnetization reversal,1,2,8,9and persistent magnet- ization precession have been achieved.10–12These results are motivated by technological interest, aimed to design the next generation of nanomagnetic logic gates, magnetic memoriesand nanoscale oscillators. In this Letter, we focus on this lat- ter device category. The persistent magnetization precession driven by spin-orbit interactions in Pt/Py bilayer was ﬁrstmeasured by Demidov et al. 10where the torque from spin- orbit coupling, mainly spin-Hall effect, originating from a bias current ﬂowing in the Pt layer was large enough to com-pensate the magnetic losses and to excite a persistent mag- netization precession in the Py layer. Demidov et al. 10also demonstrated the nature of the excited mode to be a non-propagating spin wave with localization region of less than 100 nm. Here, we performed a systematic study of that ex- perimental system 10to understand the physical origin of the excited modes. For in plane-ﬁelds, our computations repro- duce quantitatively the experimental oscillation frequency as a function of current and the localization of the excitedmode. For out-of-plane-ﬁelds, we predict the excitation of propagating spin waves with an asymmetric propagation pat- tern. Our ﬁndings show this device geometry is prototypefor spin-wave emitters for magnonic applications, where spin waves are used for transmission and processing infor- mation on nanoscale. 13,14 The device is a bilayer composed by Pt(8)/Py(5) (thick- nesses in nm). The bias current is injected in the center of the Py layer by means of two triangular Gold (Au) contacts(thickness of 150 nm) at a nominal distance d. A sketch of the device is displayed in Fig. 1(a). A Cartesian coordinatesystem is introduced, where the x-axis is parallel to the direc- tion of the injected current, while the yandzaxes are the other in-plane and the out-of-plane directions, respectively. The in-plane ﬁeld has been applied along the y-direction to saturate the magnetization along that direction. The out-of-plane ﬁeld has been applied in the yz-plane with a 15 /C14angle with respect to the z-axis. In general, the magnetization dy- namics should be studied by the Landau-Lifshitz-Gilbertequation which takes into account: the adiabatic s A/C0STand non-adiabatic sNA/C0STspin-transfer-torques due to the current which ﬂows into the ferromagnet, the spin-orbit torques fromthe spin Hall s SHEand the Rashba sREeffects8,15–19 dM dt¼/C0c0M/C2HEFFþa MSM/C2dM dtþsSHE þsREþsA/C0STþsNA/C0ST (1) where MandHEFFare the magnetization and the effective ﬁeld vectors of the ferromagnet, respectively. HEFFtakes into account the exchange, the magnetocrystalline anisot-ropy, the external, the self-magnetostatic ﬁelds, and the Oersted ﬁeld. a,M S,a n d c0are the Gilbert damping, the sat- uration magnetization, and the gyromagnetic ratio, respec-tively. The ﬁrst step is the computation of the spatial distribution of the current density in the Py and Pt layers by means of the Ohm’s law J¼q /C01E,w h e r e Eis the local electric ﬁeld and qis the resistivity of the material. The electric ﬁeld can be computed as the gradient ﬁeld of the electrostatic potential V, so that E¼/C0 r V. For the charge conservation law that yields r/C1J¼0, we have r/C1ð q/C01rVÞ¼0. This kind of Elliptic differential equation is numerically solved with the Finite Element Method (1storder tetrahedral, employing more than 3 /C210 6elements and about 6.7 /C2105nodes)20with the boundary condition @V=@n¼0( w h e r e nis the normal to the geometrical boundary) except for the contacts, where the current density is injected. 0003-6951/2014/105(4)/042412/5/$30.00 VC2014 AIP Publishing LLC 105, 042412-1APPLIED PHYSICS LETTERS 105, 042412 (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: 129.101.79.200 On: Mon, 01 Sep 2014 16:04:05Figure 1(b) shows a reduced region (between 1000 nm and 3000 nm) of the spatial distribution of the current densityin the Pt layer as computed for a disk of 4 lm in diameter, the material conductivities used for the computation are 4.1 /C210 8(Xm)/C01, 5.1/C2107(Xm)/C01, and 6.4 /C2106(Xm)/C01 for the Gold, Platinum, and Permalloy, respectively. The cur- rent ﬂows almost totally in the two Gold contacts up to the position where it is injected in the Py/Pt bilayer. In betweenthe two contacts, around 90% of the current ﬂows in the Pt layer. In terms of spatial distribution, the current is localized mainly in the center of the system with a symmetric spreadaround the y-direction (perpendicular to an ideal connection line of the two contacts). The current proﬁle for two particu- lar sections (A blue line and B black line) of the device aredisplayed in Fig. 1(c). In the section A ( y-direction perpen- dicular to the current ﬂow) the current density exhibits a maximum value in the center. On the other hand, the resultsrelated to the section B (parallel to the current ﬂow) indicate the presence of two maxima near the boundary with the two contacts at the position where the current starts and stops theﬂow in the Py/Pt bilayer. The s A/C0ST,sNA/C0ST, and sREtorques, being proportional to the current density ﬂowing into the ferromagnet (less than10%), are negligible with respect to the s SHE, which is pro- portional to the current which ﬂows into the Pt layer (more than 90%).21With this in mind, Eq. (1)can be simpliﬁed as follows: dM dt¼/C0c0M/C2HEFFþa MSM /C2dM dt/C0lBaHJPtx;yðÞ eM2 StPyM/C2M/C2r (2) where JPtðx;yÞis the spatial distribution of the modulus of the current density in the Pt layer considering the same sign of the applied current, lBis the Bohr Magneton, eis the elec- tric charge, and tPyis the thickness of the Py-layer. ris the polarization of the spin current due to the spin-dependentscattering in the Pt layer. For each computational cell, the current density vector JPt,r, and the z-axis satisfy the rela- tionship r¼z/C2JPt jJPtj.4aHis the spin Hall angle given by the ratio between the amplitude of the transverse spin current density generated in the Pt and the charge current densityﬂowing in it. The parameters used for our numerical simula- tions are: exchange constant 1.3 /C210 /C011J/m, spin-Hall angle 0.08, Gilbert damping 0.02, and saturation magnetiza-tion 650 /C210 3A/m.25While the current density distribution and the Oersted ﬁeld have been computed by considering a disk with a diameter of 4 lm, the micromagnetic computa- tions have been performed for a disk diameter of 2.5 lmt o reduce the computational time of the systematic study.22The cubic discretization cell is 5 nm in side, which is smallerthan the exchange length for the Py ( /C257 nm). The effects of the thermal ﬂuctuations have been taken into account as a random thermal ﬁeld H thadded to the effective ﬁeld for each computational cell.23–25 The ﬁrst step of our analysis is to understand the origin of the persistent magnetization precession measured experi-mentally by Demidov et al. 10A systematic study has been performed as a function of the current and the magnetic ﬁeld applied along the y-direction and for different distances d between the two Au contacts (see Fig. 1(a)). For each mag- netic ﬁeld value, the initial conﬁguration of the magnetiza- tion has been computed by solving Eq. (2)with JPtðx;yÞ equal to zero up to reach the condition that Mis parallel to HEFF(dM dt/C250). Starting from that static state, the dynamical response of the magnetic device has been computed bysweeping the current back and forth from 0 up to 20 mA. For increasing current, a critical value I ONexists where the mag- netic conﬁguration becomes unstable and a self-oscillation isthen excited. On the other hand, for decreasing current the self-oscillation is switched off at I OFF<ION. Fig. 1(d) sum- marizes IOFFandIONas a function of the external ﬁeld for d¼100 nm (solid and dotted lines with square) and 200 nm (solid and dotted line with circles). At d¼200 nm and for the whole range of the applied ﬁeld, both IOFFandIONare FIG. 1. (a) Schematic view of the studied device, dis the distance between the Au contacts. Thicknesses of the layer are expressed in nm. The coordinate axes are also shown. (b) Example of spatial distribution of the current density as computed numeri-cally (reduced square region of 2000 /C22000 nm 2of a disk of 4 lmi nd i a m - eter), (c) current proﬁles for the sec- tions A and B as indicated in (b). (d) Critical currents IOFF and IONas a function of the applied ﬁeld for two different electrodes distances ford¼100 (solid and dotted lines with circles) and 200 nm (solid and dotted lines with squares).042412-2 Giordano et al. Appl. Phys. Lett. 105, 042412 (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: 129.101.79.200 On: Mon, 01 Sep 2014 16:04:05larger than the one at d¼100 nm. This is because at d¼200 nm the spatial distribution of the current density is widely spread when compared to the conﬁguration at d¼100 nm, and consequently it should be larger to compen- sate the losses due to the Gilbert damping. We also com- puted IONforH¼100 mT at d¼150 nm ﬁnding a value of 16 mA which is in between the ones at d¼100 nm (14 mT) and at d¼200 nm (18 mT). Fig. 2(a) summarizes the oscillation frequencies (d¼100 nm) for two currents ( I¼14 and 16.5 mA) as a func- tion of the ﬁeld. At ﬁxed ﬁeld, the oscillation frequency exhibits red shift similarly to what observed for spin-transfer- torque (STT) oscillators and as expected for self-oscillationswith an in-plane oscillation axis. 26The dynamical state com- puted at H¼100 mT, d¼100 nm, and I¼14 mA is described in detail (point A of Fig. 2(a)), but qualitative similar results are observed for other currents. Fig. 2(b) shows the temporal evolution of the y-component of the spatial average of the normalized magnetization precession as computed for thewhole Py-layer. As can be observed, the magnetization oscil- lation is only 2% of the possible maximum oscillation and it is localized near the gold contact region, see a zoom of thesnapshots in Fig. 2(d) related to the points 1–4 as indicated in Fig.2(b). At the critical current I ON, the magnetization close to the gold contacts starts to oscillate around an oscillationaxis which is reversed (along /C0y-direction) with respect to the equilibrium conﬁguration ( þy-direction). Fig. 2(c) (main panel) shows the power spectrum of the self-oscillationachieved for H¼100 mT, d¼100 nm, and I¼14 mA as computed with the micromagnetic spectral mapping technique. 27,28The excited mode P1is characterized by anoscillation frequency of 9.98 GHz and a uniform spatial distri- bution, as displayed in the inset of Fig. 2(c). The mode is strongly localized in the central region of the device where the current is injected into the bilayer. This result reproducesthe experimental ﬁnding of the spatial distribution of the mode measured in Ref. 10(compare the inset of Fig. 2(c) with Fig. 3 of Ref. 10). In addition, we test the prediction of this model directly to the experimental data by performing the micromagnetic simulations at T¼300 K. The hysteretic behavior of the crit- ical current disappears at room temperature, and the critical currents as a function of the ﬁeld are coincident to the I OFF of Fig. 1(d) and in agreement to the trend measured in the experimental data (compare the IOFFcurve in Fig. 1(a) and Fig. 4(a) in Ref. 10). Fig. 3(a) shows a comparison between the experimental oscillation frequencies (red line in Fig. 2(d)of Ref. 10) and the ones computed micromagnetically as a function of the current ( H¼90 mT, d¼100 nm). As can be observed, a good agreement is found pointing out that theapproximations used to simplify Eq. (1)into Eq. (2)are con- sistent within this experimental framework. In this study, we considered the magnetization dynamics in high ﬁeld regime(H/C2190 mT), where a single mode is excited, and the devices can be used as harmonic microwave oscillators. At lower ﬁeld, the oscillation is characterized by multimode powerspectra similarly to what observed in Refs. 12and29. In this region, the Oersted ﬁeld has a key role as also recently dem- onstrated in STT oscillators; 30,31however, those results are out of the aim of this work and will be discussed elsewhere. Fig. 3(b) displays the linewidth as a function of the current (H¼90 mT, d¼100 nm), a minimum of 142 MHz is FIG. 2. (a) Oscillation frequencies as a function of the applied ﬁeld for I¼14 mA (dotted line with squares) and I¼16.5 mA (solid line with circles) at d¼100 nm. (b) Time domain trace of the average y-component of the normalized magnetization computed for the whole cross section of the Py layer for I¼14 mA and H¼100 mT. (c) Main panel: Power spectrum of the self oscillation excited at I¼14 mA, H¼100 mT as computed with the micromagnetic spectral mapping technique. Inset: spatial distribution of the excited mode P1(the power increases from white to black, the yellow lines indicate the location of the Gold contacts). (d) Snapshots of the magnetization (reduced square region of 500 /C2500 nm2) as computed at the time indicated with the points 1–4 in Fig. 2(b) (the color is related to the ycomponent of the magnetization mY, while the arrows indicate the in-plane component of the magnetization). (Multimedia view). [URL: http://dx.doi.org/10.1063/1.4892168.1 ]042412-3 Giordano et al. Appl. Phys. Lett. 105, 042412 (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: 129.101.79.200 On: Mon, 01 Sep 2014 16:04:05observed at I¼16.3 mA. The lineshape of the power spec- trum is well approximated by a Lorentzian function and thelinewidths have been computed as the full width at half max- imum of the Lorentzian ﬁtting of the power spectra. The same parameters have been used to study the behav- ior in out-of plane ﬁelds ( d¼100 nm). In particular, for ﬁelds larger than 700 mT, our results predict a qualitative different behavior compared to the case of the in-plane conﬁguration.Fig. 3(c) shows the oscillation frequencies as a function of the current for H¼800 mT ( d¼100 nm), the tunability has a different sign than the case of in-plane ﬁelds, it changes fromred to blue shift ( df/dI/C25120 MHz/mA). Our results indicate that a propagating mode is excited and a reduction of one order of magnitude of the linewidth is also predicted (com-pare Fig. 3(b) with Fig. 3(d)). One reason for this can be attributed to the different origins of the excited mode. For a non-linear oscillator, the linewidth is inversely proportionalto the magnetic volume involved in the dynamics, 32and here the magnetic volume of the propagating mode is larger than the one of the localized mode. Differently from whatobserved in point contact geometries, where exchange- dominated cylindrical spin-wave modes are excited (namely, linear-Slonczewski mode), here the proﬁle of the propagationwave is strongly asymmetric. 33This asymmetry is related to the spatial distribution of the current density. In other words, the current density near the contact region is large enough tocompensate the Gilbert damping, exciting a self oscillation. On the other hand, the current density spreads (see Fig. 1(b)) when ﬂowing between the two Gold contacts and from a criti-cal distance it gives rise to a negative damping which com- pensates the Gilbert damping only partially. An additional source of asymmetry is the Oersted ﬁeld and the y-component of the torque from spin-Hall effect which is an odd function if considering the center of the disk as the origin of the Cartesian coordinate system r yðx;yÞ¼ryðx;/C0yÞ. Fig. 4shows an example of the snapshot of the magnetization, where a clear asymmetric path of propagation can be observed. We estimated a wavelength kR¼32565 nm along the dashed line as displayed in Fig. 4. The geometry of this spin-Hall oscillator can be qualita- tively compared to a STT nanocontact oscillator (STNO). Inan STNO, a localized spin-polarized current density is injected via a nano-aperture in an extended ferromagnet, the excited mode depends on the direction of the external ﬁeld,and in fact a localized “Bullet” and a linear propagating Slonczweski mode are excited for in-plane and out-of-plane conﬁguration, respectively. 13,29,34The tunability of the oscil- lation frequency also changes as a function of current from red to blue shift. However some differences can be under- lined. In STNOs, the bullet mode is characterized by a uni-form precession (same phase) of the spins below the nanocontact, differently the localized mode observed here presents spins which oscillate at the same frequency but withdifferent phase (for instance see Fig. 2(d), spin #3). This dif- ference is related to the non-uniform torque due to the spatial conﬁguration of the current density, in fact this dephasing ismore evident for oscillator with Gold contacts at larger dis- tance. Also the spatial structure of the propagating mode is different. While the STNO can be seen as two-dimensionalsystem, in which magnetic excitations can propagate in the whole plane, 33here the propagation is asymmetric in the plane with the advantage that the spin waves can propagatefor a longer distance compared to the STNO. While ﬁeld tun- able radiation patterns have been measured for STNO, 35here the direction of the ﬁeld plays a crucial role only for the na-ture of the excited spin wave being the polarization of the spin current independent on the ﬁeld itself. In other words, to observe dynamical precession of the magnetization, thedirection of the current, the in-plane component of the out-of plane ﬁeld and the out of plane direction should form a right hand set of orthogonal vectors. FIG. 4. Example of spatial distribution of the magnetization as computed by means of micromagnetic simulations for out-of-plane ﬁeld H¼800 mT (I¼37 mA) (the color is related to the xcomponent of the magnetization, the arrows indicate the in-plane component of the magnetization). The dashed line shows the ideal path for the estimation of the wavelength kR. (Multimedia view). [URL: http://dx.doi.org/10.1063/1.4892168.2 ]). FIG. 3. (a) A comparison between experimental oscillation frequencies from Fig. 2(d) of Ref. 10(empty circles) and the micromagnetic computations (solid line with circles), and (b) predicted micromagnetic linewidths (in- plane H¼90 mT, d¼100 nm, and T¼300 K) as a function of the current. (c) Predicted oscillation frequencies and (d) linewidths as a function of the current for an out of plane ﬁeld H¼800 mT, d¼100 nm, and T¼300 K.042412-4 Giordano et al. Appl. Phys. Lett. 105, 042412 (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: 129.101.79.200 On: Mon, 01 Sep 2014 16:04:05While the critical current are comparable for in-plane ﬁeld, the spin-Hall oscillator has the disadvantage to need larger currents (30 mA compared to 10 mA) to excite propagat- ing spin waves. There are at least two reasons for this. First,the Pt has a spin-Hall angle of 0.08 which is smaller than typi- cal values of spin-polarization /C250 . 3 5f o rP yi nS T N O . Second, the negative damping due to the spin-Hall effect hasthe polarization always directed in the plane, while an out-of- plane ﬁeld tilts the STNO polarizer out of-plane. This out of plane component can reduce the critical current signiﬁcantly. In summary, this Letter introduces a micromagnetic framework able to describe recent experiments of magnetiza-tion dynamics driven by an in-plane current in heavy metal/ ferromagnet bilayer. Similar to that observed in STNO, it is possible to identify two different regimes of dynamicalbehavior, localized and propagating modes for in-plane and out-of-plane ﬁeld direction, respectively. For in-plane ﬁelds, the oscillation frequency and the spatial distribution of theexcited modes are in agreement with the experimental data as reported in Ref. 10. For out-of plane ﬁelds, our ﬁndings show that this device geometry is a possible candidate fornanoscale spin-wave emitters for magnonic applications where spin waves are used for transmission and processing information on nanoscale. 13,36,37Although the critical cur- rents to excite propagating spin waves in spin-Hall oscilla- tors are larger than the ones in STNOs, we believe that can be reduced by optimizing materials and geometrical proper-ties, for instance, by considering the giant spin-Hall angle of Tungsten (W) in W/CoFeB bilayer which is at least three times larger than the one measured in Pt/Py. 38 This work was supported by Project MAT2011-28532- C03-01 from Spanish government and MIUR-PRIN 2010–11 Project 2010ECA8P3 “DyNanoMag.” 1I. M. Miron, K. Garello, G. Gaudin, P.-J. Zermatten, M. V. Costache, S. Auffret, S. Bandiera, B. Rodmacq, A. Schuhl, and P. Gambardella, Nature 476, 189–193 (2011). 2L. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and R. A. 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Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.101.79.200 On: Mon, 01 Sep 2014 16:04:05
1.4816089.pdf	Comparison of microwave absorption properties of SrFe12O19, SrFe12O19/NiFe2O4, and NiFe2O4 particles M. Mehdipour and H. Shokrollahi Citation: J. Appl. Phys. 114, 043906 (2013); doi: 10.1063/1.4816089 View online: http://dx.doi.org/10.1063/1.4816089 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v114/i4 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 03 Aug 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_permissionsComparison of microwave absorption properties of SrFe 12O19, SrFe 12O19/NiFe 2O4, and NiFe 2O4particles M. Mehdipour1,a)and H. Shokrollahi2 1Department of Materials Engineering, Faculty of Mechanical Engineering, University of Tabriz, Tabriz 5166-16471, Iran 2Electroceramics Group, Department of Materials Science and Engineering, Shiraz University of Technology, Shiraz 71555-313, Iran (Received 5 May 2013; accepted 2 July 2013; published online 24 July 2013) In this study, ferrimagnetic (SrFe 12O19, SrFe 12O19/NiFe 2O4, and NiFe 2O4) nanostructure particles were synthesized by the co-precipitation of chloride salts using the sodium hydroxide solution.The resulting precursors were heat-treated at 1100 /C14C for 4 h. After cooling in the furnace, the ferrite powders were pressed at 10 bars and then sintered at 1200/C14C for 4 h. The saturation magnetization was increased and the coercivity was decreased by sintering (because ofmorphology changing) and alternating of the ferrite kind. For example, at SrFe 12O19, the saturation magnetization was increased from 291 G to 300 G and the coercivity was decreased from 2.8 kOe to 1.8 kOe by sintering. The microwave absorption properties of the nanostructureparticles were studied by ferromagnetic resonance and transmit-line theories, as well as Reﬂection Loss plots. Before sintering, the RL spectra of SrFe 12O19and the composite were below /C03 dB, but they reached /C06 dB at 11.1 GHz for NiFe 2O4. The RL spectra of the samples were increased by sintering due to reduction of porosity and damping factor. The maximum microwave absorption reached /C035 dB (at resonance frequency) for the NiFe 2O4state.VC2013 AIP Publishing LLC . [http://dx.doi.org/10.1063/1.4816089 ] I. INTRODUCTION The absorbing materials are used to minimize the elec- tromagnetic interference (EMI), which is a speciﬁc type of environmental pollution1and for the development of radar absorbing materials that are fundamental in stealth technol- ogy2To overcome EMI problems, it is suggested that the EM absorbing materials with the capability for absorbingunwanted EMI signals be used. Recently, demands for vari- ous EMI absorbers have been increased dramatically; thus, no single material can fulﬁll the demands for large absorp-tion peak(s), less coating thickness, and wide working fre- quency bandwidth 3In general, practical microwave absorbers are divided into two basic types according to peakpattern: the ﬁrst type is resonant absorber (e.g., ferrites) and the second type is graded dielectric absorber (e.g., foams). The second type is impractical for EMI or radar cross-section (RCS) reduction because of high thickness. 4,5 Therefore, to improve the EM absorption properties of the RCS materials, many studies have focused on the resonantabsorbers, for example, ferrites. 6,7 Ferrites are classiﬁed into three groups according to their crystalline structures (hexaferrite, spinel, and garnet).Hexaferrites are classiﬁed into six classes according to their compositions and crystalline structures. They are M, W, X, Y, Z, and U type hex-ferrites. 8,9The M-type strontium hexa- ferrite (SrFe 12O19), with a hexagonal structure and hard mag- netic properties, has also been the subject of continuous interest as an electro-magnetic (EM) absorber for severaldecades due to its applicability as a dielectric or magnetic ﬁl- ler in electro-magnetic attenuation.8The nickel ferrite (NiFe 2O4), which has a cubic spinel structure, is a kind of magnetic material and has been studied as a microwaveabsorbing material. 10Traditional ferrite materials produced large electric or magnetic loss as advantage factor, but their fatal disadvantages restricted their widespread applications.For example, they have relatively large density (e.g., the den- sity of SrFe 12O19is about 5 g/cm3). Recently, many studies have also been focused on new composite materialsystems. 11,12 As compared to the literature in this ﬁeld, the present investigation deals with the synthesis of the nanostructureparticles of SrFe 12O19, SrFe 12O19/NiFe 2O4, and NiFe 2O4by the co-precipitation method, as well as analyzing the effect of the sintering and ferrite kind. This method is a low-costtechnique suitable for mass production, as compared with the other methods. 10The microwave absorption properties for nanostructure particles were studied by Reﬂection Lossplots from Network Analyzer measurement, ferromagnetic resonance and transmit-line theories. II. EXPERIMENTAL In the present investigation, analytical-grade ferric chlo- ride (FeCl 3,6 H 2O), strontium chloride (SrCl 2,6 H 2O), nickel chloride (NiCl 2), and NaOH were used for the synthesis of (SrFe 12O19, SrFe 12O19/NiFe 2O4, and NiFe 2O4) nanostructure particles by co-precipitation. Stoichiometric amounts of strontium, ferric, and nickel chlorides dissolved completely in ultrapure water to make an aqueous solution. Thebrownish-colored ferrite particles were precipitated from thisa)Electronic mail: mostafa_mehdipour67@yahoo.com. Tel.: þ989417380386. Fax:þ984112372188. 0021-8979/2013/114(4)/043906/7/$30.00 VC2013 AIP Publishing LLC 114, 043906-1JOURNAL OF APPLIED PHYSICS 114, 043906 (2013) Downloaded 03 Aug 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_permissionsmixture by gradually adding the sodium hydroxide (NaOH) solution at room temperature (PH ¼12). The aqueous sus- pension was stirred gently for 15 min to achieve good homo-geneity. The resulting precipitates were ﬁltered off, washed with water, and dried at 100 /C14C overnight. The as- synthesized particles were heat-treated at 1100/C14C for 4 h with 10/C14C/min. After the ferrite powders were cooled in the furnace, they were pressed at 10 bars, heat-treated again and sintered at 1200/C14C for 4 h. After the bulk samples were cooled in the furnace again, they were crushed in order to prepare the sintered ferrites powders. The phase identiﬁcation of annealed samples was car- ried out by x-ray diffraction (XRD) using a diffractometer (Siemens, D5000) with Cu k aradiation. The morphological study was conducted by scanning electron microscopy(SEM, Hitachi S4160). Magnetic measurements were made at room temperature in the applied ﬁeld range of /C010 kOe to 10 kOe by means of a magnetometer (Magnet-Physik,C-300). To study absorption properties, all the samples (70 wt.%) were mixed with epoxy resin and 3% hardener. The resulting ferrite epoxy mixture was cast into a rectangu-lar pellet (21.7 /C210 mm 2) of thickness 2.5 mm and then cured at room temperature for 24 h. The prepared composite was polished and mounted on an aluminum foil (to obtain asingle-layer metal-backed absorber) to exactly ﬁt into the measuring waveguide. The Reﬂection Loss measurements were carried out using a Network Analyzer (ST8410-C) inthe X-band from 8 GHz to 12 GHz at room temperature. III. RESULTS AND DISCUSSION A. XRD and SEM results The indexed XRD patterns of the nanostructure ferrite par- ticles (SrFe 12O19,S r F e 12O19/NiFe 2O4,a n dN i F e 2O4)b e f o r e and after sintering (1100/C14C for 4 h) have been shown in Fig. 1. Before sintering, the phase formation occurred, but after sinter- ing there was no change in the phase. The phase formation was conﬁrmed by the data-base SrFe 12O19(2h¼34:198/C14;J C P D S card no. 01-072-0739) and NiFe 2O4(2h¼35:452/C14;J C P D Scard no. 00-044-1485). These results are also observed for par- ticles after sintering (Fig. 1(II)). Thus, the kind of phases will not be changed by sintering. As expected, the degree of crystal-line and the amount of phases are further increased. The crys- tallite size of the SrFe 12O19(Fig. 1from employing Scherer’s formula) phase was found to increase by sintering from 33 nmbefore sintering to 54 nm after sintering. The SEM micrographs of the nanostructure particles heat-treated (before the sintering) at temperature of 1100 /C14C are shown in Fig. 2. By altering the ferrite kind, the particles exhibit multiple morphologies (Figs. 2(Ia) –2(Ic) ). The reac- tion rate of a solid-state transformation obeying the Jonhson-Mehl-Avrami (JMA) kinetic model can be written in the following form: da dt¼Anð1/C0aÞln1 ð1/C0aÞðn/C01Þ=nexp/C0E RT/C18/C19 "# ; (1) where ais the reacted fraction, t is time for transformation, T is the absolute temperature, R is the gas constant, n is the Avrami index parameter, which depends on the materialkind and controls the particles grown and morphology, E is the activation energy, and A is the pre-exponential factor. So, the Avrami index parameter is changed by altering theferrite kind as follow the particles morphology is also changed. 13 After sintering, by changing the ferrite kind, the particles also exhibit multiple morphologies (Figs. 2(IIa) –2(IIc) ). The average particles sizes are about 6 lm before sintering and increase to about 20 lm (after sintering and crushing). During sintering, the densities of particles are increased due to usual processes (e.g., decreasing the pore volume).14The changing of densities of the particles is shown by the sinteringin Table I. B. Magnetic properties The magnetic properties of nanostructure particles are shown before and after sintering in Fig. 3. Their saturation FIG. 1. XRD patterns of nanostructure particles before sintering (I) SrFe 12O19(a), SrFe 12O19/NiFe 2O4(b), NiFe 2O4(c), and nanostructure particles after sinter- ing (II) SrFe 12O19(a), SrFe 12O19/NiFe 2O4(b), NiFe 2O4(c).043906-2 M. Mehdipour and H. Shokrollahi J. Appl. Phys. 114, 043906 (2013) Downloaded 03 Aug 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_permissionsmagnetization was found to depend on the ferrite kind and sintering, increasing from 291 G (for SrFe 12O19) to 305 G (for NiFe 2O4) before sintering. This increase in the satura- tion magnetization can be attributed to the increased forma- tion of the soft phase (nickel ferrite as ferrimagnetic), asconﬁrmed by the X-ray study of the powder heat-treated at 1100 /C14C (Figs. 1(I) and 1(II)). This change can also be observed for the ferrites particles after sintering. In addition,the change in morphology may also affect the magnetic properties. 15 The coerctivity (before sintering) of 2.8 kOe was observed for the SrFe 12O19, decreasing, thereafter, to 0.02 kOe for the NiFe 2O4. A decrease in the coercivity (at the composite state) can be attributed to an increase in exchangecoupling interaction. 16–18The other information about mag- netic properties by changing the ferrite kind and sintering is shown in Table II. C. Microwave absorption properties Fig.4shows the Reﬂection Loss (RL) spectra of nano- structure particles before sintering. It is seen that the RL of the samples alters by changing the ferrite kind. In this state,the RL spectra of SrFe 12O19and SrFe 12O19/NiFe 2O4are below /C03 dB, but it can reach /C06 dB at 11.1 GHz for NiFe 2O4. Generally, microwave magnetic losses of magnetic par- ticles originate from hysteresis, domain wall resonance, eddy current, and ferromagnetic resonance. In the current case, thehysteresis loss is negligible in the weak applied ﬁeld. The domain wall displacement loss occurs only in the MHz rangerather than that of GHz. Therefore, the contribution of do- main wall resonance can also be excluded. The eddy current loss, which is related to the thickness and electric conductiv-ity, can be ignored (ferrites are nonconductive). In our study, the magnetic loss contains only the ferromagnetic resonance within the X-band. 19,20Hence, the dielectric characterization can seem constant. The theory of ferromagnetic resonance has been dis- cussed extensively in several books.21–24The loss (micro- wave absorption) has been expressed in the equations of motion by two main forms: the Bloch-Bloembergen25and Landau-Lifshitz26,27forms. The Landau-Lifshitz (L-L) form will be considered here. The L-L equation can be written in the form proposed by Gilbert (in Gaussian units)28 dM dt¼cð1þ12ÞðM/C2HÞ/C01 jMjM/C2dM dt; (2) where1,c, M, and H are the dimensionless damping factor, the absolute value of the electron gyromagnetic ratio, the magnetization, and the total effective magnetic ﬁeld in thesample, respectively. The damping factor (in Eq. (1)) opposes the processional motion to relax the magnetization back to the steady-state equilibrium and it is related toimpurities (as the nonintrinsic process is often termed “two magnon-scattering”). 25A typical value of cis 2.8 GHz/kOe in the Gaussian unit for the ferrites. The net magnetic ﬁeld Hconsists of the applied ﬁelds H ext(here it is equal to zero) and coercive ﬁeld. The latter one results from the sample demagnetization and non-Maxwellian effective ﬁelds, aswell as magnetocrystalline anisotropy (2 k/M s). In general, the coercivity can attribute to the magnetization and magne- tocrystalline anisotropy by modiﬁed Brown’s equation thatis inserted at Eq. (2), 29TABLE I. The changing of density by the ferrite kind before and after sintering. Density (gr/cm3) SrFe 12O19 SrFe 12O19/NiFe 2O4 NiFe 2O4 Before the sintering 4.85 4.89 4.91 After the sintering 5.01 5.06 5.09 FIG. 2. SEM micrographs of nanostructure particles heat-treated (before sintering) at temperature of 1100/C14C (I-a) SrFe 12O19, (I-b) SrFe 12O19/NiFe 2O4, (I-c) NiFe 2O4and after sintering at 1200/C14C and crushing (II-a) SrFe 12O19, (II-b) SrFe 12O19/NiFe 2O4, (II-c) NiFe 2O4.043906-3 M. Mehdipour and H. Shokrollahi J. Appl. Phys. 114, 043906 (2013) Downloaded 03 Aug 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_permissionsHc¼2k Msa1aexak/C0NzMs; (3) where a1is the orientation distribution of the grains (the ran- domly oriented structure is equal to 0.5), akis the coefﬁcient which takes into account the reduction in anisotropy in the region near the internal surface as grain boundaries and inter-phases, and a exis used to describe the effect of the exchange coupling between the neighboring grains on the coercivity ﬁeld of the magnet. K and N zare the magnetocrystalline ani- sotropy and demagnetization factor, respectively. Thus, the net magnetic ﬁeld can be attributed to the coercivity. By solving Eq. (1)and using the Maxwell equations, permeability ðlÞcan be written by Eqs. (4)–(9),28l¼l/C1þil/C1/C1(4) l/C1¼1þ4pxmðx0/C0xÞ ðx0/C0xÞ2þ12x2(5) l/C1/C1¼4p1xx m ðx0/C0xÞ2þ12x2(6) x0¼cH; (7) xm¼cMs; (8) 1¼3pMs# HV: (9) In Eq. (2),l/C1andl/C1/C1are the real and imaginary parts, respec- tively. In Eqs. (3)and(4),xandx0are the angular fre- quency and the Larmor frequency, respectively, the second deﬁnes the natural precession frequency under a static mag- netic ﬁeld.28In Eq. (8),#and V are the pore volume and particles volume, respectively, which can be attributed to the density. The RL of a microwave absorption layer backed by a pre- fect conductor was calculated by means of the transmit-line FIG. 3. Effect of the ferrite kind on the hysteresis loops of nanostructure particles (a) before and (b) after sintering. TABLE II. The changing of magnetic properties by the ferrite kind before and after the sintering. SrFe 12O19 SrFe 12O19/NiFe 2O4 NiFe 2O4 Hc(kOe) M s(G) H c(kOe) M s(G) H c(kOe) M s(G) Before the sintering 2.8 291 0.7 302 0.02 305 After the sintering 1.8 300 0.2 345 0.02 350 FIG. 4. The changing of the RL spec- tra of nanostructure particles with altering the ferrite kind in the fre- quency range of 8–12 GHz before sintering.043906-4 M. Mehdipour and H. Shokrollahi J. Appl. Phys. 114, 043906 (2013) Downloaded 03 Aug 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_permissionstheory using the relative complex permittivity and permeability,30–32 RL¼20 logZin/C01 Zinþ1; (10) Zin¼ﬃﬃﬃl er/C20 tanh2pxd c/C18/C19 ﬃﬃﬃﬃﬃlep/C21 ; (11) where RL is a ratio of the reﬂected power to the incident power in dB, Z inis the input impedance of the absorber, d is the thick- ness of the absorber, and c is the velocity of the light. eis the permittivity which its value has been supposed about 8.8 By recalling Eqs. (2)–(10) and using the magnetic prop- erties of ferrites (according to Fig. 3(I)) at the microwave range, the 3–D models of the RL-M s-x, and RL-H c-xare shown in Fig. 5. Generally, RL is increased by decreasing the coercivity and increasing the magnetization. Thus, changes in RL (Fig. 4) can be explained due to the decrease in the coerciv- ity and an increase in the saturation magnetization forferrites particles according to the ferromagnetic resonance and transmit-line theories. Fig. 6shows the RL spectra of nanostructure particles after sintering. It is seen that RL of nanostructure particles alters again by changing the ferrite kind and shows multi- peak because of multi saturation magnetization originatedfrom distribution particles size that is usual at co- precipitation method. This behavior was studied completely in Ref. 19. In addition, the RL spectra of SrFe 12O19and SrFe 12O19/NiFe 2O4reach /C015 dB and /C018 dB at resonance frequencies, but it reached /C035 dB at 11 GHz for NiFe 2O4 with RL over /C05 dB at the whole X-band. These changes in RL can also be explained due to the decreasing of the coer- civity and the increasing of the saturation magnetization for ferrites particles. In Figs. 6and7, the nanostructure particles exhibit more RL (microwave absorption) after sintering. These obvious increases can be explained as follows due to the dampingfactor: the density of the particles increased by sintering (Table I) and then the damping factor (from Eq. (7)) decreased. As a result, the 3-D model of the changing of RL FIG. 5. (a) 3-D model of RL-M s-xand (b) 3-D model of RL-H c-x. FIG. 6. The changing of the RL spec- tra of nanostructure particles with the ferrite kind in the frequency range of 8–12 GHz after sintering.043906-5 M. Mehdipour and H. Shokrollahi J. Appl. Phys. 114, 043906 (2013) Downloaded 03 Aug 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(from theories) by damping factor and frequency (RL- 1-x) is obtained according to Fig. 7. Generally, RL is increased by decreasing the damping factor at the X-band. During sintering, the magnetic proper-ties are also improved in order to increase RL. The saturation magnetization was increased from 300 G for SrFe 12O19to 305 G for NiFe 2O4before the sintering; this change for the saturation magnetization was also observed for SrFe 12O19 before and after the sintering but the RL was increasedgreatly after the sintering. So, it is seemed the damping fac-tor had a greater effect on the RL than the magnetic proper- ties in comparison between Figs. 6and7. IV. CONCLUSION The uniform hexagonal plate and pyramidal shaped nanostructure particles (SrFe 12O19, SrFe 12O19/NiFe 2O4, and NiFe 2O4) have successfully been synthesized by the co-precipitation method. Before sintering, the Sr-ferritepure particles had lower saturation magnetization of 291 G and higher intrinsic coercivity (2.8 KOe), as compared to the Ni-ferrite and the composite. These changes in mag-netic properties were also exhibited for the ferrite particles after sintering. In the study of microwave absorption prop- erties, the optimum condition fell in the sintered pure softphase (NiFe 2O4). Hence, the sintering and the ferrite with more saturation magnetization and lower coercivity can seem as good ways to increase RL for the synthetic work-ing conditions (the thickness of 2.5 mm, X-band, and pro- duction process). The changes of RL were successfully studied by the ferromagnetic resonance and transmit-linetheories. ACKNOWLEDGMENTS The authors are grateful to Mr. Abotrab and Dr. Farshbaf, respectively, for their courtesies of providing con- venience for the VNA tests in the Antenna laboratory ofKhaje Nasiri University.1B. Vishwanathan and V. Murthy, Ferrite Materials Science and Technology (Narosa, New Delhi, 1990). 2M. Gama, C. Rezende, and C. Dantas, “Dependence of microwave absorption properties on ferrite volume fraction in MnZn ferrite/rubber ra- dar absorbing materials,” J. Magn. Magn. Mater. 323, 2782–2785 (2011). 3L. Chen, Y. Duan, L. Liu, J. Guo, and S. Liu, “Inﬂuence of SiO 2ﬁllers on microwave absorption properties of carbony1 iron/carbon black double- layer coating,” Mater. Des. 32, 570–574 (2011). 4F. Qin and C. Brosseay, “A review and analysis of microwave absorption in polymer composites ﬁlled with carbonaceous particles,” J. Appl. Phys. 111, 061301 (2012). 5X. C. Tong, Advanced Materials and Design for Electromagnetic Interference Shielding (CRC Press, 2009). 6C. Singh, S. BidraNarang, and M. Koledintseva, “Microwave absorption characteristics of substituted Ba .5Sr.5MxFe(12/C02x)O19(M¼Co-Zr and Co- Ti),” Microwave Opt. Technol. Lett. 54, 1661–1665 (2012). 7G. Mu, N. Chen, X. Oan, H. Shen, and MingyuanGu, “Preparation and microwave absorption properties of barium ferrite nanorods,” Mater. Lett. 62, 840–842 (2008). 8R. C. Pullar, “Hexagonal ferrites: A review of the synthesis, properties and applications of hexaferrite ceramics,” Prog. Mater. Sci. 57, 1191–1334 (2012). 9B. Cullity and C. Graham, Introduction to Magnetic Materials , 2nd ed. (IEEE Press Editorial Board, 2009). 10S. Tyagi, V. Agarwala, and T. Shami, “Synthesis and characterization ofSrFe 11.2Zn.8O19nanoparticles for enhanced microwave absorption,” J. Electron. Mater. 40, 2004–2014 (2011). 11S. Tyagi, H. Baskey, R. Agarwala, V. Agrwala, and T. Shami, “Development of hard/soft ferrite nanocomposite for enhanced microwaveabsorption,” Ceram. Int. 37, 2631–2641 (2011). 12Y. Qing, W. Zhou, F. Luo, and D. Zhu, “Microwave electromagnetic prop- erties of carbony1 iron particles and Si/C/N nano-powder ﬁller epoxy- silicone coating,” Physica B 405, 1181–1184 (2010). 13L. Wang, F. He, and Y. Wan, “Facile synthesis and electromagnetic wave absorption of magnetic ﬁber coated with Fe-Co alloy electroplating,” J. Alloys Compd. 509, 4726–4730 (2011). 14F. Liu, F. Sommer, C. Bos, and E. J. Mittemijer, “Analysis of solid state phase transformation linetics:models and recipes,” Int. Mater. Rev. 52, 193–212 (2007). 15M. Barisokatan, Microstructure Development in Nickel Zinc Ferrites (The Graduate School of Natural and Applied Middle East Technical University, 2005). 16M. Rangolia, M. Chhanthbar, A. Tanna, K. Modi, G. Baldha, and H. HJosi, “Magnetic behavior of nano sized and coarse powder of Cd-Ni fer- rites synthesized by wet-chemical route,” Indian J. Pure Appl. Phys. 46, 64–66 (2008). 17H. Zeng, J. Li, J. Liu, and S. Sun, “Exchange-coupled nanocomposite mag-nets by nanoparticle self-assembly,” Nature (London) 420, 395–398 (2002). 18R. C. O’ Handley, Modern Magnetic Materials Principle and Applications (Wiley Interscience Publication, 2000). 19Z. Liu, D. Zeng, R. Ramanujan, X. Zhong, and H. Davies, “Exchange interaction in rapidly solidiﬁed nanocrystalline RE-(Fe/Co)-B hard mag- netic alloys,” J. Appl. Phys. 105, 07A736 (2009). 20E. Hoseeinkhani, M. Mehdipour, and H. Shokrollahi, “Comparison of direct and indirect measurements of the saturation magnetization of barium hexa- ferrite synthesized by coprecipitation,” Electron. Mater. 42, 739–744 (2013). 21Z. Ma, Y. Zhang, C. T. Cao, J. Yuan, Q. Liu, and J. Wang, “Attractive microwave absorption and the impedance match effect in zinc oxide and carbony1 iron composite,” Physica B 406, 4620–4624 (2011). 22R. Soohoo, Theory and Application of Ferrites (Prentice-Hall, 1960). 23B. Lax and K. Button, Microwave Ferrites and Ferrimagnetics (McGraw- Hill, New York, 1962). 24S. Vonsovslii, Ferromagnetic Resonance (Pergamon Press Ltd., 1966). 25A. Gurevich and G. Melkov, Magnetization Oscillation and Waves (CRC Press Inc., 1996). 26M. Hurben and C. Patton, “Theory of two magnon scattering microwaverelaxation and ferromagnetic resonance linewidth in magnetic thin ﬁlms,” J. Appl. Phys. 83, 4344–4360 (1998). 27R. Ramprasad, P. Zurcher, M. Petras, and M. Miller, “Magnetic properties of metallic ferromagnetic nanoparticle composites,” J. Appl. Phys. 96, 519–529 (2004). 28M. Hurben, D. Franklin, and C. Patton, “Angle dependence of ferromag-netic resonance line width in easy-axis and easy-plane single crystal hex-agonal ferrite disks,” J. Appl. Phys. 81, 7458–7467 (1997). FIG. 7. 3-D model of RL- 1-x.043906-6 M. Mehdipour and H. Shokrollahi J. Appl. Phys. 114, 043906 (2013) Downloaded 03 Aug 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_permissions29U. Ozgur, Y. Alivov, and H. Morkoc, “Microwave ferrites, Part 1: Fundamental properties,” J. Mater. Sci.: Mater. Electron. 20, 789–834 (2009). 30H. Bertorello, P. Bercoff, and M. Oliva, “Model of interactions in nano-metric particles of barium hexaferrite,” J. Magn. Magn. Mater. 269, 122–130 (2004).31X. Meng, Y. Wan, Q. Li, J. Wang, and H. Luo, “The electrochemical prep-aration and microwave absorption properties of magnetic carbon ﬁbers coated with Fe 3O4ﬁlms,” Appl. Surf. Sci. 257, 10808–10814 (2011). 32L. Xi, X. Shi, Z. Wang, Y. Zuo, and I. Du, “Microwave absorption properties of Sr 2FeMnO 6nanoparticles,” Physica B 406, 2168–2171 (2011).043906-7 M. Mehdipour and H. Shokrollahi J. Appl. Phys. 114, 043906 (2013) Downloaded 03 Aug 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
1.4994972.pdf	Damping constant measurement and inverse giant magnetoresistance in spintronic devices with Fe 4N Xuan Li , Hongshi Li , Mahdi Jamali , and Jian-Ping Wang Citation: AIP Advances 7, 125303 (2017); View online: https://doi.org/10.1063/1.4994972 View Table of Contents: http://aip.scitation.org/toc/adv/7/12 Published by the American Institute of Physics Articles you may be interested in Research on c-HfO 2 (0 0 1)/ -Al2O3 (1 -1 0 2) interface in CTM devices based on first principle theory AIP Advances 7, 125001 (2017); 10.1063/1.5001904 Temperature dependence of magnetically dead layers in ferromagnetic thin-films AIP Advances 7, 115022 (2017); 10.1063/1.4997366 Possible observation of the Berezinskii-Kosterlitz-Thouless transition in boron-doped diamond films AIP Advances 7, 115119 (2017); 10.1063/1.4986315 Inverse giant magnetoresistance found in thin-film device Scilight 2017 , 240001 (2017); 10.1063/1.5013157 Laser pulse shape dependence of poly-Si crystallization AIP Advances 7, 125102 (2017); 10.1063/1.4998221 Angle and rotational direction dependent horizontal loop shift in epitaxial Co/CoO bilayers on MgO(100) AIP Advances 7, 115223 (2017); 10.1063/1.4985032AIP ADV ANCES 7, 125303 (2017) Damping constant measurement and inverse giant magnetoresistance in spintronic devices with Fe 4N Xuan Li,1Hongshi Li,2Mahdi Jamali,1and Jian-Ping Wang1,2,a 1Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, Minnesota 55455, United States 2Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455, United States (Received 9 July 2017; accepted 10 October 2017; published online 4 December 2017) Fe4N is one of the attractive materials for spintronic devices due to its large spin asymmetric conductance and negative spin polarization at the Fermi level. We have successfully deposited Fe 4N thin ﬁlm with (001) out-of-plane orientation using a DC facing-target-sputtering system. A Fe(001)/Ag(001) composite buffer layer is selected to improve the (001) orientation of the Fe 4N thin ﬁlm. The N 2partial pressure during sputtering is optimized to promote the formation of Fe 4N phase. Moreover, we have measured the ferromagnetic resonance (FMR) of the (001) oriented Fe 4N thin ﬁlm using coplanar waveguides and microwave excitation. The resonant ﬁelds are tested under different microwave excitation frequencies, and the experimental results match well with the Kittel formula. The Gilbert damping constant of Fe 4N is determined to be = 0.0210.02. We have also fabricated and characterized the current-perpendicular- to-plane (CPP) giant magnetoresistance (GMR) device with Fe 4N/Ag/Fe sandwich. Inverse giant magnetoresistance is observed in the CPP GMR device, which suggests that the spin polarization of Fe 4N and Fe 4N/Ag interface is negative. © 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/10.1063/1.4994972 I. INTRODUCTION Fe4N has been attracting attentions in spintronics1–3because it exhibits a highly spin-polarized electrical conductance and a negative spin polarization at the Fermi level.4Materials with large spin polarization are highly desired since they can improve the magnetoresistance (MR) ratio of tunneling/giant magnetoresistive devices. It is theoretically predicted that Fe 4N has a large spin polarization due to the interactions between the 3d electrons of Fe and the 2s, 2p electrons of N.4The spin polarization of Fe 4N has been experimentally determined to be 0.59 by point contact Andreev reﬂection.5Magnetic tunnel junctions with Fe 4N ferromagnetic electrodes have been reported, and their MR ratios are as large as 75%.6,7Furthermore, different from the conventional ferromagnetic electrode materials such as CoFeB and CoFe, the spin polarization of Fe 4N has been predicted to be negative,4namely the conductivity of the minority spin electrons is higher than the majority spin electrons. This characteristic is conﬁrmed by both anisotropic magnetoresistance8–10and spin- resolved photoelectron spectroscopy11results. The negative spin polarization of Fe 4N provides a pathway for a group of novel spintronic logic devices.12 Recently, the current induced magnetization switching has been realized in Fe 4N based mag- netic tunnel junctions.1According to Slonczewski’s spin transfer torque (STT) model,13the critical switching current densities of GMR or tunnel magnetoresistance (TMR) devices are proportional to the damping constants of the magnetic free layers. Thus it is of great interest to study the damping constant () of Fe 4N to understand its limitation for STT induced magnetization switching. Until aAuthor to whom correspondence should be addressed: jpwang@umn.edu 2158-3226/2017/7(12)/125303/6 7, 125303-1 ©Author(s) 2017 125303-2 Li et al. AIP Advances 7, 125303 (2017) now, only a few articles have reported the damping constant of Fe 4N.14,15Those results are subject to strong spin pumping effect which tends to overestimate the damping constant. In this work, we have deposited Fe 4N thin ﬁlms with (001) out-of-plane orientation on Fe/Ag underlayers. The Ag layer in contact with Fe 4N works natively as the bottom electrode for tunnel- ing/giant magnetoresistive devices. The damping constant of the (001) oriented Fe 4N is measured by the ferromagnetic resonance in coplanar waveguides. Due to the weak spin pumping effect of Ag,16our stack structure does not artiﬁcially increase the effective damping of the Fe 4N thin ﬁlm. Furthermore, the CPP GMR with Fe 4N/Ag/Fe sandwich is fabricated and characterized. Inverse giant magnetoresistance is observed in the CPP GMR device, which proves the negative spin polarization of Fe 4N and Fe 4N/Ag interface. II. EXPERIMENTS Fe4N multilayer stacks with a structure of MgO(001)/Fe(5nm)/Ag(35nm)/Fe 4N(17nm)/Ru(5nm) are deposited by a target-facing-target sputtering system with three pairs of DC facing sputter- ing sources. The base pressure of the sputtering chamber is lower than 4.0 10-8Torr. During the multilayer stack preparation, the Fe/Ag bilayer is ﬁrst grown onto a MgO (001) single crys- tal substrate at room temperature. Thereafter, the substrate is heated to 285oC and the Fe 4N thin ﬁlm is deposited by reactive sputtering in a Ar and N 2gases mixture. To obtain stoichiometric Fe4N thin ﬁlms, the N 2partial pressure varies from 0.35 mTorr to 0.6 mTorr while the total gas pressure is maintained at 2.5 mTorr during the sputtering process. The Ru capping layer is then deposited on top of the Fe 4N layer. After optimizing the grown conditions of the Fe 4N thin ﬁlm, stacks of MgO substrate/Fe(5)/Ag(50)/Fe 4N(7)/Ag(5)/Fe(7)/Ag(5)/Ru(8) (nm) are deposited for the CPP GMR nanoscale devices fabrication. In order to characterize the Fe 4N phase and out-of-plane crystal orientations, -2X-ray diffrac- tion (XRD) measurements are performed on a Simens Bruker D5005 system with Cu K radiation. The rocking curves of the Fe 4N thin ﬁlms are measured by a Panalytical X’Pert Pro System. X-ray photoelectron spectroscopy (XPS) is performed on a Surface Science SSX-100 system to further conﬁrm the stoichiometry. The surface roughness is characterized by an Agilent 5500 atomic force microscope (AFM). Magnetic properties of the samples are examined by a Princeton Measurements vibrating sample magnetometer (VSM). We measure the ferromagnetic resonance (FMR) of the Fe 4N thin ﬁlms utilizing coplanar waveguides for the magnetization excitation. The FMR results are ﬁtted by the Kittel formula, and the damping constant is determined from the linewidth of the resonance ﬁeld. We also fabricate the Fe 4N/Ag/Fe CPP GMR by an electron beam lithography and Ar+ion etching combined process. The magnetoresistance is measured with a four-point-probe method. III. RESULTS AND DISCUSSION A. Structural properties of Fe 4N Figure 1(a) shows the out-of-plane X-ray diffraction (XRD) patterns of the MgO substrate/Fe(5nm)/Ag(35nm)/Fe-N(17nm)/Ru(5nm) stacks. Fe 4N has the face-centered-cubic Fe lat- tice with N atom located at the body center. The in-plane lattice constants of Fe 4N and Ag are 3.795Å and 4.079Å respectively, thus there is a 7% lattice mismatch between the Fe 4Nf001gand Ag f001g planes. In our experiment, the substrate temperature is maintained at 285C for all the growth condi- tions. Based on the Fe-N phase diagram, several iron nitride compounds can be formed with different Fe:N compositions, therefore the N 2partial pressure during sputtering needs to be optimized. As the N 2partial pressure is gradually tuned from 0.35 mTorr to 0.6 mTorr, the nitrogen composition increases continuously in the Fe-N thin ﬁlms. In Figure 1(a), Fe 4N (002) diffraction peaks can be observed for all the growth conditions, because the high substrate temperature promotes the formation of Fe 4N phase. The Fe 4N (001) diffraction peak is absent in our samples due to the small ﬁlm thick- ness. When the N 2sputtering partial pressures are 0.5 mTorr and 0.6 mTorr, besides the Fe 4N (002) peak, two small peaks of Fe 3N near 39.2and 57.5are present in the XRD pattern, which indicates that the N:Fe atomic ratio is larger than 1:4. As we decrease the N 2partial pressure to 0.4 mTorr,125303-3 Li et al. AIP Advances 7, 125303 (2017) FIG. 1. (a) XRD -2scans of the 17nm Fe-N thin ﬁlms deposited with different N 2partial pressures. The Fe(002) diffraction signals are from the Fe buffer layer; (b) The rocking curve measured on Fe 4N (002) peak of the sample with 0.4 mTorr N 2 partial pressure; (c) XPS spectrum of the same Fe 4N sample. the peaks of Fe 3N disappear, and the Fe 4N (002) peak becomes very pronounced. When we further decrease the N 2partial pressure to 0.35 mTorr, the intensity of the Fe 4N (002) peak is weaker, and the Fe (002) peak appears. Therefore we conclude that the 0.4 mTorr N 2partial pressure is the optimized condition for the growth of Fe 4N thin ﬁlms. The substrate temperature cannot be further increased in the deposition process due to the limitation of the sputtering system. Alternatively, we transfer the optimized sample to another chamber without breaking the vacuum, and post-annealed the sample at 325C for 2 hours. No noticeable change is seen in the XRD result of the annealed sample. Although the post-annealing does not promote the crystallinity, it improves the surface ﬂatness of the sample. The root mean square (RMS) roughness reduces from 0.75 nm of the as-deposited sample to 0.34 nm with the post-annealing process. The rocking curve is measured on the Fe 4N (002) diffraction peak of the optimized sample, as shown in Figure 1(b). Considering that the Fe 4N layer is as thin as 17nm, the FWHM ( !=2.26) of the rocking curve indicates that the ﬁlm has strong (001) out-of-plane orientation. In order to conﬁrm the stoichiometry of the Fe 4N thin ﬁlm, we perform X-ray photoelectron spectroscopy (XPS) measurement on the optimized Fe 4N sample. In Figure 1(c), Fe 2p3/2, Fe 2p1/2, and N 1s peaks are presented in the XPS spectrum. By integrating the areas under these peaks and dividing them by their sensitive factors, the N/Fe atomic ratio of 0.22 0.025 is obtained. This result indicates that the Fe 4N thin ﬁlm with 0.4 mTorr N 2sputtering partial pressure is nearly stoichiometric. The saturation magnetization of the post-annealed Fe 4N sample is measured to be 1050 emu/cm3by vibrating sample magnetometry. B. Damping constant of Fe 4N In order to measure the ferromagnetic resonance (FMR) of the (001) oriented Fe 4N thin ﬁlm in the developed stack, the 5nm Fe underlayer needs to be excluded from the stack to eliminate the interference of FMR signals. We initially tried to deposit a MgO substrate/Ag/Fe 4N stack without the125303-4 Li et al. AIP Advances 7, 125303 (2017) Fe underlayer. However both the ﬁlm adhesion and crystal quality of the samples are poor. It has been reported that (001) oriented Fe 4N thin ﬁlms can be grown epitaxially on MgO substrates,2though the lattice mismatch between Fe 4N and MgO is as large as 9.6%. Therefore, we modify the thin ﬁlm stack with a structure of MgO substrate/Fe 4N 17/Ag 17/Ru 5 (nm). After that, the sample is patterned into micrometer scale coplanar waveguides for the magnetization excitation. The details of coplanar waveguides and magnetization excitation methods are discussed in our previous publication.17 Next, we inject the radio frequency (RF) signals into the waveguides and measure the resonance ﬁeld of the magnetization excitation. The frequency of the RF signal ranges from 4GHz to 17GHz. The excitation frequency versus the resonance ﬁeld relationship matches well with the Kittel formula f= 2pH(H+Ms), as shown in Figure 2. By the curve ﬁtting, we extract the saturation magnetiza- tion to be1000 emu/cm3, which is very close to the vibrating sample magnetometry measurement result. The gyromagnetic ratio is ﬁtted to be 2.88 105rad/(A/m). The Gilbert damping constant of Fe4N is calculated from =p 3 H 2!, and it is determined to be about = 0.019 for the excitation fre- quency of 16GHz. The damping constant of Fe 4N deduced from the full excitation frequency range is = 0.0210.02, which is comparable to the other soft magnetic materials, such as NiFe,18CoNi19and annealed CoFeB.17The relatively small damping constant of Fe 4N may give a low critical switching current in GMR/TMR devices according to Slonczewski’s spin transfer torque model.13 C. Inverse magnetoresistance of Fe 4N/Ag/Fe CPP GMR Based on the (001) oriented Fe 4N thin ﬁlms that we have developed, we further prepare a current- perpendicular-to-plane (CPP) giant magnetoresistance (GMR) stack with a multilayer structure of MgO substrate/Fe(5)/Ag(50)/Fe 4N(7)/Ag(5)/Fe(7)/Ag(5)/Ru(8) (nm). A in-vacuum annealing pro- cess mentioned above is applied on the Fe 4N layer prior to depositing the rest layers of the stack. A reference multilayer stack is also prepared by replacing the Fe 4N (7nm) layer with another Fe (7nm) layer. These two samples are subsequently fabricated into 100nm nanopillar devices by elec- tron beam lithography and Ar+ion beam etching combined processes. The lateral dimensions of the elliptical nanopillars are 160 100nm2and 140100nm2respectively for the CPP GMR devices with Fe4N/Ag/Fe and Fe/Ag/Fe sandwiches. Both the Fe 4N and the Fe nanometer scale magnets of the as-fabricated devices have their easy magnetic axis along the in-plane long axes of the ellipses. Giant magnetoresistance signals of the devices are measured by a four-point-probe method. TheR-H loops of the Fe 4N/Ag/Fe CPP GMR and Fe/Ag/Fe CPP GMR are given in Figure 3(a) and Figure 3(b) respectively. Since there is no pinned magnetic layer in both the CPP GMR devices, the magnetizations of the two magnetic layers are anti-parallel coupled by dipolar magnetic interactions in remanence states. When a relatively large magnetic ﬁeld is applied along the long axis of the 100nm ellipses, the two magnetic layers are parallel aligned by the external ﬁeld. Figure 3(a) shows that the Fe 4N/Ag/Fe CPP GMR device has lower resistance in the anti- parallel state and higher resistance in the parallel state, namely the inverse giant magnetoresistance FIG. 2. The resonant magnetic ﬁelds for different excitation frequencies which are overlaid with the Kittel formula curve ﬁtting. The inset shows the FMR line width of the Fe 4N thin ﬁlm measured at 16GHz.125303-5 Li et al. AIP Advances 7, 125303 (2017) FIG. 3. The giant magnetoresistance signals of (a) Fe 4N/Ag/Fe CPP GMR; (b) Fe/Ag/Fe CPP GMR as a function of in-plane magnetic ﬁeld. is observed. This behavior is quite different from the typical giant magnetoresistance which presents in the CPP GMR devices with the same ferromagnetic free and ﬁxed layers, as seen in the Fe/Ag/Fe CPP GMR device in Figure 3(b). We attribute this unique inverse giant magnetoresistance behavior to the negative spin polarization of Fe 4N and the negative spin scattering asymmetry at the Fe 4N/Ag interface. The Fe 4N/Ag/Fe CPP GMR that has inverse giant magnetoresistance may lead to novel spin-logic devices. GMR/TMR elements with low switching current densities are desired in the spin- logic applications, where Fe 4N may show advantages due to its relatively small damping constant. In addition, the magnetoresistance R of the Fe 4N/Ag/Fe CPP GMR is observed to be smaller than that of the Fe/Ag/Fe device with similar lateral dimensions. It is known that the magnetoresistance signal of GMR is mainly contributed by the interface spin dependent scattering. This result suggests that the majority/minority spin contrast of the Fe 4N/Ag interface is not as large as that of the Fe/Ag interface. To further improve the magnetoresistance signal, a spacer that pairs well with both Fe 4N and Fe needs to be discovered. IV. CONCLUSIONS Fe4N thin ﬁlms with (001) out-of-plane orientation are prepared on Fe(001)/Ag(001) buffer layers by facing-target-sputtering. The N 2partial pressure during sputtering is optimized to promote the formation of Fe 4N phase. Moreover, the Gilbert damping constant ( ) of the Fe 4N thin ﬁlm in contact with Ag is measured by ferromagnetic resonance and is extracted from the damping-linewidth relationship. The Fe4N is determined to be 0.021 0.02. Inverse giant magnetoresistance is observed in the CPP GMR device with Fe 4N for the ﬁrst time. This unique magnetoresistance behavior can be explained by the negative spin polarization of Fe 4N and the negative spin scattering asymmetry at Fe 4N/Ag interface. ACKNOWLEDGMENTS This work was supported by the C-SPIN center, one of six STARnet program research centers, a Semiconductor Research Corporation program, sponsored by MARCO and DARPA. Device fabri- cation was performed at the University of Minnesota Nanofabrication Center, which receives support from the National Science Foundation (NSF) through the National Nanotechnology Infrastructure Network program. Thin ﬁlm characterization was performed at the University of Minnesota Charac- terization Facility, which has received capital equipment funding from the NSF through the Materials Research Science and Engineering Center. 1S. Isogami, M. Tsunoda, Y . Komasaki, A. Sakuma, and M. Takahasi, Appl. Phys. Express 3, 103002 (2010). 2W. B. Mi, Z. B. Guo, X. P. Feng, and H. L. Bai, Acta Mater. 61, 6387 (2013). 3H. Xiang, F. Y . Shi, M. S. Rzchowski, P. M. V oyles, and Y . A. Chang, J. Appl. Phys. 109, 07E126 (2011). 4S. Kokado, N. Fujima, K. Harigaya, H. Shimizu, and A. Sakuma, Phys. Rev. B 73, 172410 (2006). 5A. Narahara, K. Ito, T. Suemasu, Y . K. Takahashi, A. Ranajikanth, and K. Hono, Appl. Phys. Lett. 94, 202502 (2009). 6K. Sunaga, M. Tsunoda, K. Komagaki, Y . Uehara, and M. Takahashi, J. Appl. Phys. 102, 013917 (2007). 7Y . Komasaki, M. Tsunoda, S. Isogami, and M. Takahashi, J. Appl. Phys. 105, 07C928 (2009).125303-6 Li et al. AIP Advances 7, 125303 (2017) 8K. Ito, K. Kabara, H. Takahashi, T. Sanai, K. Toko, T. Suemasu, and M. Tsunoda, Japanese J. Appl. Phys. 51, 068001 (2012). 9M. Tsunoda, Y . Komasaki, S. Kokado, S. Isogami, C. C. Chen, and M. Takahashi, Appl. Phys. Express 2, 083001 (2009). 10M. Tsunoda, H. Takahashi, S. Kokado, Y . Komasaki, A. Sakuma, and M. Takahashi, Appl. Phys. Express 3, 113003 (2010). 11K. Ito, K. Okamoto, K. Harada, T. Sanai, K. Toko, S. Ueda, Y . Imai, T. Okuda, K. Miyamoto, A. Kimura, and T. Suemasu, J. Appl. Phys. 112, 013911 (2012). 12S. Isogami and T. Owada, IEEJ Trans. Electr. Electron. Eng. 9, S73–S75 (2014). 13J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1–L7 (1996). 14S. Isogami, M. Tsunoda, M. Oogane, A. Sakuma, and M. Takahasi, Appl. Phys. Express 6, 063004 (2013). 15S. Isogami, M. Tsunoda, M. Oogane, A. Sakuma, and M. Takahashi, Jpn. J. Appl. Phys. 52, 073001 (2013). 16E. Barati, M. Cinal, D. M. Edwards, and A. Umerski, Phys. Rev. B 90, 014420 (2014). 17M. Jamali, A. Klemm, and J. P. Wang, Appl. Phys. Lett. 103, 252409 (2013). 18N. Inaba, H. Asanuma, S. Igarashi, S. Mori, F. Kirino, K. Koike, and H. Morita, IEEE Trans. Magn. 42, 2372 (2006). 19J. M. L. Beaujour, W. Chen, K. Krycka, C. C. Kao, J. Z. Sun, and A. D. Kent, Eur. Phys. J. B 59, 475 (2007).
1.30531.pdf	High speed bubble garnets based on large gyromagnetic ratios (High g) R. C. LeCraw , S. L. Blank , G. P. Vella‐ Coleiro , and R. D. Pierce Citation: AIP Conference Proceedings 29, 91 (1976); doi: 10.1063/1.30531 View online: http://dx.doi.org/10.1063/1.30531 View Table of Contents: http://aip.scitation.org/toc/apc/29/1 Published by the American Institute of PhysicsSection 7 Bubble Materials R.M. Josephs, Chairman 91 HIGH SPEED BUBBLE GARNETS BASED ON LARGE GYROMAGNETIC RATIOS (HIGH g) R. C. LeCraw, S. L. Blank, G. P. Vella-Coleiro and R. D. Pierce Bell Laboratories, Murray Hill, New Jersey 07974 ABSTRACT An approach to overcoming the problem of dynamic conversion in high-mobility bubble garnets is described based on large gyro- magnetic ratios (high-g factors). In a film of Eu I 45Y0 45Cai iFe3 QSi 0 6Ge0 qO12 , a g factor°greater tN~n 30"~as ~een 65rained, which increases the usable domain wall velocity before onset of dynamic conversion by more than an order of magnitude over comparable bubble garnet films with g approximately 2. The tem- perature dependence of the important bubble parameters has been measured and a simple bias magnet constructed which matches the steeper than usual variation of the bubble collapse field with temperature. Two different methods of hard bubble suppression are described, one involving short oxygen anneals at ~I050°C, and the other ion implantation. When useful bubble garnet materials with relatively high mobilities. ~i000 cm/secOe or greater, became available, I-3 another limita- tion on achieving high bubble velocities was observed. Experimentally, erratic propagation of bubble domains was observed during repeti- tive bubble transport measurements. # This was attributed to the conversion of a normal bubble domain into a relatively complex state, similar to a hard bubble, during rapid displace- ment. A model for this effect, called dynamic conversion, was given by Hagedorn, s who extend- 6 7 ed previous work of Slonczewski and Thiele. The critical or limiting velocity is given by G V = 24 YA/hK ½ (i) p u where Y = ge/2mc is the gyromagnetic ratio, A is the exchange constant, h is the film thick- ness, and K u is the uniaxial anisotropy con- stant. (In a ferrimagnet the g in Y is actu- ally the effective g, geff, but for conveni- ence g will be used here.) Thus far attempts to maximize Vp have in- volved keeping h and K u as small as possible consistent with other bubble requirements, and using as little diamagnetic substitution as possible to achieve the required reduced mag- netization, e.g., Ge in preference to Ga, 8,9 This has the effect of keeping the exchange constant A as large as possible. Another attempt was a three-layer film described by Hagedorn, 5 the purpose of the thin middle diamagnetic layer being to suppress undesir- able motions of Bloch lines which were be- lieved to lead to the dynamic instability. Some success in suppressing dynamic con- version has been achieved by keeping the thickness small,1°but this is a severely limit- ing boundary condition. The elimination of dynamic conversion of bubbles by using Permalloy-coated garnet films has also been proposed.llHowever, we believe these latter results are not yet conclusive, particularly in light of the discovery of the bubble over- shoot effect. 12 It was finally realized that the gyro- magnetic ratio y = ge/2mc in Eq. (i) had not been considered sufficiently for increasing the critical velocity, possibly because of the customary assumption that g z 2. However, a garnet system involving Eu together with a diamagnetic substitution on tetrahedral sites was shown by LeCraw, Remeika, and Matthews 13 to produce very large values of g where g = (MEn + MFe) I MEn + MFe~-I.- gEu gFe/ \ (2) Here MEu and MFe are the magnetizations of the Eu and total Fe sublattices, respectively, and gEu and gFe are the g factors for the Eu and Fe ions, respectively. Because of the J = O ground state of Eu, gEu >> 2, and hence g in Eq. (2) becomes very large as Mp e ÷ O. The total moment does not vanish as MFe ÷ O be- cause of the induced Eu moment, which results from its exchange coupling almost exclusively to tetrahedral Pe ions. These results are shown in Figs. 3-5 of Ref. 13. Only Eu of the rare earth (RE) series is effective in this way. The other possibly usable rare earth iron garnets, those with line widths no greater than Sm, i.e., Sm, Gd, Er, Tm, Yb have gRE ~ 2. Thus the denominator in Eq. (2) can become zero (high g) only at very nearly the same point at which the numerator becomes zero (zero total moment). The expected influence of the mobility U and g on wall velocity is shown schematically in Fig. I. This shows that even with large mobilities, high-g factors are necessary to achieve large usable velocities. Experi- mentally we have found that U is essentially independent of g, which indicates that if the simple model for domain wall motion is used, in the Gilbert equation is proportional to g. This observation is consistent with reson- ance linewidth measurements on films with widely differing g factors, where AH is ob- served to be essentially independent of g. It is also consistent with an expression de- rived by Fierce and LeCraw I# for the effec- tive phenomenological damping constant of a multiple sublattice system in which one sub- lattice contains ions with relatively large damping. 8000 o w 6000 o 4000 O w > w 2000 0 0 I I/ i g =I0 #/ F - ~ g=2 - 2.5 5.0 7.5 I 0.0 EFFECTIVE DRIVE FIELD (OE) Fig. i Schematic representation showing how both mobility and g influence the bubble velocity. 92 TABLE I Pertinent Bubble Properties of a Film of EUl.45Y0.45Cal.lFe3.gSi0.6Ge0.5012 Thickness 4.23 pm Demagnetized Strip Width 5.18 ~m Collapse Field 100.20e Material Length 0.64 ~m Curie Point 466°K Coercivity 0.4 Oe 4~M 218 G o Lattice Constant 12.385 A 2Ku/M 1500 Oe Mobility 1500 cm/secOe g >30 Thus a series of LPE films was grown con- taining Eu, with Ge-Ca instead of Ga used to reduce MFe to study the effects of high g factors on bubble velocities. Is Brief details of the growth conditions are given in Ref. 15, and the details of the phase equilibria 16 and the growth kinetics 9'16 in systems containing divalent-tetravalent ion substitutions are dis- cussed elsewhere. Table I shows the properties measured on a typical high-g film. It should be noted that 2Ku/M and g given in Table I were measured by microwave resonance techniques, 17 the lower limit on g being deter- mined by the microwave frequency of 17.5 GHz. It should also be pointed out that these large g factors are not strongly temperature depend- ent since when MFe is compensated to he ap- proximately zero by diamagnetic tetrahedral site substitution, the small remaining MFe is only slowly varying with T. Using such a high-g film, the propagation data at 1 and 2 MHz shown in Fig. 2 were ob- tained with a TX-type circuit having a period of 28.8 ~m. The parallel margins indicate that there is no discernible limit in the number of error-free propagation steps. Two IlO I I I I I 105 0 -- I00 Q _1 I,i.I I.i. 95 90 O ,O C ,0 ,,0 ~ ~ I TX TX 2 MHZ I MHZ 18 OE DRIVE 25 OEDRIVE 85 I I I I I 0 I0 2 104 106 108 lOt° 1012 NUMBER OF STEPS Fig. 2 Bias field margins vs. number of steps propagated at i and 2 MHz rates for a film of EUl.45Y0.45CaI.IFe3.9Si0.6Ge0.5 O12" MHz was the limit of the operating range of the propagation drive circuitry. At this frequency the available rf drive was 18 Oe compared to 25 Oe at 1 MHz. For the 28.8 ~m circuit period, 2 MHz is approaching the mobility-limited operating frequency at this drive. However, from Eq. (I) it can be cal- culated that dynamic conversion effects for g > 30 would not have occurred until above i0 MHz. A striking confirmation of the effect of g on the critical domain wall velocity has been observed on a 9-~m-thick film of the same composition as that in Table I together with a film similar in all other parameters but with less Ca and a g of 1.07. For the latter film the critical velocity is zl000 cm/sec, whereas for the 9-pm film with g > 30, Vp z30,000 cm/ sec. Velocities as high as 60,000 cm/sec were observed on the high-g film, which is probably the largest domain wall velocity yet observed in a magnetic garnet. TM Thus far we have reviewed briefly what has been published previously on high-g bubble gar- nets. We will now consider later developments: Another confirmation of the greatly in- creased suppression of dynamic conversion by high g factors has been observed recently by G. P. Vella-Coleiro 19 in noting the absence of bubble overshoot in a high-g film during his investigations of bubble motion using very high speed photography. This result together with the 2 MHz bubble propagation rate with flat bias margins out to i0" steps (Fig. 2), and the 60,000 cm/sec domain wall velocity all com- bine to give evidence which strongly supports the effect of high g factors on dynamic con- version. HARD BUBBLE SUPPRESSION In the absence of dynamic conversion it was expected from theoretical considerations that a material with g ~ 20 would not exhibit hard bubbles. Compared to the usual YSmCa- type garnets it was much more difficult to produce hard bubbles by the usual pulsing or rapid demagnetization techniques in the high g garnets. Yet some hard bubbles were pro- duced in all of the as-grown films. When the high-g samples were annealed in 02 at i050°C for 0.5 h, however, the hard bubbles were eliminated. This was repeated on several different high-g samples in both 02 and N 2. With the sample used for the data in Table I and Fig. 2, the annealing time was 0.75 h in 02 at i050°C. (Extra time was used to be certain). These short anneals may relieve some highly localized strains acting as pinning points with which the domain walls interact in a complicated way to produce hard bubbles as the domain walls "snap off" these points. How- ever, only a very small decrease in overall coercivity is observed for this short anneal at a temperature which is only slightly above the growth temperature. Etching off 3 ~m of the film in 1 ~m steps was tried to determine if the surface played a role, but no differ- ences were seen. It should be noted that the above short 02 anneal does not eliminate hard bubbles in the same class of EuYCa-type samples but with less Ca and g = 1.07. Even several hours in 02 at i050°C did not suffice. Further work needs to be done to understand this strik- ing effect. The short 02 anneals which effectively eliminate hard bubbles in high-g samples should not be confused with the recently described technique of suppression of hard bubbles in the usual type of bubble garnets by inert atmosphere annealing. There the temperatures are higher and the times are longer. This process is 93 reasonably well understood. 20 Ion implantation has also been used suc- cessfully to suppress hard bubbles in high-g films~ although the range of dosage, 1-3×1014 Ne/cm at I00 keV is only about one-third as wide as with the usual YSmGa and YSmCa-type films. 21 This is qualitatively consistent with the negative magnetostriction constant, esti- mated to be %111 ~ -0.3×10 -6 , using the method of R. L. White. 22 This value is almost an order of magnitude lower than %111 for the usual bubble garnets not containing Eu. By using Ferrofluid we have confirmed the exist- ance of a thin top layer with planar magneti- zation, as has been observed in other ion implanted garnets with negative magnetostric- tion. 23 TEMPERATURE DEPENDENCE In order to effectively utilize the con- siderable increase in possible bubble device speed with high-g films, it is necessary to know and allow for the temperature dependence of the important bubble parameters. Figures 3 and 4 show the temperature dependence of the collapse field Ho, the anisotropy field Hk, the magnetization 4~M, the material length %, the demagnetized strip width w, and the quality factor Q - Hk/4~M from 0 to 100°C. These data were obtained on a typical high-g film of the approximate composition given in Table I. The film was 6.6 ~m thick. The quantities direct- ly measured were Ho, Hk, w and the film thick- ness t, from which the other parameters were calculated using well-known relationships. Values of the wall energy o, the uniaxial an- isotropy constant Ku, and the exchange con- stant A can also be calculated at each tempera- ture. At 25°C they are, respectively, 0.23 erg/cm 2, 1.2×104 erg/cm 3, and 2.6×10 -7 erg/cm. The temperature dependence of H o is par- ticularly important since the permanent magnet which provides the bias field for a bubble device must track this field. Figure 3 shows that Ho(T ) is very nearly linear with a tem- perature coefficient at 50°C of -0.58%/°C. Similar high-g samples grown from different melts show very nearly the same linearity and slope. While Ho(T ) is steeper than would ordinarily be desired, another important de- 250~....~ r ( f i 2.5 200~..,,~ 2.0 <~-- 4~M ~ I 50 } ~ 100 % -r 5O 0 -- 0 I ] I I 0 20 40 60 80 I00 T(°C) LO Fig. 3 The collapse field Ho, the anisotropy field Hk, and the magnetization 4~M vs. tem- perature for a high-g film of EUl.45Y0.45Cal.iFe3.gSio.6Geo.5O12. (~ n- O =L 2 8 .~--. Q 6 ~il.O ,O-~W 0.8 ~.6 TM 0.5 I I [ I 0 20 80 I01 40 60 T(°C) Fig. 4 The material length ~, the demagnet- ized strip width w, and the quality factor Q = Hk/4~M vs. temperature for the same film as in Fig. 3. vice parameter, Q, remains nearly constant instead of decreasing strongly with T as ob- served in most bubble materials. This be- havior occurs because 4FM decreases rapidly enough to compensate for the usual rapid de- crease of K u with T. In a typical YSmCa-type film with a similar Curie point the variation of H o with temperature is much slower than -0.58%/°C. We believe the difference is because of the fol- lowing: In the present material with g > 20, the net magnetization of the iron sublat~ices is approximately zero and hence the net moment is essentially all from the Eu ions. LeCraw, Remeika and Matthews 13 showed that the exchange field acting on the Eu is almost entirely due to the tetrahedral iron ions. Thus one would expect, based on the temperature independent paramagnetie susceptibility XEu of Eu, an ob- served moment proportional to Mtet, which varies much more slowly than -0.58%/°C. Such behavior can be expected at lower temperatures. However, the J = 1 multiplet levels of Eu, which average %500OK above the J = 0 ground state, begin to be occupied at room temperature and XEu becomes temperature dependent. Thus the temperature dependence of H o and 4~M shown in Fig. 3 are determined by the combined ef- fects of Mtet(T ) and XEu(T), making them steeper than in the usual YSmCa-type films where the net moment is dominated by the tetrahedral iron. Several methods of constructing a bias magnet to match the slope of H o have been con- sidered. One such method uses the following principle. If two permanent magnets with wide- ly different linear temperature coefficients (TC) are combined in opposing fashion, the TC of the resultant field can be made larger than the TC of either magnet. This is possible be- cause the magnets can be adjusted to exactly cancel at some arbitrary temperature. Then at other temperatures, the TC of the net field is determined by the temperature dependence of the difference in the magnitudes of the two component fields. Thus, a magnet system with a desired TC can be constructed by properly selecting the magnitudes of the opposing fields and the cancellation temperature. Satisfactory operation from 23 to 100°C of a high-g chip has been achieved using a magnet structure of this type. 24 However, a single permanent magnet material with the 94 desired temperature coefficient results in a less cumbersome and physically simpler device. Such a ferrite material has recently been developed at these laboratories by F. J. Schnettler, E. M. Gyorgy and R. D. Pierce and will be reported on separately. A device module using this new ferrite material and a high-g chip has be&n assembled and tested at i00 kHz, yielding highly satisfactory results. The temperature coefficient measured in the air gap of the ferrite bias magnet structure was -0.6%/°C at 50°C compared to -0.58%/°C for the bubble collapse field H o in Fig. 3, and like Ho, the slope was quite linear from 0 to 100°C. 80 o 60 u w ~ 4o -..I -- z ~ 2o I.- x w lo o 0 0 o o I I I 25 50 75 T(°C) 100 Fig. 5 External bias values for operation at i00 kHz of the module containing the high-g chip. The external margins are much larger than the chip margin of ~12 Oe, because of the shielding effect of the U-shaped metallic mag- netic yoke of the ferrite bias magnet structure. The almost flat margins show that the ferrite bias field at the chip is tracking closely the fall-off of the bubble collapse field. (The zero not being in the middle of the margins indicates only that the ferrite bias magnet was set a few Oe too low). Figure 5 shows that the module will operate over the temperature range with no external field. It also shows the measured values of external field over which the module will operate. The external field margins shown are much larger than the actual margins, ~12 Oe, for a complete circuit on this chip, because of the shielding effect from the U- shaped metallic magnetic yoke which supports the ferrite permanent magnet pieces to form the air gap for the chip. The approximately horizontal external field margins show that the ferrite bias field at the chip is tracking closely the fall-off of the bubble collapse field. The fact that the zero external field point is not in the middle of the range in- dicates only that the ferrite bias magnet was set a few Oe too low. This can be adjusted easily. These data are all taken for steady state operation. Transient operation of the device will require further study. CONCLUSIONS High g factors arising from Eu in the iron garnets have been shown to suppress dynamic conversion in bubble garnets. The hard bubble problem can be eliminated either by short oxygen anneals at ~I050°C or by ion implantation. A ferrite permanent magnet material is now available which matches the steeper than usual temperature dependence of the bubble collapse field. There a~e two principal limitations: For g ~ 20, the minimum bubble diameter is ~5 ~m. This corresponds to Ca = I.i, or MFe = 0, in the system (EuYCa)3(FeGeSi)5012. Bubble diameters of ~3.5 ~m have been achieved with g = 6, which yields a three times higher dynamic conversion frequency than usual bubble materials. This was done by increasing the Ca > i.i which increases M s by adding to the Eu moment some moment from the net iron lat- tice, since now the net iron moment is domin- ated by the octahedral sites instead of the tetrahedral sites as when Ca < i.i. The maximum mobility for a high-g garnet with a bubble size of ~5 ~m is ~1600 cm/secOe, this being determined primarily by the Eu damping. The amount of Eu cannot be greatly decreased, for although g can still he high, M s would decrease which would increase the bubble size. Because of the high dynamic conversion frequencies, however, this system offers the highest presently obtainable bubble device speeds for bubbles in the 3 to 6 ~m range. ACKNOWLEDGMENTS We wish to thank A. D. Butherus for the permanent magnet designs and W. Strauss for the measurements of the device margins vs. temperature. We also thank W. B. Venard, W. A. Biolsi and R. J. Peirce for very helpful technical assistance, and R. Wolfe, J. W. Nielsen and F. B. Hagedorn for their continu- ing useful discussions. REFERENCES i. E. A. Giess, B. A. Calhoun, E. Klokholm, T. R. McGuire and L. L. Rosier, Mat. Res. Bull. 6, 317 (1971). 2. W. A. Bonner, J. E. Geusic, D. H. Smith, F. C. Rossol, L. G. Van Uitert and G. P. Vella-Coleiro, J. Appl. Phys. 43, 3226 (1972). 3. J. W. Nielsen, S. L. Blank, D. H. Smith, G. P. Vella-Coleiro, F. B. Hagedorn, R. L. Barns and W. A. Biolsi, J. Electron. Mat'l. 3, 693 (1974). 4. G. P. Vella-Coleiro, F. B. Hagedorn, Y. S. Chen and S. L. Blank, Appl. Phys. Lett. 22, 324 (1973). 5. F. B. Hagedorn, J. Appl. Phys. 45, 3129 (1974). 6. J. C. Slonczewski, J. Appl. Phys. 44, 1759 (1973). 7. A. A. Thiele, Phys. Rev. B7, 391 (1973). 8. S. Geller, H. J. Williams, G. P. Espinosa and R. C. Sherwood, Bell Syst. Tech. J. 43, 565 (1964). 9. W. A. Bonner, J. E. Geusic, D. H. Smith, L. G. Van Uitert and G. P. Vella-Coleiro, Mat'l. Res. Bull. 8, 1223 (1973). i0. F. B. Hagedorn, S. L. Blank and R. J. Peirce, Appl. Phys. Lett. 26, 206 (1975). ii. R. Suzuki and Yutaka Sugita~, Appl. Phys. Lett. 26, 587 (1975). 12. A. P. Malozemoff and J. C. DeLuea, Appl. Phys. Lett. 26, 719 (1975). 95 13. R. C. LeCraw, J. P. Remeika and H. Matthews, J. Appl. Phys. 36, 901 (1965). 14. R. D. Pierce and R. C. LeCraw (unpublished). 15. R° C. LeCraw, S. L. Blank and G. P. Vella-Coleiro, Appl. Phys. Lett. 26, 402 (1975). 16. S. L. Blank, J. W. Nielsen and W. A. Biolsi, presented at the Annual Meeting of the Electrochemical Society, Dallas, Texas, October 1975 and submitted for publication. 17. R. C. LeCraw and R. D. Pierce, AlP Conf. Proc. 5, 200 (1972). 18. G. P. Vella-Coleiro, S. L. Blank and R. C. LeCraw, Appl. Phys. Lett. 26, 722 (1975). 19. G. P. Vella-Coleiro, paper this conference. 20. R. C. LeCraw, E. M. Gyorgy and R. Wolfe, Appl. Phys. Lett. 24, 573 (1974). 21. R. Wolfe, J. C. North and Y. P. Lai, Appl. Phys. Lett. 22, 683 (1973). 22. R. L. White, IEEE Trans, Mag. MAG-9, 606 (1973). 23. R. Wolfe and J. C. North, Appl. Phys. Lett. 25, 122 (1974). 24. A. D. Butherus and W. Strauss, private communication. STUDY OF DEFECTS IN REDUCED LPE BUBBLE GARNET FILMS R. C. LeCraw, E. M. Gyorgy and R. Wolfe Bell Laboratories, Murray Hill, New Jersey ABSTRACT Aluminum deposited on LPE bubble garnet films, which are then heated for 0.5 hour at 450°C, has been used to study reduction- associated defects in garnets. This treatment leaves the garnet darkened but magnetically unchanged. Subsequent heating at %600°C with the AI removed, causes a controllable reduc- tion in magnetization due to Ga-Fe redistribu- tion, accelerated by the defects introduced at 450°C. Defects introduced into the same garnet films by annealing in nitrogen without aluminum at 1250°C have distinctly different character- istics. Thus the existance of at least two different types of reduction-associated defects in garnets has been demonstrated. It has been shown that when Si is deposit- ed on Ga-containing LPE bubble garnet films which are then heated in the range 600 to 700°C, rapid changes in the magnetization, Ms, occur under the Si. This was attributed to oxygen vacancies created at the Si-garnet interface. The oxygen vacancies or some other kind of reduction-associated defects then diffuse through the film. The defects cause local distortions of the lattice which accelerate the interchange of Ga and Fe ions between octa- hedral and tetrahedral sites, thus changing M s . Highly localized control of M s was obtained in this way. We have found that for studying the nature of this effect, AI is much more useful than Si because defects can be produced readily at the Al-garnet interface at 400-450°C rather than 600-700°C. These defects diffuse through the film thickness in times of the order of hours at substantially lower temperatures (~425°C) than are required (~600°C) for the Ga-Fe re- distribution to occur. In a typical experiment, a 5000 ~ layer of A1 was deposited by e-beam evaporation on a 6 ~m thick film of Y2.6Sm0.4Fe3.SGal.2012. Three samples from this wafer were initially given the same heat treatment, i.e., 0.5 h at 450°C in nitrogen. The AI was then etched off of each sample. Sample 1 was then heated at 600°C in N2, which greatly decreased M s as shown by the large decrease in the room tem- perature bubble collapse field, Ho, in Fig. l(a). The sample was then put in 02 for 3 h at 1000°C, and H o returned very nearly to its original value. With Sample 2, Fig. l(b), after the initial AI treatment, 1.6 ~m was etched off by hot phosphoric acid. H o decreased by 90e due to the decrease in film thickness. The sample 07974 I00 50 50°, N2 1000°'02~ - O0 °, N 2 / (a) I I I I I --.I00 I¢I 0 Q ._I hl " 50 IJJ If) .J 0 0 0 f 450 , N 2 -L 1.6F m ETCH ~600 °, N z (b) I I I \| I oo! J 450 °, N 2 50 (c) 0 I I I I I 0 4 8 I; = 16 20 TIME ( HOURS ) Fig. i Bubble collapse field, Ho, vs. heating time for three samples from a 6 ~m thick wafer of composition Y2.6Sm0.4Fe3.8G~I.~OI2. Each of the samples was coated with 5000 N of A1 and heated for 0.5 h in nitrogen. Then the A1 was etched off each sample and the heat treatments shown in the figure were continued. The two vertical dark bars in (b) and (c) show the drops in collapse field upon etching the samples. The curves are best fits to the ex- perimental points.
1.4906843.pdf	Magnetic tunnel junctions using Co/Ni multilayer electrodes with perpendicular magnetic anisotropy Ia. Lytvynenko, C. Deranlot, S. Andrieu, and T. Hauet Citation: Journal of Applied Physics 117, 053906 (2015); doi: 10.1063/1.4906843 View online: http://dx.doi.org/10.1063/1.4906843 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/117/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Influence of magnetic electrodes thicknesses on the transport properties of magnetic tunnel junctions with perpendicular anisotropy Appl. Phys. Lett. 105, 052408 (2014); 10.1063/1.4892450 Co/Pt multilayer-based magnetic tunnel junctions with perpendicular magnetic anisotropy J. Appl. Phys. 111, 07C703 (2012); 10.1063/1.3670972 Effect of annealing on the magnetic tunnel junction with Co/Pt perpendicular anisotropy ferromagnetic multilayers J. Appl. Phys. 107, 09C711 (2010); 10.1063/1.3358249 Co ∕ Pt multilayer based magnetic tunnel junctions using perpendicular magnetic anisotropy J. Appl. Phys. 103, 07A917 (2008); 10.1063/1.2838754 Magnetoresistance in Co ∕ Pt based magnetic tunnel junctions with out-of-plane magnetization J. Appl. Phys. 103, 07A918 (2008); 10.1063/1.2838282 [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: 198.91.36.79 On: Wed, 04 Feb 2015 10:09:32Magnetic tunnel junctions using Co/Ni multilayer electrodes with perpendicular magnetic anisotropy Ia. Lytvynenko,1C. Deranlot,2S. Andrieu,3and T. Hauet3 1Sumy State University, 40007 Sumy, Ukraine 2Unit/C19e Mixte de Physique CNRS/Thales, associ /C19ee/C18a l’Universit /C19e Paris-Sud, 91767 Palaiseau 3Institut Jean Lamour, UMR CNRS 7198, Nancy-Universit /C19e, 54506 Vandoeuvre le `s Nancy, France (Received 2 December 2014; accepted 16 January 2015; published online 3 February 2015) Magnetic and magneto-transport properties of amorphous Al 2O3-based magnetic tunnel junctions (MTJ) having two Co/Ni multilayer electrodes exhibiting perpendicular magnetic anisotropy (PMA) are presented. An additional Co/Pt multilayer is required to maintain PMA in the top Co/Ni electrode. Slight stacking variations lead to dramatic magnetic changes due to dipolar interactionsbetween the top and bottom electrodes. Tunnel magneto-resistance (TMR) of up to 8% at 300 K is measured for the MTJ with two PMA electrodes. The TMR value increases when the top PMA electrode is replaced by an in-plane magnetized Co layer. These observations can be attributed tosigniﬁcant intermixing in the top Co/Ni electrode. VC2015 AIP Publishing LLC . [http://dx.doi.org/10.1063/1.4906843 ] I. INTRODUCTION Magnetic tunnel junctions (MTJs) having electrodes with perpendicular magnetic anisot ropy (PMA) have attracted con- siderable interest because they are promising candidates for spin transfer torque magnetic random access memories (STT- MRAM).1–3Finding PMA materials that simultaneously show large tunnel magneto-resistance (TMR), low damping, low switching current density, and high thermal stability remains a challenge in implementing S TT-MRAM. One of the most im- portant systems investigated to date is the CoFeB/MgO/CoFeBstack where perpendicular ani sotropy is created at the CoFeB/ MgO and MgO/CoFeB interfaces. 1,4Appropriate choices of buffer and capping layers can enhance the interface PMA.5,6 The main limitation of using CoFeB is the thinness of the elec- trodes ( <1.6 nm) which is necessary for PMA, as at this thick- ness thermal stability can be an issue.7Rare-earth/transition metal ferrimagnet alloys,8[Fe1-xCox/Pt] multilayers (MLs),9,10 and L1 0(Fe,Co)Pt alloys11have also been tested since they have large PMA. However, both these systems have largedamping and low spin polarization. 12Spin polarization can be improved by inserting CoFeB at the MgO barrier interface.13 Co/Ni MLs have attracted considerable attention for spin-transfer applications since they meet the requirementspreviously listed. In terms of anisotropy, the Co/Ni interface produces PMA as large as few MJ/m 3.14Changing the thick- ness of Co allows the magnitude of the PMA to be easily tuned. The saturation magnetization of Co/Ni MLs is approx- imately 700 kA/m, although the exact value will depend on the individual Co and Ni layer thicknesses. Gilbert damping of Co/Ni MLs mostly ranges between 0.01 and 0.02, depend-ing on the composition. 15,16Finally, high spin polarization has been deduced as a result of spin transfer induced domain wall motion experiments.17The importance of such a set of characteristics for achieving low critical current and sub- nanosecond switching time has been already demonstrated in metallic Co/Ni-based spin-valves nanopillars.3,18However, no magneto-resistance or spin-transfer torque experimentshave yet been reported for MTJs using PMA Co/Ni electro- des and a tunnel barrier. The difﬁculties of growing a bccMgO (100) barrier on top of fcc Co/Ni (111) stack, as wellas Co/Ni on a MgO barrier is the limiting factor. 19Recently, You et al.19succeeded in growing MgO-based MTJs using two PMA Co/Ni electrodes but only magnetometry measure-ments were provided. In this letter, we report an investigation of magnetic and magnetotransport properties of two amorphous Al 2O3-based magnetic tunnel junctions having one or two fcc (111) Co/Ni PMA electrodes. Magnetometry measurement reveals thatsubtle variations of magnetization or anisotropy in the topelectrode can strongly affect its magnetic reversal propertiesdue to dipolar coupling between electrodes. Magneto-transport measurements demonstrate up to 8% TMR at RTfor a MTJ with two Co/Ni PMA electrodes. Here, TMR isdeﬁned as the normalized difference between parallel andanti-parallel alignment of the two electrodes magnetization.The TMR increases to 16% at 20 K. Replacing the top PMAsoft electrode by an in-plane magnetized Co (15 nm) layerincreases the TMR by a factor of two at room temperature.We discuss our results in terms of the structural features ofthe electrodes. II. EXPERIMENTAL METHODS The samples were prepared on thermally oxidized silicon substrates, where the oxide layer thickness was 400 nm, usingmagnetron sputtering with a base pressure of 5 /C210 /C08mbar. The deposition was performed at room temperature. Co/Niand Co/Pt MLs, as well as the Ta and Pt layers, were grownby dc-magnetron sputtering. Three MTJ samples were pro-duced which all have (i) the same bottom PMA electrodeTa(5)/Pt(10)/Co(0.6)/[Ni(0.6)/Co(0.3)]3 (thicknesses innm) and (ii) the same Al2O3 (2.5 nm) barrier obtainedthrough the deposition of 1.5 nm Al layer following by anoxidation in a Ar þO 2plasma. The three samples then had different top electrodes deposited; sample A had a PMA top 0021-8979/2015/117(5)/053906/4/$30.00 VC2015 AIP Publishing LLC 117, 053906-1JOURNAL OF APPLIED PHYSICS 117, 053906 (2015) [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: 198.91.36.79 On: Wed, 04 Feb 2015 10:09:32electrode consisting of [Co(0.2)/Ni(0.6]3/Pt(1)/[Co(0.6)/ Pt(1)]3, Sample B had a PMA top electrode consisting of[Co(0.3)/Ni(0.6]3/Pt(2)/[Co(0.6)/Pt(1)]3 and Sample C topelectrode consists in an in-plane magnetized Co(15) singlelayer (again all thicknesses in nm). UV lithography was usedto pattern samples B and C into MTJ devices with junctionssize from 10 /C210lm 2up to 50 /C250lm2having 1 G X.lm2 RA product. Magnetic characterization was performed at 300 K using an Alternative Gradient Field Magnetometer(AGFM). The transport properties were measured using aPhysical Properties Measurement System (PPMS) cryostatover a temperature range from 20 to 300 K. III. RESULT AND DISCUSSION Fig.1shows normalized magnetiz ation curve measured on samples A and B, using an AGFM. In the case of sample B, asthe ﬁeld is applied perpendicularly to the layers, we observeloops with full remanent magne tization and two successive jumps at reverse ﬁelds of 130 Oe and 270 Oe, respectively. Theﬁrst magnetization jump has a l arger magnitude than the second one. This indicates that the top electrode with the largest totalmoment [Co(0.2)/Ni(0.6]3/Pt(2 )/[Co(0.6)/Pt(1)]3, is softer than the bottom Pt(10)/Co(0.6)/[Ni(0.6/Co(0.3)]3. This result is counter-intuitive since Co/Pt ML is expected to have a much larger PMA than Co/Ni ML. 20However, the well established layer by layer growth of the bottom Co/NiML on a smooth (111) textured Pt buffer 21has to be compared with the island-like growth process of the top ML on a Al 2O3 oxide barrier.22Moreover, the top ML may not be well (111) textured on the amorphous barrier, and it has been found that(100) and (110) grain signiﬁcantly reduce PMA in Co/NiML. 23The tail of the ﬁrst magneti zation jump is typical of the dipolar interactions (so-called demagnetization ﬁeld) in PMAﬁlm thicker than few nanometers 24but that a fully anti-parallel state is reached before the second step. Sample A shows a dif-ferent behavior, a slight decrea se of the Pt interlayer thickness in the top layer as compared with sample B leads to a drasticchange in the normalized magnetization versus ﬁeld loop withthe disappearance of the an ti-parallel plateau (Fig. 1). The fact that top and bottom layers rever se together is due to dipolarinteractions which effectiv ely couples the two layers. 25,26As a consequence, one has to carefully tune the electrodes not only to insure PMA but also limit the inter-layer dipolar coupling. Magneto-resistance measur ements performed on pat- terned sample B are shown in Fig. 2. A signiﬁcant TMR was measured in the MTJ with two Co/Ni PMA electrodes.TMR values of 8% at 300 K and 16% at 20 K were meas- ured for a 50 mV bias voltage . These values are smaller than the best reported TMRs (about 80%) for CoFeB/ Al 2O3-based MTJ.27Nevertheless, it is of the same order of magnitude as the previously reported for Al 2O3-based MTJs with PMA electrodes.9,10Fig. 2(b) shows that the temperature dependence of TMR ﬁts well with the (1- aT3/2) dependence usually reported and linked to the spin- polarization decrease and increase of the inelastic processes as the temperature increases.28 The Brinkman model9,29that describes the bias-voltage dependence of tunnel current can be employed to provide in-formation on the barrier features. Room temperature I(V) curve measured in parallel state is presented in Fig. 3(a)and compared with a ﬁt to the Brinkman model using a 2.5 nm barrier width. A good match is obtained with a 1.18 eV zero bias barrier height and no barrier asymmetry. This barrierheight value conﬁrms the average quality of our Al 2O3layer. As barrier values of up to 3 eV can be achieved there is con- siderable scope for much larger TMR if we further improveour Al 2O3barrier layer.30Interestingly, no barrier asymme- try is needed in the Brinkman ﬁt. This indicates that the bot- tom and top interfaces are similar. The same conclusion can be drawn from the voltage dependence of TMR (Fig. 3(b)). FIG. 1. Room temperature normalized magnetization vs ﬁeld measurements of Co(0.6)/[Ni(0.6/Co(0.3)]3/AlOx(2.5)/[Co(0.2)/Ni(0.6]3/ Pt(y)/[Co(0.6)/ Pt(1)]3 MTJ under out-of-plane applied magnetic ﬁeld with y ¼1( r e do p e n circle, sample A) and 2 nm (black solid square, sample B). FIG. 2. (a) MR vs out-of-plane ﬁeld for sample B with both PMA hard [Co/Ni] and soft [Co/Ni][Co/Pt] electrodes measured under 50 mV bias voltage, at the 300 K (black line) and 20 K (red line). (b) For the same sample, TMR (black points) and resistance (blue triangle) vs temperature. The red line cor- responds to the theoretical (1- aT 3/2) dependence of the TMR. The blue line is a guide for the eye.053906-2 Lytvynenko et al. J. Appl. Phys. 117, 053906 (2015) [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: 198.91.36.79 On: Wed, 04 Feb 2015 10:09:32The decrease of TMR with increasing bias voltage mostly due to inelastic scattering by magnons, excitations and the shape of the electronic density of states,28is symmetric. It might have been anticipated that the TMR(V) curves would be asymmetric since the bottom and top electrode stacks are different, leading to a different density of states and thusspin polarization at the Fermi energy. The lack of TMR(V) asymmetry observed here suggests that diffusive processes strongly affect the magneto-transport properties. Such diffu- sive processes are well-known to be enhanced by interface roughness and structural defects in the layers. Consequently,increasing the crystalline quality of the layers should lead to larger TMR. Fig. 4shows magneto-resistance measurements per- formed on sample C which differs from sample B as the top Co electrode has in-plane magnetized allowing an orthogonal conﬁguration between the two electrodes at remanence.Interest in such an orthogonal magnetic geometry has grown in the past years because of its possible use in sensors, 31,32 OST-MRAM,33and RF oscillators.34,35In sample C, when the external magnetic ﬁeld is large enough, the magnetizationsof both electrodes are aligned along the ﬁeld (applied either in-plane or out-of-plane). At zero applied ﬁeld, the Co/NiML moment is perpendicular, whereas Co moment lies in- plane. Hysteresis occurs for the case when the applied ﬁeld is out-of-plane as the bottom Co/Ni ML magnetizationreverses. The coercive ﬁeld is 270 Oe, identical to the value obtained from magnetometry measurements shown in Figs. 1and2for sample B. In the case of an in-plane applied ﬁeld, no hysteresis is observed for the Co layer. Note thatthe in-plane ﬁeld curve provides a measure of the anisot- ropy ﬁeld of the bottom Co/Ni ML which is approximately 12 kOe, in agreement with previous measurements. 14,36We note that the difference in resistance between the saturated state and the remanent state is about 8% at 300 K for this orthogonal magnetic conﬁguration. This corresponds to aTMR (i.e., the normalized difference between parallel andanti-parallel alignment of the two electrodes magnetization) of 16% between the parallel and a hypothetical anti-parallel state. Hence, at room temperature sample C would in prin-cipal have two times higher TMR than sample B. This result is different to expectations since in recent spin- resolved photo-emission sp ectroscopy experiments, we observed that spin-polarization at the Fermi level for epi- taxial [Co(x)/Ni(0.6)] ML with 0.1 nm <x<0.6 nm, is larger than for pure Co. 37Since the bottom electrode and barrier quality are expected to be the same for both sam-ples, it most probably indicates intermixing in the top Co/ Ni ML that leads to lower than expected polarization and PMA, compared to a well layered stack. Indeed, the spin-polarization for a CoNi alloy is expected to be lower than for pure Co. 38 IV. CONCLUSION In summary, our work demonstrates the potential of Al2O3-based magnetic tunnel junction (MTJ) with one or two perpendicular anisotropy (PMA) Co/Ni electrodes forfuture spin electronics device (MRAM, sensors, RF oscilla-tors). Due to the island growth of Co/Ni on the Al 2O3bar- rier, the top Co/Ni electrode has to be covered by a Co/Pt stack in order to maintain the PMA in the top electrode. At300 K, 8% TMR at 300 K was measured in the full PMA Pt/ Co[Ni/Co] 3/Al2O3/[Co/Ni] 3/Pt/[Co/Pt] 3. Study of the tunnel barrier characteristics showed that our Al 2O3layer crystal- line quality can be improved. Moreover, comparison withorthogonal anisotropy Pt/Co[Ni/Co] 3/Al2O3/Co MTJ indi- cated that intermixing must exist in the top Co/Ni electrode of the full PMA MTJ which lowers its polarizationand PMA. Overall, this provides encouragement that it will be possible to achieve larger PMA and TMR values for Co/Ni-based magnetic t unnel junction with both PMA electrodes. ACKNOWLEDGMENTS The authors thank G. Lengaigne for patterning the magnetic tunnel junctions, S. Suire for help with transportexperiments, and T. Thomson for improving the manuscript. This work was partially funded by the Region Lorraine and French embassy in Ukraine. FIG. 3. (a) Experimental current vs bias voltage (black points) for sample B at 300 K compared with Brinkman ﬁt (dashed white line). (b) TMR versus bias voltage measured on sample B at 300 K. FIG. 4. (a) MR of sample C, i.e., Pt/Co(0.6)/[Ni(0.6)/Co(0.3)]3/AlOx(2.5)/ Co(15) MTJ measured applying the magnetic ﬁeld in-plane (black line) and out-of-plane (red line) under 50 mV bias voltage at 300 K. (b) Corresponds to a zoom around zero ﬁeld and highlights the reversal of PMA Co/Ni bottom electrode magnetization at /C0270 Oe.053906-3 Lytvynenko et al. J. Appl. Phys. 117, 053906 (2015) [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: 198.91.36.79 On: Wed, 04 Feb 2015 10:09:321S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan, M. Endo, S. Kanai, J. Hayakawa, F. Matsukura, and H. 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1.4757906.pdf	Tuning the direction of exchange bias in ferromagnetic/antiferromagnetic bilayer by angular-dependent spin-polarized current XiaoLi Tang, Hua Su, Huai-Wu Zhang, Yu-Lan Jing, and Zhi-Yong Zhong Citation: Journal of Applied Physics 112, 073916 (2012); doi: 10.1063/1.4757906 View online: http://dx.doi.org/10.1063/1.4757906 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/112/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Instability of exchange bias induced by an overlaid superconductor tab in antiferromagnet\ferromagnet bilayers Appl. Phys. Lett. 100, 242602 (2012); 10.1063/1.4729031 Spin torque-driven switching of exchange bias in a spin valve J. Appl. Phys. 106, 073906 (2009); 10.1063/1.3236572 Uncompensated antiferromagnetic spins at the interface in Mn–Ir based exchange biased bilayers J. Appl. Phys. 101, 09E510 (2007); 10.1063/1.2710216 Universal time relaxation behavior of the exchange bias in ferromagnetic/antiferromagnetic bilayers J. Appl. Phys. 99, 033910 (2006); 10.1063/1.2169876 Micromagnetic calculations of bias field and coercivity of compensated ferromagnetic antiferromagnetic bilayers J. Appl. Phys. 93, 8618 (2003); 10.1063/1.1557859 [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: 131.181.251.130 On: Sun, 23 Nov 2014 18:01:47Tuning the direction of exchange bias in ferromagnetic/antiferromagnetic bilayer by angular-dependent spin-polarized current XiaoLi Tang, Hua Su,a)Huai-Wu Zhang, Yu-Lan Jing, and Zhi-Y ong Zhong State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, China (Received 15 May 2012; accepted 29 August 2012; published online 8 October 2012) The angular dependence of an external magnetic ﬁeld applied with an in-plane alternating pulse that induces a new direction of exchange bias (EB) is observed in an NiFe/IrMn bilayer.Depending strongly on the direction of the external ﬁeld, EB ﬁeld gradually orients along the external ﬁeld with the increase in current pulse, and the new direction of EB is ﬁnally established. Furthermore, the new direction of EB can also be induced along the external ﬁeld at once when theapplied pulse is larger than the critical current. Because the strength and direction of the EB is highly correlated with the micromagnetic state distribution of the antiferromagnet, the observations are explained by the spin torque exerted on the antiferromagnetic moments and provide evidencefor the prediction of spin transfer and current-induced switching in antiferromagnets. VC2012 American Institute of Physics .[http://dx.doi.org/10.1063/1.4757906 ] I. INTRODUCTION Slonczewski and Berger proposed in 1996 that the orien- tation of a magnetic layer can be switched by a spin-polarized current.1,2Recently, the concept of spin torque (ST) affecting the magnetic state of ferromagnetic (FM) metals has been fur- ther extended to antiferromagnetic (AFM) metals.3–5Then, a series of experiments were carried out in exchange-biasedspin valves (EB-SV), and they all produced indirect evidences for ST effects in antiferromagnets. 6–9Because the EB-SV has two FM layers,10it must be distinguished whether the spin torque is transferred to an AFM layer and not between two FM layers, as has been discussed in previous studies.3,7,8 Therefore, in our present work, we used an FM/AFM bilayer, which had only one FM and AFM layer. In this way, the effect of ST on the AFM material could be observed more distinctly. In addition, in previous studies, only the directionsof the spin-polarized electrons parallel or antiparallel to the initial EB direction were considered in researching the effect of ST on AFM material. 6–9However, according to the physi- cal mechanism of ST, the orientation of spin-polarized electrons is important in its effect on the local moments.10 Therefore, for the present paper, our chief aim was to study the effect of different orientations of spin-polarized current acting on the FM/AFM bilayer. II. EXPERIMENTAL PROCEDURE The basic structure of the EB bilayer used in this study was NiFe (15 nm)/IrMn (15 nm) fabricated on a 10 /C210 mm2 Si substrate. A constant magnetic ﬁeld of /C24300 Oe along the substrate surface was applied during ﬁlm growth to develop the EB. The effects of angular-dependent spin-polarized cur-rent on EB were studied via the magnetization by hysteresisloops measured using a BHV-525 vibrating sample magne- tometer (VSM). The experiments were carried out in the following way. First, the sample was mounted on the rotatable holder in an external magnetic ﬁeld. The initial direction of the EB ﬁeld lay along the ﬁeld applied during deposition. By physicallyrotating the sample, the external magnetic ﬁeld H pwas applied in the ﬁlm plane at an angle hwith respect to the ini- tial direction of the EB ﬁeld, as shown in Figure 1. Then a 100 ms pulse of current Ipwas applied through two probes. A 1.5 kOe external ﬁeld Hpwas applied at an angle h. The large Hpwas used to suppress the current-induced magnetic ﬁeld and to keep the magnetic moments of the pinned NiFe layer along the Hp. In this way, the spin orientations of electrons ﬂowing in the NiFe layer were polarized at an angle hwith respect to the initial direction of the EB. In the end, after the pulse was applied with Hp, the hysteresis loops were meas- ured at various hangles. For convenience of measurement, the experiments were performed in different samples, which were fabricated at the same time. III. RESULTS AND DISCUSSION In investigating the angular-dependent current on the EB bilayer, the angle hforHpwas ﬁrst set at 45/C14. The pro- cess was repeated for pulses of 100, 200, 250, 300, and 350 mA. After every pulse was applied, the hysteresis loops weremeasured in the initial direction of EB ( h¼0 /C14) and the direc- tion of Hp(h¼45/C14), as shown in Figure 2. The results showed that the EB ﬁeld Hexand the square- ness ratios of the magnetization curves decreased with an increase in the pulse at h¼0/C14. For the magnetization curves measured at 45/C14, their squareness ratios increased with the increase in the pulse. When the pulse was increased to 350 mA, the sample achieved the largest squareness and its Hex reached 37 Oe. According to the research of the angular de- pendence of EB, along the direction of the EB, the sample had the best squareness ratio. Furthermore, the smallesta)Author to whom correspondence should be addressed. Electronic mail: uestcsh@163.com. 0021-8979/2012/112(7)/073916/5/$30.00 VC2012 American Institute of Physics 112, 073916-1JOURNAL OF APPLIED PHYSICS 112, 073916 (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: 131.181.251.130 On: Sun, 23 Nov 2014 18:01:47Hex¼0 occurred perpendicular to the direction of the EB ﬁeld.11,12In order to conﬁrm that the direction of EB was laid in 45/C14, we also measured the magnetization hysteresis loop along h¼135/C14after the pulse was applied. Indeed the Hex along 135/C14was nearly zero after the pulse 350 mA was applied. Therefore, we conﬁrmed that the direction of the EBﬁeld deviated from its initial direction and that a direction of EB at 45 /C14was achieved. Furthermore, after the tuning, the magnetic ﬁeld along the 45/C14was cycled 20 times. The resulting magnetization curves are displayed in Figure 3. We observe no obvious changes in run-to-run measurements. This indicates that thereorienting of the EB was in a stable state. The angle hforH pwas also set at 90/C14and 135/C14. The process was the same as h¼45/C14. As shown in Figure 4(a),i n the initial state, the direction of the EB ﬁeld was along 0/C14. Therefore, the large EB occurred at h¼0, whereas the small- estHex¼0 Oe occurred at h¼90/C14. After the pulses were applied with Hpat 90/C14, the EB ﬁeld tended to zero at h¼0/C14, and the largest EB ﬁeld was achieved at h¼90/C14. This means that the EB ﬁeld changed from its initial direction h¼0/C14to90/C14. For the condition of the pulse set with Hpat 135/C14, the EB ﬁeld at h¼0 was positive, and it was negative at h¼135/C14in the initial state. This is in accordance with obser- vations of the angular dependence of EB.11After the pulses were applied with Hpat 135/C14, the EB ﬁeld along h¼0 and h¼135 had all changed their direction. Based on the obser- vations in Figure 4(b), the direction of the EB was ﬁnally achieved along Hp(h¼135/C14). From the descriptions above, an interesting feature could be observed: the direction of EB could be re-oriented along the external ﬁeld Hpwhen a pulse was applied. This means that, using only the pulse of current and the external ﬁeld, the direction of the EB ﬁeld could be tuned as required. In changing or achieving EB, the process of annealing the sample in a magnetic ﬁeld is always adopted.13It is easy to assume that Joule heating in our samples was generatedby the pulse. However, based on our previous work, we revealed that Joule heating plays a minor role in current pulse experiments. 14,15Furthermore, we measured the resist- ance of the sample with different current. The results are listed in Table I. We observed that the resistance of the exchange-biased sample is not very large; it is only 10 X.I n addition, the resistance is independent of the current. The minor difference may be a testing error. Therefore, we con- sidered that Joule heating weak in our experiments. It maynot be the real reason for our observations. As mentioned earlier, a natural explanation of our data is the idea of spin transfer and current-induced switching in the AFM layer. During the pulse application process, the external mag- netic ﬁeld H pwas applied simultaneously at an angle h.T h e spin orientations of electrons ﬂowing from the NiFe layer tothe IrMn layer were polarized at an angle hwith respect to the initial direction of the EB ﬁeld. According to the interfacial uncompensated spin model, 16,17the net uncompensated spins at the FM/AFM interface induce an energy barrier for spin re- versal of the FM layer, which produces EB. Because the moments of electrons ﬂowing from the FM layer to the AFMlayer were oriented at an angle hto the interfacial net spins, they could induce torques on the uncompensated moments and force the uncompensated spins to take the orientation ofthe spin-polarization of the conduction electrons, as displayed in Figure 5. Because the orientation of the FM/AFM FIG. 1. Schematic illustration of the pulse of current Ipand external mag- netic ﬁeld Hpapplied in the current induced experiments. FIG. 2. Typical variation in magnetization hysteresis loops measured at (a) h¼0 and (b) h¼45/C14after the pulses were applied. FIG. 3. Hysteresis loops of NiFe/IrMn after 45/C14tuning. Virgin (cycle 1) and trained (cycles 5, 10, and 20) magnetization curves are shown.073916-2 Tang et al. J. Appl. Phys. 112, 073916 (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: 131.181.251.130 On: Sun, 23 Nov 2014 18:01:47interfacial uncompensated spin changed, the direction of the EB was re-oriented. In addition, according to Eq. (1), the threshold current Ic for ﬂipping the magnetic moment due to ST is proportional to the angle ubetween the polarized electron and the mag- netic moment m.18 Ic¼2eamVðHkþHÞ=g/C22hjcosuj; (1) where eis the electron charge; mis the magnetic moment; V is the volume of the magnet; ais the Gilbert damping param- eter; Hkis the anisotropic ﬁeld; His the applied ﬁeld; and g is the spin-polarization factor. If we deﬁne uc1as the angle that can be switched by Ic1, any angle larger than uc1cannot be switched by Ic1. Because the net uncompensated spins in the interface have different angles, a pulse of lower magnitude than Ic1can only induce the switching of some of the uncompensated spins. There- fore, as observed in Figures 2and4, a pulse of lower magni- tude cannot completely orient the direction of EB along theexternal ﬁeld. With an increase of the pulse to a larger value, most of the uncompensated spins oriented along the external ﬁeld, and the EB was re-oriented. For further studies, we also applied one pulse as large as 400 mA along 45 /C14,9 0/C14, and 135/C14to tune the direction of the EB. Their hysteresis loops are displayed in Figure 6.I ti sobvious that the direction of the EB has been induced along the external ﬁeld at once after the 400 mA pulse was applied. This result also provided evidence that the observa-tions in Figures 2and4were not a cumulative effect. The direction of EB could be achieved along the external ﬁeld when the pulse was larger than the critical current I c,w h i c h was large enough to switch most of the interfacial uncom- pensated spins. In addition, we also decreased the external ﬁeld to 250 Oe, which is only a little larger than the satu-rated ﬁeld, to tune the EB along 90 /C14. The magnetization curve measured after the tuning is almost the same as the 1.5 kOe external ﬁeld applied, shown in Figure 7. It indi- cates that the external ﬁeld (given that it is sufﬁcient to align the FM moments along the tuning direction) can produce the same result. FIG. 4. Comparison of magnetization hysteresis loops measured in the initial direction of exchange bias and the angle for setting Hpafter every pulse was applied. (a) Hpset at 90/C14; (b) Hpset at 135/C14. TABLE I. The resistance for NiFe(15 nm)/IrMn(15 nm) measured with dif- ferent current. Testing current 1 mA 100 mA 200 mA 250 mA 300 mA 350 mA Resistance ( X) 10.12 10.02 10.05 10.15 10.07 10.10 FIG. 5. Schematic illustration of the angular dependence of spin-polarized current induced torque on interfacial uncompensated spins (m i: the initial net interfacial uncompensated spins; m t: the new interfacial uncompensated spins induced by the spin transfer).073916-3 Tang et al. J. Appl. Phys. 112, 073916 (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: 131.181.251.130 On: Sun, 23 Nov 2014 18:01:47Using pulses of current with an external magnetic ﬁeld to tune the direction of the EB gave us a convenient way totune EB after deposition. To date, many other post- deposition strategies have been used for EB tuning. 19–25 However, many of these approaches have disadvantages in material, magnitude of applied ﬁeld, and so on. In the present study, only a moderate magnetic ﬁeld and pulse were required to tune Hexafter deposition. Furthermore, the tuning has been realized in the materials (NiFe and IrMn) that are typically used in spintronic devices. This strategy should be useful for the research and design of devices based on ST inAFM materials.IV. CONCLUSIONS In summary, it was observed that the direction of EB could be tuned by pulses of current with an external ﬁeld.The tunable direction of EB was correlated with the direction of the external ﬁeld. The observations provided evidence to support the prediction of ST and current induced switchingin AFM material, and the technique should prove very useful for the fabrication and design of spintronic devices. It also provides a way to account in detail for the EB and the do- main structure at an AFM/FM interface. ACKNOWLEDGMENTS This work was supported by the Innovative Research Groups of the NSFC (Grant No. 61021061), NSFC (Grant No. 51171038), and Fundamental Research Funds for the Central Universities. 1J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). 2L. Berger, Phys. Rev. B 54, 9353 (1996). 3A. S. N /C19u~nez, R. A. Duine, P. Haney, and A. H. MacDonald, Phys. Rev. B 73, 214426 (2006). 4H. V. Gomonay and V. M. Loktev, Phys. Rev. B 81, 144427 (2010). 5E. V. Gomona K%and V. M. Loktev, Low Temp. Phys. 34, 198 (2008). 6X. L. Tang, H. W. Zhang, H. Su, Z. Y. Zhong, and Y. L. Jing, Appl. Phys. Lett. 91, 122504 (2007). 7S. Urazhdin and N. Anthony, Phys. Rev. Lett. 99, 046602 (2007). 8Z. Wei, A. Sharma, A. S. Nunez, P. M. Haney, R. A. Duine, J. Bass, A. H. Macdonald, and M. Tsoi, Phys. Rev. Lett. 98, 116603 (2007).FIG. 6. Magnetization hysteresis loops measured before and after a single pulse of 400 mA was applied with Hpat angle (a)h¼45/C14, (b) h¼90/C14, and (c)h¼135/C14. FIG. 7. 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1.3248220.pdf	Detection of the static and kinetic pinning of domain walls in ferromagnetic nanowires Sung-Min Ahn, Kyoung-Woong Moon, Dong-Hyun Kim, and Sug-Bong Choe Citation: Applied Physics Letters 95, 152506 (2009); doi: 10.1063/1.3248220 View online: http://dx.doi.org/10.1063/1.3248220 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/95/15?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Thermally activated stochastic domain-wall depinning in ferromagnetic nanowires J. Appl. Phys. 114, 103901 (2013); 10.1063/1.4819808 Geometric dependence of static and kinetic pinning of domain walls on ferromagnetic nanowires J. Appl. Phys. 111, 07D309 (2012); 10.1063/1.3677872 Magnetic imaging of the pinning mechanism of asymmetric transverse domain walls in ferromagnetic nanowires Appl. Phys. Lett. 97, 233102 (2010); 10.1063/1.3523351 High efficiency domain wall gate in ferromagnetic nanowires Appl. Phys. Lett. 93, 163108 (2008); 10.1063/1.3005586 Domain wall pinning and potential landscapes created by constrictions and protrusions in ferromagnetic nanowires J. Appl. Phys. 103, 114307 (2008); 10.1063/1.2936981 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: 140.232.1.111 On: Fri, 19 Dec 2014 12:34:05Detection of the static and kinetic pinning of domain walls in ferromagnetic nanowires Sung-Min Ahn,1Kyoung-Woong Moon,1Dong-Hyun Kim,2and Sug-Bong Choe1,a/H20850 1Department of Physics, Seoul National University, Seoul 151-742, Republic of Korea 2Department of Physics, Chungbuk National University, Cheongju 361-763, Republic of Korea /H20849Received 24 July 2009; accepted 24 September 2009; published online 13 October 2009 /H20850 Two distinct pinning mechanisms named as kinetic and static pinning of magnetic domain wall /H20849DW /H20850are experimentally resolved. Both the pinning situations are realized at an artiﬁcial notch on U-shaped Permalloy nanowires, depending on the initial DW states, moving or pinned. The kineticdepinning ﬁeld—a critical ﬁeld for a moving DW to be trapped at a notch—is revealed to bedistinguishably smaller than the static depinning ﬁeld—a critical ﬁeld to depin a trapped DW at thenotch. Based on one-dimensional collective model, the discrepancy is explained by the tilting angleof the moving DW. © 2009 American Institute of Physics ./H20851doi:10.1063/1.3248220 /H20852 Magnetic domain wall /H20849DW /H20850in nanowires has been fo- cused due to the promising applications such as magneticlogic and memory devices. 1,2Since the DW carries the logic and/or memory information, it is essential to precisely ma-nipulate the DW positions, practically by introducing artiﬁ-cial constraints such as notches. 3–7The DW shift between notches is accomplished by two successive processes: /H20849i/H20850de- pinning of a trapped DW from a notch and /H20849ii/H20850pinning of a moving DW at another notch. Thus, one has to distinguishtwo pinning mechanisms depending on the initial states ofDWs, either trapped or moving. We denote the former andthe latter as the static and kinetic pinning processes, respec-tively. All the previous studies have examined only the staticpinning process but the kinetic pinning process has not beenexperimentally demonstrated yet, despite a micromagneticprediction. 8In this letter, we present an experimental proof that the kinetic pinning process is distinct from the staticpinning process, by exhibiting the noticeably differentstrengths of the depinning ﬁelds. For this study, 20-nm-thick Ni 81Fe19ﬁlms are deposited onto Si /H20849100 /H20850substrates by dc-magnetron sputtering under 2 mTorr Ar pressure. U-shaped nanowire structures are thenpatterned by the electron beam lithography followed by re-active ion etching. Several structures are realized with differ-ent widths—350, 620, and 1170 nm, respectively. The sec-ondary electron microscopy image of the 620-nm-widenanowire structure is depicted in Fig. 1. A notch is placed in the middle of the structure as designated by the arrows in theﬁgure. The notch is composed of two symmetric triangles,which exhibits a unique static depinning ﬁeld 9irrespective of the DW chirality and propagation direction,10unlikely to the single notches exhibiting complex pinning mechanisms.11 The notch depths are 90, 170, and 350 nm, respectively, foreach nanowires, which are roughly 30% of the nanowirewidths. The DW propagation along the nanowires is then mea- sured by a longitudinal Kerr effect measurement system witha laser spot of /H11011500 nm in diameter by use of a 405 nm laser and an objective lens of the numerical aperture 0.9. Thelaser spot is placed at the left side of the notch as shown bythe circle in Fig. 1. The measurement scheme is as follows./H208491/H20850An external magnetic ﬁeld H sat/H20849/H11011400 Oe /H20850is ﬁrst ap- plied to the structure with an angle /H9258/H20849/H1101160° /H20850and thus, a DW is created at the left corner after turning-off the magnetic ﬁeld. /H208492/H20850The magnetic ﬁeld is then applied rightward up to Hsweep in the horizontal direction, to bring the DW from the left corner to the right. /H208493/H20850Finally the magnetic ﬁeld is swept leftward to bring the DW back to the left corner. Depending on the strength of Hsweep, the DW is brought to the different positions as pointed by A,B,C, and D, respectively in Fig. 1, which in turn generates four different hysteresis loops. We denote three depinning ﬁelds as the de- pinning ﬁeld Hleft→from the position A, the depinning ﬁeld Hnotchl→from the position B, and the depinning ﬁeld Hnotchr→ from the position C. The values of the depinning ﬁelds are listed in Table I. In Regime I with Hsweep/H11021Hleft→, no change in the Kerr signal is observed as shown in Fig. 2/H20849a/H20850, since the DW is kept pinned at the natural edge roughness of the left corner. Note that Hleft→is set to be a small value by adjusting /H9258and Hsat. In Regime II, the DW is depinned from the left corner and then, pinned at the notch. There are two possible pinningpositions, either the left or the right sides of the notch, asdesignated by the positions BandC, respectively. We thus classify Regime II into two subregimes. For Regime IIa with H left→/H11021Hsweep/H11021Hnotchl→, the DW is pinned at B. By reversing the sweeping ﬁeld, the DW is depinned leftward under the a/H20850Electronic mail: sugbong@snu.ac.kr. Hsweep Hθ 1 Hsat1μm 5μmμmABCBCD FIG. 1. Secondary electron microscope image of 620-nm-wide Permalloy U-shaped nanowire structure with a notch. The circle on the wire shows theposition of the probing laser spot for the Kerr effect measurement. Typicalpinning positions are designated by A,B,C, and D, respectively. The inset shows the high resolution image of the notch. The arrows indicate the mag-netic ﬁeld directions of H satand Hsweep.APPLIED PHYSICS LETTERS 95, 152506 /H208492009 /H20850 0003-6951/2009/95 /H2084915/H20850/152506/3/$25.00 © 2009 American Institute of Physics 95, 152506-1 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: 140.232.1.111 On: Fri, 19 Dec 2014 12:34:05depinning ﬁeld Hnotchl←, as plotted in Fig. 2/H20849b/H20850. On the other hand, in Regime IIb with Hnotchl→/H11021Hsweep/H11021Hnotchr→, the DW is pinned at Cand then, depinned leftward under Hnotchr←as shown in Fig. 2/H20849c/H20850. This regime appears only when Hnotchl→ /H11021Hnotchr→. These two depinning ﬁelds can be tuned indepen- dently by adjusting the depth and the slope of the notch.12 Note that these two leftward depinning ﬁelds Hnotchl←and Hnotchr←are governed by the static pinning process, since the DWs in these cases are initially trapped at the notch. The kinetic process is realized in Regime III with Hnotchr→/H11021Hsweep. In this regime, the hysteresis loop shown in Fig. 2/H20849d/H20850exhibits much smaller depinning ﬁeld compared with those in Regime II. In this regime, Hsweep is strong enough to bring the DW to the position D. By sweeping a negative magnetic ﬁeld, the DW is depinned from Dunder the depinning ﬁeld Hright←. Note that Hright←is much smaller than Hnotchl←and Hnotchr←as listed in Table I. However, it is quite interesting to see that once depinned from the right corner,the kinetic DW continues to pass through the notch. One thusconjectures that the kinetic DW experiences much smallerpinning ﬁeld in comparison with the static DW. Figure 3summarizes the depinning ﬁelds with respect to the strength of H sweep. Note that Hsweep is the maximum ﬁeld swept horizontally to the rightward; the maximum ﬁeld tothe leftward is ﬁxed to /H11002300 Oe. Each symbol is obtained by averaging more than ten times repeated measurements. Allthe nanowires with different widths, /H20849a/H20850620, /H20849b/H20850350, and /H20849c/H20850 1170 nm, exhibit basically the same behavior. It is clearlyseen from the plots that there exist three /H20849or four /H20850regimes with distinct depinning ﬁelds. The depinning ﬁeld insideeach regime is almost constant irrespective of H sweep. The threshold values in the abscissa are the rightward depinning ﬁelds i.e., Hleft→,Hnotchl→, and Hnotchr→for each position as denoted in the plot. The ordinate corresponds to the leftward depin- ning ﬁelds i.e., Hnotchl←,Hnotchr←, and Hright←. The values are listed in Table I. One-dimensional collective model13of the DW is adopted to explain the present results. In this model, by as-suming a rigid DW, the DW motion is described by the twoparameters, the position qand the tilting angle /H9274of the mag- netization inside the DW. The equation of motion is thengiven by 1+ /H92512 /H9251/H9253/H9004q˙=H−1 2MS/H9255/H11032/H20849q/H20850+1 /H9251HK 2sin/H208492/H9274/H20850, 1+/H92512 /H9253/H9274˙=H−1 2MS/H9255/H11032/H20849q/H20850−/H9251HK 2sin/H208492/H9274/H20850, /H208491/H20850 where /H9251is the Gilbert damping constant, /H9253is the gyromag- netic ratio, /H9004is the DW width, MSis the saturation magne- tization, His the strength of the external magnetic ﬁeld, and HKis the shape anisotropy ﬁeld of the DWs. The energy function /H9255/H20849q/H20850describes the pinning potential around the notch and /H9255/H11032denotes /H11509/H9255//H11509q. For the static pinning case, the DW is initially placed at the position q0for minimum potential energy /H9255/H20849q0/H20850and the zero tilting angle, /H9274=0. With gradual increment of H, the DW is gradually shifted inside the potential to the positionq Hfor/H9255/H11032/H20849qH/H20850=2MSHwith maintaining /H9274=0. The DW is ﬁnally depinned from the notch, just when the external mag- netic ﬁeld exceeds the maximum pinning ﬁeld. The static depinning ﬁeld is thus given by Hdp‘Static’=/H20851/H9255/H11032/H20852MAX /2MS.O n the other hand, for the kinetic pinning case, the DW is ini- tially moving. Let us consider that it moves in + qdirection with positive H. For this case the DW has nonzero tilting angle/H9274. Thus, the DW can stop /H20849i.e., q˙=0 and /H9274˙=0/H20850only when the condition H/H11021Hdp‘Static’−HKsin/H208492/H9274/H20850/2/H9251holds for all the time in the whole notch area. The DW thus has a chance to pass through the notch under a ﬁeld smaller than Hdp‘Static’. Note that the term sin /H208492/H9274/H20850initially has a positive value for a ﬁeld below the Walker breakdown ﬁeld14or has an alternat- ing value between /H110061 above the Walker breakdown ﬁeld.TABLE I. The leftward and rightward depinning ﬁelds of the notches in several nanowires with different widths. The ﬁeld unit is oersted. Note that/H11003indicates that the depinning is forbidden since H notchl→/H11022Hnotchr→. 350 nm 620 nm 1170 nm Hleft→70.0/H110064.0 35.6 /H110060.4 20.0 /H110064.0 Hnotchl→252.0/H110064.0 66.8 /H110060.4 44.0 /H110064.0 Hnotchr→/H11003 146.8/H110060.4 98.0 /H110062.0 Hright←102.7/H110063.5 34.6 /H110064.2 12.9 /H110063.1 Hnotchr←/H11003 96.9/H110066.3 58.1 /H110067.0 Hnotchl←128.0/H110062.4 67.5 /H110063.5 15.1 /H110060.7 1 -101 01(a) (b)malized )←lHnotch→ leftH -10 -101 (c)Kerr voltage (norm ←r notchH -200 -100 0 100 200-101K (d) Magnetic Field (Oe)← rightH Magnetic Field (Oe) FIG. 2. Longitudinal Kerr hysteresis loops of the nanowire structure shown in Fig. 1. Depending on the strength of Hsweep, typical loops are shown for the four different regimes: /H20849a/H20850Regime I, /H20849b/H20850Regime IIa, /H20849c/H20850Regime IIb, and /H20849d/H20850Regime III, respectively.50100(a)ld(Oe)→ leftH→lHnotch→rHnotch ←r notchH ←lHnotch ←H 0 50 100 150 2000 50100Depinning fiel (b) (c)rightH IIIaIIa IIbIIb IIIIII IIIaIIaIIbIIbIIIIII 0 100 200 300050 0 50 100 Sweepin gfield(Oe)III IIIIII FIG. 3. Depinning ﬁelds with respect to Hsweep for several nanowire struc- tures with different widths: /H20849a/H20850620, /H20849b/H20850350, and /H20849c/H208501170 nm, respectively.152506-2 Ahn et al. Appl. Phys. Lett. 95, 152506 /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: 140.232.1.111 On: Fri, 19 Dec 2014 12:34:05For the simplest case of the pinning potential as given by /H9255/H20849q/H20850=/H209020, for q/H113490 2MSH0q, for 0 /H11349q/H11349/H9254. 2MSH0/H9254, for q/H11350/H9254/H20903/H208492/H20850 Equation /H208491/H20850can be analytically solved for a small tilting angle i.e., sin /H208492/H9274/H20850/H110612/H9274. Here, /H9254is the lateral size of the pinning potential and H0is the pinning ﬁeld. The solution is q/H20849t/H20850=/H9253/H9004 /H9251/H20849H−H0/H20850t+/H9004 /H92512H0 HK/H208751 − exp/H20873−/H9251/H9253 1+/H92512HKt/H20874/H20876, /H9274/H20849t/H20850=H0 /H9251HKexp/H20873−/H9251/H9253 1+/H92512HKt/H20874−H0−H /H9251HK, /H208493/H20850 for 0/H11349q/H11349/H9254. The maximum value of q/H20849t/H20850is given by qmax=/H9004 /H92512HK/H20877H0+/H208491+/H92512/H20850/H20849H0−H/H20850/H20873log/H20875/H208491 +/H92512/H20850H0−H H0/H20876−1/H20874/H20878. /H208494/H20850 Under the approximation that 1+ /H92512/H110611 since /H9251/H112701, it be- comes qmax /H11011/H9004 /H92512H0 HK/H20858 n=1/H11009/H20849H/H0/H20850n+1 n/H20849n+1/H20850. /H208495/H20850 The DW is pinned if qmax/H11349/H9254, otherwise it is depinned from the notch. Therefore, the kinetic depinning ﬁeld Hdp‘kinetic’is determined by the condition qmax=/H9254. Expanding the summa- tion in Eq. /H208495/H20850up to n=4, the valid root for the kinetic depinning ﬁeld is ﬁnally obtained as Hdp‘kinetic’/H11011/H9251/H208812/H9254HKH0//H9004−/H92512/H9254HK/3/H9004+O/H20849/H92513/H20850. /H208496/H20850 In contrast, the static depinning ﬁeld in this case is readily obtained as Hdp‘static’=H0. In Permalloy nanowires, the valuesof the parameters in Eq. /H208496/H20850are typically in the orders of magnitudes— /H9251/H110110.01,/H9254/H11011/H9004, and HK/H11011a few kilo-oersted.13 Therefore, the kinetic depinning ﬁeld of the notches in our samples is estimated to be about a few oersted, which issigniﬁcantly smaller than the static depinning ﬁeld of a fewtens of oersted. In our experiments, we prove the existence ofthe two distinct pinning mechanisms by demonstrating thatthe kinetic depinning ﬁeld is smaller than the static depin-ning ﬁeld. The upper bound of the kinetic depinning ﬁeld isgiven in the experiments and the exact kinetic depinningﬁeld measurement can probe the realistic pinning potentialproﬁles. This study was supported by the KOSEF through the NRL program /H20849Grant No. R0A-2007-000-20032-0 /H20850. 1S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320,1 9 0 /H208492008 /H20850. 2D. A. Allwood, G. Xiong, C. C. Faulkner, D. Atkinson, D. Petit, and R. P. Cowburn, Science 309, 1688 /H208492005 /H20850. 3D. A. Allwood, G. Xiong, and R. P. Cowburn, Appl. Phys. Lett. 85, 2848 /H208492004 /H20850. 4M. T. Bryan, T. Schreﬂ, and D. A. Allwood, Appl. Phys. Lett. 91, 142502 /H208492007 /H20850. 5M. Tsoi, R. E. Fontana, and S. S. P. Parkin, Appl. Phys. Lett. 83, 2617 /H208492003 /H20850. 6J. Grollier, P. Boulenc, V. Cros, A. Hamzi, A. Vaurès, A. Fert, and G. Faini, Appl. Phys. Lett. 83, 509 /H208492003 /H20850. 7C. K. Lim, T. Devolder, C. Chappert, J. Grollier, V. Cros, A. Vaurès, A. Fert, and G. Faini, Appl. Phys. Lett. 84, 2820 /H208492004 /H20850. 8S.-M. Ahn, D.-H. Kim, and S.-B. Choe, IEEE Trans. Magn. 45, 2478 /H208492009 /H20850. 9S.-B. Choe, J. Magn. Magn. Mater. 320, 1112 /H208492008 /H20850. 10L. K. Bogart and D. Atkinson, Phys. Rev. B 79, 054414 /H208492009 /H20850. 11D. Atkinson, D. S. Eastwood, and L. K. Bogart, Appl. Phys. Lett. 92, 022510 /H208492008 /H20850. 12K.-J. Kim, C.-Y. You, and S.-B. Choe, J. Magn. 13, 136 /H208492008 /H20850. 13L. Thomas, M. Hayashi, X. Jiang, R. Moriya, C. Rettner, and S. S. P. Parkin, Nature /H20849London /H20850443, 197 /H208492006 /H20850. 14A. Mougin, M. Cormier, J. P. Adam, P. J. Metaxas, and J. Ferré, Europhys. Lett. 78, 57007 /H208492007 /H20850.152506-3 Ahn et al. Appl. Phys. Lett. 95, 152506 /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: 140.232.1.111 On: Fri, 19 Dec 2014 12:34:05
1.337163.pdf	Growthinduced anisotropy in bismuth: Rareearth iron garnets V. J. Fratello, S. E. G. Slusky, C. D. Brandle, and M. P. Norelli Citation: J. Appl. Phys. 60, 2488 (1986); doi: 10.1063/1.337163 View online: http://dx.doi.org/10.1063/1.337163 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v60/i7 Published by the American Institute of Physics. Related Articles Field dependent magnetic anisotropy of Fe1−xZnx thin films J. Appl. Phys. 113, 17A920 (2013) Magnetism and multiplets in metal-phthalocyanine molecules J. Appl. Phys. 113, 17E130 (2013) Ab initio study on magnetic anisotropy change of SrCoxTixFe12−2xO19 J. Appl. Phys. 113, 17D909 (2013) Threshold current for switching of a perpendicular magnetic layer induced by spin Hall effect Appl. Phys. Lett. 102, 112410 (2013) A first-principles study of magnetism of lithium fluorosulphate LiFeSO4F J. Appl. Phys. 113, 17B302 (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 26 Mar 2013 to 142.51.1.212. 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_permissionsGrowth-induced anisotropy in bismuth: Rare-earth iron garnets v. J. Fratelio, S. E. G. Slusky, C. D. Brandle, and M. P. Norelli AT&T Bell Laboratories. Murray Hill. New Jersey 07974 (Received 5 May 1986; accepted for publication 25 June 1986) The bismuth-doped rare-earth iron garnets, (R3 _x_yBixPb y )Fe5012 (Bi:RIG, R = Pr, Nd, Sm, Eu, Gd, Th, Dy, Ho, Er, Tm, Yb, Lu, and Y), were prepared under constant growth conditions to investigate the influence of ionic species on the bismuth-based growth-induced uniaxial anisotropy K ~. The effect of ionic species on growth-induced anisotropy in Bi:RIG was not consistent with the ionic size model of site ordering. In particular, Bi:SmIG, Bi:EuIG, and Bi:ThIG displayed high growth-induced anisotropies, up to 331 000 erg/cm3 at room temperature for x;::::O.5. The temperature dependence of these K ~ 's was somewhat higher than that of the wen studied Bi:YIG. The site ordering ofBi can be modeled by assuming that small, 10w-oxygen-coordination BiO: 3 -2w melt complexes have a strong site selectivity for small, high-oxygen coordination sites at the growth interface. I. INTRODUCTION Bismuth-doped yttrium and rare-earth iron garnets have long been of interest for magnetic bubble and magneto optic applications. They are particularly useful for high-den sity wide-tern perature-range magnetic bubble devices be cause they can be grown with large growth-induced uniaxial anisotropies that have small temperature derivatives. I How ever, the origins of this growth-induced anisotropy and its dependence on other dodecahedral cations are not complete ly understood. The Bi-based growth-induced anisotropy is proportion al to the undercooling of the melt below its saturation tem perature.2.3 To achieve the large anisotropies required for small bubbles, large undercoolings are required. These can cause homogeneous nucleation in the melt resulting in inclu sion defects in the film. Therefore, to achieve low-defect den sities in films, it is important to increase the amount ofuniax ial anisotropy induced per degree of undercooIing. The systems that have been most studied for magneto optic applications are the Bi:Y,2-4 Bi:Gd,5.6 and Bi:Lu3.4.7 iron garnets. Bi, a large ion [ionic radius, rj = 1.13 A (Ref. 8) 1, yields approximately equal growth-induced anisotro pies with Y [rj = 1.019 A (Ref. 9) 1 and Gd [rj = 1.053 A (Ref. 9) J and a much lower growth-ind.uced anisotropy with Lu [rj = 0.977 A (Ref. 9) J. The conventional site pre ference model. of growth-induced anisotropy predicts that ordering on the crystallographically inequivalent dodecahe dral sites at the growth interface should scale with the differ ence in ionic radii. 10.11 This model seems to be inconsistent with the results, so a new site ordering model is required for Bi ions. Therefore, we studied the effect of the principal dodecahedral ionic species R on the growth induced anisotropy K ~ in the Bi-substituted garnets (R3_x_yBixPby)FesOI2 (Bi:RIG, R=Pr, Nd, Sm, Eu, Gd, Th, Dy, Ho, Er, Tm, Yb, Lu, and V). Most of these garnets had not been previously prepared with the addition of Bi. The Bi:TmI2-15 and Bi:Ybls iron garnets had been previously investigated for magneto-optic and magnetic bubble applications. The magneto-optic prop erties of bulk, flux-grown (BiSm)3(FeGa)SOI2 had been studied,I6 but no measurement of K! was made. Kravt chenko et al. had seen that Pr causes a negative or in-plane anisotropy in (BiPrGdYbh(FeAl)sOI2 filmS.17 Mixed (BiSmLu) 3Fe5012 garnets had been tested for bubble device applications,18 but the multiple ion pairs obscured the sources of the anisotropy-the additional growth-induced anisotropy resulting from the addition of Bi had been as cribed to the Bi:Lu pair. In our studies we isolated the vari able of ionic species by growing bismuth-doped, single rare earth, unsubstituted iron garnets under nearly identical growth conditions. II. EXPERIMENT A. Growth The films were grown by standard LPE techniques from a PbO/Bi203/v 205 flux. 19 The molar fraction of garnet ox ides in the melt, R4 = ~ garnet oxides/ (~ garnet oxides + ~ flux oxides), was chosen for each melt to give a satura tion temperature Ts = 900 ± 2 ·C. The molar ratio R I = Fe203iR203 was fixed at 40 to maximize the ratio ofBi to rare earth (and hence Bi incorporation) without getting too close to the phase field of magnetoplumbite, which oc curs at high values of R I' A bismuth:gadolinium iron garnet melt was formulated to check the requirements for growth and characterization of these tUrns. The saturation temperature T. of this melt was 899 ·C and films were grown at temperatures of 800 to 896 ·C on MgCaZr substitutedGGG substrates20 (lattice parameter, ao = 12.497 A) that nearly match the film lattice parameter of Bi:GdIG. Figure 1 shows the variation of the growth rate f with growth temperature Tg• The curve was drawn by fitting Van Erk's function21 to a simple Arrhenius exponential [(T. -Tg )/T.Tg J{l//[(C.lCL) - 1 n =Ae-G1RT, (1) where Cs and CL are the concentrations of garnet oxides in the solid and liquid, respectively, and A and G are the fitting parameters. This relation would be expected if the growth rate is mainly either diffusion or interface controlled. The 2488 J. Appl. Phys. 60 (7). 1 October 1986 0021-8979/86/192488-10$02.40 @ 1986 American Institute of Physics 2488 Downloaded 26 Mar 2013 to 142.51.1.212. 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:z i ..... E ::t. w f- <I: II:: :I: I- ~ 0 II:: <.!> 0.6 0,5 0.4 0,3 0.2 0,1 O~~~--~ __ L-~ __ -L __ ~~~~ __ ~~ 800 810 820 830 840 850 860 870 880 890 900 T 9 (OC) FIG. 1. Growth rate ofBi:GdIG as a function of growth temperature. fitted function simplifies to f = 250 /tmK (_1 ___ 1_)e -1.16x 10" K (lIT, -liT,>. s Tg T. (2) The results on (Gd3_x_yBixPby )FeSOI2 (Bi:GdIG) films were used to establish growth and characterization conditions. Then melts were prepared to grow (R3_x_yBixPby)Fes012 (Bi:RIG) films, where R=Pr, Nd, Sm, Eu, Th, Dy, Ho, Er, Tm, Yb, Lu, and Y. Bi:PrIG and Bi:NdIG do not nucleate homogeneously,but they can be prepared by LPE on a substrate of suitable lattice param eter.22 For each gamet, the film was matched to the best available substrates, including rare-earth gallium garnets and garnets substituted with Sc (Ref. 23), CaZr (Ref. 24), and MgCaZr (Ref. 20) that spanned the lattice parameter range of 12.295-12.640 A. In cases where a dose match was not possible, films were grown on two different substrates with lattice parameters that bracketed that of the film so that any effect of the mismatch tended to average out. All sub strates were oriented in the ( 111) direction. Garnets containing sroalJ (rl < 1.01 A) or large (rl > 1.07 A) rare-earth ions were more soluble in the flux than garnets containing intermediately sized ions. These garnets have lower free energies offormation2S as a result of distortions in the structure by non optimally sized dodecahe dral ions. To maintain a constant saturation temperature T. = 900 °C, the concentration of garnet oxides in the melt R4 had to be varied as is shown in Fig. 2. From each melt we grew four samples, two at 850 ± 2 ·C and two at 875 ± 2 ·C. The data for each pair were consistent within the experimental uncertainties and were averaged to give the results. B. Characterization The Bi and Pb concentrations in the (R3 _ x _ y :Six Pby ) FeS012 films were measured by x-ray fluorescence (XRF) with a nondispersive x-ray milliprobe spectrometer using Cr radiation and a Princeton Gamma-Tech x-ray analyzer. Data from sequentially etched samples showed that the thin film approximation I}=I~ (l-e-bt);:::;Iloobt;:::;k/p:"t (3) could be used for samples -l/tm thick or less. I} and I 100 are the XRF intensities of element i from a thin film and an 2489 J. Appl, Phys., Vol. 50, No.7, 1 October 1986 0,25 I I T 0.24 f-Ts = 900·C Q 0,23 - 0.22 - 0 0,21 - 0 v 0,20 - 0: 0 0.19 f- - 0 0,18 - - 0 0,17 I- - 0 0.16 0 - 0 0 0 0 0.15 I-0 - I I I 1.00 1.05 1.10 . IONIC RADIUS (A) FIG. 2. Melt concentration of garnet oxides R. required to maintain a satu- ration temperature T, = 9OO'C as a function of the ionic radius of the rare earth ion R in Bi:RIG melts. infinitely thick sample respectively, b is a wavelength-depen dent x-ray absorption coefficient, t is the sample thickness, P:" is the molar density of the constituent of interest, and kj is an XRF efficiency constant, independent of the material properties, for element i in the x-ray milliprobe. This is a fair approximation even for multicomponent films and for poly chromatic radiation though the factor of proportionality kj becomes complicated. Bulk and thin-film standards of Fe, Bi, and Pb were used to determine the fluorescence efficiencies kl of the Fe Ka, Bi La, and PbL{3 fluorescence peaks. The Bi and Pb concentra tions in the garnet films were then determined relative to the Fe content C Fe' For all these films we considered C Fe = 5 per formula unit. This ignores the small octahedral incor poration ofPb4+ 26 and Pt, which should not exceed 1 %. Pt cannot be detected by XRF because of spectra11ine interfer ence from the major constituents. Thus x = 5UJiII?)(k Fe1k Bi) (4) and y = 5Ur/I?) (k Fe1k Ph)' (5) The uncertainty in counting and cailibration yie]lds an uncer tainty in the concentrations of ± 10 percent with a mini mum uncertainty of ± 0.01 per formula unit resulting from background and diffraction effects. The prism coupling technique27.28 was used to measure the index of refraction n at 633 nm and the sample thickness t. Substrate lattice parameters ao were determined by the Bond method.29 Film lattice parameters at were then ob tained by difference, measuring the displacement of the (888) Bragg reflection from that of the substrate and cor recting for mismatch strains. The effective anisotropy field Hie = Hk -4TrM., the Fratello et at. 2489 Downloaded 26 Mar 2013 to 142.51.1.212. 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_permissionscubic anisotropy, K J M" and the gyromagnetic ratio r were measured by ferromagnetic resonance (FMR) spectra taken at multiple orientations.30 The Gilbert damping parameter a was calculated using r and the linewidth of the resonance mode with the applied field perpendicular to the film.31 These measurements were made at -60, 25, and 14O·C to ascertain the temperature dependence of these quantities. Several of the garnets (Th at an temperatures; Dy and Ho at room temperature and below; Sm at low temperature) were too highly damped for FMR measurements to be made. In addition, the low gyromagnetic ratios of the Dy and Ho based garnets resulted in resonance fields beyond the limits of our equipment. Low-temperature data could not be taken for Bi:GdIG because of the proximity of its compensation point. In the cases where FMR data could not be taken at 25 or 140 ·C, magnetometer data were used to estimate H k as is discussed in the Appendix. A vibrating sample magnetometer (VSM) was used to measure the saturation magnetization 41rMs of the samples at 25 and 140 ·C, and to determine the Curie temperature T c. The accuracy of the 41rMs measurements is limited by both the magnetometer accuracy ( ± 20 Oe) and the accu racy of the sample thickness measurement and uniformity ( ± 5%). 41rMs at -60·C was calculated from molecular field data. It was possible to use the VSM to make measure ments of Hk (if positive) and/or H k (if negative) by noting the field at which the sample saturates in the parallel and perpendicular directions, respectively (see Appendix). These values were corrected for K I and they correlate within 10% with the FMR values, so they were used in the cases when FMR data could not be taken. The H k and H k data were used with the 41rMs values to calculate the uniaxial anisotropy Ku = (HkMs/2) = [(Hic + 41rMs)M.J2]. (6) The portion of this anisotropy that arises from stress must be calculated. In garnets with no diamagnetic substitu tion on the iron sites, the magnetostriction coefficient ..1.111 is simply a ljnear combination of the ..1.111' s of the dodecahedral components added to the..1.111 of the Fe lattice. The magneto strictions were determined from measured data on pure rare earth garnets taken at room temperature and below. These data were interpolated to determine the magnetostrictions at -60 ·C, and extrapolated to 14O·C. 32-37 Magnetostriction TABLE I. Properties of (Gd, .. x-,Bix Pb, )Fe5012 films. .iT x y Of n ('C) (± 10%) (± 10%) (±O.OOI A) ( ±0.OO5) 3 0.10 0.01 12.480 2.353 11 0.17 0.03 12.482 2.364 23 0.24 0.04 12.485 2.376 25 0.27 0.05 12.486 2.379 37 0.33 0.07 12.491 2.392 49 0.40 0.07 12.497 2.405 49 0.40 0.07 12.497 2.407 62 0.51 0.10 12.501 2.422 74 0.59 0.11 12.505 2.436 99 0.73 0.14 12.514 2.464 2490 J. Appl. Phys., Vol. 50, No. 7,1 October 1986 O.B ~ ~ z 0.7 ::> <[ ..J ::> 0.6 :::E 0:: 0 "-0.5 ..... <II :::E 0 0.4 f- :'!. z 0.3 0 f= <[ 0.2 0:: f- Z w 0.1 u z 0 u 0 0 o Bi • pb 10 20 30 40 50 60 70 BO 90 100 l1T (OC) FIG. 3. Incorporation ofBiand Pbin (Gd, _x_,BixPb, )Fe5012 asa func tion of the melt undercooling .iT = T, -T •. data for Bi (Ref. 2), Pb (Ref. 38), and Nd (Ref. 39) were extrapolated from mixed garnets. The magnetostrictive ef feet ofPb was only measured at room temperature, but, since it has the same magnitude as that ofBi at room temperature, it was assumed that A. r~1 =..1.~: I at all temperatures. No magnetostriction data were available for LuIG or PrIG so the YIG data were used for these samples. The stress-induced uniaxial anisotropy was then calcu lated from KS = 1. E (af -ao) A. u 2 (l-f..L) af I'" (7) where Young's modulus E for iron garnets is -2 X 1012 dyne/cm2 and Poisson's ratio f..L = 0.29 (Ref. 40). This, in tum, allows the computation of the growth induced contribution to the uniaxial anisotropy m. RESULTS AND DISCUSSION A. Bi:GdIG (8) Table I gives the results of the baseline study on (Gd3 _" _yBi" Pby )FeSOI2• Figure 3 shows that the incor poration of Bi and Pb increases linearly with supercooling AT, in spite of the nonlinearity of the growth rate (see Fig. 47rM, Tc K~ (±200e±5%) ( ±I'C) ( ± 2000 erg/em' ± 10%) 142 286 6000 180 288 12000 239 289 25000 280 289 35000 291 291 47000 334 292 61000 331 292 68000 397 295 79000 414 297 94000 525 301 120000 Fratello et al. 2490 Downloaded 26 Mar 2013 to 142.51.1.212. 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_permissions1 ). Although the Pt content could not be measured, it should also increase with increased undercoaling. Both the Bi and Pb concentrations have a positive intercept. For the Bi con centration x, x = 0.0066AT + 0.09, and for the Pb concentrationy, y = 0.OO14AT + 0.01. (9) (10) These intercepts, Xo = 0.09 and Yo = 0.01 per formula unit, represent the equilibrium concentrations ofBi and Pb at the saturation temperature. Other researchers have seen posi tive xo's in Bi:GdIG (Refs. 5 and 6) and in Bi:YIG grown from a pure Bi203 flux.4 The physical and magnetic properties are all functions of x and y. Unfortunately, the effects of these variables can not be separated statistically because it was found that the Pb concentration was proportional to the Bi concentration, i.e., y-0.18x. The method ofStrocka, Holst, and Tolksdorf' I and the data of Shannon and Prewitt8,9 were used to calculate the lattice parameters of these garnets. It was assumed that the Pb mainly occupies the dodecahedral site as Pb2+ _Pb4+ pairs to preserve stoichiometry, but that one third of the Pb4+ goes on octahedral sites.26 Using the incorporation data from Eqs. (9) and (10), the calculation predicts dafldAT= 3.2X 10-4 Ye, which is in good agreement with the measured value of 3.5 X 10-4 AiC. The difference may result from Pt incorporation. The index of refraction data were fitted to n = 2.336 + 0.145(x + y) (11 ) in Fig. 4. This data is similar to that taken by Moriceau et al.,42 though they did not account for the effect ofPb. Similarly, the Curie temperature data shown in Fig. 5 were also fitted to a straight line. Tc =283.8+ 19.1(x+y)·C. (12) The intercept is lower than the value of 291 ·C given for pure bulk GdIG by Bertaut and Pauthenet,43 possibly as a result of the incorporation of octahedral Pt from the crucible or the incorporation of off-stoichiometric octahedral Gd as is seen in GGG.44 An octahedral substitution of 0.04 would ac- 2.48 2.46 2.44 2.42 2.38 2.36 2491 o 0.2 0.4 0.6 0.8 1.0 x+y FIG. 4. Variation of the in dex of refraction in (Gd) -x _yBixPb y )Fe,O'2 films with Bi and Pb content. J. Appl. Phys., Vol. 50, No.7, 1 October 1986 o 280~~--~-i--~~--~~--~~ o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x+y Fig. 5. Variation of the Curie temperature in (Gd) _ x-yBix Pby )Fe,O'2 films with Bi and Pb content. count for this decrease in T c (Ref. 45). The slope of 19.1 ·C per formula unit in Eq. (12) is a combination of the positive effects of dodecahedral Bi3+ ( + 36·C per formula unit46) and Pb2+ and the negative effects of octahedral Pb4+ and Pt4+ (-166·C per formula unit4s). Since dodecahedral Pb4+ is a small ion, it probably reduces Tc slightly. Finally, the growth-induced uniaxial anisotropy K! was found to vary linearly with the Bi concentration, x (see Fig. 6). K! = 187 OOO(x -0,08) (erg/cm3) . (13) The relatively small contribution of the small Pb content is negligible. Note that the x intercept is virtually identical to the equilibrium Bi concentration Xo found in Fig. 3 and Eq. (9). This suggests that the equilibrium Bi is distributed ran domly in the lattice and does not contribute to the growth induced anisotropy. Thus, the relation K! cr:.AT is ob served. 2.3 125,000 100,000 ;; E 75,000 u .... 0 ~ ~ "'~ 50,000 "" 25,000 o o O~~L-~~~~~~~~~~~~~ o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 O.B FIG. 6. Relation of the growth-induced anisotropy in (Gd) _x_yBixPb y) Fe,OI2 to the Bi content. Fratello et al. 2491 Downloaded 26 Mar 2013 to 142.51.1.212. 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_permissionsTABLE II. Properties of (R3 _ x _yBixPby )Fe~OI2 films. x y af n R (± 10%) (± 10%) (±O.OOI A) ( ±0.OO5) Pr 0.70 0.15 12.644 2.478 Nd 0.68 0.11 12.606 2.459 Sm 0.51 0.09 12.549 2.437 Eu 0.48 0.08 12.521 2.422 Gd 0.40 0.07 12.497 2.406 10 0.34 0.06 12.460 2.405 Dy 0.34 0.06 12.433 2.401 Ho 0.32 0.04 12.406 2.392 Er 0.31 0.06 12.379 2.386 Tm 0.30 0.04 12.354 2.385 Yb 0.31 0.04 12.332 2.381 Lu 0.38 0.04 12.315 2.374 Y 0.34 0.05 12.404 2.367 B.Bi:RIG Table II gives the data for (R3 _x_yBixPb y )Fe5012' Only the data for the samples with !l. T = 50 ·C are included. The !l. T = 25 ·C data are similar, but the changes are smaller and closer in magnitude to the experimental uncertainties. The large Bi and Pb ions are most readily incorporated in the large lattice parameter garnets. In Fig. 7 the distribu tion coefficients increase with increasing ionic radius of the R ion. We cannot determine whether the distribution coeffi cient of Hi with respect to R in Bi:RIG is dependent on the lattice parameter or the ionic radius 'j of the rare earth R. However, Mada and Yamaguchi 18 found that Bi substituted preferentially for Sm in (SmLu)3Fe50,2' which is consistent with the latter explanation. The room temperature growth-induced anisotropy data shown in Fig. 8 have several interesting features. Gd and Y are the dodecahedral ions most commonly paired with Bi, 0.25 r----,I----.----Ir------.c 0.20 I- f- Z lJ.J U G: u.. 0.151-lJ.J 0 u Z 0 i= ::l 0.10 f-al a: f-If) c 0.05 I- 0 0 o Bi • pb 00 ••• 0 • I 1.00 o - 0 0 - 0 0 0 0 0 - .I • • • • , • • I I 105 1.10 0 IONIC RADIUS (Al FIG. 7. Incorporation of Bi and Pb in (R3_x_yBixPby)Fe~O'2 films (Il T = 50'C) as a function of the rare earth ionic radius. 2492 J. Appl. Phys., Vol. 50, No.7, 1 October 1986 41TM, Tc K~ (± 200e± 5%) ( ±2'C) ( ± 2000 erg/em3 ± 10%) 2250 319 -139000 2110 311 -0 1850 301 331000 1380 303 199000 330 292 65000 470 292 156000 660 287 59000 1110 285 26000 1290 279 12000 1550 280 9000 1660 274 21000 1860 270 17000 1920 281 69000 but they do not yield the highest growth-induced anisotro pies. Bi:Eu and Bi:Th are considerably higher and Bi:Sm is five times better than Bi:Y. We did not plot the growth induced anisotropy per formula unit ofBi K ~/x because the zero intercept Xo varies with the rare-earth species, but the results would be qualitatively the same as in Fig. 8. These data do not fit well with the conventional model of growth-induced anisotropy resulting from atomic order ing that is most frequently applied to mixed rare-earth iron garnets.to •• 1 This model predicts that large ions should order best with small ions. Our data shows that Bi, a large ion, gives rise to the highest growth-induced anisotropy with large anisotropy-producing ions such as Sm and Eu, and yields small growth-induced anisotropies with small ions. When compared to the data of Gyorgy et al. II from the (110) faces of (R.R2)IG [and that of Hansen46 for the Pr:Y pair, which behaves differently in the (111) direction}, Bi I 300,000 t- 200,000 - '" E 0 .... 01 .... 100,000 I-~ 0>::> Y Dy :.:: 0 0 L)5 YbTm f60 o 0 oEr 0 -100,000 I- I 100 IONIC I Tb 0 Gd 0 105 Eu 0 I sm o 1.10 RADIUS (Al - - Nd - p~ FIG. 8. Growth-induced anisotropy of Bi:RIG films (IlT = 50 'C) plotted against the ionic radius of R. Fratelio ef al. 2492 Downloaded 26 Mar 2013 to 142.51.1.212. 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_permissionsbehaves most like Lu and Y, ions with which it shares the characteristic of a filled (or absent) /shell. The similarity departs with Nd, which has a negative K ~ in combination with Lu or Y, but has zero growth-induced anisotropy in Bi:NdIG for reasons that are not completely understood. It is not surprising that Bi orders differently from the rare earths as it is different chemically, having a much lower oxygen bond energy. The theory ofvan Erk47 and cryoscopic (freezing point depression) studies48 suggest that the garnet oxides exist in the melt as charged, single-cation complexes, MO: v -2w, where u is the valence of M and w is its oxygen coordination. Wolfe et al.49 pointed out that chemically dif ferent ions such as Ca2+ would form different oxide com plexes in the melt than the rare earths, thus resulting in dif ferent ordering processes. Flux oxides such as PbO and Bi203 are at least partially dissociated in the melt to metal and oxygen ions. so Bi may exist in the melt as one or more of the species Be + , BiO + , BiOi, etc. The equilibrium balance of species depends on the flux chemistry, growth tempera ture,and growth atmosphere. The Bi complex with the same oxygen coordination number w as that of the dissolved rare earth oxide RO: 3 -2w, should be incorporated normally in the growing garnet. This may be the source of the equilibri um Hi, which is distributed randomly because this complex has a shape and size similar to the rare earth complex. The lower coordination complexes of Bi may be effectively "off stoichiometry" and their incorporation would require com pensation by dissolved oxygen in the melt or might even cause oxygen vacancies. Thus, they would have small equi librium distribution coefficients. However, they could be in corporated kinetically with a high site selectivity for the smallest site at the growth interface with the highest oxygen coordination. This mode! would expl.cin many of the unusual features of the Bi-based growth·.jnduced anisotropy. Anisotropy would arise only 11'1>1)1 kinetic incorporation of low-oxygen coordination BiO~ 3·-W complexes, the concentration of which should be proportional to the undercooling. Klages and Tolksdorffound the following relation6: x = xo + 0.0128x0-6.T + 0.00423dT (14) Our model suggests that the first term represents the high oxygen-coordination equilibrium Hi, the second term the ki netic incorporation of the same complex, and the third term the kinetic incorporation of the lower oxidation complexes that cause the growth-induced anisotropy. Thus the slope of K ~ vs dT is nearJy constant for a variety of melt composi tions for any given ionic species R. The ordering does not peak as expected at x = 1.5 because the ordering species is not the total Bi concentration x but only the fraction repre sented by the last term in Eq. (14). The high K ~ 's of the Hi garnets have previously been attributed to higher site selectivity.46 The site selectivity of these small, positively-charged complexes would naturally be strong for a small site with high oxygen coordination. The preference for small sites explains the behavior of K ~ versus the ionic species R as Bi will order best with large ions and compete for the small sites with small ions. The ordered ions should interact with the other rare earths in a similar fashion 2493 J. Appl. Phys., Vol. 60, No.7, 1 October 1986 to Y and Lu, with which Bi shares a similar electronic struc ture, so the highest degree of anisotropy will be generated by large, anisotropy-producing ions such as Sm and Eu. If the incorporation of these complexes does result in permanent bound oxygen vacancies, then these may even be the cause of the growth-induced anisotropy. Nov8.k.S1 has theorized that Bi-based anisotropy results from modification of the single ion Fe3 + anisotropy through the neighboring oxygens. Oxy gen vacancies would certainly provide a strong effect. This model suggests a reason for the similarity between Bi-and Pb-based growth-induced anisotropy. For example, the Pb-Eu pair yields an exceptionally high growth-induced anisotropy52 compared to Ph-Y and Pb-Gd (similar to Bi Eu). Additionally, we earlier hypothesized that the Pb based growth-induced anisotropy might arise solely from Pb2+ (Ref. 38), which is isoelectronic with Be+ and also seems to have an equilibrium concentration. Pb2+ might be expected to exist in the melt as a single ion or a small neutral PbO cluster, while the more highly charged Pb4+ would have a tendency towards a higher oxygen coordination, pos sibly Pb02. c. Temperature dependence The way in which magnetic bubble material parameters vary with temperature is important for wide-temperature range device operation. Table III shows how the FMR data for the cubic anisotropy K1, gyromagnetic ratio y, and Gil bert damping parameter a vary with temperature. As was previously mentioned, it was not possible to take data on films containing Sm and Gd at low temperatures, Dy and Ho at room temperature, and Tb at any temperature. K 1 data for Pr-based samples could not be calculated consistently. An important feature of (YBiCa)3(FeGeSi) sOI2 bub ble materials is the small temperature derivative of the growth-induced anisotropy compared to that of convention al (YSmLuCa)3(FeGe)SOI2 materials. 1 lfthe high growth induced anisotropies observed here are to be useful, we must determine if their temperature derivatives are also low. Ta ble IV shows the values of K ~ and the ratios of the high-and low-temperature values to the room-temperature data. The uncertainties in 41TM, and K~ are much greater at these tem.perature extremes. The uncertainties become compara ble to the magnitude of the data for Er, Tm, and Yb and produce considerable scatter, so these materials are not in duded in the table. The temperature dependence of Bi:Lu is similar to that ofBi:Y. Lu has a filled/she!] and both Y and Lu have d Ii! valence shells. The other rare-earth ions have partially filled /shells and display stronger temperature dependences. This effect peaks at Bi:TbIG, which has an exceptionally strong temperature dependence. To compare these effects to bubble materials, in 2-J.Lm bubble (BiYCa)3(FeGeSi)5012 fiJms K~ ( -60 ·C)/ K~ (25 ·C) = 1.4andK~ (25 ·C)/K~ (140 ·C) = 2.1, while m 2-J.Lm bubble (YSmLuCah(FeGe)SOI2 films K~ ( -60 ·C)/K~ (25 ·C) = 2.2 (60% higher) and K~ (25 ·C)/K~ (140 ·C) = 3.9 (90% higher). The in creases seen in the temperature dependence of the Bi:RIG samples over that ofBi:YIG are 0-20% at low temperatures Fratello et al. 2493 Downloaded 26 Mar 2013 to 142.51.1.212. 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_permissionsTABLE Ill. Variation of FMR measurements with temperature. K, ( ± SOO erg/em3 ± 2S%) R -60°C 2SoC 140°C -60°C Pr 3.3 Nd -67000 -ISOOO -2300 3.4 Sm -13000 -2600 Eu -84000 -31 000 -S600 1.94 Gd -7800 -2600 Dy -1500 Ho -2200 Er -18000 -6900 -1500 1.96 Tm -17000 -7000 -1500 2.22 Yb -7300 -3400 -800 2.68 Lu -11 000 -5200 -900 2.83 y -12000 -6000 -1100 2.83 and 0-30% at high temperatures (except for Bi:TbIG). These Bi:RIG samples have much smaller temperature de rivatives than materials with SmLu-based anisotropy, but are significantly worse than Bi:YIG based materials at high temperatures. IV. CONCLUSIONS This study showed that the bismuth:rare-earth growth induced anisotropy varies considerably with ionic species. Ionic size does not seem to be important in the ordering process, but ions such as Sm and Eu, which produce large growth-induced anisotropies when paired with Y or Lu, also do so with Hi. This behavior can be modeled by considering that the Bi-based growth-induced anisotropy arises from or dering of small, low-oxygen-coordination BiO: 3 -2w com plexes on small, high-oxygen-coordinated interface sites. Three of the rare earths, Sm, Eu, and Th, produce a larger growth-induced anisotropy in combination with Bi than Y does. This suggests that inclusion of one or more of these elements in Bi:YIG-based magnetic bubble materials would reduce the undercooling necessary to achieve K ~'s large enough for device operation and, thus, yield lower defect densities. In general, the growth-induced anisotropy in TABLE IV. Variation of growth-induced anisotropy with temperature. K~( -60°C) K! (2S 0c) r a ( ± 0.01 MHz/Oe) ( ±3%) 2SoC 140°C -60°C 2SoC 140°C 3.18 3.07 0.14 0.092 0.063 3.23 3.06 0.26 O.IS 0.090 2.70 2.65 0.30 0.16 2.05 2.14 0.051 0.041 0.034 3.11 2.85 0.044 0.013 1.41 0.18 1.81 0.2S 2.13 2.22 0.206 0.111 0.068 2.35 2.41 0.030 0.016 0.011 2.68 2.66 0.102 0.034 0.020 2.82 2.80 <0.001 .;;0.001 <0.001 2.82 2.80 .;;0.001 .;;0.001 <0.001 Bi:RIG has a somewhat stronger temperature dependence than that of Bi:YIG. This suggests that magnetic bubble de vices made on garnets with growth-induced anisotropy de rived solely from a bismuth:rare-earth pair would not oper ate over as wide a temperature range as materials based on Bi:YIG. APPENDIX: MEASURING UNIAXiAL ANISOTROPY WITH A VIBRATING SAMPLE MAGNETOMETER Vibrating sample magnetometers are commercially available instruments that measure magnetization. In such an apparatus, a monotonically varying magnetic field is ap plied to the sample while the net magnetic moment of the sample is measured by reference to the response of nearby coils to the vibration of the sample. Examination of the re sulting magnetic moment versus applied field curve shows that the magnetic moment rises with increasing field until the saturation magnetization is reached. Information on the uniaxial anisotropy can be extracted from the same curve, specifically when the samples in question are thin films of material with [111] uniaxial. anisotropy. When the value of the effective uniaxial anisotropy field H k = 2KulMs -41T'Ms is negative and the applied field H K~(l4O°C) K~(-60"C) K!(2S0C) K~(25°C) K!(l40°C) R ( ± 2000 erg/ern3 ± 20%) ( ± 2000 erg/ern3 ± 10%) ( ± 2000 erg/em3 ± 20%) ( ±0.3) ( ±0.3) Pr -240000 -139000 -72000 1.7 1.9 Nd -0 -0 -0 Sm 331000 13l 000 2.5 Eu 285000 199000 97000 1.4 2.1 Od 65000 27000 2.4 Tb IS6000 40000 3.9 Dy 63000 30000 2.1 Ho 26000 15000 1.7 Lu 23000 17000 11000 1.4 1.6 Y 89000 69000 36000 1.4 1.9 2494 J. Appl. Phys., Vol. 50, No.7, 1 October 1986 Fratello et al. 2494 Downloaded 26 Mar 2013 to 142.51.1.212. 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_permissions1\ Z [111J 8 M 1\ x [ffO] FIG. 9. The coordinate system used in the analysis of uniaxial anisotropy determination from VSM data. is normal to the film (a situation that we will call case 1), the field at which the sample saturates is approximately equal to Hie. When the value of the uniaxial anisotropy field Hk == 2KulMs is positive and the applied field is in the plane of the film (caned case II), the field at which the sam ple saturates is approximately equal to H k' We derive here a theory for these empirical observations, including the cor rection for the first-order crystalline anisotropy constant. This derivation involves an examination of the energetic consequences of the magnetization of a thin film and is simi lar in approach to an analysis by Dillon.s3 The coordinate system used is shown in Fig. 9. The x, y, and z axes are the [ 1 TO], [112], and [111] axes of the crystal, respectively. The film normal is the [111] direction. The film uniaxial anisotropy is directed along the film normal. Consider Ms to have magnitude Ms and be directed along the direction de fined by 8 and </1. In this derivation, we shall ignore the wall energy and end effects. A. Case I: Hie < 0 and applied normal to film SubcaselA:H k <0 With zero applied field, Ms of the domains in the film will be oriented with 8-;::;'1T/2. The azimuthal orientation of Ms ,tP, will differ from domain to domain. As H increases normal to the film, () in all domains will decrease until aU the domains merge into one with 8 = 0 at saturation. In this system the energy density of a single domain is the sum of the Zeeman energy Ez = -Ms·H = -MsH cos 8, (15) the demagnetizing energy ED = 21TM; cos2 8, the uniaxial anisotropy energy Eu = -Ku cos28, (16) (17) and the crystalline anisotropy energy, of which the first term is Ec = K) (! cos4 8 +! sin4 8 + v2 cos 8 sin3 8 X (j sin3 tP -cos2 tP sin tP) ] in this coordinate system. (18) The equilibrium value of tP is defined by the minima of Ec. For K) negative, the typical situation in materials we see, the minima occur at tP = (1T!2), (71T/6), and (111T/6). 2495 J. Appl. Phys., Vol. 60, No.7, 1 October 1986 These are three equivalent [112] directions in the crystal. Ms will be oriented in one of these three directions in all the domains. For K) positive, the minima occur at tP = (31T12), (1T16), and (51T/6). At any ofthese energy minima, thecrys talline energy is Ec = K) q cos4 () +! sin4 0) -(v'113) IK)I cos 8 sin3 O. (19) Differentiating the total energy density with respect to cos 8 yields (aE /a cos 8) = -MsH -2Ku cos () + 41rM; cos () + K) (~COS3 () -sin2 () cos 8) -(v'113)IK)I( sin38-3cos28sin(). (20) At equilibrium, where this derivative is zero H = -Hie cos () + (K)IMs) (~COS3 () -sin2 8 cos 8) -(v'1/3){ IK) /lMs )(sin3 8 - 3 cos2 8 sin 8). (21) Therefore, at saturation, when sin 8 = 0, (22) Thus, saturation occurs approximately at a field equal to the absolute value of H Ie with a small correction for K). Subcase I B: Hk > 0 The zero-field domain pattern is of small domains, ei ther stripes or bubbles, with lateral dimensions generally within an order of magnitude of the film thickness. The mag netization is oriented with 8 either zero or 1T in these do mains, with the total volume of the 8 = 0 domains equal to the volume of the () = 1T so that the total magnetic moment is zero. As a magnetic field is applied normal to the film, Ms in the domains with Ms antiparallel to H is pulled around to ward H and domain walls shift until, when the magnitude of the applied field is equal to Hcolla~' the magnetization is oriented with 8<1T12 throughout the film. If Hcolla~ < -Hie + (4/3) (K)/M s)' the subcase I A treatment then holds for this system. lB. Case II: Hk > 0 and H applied In the plane of the film Consider that H is applied at azimuth all angle tP H' With zero applied field, the film win be broken up into small do mains, as described for subcase I B. Half the volume will be made up of domains with 8 = 0, the other half with 8 = 1T. In this situation the demagnetizing energy for a thin-film do main quoted in Eq. (16) no longer applies. In fact, the de magnetizing energy, as well as the net magnetic moment, is zero. As H increases in the plane of the film, the moments in the small domains will cant in the plane defined by the z-axis and H as shown in Fig. 10, until at saturation they merge into a single domain magnetized in the H direction [0;::: ( 11"12) , tP~tPh J. still with zero demagnetizing energy. The Zeeman energy becomes -Ms H sin () cos (tP -tP H ). If there are always equal volumes of material with 8 equal to t and (1T12) -t, where t is the angle shown in Fig. 10, any terms in odd powers of cos () sum to zero. This Fratello et at. 2495 Downloaded 26 Mar 2013 to 142.51.1.212. 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_permissionsFIG. 10. Magnetization vectors in the domain structure described in case II. An external magnetic field is being applied in the plane of the film and pointing toward the right. The magnitude of the external field is insufficient to saturate the film. is strictly true only for H applied along a [110] direction. For H applied in any other direction, there is an asymmetry in the energy for up and down domains as the magnetization rotates toward the plane. This yields a difference in total domain size and a difference in angle 8 for up and down domains. 54 The effect on the total energy of the system is second orderinK I and is dependent on the ratioKIIKu• For small KIIKu it can be neglected. This condition holds well for Bi:ThIG and Bi:DyIG, but the approximation is some what less accurate for Bi:HoIG. Therefore the total energy is the energy density E = -MsH sin 8 cos (¢ -¢H) -Ku cos28 + Klq cos4 8 +! sin4 (J) (23) summed over all the small domains in the film. Differentiat ing with respect to ¢ then yields aE = -MsHsinOsin(¢-ifJH)' (24) a¢ At equilibrium, when H is nonzero, ¢ = ¢ H' Differentiating with respect to sin 8 yields ---= -MsH + 2Ku sin 8 a sin (J + KI ( -j cos2 8 sin (J + sin3 0). (25) At equilibrium H = Hk sin 8 + (KIIMs) ( -j cos2 (J sin (J + sin3 8) (26) and at saturation, when (J = 1T12, H=Hk + (K/M s)' (27) Thus, as expected, saturation occurs when H is approxi mately equal to H k with a smaIl correction for K I' Thus we have shown that for H Ie negative and H applied normal to the film, saturation occurs at an applied field of -Hie + (413 )(Kl/iI1s ). In addition, we have shown that for Hk positive andH applied in the plane of the film, satura tion occurs at an applied field of H k + (K II Ms ). This deri vation permits us to calculate uniaxial anisotropy fields from vibrating sample magnetometer data, if we have information on crystalline anisotropy or if we can assume that the K I contribution is negligible. ACKNOWLEDGMENTS We would like to thank S. M. Vincent who took the XRF data and R. B. van Dover who sputtered the XRP tUm standards. R. D. Pierce provided many helpful insights on 2496 J. Appl. Phys., Vol. 50, No.7, 1 October 1986 the use of the VSM to determine anisotropies. This work was partially supported by Tri-ServicelNASA contract F33615- 81-C-1404. 'R. C. LeCraw, L. C. Luther, and E. M. Gyorgy, J. Appl. Phys. 53, 2481 ( 1982). 2p. Hansen, K. Witter, and W. Tolksdorf, Phys. Rev. B 27,6608 (1983). 3p. Hansen and K. Witter, J. Appl. Phys. 58, 454 (1985). 4L. C. Luther, R. C. LeCraw, and F. Ermanis, 1. Cryst. Growth 58, 95 (1982). 'Po Hansen, K. Witter, and W. Tolksdorf, Phys. Rev. B 27, 4375 (1983). 6C._P. Klages and W. Tolksdorf, 1. Cryst. 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Cerceau, Mat- er. Res. Bull. 14, 559 (1979). 181. Mada and K. Yamaguchi, 1. Appl. Phys. 57, 3882 (1985). 19L. C. Luther, V. Y. S. Rana, S. 1. Licht, and M. P. Norelli (unpublished). 2°D. Mateika, R. Laurien, and Ch. Rusche, 1. Cryst. Growth 56, 677 (1982). 21W. van Erk, 1. Cryst. Growth 43, 446 (1978). 22y. 1. Fratello, C. D. Brandle, and S. E. G. Slusky, 1. Cryst. Growth 75,281 (1986). 23c. D. Brandle and R. L. Barns, J. Cryst. Growth 20, I (1973). 24D. Mateika, 1. Hermring, R. Rath, and Ch. Rusche,l. Cryst. Growth 30, 311 (1975). 2'N. Kimizuka, A. Yamamoto, H. Ohashi, T. Sugihara, and T. Sekine, J. Solid State Chern. 49,65 (1983). 26G. B. Scott and 1. L. Page, 1. Appl. Phys. 48, 1342 (1977). 27M. 1. Sun, Appl. Phys. Lett. 33, 291 (1978). 211'. W. Hou and C. 1. Mogab, Appl. Opt. 20, 3184 (1981). 2"W. L. Bond, Acta Cryst. 13,814 (1960); A31, 698 (1975). :.or.. C. Hsia, P. E. Wigen, R. L. Bergner, and R. D. Henry, IEEE Trans. Magn. MAG-l7, 2559 (1981). 31 A. H. Morrish, The Physical Principles of Magnetism (Wiley, N ew York, 1965), p. 568. 32S. lida, J. Phys. Soc. Ipn. 22, 1201 (1967). 33p. Hansen, 1. Appl. Phys. 45, 3638 (1974). 34A. E. aark, B. F. DeSavage, N. Tsuya, and S. Kawakami. 1. Appl. Phys. 37, 1324 (1966). 3'A. E. Clark, 1. J. Rhyne, and E. R. Callen. 1. Appl. Phys. 39, 573 (1968). ""K. P. Belov, A. K. Gapeev, R. Z. Levitin, A. S. Markosyan, and Yu. F. Popov, Sov. Phys. IETP 41,117 (1975) [Zh. Eksp. Tear. Fiz. 68, 241 (1975) ]. 37M. Guillot and E. du Tremolet de Lacheisserie. Z. Phys. B 39, 109 (1980). 38y. 1. Fratello, S. E. G. Slusky, V. V. S. Rana, C. D. Brandle, and J. E. Ballintine. 1. Appl. Phys. 59, 564 (1986). 3"R. F. AI'mukhametov, K. P. Belov, and N. Yolkova, Sov. Phys. Solid State 25, 861 (1983) [Fiz. Tverd. Teia25, 1499 (1983)J. 4OG. Winkler, Magnetic Garnets, Vol. 5 of Vieweg Tracts in Pu,.e and Ap plied Physics. (Friedr. Yieweg and Sohn, Braunschweig, W. Germany, 1981), p. 143. 4lR. Strocka, P. Holst, and W. Tolksdorf, Philips 1. Res. 33, 186 (1978). Fratello et al. 2496 Downloaded 26 Mar 2013 to 142.51.1.212. 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_permissions42H. Moriceau, B. Ferrand, M. F. Armand, M. Olivier, D. Challeton, and J. Daval, IEEE Trans. Magn. MAG-20, 1004 (1984). 43F. Bertaut and R. Pauthenet, lEE Conf. Publ. (London) Part B Suppl. 104,261 (1957) . .. C. D. Brandle and R. L. Bams, J. Cryst. Growth 26,169 (1974). 4sM. A. Gilleo and S. Geller, Phys. Rev. 110,73 (1958). 46p. Hansen, W. Tolksdorf, K. Witter, and 1. M. Robertson, IEEE Trans. Magn. MAG-20, 1099 (1984). 41W. van Erk, 1. Cryst. Growth 46,539 (1979). 4·K. Fischer, Krist. Tech. 14, 835 (1979). 2497 J. Appl. Phys., Vol. 50, No.7, 1 October 1986 4"R. Wolfe, R. C. LeCraw, S. L. Blank, and R. D. Pierce, Magnetism and Magnetic Materials. 1976. edited by 1.1. Becker and G. H. Lander, AlP Conf. Proc. 34 (AlP, New York, 1976), p. 172. ,0D. Linzen, E. Sinn, K. Fischer, Th. Klupsch, and S. Bommann, Cryst. Res. Tech. 18,165 (1983). 'Ip. Novak, Phys. Lett. I04A, 293 (1984). '2T. S. Plaskett, E. Klokholm, D. C. Cronemeyer, P. C. Yin, and S. E. Blum, Appl. Phys. Lett. 25, 357 (1974). '3J. F. Dillon, Jr., J. Phys. Soc. Jpn. 19, 1662 (1964). s4S._K. Chung and M. W. Muller, 1. Mag. Magn. Mater. I, 114 (1975). Frateilo et al. 2497 Downloaded 26 Mar 2013 to 142.51.1.212. 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
5.0010295.pdf	J. Chem. Phys. 152, 244118 (2020); https://doi.org/10.1063/5.0010295 152, 244118 © 2020 Author(s).A simple molecular orbital picture of RIXS distilled from many-body damped response theory Cite as: J. Chem. Phys. 152, 244118 (2020); https://doi.org/10.1063/5.0010295 Submitted: 09 April 2020 . Accepted: 17 May 2020 . Published Online: 25 June 2020 Kaushik D. Nanda , and Anna I. Krylov ARTICLES YOU MAY BE INTERESTED IN Modern quantum chemistry with [Open]Molcas The Journal of Chemical Physics 152, 214117 (2020); https://doi.org/10.1063/5.0004835 Reduced scaling extended multi-state CASPT2 (XMS-CASPT2) using supporting subspaces and tensor hyper-contraction The Journal of Chemical Physics 152, 234113 (2020); https://doi.org/10.1063/5.0007417 The ORCA quantum chemistry program package The Journal of Chemical Physics 152, 224108 (2020); https://doi.org/10.1063/5.0004608The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp A simple molecular orbital picture of RIXS distilled from many-body damped response theory Cite as: J. Chem. Phys. 152, 244118 (2020); doi: 10.1063/5.0010295 Submitted: 9 April 2020 •Accepted: 17 May 2020 • Published Online: 25 June 2020 Kaushik D. Nandaa) and Anna I. Krylova) AFFILIATIONS Department of Chemistry, University of Southern California, Los Angeles, California 90089-0482, USA a)Author to whom correspondence should be addressed: kaushikdnanda@gmail.com and krylov@usc.edu ABSTRACT Ab initio calculations of resonant inelastic x-ray scattering (RIXS) often rely on damped response theory, which prevents the divergence of response solutions in the resonant regime. Within the damped response theory formalism, RIXS moments are expressed as the sum over all electronic states of the system [sum-over-states (SOS) expressions]. By invoking resonance arguments, this expression can be reduced to a few terms, an approximation commonly exploited for the interpretation of computed cross sections. We present an alter- native approach: a rigorous formalism for deriving a simple molecular orbital picture of the RIXS process from many-body calculations using the damped response theory. In practical implementations, the SOS expressions of RIXS moments are recast in terms of matrix ele- ments between the zero-order wave functions and first-order frequency-dependent response wave functions of the initial and final states such that the RIXS moments can be evaluated using complex response one-particle transition density matrices (1PTDMs). Visualization of these 1PTDMs connects the RIXS process with the changes in electronic density. We demonstrate that the real and imaginary compo- nents of the response 1PTDMs can be interpreted as contributions of the undamped off-resonance and damped near-resonance SOS terms, respectively. By analyzing these 1PTDMs in terms of natural transition orbitals, we derive a rigorous, black-box mapping of the RIXS pro- cess into a molecular orbital picture. We illustrate the utility of the new tool by analyzing RIXS transitions in the OH radical, benzene, para -nitroaniline, and 4-amino-4′-nitrostilbene. These examples highlight the significance of both the near-resonance and off-resonance channels. Published under license by AIP Publishing. https://doi.org/10.1063/5.0010295 .,s I. INTRODUCTION Resonant inelastic x-ray scattering1–5(RIXS) is a two-photon process, wherein a resonant x-ray photon is absorbed and another x-ray photon of lower energy is emitted; thus, it can be described as the resonant Raman scattering of an x-ray photon. RIXS is a coherent process, which involves photoexcitation of a core elec- tron to a virtual core-excited state and simultaneous filling of this core hole by radiative decay of a valence electron, as shown in Fig. 1. Thus, the overall transition is from the ground state to a valence excited state, and the difference in the photon energies equals the energy gap between the initial and the final states of the system.For pedagogical simplicity, the RIXS process is often described as a two-step process comprising x-ray absorption and x-ray emis- sion. Owing to its two-photon nature, RIXS transitions obey differ- ent selection rules than one-photon transitions. Thus, RIXS provides information complementary to that delivered by x-ray absorption and emission spectroscopies (XAS and XES, respectively). Similar to XAS and XES, RIXS exploits the elemental specificity of x rays, the compact nature of the core orbitals, and the strong sensitiv- ity to the local environment and bonding pattern around a specific atom as well as its oxidation state.6RIXS has been used exten- sively for probing charge-transfer and crystal-field transitions in metal oxides.4,5With the advent of high-brilliance radiation sources, RIXS is now also used to study excited-state nuclear dynamics7 J. Chem. Phys. 152, 244118 (2020); doi: 10.1063/5.0010295 152, 244118-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1 . Schematic representation of the RIXS process. (a) Molecular orbital picture: one electron is excited to an unoccupied orbital, leaving behind a hole, and another electron is coherently de-excited, filling the hole. (b) Many- body states picture: the dashed line depicts the virtual intermediate state, often assumed to be a core-excited state resonant with the incoming photon’s energy. and to detect transient species in ultrafast reactions in complex environments.8 Theoretical modeling of RIXS spectra is critical for connect- ing the measured RIXS spectra with the electronic structure of the molecule. Ultimately, the mechanistic interpretation of a spectrum hinges on our ability to describe the underlying process in terms of transitions between molecular orbitals. The orbital picture of one-photon UV–visible (UV–vis), XAS, and XES transitions can be extracted from many-body wave func- tions by using reduced quantities, such as one-particle transi- tion density matrices (1PTDMs) and natural transition orbitals (NTOs).9–161PTDMs contain essential information about electronic transitions needed to compute the observables (e.g., cross sections). NTOs, computed by the singular value decomposition (SVD) of 1PTDMs, provide the most compact representation of the transition in terms of the hole and particle states. In contrast to wave-function amplitudes, NTOs are invariant with respect to all allowed orbital rotations, which makes them insensitive to the basis-set choice. Since 1PTDMs and NTOs are directly mapped to the experimen- tal observables, they provide a rigorous and robust framework for wave-function analysis. The concept of NTOs has been generalized to two-photon absorption (2PA) transitions, enabling the analysis of the corre- sponding response 1PTDMs and characteristic 2PA virtual states;17 to non-Hermitian quantum mechanics, enabling the analysis of the complex-valued 1PTDMs and transitions involving states in the continuum;18and to spinless 1PTDMs, enabling the analysis of ten- sorial properties (spin–orbit couplings) and spin-forbidden transi- tions.19Here, we extend the concept of NTOs to RIXS transition moments. The main challenge in interpreting the RIXS process in terms of molecular orbitals stems from its nonlinear (two-photon) nature. Because of it, the scattering moments are given by cumbersomesum-over-states (SOS) expressions20–22and not by matrix elements between the initial and final states, as in the case of UV–vis, XAS, and XES transitions. This dependence on all electronic states of the system makes the analysis of RIXS moments more difficult than the analysis of one-photon moments (matrix elements of the dipole operator between the initial and final states). Furthermore, the RIXS moments are complex-valued and tensors of rank two (3×3 matrices), in contrast to the one-dimensional one-photon moments, which are real-valued vectors with components along the three Cartesian coordinates. Because of its resonant nature, the qualitative picture of the RIXS transition is traditionally derived using approximate few-state models—in particular, a three-state model—involving few near- resonant core-excited states along with the initial and final states. In a few-state model, the orbitals involved in the transitions from the initial state to different intermediate states and from these inter- mediate states to the final state are computed and stitched together to construct the orbital picture of the RIXS process. For example, in the three-state model, the virtual state (see Fig. 1) of the two-photon RIXS process corresponds to the core-excited state for which the XAS peak is resonant with the incoming photon’s energy. Although being physically justified, such an approach involves arbitrariness and is prone to potential loss of accuracy, because it is not always easy to identify the important intermediate states that need to be included in these few-state models. In this approach, the orbital character of the virtual state of the RIXS process is determined by the (somewhat arbitrary) choice of the intermediate states picked in the few-state model. The loss of accuracy can occur when off- resonance channels make non-negligible contributions to the RIXS cross sections. Here, we overcome these challenges using a novel approach of deriving the mechanistic details of the RIXS transitions by means of NTOs computed directly from the complex-valued damped response 1PTDMs that enter the expressions of RIXS moments. This leads to a rigorous and black-box procedure of mapping the computed scattering moments into molecular orbitals. In contrast to traditional approaches, our scheme does not invoke arbitrary truncation of the SOS expressions and is orbital invariant. We dis- cuss the meaning of the real and the imaginary components of these 1PTDMs and the corresponding NTOs by analyzing RIXS transitions in the OH radical, benzene, para -nitroaniline (pNA), and 4-amino-4′-nitrostilbene (4A4NS). The pNA and 4A4NS exam- ples illustrate the importance of off-resonance RIXS channels and highlight the advantages of fully analytic calculation and analysis of the RIXS moments over approximate treatments by few-state models. We also illustrate how a quantitative metric for the extent of delocalization of electronic density during the RIXS transition can be computed using these response 1PTDMs. While this approach builds upon our prior work on 2PA tran- sitions,17the novelty lies in the interpretation of complex-valued RIXS 1PTDMs (and their NTOs) instead of the real-valued 2PA 1PTDMs. II. THEORY The RIXS scattering moments are given by the Kramers– Heisenberg–Dirac formula as SOS expressions,20–23 J. Chem. Phys. 152, 244118 (2020); doi: 10.1063/5.0010295 152, 244118-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Mxy g→f(ω1x,−ω2y,ϵ)=−∑ n(⟨Ψf∣μy∣Ψn⟩⟨Ψn∣μx∣Ψg⟩ Ωng−ω1x−iϵn +⟨Ψf∣μx∣Ψn⟩⟨Ψn∣μy∣Ψg⟩ Ωng+ω2y+iϵn) (1) and Mxy f→g(−ω1x,ω2y,−ϵ)=−∑ n(⟨Ψg∣μx∣Ψn⟩⟨Ψn∣μy∣Ψf⟩ Ωng−ω1x+iϵn +⟨Ψg∣μy∣Ψn⟩⟨Ψn∣μx∣Ψf⟩ Ωng+ω2y−iϵn), (2) where Ω ng=En−Egandω1xandω2yare the absorbed and emit- ted frequencies (polarized along x- and y-directions), respectively, satisfying the RIXS resonance condition, ω1−ω2=Ωfg. (3) Note that in Hermitian theories, the two scattering moments are complex conjugates of each other, i.e., Mxy f→g(−ω1x,ω2y,−ϵ)=(Mxy g→f(ω1x,−ω2y,ϵ))∗ , (4) where∗denotes complex conjugation. However, for coupled- cluster methods, this is not the case.21–23ϵnis the inverse life- time parameter for state n. If the lifetimes of all states are infi- nite (ϵn= 0∀n), in the case of RIXS, at least for one state k, the denominator in the SOS term is zero (Ω kg−ω1= 0). In other words, the RIXS moments have first-order poles at Ω ngs. In practice, this means that attempts to calculate RIXS moments assuming infinite lifetimes of excited states encounter divergent solutions. Most theoretical formulations for calculating RIXS moments use empirical non-zero inverse lifetimes for all states, which are assumed to have the same nonzero value ϵ.20–26The introduction of this imaginary phenomenological (damping) parameter iϵbrings the poles due to resonances into the complex plane. The impact of introducing iϵon individual SOS terms depends on whether \|Ωng−ω1\| is less than or greater than \| ϵ\|, as shown in Fig. 2. The contribution of the SOS terms that have \|Ω ng−ω1\|<\|ϵ\| (nearly resonant SOS terms) is dominated by their imaginary components. The real components of these terms are smaller than the imaginary components. In particular, for the SOS terms with \|Ω ng−ω1\| = 0, the real components are zero and the imaginary components equal ⟨Ψf∣μy∣Ψn⟩⟨Ψn∣μx∣Ψg⟩ ϵ(or⟨Ψg∣μx∣Ψn⟩⟨Ψn∣μy∣Ψf⟩ ϵ); thus, the absolute contribution of each of these SOS terms is effectively damped from infinity to a finite value by virtue of iϵ. On the other hand, the real compo- nents are larger than the imaginary ones for the SOS terms for which \|Ωng−ω1\|>\|ϵ\|. In short, the damping puts the near-resonance and off-resonance contributions in the imaginary and real components of the RIXS moments, respectively. Computing the full set of electronic states for calculating the RIXS moments via Eqs. (1) and (2) is obviously impractical. Often, an approximated (truncated) SOS can provide a qualitatively cor- rect value of the RIXS moment; however, the error introduced due to truncation is difficult to evaluate a priori . Nevertheless, many studies employ such truncations for computing the RIXS FIG. 2 . Impact of the imaginary phenomenological damping ( iϵ) on the RIXS moments. Individual SOS terms are given by⟨Ψg∣μ∣Ψn⟩⟨Ψn∣μ∣Ψf⟩ Ωng−ω−iϵ =μgnμnfUϵ(Ωng−ω), where Uϵ(Ωng−ω)is related to the Green’s function: G+(ω)=limϵ→0+[H−ω−iϵ]−1=limϵ→0+∑n∣Ψn⟩Uϵ(Ωng−ω)⟨Ψn∣. The poles of Uare also poles of G+.Ucan be written as Uϵ(Ω−ω)=(1 Ω−ω−iϵ). The black hyperbola represents Uwithout the imaginary damping, i.e., U0(Ω−ω)=(1 Ω−ω). When Ω−ω= 0, this function has a pole and is indeter- minate. The red and blue curves represent the imaginary and real components of damped U,ImUϵ(Ω−ω)=ϵ (Ω−ω)2+ϵ2andReUϵ(Ω−ω)=Ω−ω (Ω−ω)2+ϵ2, respectively. When Ω−ω= 0,ImUϵ=1 ϵandReUϵ=0. Effectively, the imaginary damping brings the damped contribution of U0predominantly into the imaginary component of UϵforΩ−ω<ϵ(near resonance). For Ω−ω>ϵ(off resonance), the damping has a smaller impact such that U0andReUϵhave a similar magnitude for large Ω−ω. moments and other nonlinear properties. An alternative, more rig- orous strategy involves recasting the SOS expressions into a closed form using damped response theory.20–23,27–32By doing so, one circumvents the need to compute the wave functions and ener- gies of all electronic states. Instead, only a handful of response wave functions need to be computed.27The RIXS moments com- puted with the damped response theory approach are formally and numerically equivalent to the full SOS result. This strategy for RIXS calculations has been exploited in the analytic implemen- tations based on algebraic diagrammatic construction,20coupled- cluster,21,23and equation-of-motion coupled-cluster22methods. Importantly, the damped response theory approach can be for- mulated in terms of response transition density matrices, which can be exploited to obtain a concise description of the RIXS transition. Within the damped response theory framework with ϵn=ϵ∀n, Eqs. (1) and (2) are rewritten using Eq. (3) as Mxy g→f(ω1x,ϵ)=−∑ n(⟨Ψf∣μy∣Ψn⟩⟨Ψn∣μx∣Ψg⟩ Ωng−ω1x−iϵ +⟨Ψf∣μx∣Ψn⟩⟨Ψn∣μy∣Ψg⟩ Ωnf+ω1x+iϵ) =−(⟨Ψf∣μy∣Xϵ,ω1xg⟩+⟨˜Xϵ,ω1x f∣μy∣Ψg⟩) (5) J. Chem. Phys. 152, 244118 (2020); doi: 10.1063/5.0010295 152, 244118-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp and Mxy f→g(ω2y,−ϵ)=−∑ n(⟨Ψg∣μx∣Ψn⟩⟨Ψn∣μy∣Ψf⟩ Ωnf−ω2y+iϵ +⟨Ψg∣μy∣Ψn⟩⟨Ψn∣μx∣Ψf⟩ Ωng+ω2y−iϵ) =−(⟨Ψg∣μx∣X−ϵ,ω2y f⟩+⟨˜X−ϵ,ω2y g∣μx∣Ψf⟩), (6) respectively, where ∣Xϵ,ωx k⟩and⟨˜Xϵ,ωx k∣are the complex right and left first-order response wave functions of state kdue to the perturbing electric field of frequency ω1xpolarized along the xdirection. These response wave functions depend parametrically on ϵand are given, according to first-order perturbation theory, as17,22,29 ∣Xϵ,ω1x k⟩=∑ n∣Ψn⟩⟨Ψn∣μx∣Ψk⟩ Ωnk−ω1x−iϵ(7) and ⟨˜Xϵ,ω1x k∣=∑ n⟨Ψn∣μx∣Ψk⟩ Ωnk+ω1x+iϵ⟨Ψn∣. (8) These first-order many-body response functions are computed iter- atively by solving the following response equations: (H−Ek−ω1x−iϵ)Xϵ,ω1x k=⟨Φν∣μx∣Ψk⟩ (9) and ˜Xϵ,ω1x k(H−Ek+ω1x+iϵ)=⟨Ψk∣μx∣Φν⟩, (10) where { Φν} is the set of ν-tuply excited Slater determinants from the target-state manifold; e.g., in EOM-CCSD damped response the- ory, {Φν} spans the reference, singly excited, and doubly excited determinants. We recast Eqs. (5) and (6) as Mxy g→f(ω1x, +ϵ)=∑ pqγϵ,x pqμy pq (11) and Mxy f→g(ω2y,−ϵ)=∑ pq˜γ−ϵ,y pqμx pq, (12) respectively, where γxand ˜γyare the complex response reduced 1PTDMs given by γϵ,x pq≡γ+ϵ,ω1xpq=−(⟨Ψf∣ˆp†ˆq∣Xϵ,ω1xg⟩+⟨˜Xϵ,ω1x f∣ˆp†ˆq∣Ψg⟩) (13) and ˜γ−ϵ,y pq≡˜γ−ϵ,ω2y pq=−(⟨Ψg∣ˆp†ˆq∣X−ϵ,ω2y f⟩+⟨˜X−ϵ,ω2y g∣ˆp†ˆq∣Ψf⟩), (14) where ˆp†and ˆqare the creation and annihilation operators in molec- ular orbitals ϕpandϕq, respectively. Following our previous work,17 we useωDM to denote the individual components of 1PTDMs on the RHS of Eqs. (13) and (14) between a frequency-dependent response state and a zero-order state. Thus, γϵ,xis the sum of a ωDM between the final state and a response ground state and another ωDM between a response final state and the initial state. For one-photon transitions, the reduced 1PTDM can be inter- preted as the exciton’s wave function according to Ψexc(rh,re)=∑ pqγpqϕp(re)ϕq(rh), (15)where rhandreare the hole and electron (particle) coordinates12,14,33 [in terms of ˜γ,Ψexc(rh,re)=∑pq˜γpqϕp(rh)ϕq(re)]. For the second-order (two-photon) RIXS process, the exci- ton’s wave function (as well as response 1PTDMs) has polarized components along the three Cartesian components ( ˆx,ˆy, and ˆz), Ψϵ,x exc(rh,re)=∑ pqγϵ,x pqϕp(re)ϕq(rh) (16) and Ψϵ exc(rh,re)=Ψϵ,x exc(rh,re)ˆx+Ψϵ,y exc(rh,re)ˆy+Ψϵ,z exc(rh,re)ˆz. (17) In spatial representation, the exciton’s wave function provides a visual map of how the electronic distribution changes upon the tran- sition.12,14,34,35It can also be used to compute various physical quan- tities, such as the correlation between the hole and electron and their average separation,12,15 dexc=√ ⟨Ψexc(rh,re)∣(rh−re)2∣Ψexc(rh,re) ⟩ =⌟roo⟪⟪op ⌟roo⟪mo⟨⌟roo⟪mo⟨⌟roo⟪mo⟨⌟roo⟪⟨o⟪∥γϵ,x∥2(dϵ,x exc)2+∥γϵ,y∥2(dϵ,y exc)2+∥γϵ,z∥2(dϵ,z exc)2 ∥γϵ,x∥2+∥γϵ,y∥2+∥γϵ,z∥2, (18) where dϵ,x exc=√ ⟨Ψϵ,x exc(rh,re)∣(rh−re)2∣Ψϵ,x exc(rh,re) ⟩ =⌟roo⟪⟪op ⌟roo⟪mo⟨⌟roo⟪mo⟨⌟roo⟪mo⟨⌟roo⟪⟨o⟪∥γϵ,x,Re∥2(dϵ,x,Re exc)2+∥γϵ,x,Im∥2(dϵ,x,Im exc)2 ∥γϵ,x,Re∥2+∥γϵ,x,Im∥2, (19) dϵ,x,Re/Im exc=√ ⟨Ψϵ,x,Re/Im exc (rh,re)∣(rh−re)2∣Ψϵ,x,Re/Im exc (rh,re) ⟩, (20) and ∥γϵ,x∥2=∥γϵ,x,Re∥2+∥γϵ,x,Im∥2. (21) These exciton descriptors facilitate the assignment of the transitions in terms of valence, Rydberg, or charge-transfer character.12,15,16 The description of exciton’s wave function for a one-photon transition is the most concise in terms of NTOs, which are com- puted by means of unitary orbital transformations.12,13,16,17This is achieved by the singular value decomposition (SVD) of the 1PTDM as follows: γ=VΣUT, (22) where Σis the diagonal matrix of singular values, σKs, and matrices VandUcontain the hole and particle NTOs according to ψe K(r)=∑ qVqKϕq(r) (23) and ψh K(r)=∑ qUqKϕq(r). (24) The squares of σKs can be interpreted as the weights of the respective NTO pair when divided by the square of the Frobenius norm ofγ, ∥γ∥2≡∑ pqγ2 pq=∑ kσ2 K. (25) J. Chem. Phys. 152, 244118 (2020); doi: 10.1063/5.0010295 152, 244118-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp In the analyses below, we report such normalized singular values, σ′ K=σK ∥γ∥, (26) which are equivalent to using normalized 1PTDMs in the wave- function analysis. The SVD procedure removes the arbitrariness associated with the orbital choice. Since only a handful of σKs are non-negligible, the NTO representation enables the most compact molecular orbital representation of any transition, including transitions between multi-configurational correlated wave functions. In terms of NTOs, the exciton’s wave function for a one-photon transition is given by Ψexc(rh,re)=∑ KσKψe K(re)ψh K(rh). (27) In contrast to the real-valued exciton’s wave functions and 1PTDMs for one-photon and two-photon absorption pro- cesses,12,13,16,17the exciton’s wave function and response 1PTDMs for the RIXS process are complex because the response wave func- tions become complex within the damped response theory formal- ism. Below, we explain how to interpret these complex 1TPDMs. Rewriting the response 1PTDM in Eq. (16), we obtain Ψϵ,x exc(rh,re)=∑ pqγϵ,x,Re pqϕp(re)ϕq(rh) +i∑ pqγϵ,x,Im pqϕp(re)ϕq(rh). (28) Since the real and imaginary components of the RIXS scattering moments accumulate the off-resonance and near-resonance SOS terms, respectively, the corresponding real and imaginary response 1PTDMs provide the cumulative orbital information of these off- resonance and near-resonance terms. We reformulate Eq. (28) to Ψϵ,x exc(rh,re)=∑ Kσϵ,x,Re Kψϵ,x,Re K(re)ψϵ,x,Re K(rh) +i∑ Lσϵ,x,Im Lψϵ,x,Im L(re)ψϵ,x,Im L(rh) (29) by performing SVD on the real and imaginary response 1PTDMs separately, so that two sets of real NTOs [ {ψRe K(re),ψRe K(rh)}and {ψIm L(re),ψIm L(rh)}] are obtained and used for visualization and interpretation. The relative significance of the off-resonance and near- resonance terms for a given RIXS moment can be estimated from the norms of the real and imaginary response 1PTDMs, ∥γϵ,x,Re∥and ∥γϵ,x,Im∥. For example, if υx,Im=∥γϵ,x,Im∥2 ∥γϵ,x∥2≈1 orυx,Re=∥γϵ,x,Re∥2 ∥γϵ,x∥2≈0, the corresponding xcomponent of the exciton’s wave function can be approximated by just the imaginary near-resonance con- tributions. Similarly, the relative significance of the off-resonance and near-resonance terms for the overall RIXS transition can be estimated using the norms of response 1PTDMs along the three Cartesian coordinates. For example, if ΥIm=Υx,Im+Υy,Im+Υz,Im ≈1 orΥRe=Υx,Re+Υy,Re+Υz,Re≈0, where Υx,Im=∥γϵ,x,Im∥2 ∑x,y,z∥γϵ,x∥2 andΥx,Re=∥γϵ,x,Re∥2 ∑x,y,z∥γϵ,x∥2, then the RIXS transition has predominant contributions from near-resonance channels. In the discussion that follows, we drop the index ϵinγϵ,x,Reandγϵ,x,Imfor brevity.III. COMPUTATIONAL DETAILS Using the existing infrastructure of the libwfa library15for wave-function analysis,12,13,16we implemented the calculations of the NTOs for RIXS 1PTDMs in the Q-Chem package.36,37Below, we illustrate the utility of this orbital analysis for the RIXS transi- tions in the OH radical, benzene, para -nitroaniline, and 4-amino-4′- nitrostilbene. In all calculations, we employ the recently developed implementation22of RIXS calculations within the fc-CVS-EOM-EE- CCSD framework.38For the OH radical, benzene, and pNA, we use the 6-311(2+,+)G∗∗basis set with the uncontracted core (uC) func- tions.39For the OH radical, we use the experimental bond length of 0.9697 Å. We use the geometries from Refs. 17 and 22 for benzene and pNA, respectively. For 4A4NS, we use the B3LYP/6-311G∗∗ optimized geometry and the 6-31+G∗basis set with the uncon- tracted core functions39for XAS, XES, and RIXS calculations. The relevant Cartesian coordinates are provided in the supplementary material. The phenomenological damping parameter ϵwas set to 0.005 (OH), 0.01 (benzene and pNA), and 0.03 (4A4NS). We use Q-Chem’s symmetry notations throughout this paper (more details can be found in Refs. 17 and 22 and at http://iopenshell.usc.edu/resources/howto/symmetry/). We use the C2vsymmetry group for OH and pNA, D2hsymmetry group for benzene, and Cssymmetry group for 4A4NS. The NTOs and canon- ical MOs were visualized with Gabedit40and IQMol,41respectively. IV. RESULTS AND DISCUSSION A. Wave-function analysis of RIXS transition in OH In the OH radical, the πyorbital is singly occupied (see Fig. 3), so the lowest resonant x-ray absorption involves the excitation of the 1sOelectron to fill the valence πhole.42This is shown by the NTO analysis of the XB 2→c1A 1transition presented in Fig. 4 and the supplementary material. Here and below, the prefix “c” denotes a core-excited state or a core orbital. The lowest valence excited state is the 1A 1state at 4.15 eV. Within the three-state model, the XB 2→1A1RIXS transition, with the incoming photon’s energy tuned at the XB 2→c1A 1resonance, entails x-ray absorption from the ground state to the c1A 1state and FIG. 3 . Molecular orbitals and ground-state electronic configuration of the OH radical. J. Chem. Phys. 152, 244118 (2020); doi: 10.1063/5.0010295 152, 244118-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 4 . NTO analysis of the XB 2→1A1RIXS transition in the OH radical. Com- parison of important NTO pairs computed separately for the x-ray absorption (XB 2→c1A 1) and x-ray emission (c1A 1→1A1) transitions with the NTO pairs computed from the RIXS 1PTDMs show that the three-state model is appropriate for this transition. x-ray emission from the c1A 1state to the final valence 1A 1state. This transition is the dominant inelastic feature in the RIXS spec- trum of aqueous OH.8The NTO analysis for the c1A 1→1A1emis- sion, which is shown in Fig. 4, indicates σz→1scharacter. Based on this three-state model, one can now identify the σz→1s→πy orbital channel as the dominant pathway in the XB 2→1A1RIXS transition. Let us now compare this approximate analysis based on the three-state model with the NTO analysis of the RIXS 1PTDMs. In the analytic RIXS calculations, we find that the imaginary Myz components—and thus the near-resonance orbital channels—are dominant for the XB 2→1A1RIXS transition and, therefore, have the dominant contribution to the cross section. The RIXS 1PTDMs corresponding to the imaginary Myz g→fandMyz f→gcomponents are the γy,Imand ˜γz,Im1PTDMs, respectively. The detailed NTO analyses of these RIXS 1PTDMs are given in the supplementary material. By inspecting the norms of the imaginary RIXS 1PTDMs and of the respective ωDMs, we note that only the first terms in Eqs. (13) and (14) provide significant contributions. Thus, the NTOs of the full imaginary RIXS 1PTDMs can be explained by interpreting these imaginaryωDMs. The first imaginary ωDM in Eq. (13) reflects the transition from the “virtual” Xgstate (first-order response ground state) to the final state, and so the NTO pairs correspond to the transition that fills the core hole (emission). Complementary to this orbital transition, the NTO pairs from the first imaginary ωDM in Eq. (14) reflect the transition from the Xf“virtual” state (first-order perturbed final state) to the initial state, i.e., the reverse of core-hole formation in the absorption step. By joining these two sets of NTOs together, the orbital picture of the RIXS transition is constructed. For the XB 2→1A1RIXS transition, each set consists of one dominant NTO pair (see the supplementary material). The analysis of γy,Im identifies the σzhole and 1 sparticle NTOs; the analysis of ˜γz,Imiden- tifies theπyhole and 1 sparticle NTOs. Using the norms of response1PTDMs, we obtain ΥIm= 1.00 for this transition. Thus, the dom- inant RIXS channel is resonant and given by σz→1s→πywith its 1s→πyexcitation and σz→1sde-excitation components. This is consistent with the three-state model described above, indicating that for this RIXS transition, the three-state model provides a good approximation to the full SOS expression. B. Wave-function analysis of RIXS transitions in benzene Figure 5 shows the occupied molecular orbitals of benzene. The six 1sCcore orbitals form six nearly degenerate delocalized molecu- lar orbitals. Depending on the symmetry of the target orbital, dif- ferent core orbitals are active in the XAS transitions.22The relevant virtual molecular orbitals (not shown) are doubly degenerate π∗ LUMO and diffuse sandpRydberg orbitals. The two dominant features in the XAS spectrum of ben- zene43–50are peak A and peak B at 285.97 eV and 287.80 eV, FIG. 5 . Molecular orbitals and ground-state electronic configuration of benzene. J. Chem. Phys. 152, 244118 (2020); doi: 10.1063/5.0010295 152, 244118-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp respectively [theoretical values22computed with fc-CVS-EOM-EE- CCSD/uC-6-311(2+,+)G∗∗]. When the incoming photon’s energy is tuned to the peak-A resonance, the dominant inelastic feature is the energy-loss peak at 10.67 eV, characterized by equal contributions from the degenerate XA g→13B 2gand XA g→12B 3gtransitions. In contrast, when the incoming photon’s energy is tuned to the peak- B resonance, the dominant inelastic feature is the energy-loss peak at 6.45 eV, characterized by equal contributions from the degener- ate XA g→1B2gand XA g→1B3gtransitions. Below, we show the NTO analysis of only the XA g→13B 2gand XA g→1B2gRIXS tran- sitions with incoming photon energies tuned at peak-A and peak-B resonances, respectively. The NTO analyses for the two transitions are similar, except for the differences in symmetry labels of the orbitals. The NTO analysis for the dark one-photon XA g→13B 2gtransi- tion (given in the supplementary material) suggests that this valence transition is made up of two orbital transitions: b2u→auandb3u→b1u. Similarly, the NTO analysis of XAS peak A (XA g→c2B 1u) transition in Fig. 6 shows two dominant orbital transitions: cb1g →auand cag→b1u. Similarly, the NTO analysis of the c2B 1u →13B 2gx-ray emission shows two dominant orbital transitions: b2u →cb1gandb3u→cag. Based on these analyses, the three-state model for the XA g→13B 2gRIXS transition identifies two important orbital channels: b2u→cb1g→auandb3u→cag→b1u. Theg→fandf→gRIXS moment tensors are dominated by the imaginary zxcomponents. This is also reflected in the norms of the imaginary 1PTDMs given in the supplementary material, which are a few orders of magnitude larger than those of real 1PTDMs ( ΥIm= 1.00). The NTO analyses of the γz,Imand ˜γx,Im RIXS 1PTDMs of the XA g→13B 2gshown in Fig. 6 and in the supplementary material identify two dominant near-resonance orbital channels: b2u→cb1g→auandb3u→cag→b1u. In other words, cb1gandcagare the intermediate core orbitals that facilitate the two-photon inelastic scattering, driving the electronic density FIG. 6 . NTO analysis of the XA g→13B 2gRIXS transition in benzene. Comparison of important NTO pairs computed separately for the x-ray absorption (XA g→c2B 1u) and x-ray emission (c2B 1u→13B 2g) transitions with the NTO pairs computed from the RIXS 1PTDMs shows that the three-state model is adequate for this RIXS transition. Both orbital channels contribute significantly into this transition. FIG. 7 . NTO analysis of the 1A g→1B2gRIXS transition in benzene. Comparison of important NTO pairs computed separately for the x-ray absorption (1A g→c2B 3u) and x-ray emission (c2B 3u→1B2g) transitions with the NTO pairs computed from the RIXS 1PTDMs shows that the three-state model is adequate for this RIXS transition. Orbital channel 1 provides dominant contributions. J. Chem. Phys. 152, 244118 (2020); doi: 10.1063/5.0010295 152, 244118-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp from the b2uorbital to the 2 auorbital and from the b3uorbital to the 4b1uorbital, respectively. This orbital analysis of the RIXS 1PTDMs is consistent with the approximate analysis from the three-state model. The one-photon XA g→1B2gtransition is dark; its NTO analysis given in the supplementary material reveals its dominant HOMO–LUMO character ( b2g→ag) and a miniscule contribution from the b1u→b3utransition. The NTO analysis of XAS peak B transition (XA g→c2B 3u) shown in the supplementary material and Fig. 7 indicates that this core excitation has predominantly cb3u→ag character, with a small contribution from the cag→b3utransition. The NTO analysis of the c2B 3u→1B2gx-ray emission transition has a predominantly b2g→cb3ucharacter with a small contribution from the b1u→cagtransition. Thus, within the three-state model, the NTO analyses of the XA g→c2B 3ux-ray absorption and c2B 3u →1B2gx-ray emission identify the orbital character as b2g→cb3u→ ag, with a small contribution from the b1u→cag→b3uchannel. The RIXS moment tensor for the XA g→1B2gtransition is dominated by the imaginary component of Mxzmoments. Here, we perform NTO analyses of the γx,Imand ˜γz,ImRIXS 1PTDMs cor- responding to the Mxz g→fandMxz f→gcomponents, respectively. The analysis of γx,Imidentifies the b2g→cb3uNTO pair as dominant, with a miniscule contribution from the b1u→cagNTO pair. The analysis of ˜γz,Imidentifies the dominant ag→cb3uNTO pair and a less important b3u→cagNTO pair. Combining these two analyses, the dominant orbital channel is b2g→cb3u→ag. We obtain ΥIm = 0.98 for this RIXS transition, which is consistent with the analysis from the three-state model discussed above. Similarly, the dominant RIXS channel is resonant and given by b3g→cb2u→agfor the XA g →1B3gtransition. C. Wave-function analysis of RIXS transitions in para -nitroaniline The orbital analysis of the selected RIXS transitions in the OH radical and benzene supports the notion that the dominant orbital channel in RIXS is (nearly) resonant and that three-state models are sufficient for determining the important orbitals involved in the RIXS transition. In this section, we present a counterexample illus- trating the limitations of few-core-excited-states models. We con- sider RIXS transitions in para -nitroaniline (pNA) and show that for this system, the predominant channel driving the electronic den- sity in the course of inelastic scattering may or may not be (nearly) resonant in character. Figure 8 shows the occupied molecular orbitals of pNA. The special feature of this molecule is that the lowest excited state has strong intramolecular charge-transfer character.51–54This state cor- responds to the lowest fully symmetric XA 1→2A1transition with a large oscillator strength ( f= 0.4). The NTO analysis (provided in the supplementary material) shows that this transition can be described as HOMO–LUMO excitation: π(b2)→π∗(b2). The strong charge-transfer character of this transition ( Δμ = 3.2 a.u.) has non-trivial consequences on the character of the 2PA transition, as discussed in Ref. 17. Specifically, we have shown that the 2PA moments for the XA 1→2A1transition in pNA can be described by the two-state model involving just the initial and final states, in contrast to other 2PA examples involving specific FIG. 8 . Molecular orbitals and ground-state electronic configuration of para- nitroaniline. J. Chem. Phys. 152, 244118 (2020); doi: 10.1063/5.0010295 152, 244118-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp virtual states giving dominant contributions to the cross sections. Thus, for this transition, the 2PA transition moments are given according to Mxy g→f≈−∑ n=g,f(⟨Ψf∣μy∣Ψn⟩⟨Ψn∣μx∣Ψg⟩ Ωng−ω1x−iϵ+⟨Ψf∣μx∣Ψn⟩⟨Ψn∣μy∣Ψg⟩ Ωng−ω2y−iϵ) ≈−⟨Ψf∣μy∣Ψg⟩⟨Ψf∣μx∣Ψf⟩−⟨Ψg∣μx∣Ψg⟩ ω1x+iϵ −⟨Ψf∣μx∣Ψg⟩⟨Ψf∣μy∣Ψf⟩−⟨Ψg∣μy∣Ψg⟩ ω2y+iϵ(30)and Mxy f→g≈−∑ n=g,f(⟨Ψg∣μx∣Ψn⟩⟨Ψn∣μy∣Ψf⟩ Ωng−ω1x−iϵ+⟨Ψg∣μy∣Ψn⟩⟨Ψn∣μx∣Ψf⟩ Ωng−ω2y−iϵ) ≈−⟨Ψg∣μy∣Ψf⟩⟨Ψf∣μx∣Ψf⟩−⟨Ψg∣μx∣Ψg⟩ ω1x+iϵ −⟨Ψg∣μx∣Ψf⟩⟨Ψf∣μy∣Ψf⟩−⟨Ψg∣μy∣Ψg⟩ ω2y+iϵ. (31) Similarly to the one-photon transition, this 2PA transition also has intramolecular charge-transfer character; its large2PA moments FIG. 9 . NTO analysis of the XA 1→2A1RIXS transition in para-nitroaniline using the three-state model involving the core-excited (top panel) c6B 2state, (middle panel) c1A 1 state, and (bottom panel) c1B 1state. J. Chem. Phys. 152, 244118 (2020); doi: 10.1063/5.0010295 152, 244118-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp result from the large one-photon transition dipole moment and the large difference in the dipole moments between the initial and final states. As discussed in Ref. 17, these two quantities are present in the numerators of Eqs. (30) and (31). On the other hand, it is the pole structure (which comes from the denomina- tors) of Eqs. (1) and (2) that imparts the resonant character to a RIXS transition. Thus, one can potentially identify two-photon RIXS transitions involving the XA 1and 2A 1states in pNA for which both the near-resonance (involving the intermediate core states) and off-resonance (involving the initial and final valence states) orbital channels are important. Below, we provide such an exam- ple by considering the XA 1→2A1RIXS transition in pNA for which the incoming photon frequency is tuned to its XA 1→c6B 2 C-edge resonance at 288.01 eV. We compute the XA 1→2A1RIXS cross section using a modified fc-CVS-EOM-EE-CCSD method in which the SOS includes the CVS states plus the initial and the final states, so that the RHS terms in Eqs. (30) and (31) are also incorporated. The XA 1→c6B 2core excitation is dark due to symme- try, but it is nearly degenerate with the bright (near-)degenerate XA 1→c1A 1and XA 1→c1B 1transitions at 287.96 eV (see the supplementary material). The NTO analysis for XA 1→c6B 2core excitation given in the supplementary material and Fig. 9 reveals two dominant orbital transitions: ca1→b2andcb1→a2. From the NTO analyses of the XA 1→c6B 2x-ray absorption and c6B 2→2A1x- ray emission, the three-state model suggests that the b2→ca1→b2 near-resonance channel should dominate for the γy,Imand ˜γy,Im 1PTDMs. For this RIXS transition, the Mxx,Myy, and MzzRIXS moments are comparable. Myy,Imand Mxx,Imhave larger magnitudes, indi- cating that the near-resonance channels along the yand xaxis have the largest contribution. The NTO analyses of γyand ˜γy RIXS 1PTDMs are given in the supplementary material. The near- resonance mechanism of electronic density transfer in the inelastic scattering obtained from analyzing γy,Imand ˜γy,Imis not what is expected from the three-state model (Fig. 10); the obtained interme- diate core orbitals are a linear combination of the six 1 sCmolecular FIG. 10 . NTO analysis of the real and imaginary γyand ˜γy1PTDMs for the XA1→2A1RIXS transition in para-nitroaniline. Here, the off-resonance chan- nel is dominant and the near-resonance channel is not the one predicted by the three-state model. For both channels, the core-hole NTOs are given as a linear combination of the six 1 sCmolecular orbitals as the large numerators of some of the off-resonance SOS terms make their contribution larger than that of the near-resonant SOS terms even in the imaginary components of these 1PTDMs.orbitals. This reflects that the SOS resonant term with Ω ng=ω1 does not provide the dominant contribution to the RIXS moment. In fact, the damped contributions from other off-resonance terms, which are collected in the γy,Imand ˜γy,Im1PTDMs, contribute more than the near-resonance term primarily due to larger transition dipole moments than the ones forming the near-resonance term (i.e., the transition dipole moments for XA 1→c6B 2and 2A 1→c6B 2 transitions). The fact that the off-resonance terms are dominant is also reflected in the larger norms of the γy,Reand ˜γy,Re1PTDMs than theγy,Imand ˜γy,Im1PTDMs ( υy,Im= 0.03). Furthermore, the rms electron–hole distances ( dy exc) computed for these compo- nents of the RIXS transition ( ≈2.4 Å) are smaller than the 3.6 Å value computed for the one-photon XA 1→2A1transition (see the supplementary material), highlighting the local character of the RIXS transition along the ydirection. Since the XA 1→c6B 2core excitation is nearly degenerate with XA 1→c1A 1and XA 1→c1B 1core excitations, the latter two tran- sitions with larger oscillator strengths open near-resonance orbital channels and impact the imaginary MzzandMxxRIXS moments. As shown in Fig. 9, the ca1→a1and cb1→b1transitions are important in the XA 1→c1A 1core excitation and the cb1→a1and ca1→b1transitions are important in the XA 1→c1B 1core excitation. These orbital transitions also dominate the NTO analysis of ˜γz,Imand ˜γx,Im1PTDMs, consistent with the three-state models constructed with the NTO analyses of these two XAS peaks. Furthermore, the rms electron–hole distances for these 1PTDMs are smaller than the one computed for the one-photon XA 1→2A1transition, indicat- ing that these near-resonance channels are local and confined to the respective active core-hole orbitals. For the MxxRIXS moments, the imaginary components are larger than the respective real components ( υx,Im= 0.71), con- sistent with larger norm for the γx,Imand ˜γx,Im1PTDMs than the respective γx,Reand ˜γx,Re1PTDMs (Fig. 11). On the other hand, the real components are larger than the imaginary com- ponents for the Mzzmoments ( υz,Im= 0.01), consistent with the larger norms for γz,Reand ˜γz,Re1PTDMs than those for γz,Imand ˜γz,Im1PTDMs. The NTO analyses of the γz,Reand ˜γz,Re1PTDMs FIG. 11 . NTO analysis of the real and imaginary γxand ˜γx1PTDMs for the XA1→2A1RIXS transition in para-nitroaniline. Here, the near-resonance chan- nel 1 is dominant. The two near-resonance channels are also consistent with the predictions by the three-state model. J. Chem. Phys. 152, 244118 (2020); doi: 10.1063/5.0010295 152, 244118-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 12 . NTO analysis of the real and imaginary γzand ˜γz1PTDMs for the XA1→2A1RIXS transition in para-nitroaniline. Here, the off-resonance channel is dominant and features the same π→π∗intramolecular charge-transfer chan- nel that characterizes the one-photon XA 1→2A1transition. The near-resonance channel is the one predicted by the three-state model. show that the b2(π)→b2(π∗)transition is the significant off- resonance RIXS channel, indicating that this two-photon process has some intramolecular charge-transfer character (Fig. 12). This is further supported by the larger rms electron–hole distances for theγz,Reand ˜γz,Re1PTDMs than the other RIXS 1PTDMs and comparable to the rms electron–hole distance for the one- photon XA 1→2A1transition. This intramolecular π→π∗charge- transfer channel is, however, not the dominant off-resonance chan- nel in the overall RIXS process as Υz,Re<Υy,Re, even though the overall character of this RIXS transition is not resonant (ΥIm= 0.03). D. Wave-function analysis of RIXS transitions in 4-amino-4′-nitrostilbene Similar to pNA, 4-amino-4′-nitrostilbene (4A4NS) is a push– pull chromophore. Its one-photon XA′→2A′transition has an even larger oscillator strength than the XA 1→2A1transition in pNA with strong intramolecular charge-transfer character ( f= 1.22, Δμx= 5.0 a.u.; see the NTO analysis of this transition in the sup- plementary material). For this molecule, we pick the XA′→2A′ RIXS transition and the incoming photon frequency that is reso- nant with its lowest N-edge XAS peak (XA′→c1A′) computed at 404.01 eV. We use the modified fc-CVS-EOM-EE-CCSD method that was used in the pNA example discussed above. For the XA′ →2A′RIXS transition, only the real Mxxmoments and the real and imaginary Mzzmoments are important. The NTO analysis in Fig. 13 of the γx,Reand ˜γx,ReRIXS 1PTDMs identifies its dom- inant off-resonance intramolecular charge-transfer channel ( υx,Re = 1.00), which also describes the one-photon XA′→2A′transition. The NTO analyses of the real and imaginary components of γzand ˜γz1PTDMs show similar orbitals (see the supplementary material), with the norms of the real 1PTDMs larger than the imaginary 1PTDMs (υz,Im= 0.28). This indicates that important orbital chan- nels along the zaxis are off-resonance, originating from the large numerators in the SOS off-resonance terms, which also dominate FIG. 13 . NTO analysis of the x- and z-component 1PTDMs for the XA′→2A′ RIXS transition in 4-amino-4′-nitrostilbene. Here, the off-resonance channels are dominant. The three-state model predicts near-resonance channels; thus, it is inadequate for this transition. the imaginary 1PTDMs. This is not surprising because the low- est N-edge XAS peak (at which the incoming photon energy is tuned) is separated by more than 1 eV from other XAS transitions. Clearly, a few-core-excited-state model would be inadequate for this transition dominated by off-resonance channels—in particular, the intramolecular charge-transfer channel—and Υx,Re= 0.50. This example provides another illustration of the merits of the analytic approach for characterizing RIXS transitions. V. CONCLUSION We presented a novel black-box approach for deriving the molecular orbital picture of RIXS transitions based on the corresponding response 1PTDMs and their NTOs. This is the first example of the generalization of the concept of NTOs to coher- ent nonlinear x-ray processes. This new tool for analyzing RIXS transitions relies on the rigorous and compact formalism of the response 1PTDMs based on damped response theory. The NTOs computed with this approach facilitate the visualization of RIXS transitions in terms of orbital channels but without crude sim- plifying assumptions. 1PTDMs are also useful for computing the physical quantities related to the spatial extent of the RIXS tran- sitions such as the average electron–hole separation. This ana- lytic approach is superior to the traditional few-state treatments, which relies on computing few intermediate core-excited states for a qualitative orbital picture of RIXS. The few-state treatment inher- ently suffers from the arbitrariness of the choice of intermediate states and potential loss of accuracy; it also ignores the coher- ent nature of RIXS. We demonstrate the utility of the new anal- ysis tool by calculating the orbital picture of RIXS transitions in the OH radical, benzene, pNA, and 4A4NS molecules. The RIXS transitions in the latter two systems have significant contributions from off-resonance orbital channels, which are difficult to cap- ture with the few-state models, illustrating the merits of the rig- orous analytic approach for analyzing RIXS transitions. For chro- mophores in complex environments, ab initio methods augmented J. Chem. Phys. 152, 244118 (2020); doi: 10.1063/5.0010295 152, 244118-11 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp with our analysis tool can help elucidate the role of molecular struc- ture and intermolecular interactions on the RIXS spectra and pro- vide rigorous assignments of the features in experimental RIXS spectra. 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Phys. 152, 244118 (2020); doi: 10.1063/5.0010295 152, 244118-13 Published under license by AIP Publishing
1.1689211.pdf	Quasiperiodic magnetization dynamics in uniformly magnetized particles and films Claudio Serpico, Massimiliano d’Aquino, Giorgio Bertotti, and Isaak D. Mayergoyz Citation: Journal of Applied Physics 95, 7052 (2004); doi: 10.1063/1.1689211 View online: http://dx.doi.org/10.1063/1.1689211 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/95/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Effects of Magnetic Interactions in Superparamagnetic Granular Films AIP Conf. Proc. 795, 197 (2005); 10.1063/1.2128328 Magnetics of ultrathin FePt nanoparticle films J. Appl. Phys. 95, 1481 (2004); 10.1063/1.1635669 Thickness dependence of magnetic blocking in granular thin films with interacting magnetic particles J. Appl. Phys. 93, 9208 (2003); 10.1063/1.1569993 Magnetic measurements and numerical simulations of interacting nanometer-scale particles J. Appl. Phys. 89, 7472 (2001); 10.1063/1.1360393 Magnetic behavior of nanostructured Fe films measured by magnetic dichroism J. Appl. Phys. 88, 3414 (2000); 10.1063/1.1288230 [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: 193.0.65.67 On: Wed, 10 Dec 2014 16:06:59Quasiperiodic magnetization dynamics in uniformly magnetized particles and ﬁlms Claudio Serpicoa)and Massimiliano d’Aquino Department of Electrical Engineering, University of Napoli ‘‘Federico II,’’Napoli, Italy Giorgio Bertotti IEN, Galileo Ferraris, Strada delle Cacce, 91 I-10135 Torino, Italy Isaak D. Mayergoyz Department of Electrical and Computer Engineering, University of Maryland, College Park, Maryland 20742 ~Presented on 8 January 2004 ! We study Landau–Lifshitz–Gilbert ~LLG!magnetization dynamics in uniformly magnetized uniaxial particles and ﬁlms subject to circularly polarized electromagnetic ﬁelds. Rotationalinvariance of the system and the introduction of an appropriate rotating reference frame permit oneto reduce the problem to the study of an autonomous dynamical system on the unit sphere.Quasiperiodic magnetization dynamics correspond to limit cycles of this reduced dynamical system.A perturbation technique based on the Poincare ´–Melnikov method is applied to predict the existence, the number, the shape, and the stability of these limit cycles for small value of thedamping constant in the LLG equation. © 2004 American Institute of Physics. @DOI: 10.1063/1.1689211 # Magnetization dynamics in magnetic thin ﬁlms and par- ticles has lately been the focus of considerable research inconnection with recent developments in the area of magneticdata storage technologies. The fundamental equation govern-ing magnetization dynamics is the Landau–Lifshitz–Gilbert~LLG!equation which, in most studies on the subject, is solved numerically. In fact, exact analytical solutions can bederived in few cases and are generally obtained by lineariz-ing the equation around certain given states. In an approachrecently proposed, 1exact analytical solutions were derived for the full nonlinear LLG equation with damping in the casewhen the magnetic body is an ellipsoidal particle with rota-tional symmetry around a certain axis and the external ﬁeldis circularly polarized. In this case, it is possible to study theproblem in an appropriate rotating reference frame where theapplied ﬁeld does not explicitly depend on time. In this ref-erence frame, magnetization dynamics is described by au-tonomous dynamical system on the unit sphere that may ex-hibit various phase portraits characterized by equilibriumpoints and limit cycles. 1Equilibria in the rotating frame cor- responds to uniform periodic solutions in the laboratory frame. Limit cycles in the rotating frame correspond to uni-form quasiperiodic 2magnetization motions in the laboratory frame, deriving from the combination of the rotation of theframe and the periodicity of the limit cycle ~see Fig.1 !. The purpose of this article is to present a technique to predict theexistence, the number, the shape and the stability of theselimit cycles ~and therefore of the quasiperiodic magnetiza- tion modes !in the special case, often encountered in the applications, of small values of the damping constant in theLLG equation. The analysis is carried out by using an appro- priate perturbation technique which is generally referred toas Poincare ´–Melnikov function technique. 3 We consider an uniformly magnetized thin ﬁlm or sphe- roidal particle subject to a time-varying external magneticﬁeld. The magnetization dynamics is governed by the LLGequation which is written in the following dimensionlessform: dm dt2am3dm dt52m3heff~t,m!, ~1! wherem5M/Ms,Mis the magnetization, Msis the satura- tion magnetization, heff5Heff/Msis the normalized effective ﬁeld, time is measured in units of ( gMs)21,gis the absolute value of the gyromagnetic ratio, and ais the damping con- stant. The effective ﬁeld is given by heff~t,m!52D’m’2Dzmzez1hazez1ha’~t!, ~2! whereezis the unit vector along the symmetry axis z, the subscript ‘‘ ’’’ denotes components normal to the symmetry axis andD’,Dzdescribe ~shape and crystalline !anisotropy of the body. The applied ﬁeld has the dc component haz along the zaxis and the time-harmonic component ha’(t) uniformly rotating with angular frequency vin the plane normal to the symmetry axis: ha’(t)5ha’@cos(vt)ex 1sin(vt)ey#, whereha’is the amplitude of the rotating ﬁeld, andex,eyare the unit vectors along the axis xandy, re- spectively. The dynamical system deﬁned by Eq. ~1!is non- autonomous ( heffexplicitly depends on time !and it is char- acterized by magnetization dynamics with umu51. In other words, Eq. ~1!deﬁnes a nonautonomous vector ﬁeld on the unit sphere. The analysis of this system is greatly simpliﬁedwhen Eq. ~1!is studied in the reference frame rotating ata!Author to whom correspondence should be addressed; electronic mail: serpico@unina.itJOURNAL OF APPLIED PHYSICS VOLUME 95, NUMBER 11 1 JUNE 2004 7052 0021-8979/2004/95(11)/7052/3/$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: 193.0.65.67 On: Wed, 10 Dec 2014 16:06:59angular velocity varound the symmetry axis ez. By choos- ing an appropriate origin of the time, we can obtain that inthe rotating frame h a’5ha’exand heff~m!52D’m’2Dzmzez1hazez1ha’ex. ~3! In addition, in passing to the new frame, the derivative of m(t) transforms according to the rule dm/dt°dm/dt 1vez3m, and thus Eq. ~1!takes the following autonomous form: dm dt2am3dm dt52m3~heff~m!2vez1avm3ez!. ~4! Equation ~4!describes an autonomous dynamical system evolving on the surface of the unit sphere umu51. It is inter- esting to notice that equilibria in the rotating frame corre-spond to periodic solutions in the laboratory frame whilelimit cycles in the rotating frame correspond to quasiperiodicmagnetization solutions in the laboratory frame. 1The quasi- periodicity derives from the combination of the rotation ofthe frame with angular frequency vand the periodicity of the limit cycle in the rotating frame with angular frequency self-generated by the dynamical system ~and in general not com- mensurable with v!. Notice also that chaos is not permitted in this dynamical system, despite the presence of a drivingsinusoidal ﬁeld, due to the rotational symmetry and the con-sequent reduction to an autonomous dynamical system on atwo dimensional ~2D!manifold. 2 Let us focus our analysis on the quasiperiodic solutions ~limit cycles in the rotating frame !. In order to establish the existence, the number and the locations of the limit cycleswe can exploit the fact that ais generally a small parameter (a’102341022). Thus, we can start our analysis by con- sidering the case a50 which can be easily treated because the dynamical system Eq. ~4!admits the following integral of motion which can be seen as a generalized energy of thesystem: G ~m!5Dzmz2/21D’m’2/22ha’mx2~haz2v!mz.~5! It is interesting to notice that the function G(m), for a.0, satisﬁes the following equation along the trajectories of thedynamical system Eq. ~4!: dG dt5aFv~m3ez!dm dt2Udm dtU2G52aP~m!, ~6!where P~m!is an ‘‘absorbed power’’ function which is de- ﬁned by the opposite of the above expression in squarebrackets. This function will be instrumental in the followingto give an energy interpretation of limit cycles. The phase portrait for a50 is given by the contour lines of the function G(m). To give a planar representation of the phase portraits, we use the stereographic projection vari-ables:w 15mx/(11mz),w25my/(11mz). In Fig. 2, the phase portrait is represented on the ( w1,w2) plane for the case of a thin ﬁlm. This phase portrait is characterized bythree centers C 1,C2andC3~outside Fig. 1 !and a saddle S with two homoclinic orbits G1andG2. When the small damping is introduced, almost all closed trajectories aroundcenters are slightly modiﬁed and collectively form spiral-shaped trajectories toward attractors. There are only specialtrajectories which remain practically unchanged under theintroduction of the small damping. Two of these trajectoriesare indicated in Fig. 2 and correspond to the values G 5g Q1andG5gQ2. These special values of Gand the cor- responding trajectories can be found by using a perturbation technique which is generally referred to as the Poincare `– Melnikov function method.3In order to apply this technique, it is convenient to transform Eq. ~4!in the following pertur- bative form ~ais a small parameter !: dm dt5f1~m!1af2~m,a! ~7! where f1~m!52m3~heff2vez!5m3„mG~m!, ~8! f2~m,a!5a 11a2m3heff21 11a2m3~m3heff!.~9! Let us start with the case a50.As we already observed, the dynamical system is integrable and the trajectories are givenimplicitly by the equation G(m)5g, withgvarying in the appropriate range. In addition, the vector ﬁeld f 1(m)i s Hamiltonian2as it can be derived from Eq. ~8!. The Poincare´-Melnikov technique is based on the Taylor expan- FIG. 1. Quasiperiodic trajectories of magnetization on the unit sphere in the laboratory ~left!and the rotating frames ~right!. Value of the parameters: a 50.05,Dz51,D’50,haz50.6,ha’50.15, and v51.1. FIG. 2. Phase portrait of conservative system on the stereographic plane w15mx/(11mz),w25my/(11mz). Value of the parameters: a50,Dz 51,D’50,haz50.6,ha’50.15, and v51.1.7053 J. Appl. Phys., Vol. 95, No. 11, Part 2, 1 June 2004 Serpico 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: 193.0.65.67 On: Wed, 10 Dec 2014 16:06:59sion of an appropriate Poincare `map of the perturbed dy- namical system in terms of the perturbation parameter a, around a50. The zero order term of this expansion is the identity, while the ﬁrst order term of the expansion is pro-portional to the Melnikov function which, in the case ofHamiltonian unperturbed vector ﬁeld, is given by the follow-ing integral along the trajectories of the unperturbed system~for details see Ref. 3 !. M ~g!5E 0Tgf1~mg~t!!‘f2~mg~t!,0!dt, ~10! wheremg(t) is the trajectory of the unperturbed system with G(mg(t))5g,Tgis its period and f1(m)‘f2(m,0)5m (f1(m)3f2(m,0)). By using the expressions of f1(m) and f2(m,0) given in Eqs. ~8!–~9!and appropriate algebraic ma- nipulations one can derive that M~g!52E 0TgFv~mg3ez!dmg dt2Udmg dtU2Gdt. ~11! The last equation can also be transformed in the following line-integral form which permits one to compute M(g) with- out deriving the time dependence of mg(t) M~g!52R G5gm3heffdm. ~12! Periodic orbits of the perturbed system are given by ﬁxed points of the Poincare ´map which, for sufﬁciently small a, can be found from the zeros of the Melnikov function. InFig. 3, the Melnikov function computed from Eq. ~12!is plotted versus the value of Gand the zeros of M(g), which correspond to the trajectories labeled as G5g Q1andG 5gQ2in Fig. 2, are emphasized. In Fig. 4, by sketching the phase portrait for the dissipative case we have then veriﬁed that the limit cycles, indicated with Q1~stable !andQ2~un-stable !, associated to the zeros of the Melnikov function, are preserved under the introduction of the damping with a 50.05. Let us notice that the introduction of damping trans- formed centers in foci F1(unstable), F2(stable), and F3(unstable), and disconneted the homoclinic trajectories that now have become the open spiraling separatrices L1and L2. It is interesting to notice that the Melnikov function given by Eq. ~11!can be rewritten as M(g) 5*0TgP(mg(t))dt. In this respect, it is possible to give a physical interpretation of limit cycles: the limit cycles arise from those unperturbed trajectories on which there is an av-erage balance between ‘‘absorption’’ ( P(m)>0) and ‘‘gen- eration’’ ( P(m)<0) of the generalized energy function G(m). By using the technique we have just illustrated that it is possible to predict the existence and the number of the limitcycles in a certain interval of values of aaround a50. The stability of the limit cycles can be obtained by studying thesign of the derivative of the Melnikov function at its zeros: 3 a limit cycle is stable for negative derivative, unstable for thepositive derivative. Finally the shape of the limit cycles canbe estimated by taking, as ﬁrst order approximation, the un-perturbed trajectories corresponding to the values of the en-ergy function G(m) where the Melnikov function vanishes. This work has been supported by the Italian MIUR- FIRB under Contract No. RBAU01B2T8. 1G. Bertotti, C. Serpico, and I. D. Mayergoyz, Phys. Rev. Lett. 86,7 2 4 ~2001!. 2A. J. Lichtenberg and M. A. Lieberman, Regular and Chaotic Dynamics ~Springer, New York, 1992 !. 3L. Perko, Differential Equations and Dynamical Systems ~Springer, Berlin, 1996!. FIG. 3. Two branches of the Melnikov function vs the value of G(m):gQ1 andgQ2correspond to Q1andQ2in Fig. 2. FIG. 4. Phase portrait of dissipative system. The parameters are the same as in Fig. 2 except for a50.05.7054 J. Appl. Phys., Vol. 95, No. 11, Part 2, 1 June 2004 Serpico 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: 193.0.65.67 On: Wed, 10 Dec 2014 16:06:59
1.2838490.pdf	Spin transfer precessional dynamics in Co 60 Fe 20 B 20 nanocontacts W. H. Rippard, M. R. Pufall, M. L. Schneider, K. Garello, and S. E. Russek Citation: Journal of Applied Physics 103, 053914 (2008); doi: 10.1063/1.2838490 View online: http://dx.doi.org/10.1063/1.2838490 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/103/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Current-perpendicular-to-the-plane giant magnetoresistance of an all-metal spin valve structure with Co 40 Fe 40 B 20 magnetic layer Appl. Phys. Lett. 96, 142503 (2010); 10.1063/1.3385730 Exchange bias of spin valve structure with a top-pinned Co 40 Fe 40 B 20 Ir Mn Appl. Phys. Lett. 93, 012501 (2008); 10.1063/1.2956680 Effect of high annealing temperature on giant tunnel magnetoresistance ratio of Co Fe B Mg O Co Fe B magnetic tunnel junctions Appl. Phys. Lett. 89, 232510 (2006); 10.1063/1.2402904 Valve behavior of giant magnetoimpedance in field-annealed Co 70 Fe 5 Si 15 Nb 2.2 Cu 0.8 B 7 amorphous ribbon J. Appl. Phys. 97, 10M108 (2005); 10.1063/1.1854891 Depth profiles of magnetostatic and dynamic characteristics in annealed Co 66 Fe 4 B 15 Si 15 amorphous ribbons J. Appl. Phys. 93, 7214 (2003); 10.1063/1.1540042 [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: 72.28.241.99 On: Mon, 07 Apr 2014 13:22:15Spin transfer precessional dynamics in Co 60Fe20B20nanocontacts W. H. Rippard,a/H20850M. R. Pufall, M. L. Schneider, K. Garello, and S. E. Russek Electromagnetics Division, National Institute of Standards and Technology, Boulder , Colorado 80305, USA /H20849Received 6 June 2007; accepted 3 December 2007; published online 13 March 2008 /H20850 We report on the precessional dynamics in spin transfer oscillators having Co 60Fe20B20free layers as a function of annealing time at 225 °C. Repeated annealing reduces the critical current Icby roughly a factor of 3 and increases the tunability of the oscillation frequency with current df/dI. The decrease in Iccorrelates with an increasing giant magnetoresistance /H20849GMR /H20850during the ﬁrst 3 h of annealing. For longer times, df/dIcontinues to increase, although the GMR does not. The variations in the macroscopic Co 60Fe20B20magnetization parameters and contact dimensions with annealing are not sufﬁcient to account for the later changes. © 2008 American Institute of Physics . /H20851DOI: 10.1063/1.2838490 /H20852 INTRODUCTION The spin transfer effect is known to give rise to micro- wave dynamics in a variety of magnetic nanostructures andmagnetic materials. Material systems have included Co, Fe,Ni /Fe, Ni /Fe /Cu alloys, Co /Fe based alloys, and CoFeB alloys. 1–5In CoFeB tunnel junction structures, temperature annealing has been shown to be particularly important inoptimizing device characteristics, such as the tunnelingmagnetoresistance. 6Here, we show that the temperature an- nealing of all-metallic spin transfer nanoscale oscillators /H20849ST- NOs /H20850containing amorphous CoFeB free layers leads to sig- niﬁcantly reduced critical currents as well as to increasedtunability of the oscillation frequency with current. Over theﬁrst three anneals, the spectral output properties of the oscil-lators with temperature annealing are consistent withchanges in the giant magnetoreistance /H20849GMR /H20850of the struc- ture, although the later changes are not. AlthoughCo 60Fe20B20has a larger saturation magnetization and a damping parameter and an exchange stiffness similar to thatof Ni 80Fe20, the annealing process yields a lower value of the critical current density for the Co 60Fe20B20structures, sug- gesting a method for reducing the critical currents in spintransfer based devices. EXPERIMENT The devices discussed here consist of a nominally 50 nm diameter electrical contact made to the top of a continuous10/H1100320 /H9262m2spin-valve mesa.4The spin valve comprises Ta /H208493n m /H20850/Cu /H2084915 nm /H20850/Co90Fe10 /H2084920 nm /H20850/Cu /H208494n m /H20850/Co60Fe20B20 /H208495n m /H20850/Cu/H208493n m /H20850/Ta /H208493n m /H20850. In this structure, precessional motion is induced in the CoFeB layer, and the CoFe layer acts as the “ﬁxed” layer due to its largerthickness and saturation magnetization. The devices are dccurrent biased so that the precessional motion of the freelayer induces a microwave voltage across the device throughthe GMR effect, which is measured with a spectrum ana-lyzer. All measurements discussed here were performed atroom temperature, and all anneals were done at 225 °C inincrements of 60 min in an applied ﬁeld /H92620Hanneal =0.1 T. In our nanocontact geometry, the magnetic material around thecontact is protected from exposure to atmosphere by the Tacapping layer throughout the fabrication process and after-wards. The magnetic material which is the vicinity of thecontact is further protected from atmosphere from the cross-linked polymethylmethacrylate which forms the insulatingbarrier in the device structure. These layers act to prevent theformation of magnetic oxides and to prevent the possibilityof the annealing either oxidizing the structure or changinglocal oxidation states within it. The data shown here are for asingle device upon repeated annealing, but the qualitativefeatures discussed have been measured in tens of devices inseveral different applied ﬁeld geometries. RESULTS AND DISCUSSION The spectral output of the STNO devices, i.e., frequency, power output, linewidth, and tunability of frequency withcurrent, varies signiﬁcantly upon successive annealing, as dothe device critical current and GMR value. Figure 1/H20849a/H20850shows the precessional frequencies as a function of dc current for /H92620Happ=0.85 T applied at angle /H9258H=80° out of the ﬁlm plane for cumulative anneal times up to 6 h. The correspond-ing oscillation linewidth /H20851full width at half maximum FWHM /H20852and output power /H20849integrated area under the spectral peak /H20850are shown in Figs. 1/H20849b/H20850and1/H20849c/H20850, respectively. As seen in Fig. 1/H20849a/H20850, the annealing process results in an increase in the variation of the oscillation frequency withcurrent df/dIfrom 0.5 to 3.25 GHz /mA after 6 h of anneal- ing, roughly six times that of the as-prepared sample /H20851see also Fig. 2/H20849a/H20850/H20852. In conjunction, the critical current I c, deter- mined by the lowest current at which precessional dynamicsare measured, is reduced from 6.75 to 2.25 mA after severalanneal cycles /H20851see also Fig. 2/H20849b/H20850/H20852. For comparison, similar devices having NiFe as the free layer typically have criticalcurrents of 4–5 mA and frequency tunabilities of roughly0.5–1 GHz /mA. 7Hence, annealing results in comparatively reduced values for Icand increased values for df/dIin CoFeB structures compared to similar NiFe based devices.a/H20850Electronic mail: rippard@boulder.nist.gov.JOURNAL OF APPLIED PHYSICS 103, 053914 /H208492008 /H20850 0021-8979/2008/103 /H208495/H20850/053914/4/$23.00 © 2008 American Institute of Physics 103 , 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: 72.28.241.99 On: Mon, 07 Apr 2014 13:22:15We note that transmission electron microscopy studies have veriﬁed that the CoFeB layer remains amorphous throughoutthe annealing process. There is also a change in the variation of frequency with applied ﬁeld df/d /H92620Hduring the temperature annealing pro- cess. However, its value is more difﬁcult to quantify. Thelocations of the discontinuous jumps of frequency with cur-rent are sensitive functions of applied ﬁeld, 7and there is only a relatively small range of accessible ﬁelds over which thedynamics can be measured /H20849 /H92620Happ=0.6–1.1 T /H20850. Experi- mentally, this results in values for df/d/H92620Hranging from 10 to 35 GHz /T, depending on the particular choice for the applied bias current.For this geometry, the oscillation linewidth generally varies signiﬁcantly as a function of current.7As seen in Fig. 1/H20849b/H20850, the linewidth also generally broadens with anneal time. In addition, the annealing process also changes the deviceoutput power /H20851Fig. 1/H20849c/H20850/H20852. Upon initial anneal, the maximum output power increases. In this case, it is possible that in-creased powers from the as-prepared device could be ob-tained for larger currents since the device dynamics have notceased at I=18 mA. However, we have explicitly measured several other devices in which the precessional dynamicsturn off at the highest applied currents and still found anincrease in output power upon annealing. For the deviceshown here, all of the precessional dynamics have turned offat the highest current levels shown, except for the initialanneal and the as-prepared states. For a cumulative annealtime of up to 3 h, the maximum power remains relativelyconstant and then decreases upon further annealing. As wediscuss below, this behavior is consistent with an initial in-crease in the GMR of the device with temperature annealingand a reduction in the current being passed through the de-vice, as the device output power scales as I 2/H9004R, where /H9004Ris the change in resistance associated with the magnetic excita-tion. As discussed elsewhere, the currents applied to the de-vices result in Joule heating of a few tens of degreescelsius. 8,9Hence, heating effects are not sufﬁcient to result in changes of the device GMR as a function of current bias. The decrease in Iccorrelates with an increase in the GMR value of the CoFe /CoFeB spin valve with annealing. Current-in-plane /H20849CIP /H20850GMR measurements of a similarly prepared spin valve are shown as a function of the cumula-tive anneal time in Fig. 2/H20849b/H20850. During the ﬁrst 3 h of anneal- ing, the GMR increases 10,11from 0.3% to roughly 0.6% and then remains relatively constant upon further annealing.12 During these initial anneals, the critical current in the nano-contact is reduced by a factor of 3 from 6.75 to 2.25 mA andthen remains relatively constant during subsequent anneals.The correlation between the changes in the GMR and thevalues of the critical current suggests that changes in thespin-dependent transport in the device occur during the ini-tial anneals and are responsible for an increased spin torqueefﬁciency and the reduced values of I c, in accordance with Ref. 13. The fvsIcurves do not follow a universal dependence since normalizing the bias current by the critical current for agiven data set does not give agreement among the preces-sional frequencies for the various anneal times. This suggeststhat the precessional trajectories themselves are changing.The closest agreement occurs among the ﬁrst three annealsand is signiﬁcantly worse for longer anneal times. This canbe seen in Fig. 1/H20849a/H20850. For anneal times longer than 3 h, the critical current does not change. Hence, normalizing the biascurrent by I cwill not change the relative frequency differ- ences for these data. Some of the initial increase in df/dI may result from the decreased critical currents since, for agiven absolute current, the normalized bias current I/I cis increasing during the ﬁrst three anneals. However, duringthese ﬁrst anneals, the critical current changes by a factor of3, whereas df/dIvaries by only roughly a factor of 2. One possible explanation for the decrease in critical cur- FIG. 1. /H20849Color online /H20850/H20849a/H20850Frequency vs current bias for cumulative anneal times ranging from 0 to 6 h, as labeled. /H20849b/H20850Representative data showing the oscillator linewidth /H20849FWHM /H20850vs current bias for several different anneal times, only three times are shown for clarity. /H20849c/H20850Representative data show- ing the output power vs current bias for several different anneal times. Thesolid lines are spline ﬁts to the data and are added only for visual clarity. Thesymbols in all parts correspond to the same cumulative anneal times. FIG. 2. /H20849Color online /H20850/H20849a/H20850Zero-bias device resistance and df/dIas functions of the cumulative anneal time. /H20849b/H20850Critical current and the GMR value as functions of the cumulative anneal time. The GMR values were determinedfrom CIP measurements of a similarly prepared spin-valve structure.053914-2 Rippard et al. J. Appl. Phys. 103 , 053914 /H208492008 /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: 72.28.241.99 On: Mon, 07 Apr 2014 13:22:15rent and increase in df/dIis that the effective contact area is altered by the anneals so that a constant current would notcorrespond to a constant current density. In Fig. 2/H20849a/H20850, we plot the device resistance as a function of anneal time. The as-prepared device has a resistance of roughly 8 /H9024and I c =6.75 mA. Over the ﬁrst three anneals the critical current is reduced by a factor of 3 to a value of 2.25 mA.14Assuming that the device resistance is inversely proportional to contactarea, a corresponding factor of three increase in the resis-tance would be required for the different critical currents tocorrespond to a constant current density. However, the de-vice resistance changes by less than 15%, indicating that thecontact dimensions are not altered enough to account for thefactor of 3 decrease in I c, although slight changes in the device size, might play some role. Similarly, a change incontact diameter of roughly a factor of 3 would be requiredto account for the factor of 6 increase in df/dIbased on contact size effects. 15,16Together, these measurements indi- cate that changes in the spin-dependent transport in the con-tact that occur during the ﬁrst 3 h of annealing are morelikely responsible for the initial changes in the oscillatorcharacteristics. Another possible explanation for the decrease in I cwith thermal anneal is that the magnetic properties of the CoFeBare altered in such a way as to increase the strength of thespin torque effect for a given current density. For instance,within the macrospin approximation, the critical current gen-erally scales as I c/H11008/H20849/H9251MsMeff//H9255/H20850, where /H9251is the Gilbert damping parameter, Msis the saturation magnetization, Meff is the effective saturation magnetization which includes the out of plane anisotropy, and /H9255is the spin transfer efﬁciency.17 Through vector network analyzer based ferromagnetic reso- nance measurements18of identically prepared spin-valve structures, we have found that /H9251=0.009 /H110060.002, /H92620Meff =1.16/H110060.05, and /H92620Ms=1.44/H110060.04 throughout the anneal process. This is somewhat different from what was measuredin Ref. 18but here, the CoFeB ﬁlm is grown on a Cu seed layer instead of SiO 2and the anneals are performed at a lower temperature, which likely accounts for the discrepan-cies. Within the macrospin approximation, these variations are insufﬁcient to produce the reduced critical currents andobserved changes in df/dI. This again indicates that changes in the spin-dependent transport are responsible for thechanges in the oscillator behavior during the ﬁrst severalanneals, and that the changes that occur during the later an-neals do not result from differences in the magnetostaticproperties of the CoFeB ﬁlm. The macrospin model is only an approximation to the nanocontact geometry, as there is spinwave radiation awayfrom the contact area. Theoretical models of the nanocontactgeometry generally include a second term in the predictedcritical current that is proportional to D, the spinwave ex- change stiffness, which accounts for this radiation. 17,19While this term changes the exact functional form for the criticalcurrent, the scaling arguments above are still applicable. In-deed, when a spinwave radiation term is included in the formfor the critical current, even greater changes in the devicedimensions and material properties are required if they are toproperly account for the measured changes in the devicecharacteristics. 14Hence, the conclusions drawn from the comparison of the data with the macrospin model are justi-ﬁed. We also note that, within the context of current nano-contact theories, the constant value of the critical currentduring the later anneals precludes the possibility that changesinDcan account for the increase in df/dIduring the later anneals. The linewidth, averaged over all current values, is shown in Fig. 3/H20849a/H20850as a function of the cumulative anneal time. In the as-prepared state, the average linewidth is roughly40 MHz. During the ﬁrst several anneals, the average line-width increases and reaches a value of roughly 200 MHz fora cumulative anneal time of 3 h. For anneal times of 4 and5 h, the average linewidth slightly decreases before increas-ing by roughly a factor of 4 to 600 MHz at a cumulativeanneal time of 6 h, a factor of 15 larger than in the as-prepared device, indicating that the device has degraded.Qualitatively, the overall increase in the average linewidthwith annealing mimics the increase in df/dI. However, quan- titatively, df/dIincreases by roughly a factor of 6 during the anneal process, while the linewidth increases by roughly afactor of 15. This indicates that the linewidths here are notlimited by current noise in the system. The measurementswere stopped after the sixth anneal as the oscillator line-widths had become quite broad and the absolute outputpower relatively low /H20851see inset Fig. 3/H20849a/H20850/H20852. This is possibly due to changes in the material microstructure in the vicinityof the contact such as diffusion of CoFe or CoFeB into theCu layers or a redistribution of B in the free layer. It is alsopossible that the changes in the spin transfer induced dynam-ics are due to changes in the material microstructure in thevicinity of the contact, although any such changes wouldneed to be on a level that would not affect the device resis-tance, and such that they are not reﬂected in the macroscopicmagnetization characteristics. We note that the changes with FIG. 3. /H20849Color online /H20850/H20849a/H20850Average linewidths for the data in Fig. 1/H20849a/H20850as a function of cumulative anneal time. /H20849inset /H20850Representative spectral traces for anneal times of 1 and 6 h showing the spectral traces corresponding to themaximum device output power for both anneals. /H20849b/H20850Normalized output power as a function of bias current for several representative anneal times.The solid lines are spline ﬁts to the data and are added only for visual clarity.053914-3 Rippard et al. J. Appl. Phys. 103 , 053914 /H208492008 /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: 72.28.241.99 On: Mon, 07 Apr 2014 13:22:15annealing have been observed in all contacts measured, and the data are not simply reﬂecting anomalous changes in themicrostructure of a speciﬁc device. The output power, normalized by I 2/H9004R, as a function of current is shown in Fig. 3/H20849b/H20850. If the precessional trajectories for the various anneal times were nearly identical, then thenormalized output power, which accounts for differences inthe GMR and bias currents, should also be very similar be-tween the different anneals. However, as seen in Fig. 3/H20849b/H20850, this normalization process does not give good agreement be-tween the different anneals, indicating that the mode struc-ture of the oscillations is changing. This can also be seen bycomparing the data for the different anneal times shown inFig. 1/H20849a/H20850. For the longest anneal time, the maximum excited frequency is roughly 27 GHz, whereas the maximum fre-quency is less than 20 GHz in the as-prepared state. As themagnetic properties of the CoFeB ﬁlm are not changing sig-niﬁcantly /H20849the magnetostatic ﬁelds within the device are rela- tively constant /H20850, the changes in the excitation frequency are most easily accounted for through changes in the preces-sional modes of the excitations. The particular changes thatare likely occurring are not presently known with certainty.They may correspond to changes in the excitation size, tochanges in the mode structure, or may result from changes inthe magnitude of the Oersted ﬁeld. However, we note that,for a given applied ﬁeld value, the frequency at which themaximum output power occurs is very similar across theanneals /H20849see Fig. 1/H20850. While the normalized output powers are not constant throughout the annealing process, this likely in-dicates that the excited modes are also not too different. SUMMARY In summary, we have measured the spectral properties of STNOs with CoFeB as the free layer as a function of suc-cessive 225 °C temperature anneals. The anneals result inchanges in the oscillator critical current, output power, fre-quency variation with bias, and oscillator linewidth. Thechanges that occur during the ﬁrst several anneals correlatewith increases in the GMR of the structure. The later changeslikely result from changes in the details of the mode excitedby the spin transfer effect. These measurements demonstratea method of reducing the critical current values in STNOs aswell as increasing their frequency tunability as a function of current as compared to more typical NiFe based devices.They also show the importance of thermal history in CoFeBbased devices. ACKNOWLEDGMENTS This paper is a contribution of NIST, not subject to copy- right. 1M. R. Pufall, W. H. Rippard, and T. J. Silva, Appl. Phys. Lett. 83,3 2 3 /H208492003 /H20850. 2For a review, see M. D. Stiles and J. Miltat, Spin Dynamics in Conﬁned Magnetic Structures , edited by B. Hillebrands and A. Thiaville /H20849Springer, Berlin, 2006 /H20850, V ol. 3. 3S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J. Schoe- lkopf, R. A. Buhrman, and D. C. Ralph, Nature /H20849London /H20850425,3 0 8 /H208492003 /H20850. 4W. H. Rippard, M. R. Pufall, S. Kaka, S. E. Russek, and T. J. Silva, Phys. Rev. Lett. 92, 027201 /H208492004 /H20850. 5K. Yagami, A. Tulapurkar, A. Fukushima, and Y . Suzuki, Appl. Phys. Lett. 85, 5634 /H208492004 /H20850. 6T. Dimopoulos, G. Gieres, J. Wecker, N. Wiese, and M. Sacher, J. Appl. Phys. 96, 6382 /H208492004 /H20850. 7W. H. Rippard, M. R. Pufall, and S. E. Russek, Phys. Rev. B 74, 224409 /H208492006 /H20850. 8E. B. Myers, F. J. Albert, J. C. Sankey, E. Bonet, R. A. Buhrman, and D. C. Ralph, Phys. Rev. Lett. 89, 196801 /H208492002 /H20850. 9I. 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. 10M. Jimbo, K. Komiyama, Y . Shirota, Y . Fujiwara, S. Tsunashima, and M. Matsuura, J. Magn. Magn. Mater. 165, 308 /H208491997 /H20850. 11T. Feng and J. R. Childress, J. Appl. Phys. 85, 4937 /H208491999 /H20850. 12We note that the absolute values of the GMR are low due to the relatively thick Cu layers included in the device structure. 13A. Manchon, N. Strelkov, A. Deac, A. Vedyayev, and B. Dieny, Phys. Rev. B 73, 184418 /H208492006 /H20850. 14We note that this method of determining Icis only approximate since it is limited to measuring the lowest current at which the oscillator powerexceeds the noise ﬂoor of the measurement. Since the output power ischanging with annealing, this makes the reported relative values of I cas a function of annealing only approximate. 15M. A. Hoefer, M. J. Ablowitz, B. Ilan, M. R. Pufall, and T. J. Silva, Phys. Rev. Lett. 95, 267206 /H208492005 /H20850; M. A. Hoefer, Ph.D. thesis, University of Colorado, 2005. 16F. B. Mancoff, N. D. Rizzo, B. N. Engel, and S. Tehrani, Appl. Phys. Lett. 88, 112507 /H208492006 /H20850. 17J. C. Slonczewski, J. Magn. Magn. Mater. 159,L 1 /H208491996 /H20850;195, L261 /H208491999 /H20850. 18C. Bilzer, T. Devolder, J.-V . Kim, G. Council, C. Chappert, S. Cardoso, and P. Freitas, J. Appl. Phys. 100, 053903 /H208492006 /H20850. 19A. N. Slavin and P. Kabos, IEEE Trans. Magn. 41, 1264 /H208492006 /H20850.053914-4 Rippard et al. J. Appl. Phys. 103 , 053914 /H208492008 /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: 72.28.241.99 On: Mon, 07 Apr 2014 13:22:15
1.1711814.pdf	Observation of near-critical reﬂection of internal waves in a stably stratiﬁed ﬂuid Thierry Dauxois,a)Anthony Didier, and Eric Falconb) Laboratoire de Physique, E´cole Normale Supe ´rieure de Lyon, UMR-CNRS 5672, 46 alle ´e d’Italie, 69007 Lyon, France ~Received 22 April 2003; accepted 10 February 2004; published online 29 April 2004 ! An experimental study is reported of the near-critical reﬂection of internal gravity waves over sloping topography in a stratiﬁed ﬂuid. An overturning instability close to the slope and triggeringthe boundary-mixing process is observed and characterized. These observations are found in goodagreement with a recent nonlinear theory. © 2004 American Institute of Physics. @DOI: 10.1063/1.1711814 # I. INTRODUCTION The reﬂection of near-critical internal waves over slop- ing topography plays a crucial role in determining exchangesbetween the coastal ocean and the adjacent deep waters. Di-rect measurements of mixing in the ocean, using tracers, 1 have vindicated decades of phenomenological and theoreti-cal inferences. In particular, these measurements have shownthat most of the vertical mixing is not taking place inside theocean, but close to the boundaries and topographic features. 2 These results have directed attention to the possible role ofinternal wave reﬂection in the boundary-mixing process. Internal waves have different properties of reﬂection from a rigid boundary than do sound or light waves. 3Instead of following the familiar Snell’s law, internal waves reﬂectoff a boundary such that the angle with respect to gravitydirection is preserved upon reﬂection ~Fig. 1 !. This peculiar reﬂection law leads to a concentration of the reﬂected energydensity into a narrow ray tube upon reﬂection as displayed inFig. 1. Theoretical descriptions of this reﬂection processhave been framed largely in terms of linear and stationarywave dynamics. 3,4However, when the slope angle, g,i s equal to the incident wave angle, b, these restrictions lead to an unrealistic prediction: The reﬂected rays lie along theslope with an inﬁnite amplitude and a vanishing group ve-locity. Theoretical results have recently healed this singular-ity by taking into account the role of transience andnonlinearity. 5 Following preliminary oceanographic measurements,6,7 Eriksen8has beautifully observed, near the bottom of a steep ﬂank of a tall North Paciﬁc Ocean seamount, an internalwave reﬂection process leading to a clear departure from aGarett–Munk model 9for wave frequencies at which ray and bottom slopes match. Several experimental facilities10–15 have therefore been dedicated to the understanding of theinternal wave reﬂection and associated instabilities. However, results of a controlled laboratory experiment close to the critical conditions are still lacking since previousones with a moving paddle 10,11,13at one end of a very long tank, or by the vibration of the tank itself,14does not gener- ate a clear incident wave beam as needed for a careful study.In addition, a direct comparison with the recent and completenonlinear theory near the critical reﬂection would be pos-sible. Finally, the goal is to improve the understanding of thepossible mixing mechanism near the sloping topography ofocean as very recently initiated by MacPhee and Kunze 16by exhibiting the instability mechanisms leading to mixing. Thepaper is organized as follows. The experimental setup is de-scribed and carefully explained in Sec. II. The main experi-mental results are presented in Sec. III, and comparisonswith the theory is also provided. Finally, Sec. IV containsconclusions and perspectives. II. EXPERIMENTAL SETUP The experimental setup consists of a 38 cm long Plexi- glas tank, 10 cm wide, ﬁlled up to 22 cm height with alinearly salt-stratiﬁed water obtained by the ‘‘double bucket’’method. 17The choice of salt, sodium nitrate (NaNO 3) snow, is motivated due to its highly solubility in water leading to asalty water viscosity close to the fresh one. Moreover, thissalt allows to reach a strong stratiﬁcation: The ﬂuid density @1&r(z)<1.2 g/cm3#measured at different altitudes (22>z.0 cm) by a conductimetric probe leads to a linear vertical density proﬁle of slope dr/dz.20.0104 g/cm4.A rectangular Plexiglas sheet, 3 mm thick and 9.6 cm wide ~to allow exchange of water !is introduced at one end of the tank with an angle g535° with respect to the horizontal tank bottom ~see Fig. 1 !to create the reﬂective sloping boundary. Internal waves are generated by a sinusoidal excitation provided by the vertical motion of a horizontal PVC plung-ing cylinder ~3.1 cm in diameter and 9.4 cm long !. The cyl- inder is located roughly midway between the base tank andthe free surface. This wave maker is driven by an electro-magnetic vibration exciter powered by a low frequencypower supply. Optical measurements conﬁrm 18that the cyl- inder motion is sinusoidal without distortions for vibrationalfrequencies 0.2 <f<0.5 Hz and maximal displacement am- plitudes,A pp, up to 8.5 mm ~peak to peak !.a!Electronic mail: thierry.dauxois@ens-lyon.fr b!Electronic mail: eric.falcon@ens-lyon.frPHYSICS OF FLUIDS VOLUME 16, NUMBER 6 JUNE 2004 1936 1070-6631/2004/16(6)/1936/6/$22.00 © 2004 American Institute of Physics The well-known and nonintuitive dispersion relation of internal waves in an incompressible, inviscid and linearlystratiﬁed ﬂuid reads 19 v56Nk’ uku56Nsinb, ~1! wherek’is the component of kperpendicular to the zaxis, N5A2(g/r0)]r/]zis the constant buoyancy ~or Brunt– Va¨isa¨la¨!frequency, r(z) the ﬂuid density at altitude z,g 5981 cm/s2the acceleration of gravity, and r0.1 g/cm3a reference density. Thus, from Eq. ~1!, the wave frequency, v52pf, determines the inclination angle bof the phase surfaces with the vertical, and not the magnitude of the wavevectork. From Eq. ~1!, bis also the angle between the group velocitycg5]v/]kand the horizontal, since cg’k. Thus, for a given stratiﬁcation and frequency, internal waves of low~higher !frequency propagate at low ~steeper !angle. The outward radiation of energy is thus along four beams oriented at an angle bwith the horizontal, the familiar St. Andrews Cross structure.20The beam propagating di- rectly toward the slope has been singled out by adding a gridon the surface of water. This grid strongly damps the threeother wave beams that propagate towards the free surface,and thereby prevents reﬂection of such beams.Aplanar wavepattern consisting of parallel rays is thus generated from thewave maker.Although it is well-known that the spatial spec-trum of waves generated by an oscillating cylinder islarge, 15,21,22it has been experimentally checked that the dominant wavelength is approximately equal to the cylinderdiameter ~see later !. Moreover, measurements conﬁrm that the low vibrational amplitudes of cylinder do not stronglyinﬂuence the wavelength generated. Different visualization methods are used to study the re- ﬂection mechanism of such internal gravity waves by aboundary layer. First, the usual shadowgraph technique 23al- lows visualization of the qualitative and global two-dimensional evolution of the incident and reﬂected waves. Itinvolves projecting a point source of light through stratiﬁedwater onto a screen behind the tank. The optical refractiveindex variations induced by the ﬂuid density variations, al-lows one to observe isodensity lines ~or isopycnals !on the screen, located perpendicularly to the light source and paral-lel to the longest wall tank. It is therefore possible to mea- sure the group velocity angle, b, and the phase velocity vwof the incident wave. For various frequencies of excitation,0.2<f<0.5 Hz, the angle bof the St. Andrews Cross is measured on a screen leading to a linear relation between v and sin bas predicted by Eq. ~1!with a slope of N53.1 60.1 rad/s, for the stratiﬁed ﬂuid prepared as earlier. This value is in good agreement with the earlier static one ob-tained from the density proﬁle with a conductimetric probe.The cutoff frequency is then f c5N/(2p).0.5 Hz. By time of ﬂight measurements between signals delivered by twophotodiodes, 1 cm apart, each of 7 mm 2area, located behind the tank along the propagation direction, a value vw50.6 60.4 cm/s was obtained for an excitation frequency f 50.25 Hz. When these signals are crosscorrelated by means of a spectrum analyzer, the averaged dephasing time leads to vw50.760.3 cm/s, close to the previous value. The wave- length of the incident wave is thus l5vw/f.3 cm, corre- sponding as expected to the oscillating cylinder’s diameter.However, neither this shadowgraph technique nor one 23us- ing passive tracers ~ﬂuorescein dye !is sufﬁciently sensitive to observe quantitative and local internal wave propertiesclosed to the reﬂective boundary layer. Accordingly, the classical Schlieren method 23,24of visu- alization has been used. Let us just note that behind the tank,the light beam is refocused by a lens a small distance after aslit~instead of the usual knife blade to increase contrast !to ﬁlter the rays. The slit is oriented orthogonally to the slopegenerating straight horizontal fringe lines in the case of noexcitation. The image of the observation ﬁeld ~strongly de- pendent of the 7.5 cm diameter lens !is focused on the screen by a last lens. The internal wave, producing density distur-bances, causes lines to distort, this distorting line patternbeing recorded by a camera. Note that this experimentaltechnique is sensitive to the index gradient, and therefore tothe density gradient, orthogonal to the slit, i.e., parallel to the slope. III. EXPERIMENTAL RESULTS AND COMPARISON WITH THEORY The Schlieren technique allows one to carefully observe quantitative and local internal wave properties during thereﬂection process. Critical reﬂection arises when an incidentwave beam with an angle of propagation breﬂects off the slope of angle g.b, the reﬂected wave being then trapped along the plane slope. This corresponds to a critical fre-quencyf c5Nsing.0.2860.01 Hz, Nbeing equal to 3.1 60.1 rad/s for all experiments. It is possible to observe that the isodensity lines ~isopycnals !, initially horizontal without excitation, are bent for an excitation near fc(0.78 <f/fc <1.14), and fold over themselves along the length of the slope. Figure 2 shows a time sequence of constant densitysurfaces, depicting sequential snapshots of the ﬂow through-out one period of its development. These pictures strikingly show the distortion of the isopycnals in the slightly subcritical case with f/f c50.78. In panel ~a!, the density disturbance is very small and one can observe essentially the initial horizontal background stratiﬁ- FIG. 1. Schematic view of the reﬂection process when the incident wave nearly satisﬁes the critical condition g’b. The group velocity of the re- ﬂected wave makes a very shallow angle with the slope. cgindicates the incident and reﬂected group velocities. The reﬂection law leads to a concen-tration of the energy density into a narrow ray tube.1937 Phys. Fluids, Vol. 16, No. 6, June 2004 Observation of near-critical reﬂectioncation. This background stratiﬁcation, usually obtained with dye ﬂuorescein, should be invisible with this Schlierenmethod. However, as previously reported by MacPhee andKunze, 16the comparison between shadowgraph and ﬂuores- cein dye visualizations allows to identify these lines withisopycnals. In panel ~b!, the disturbance generated by the incident wave breaking against the slope begins to ‘‘fold-up’’the isopycnals. As time progresses @see panel ~c!#, wave overturning develops around a front: The buoyancy becomesstatically unstable. This overturned region climbs along the FIG. 2. ~Color !Schlieren pictures showing the slightly subcritical reﬂection ( f/fc50.78) of an internal wave on a slope, during one incident wave period T. The slope ~thick black line !h a sa na n g l e g535°. The incident wave plane comes in from the left ~inclined black region near the top left corner between blue and yellow regions !. The reﬂected wave plane is hardly noticeable. Wave maker vibrational frequency and peak to peak amplitude are, respectively, f 50.22 Hz and App56.7 mm.1938 Phys. Fluids, Vol. 16, No. 6, June 2004 Dauxois, Didier, and Falconslope as time continues, and the folded isopycnals collapse into turbulence that mixes the density ﬁeld within the break-ing region @see panel ~d!#. The maximum thickness of this reﬂected disturbance is of the order of 5 mm, and decreaseswith time as theoretically predicted. 5Finally, the ﬂow begins to relaminarize @panel ~e!#. One can check that panels ~a!and ~f!are almost identical, showing that the ﬂow is entirely restratiﬁed @panel ~f!#after one period of excitation. Figure 2 has been analyzed with an image processing software ~Scion-Image !to extract the isopycnals from the pictures. A typical experimental result for the distortion ofisopycnals is reported in Fig. 3 ~a!, and is compared with a theoretical result in Fig. 3 ~b!. The analytic solution of the density ﬁeld of the initial value problem in the critical casereads 5 r5r0H12N2 gFzcosg2cBAukusin2~2b! 2v1GJ, ~2! where B5At zJ1sin@v1t2ukusin~b1g!x#, ~3! J1[J1~2A2v1cos2bukutz!, ~4! J1() being the Bessel function, v1is the positive solution of Eq. ~1!,cis the maximum amplitude of the streamfunc- tion, and xis the horizontal coordinate. Figure 3 shows a good qualitative agreement between experimental and theo-retical results, the value for the time tbeing arbitrarily cho- sen. Far from the slope, the density disturbance is very smalland one sees essentially the initial background stratiﬁcation.Closer to the slope, the disturbance folds up the isopycnals,and this produces a region of static instability. Recording several isopycnals and using image process- ing, it is also possible to follow the temporal evolution of asingle isopycnal during its overturning. As this phenomenon is periodic with a period T51/f, it is possible to reconstruct from this temporal evolution the density proﬁle picture at agiven time t. This allows one to follow the position, and therefore the propagation velocity of the front along theslope at different times. The front is deﬁned as the inﬂexionpoint @represented by the star in Fig. 3 ~b!#of the followed isopycnal. Figure 4 ~a!shows, during two periods, the isopy- cnal front position, x f, along the slope as a function of time. The periodic evolution of this front position is clearly ob-served, and the local slope of the curves in Fig. 4 ~a!allows one to roughly measure the front velocity, vf, as a function of time, as reported in Fig. 4 ~b!. The front velocity from its creation to its collapsing in- creases from 0.5 up to 3 cm/s. The front has thus travelledroughly 4 cm in one period ( ;4.5 s). This leads to an aver- aged front velocity in agreement with the phase speed mea-surement obtained from the shadowgraph visualizations ~see Sec. II !. The wave maker frequency is now increased up to f 50.32 Hz, to have an incident plane wave tilted with an angle bsteeper than the slope angle g. In this slightly super- critical case ( f/fc51.14), intrusions are still observed, but the density ﬁeld does not fold up so abruptly and does notlead anymore to overturning instability ~see Fig. 5 !. Except the value of the frequency f, all others parameter values have been kept identical to the ones in Fig. 2. The instant ofthis snapshot has been chosen when the isopycnal distortionis the largest. This distortion is clearly far from leading to anoverturning instability as encountered in the subcritical caseof Fig. 2 ~c!: The reﬂected wave is not trapped along the slope in the boundary layer, and consequently the isopycnalsare not overturned. Moreover the density front velocity ismeasured roughly constant, 0.5 cm/s, during two periods ofvibration. Both differences conﬁrm that the singularity ap-pears only in the critical case. 5 FIG. 3. ~a!Experimental isopycnals extracted from a region of Fig. 2 ~c!.~b! Theoretical isopycnals from Eq. ~2!. The star indicates the position of the front. FIG. 4. Temporal evolution of the isopycnal front position ~a!and velocity ~b!along the slope, during two periods of vibration. T.4.5 s,f/fc50.78, andApp56.7 mm.1939 Phys. Fluids, Vol. 16, No. 6, June 2004 Observation of near-critical reﬂectionIV. CONCLUSIONS AND PERSPECTIVES The Schlieren technique allows us to study the spa- tiotemporal evolution of the internal waves reﬂection closeto the critical reﬂection, where nonlinear processes occur.The dynamics of isopycnals is then found in agreement witha recent nonlinear theory. 5Moreover, this experiment con- ﬁrms the theoretically predicted scenario for the transition toboundary-layer turbulence responsible for boundary mixing:The growth of a density perturbation produces a staticallyunstable density ﬁeld which then overturns with small-scaleﬂuctuations inside. 16Panel ~d!of Fig. 2 is characteristic of the onset of turbulence triggered by overturning instabilitynear the slope. This turbulent mechanism is likely responsible for the formation of highly ‘‘stepped’’ temperature proﬁles nearsteep slopes in lakes. 25Moreover, the formation of sus- pended sediment layers, called nepheloid layers, at continen-tal slopes has been linked to critical angle reﬂection of inter-nal waves, 16,26and suggests that this reﬂection process plays an important role in the seawards transport of sediments inthe ﬂuid. A possible extension of this work, with a smallerslope angle to be closer to real oceanographic situations,would be to study and characterize the long time behaviorand diffusion process of such particle layers toward the ﬂuidinterior. ACKNOWLEDGMENTS The authors warmly thank J.-C. Ge ´minard and J. Som- meria for helpful suggestions. This work has been partiallysupported by the French Ministe `re de la Recherche grant ACI jeune chercheur-2001 No. 21-31. 1J. R. Ledwell, A. J. Watson, and C. S. Law, ‘‘Evidence for slow mixing across the pycnocline from an open ocean tracer release experiment,’’Nature ~London !364,7 0 1 ~1993!. 2K. L. Polzin, J. M. Toole, J. R. Ledwell, and R. W. Schmitt, ‘‘Spatialvariability of turbulent mixing in the abyssal ocean,’’ Science 76,9 3 ~1997!. 3O. M. Phillips, The Dynamics of the Upper Ocean ~Cambridge University Press, Cambridge, 1966 !. 4D. A. Gilbert, ‘‘Search for evidence of critical internal wave reﬂection on the continental rise and slope off Nova Scotia,’’ Atmos.-Ocean. 31,9 9 ~1993!. 5T. Dauxois and W. R. Young, ‘‘Near-critical reﬂection of internal waves,’’ J. Fluid Mech. 390, 271 ~1999!. 6H. Sandstrom, ‘‘The importance of the topography in generation and propagation of internal waves,’’Ph.D. thesis, University of California, SanDiego, La Jolla, 1966. 7C. C. Eriksen, ‘‘Observations of internal wave reﬂection off sloping bot-toms,’’ J. Geophys. 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FIG. 5. ~Color !Schlieren picture showing the isodensity lines duringthe slightly supercritical reﬂection (f/f c51.14) of an internal wave on a slope. The slope ~thick black line !has an angle g535°. The incident wave plane comes in from the left ~inclined white region near the top left cornerbetween blue and yellow regions !. The reﬂected wave plane is hardly notice- able. Vibrational parameters: f 50.32 Hz and A pp56.7 mm. This picture should be compared with Fig.2~c!.1940 Phys. Fluids, Vol. 16, No. 6, June 2004 Dauxois, Didier, and Falcon20D. E. Mowbray and B. S. H. Rarity, ‘‘A theoretical and experimental investigation of the phase conﬁguration of internal waves of small ampli-tude in a density stratiﬁed liquid,’’ J. Fluid Mech. 28,1~1967!. 21J. C. Appleby and D. G. Crighton, ‘‘Internal gravity waves generated by oscillations of a sphere,’’ J. Fluid Mech. 183, 439 ~1987!. 22B. Voisin, ‘‘Limit state of internal wave beams,’’J. Fluid Mech. 496, 243 ~2003!. 23W. Merzkirch, Flow Visualization ~Academic, New York, 1974 !.24S. B. Dalziel, G. O. Hughes, and B. R. Sutherland, ‘‘Whole-ﬁeld density measurements by ‘synthetic Schlieren,’’’ Exp. Fluids 28,3 2 2 ~2000!;F . Peters, ‘‘Schlieren interferometry applied to a gravity wave in a density-stratiﬁed liquid,’’ ibid.3, 261 ~1985!. 25D. R. Caldwell, J. M. Brubaker, and V. T. Neal, ‘‘Thermal microstructure on a lake slope,’’ Limnol. Oceanogr. 23, 372 ~1978!. 26S.A.Thorpe and M.White, ‘‘Adeep intermediate nepheloid layer,’’Deep- Sea Res., Part A 35, 1665 ~1988!.1941 Phys. Fluids, Vol. 16, No. 6, June 2004 Observation of near-critical reﬂection
1.4863936.pdf	Spin wave based parallel logic operations for binary data coded with domain walls Y. Urazuka , S. Oyabu , H. Chen , B. Peng , H. Otsuki , T. Tanaka , and K. Matsuyama Citation: J. Appl. Phys. 115, 17D505 (2014); doi: 10.1063/1.4863936 View online: http://dx.doi.org/10.1063/1.4863936 View Table of Contents: http://aip.scitation.org/toc/jap/115/17 Published by the American Institute of Physics Spin wave based parallel logic operations for binary data coded with domain walls Y . Urazuka, S. Oyabu, H. Chen, B. Peng, H. Otsuki, T. Tanaka,a)and K. Matsuyama ISEE, Kyushu University, Motooka 744, Nishi-ku, Fukuoka 819-0395, Japan (Presented 7 November 2013; received 22 September 2013; accepted 1 November 2013; published online 3 February 2014) We numerically investigate the feasibility of spin wave (SW) based parallel logic operations, where the phase of SW packet (SWP) is exploited as a state variable and the phase shift caused by the interaction with domain wall (DW) is utilized as a logic inversion functionality. A designedfunctional element consists of parallel ferromagnetic nanowires (6 nm-thick, 36 nm-width, 5120 nm-length, and 200 nm separation) with the perpendicular magnetization and sub- lm scale overlaid conductors. The logic outputs for binary data, coded with the existence (“1”) or absence(“0”) of the DW, are inductively read out from interferometric aspect of the superposed SWPs, one of them propagating through the stored data area. A practical exclusive-or operation, based on 2 p periodicity in the phase logic, is demonstrated for the individual nanowire with an order ofdifferent output voltage V out, depending on the logic output for the stored data. The inductive output from the two nanowires exhibits well deﬁned three different signal levels, corresponding to the information distance (Hamming distance) between 2-bit data stored in the multiple nanowires. VC2014 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4863936 ] I. INTRODUCTION Concept of the spin wave (SW) based logic is considered as a possible solution to overcome the fundamental performance limit in charge based CMOS devices.1,2The sim- ple device structure and charge less data processing in the SW devices have extent potential for low power consumption and ultimate downsizing. In addition, implementation of thedomain wall (DW) enables nonvolatile and reconﬁgurable functionarity. 3,4However, further unique advantage will be desirable to explore outstanding performance, exceeding theconventional devices. Parallel logic operation is one of prom- ising performances, which cannot be realized with transistor based circuits. Recently, multi-frequency SW logic circuitshave been proposed for the parallel logic operation. 5In the present report, we describe and analyze another type of spatially parallel logic operations utilizing multiple SW wave-guides with perpendicular anisotropy. Inductive coupling between multiple nanowires and strip conductors are utilized as input/output elements array in parallel logic operations. II. NUMERICAL MODEL Figure 1shows a schematic ﬁgure of the designed device structure, optimized through preliminary numerical simula- tions. Perpendicularly magnetized nanowires with dimensionsof 6 nm-thick, 36 nm-width, 5120 nm-length, and 200 nm sep- aration are assumed as multiple spin wave guides. The wave guide size was determined, considering the progressing litho-graphic technique and high quality thin ﬁlm processing for metallic ferromagnets. The wire is divided into one dimen- sional numerical cell with the discretization length of1.25 nm so that the spatial proﬁles of the spin wave and the domain wall ( /C2410 nm) can be treated as a quasi continuous model. The effective demagnetizing ﬁelds are calculated byintegrating the apparent surface charge at individual numeri- cal cells. The successive logic operations were analyzed by solving the Landau-Lifshitz-Gilbert equation with a ﬁnite dif-ferential method. The following material parameters are assumed in the simulation; the exchange stiffness constant A¼3.0/C210 /C07erg/cm, Gilbert damping constant a¼0.01, the gyromagnetic constant c¼1.76/C2107rad/s/Oe. The val- ues of saturation magnetization Msand the perpendicular ani- sotropy ﬁeld Hkare optimized by comparing the device performance, particularly the DW induced logic inversion functionality as described in Sec. III. The SW packet (SWP) is generated with non-uniform Oersted ﬁelds induced withone cycle application of the pulsed microwave current through the overlaid conductor strips, noted as spin wave generators (GE1, GE2) in the ﬁgure. The pulse amplitude andthe frequency are chosen to be 1 mA and 5 GHz, respectively. The excitation frequency was chosen to be 5 GHz at which the most efﬁcient SWP emission was achieved. The conduc-tor width and the spacing from the SW guide surface were designed as 100 nm and 30 nm, respectively. The inductive output voltage ( V out) from the SWP was evaluated from the time derivative of the magnetic ﬂux in the detection area with lateral dimension of 100 /C2472 nm2, as noted by broken line in the ﬁgure. A pair of Neel type domain walls, a binary in-formation carrier, is stabilized by using of artiﬁcial pinning sites. The local 10% reduction of H kwith lateral length of 230 nm was assumed as a pinning site. III. RESULTS AND DISCUSSION One of key functions in the proposed logic concept is realizing an accurate p-phase shifter with the DW. The Neela)Author to whom correspondence should be addressed. Electronic mail: t-tanaka@ed.kyushu-u.ac.jp. 0021-8979/2014/115(17)/17D505/3/$30.00 VC2014 AIP Publishing LLC 115, 17D505-1JOURNAL OF APPLIED PHYSICS 115, 17D505 (2014) type domain wall is the ground state in the nanowire with the assumed lateral dimension and the investigated range of ma- terial parameters. Figure 2presents values of phase shift D/ induced with a pair of Neel walls investigated for variousvalues of M sandHk. It can be seen in the ﬁgure that the D/ is varied from 0.8 pto 1.2 p, depending on the value of Ms andHk. The results suggest that the probable origin of the phase shift is attributable to internal ﬁelds modulation inside the DW, which is related to the values of MsandHk.A s shown in the ﬁgure, the D/can be adjusted as pby adopting the values of Ms¼500 emu/cm3andHk¼6.4 kOe for the assumed dimension of the nanowire. Resultantly, a pair of Neel walls can be utilized as a p-phase shifter, offering logic inversion functionality. The logic operation starts from the simultaneous emis- sion of a pair of SW packets from the SW generators (GE1,GE2) and following spontaneous propagation to the detec- tion area. The micromagnetic conﬁguration of the super- posed SW packets at the detection area is modulated by theNeel wall pair located at one side of the propagation path. Snap shots of the micromagnetic proﬁle of the superposedSW packets at the detection area are presented in the upper panel of Fig. 3. When the DW-pair is located at one of the two pinning sites, corresponding to binary data (0,1) or (1,0), relative phase shift for the SW packets becomes p, and the superposed SW packets exhibit node at the center of thedetection area, as shown in Fig. 3(a). On the other hand, when the two DW-pairs are located at both of the pinning sites or no DW-pair, corresponding to (1,1) and (0,0), no dis-tinguished phase difference is observed between the SWPs. In this case, an anti-node appears at the detection area (Fig. 3(b)). The lower panels in the ﬁgure present the transients of the out of plane component of the magnetic ﬂux / z,e v a l - uated at various points by artiﬁcially shifting the detection area along the nanowire. As shown in the ﬁgure, the alternative FIG. 1. Schematic of a designed parallel logic device, where a pair of DWs is exploited as a binary information, acting as a logic inversion element for propagating SW packets. The interferometric logic outputs in the multiple wave guides are inductively read out through overlaid ﬂux detection area. FIG. 2. DW induced phase shift D/of propagating SW packet evaluated for various perpendicular anisotropy ﬁelds and saturation magnetization. FIG. 3. (a) and (b) Snapshots of the micromagnetic conﬁguration for SW packets superposed at the detection area. (c) and (d) Transient of time deriv-ative for / z, out of plane component of the ﬂux, evaluated at various points along the nanowire. FIG. 4. The inductive output voltage Voutfor various logic inputs coded with existence (“1”) and absence (“0”) of the DW pair. The signiﬁcant difference in the Voutdemonstrates the exclusive-OR operations for the stored data.17D505-2 Urazuka et al. J. Appl. Phys. 115, 17D505 (2014) d/z/dtchange takes minimum (maximum) amplitude at the anti-node (node) of the superposed SWPs. The above mentioned signiﬁcant effect of the DW on the interferometric interaction between the SWPs can beapplicable to the exclusive-OR (EXOR) logic operation, as demonstrated in Fig. 4. The binary data “0” and “1” are coded with the absence and existence of the DW-pair in thepinning sites. An order of different inductive output voltage V outcorresponds to the EXOR logic output for the two-bit data stored in the nanowire, which reﬂects the difference inthe magnetization conﬁguration on the superposed SW pack- ets, as shown in Fig. 3. Somewhat enlarged tail in the V outis ascribed to the inﬂuence from the SWP reﬂection at the wireends. A successful parallel logic operation performed with the multiple wave guides is presented in Fig. 5. The results indi- cate that the logic output from the exclusive-OR operation in the individual nanowire is inductively integrated, which leads to the distinguished three different output levels ofV out. Consequently, the amplitude of Voutis related to aninformation distance (Hamming distance) between the binary data train, noted as data_A and data_B (shown in Fig. 1). Unfavorable residual Vout, shown in Fig. 5(a), is attributable to the stray ﬁelds from the neighboring DWs. The magneto- static interference is a reduction limit of the inter-wire sepa- ration, chosen to be 200 nm in the present design. Theprinciple of the proposed parallel logic can be extended to a big data train by increasing the number of wave guides, which is promising for image processing, speech recognition,data mining, etc. IV. CONCLUSION Feasibility of the SW based parallel logic operations were investigated through micromagnetic simulations. A designed prototype device consists of perpendicularly magnetized multiple ferromagnetic nanowires (6 nm-thick, 36 nm-width, 5120 nm-length, and 200 nm separation) for SW guides and overlaid conductors for Input-Output opera-tion. A pair of Neel walls was utilized as a binary informa- tion carrier, which causes pphase shift for the SW packet propagating over the wave guide with optimized material pa-rameters. The exclusive-OR operation results for the individ- ual wave guide were integrated with inductive coupling. Resultantly, the output voltage for the multiple wave guidesis related to an information distance (Hamming distance) between the binary data train stored in them. The obtained simulation results present a possibility for parallel data proc-essing for further large data train stored in wave guides array. 1M. P. Kostylev, A. A. Serga, T. Schneider, B. Leven, and B. Hillebrands, Appl. Phys. Lett. 87, 153501 (2005). 2A. Khitun, M. Bao, and K. L. Wang, Superlattices Microstruct. 38, 184 (2005). 3R. Hertel, W. Wulfhekel, and J. Kischner, Phys. Rev. Lett. 93, 257202 (2004). 4K. Nagai, Y. Cao, T. Tanaka, and K. Matsuyama, J. Appl. Phys. 111, 07D130 (2012). 5A. Khitun, J. Appl. Phys. 111, 054307 (2012). FIG. 5. Parallel logic operation results with two wave guides. Distinguished three different amplitudes of Voutdepend on the information distance between the two-bit data train stored in the nanowire.17D505-3 Urazuka et al. J. Appl. Phys. 115, 17D505 (2014)
5.0005472.pdf	J. Appl. Phys. 127, 133906 (2020); https://doi.org/10.1063/5.0005472 127, 133906 © 2020 Author(s).Reduction of the switching current in perpendicularly magnetized nanomagnets using an antiferromagnetic coupling structure Cite as: J. Appl. Phys. 127, 133906 (2020); https://doi.org/10.1063/5.0005472 Submitted: 21 February 2020 . Accepted: 20 March 2020 . Published Online: 06 April 2020 Keisuke Yamada , Keisuke Kubota , and Yoshinobu Nakatani ARTICLES YOU MAY BE INTERESTED IN Dissipative couplings in cavity magnonics Journal of Applied Physics 127, 130901 (2020); https://doi.org/10.1063/1.5144202 Temperature-FORC analysis of a magnetocaloric Heusler alloy using a unified driving force approach (T*FORC) Journal of Applied Physics 127, 133902 (2020); https://doi.org/10.1063/5.0005076 Spintronics with compensated ferrimagnets Applied Physics Letters 116, 110501 (2020); https://doi.org/10.1063/1.5144076Reduction of the switching current in perpendicularly magnetized nanomagnets using an antiferromagnetic coupling structure Cite as: J. Appl. Phys. 127, 133906 (2020); doi: 10.1063/5.0005472 View Online Export Citation CrossMar k Submitted: 21 February 2020 · Accepted: 20 March 2020 · Published Online: 6 April 2020 Keisuke Yamada,1,a) Keisuke Kubota,2and Yoshinobu Nakatani2,b) AFFILIATIONS 1Department of Chemistry and Biomolecular Science, Faculty of Engineering, Gifu University, Gifu 501-1193, Japan 2Graduate School of Informatics and Engineering, University of Electro-Communications, Chofu, Tokyo 182-8585, Japan a)Author to whom correspondence should be addressed: yamada_k@gifu-u.ac.jp b)Email: nakatani@cs.uec.ac.jp ABSTRACT This paper reports a current-induced magnetization switching with a nanosecond-scale pulse current in a nanomagnet using a perpendicu- larly magnetized synthetic antiferromagnetic coupling (p-AFC) structure. The results indicate that the magnetization switching current in the p-AFC structure is less than that in the single-nanomagnet structure with perpendicular anisotropy when the differences in thickness and saturation magnetization between the upper and lower layers of the p-AFC structure are small and the Gilbert damping constant is alsosmall. The results also show that the p-AFC structure can reduce the switching current when the pulse duration is short and its structure iseffective for a high-speed switching. The results of this study shall be useful in the design of spin-transfer torque random access memory. Published under license by AIP Publishing. https://doi.org/10.1063/5.0005472 I. INTRODUCTION Magnetization switching in nanomagnets using spin-transfer torque (STT) is a novel technique for memory development. 1,2 Several related studies3–20have been conducted, as this technology can be used to develop next-generation nonvolatile STT magnetic random access memory (STT-MRAM) with low-power consump- tion and high-speed magnetization switching. When a high-density magnetic nonvolatile memory is realized, a magnet for storing information is required to be easily written, that is, to have a low switching current ( Isw) and reduce the element size. However, when the element size is reduced, the thermal stability factor ( Δ)o f the magnet for retaining the information is also reduced; therefore, an element structure with a small element size and a high thermal stability factor is required.8–16We have previously shown that decreasing the Gilbert damping constant ( α) is effective with a decreasing Iswwhen the pulse current duration is long. However, when the pulse current duration with nanosecond order is used, Isw does not decrease no matter how much the Gilbert damping cons- tant decreases.21–23Therefore, a technique for reducing Iswis neces- sary, particularly for a short duration of pulse current.To satisfy a low Iswand high thermal stability requirement, exchange-coupled composite (ECC) structures24–26and synthetic antiferromagnetic coupling (AFC) structures27–33have been pro- posed as device structures. The ECC structure is a structure in which a soft magnetic layer with a small anisotropy magnetic field (Hk), and a hard magnetic layer with a high Hkare ferromagneti- cally coupled. ECC structures have been used to reduce the switch-ing field and achieve high thermal stability in a hard disk drive(HDD). 24–26The AFC structure has been proposed for STT- MRAM with in-plane magnetization type, and the AFC has a structure using a magnetic material for the upper and lower layersand a non-magnetic material (e.g., Ru) for the intermediatelayer. 27–29,33In the in-plane STT-MRAM type, a complex magnetic structure appears at the time of magnetization switching due to the magnetic pole appearing at the end of the storage element, which increases the switching current. The AFC structure has been pro-posed as a structure that can reduce the switching current byreducing the number of magnetic poles appearing at the edge ofthe element because magnetic layers with different magnetization directions are stacked. Both ECC and AFC structures have improved thermal stability and reduced switching current.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 133906 (2020); doi: 10.1063/5.0005472 127, 133906-1 Published under license by AIP Publishing.In the studies of the AFC structure with perpendicular magnetic anisotropy (p-AFC) reported so far, nanostructures with the p-AFC structure and free layers are combined to eval-uate the magnetic structure of the structure experimentally andto measure the conduction characteristics such as magnetoresis-tance (MR). 34–36In addition, there are reports on high- frequency magnetization pre cession in the p-AFC layer using calculations to improve the magnetization reversal of the freelayer. 37However, there is scant literature on the switching current in the p-AFC structure, especially on the effect of mag-netic material parameters of the layer thickness, the Gilbert damping constant, and the saturation magnetization on the switching current.In this study, we investigated the change in the magnetization switching current with the nanosecond-scale duration of pulse current in nanomagnets of a p-AFC structure using simulations.The change in the switching current with the thicknesses of theupper and lower layers was examined. Then, the saturation magne-tization effect of the upper and lower layers on the switching current was determined. Finally, an empirical formula for the switching current in the p-AFC structure was derived. II. MODEL DEFINITION A micromagnetic model was used in the simulations. 3,4,21The magnetization motion was calculated using the Landau –Lifshitz – FIG. 1. (a) Illustrations of the simula- tion model. Left: the p-AFC structurehas a diameter of 30 nm. Spin currentis injected from the lower layer with respect to the z-axis. Right: the calcu- lation model of each layer divided bythe calculation cell dimensiondz= 0.25 nm when h 1= 2 nm and h2- = 1.0 nm. Exchange stiffness constants that acted on each layer are indicatedbyAand A IL. (b) Result of required anisotropy constant (lower layer) for upper layer thickness and saturation magnetization. Top and bottom axeshave as reference model (1) andmodel (2), respectively. (c) Magnetic parameters used in each model. (1) The film thickness dependence model,(2) the saturation magnetization depen-dence model, and (3) the single-layer model.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 133906 (2020); doi: 10.1063/5.0005472 127, 133906-2 Published under license by AIP Publishing.Gilbert (LLG) equation with a spin-transfer torque term, d~m dt¼/C0 j γj(~m/C2~Heff)þα~m/C2d~m dt/C0μBgPI 2eMsV~m/C2(~m/C2~nS), (1) where γ,~m,~Heff,α,μB,g,P,I,e,Ms,V,a n d ~nSare the gyro- magnetic ratio, magnetic moment, effective magnetic field,Gilbert damping constant, Bohr magneton, g-factor, spin polari-zation, current, electron charge, saturation magnetization, magnet volume, and unit vector of the spin-transfer torque, respectively. As a simple model, the p-AFC structure was adouble-layer structure [thickness: upper layer ( h 2) and lower layer ( h1)], consisting of two circular plates with antiferromag- netic coupling, as shown in Fig. 1(a) . Usually, the Ru layer is inserted between two layers to realize antiferromagnetic cou- pling. However, we neglected the Ru layer in our model forsimplicity. The diameter of each layer was d= 30 nm. Each layer was divided in the film thickness direction [one-dimensional (1D) model]. The calculation cell dimension dzwas 0.25 nm. In the simulation, it was assumed that the spin diffusion lengthwas less than the film thickness, and the spin torque acts only on the lower layer [ Fig. 1(a) ]. The pulse current with a rectangular waveform had a duration oft p= 1.0 ns and a rise and fall time of zero. The initial magnetiza- tion had a magnetization angle θinit¼7:40/C14of the lower layer to thez-axis (172.6° for the upper layer). The current with spin polar- ization P= 1.0 flowed along the z-axis. When the angle of the lower layer magnetization reached θcrit¼172:6/C14with respect to the z-axis (7.4° for the upper layer), it was regarded as magnetization switching and estimated the switching current density ( Jsw). The Oersted field generated by the current was ignored. All the simula- tions were performed at a temperature T=0 . In this study, we investigated the changes in the switching current using two models: (1) film thickness dependence and (2)saturation magnetization dependence. In model (1), h 1= 2 nm was fixed, and h2was changed to 0.25 –2 nm. The saturation magnetiza- tion was Ms1,2= 600 emu/cm3at both layers, the perpendicular FIG. 2. (a) and (b) Magnetization switching of each layer in the p-AFC structure in model (1) represents an average value of magnetization along the z-axis. (a) h2= 1.0 nm, α= 0.001, current density J= 5.00 × 1010A/m2and (b) h2= 2.0 nm, α= 0.001, J= 1.00 × 1012A/m2. FIG. 3. (a) Variation in the switching current density ( Jsw) with the interlayer exchange constant AILfor various αin model (1) when h2= 1.0 nm. (b) Variation in the optimized interlayer exchange constant AILoptwithαfor each h2in model (1).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 133906 (2020); doi: 10.1063/5.0005472 127, 133906-3 Published under license by AIP Publishing.magnetic anisotropy constant Kuhad the same value at both the upper and lower layers, and Kuwas estimated when the thermal stability factor of combined layers was Δ= 60. The thermal stability factor was calculated from the minimum energy barrier when themagnetization was switched. In model (2), the saturation magnetization was fixed at M s1= 600 emu/cm3for the lower layer and Ms2= 150 –525 emu/cm3 at the upper layer. Here, h1=h2= 2 nm was fixed, the perpen- dicular magnetic anisotropy magnetic field Hkwas the same for the upper and lower layers ( Hk1=Hk2), and the value of Ku was estimated when the thermal stability factor of combined layers was Δ= 60. Figure 1(b) shows the changes in the required anisotropy constant of the lower layer ( Ku1)f o rt h e upper layer ( h2) thickness and saturation magnetization in each model. In model (1), the required anisotropy constantd e c r e a s e da st h ef i l mt h i c k n e s so ft h eu p p e rl a y e ri n c r e a s e d , because the total volume increased. In model (2), the required anisotropy constant was smaller than that of model (1). Sincewe assumed that H k1=Hk2in model (2), Ku2is larger than Ku1 for small Ms2. The switching current by a single-layer (SL) model (3) was also examined by comparing the switching currents of models (1) and(2) [h= 2 nm, K u=3 . 4 8M e r g / c m3(Δ=6 0 ) , Ms= 600 emu/cm3]. The following magnetic parameters were used in the simulations: gyro- magnetic ratio γ= 17.6 Mrad/(Oe s) and exchange stiffness constant A=1 . 0 μerg/cm (exchange constant within each layer). For the inter- layer exchange constant AIL(exchange constant at the interface between the upper and lower layers), a value that minimizes theswitching current under each condition was used [ Aand A ILfor each divided layer are shown in the right illustration of Fig. 1(a) ]. The results for each model are summarized in Fig. 1(c) . III. 1D SIMULATION RESULTS A. Behavior of magnetization switching of each layer by applying a current The state of magnetization switching of each layer in the p-AFC structure in model (1) is shown in Figs. 2(a) and2(b). The conditions in Figs. 2(a) and 2(b) are as follows: (a) h2= 1.0 nm, α= 0.001, and current density J= 5.00 × 1010A/m2; (b) h2= 2.0 nm, α= 0.001, and J= 1.00 × 1012A/m2.I nFig. 2(a) , the magnetizations of each layer gradually start to switch at time t= 0.1 ns. At t∼0.65 ns, the mutual magnetizations are parallel to the x–yplane, and at t∼0.99 ns, they mutually switch. Under the condition in Fig. 2(a) , the magnetization switching occurs in each layer while maintaining the p-AFC structure. For the condition in Fig. 2(b) , although the magnetization of the lower layer (first) is switched,the upper layer (second) is not switched when the p-AFC structureis not maintained. From these results, it is found that the thickness ofh 2needs to be thinner than h1due to induced magnetization switching while maintaining the p-AFC structure. B. Estimation and optimization of the interlayer exchange constant Figure 3(a) shows the change in the switching current due to the interlayer exchange constant AILand the Gilbert damping cons- tantαin model (1) when h2= 1.0 nm. When αis large, the switch- ing current decreases with AIL. However, when αis small, the switching current slightly increases as AILdecreases. When AILis very small in either case, the antiferromagnetic structure is notmaintained, and the magnetization switching does not occur. From these results, it can be seen that the optimized interlayer exchange constant ( A ILopt) that minimizes the switching current is different for each α.Figure 3(b) shows variations in AILoptwith αfor various h2 values in model (1). AILoptchanges not only with αbut also with h2. In addition, it is small when αis large; however, AILoptincreases with α.AILoptis a value in the same order as the value obtained by experi- ments.33(Note that the Ru layer is inserted between two layers to realize antiferromagnetic coupling. The exchange coupling value,A IL, can be controlled by changing the thickness of the Ru layer.) C. Film thickness ( h2) dependence on Jswandα Figure 4(a) shows the variation of switching current density (Jsw) with h2andα, using AILopt(tp= 1.0 ns) in model (1). When α FIG. 4. (a) Variation in Jswwithαfor each h2when tp= 1.0 ns and AILoptin model (1). The black filled circle is Jswin the SL structure. (b) Results of Jswfor h2when α= 0.001 and 1. The black dotted line indicates a linear line.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 133906 (2020); doi: 10.1063/5.0005472 127, 133906-4 Published under license by AIP Publishing.is small, Jswdecreases more than the SL structure, especially near α< 0.01, and Jswdecreases as h2thickness increases. In the region, where αis large ( α> 0.03), Jswis almost consistent with the Jswof the SL structure. When h2= 1.5 nm and α< 0.001, Jswdecreases by approximately 66% compared to the SL structure. From theseresults, J swbecomes smaller when αandh2are smaller and thicker, respectively. In other words, the p-AFC structure is more superior than the SL structure. Figure 4(b) summarizes the variations in Jsw with αand h2when α= 0.001 and 1. When αis small ( α= 0.001), Jswdecreases almost linearly with h2. (Note that there is a deviation in linearity with h2= 1.5 nm. The reason for this is that as the thickness of h2increases, the difference in the trajectories of the magnetization reversal of the upper and lower layers increases, anda larger current is required compared with the ideal case.) Based on these findings, it can be considered that the switching current is proportional to the difference in the film thickness of each layer (Jsw∝h1–h2), and Jswbecomes constant regardless of the value of h2when αis large ( α= 1). D. Pulse duration dependence on Jswandαat h2= 1.0 nm Figure 5(a) shows the change in Jswwith αath2= 1.0 nm for various pulse durations ( tp= 0.1, 1.0, and 10 ns), using AILopt(each tp) in model (1). Jswis almost the same as that of the SL structure (the black dotted line) when the pulse duration is long; that is, FIG. 5. (a) Variation in Jswwithαfor each pulse duration ( tp= 0.1, 1.0, and 10 ns) when h2= 1.0 nm and AILoptin model (1). The black dotted line shows the result of the SL structure for each pulse duration ( tp= 0.1, 1.0, and 10 ns). (b) and (c) Results of Jswfortpwhen (b) α= 0.001 and (c) α= 0.01 when h2= 1.0 nm.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 133906 (2020); doi: 10.1063/5.0005472 127, 133906-5 Published under license by AIP Publishing.there is no advantage of the p-AFC structure. However, Jswis smaller than that of the SL structure when the pulse duration is short. Furthermore, as the pulse duration is shorter, the value of α increased where the effect of switching current reduction isobtained. This indicates that the p-AFC structure is effective in theshort-pulse duration region, that is, it shows that the p-AFC struc- ture can reduce the switching current by a single short-pulse dura- tion. In addition, it is unnecessary to use a material with a small α as the pulse duration decreases. This could be resolved that it hasbeen a problem with the conventional single-layer structure, evenin a region where αis small. Figures 5(b) and 5(c) show the effect of t ponJswwith α= 0.001 and 0.01, respectively, when h2= 1.0 nm. Jswof the p-AFC structure is smaller than that of the SL structure, and Jsw decreases particularly as tpis shorter. In the case of α= 0.001, Jsw decreases about half when tpis shorter than 2 ns. In the case of α= 0.01, Jswof the p-AFC is larger than that of the SL structurewhen tpis longer than 2 ns. However, Jswdecreases when tpis shorter than ∼2 ns and decreases to about 60% when tp= 0.5 ns. E. Saturation magnetization ( Ms2) dependence on Jsw andα Figure 6(a) shows Jswas a function of αfor the saturation magnetization in model (2) at tp= 1.0 ns. Here, the optimized inter- layer exchange constant ( AILopt) that minimizes the switching current was also used. The result shows the same tendency as thatof model (1). When αis small ( α< 0.01), J swsignificantly decreases in proportion to the value of Ms2.Jswreduction of approximately 65% compared to that in the SL structure was obtained forM s2= 450 emu/cm3when α= 0.001. In the region, where αis large (α> 0.03), Jswbecomes a constant irrespective of the value of Ms2. Figure 6(b) summarizes the results of JswforMs2when α= 0.001 and 1, similar to Fig. 4(b) . When αis small ( α= 0.001), Jsw decreases almost linearly with Ms2, and a tendency of Jsw∝ Ms1–Ms2is obtained. [The trajectories of the magnetization reversal of the upper and lower layers are different when Ms2= 450 emu/cm3, as observed in Fig. 4(b) . The result deviates from linearity.] When α is large ( α=1 ) , Jswis almost constant regardless of Ms2. F. Pulse duration dependence on Jswandαat Ms2= 300 emu/cm3 Figure 7(a) shows the change in Jswwith αatMs2=3 0 0e m u / c m3 for various pulse durations ( tp= 0.1, 1.0, and 10 ns), using AILopt (each tp) in model (2). The relationship between Jswandαfortpis the same as in model (1), as described in Sec. III D . The result shows that the p-AFC structure can reduce the switching current by a single short-pulse duration. Figures 7(b) and 7(c) show the effect of tponJswwith α= 0.001 and 0.01, respectively, when Ms2= 300 emu/cm3. These results are also the same as in Figs. 5(b) and 5(c). In the case of α= 0.001 ( α= 0.01), Jswdecreases about half (60%) when tpis shorter than 2 ns (0.5 ns) as shown in Figs. 7(b) and7(c). G. Derivation of empirical equation and comparison with simulation results The equation of the switching current in the AFC structure is examined. To analytically derive the equation of the switchingcurrent in the AFC structure, it is necessary to solve two LLG equa- tions whose polar angle and azimuth angle are unknown variables. Therefore, the analysis of the switching current is difficult. Here,we propose an equation for the switching current based on theresults obtained from the simulation and empirical equation previ-ously obtained for SL structures. 21 The switching current ( Isw) in the SL structure previously obtained is expressed by the following equation:21 Isw¼2eMsV μBgPHeff KαγþC1 tp/C20/C21 : (2) The first term on the right-hand side of the equation is propor- tional to α, and the second term is inversely proportional to the pulse duration tp. When αis large, the first term on the right-hand FIG. 6. (a) Results of Jswfor various values of αfor each Ms2when tp= 1.0 ns and AILoptin model (2). The black filled circle is Jswin the SL structure. (b) Results of JswforMs2when α= 0.001 and 1. The black dotted line indicates a linear line.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 133906 (2020); doi: 10.1063/5.0005472 127, 133906-6 Published under license by AIP Publishing.side is larger than the second term. For this reason, Iswdecreases due to the reduction of αin a region where αis large. When αis small, the second term on the right-hand side is larger than thefirst term. Therefore, when αis small, I swdoes not decrease even if αis further reduced. Furthermore, Iswincreases when tpis short- ened. This shortening occurs because even when αis small and no loss occurs for the spin dynamics, the magnetization reversalrequires a current proportional to the spin angular momentum ofthe element. Moreover, a large current is required to inject therequired angular momentum within a short time. In model (1), within the limit of α< 0.001, the simulation result represents the relationship of J sw∝h1–h2(=V1–V2), as shown in Fig. 4(b) . Similar to model (2), the Jsw∝Ms1–Ms2relationship is presented in Fig. 6(b) . In each model, Iswin the region, where αis large, is consistent with Iswin the SL structure. As described in Eq. (2), the first term on the right-hand side of the equation represents Iswin the region where αis large, and the second term on the right-hand side represents Iswwhere αis small. From the simulation results obtained in this study, we find the equation that is considered as Iswof the AFC structure as follows: Isw¼2e μBgP(Meff sVeffHeff K)αγþ(Ms1V1/C0Ms2V2)C1 tp/C20/C21 ¼2e μBgP2ΔkTαγþ(Ms1V1/C0Ms2V2)C1 tp/C20/C21 , (3) FIG. 7. (a) Variation in Jswwithαfor each pulse duration ( tp= 0.1, 1.0, and 10 ns) when Ms2= 300 emu/cm3andAILoptin model (2). The black dotted line shows the result of the SL structure for each pulse duration ( tp= 0.1, 1.0, and 10 ns). (b) and (c) Results of Jswfortpwhen (b) α= 0.001 and (c) α= 0.01 when Ms2= 300 emu/cm3.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 133906 (2020); doi: 10.1063/5.0005472 127, 133906-7 Published under license by AIP Publishing.where C1is 5.48.21In the region where αis large, Iswof the AFC structure is consistent with the Iswof the SL structure with the same thermal stability factor. Therefore, the first term in Eq. (3) expresses the thermal stability factor or the product of effectiveM eff s,Veff, and Heff Kfor the entire AFC structure. In the region where αis small, because Iswis the difference in Msor volume of each layer, the second term in Eq. (3)is the difference in MsVin each layer. Ac o m p a r i s o no fE q . (3)and the simulation results is pre- sented in Figs. 8(a) and 8(b) when tp= 1.0 ns. Equation (3)is in considerable agreement with the simulation results and shows the effectiveness of the proposed model. However, the differencebetween Eq. (3)and the simulation results becomes large when h 2=1 . 5n m a n d Ms2= 450 emu/cm3. As shown in Fig. 2(b) ,t h e magnetization switching does not occur when h1=h2(or Ms1=Ms2), which indicates that Eq. (3)deviates from the simula- tion results in the condition. This result is also consistent with thesimulation results in the case of α= 0.001, which deviate from linearity (the black dotted line shows a linear line) as shown in Figs. 4(b) and6(b). In a region where αis small, current proportional to the spin angular momentum of the element is required for themagnetization reversal. In the AFC structure, the total spin angular momentum reversed by the magnetization reversal is not the sum of M sVof both layers but is only proportional to the difference between MsVof both layers. From this consider- ation, the second term in Eq. (3)is taken to be the difference between the two values of MsV.It can be also considered that in the AFC structure, the transfer of the spin angular momentum is performed without loss in the upper and lower layers duringthe magnetization reversal. IV. 3D SIMULATION RESULTS Until Sec. III G , the simulations at T= 0 K have been per- formed using a simple one-dimensional (1D) micromagneticmodel, which is a simple model because the size of the targetmagnetic material is small. However, in a real magnetic material, the direction of magnetization is not uniform, and the tempera- ture can influence the switching time and current. The three-dimensional (3D) micromagnetic simulations were performedunder T= 300 K, when t p= 1.0 ns and h2= 1.0 nm in model (1), to confirm the validity of the proposed p-AFC structure. Furthermore, the change in the magnetization switching probabil- ity with the switching current, when thermal fluctuation ispresent, was also investigated. In the 3D simulations, we examined two models: model (1) and the SL structural model. The diameter of each layer was d=3 0n m , which was divided into rectangular prisms with dimensions of 1.875 × 1.875 × 1.0 nm 3. The following magnetic parameters were used in the 3D simulation in model (1): Ms1,2= 600 emu/cm3, Ku1,2=3 . 1 8M e r g / c m3,h1= 2 nm, h2= 1 nm, A=1 . 0 μerg/cm, and AILused the optimal value AILopt, which is similar to the value of the 1D model, and in the SL structural model: Ms= 600 emu/cm3, Ku=3 . 4 4M e r g / c m3,h1=2n m , a n d A=1 . 0 μerg/cm. In the finite temperature calculation, the effect of the thermal fluctuation of mag-netization ( T= 300 K) was calculated by adding it into Eq. (1). 38The simulation was performed 300 times at each current density (the number of iterations is 300 times), and the switching probability Psw was calculated. The switching current density Jswwas defined when Pswis 0.5. Figure 9(a) shows the change in Jswwith αath2= 1.0 nm for tp= 1.0 ns, using AILoptin model (1). The results are similar to those of the 1D simulations. Jswis similar to that of the SL structure (the black dotted line) in the region where αis large ( α> 0.03). When α is small, Jswdecreases when compared to the SL structure, particu- larly near α< 0.01, and Jswdecreases as αdecreases. Jswdecreases by approximately 54% compared to the SL structure when α= 0.001. This result shows that the p-AFC structure can reduce the switching current in the region where αis small even at finite temperature in the 3D micromagnetic model. Figures 9(b) –9(e) show the change in Pswwith J/Jsw[Jis the applied current density and is normalized by the value of Jsw to compare model (1) with the SL structure] for various αin h2= 1.0 nm and tp= 1.0 ns. The change of Pswwith J/Jswin model (1) is very similar to that observed in the SL structure [ Figs. 9(c) –9(e)] except for very small αcase [ Fig. 9(b) ]. These results show that the p-AFC structure simply reduces the switching current. FIG. 8. Comparison of Eq. (3)and the simulation results. (a) and (b) are the simulation results (dots) and Eq. (3)(dot lines) in models (1) and (2) when tp= 1.0 ns, respectively. Black dots and the dotted line indicate the SL structure.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 133906 (2020); doi: 10.1063/5.0005472 127, 133906-8 Published under license by AIP Publishing.V. CONCLUSION We simulated the magnetization switching in the p-AFC structure, compared the Jswof the ferromagnetic SL structure and investigated its superiority. First, the dependence of Jswonα when the thickness of the lower layer ( h1) in the p-AFC structure is fixed, and the thickness of the upper layer ( h2)i sc h a n g e d ,i s investigated. From the results, when αis small (or large), the value of Jsw∝h1–h2is obtained (note that Jswis constant regardless of the value of h2). Similarly, magnetization switching simulations were performed when upper layer and lower layer thicknesses were fixed, and the saturation magnetization of theupper layer was varied. Similar to the results obtained for the filmthickness variations, when αis small (or large), the value of J sw∝Ms1–Ms2is obtained ( Jswis also constant irrespective of Ms2 value). In addition, from the simulation results of varying pulseduration, we confirm that the switching current is reduced, espe- cially for a short-pulse duration. The required αincreases as the pulse duration decreases. We also simulated the magnetizationswitching in the p-AFC structure used by the 3D micromagneticmodel with T= 300 K. The results are similar to those of the 1D model. From these results, it is demonstrated that the p-AFC structure can be used to provide a difference in the film thickness or satura- tion magnetization of each layer. Additionally, the p-AFC structure can reduce the switching current with a short-pulse duration, which has been a problem with the conventional SL structure. Because a nanomagnetic material with a p-AFC structure has a double-layer structure, the p-AFC structure can be larger in volume than a single-layer structure. Therefore, a large anisotropic constant is not required to maintain thermal stability. The p-AFC FIG. 9. (a) Variation in Jswwithαfor h2= 1.0 nm when tp= 1.0 ns and AILopt in model (1) calculated by the 3D micromagnetic simulations withT= 300 K. The black filled circle is J sw in the SL structure. The inset in (a) shows the 3D micromagnetic model. (b)–(e) Results of PswforJ/Jswwhen α= 0.001, 0.01, 0.1, and 1, respec- tively. The blue dotted line indicates Psw= 0.5.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 133906 (2020); doi: 10.1063/5.0005472 127, 133906-9 Published under license by AIP Publishing.structure is particularly effective in a finite element and is consid- ered to be a useful structure in STT-MRAM. ACKNOWLEDGMENTS We thank Dr. Tomohiro Taniguchi (AIST) for the valuable discussions on theoretical background. This study was supported inpart by the JSPS KAKENHI under Grant Nos. 15H05702 and 17H04795. We would like to acknowledge Editage ( www.editage. com) for English language editing. The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1L. Berger, Phys. Rev. B 54, 9353 (1996). 2J. C. Sloncezewski, J. Magn. Magn. Mater 159, L1 (1996). 3J. Z. Sun, Phys. Rev. B 62, 570 (2000). 4J. Miltat, G. Albuquerque, A. Thiaville, and C. Vouille, J. Appl. Phys. 89, 6982 –6984 (2001). 5J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers, and D. C. Ralph, Phys. Rev. Lett. 84, 3149 (2000). 6Y. Huai, F. Albert, P. Nguyen, M. Pakala, and T. Valet, Appl. Phys. Lett. 84, 3118 (2004). 7R. H. Koch, J. A. Katine, and J. Z. Sun, Phys. Rev. Lett. 92, 088302 (2004). 8S. Mangin, D. Ravelosona, J. A. Katine, M. J. Carey, B. D. 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1.4961927.pdf	Twisted electron-acoustic waves in plasmas Aman-ur-Rehman, , S. Ali , S. A. Khan , and K. Shahzad Citation: Phys. Plasmas 23, 082122 (2016); doi: 10.1063/1.4961927 View online: http://dx.doi.org/10.1063/1.4961927 View Table of Contents: http://aip.scitation.org/toc/php/23/8 Published by the American Institute of Physics Twisted electron-acoustic waves in plasmas Aman-ur-Rehman,1,2,a)S.Ali,3S. A. Khan,3and K. Shahzad2 1Department of Nuclear Engineering, Pakistan Institute of Engineering and Applied Sciences (PIEAS), P. O. Nilore, Islamabad 45650,Pakistan 2Department of Physics and Applied Mathematics (DPAM), Pakistan Institute of Engineering and Applied Sciences (PIEAS), P.O. Nilore, Islamabad 45650,Pakistan 3National Centre for Physics at Quaid-e-Azam University Campus, Shahdra Valley Road, Islamabad 44000, Pakistan (Received 20 June 2016; accepted 13 August 2016; published online 31 August 2016) In the paraxial limit, a twisted electron-acoustic (EA) wave is studied in a collisionless unmagne- tized plasma, whose constituents are the dynamical cold electrons and Boltzmannian hot electrons in the background of static positive ions. The analytical and numerical solutions of the plasmakinetic equation suggest that EA waves with ﬁnite amount of orbital angular momentum exhibit a twist in its behavior. The twisted wave particle resonance is also taken into consideration that has been appeared through the effective wave number q effaccounting for Laguerre-Gaussian mode pro- ﬁles attributed to helical phase structures. Consequently, the dispersion relation and the damping rate of the EA waves are signiﬁcantly modiﬁed with the twisted parameter g, and for g!1 , the results coincide with the straight propagating plane EA waves. Numerically, new features oftwisted EA waves are identiﬁed by considering various regimes of wavelength and the results might be useful for transport and trapping of plasma particles in a two-electron component plasma. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4961927 ] I. INTRODUCTION Long ago, Fried and Gould1considered the resonance and Maxwellian distribution functions to solve a set of dynamical equations for the existence of families of acoustic-like oscillations in an electron-ion hot plasma. In particular, some efforts 2were made to investigate the low-phase oscilla- tions (viz., vTc/C28x/k/C28vTh) on the cold-electron dynamical scale known as the electron-acoustic (EA) waves.2These oscillations propagate in a two-electron component plasma, whose constituents are the immobile positive ions, appearing only through the equilibrium charge-neutrality condition and the cold inertial (hot) electrons (following the Boltzmann dis- tribution) that provide inertia to maintain the wave in the pres- ence of two temperatures ( Th,Tc) and number densities ( nh0, nc0). Lots of investigations have been carried out to study the two-electron component plasmas in space3–5and laboratory devices.6,7Earth’s magnetosphere, which includes various regions, e.g., plasma sheet boundary layer, bow shock, cusp, high altitude polar magnetosphere, magnetosheath, magneto- pause, magnetotail, etc., all contain two distinct groups of electrons. Similarly, laser-induced plasmas3,6,7have also con- ﬁrmed the coexistence of hot and cold electrons. The former have energy in the range of 10–50 keV, whereas the latter have the range of 0.1–1 keV for the intensities of neodymium-glass and CO 2lasers, respectively, above the val- ues 1014W/cm2and 1012W/cm2. Gary and Tokar8studied the propagation of the EA mode in an unmagnetized uniform Maxwellian plasma and explained its existence criterion as well as damping condi- tions. Yu, Shukla, and Ong9presented the excitations of largeamplitude electromagnetic waves by coupling the EA waves to identify the growth rate and scattering instabilities in a two-electron component plasma. Berthomier et al .10 employed the pseudopotential approach to investigate the fully nonlinear EA waves and carried out the parametric anal-ysis for the proﬁles of wave potential to account for the drifted electron density and beam temperature in a four component unmagnetized plasma. In addition, El-Taibany and Moslem 11 introduced the modiﬁcations due to higher order nonlinearity and dispersion effects involving the EA waves and used the electron beam and vortex-like electron distributions.Recently, 12some investigations have been made to examine the impact of the electron superthermailty on the proﬁles of EA waves in j–distributed non-Maxwellian plasmas. Various techniques, for example, the spiral phase plates,13the cylindrical lenses,14and the synthesized holo- grams,15have been used and led to the existence of orbital angular momentum (OAM) caused by the variation of phasestructure through exp ðilhÞ, where his the azimuthal angle andlis quantum number involving the azimuthal OAM. It is now well established 16,17that photon beams can be repre- sented by the Laguerre-Gaussian (LG) mode proﬁles thatcould carry the non-zero azimuthal phase associated with the helical wavefronts. This is because of non-zero azimuthal component of the Poynting vector at every radial position inthe beam. The LG photon states are usually described by the LG functions to giving rise to twisted wave solutions instead of commonly known plane wave solutions. The usefulness ofthe LG states has been demonstrated in context of laboratoryapplications 18,19and astrophysics.20The study of plasma dynamics with ﬁnite OAM states is natural as can be seen from the recent literature survey.21,22New effects and non- trivial properties attributed to ﬁnite OAM states have beena)Electronic mail: amansadiq@gmail.com 1070-664X/2016/23(8)/082122/6/$30.00 Published by AIP Publishing. 23, 082122-1PHYSICS OF PLASMAS 23, 082122 (2016) predicted in plasmas, especially the twisted waves and the associated instabilities.23–26 Studies like plasmons23,27and phonons28,29with ﬁnite OAM states, the stimulated Raman and Brillouin backscat- terings,30the inverse Faraday effect of linearly polarized laser pulses,31and magnetic ﬁeld generation by higher order LG plasmons32have already been pointed out in plasmas. In particular, Shahzad and Ali33derived the dispersion relation for EA waves in the paraxial approximation and employedthe Gaussian and LG beam solutions using the hydrodynamicmodel. They also suggested an approximate solution for the electrostatic potential and computed the energy ﬂux of the EA waves in an unmagnetized collisionless uniform plasma.Mendonca 34employed the kinetic treatment for studying the LG plasmons with ﬁnite OAM states for the ﬁrst time and showed various unique features involving the azimuthal electron oscillations, to obtaining twisted Langmuir wavesand its Landau damping rate. The twisted plasmons withOAM states constitute nonuniform phase surfaces of the complex structure. Very recently, using kinetic description, the linearized ion vortex structures have been presented. 35 In this paper, we investigate the dispersive properties of the twisted EA waves and twisted damping rate in an unmag- netized collisionless two-electron component plasma. By uti-lizing the Vlasov-Poisson set of equations, a generalizedexpression of the dielectric function is obtained in the pres- ence of both axial and azimuthal velocity components. It is found that ﬁnite OAM states signiﬁcantly modify the charac-teristics of EA waves at the cold-electron timescale. This manuscript is organized in the following fashion. In Sec. II, we employ the linear kinetic theory to derive a dielec- tric function for the twisted EA waves in a two-electron com-ponent plasma. Section IIIdescribes the linear dispersion relation and the damping rate of the EA waves, and Sec. IV contains numerical results and its short summary. II. MODEL AND GOVERNING EQUATIONS For the derivation of dielectric constant of the electron- acoustic (EA) waves, we consider an unmagnetized colli- sionless uniform plasma, consisting of hot and cold electrons of number densities ( nh0,nc0) and temperatures ( Th,Tc) with a neutralizing background of static positive ions. At equilib- rium, the plasma demands a charge-neutrality condition as ni0¼nh0þnc0, where ns0being the equilibrium number den- sity of the sth species ( s¼hfor hot electrons, cfor cold elec- trons, and ifor positive ions). If the plasma is perturbed, the ﬂuctuations occur near the equilibrium densities and lead to the perturbed distribution function fs1¼fs–fs0. Hence, the dynamics of the EA waves can be described by the Vlasov-Poisson coupled set of equations 34,35in terms of the distribu- tion function, as @tþv/C1r ðÞ fs1þqs msE1/C1@vfs0¼0 (1) and r2V¼/C04pX sqsð fs1ðr;v;tÞdv: (2)Note that the electric ﬁeld vector E1½¼ /C0r V/C17/C0ikV/C138can be solved by using the plane wave solution, showing a con-stant phase of the wavefronts. As a result, the electric ﬁeldlines can be assumed to be straight lines. Here, Vdenotes the electrostatic potential, and q sandmsare the charge and mass of the sth species, respectively. Now taking the space-time Fourier transforms of Eqs. (1)and(2), we ﬁnally arrive at the following expression:36,37 Dk;xðÞ¼1þX svsk;xðÞ/C171þ1 k2k2 Ds1þnzsZnzsðÞ ½/C138 :(3) This is the generalized dielectric constant for the plane elec- trostatic waves in an unmagnetized collisionless plasma. Here, vsðk;xÞis the plasma susceptibility, kDs¼ ðTs=4pns0q2 sÞ1=2is the Debye length, and ZðnzsÞstands for the well-known plasma dispersion function38having the argument nzhð¼x=kvTsÞ, where vTs¼ðTs=msÞ1=2the ther- mal speed and xðkÞthe wave frequency (wave number). For the twisted electrostatic wave, the electric ﬁeld lines areassumed to follow the helical trajectories, such that the elec- tric ﬁeld vector can be described by E 1¼/C0ikef fV, with effective wave vector kef f¼/C0i Fpl@rFpl^erþl r^ehþðk/C0 i Fpl@zFplÞ^ezand ^er;^eh, and ^ezare the unit vectors. The indices pandlare the radial and angular mode numbers, respec- tively. In a cylindrical coordinate system r¼(r,h,z), one can express the electric ﬁeld components simply,23as Er¼/C0@rV/C17/C01 Fpl@rFplV;Eh¼/C01 r@hV/C17/C0ilV rand Ez¼/C0@zV/C17/C0 ikþ1 Fpl@zFpl/C18/C19 V; (4) where the LG potential having amplitude ~Vplis given by Vðr;tÞ¼ ~VplFplðr;zÞexp½iðlhþkz/C0xtÞ/C138; (5) with the LG mode functions Fplr;zðÞ¼1 2ﬃﬃﬃpplþpðÞ ! p!/C18/C191=2 XjljLjlj pexp/C0X=2ðÞ : Here, X¼r2=w2ðzÞ,w(z) being the beam waist, and Ljlj pðXÞ ¼expðXÞ p!Xjljdp dXpXjljþpexpð/C0XÞ/C2/C3 represents the associated Laguerre polynomials. See that LG potential Vdepends not only on the mode numbers pandlbut also on the azimuthal angle hvia the helical phase structure through exp ½iðlhþkz/C0xtÞ/C138.A st h e wave propagates along the z-axis with slowly varying ampli- tude, the Laplacian operator can be expressed as r2 ¼r2 ?/C0k2þ2ik@zwithr2 ?¼r/C01@rr@rþr/C02@2 h.I ti sw e l l known that potential Vsatisﬁes the paraxial equation in the limit23@2 zV/C282ik@zV,a s ðr2 ?þ2ik@zÞV¼0: (6) Thus, using r2¼r2 ?/C0k2þ2ik@zand plugging Eq. (6) into Eq. (2), we readily obtain082122-2 Aman-ur-Rehman et al. Phys. Plasmas 23, 082122 (2016) k2Vðr;tÞ¼4pX sqsð fs1ðr;v;tÞdv: (7) To solve the linearized Vlasov Eq. (1), we express the per- turbed distribution function in terms of the LG mode func- tions, such that fs1ðr;v;tÞ¼ ~fplðvÞFplðr;zÞexp½iðlhþkz/C0xtÞ/C138:(8) For ﬁnding ~fplðvÞ, we substitute Eqs. (5)and(8)into Eq. (1) and integrate over rdr, to obtain ~fplvðÞ¼X sqs msqef f/C1@vfs0 aþib~Vpl; (9) where qef f¼/C0iqr^erþlqh^ehþðk/C0iqzÞ^ez: The new parameters have been deﬁned as qr¼Ð1 0rdrF pl@rFpl;qz¼Ð1 0rdrF pl@zFpl, and qh¼Ð1 0drF2 pl.I n deriving (9), we have used x/C0v:qef f¼aþiband expressed asa¼x/C0kvz/C0lqhvhandb¼qrvr/C0qzvz. Thus, substituting Eqs. (9)and(5)into(7), the plasma dielectric constant may be written as Dx;kðÞ ¼1þX sx2 ps k2ðqef f/C1@vfs0 x/C0v:qef fdv: (10) This coincides with the usual plane wave dispersion equation (3)if we neglect the azimuthal part lqhvhand assuming jqrj;jqzj/C28j lqhj. However, Eq. (10) also shows that the Landau resonance becomes modiﬁed owing to quantum number associated with the ﬁnite OAM states, so that we have a¼6borx¼kvz/C0lqhvh6ðqrvr/C0qzvzÞ. The new resonances have vital signiﬁcance when we examine the role of real part aby neglecting the imaginary part b. This could be valid when jqrj;jqzj/C28j lqhjleads to the resonance condi- tion x¼kvzþlqhvh, behaving similar to the Landau- cyclotron resonance in magnetized plasma. Hence, Eq. (10) can be further simpliﬁed as32 Dx;kðÞ ¼1þX sx2 ps k2ðk@vzfs0þlqh@vhfs0 x/C0kvz/C0lqhvhdv: (11) Note that the susceptibilities are now modiﬁed with an addi- tional effect of azimuthal velocity vh. For evaluating the integral of Eq. (11), we consider the equilibrium distribution function to be Maxwellian and deﬁne /z¼ðx/C0lqhvhÞ=kand/h¼ðx/C0kvzÞ=lqh. Then, the modiﬁed plasma response function becomes Dx;kðÞ ¼1þX s¼c;h;i1 k2 zk2 Ds2þnzsZnzsðÞ þ nhsZnhsðÞ ½/C138 ;(12) where ZðnzsÞandZðnhsÞare the well-known axial and angu- lar dispersion functions with arguments nzsð¼/z=vTsÞand nhsð¼/h=vTsÞwith vTs¼ð2Ts=meÞ1=2. Equation (12) may be simpliﬁed for hot and cold electron species, asDx;kðÞ ¼1þ1 k2k2 Dh2þnzhZnzhðÞþnhhZnhhðÞ/C8/C9 þ1 k2k2 Dc2þnzcZnzcðÞþnhcZnhcðÞ/C8/C9 ; (13) where nzh¼x=kvTh;nzc¼x=kvTc;nhh¼x=lqhvTh, and nhc¼x=lqhvTc, while kDhandkDcdenote the hot and cold electron Debye lengths, respectively. Equation (13) gives the dielectric constant of the twisted electrostatic waves propa- gating in a two-electron component plasma in the presenceof ﬁnite OAM states owing to the LG mode proﬁles. III. TWISTED EA WAVE DISPERSION AND DAMPING RATE Here, we are interested in the investigation of twisted electron-acoustic (EA) waves (viz., vTc/C28x=k/C28vThÞat the cold electron timescale in a two-temperature electronplasma, whose constituents are the Boltzmannian hot elec-trons, dynamical cold electrons, and static positive ions. We therefore use the small and large argument expansions as n zh/C281;nzc/C291;nhh/C291, and nhc/C291 in Eq. (13), and after some straightforward steps, we eventually arrive at thesimpliﬁed expression D¼1þ1 k2k2 Dhþiﬃﬃﬃpp k2k2 Dhx kvThþgx kvThexp/C0g2x2 k2v2 Th ! () /C01 k2k2 Dck2v2 Tc 2x21þ3k2v2 Tc 2x2/C18/C19 þk2v2 Tc 2x2g21þ3k2v2 Tc 2x2g2 ! () þiﬃﬃﬃpp k2k2 Dcx kvTcexp/C0x2 k2v2 Tc ! þgx kvTcexp/C0g2x2 k2v2 Tc ! () ; (14) where gð¼k=lqhÞis the dimensionless parameter showing the helical phase structure involving the plasma oscillations.We can analyze the real ( D r) and imaginary ( Di) parts of Eq. (14) to investigate the real frequency of the twisted EA wave and its damping rate by taking into account the relationsD r¼0 and xi¼/C0Di=@xrDr. Thus, assuming x¼xrþixi, the real part from Eq. (14) can be obtained as xr¼kCea1 1þk2k2 Dh1þg2 g2þ3Tc Thnh0 nc01þg4 g21þg2 ðÞ !1=2 ; (15) where the EA speed given by Cea¼xpckDh/C17nc0 nh0Th m/C18/C191=2 : Equation (15) is the dispersion relation of the twisted EA waves in a two-electron component plasma, taking intoaccount the azimuthal wave number through the parametergð¼k=lq hÞ. For g/C291, the standard plane EA wave8is retrieved. However, for cold plasma approximation Tc¼0,082122-3 Aman-ur-Rehman et al. Phys. Plasmas 23, 082122 (2016) Eq. (15) reduces to xr/C25kCeaðð1þg2Þ=g2Þ1=2using the limit of small wavenumber (viz., kkDh/C281). This shows that the EA waves propagate with the same phase and group velocities at sufﬁciently long wavelengths.The damping conditions depend upon the velocities in the (r,h) plane and is calculated through the imaginary part of Eq. (14) by utilizing the expression xi¼/C0Di=@xrDrto obtain xi xr¼/C0ﬃﬃﬃp 8rnc0 nh0/C18/C191=21þg2 g2 !1=2 1 1þk2k2 Dh/C16/C173=21þgexp/C0g2x2 r k2v2 Th ! þnc0 nh0Th Tc/C18/C193=2 exp/C0x2 r k2v2 Tc ! þgexp/C0g2x2 r k2v2 Tc ! () 2 43 5: (16) This is the Landau damping rate of the EA waves with ﬁnite amount of OAM in an unmagnetized collisionless plasma. Italso shows that the damping is signiﬁcantly modiﬁed withthe contribution through twisted Landau resonance. Notethat the inclusion of the azimuthal velocity component leadsto the existence of the OAM parameter g, and for long wave- lengths, the wave is heavily damped owing to the presenceof hot electrons in the system. When gtends to inﬁnity, the standard damping of plane EA wave is obtained. 36In addi- tion, the damping rate is signiﬁcantly modiﬁed by otherparameters like kk Dh;nc0=nh0, and Th=Tcin a two-electron component plasma. IV. RESULTS AND DISCUSSION In this section, we solve numerically the exact disper- sion equation (12) using the normalized plasma variables as ~xr¼xr=xpe;~xi¼xi=xpe, and ~k¼kkDc. The two-electron component plasma supports the electron-acoustic (EA) oscil-lations when there are dynamical cold electrons andBoltzmannian distributed hot electrons in the background ofstationary positive ions. In particular, the ion temperature istaken to be very small as compared to the cold-electron tem-perature (viz., T i¼0:001Tc), so that the ions can be assumed to be immobile in the background plasma. Equations (15) and(16) reveal that the real and damping frequencies xrand xistrongly depend on the dimensionless twisted parameter g. For large wave numbers, the wave damping begins to decrease with the wave number. At intermediate wave num- bers, the so-called cool plasma regime appears and the Landau damping due to the hot electrons becomes insigniﬁ-cant. The dispersion relation in this regime and at even largerwave numbers gets the form x 2 r¼x2 pc1þ1=g2/C0/C12þ3k2k2 Dc1þ1=k2k2 Dh/C16/C17 1þ1=g4/C0/C1 1þ1=k2k2 Dh/C16/C17 1þ1=g2 ðÞ8 >< >:9 >= >;: This expression shows that at these wave numbers, the waves become similar to the Langmuir waves based on the coolplasma component. If kk Dhis very large, the above equation gets the following form:x2 r¼x2 pc1þ1 g2þ3k2k2 Dc1þ1=g4/C0/C1 1þ1=g2 ðÞ() : Here, xpcis the cold electron plasma frequency. The above equation is similar to the dispersion relation derived for the electron plasma waves with OAM in Ref. 34. At even larger wave numbers, the wave continues to behave like cool plasma wave. In this region, the Landau damping is signiﬁcant and is mainly due to the cold electrons in the plasma system. FIG. 1. The normalized wave frequency ðxr=xpeÞand the damping rate ðxi=xpeÞof the EA waves are plotted against the normalized wave number kkDcfor varying the number density ratio nc0¼ð0:1/C00:7Þne0with bð¼ Th=TcÞ¼100;Ti¼0:001Tcandgð¼k=lqhÞ¼1. Dotted curves are the regions where jxij>xr=2p, and continuous curves correspond to the regions where jxij<xr=2p.082122-4 Aman-ur-Rehman et al. Phys. Plasmas 23, 082122 (2016) However, similar to the above described two regions, the damping rate again depends on the value of gin the region. This behavior can be seen by looking at the second term inEquation (16), which increases with the decrease in the wave- length of the wave. The dispersion relation (15) can be expressed as x 2 r¼k2fC2 ea 1þk2k2 Dhð1þ1 g2Þþ3v2 Tc1þ1=g4 1þ1=g2g.T h i s form coincides with the ion-acoustic dispersion relation found in many elementary text books. In the long wavelength limit (viz., k2k2 Dh/C281), Eq. (15) reduces to x2 r¼k2fC2 eað1þ1 g2Þ þ3v2 Tc1þ1=g4 1þ1=g2g, representing the EA waves propagating in a dispersionless manner with the common phase speed given by xr=k¼fC2 eað1þ1 g2Þþ3v2 Tc1þ1=g4 1þ1=g2g1=2. Under the condition of weak damping, Tc/C28Th, the phase speed becomes xr=k¼Ceað1þ1 g2Þ1=2. On the other hand, for short wave- length limit, i.e., k2k2 Dh/C291, the dispersion relation becomes x2 r¼x2 pcf1þ1 g2þ3k2k2 Dc1þ1=g4 1þ1=g2g. As a consequence, the hot electrons are unable to properly shield the charge density oscillations that are set up by the cold electrons. This result is like Langmuir wave rather than the acoustic wave behavior. In all the plots, the entire EA mode displays three distinct regimes that are similar to the regimes described by the Garyand Tokar in their work.8The ﬁrst regime occurs for long wavelengths (i.e., low wave numbers). The characteristic phase speed in this regime becomes xr=k¼Ceað1þ1=g2Þ1=2 and the EA waves are strongly damped due to the presence of hot electrons xr=k/C24vTh,w h e r e vThis the hot electron ther- mal speed. In such a regime, the cold electrons have no signif-icant effects on the damping of these waves. The secondregime is the cool Langmuir-like branch of the EA wave. Thisregime is weakly damped and it occurs for intermediate val-ues of the wavelengths. The range of the wavelengths that areweakly damped depends on both the fraction of the electrondensities and the temperature ratio of the hot and cold elec-trons. In this regime, there is no strong resonance of the wavewith either of the electron species. When the wavelengthdecreases beyond the intermediate values (i.e., the secondregime), we enter into a third regime, where the EA waves arestrongly damped in the presence of the cold electrons. In this regime, the phase speed of the EA wave resonates with the cold electron thermal speed, i.e., x r=k/C24vTc. Figure 1displays the normalized real and damping fre- quencies ~xrð¼xr=xpeÞand ~xið¼xi=xpeÞagainst the nor- malized wave number ~kð¼kkDcÞfor varying the density ratios nc0¼ð0:1/C00:7Þne0with ﬁxed values of temperatures Th¼100Tcand Ti¼0:001Tc. The variation of the ratio nc0=ne0leads to the enhancement of phase speed of twisted FIG. 2. The normalized wave frequency ðxr=xpeÞand the damping rate ðxi=xpeÞof the EA waves are plotted against the normalized wave number kkDcfor different values of twisted parameter gð¼0:25;0:50;1;1:50;5;10Þ with bð¼Th=TcÞ¼100;Ti¼0:001Tcandnc0¼0:5ne0. Dotted curves are the regions where jxij>xr=2pand continuous curves correspond to the regions where jxij<xr=2p.FIG. 3. The normalized wave frequency ðxr=xpeÞand the damping rate ðxi=xpeÞof the EA waves are shown against the normalized wave number kkDcfor various values of bð¼Th=TcÞwith Ti¼0:001Tc;nc0¼0:5ne0, and g¼1. Dotted curves are the regions where jxij>xr=2pand continuous curves correspond to the regions where jxij<xr=2p.082122-5 Aman-ur-Rehman et al. Phys. Plasmas 23, 082122 (2016) EA waves, while keeping the longitudinal to azimuthal wave number ratio to be unity, i.e., g¼1. The hot electron Debye length increases as long as the hot electron density decreases;as a result, the damping rate of EA wave reduces in a twoelectron component plasma. Figures 2and3exhibit how the dispersion relation and the damping rates are affected by thepresence of twisted parameter and hot-to-cold electron tem- perature ratio. We have found that at small wavenumbers, the dispersion is acoustic and the damping is strong. Forintermediate wavelengths, the damping is weak, and at largewavelengths, the damping is very strong. Notice that thechange in the values of parameter gaffects signiﬁcantly not only the real and imaginary frequencies but also the regionwhere the EA waves are weakly damped. To summarize, we have considered a collisionless unmagnetized plasma, whose constituents are the dynamicalcold electrons and Boltzmanian distributed hot electrons inthe presence of static positive ions. Keeping in view of thekinetic theory, we have solved the Vlasov-Poisson coupled setof equations to obtain a generalized dielectric constant byusing the Laguerre-Gaussian (LG) distribution function andparaxial equation for electrostatic potential. New features ofthe twisted electron-acoustic (EA) waves involving differentregimes of wavelength are investigated in a two-electron com-ponent plasma. 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1.5064017.pdf	Investigation of the ground state domain structure transition on magnetite (Fe 3O4) A. Yani , C. Kurniawan , and D. Djuhana Citation: AIP Conference Proceedings 2023 , 020020 (2018); doi: 10.1063/1.5064017 View online: https://doi.org/10.1063/1.5064017 View Table of Contents: http://aip.scitation.org/toc/apc/2023/1 Published by the American Institute of Physics Articles you may be interested in Adsorption of organic cationic dye into Fe-doped ZnO nanoparticle coupled with montmorillonite adsorbent AIP Conference Proceedings 2023 , 020023 (2018); 10.1063/1.5064020 The influence of graphene on silver oxide synthesis through microwave assisted method AIP Conference Proceedings 2023 , 020018 (2018); 10.1063/1.5064015 Enhance UV–photocatalytic activity of Fe 3O4/ZnOZ by incorporation of noble metal silver (Ag) nanoparticles AIP Conference Proceedings 2023 , 020029 (2018); 10.1063/1.5064026 Investigation on the localized surface plasmon resonance (LSPR) of silver nanoparticles using electron energy loss spectroscopy (EELS) simulation AIP Conference Proceedings 2023 , 020016 (2018); 10.1063/1.5064013 Facile synthesis of Ag/Fe 3O4/ZrO2 composites coupled with nanographene platelets for photo- and sonophotocatalytic activities against methylene blue dye AIP Conference Proceedings 2023 , 020014 (2018); 10.1063/1.5064011 Electrical conductivity and microwave characteristics of HCl and HClO 4-doped polyaniline synthesized through chemical oxidative continuous polymerization process at various polymerization temperatures AIP Conference Proceedings 2023 , 020027 (2018); 10.1063/1.5064024 Investigation of the Ground State Domain Structure Transition on Magnetite (Fe 3O4) A. Yani 1, C. Kurniawan 2, and D. Djuhana 1, a) 1Department of Physics, Faculty of Mathematics and N atural Sciences (FMIPA), Universitas Indonesia, Depok 16424, Indonesia 2Research Center for Physics, Indonesian Institute o f Sciences (LIPI), Tangerang Selatan 15314, Indones ia a) Corresponding author: dede.djuhana@sci.ui.ac.id Abstract. We have systematically investigated the ground sta te domain structure transition on magnetite (Fe3O4) material using micromagnetic simulation. A spherical model o f magnetite nanoparticle was simulated with varied diameter size from 40 nm to 100 nm. It is observed that the domai n structure transition from single domain to multi- domain occurred between 68 nm and 69 nm. The changing of domain str ucture is followed by changing the magnetization en ergy. For a single domain particle, it is observed that the dem agnetization energy is dominant to exchange energy while the opposite occurs in multi-domain particle. Then, we have also calculated the critical diameter based on Brown an d Kittel formulas. Comparing to the formulas, our micromagnetic result s are in the range between Brown and Kittel predict ion, which are 62.9 nm and 83 nm, respectively. Therefore, the obs erved domain structure substitution from single to multi domain structure in this study is related to the critical diameter, which the magnetization energy system cha nged. Keywords: Ground state, domain structure, single domain, mul ti domain, magnetite INTRODUCTION In two decades, magnetite based nanomaterials have been attracting due to its wide application, such a s ferrofluid [1, 2], environment [3, 4], biomedical [5], magneti c resonance imaging [6], etc. Concerning the applic ation, it is important to understand the properties of magnetite . Numerous studies have been reported for both theoretical/simulation [7] and experiments [8, 9]. However, there have been little discussion about th e domain structure transition related to the critical diamet er of magnetite nanoparticles [10, 11]. In this study, we have investigated the domain stru cture of Fe 3O4 without the external field using micromagnetic simulation based on LLG equation corresponds to the diameter variation. We determined the magnetizatio n energy density such as the demagnetization and the exchang e energy. Interestingly, it is found a domain struc ture transition from single to multi domain structure under nanomet er scale (< 100 nm) of spherical diameter. In a mul ti domain structure, the exchange energy is larger than the d emagnetization energy, while in single domain, the exchange energy is close to zero. Then, we have also calculated the critical diameter based on Kittel and Brown formul as. Micromagnetic results showed that transition domain structure related to the critical diameter and enh anced the theoretical approximation. MICROMAGNETIC PROCEDURE We have systematically investigated the ground stat e of domain structure transition on Fe 3O4 nanosphere performed by public micromagnetic simulation softwa re, OOMMF based on Landau-Lifshitz-Gilbert equation [12]. Proceedings of the 3rd International Symposium on Current Progress in Mathematics and Sciences 2017 (ISCPMS2017) AIP Conf. Proc. 2023, 020020-1–020020-4; https://doi.org/10.1063/1.5064017 Published by AIP Publishing. 978-0-7354-1741-0/$30.00020020-1 ( ) eff eff dm dt m H m m H (1) where m = M/Ms is normal magnetization, α is the da mping factor, γ is the gyromagnetic ratio, Ms is th e magnetization saturation, and Heff is the effective field. We us e a spherical model with the varied diameter from 4 0 nm to 100 nm. The micromagnetic parameters are α = 0.05, Ms = 5×1 0 5 A/m, A = 1.2×10 –11 J/m is the exchange stiffness, and K = – 1.1×10 4 J/m 3 [13]. The cell sizes are 2.5×2.5×2.5 nm 3. For this purpose, we initially use a random magne tization and no external magnetic field applied in this simulati on as illustrated in Fig. 1. After the system reach es an equilibrium magnetization, then we observe the domain structure s such as a single domain (SD) or multi domain (MD) . RESULTS AND DISCUSSION Figure 2 showed the exchange and demagnetization en ergies of Fe 3O4 from the diameter D = 40 nm to D = 100 nm. We have observed that there is a magnetic domai n structure transition from SD to MD. The demagneti zation energy is dominant to exchange energy in the SD con dition, while exchange energy is larger than demagn etization energy as diameter increased in MD condition. It is found at the diameter D = 68 nm, the demagnetizati on energy decreased and the domain structure still maintained as SD. Interestingly, at the diameter D = 69 nm, t he domain structure changed to MD with the decreasing of dema gnetization energy and the increasing exchange ener gy. From this observation, we can determine a transition reg ion from SD to MD in Fe 3O4. Furthermore, we observed the FIGURE 1. The spherical model of Fe 3O4 with the diameter variation from D = 40 nm to D = 100 nm and initial random magnetization of magnetic moment is applied on the ground state condition. The color bar is the magnet ization direction. FIGURE 2. The magnetization energy density of spherical magn etite with diameter from D = 40 nm to D = 100 nm. A transition of domain structure from SD to MD is p resented by dotted line. At D = 68 nm is SD and D = 69 D is MD. 020020-2 domain structure after the diameter D = 69 nm still showed in MD with a vortex structure. At the diame ter D = 70 nm, the demagnetization energy abruptly decreased, wher eas the exchange energy sharply rised. After the di ameter D = 70 nm, the demagnetization energy moved to decease while the exchange increased. For understanding, we have also figured out the domain structure from D = 40 to D = 100 nm, which is depicted in Fig. 3. We can see th e domain structure in SD and MD and a transition region arou nd the diameter D = 68 nm and D = 69 nm also. From these results, we observed a competition energy between the demagn etization and the exchange energy in domain structu res which the demagnetization energy increased as the volume increased and the exchange energy increased that te nd to make MD domain structure. We have also calculated the critical diameter of Fe 3O4 based on Kittel formula [13-15] as follows, 2 072 kittel sADK M (2) Brown made a rigorous calculation based on analytic al micromagnetic by setting up lower bound and uppe r bound of critical diameter [10]. The lower bound is calculat ed by the equation, 27.211 2 0ADlower Ms (3) and the upper bound diameter for low anisotropy mat erial is, 29.058 2 5.615 upper o S AD M K (4) By putting the parameters of Fe 3O4 to those equations, we have calculated the approxi mation of critical diameter (D c). The comparison between theoretical/analytical an d micromagnetic results of D c are presented in the Table 1. FIGURE 3. The domain structure of Fe 3O4 represented in 3D view from the diameter D = 40 nm to D = 100 nm. The color bar is the magnetization direction. TABLE 1 . The critical diameter ( Dc) of Fe 3O4. Description Dc (nm) Kittel 83 Brown 62.9 to 72.3 Micromagnetic 68 to 69 020020-3 It can be seen from the table, that the result obtained from micromagnetic simulation was lower than Kittel approximation. Compared to Brown approximation, the micromagnetic result was closer to the upper bound. Other information about critical diameter of Fe 3O4 obtained from Coey that is 76 nm [13]. CONCLUSIONS In conclusion, we have investigated the domain structure transition of sphere Fe 3O4 by means of micromagnetic simulation. The vortex structure started to develop at D = 69 nm, while demagnetizing energy density had a close value with exchange energy at D = 75 nm. Afterward, the calculation of critical diameter as suggested by Kittel and Brown were made the results were close to the simulation. ACKNOWLEDGMENTS This work is supported by Hibah PITTA 2017 funded by Universitas Indonesia No. 639/UN2.R3.1/HKP.05.00/2017. REFERENCES 1. Y. A. Koksharov, in Magnetic Nanoparticles, edited by S. P. Gubin (Wiley-VCH, Weinheim, 2009), pp. 197- 254. 2. T. Guo, X. Bian, and C. Yang, Phys. Stat. Mech. Its Appl. 438, 560 (2015). 3. T. Sun, et al. , Appl. Surf. Sci. 416, 656 (2017). 4. C. Prasad, S. Karlapudi, P. Venkateswarlu, I. Bahadur, and S. Kumar, J. Mol. Liq. 240, 322 (2017). 5. B. D. Plouffe, et al. , J. Magn. Magn. Mater. 323, 2310 (2011). 6. C. Zhou, et al. , Mater. Sci. Eng. C. 78, 817 (2017). 7. A. Kákay and L. K. Varga, J. Magn. Magn. Mater. 272–276 , 741 (2004). 8. O. U. Rahman, S. C. Mohapatra, and S. Ahmad, Mater. Chem. Phys. 132, 196 (2012). 9. S. Upadhyay, K. Parekh, and B. Pandey, J. Alloys Compd. 678, 478 (2016). 10. A. Kákay and L. K. Varga, J. Appl. Phys. 97, 083901 (2005). 11. J. Shan, et al. , Materials Science and Technology 32, 602 (2016). 12. M. J. Donahue and D. G. Porter, OOMMF User’s Guide, Version 1.0, (National Institute of Standards and Technology, Gaithersburg, MD, 1999), available at https://math.nist.gov/oommf/ftp- archive/doc/userguide12a3_20021030.pdf. 13. J. M. D. Coey, Magnetism and Magnetic Materials (Cambridge University Press, New York, 2010). 14. A. Hubert and R. Schäfer, Magnetic Domains, The Analysis of Magnetic Microstructures (Spinger-Verlag, Berlin, 2009). 15. D. Djuhana, D. Oktri, and D. H. Kim, Adv. Mater. Res. 896, 414 (2014). 020020-4
1.5089949.pdf	Appl. Phys. Lett. 114, 212403 (2019); https://doi.org/10.1063/1.5089949 114, 212403 © 2019 Author(s).Investigation of domain wall pinning by square anti-notches and its application in three terminals MRAM Cite as: Appl. Phys. Lett. 114, 212403 (2019); https://doi.org/10.1063/1.5089949 Submitted: 24 January 2019 . Accepted: 10 May 2019 . Published Online: 31 May 2019 C. I. L. de Araujo , J. C. S. Gomes , D. Toscano , E. L. M. Paixão , P. Z. Coura , F. Sato , D. V. P. Massote , and S. A. Leonel ARTICLES YOU MAY BE INTERESTED IN Modulation of spin-orbit torque induced magnetization switching in Pt/CoFe through oxide interlayers Applied Physics Letters 114, 212404 (2019); https://doi.org/10.1063/1.5094049 Spin wave propagation in ultrathin magnetic insulators with perpendicular magnetic anisotropy Applied Physics Letters 114, 212401 (2019); https://doi.org/10.1063/1.5093265 Sub-nanosecond switching in a cryogenic spin-torque spin-valve memory element with a dilute permalloy free layer Applied Physics Letters 114, 212402 (2019); https://doi.org/10.1063/1.5094924Investigation of domain wall pinning by square anti-notches and its application in three terminals MRAM Cite as: Appl. Phys. Lett. 114, 212403 (2019); doi: 10.1063/1.5089949 Submitted: 24 January 2019 .Accepted: 10 May 2019 . Published Online: 31 May 2019 C. I. L. de Araujo,1,a)J. C. S. Gomes,2 D.Toscano,2E. L. M. Paix ~ao,2P. Z. Coura,2F.Sato,2D. V. P. Massote,2 and S. A. Leonel2,b) AFFILIATIONS 1Departamento de F /C19ısica, Laborat /C19orio de Spintr ^onica e Nanomagnetismo, Universidade Federal de Vic ¸osa, Vic ¸osa, Minas Gerais 36570-900, Brazil 2Departamento de F /C19ısica, Laborat /C19orio de Simulac ¸~ao Computacional, Universidade Federal de Juiz de Fora, Juiz de Fora, Minas Gerais 36036-330, Brazil a)Electronic mail: dearaujo@ufv.br b)Electronic mail: sidiney@ﬁsica.ufjf.br ABSTRACT In this work, we perform investigations of the competition between domain-wall pinning and attraction by antinotches and ﬁnite device borders. The conditions for optimal geometries, which can attain a stable domain-wall pinning, are presented. This allows the proposition ofa three-terminal device based on domain-wall pinning. We obtain, with very small pulses of current applied parallel to the nanotrack, a fastmotion of the domain-wall between antinotches. In addition to this, a swift stabilization of the pinned domain-wall is observed with a highpercentage of orthogonal magnetization, enabling high magnetoresistive signal measurements. Thus, our proposed device is a promising magnetoresistive random access memory device with good scalability, duration, and high speed information storage. Published under license by AIP Publishing. https://doi.org/10.1063/1.5089949 The discovery of spin valve effects 1–3and magnetic tunnel junc- tion (MTJ) measurements at room temperature4,5allowed the devel- opment of several generations of magnetoresistive random accessmemory (MRAM) devices. 6A recent demonstration of MRAM integration among metallic contacts in silicon technology7enables industrial large scale production and boosts further developments inscalability, consumption, and speed. The MRAM generations can be divided according to the principle used for magnetization switching in t h em a g n e t i ct u n n e lj u n c t i o nf r e el a y e r .I nt h ee a r l yg e n e r a t i o n s ,t h emagnetization switchings were made through Oersted ﬁelds generatedby bit lines, 8demanding large areas for the bit lines and high con- sumption due to the large currents needed. The next generation wasdeveloped with magnetization switching by spin transfer torque. 9Such an approach represented a high gain in density, once there is no needof bit lines with switching performed by the current through the stack.However, the large current density needed can cause junction thresh-old, resulting in small durability. In order to protect the junction, thenewest generations are based on three terminal devices with large cur-rents passing by just the ﬁrst ferromagnetic electrode and very smallcurrents used to measure the tunnel magnetoresistance signal. Among such technology is the spin–orbit torque MRAM, 10,11which uses heavy metals in the ﬁrst layer to split the current into spin-polarizedchannels, with high enough density to switch the ﬁrst ferromagneticlayer (FM) by spin transfer torque. Another three terminal approachcan be adapted from the original proposal of magnetic domain-wallbased MRAM, 12which is based on domain-wall motion through a very long track and pinned by triangular notches, delimiting the bit length. Alternative geometries for the bit length deﬁnition were alsoproposed. 13,14In this work, we investigate both domain-wall attraction and pinning by square antinotches, mapping best geometries for uni-form pinned domain-wall, in order to measure stable, fast, and highestvalues of tunnel magnetoresistive (TMR) signals by a magnetic tunneljunction (MTJ). The results, to be presented ahead, allowed the propo-sition of a three terminal domain-wall based MRAM, sketched in thecartoon presented in Fig. 1 . The working principle of such a device is based on a short current pulse applied in the device edges in order todetach the transverse domain-wall (TDW) from the ﬁrst antinotch tobe attracted by the second. Above the second antinotch, a MTJ will act Appl. Phys. Lett. 114, 212403 (2019); doi: 10.1063/1.5089949 114, 212403-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplin sensing, with the ﬁrst ferromagnetic layer (FM1) where the domain-wall moves, a thin insulator for the electronic tunneling, and asecond ferromagnetic layer (FM2), which is aligned orthogonal to thetrack magnetization by shape anisotropy. The tunnel magnetoresistivesignal will vary from minimum to maximum, depending on the anti-notch where the domain-wall is pinned. In order to test the best geometry to achieve high performance in the proposed device, we have performed computational simulations. AHamiltonian consisting of the isotropic Heisenberg model and theshape anisotropy can be used to describe a nanomagnet made of a softferromagnetic material H¼J/C0X hi;ji^mi/C1^mjþD JX i;j^mi/C1^mj/C03ð^mi/C1^rijÞð^mj/C1^rijÞ ðrij=aÞ3"# 8 < :9 = ;; (1) where ^miand ^mjare unit vectors which represent the magnetic moments located at the iandjsites. The ﬁrst term of Eq. (1)describes the ferromagnetic coupling, whereas the second describes the dipole-dipole interactions, which are responsible for the origin of the shapeanisotropy. In the micromagnetic approach, the renormalization ofmagnetic interaction constants depends not only on the parameters ofthe material but also on the manner in which the system is partitioned into cells. According to the micromagnetic formulation, there is an upper limit for the work-cell size. Each micromagnetic cell hosts aneffective magnetic moment ~m i¼ðMsVcelÞ^mialigned to the direction in which the atomic moments are saturated. From one cell to another,effective magnetic moments vary their directions gradually. Theseassumptions are only satisﬁed if we do not exceed the upper limit forthe work-cell size. Therefore, the volume of the micromagnetic cellV celhas to be taken very carefully. In order to choose a suitable size for the work cell, we need to estimate the characteristic lengths, which depend on the material parameters. For instance, the exchange length, k¼ﬃﬃﬃﬃﬃﬃﬃﬃ 2A l0M2sq , provides an estimate of the exchange interaction range. In the simulations, we have used typical parameters for Permalloy-79 (Ni79Fe21) with values as follows:15,16saturation magnetization Ms ¼8.6/C2105A/m, exchange stiffness constant A¼1.3/C210/C011J/m,and zero magnetocrystalline anisotropy. Thus, we have estimated kPy-79/C255.3 nm. As in many micromagnetic simulation packages, we have used in our simulations the ﬁnite difference method, which subdivides the simulated geometry into cubic cells, that is, Vcel¼a3.I n this context, the renormalization of the magnetic interaction constants is given by15J¼2aAandD J¼1 4pða kÞ2. Based on the calculation of the exchange length for Permalloy-79, we have chosen the size of the micromagnetic cell as a¼2n m <kPy-79. Thus, planar nanowires have been spatially discretized into a cubic cell grid, and the size of the work cell was chosen as Vcell¼2/C22/C22n m3, which is accurate enough for the current study. The magnetization dynamics is governed by the Landau-Lifshitz-Gilbert (LLG) equation. In order to move the domain wall from one antinotch to another, an electric current pulse is applied parallel to the nanotrack main axis. A generalized version of the LLG equation, which includes the spin torque effect, has been proposed by Zhang and Li.17Thus, the domain wall dynamics driven by the spin- polarized current applied along the x-direction can be described by @^mi @t0¼/C01 ð1þa2Þ/C26 ^mi/C2~biþa^mi/C2^mi/C2~bi/C0/C1 þ1 ð1þb2Þu ax0/C18/C19 b/C0a ðÞ ^mi/C2@^mi @x0/C20 þ1þab ðÞ ^mi/C2^mi/C2@^mi @x0/C18/C19 /C21/C27 ; (2) where the dimensionless effective ﬁeld located at the micromagnetic celliis given by ~bi¼/C0J/C01@H @^mi. The ﬁrst two terms take into account precession and damping torques, whereas the last two terms take into account the torque due to the injection of the spin-polarized electric current. The nondimensional parameters, the Gilbert damping param- etera, and the degree of nonadiabaticity bare material parameters. Typical parameters for Permalloy-79 have been used in our simula- tions, and the values are as follows:16,18a¼0.01 and b¼0.015. The inﬂuence of the ratio ( b/a) on the dynamics of magnetic domain walls has already been investigated.18–20The connection between the space- time coordinates and their dimensionless corresponding is given by Dx0¼Dx=aandDt0¼x0Dt,w h e r e x0¼ðk aÞ2cl0Msis a scale fac- tor with inverse time dimension, with c/C251.76/C21011(T s)/C01being the electron gyromagnetic ratio; for permalloy, l0Ms/C251.0 T. Thus, the product ( ax0) has the dimensions of distance divided by time (unit of velocity) as well as the term u¼jeglB 2eMs/C16/C17 P,w h e r e jeis the x- component of the electric current density vector (in our case, ~je¼je^x, so that ~u¼u^xis a velocity vector directed along the direction of elec- tron motion20). For permalloy, the constantglB 2eMs/C16/C17 /C256:7/C210/C011m3 C, where gis the Lande factor (for an electron g/C252),lBis the Bohr mag- neton, and eis the elementary positive charge. The nondimensional parameter Pis the rate of spin polarization. We used P¼0.5, which amounts to those reported in permalloy nanowires of similar thick- nesses.21We have implemented the fourth-order predictor-corrector method to solve numerically Eq. (2). For permalloy, the factor x0 /C251.33/C21012s/C01. Thus, the time step Dt0¼0:01 used in the numeri- cal simulations corresponds to Dt/C257.5/C210/C015s. In micromagnetic simulations, we have used our own computational code, which hasbeen used in several works of our group. 22,23 FIG. 1. Cartoon representing the proposed three terminal domain-wall based archi- tecture. A current pulse in the short track moves the domain-wall between two anti-notches and the magnetoresistive signal is measured by the magnetic tunnel junction above one antinotch.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 114, 212403 (2019); doi: 10.1063/1.5089949 114, 212403-2 Published under license by AIP PublishingIn the simulations, we have considered the permalloy planar nanowires with length L¼152 nm and width W¼16 nm. The anti- notch thickness is the same as that of the nanotrack thickness T.T h e antinotch parameters, that is, the antinotch length, Lnot,a sw e l la st h e antinotch width, Wnot, were varied throughout the study. The stopping criterion for the relaxation consists of integrating the LLG equation without an external agent (magnetic ﬁeld or spin-polarized current)until both the energy of the system and its magnetization vector stoposcillating. Thus, the system reaches the equilibrium magnetic state,which provides the possibility of the adjustment of the TDW width. 24 See the supplementary material for details of the relaxation simula- tions. The equilibrium conﬁguration obtained in this way has been used as the initial conﬁguration in other simulations where a singleantinotch was inserted into the nanowire. To calculate the interactionenergy DE between the TDW and the antinotch as a function of the center-to-center separation d, we ﬁx the TDW at the center of the nanowire and vary only the antinotch position along the nanowireedge, see Fig. 2(a) . For each separation d, the total energy of the system is calculated using Eq. (1), and the interaction energy has been esti- mated using the following expression: DE i¼Ei/C0E0,w h e r eE irepre- sents the total energy of the nanowire that hosts the antinotch at anyposition x,w h e r e a sE 0is the reference energy, in which the antinotch is located at the maximum possible distance from the wall, that is, atthe corner of the nanowire. Figures 2(b)–2(d) show the behavior of the interaction energy as a function of the distance dbetween the center of the antinotch and the center of the TDW, as we vary the antinotch parameters. It can be observed from Fig. 2 that the antinotches work as pinning traps for the TDW and the interaction strength increases aswe increase the antinotch width ( W not)a n dt h i c k n e s s( T), but decreases as we increase the antinotch length ( Lnot). From now on, we consider two identical antinotches equidistant from the nanowire width axis. Based on our observations, we choose a nanotrack with thickness T¼4 nm, containing a pair of square anti- notches Lnot¼Wnot¼4 nm in order to investigate the TDW magneti- zation as a function of the relative distance between the antinotchesx not. The logic states (“0” and “1”) are deﬁned according to the anti- notch magnetization if it is aligned parallel or perpendicular to thenanotrack easy axis. Therefore, any intermediate direction would hin- der the information reading in the device, decreasing the TMR signal.In some of the tested conﬁgurations, after the system reaches the relaxed magnetic state in which the TDW was located near the anti-notch on the right, we observed that due to the proximity betweenantinotches, the magnetization of the antinotch on the left was alignedwith an intermediate direction between parallel and perpendiculardirections, as can be seen in the Fig. 3(a) (Multimedia view). However, when increasing the separation between the antinotches at x not ¼18 nm, we were able to achieve at least 99.5% of magnetization aligned parallel to the easy axis for the antinotch on the left.Additionally, we consider a variety of possible candidates for the stor- age cells of the random access memory proposed in this paper. Using the initial condition of the wall close to the antinotch on the right, wehave numerically calculated the relaxed micromagnetic state of several FIG. 2. (a) Schematic view of how the dis- tance dbetween the center of the anti- notch and the center of the TDW in the nanotrack was considered. The color gra-dient in the arrows represents the mag-netic moment’s directions. We have analyzed the interaction energy as a func- tion of the distance between the center ofthe antinotch and the center of the TDWby varying (b) antinotch width ( W not), con- sidering constant antinotch length Lnot ¼4 nm and thickness T¼4 nm, (c) anti- notch length ( Lnot), considering constant antinotch width Wnot¼4 nm and thickness T¼4 nm and (d) antinotch thickness T, considering constant antinotch length Lnot ¼4 nm and width Wnot¼4 nm. FIG. 3. (a) Schematic view of a nanotrack containing two antinotches. The anti- notch on the right side has magnetization perpendicularly aligned to the easy axis.The antinotch on the left side should exhibit magnetization parallel to the easy axis, but due to its proximity to the other antinotch, its magnetization is aligned in an intermediate direction. (b) Sequence of spin-polarized current pulses applied alongthex-axis to move the TDW from one antinotch to another. Multimedia view: https:// doi.org/10.1063/1.5089949.1Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 114, 212403 (2019); doi: 10.1063/1.5089949 114, 212403-3 Published under license by AIP Publishingnanowires with different parameters of the antinotch arrangement. From these equilibrium magnetic conﬁgurations, we applied asequence of current pulses ( j e¼63/C2109A/cm2with a duration of Dt/C250.04 ns) separated by a time interval of relaxation ( je¼0w i t ha duration of Dt/C251.12 ns) in order to move the wall from one anti- notch to another as shown in Fig. 3(b) . Over a wide range of antinotch parameters and the spacing between antinotches, we numerically calculated the dynamic response of the wall under the inﬂuence of the above-mentioned current pulse sequence. The simulation results have been organized into event dia-grams (see Fig. 4 ), which show the magnetization conﬁguration of the nanowire before and after the application of the current pulse sequence. Analyzing Fig. 4 , it can be noted that the precise control of the TDW position is only possible when the geometric factors of theantinotches are adjusted properly, such as the spacing between themand the parameters of the spin-polarized current pulse simultaneously. We ﬁxed the distance between layers d lay¼2n m , w h i c h i s a good approximation for the average thickness used in general for MTJ. Due to shape anisotropy, the reference layer magnetization alwaysremains aligned parallel to the easy axis of this layer. The antinotchesare inserted into the storage-layer and their two logic states (0 and 1)corresponding to two possible magnetization orientations. The MTJ is connected to a selection transistor and, upon reading, a small electric current ﬂows through the MTJ. The information bit would correspond to the MTJ resistance. Figure 5(a) (Multimedia view) shows the TDW at the notch far from the reference layer (before the application of theﬁrst current pulse) and at the notch near the reference layer (shortlyafter the application of the ﬁrst current pulse). During the current pulse, the TDW reaches a velocity of approximately 1 km/s, as shown inFig. 5(b) , with velocity vðtÞ¼ L 2dhMxðtÞi dt,w h e r e Mxis the x-compo- nent of the system magnetization vector. The calculation of the domain wall velocity has been previously proposed.25The local tunnel magnetic conductance (TMG), which is just the inverse of the tunnelmagnetic resistance, is given by the scalar product of the facing mag- netic moments on both sides of the tunnel barrier TMG ¼P ^mi/C1^mj n, where ^miand^mjare the facing magnetic moments on the storage and the reference layers, respectively, and nis the total number of magnetic moments in the layer.6The calculated TMG evolution to the consid- ered conﬁguration, presented in Fig. 5(c) , shows variation between 0 and 1, presenting negligible signal ﬂuctuations due to a small magneti- zation oscillation during the change of states. Due to shape anisotropy, FIG. 4. TDW position and antinotch magnetization controllability diagrams, which summarize micromagnetic simulation results of a single TDW in a Permallo y planar nanowire with two identical antinotches. Before applying the sequence of current pulses, we checked if the TDW was really pinned at the antinotch on the right. A lthough the antinotches work as pinning traps, their pinning potential strength cannot be strong enough to pin the wall, so that the TDW is expelled through one of the nanowire e nds. Relaxation results in which the TDW was expelled through the right side of the nanowire are represented by blue triangles. Relaxation results in which the TDW was p inned at the anti- notch on the right are represented by black circles, however, the magnetization of the left side antinotch was not aligned with the magnetization easy axis of the nanowire, such as is shown in Fig. 3(a) . Red squares correspond to the simulation results in which the TDW was expelled from the nanowire, after the application of the current pulse sequence. Green diamonds correspond to the simulation results in which we observed the TDW position accurate control, that is, not only the TDW positi on could be controlled from one antinotch to another, but also the magnetization vectors of antinotches did present parallel (TDW absence) and perpendicular (TDW presence ) alignments with the magnetization easy axis of the nanowire, before the next current pulse is applied.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 114, 212403 (2019); doi: 10.1063/1.5089949 114, 212403-4 Published under license by AIP Publishingthe magnetization in the reference layer remains aligned to the major axis. The reference layer major axis direction is perpendicular to therecording layer easy axis, but it is parallel to the domain wall magneti-zation direction. Thus, when the wall reaches the antinotch having the reference layer (state 1), the interaction between the reference layer and the domain wall favors the parallel alignment of its magnetiza-tions, decreasing the ﬂuctuations. In conclusion, we have mapped the conditions for domain-wall pinning with or without current pulse applied as a function of a set of antinotch parameters. In addition, we found an optimal geometry as small as the dimensions used in several MRAM investigated in the lit-erature. In the investigated geometry, we observed a swift domain wallmotion between antinotches with a short current pulse with a duration ofDt/C250.04 ns. The current used is similar to the ones already used in other investigated devices 26,27which demonstrates that it would not characterize any damage to a device in such a short operational time.The observed stable pinning and magnetization stabilization in Dt /C251.12 ns allow quite fast information storage, compared to a fast MRAM described in the literature, 28and the high percentage of uni- formity in the orthogonal magnetization of domain-wall pinned in theantinotch enables maximum TMR to be measured by the MTJ. See the supplementary material for the procedure for obtaining equilibrium magnetic states, our stopping criterion for the relaxationmicromagnetic simulations, and, in particular, an example to obtain the relaxed micromagnetic state of a single transverse domain wall in a permalloy planar nanowire. This study was ﬁnancially supported in part by the Coordenac ¸~ao de Aperfeic ¸oamento de Pessoal de N /C19ıvel Superior - Brasil (CAPES) - Finance Code 001 and also supported by CNPqand FAPEMIG (Brazilian agencies). We gratefully thank our friendSaif Ullah for making the English revision of this paper. REFERENCES 1G. Binasch, P. Gr €unberg, F. Saurenbach, and W. Zinn, Phys. Rev. B 39, 4828–4830 (1989). 2M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau, F. Petroff, P. Etienne, G. Creuzet, A. Friederich, and J. Chazelas, Phys. Rev. Lett. 61, 2472–2475 (1988). 3B. Dieny, V. S. Speriosu, S. S. P. Parkin, B. A. Gurney, D. R. Wilhoit, and D. Mauri, Phys. Rev. B 43, 1297–1300 (1991). 4J. S. Moodera, L. R. Kinder, T. M. Wong, and R. Meservey, Phys. Rev. Lett. 74, 3273–3276 (1995). 5T. Miyazaki and N. Tezuka, J. Magn. Magn. Mater. 139, L231–L234 (1995). 6C. I. L. de Araujo, S. G. Alves, L. D. Buda-Prejbeanu, and B. Dieny, Phys. Rev. Appl. 6, 024015 (2016). 7Y. J. Song, J. H. Lee, H. C. 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Phys. 115, 163906 (2014). 25D. G. Porter and M. J. Donahue, J. Appl. Phys. 95, 6729 (2004). 26S.-H. Yang, K.-S. Ryu, and S. Parkin, Nat. Nanotechnol. 10, 221–226 (2015). 27R. P. Loreto, W. A. Moura-Melo, A. R. Pereira, X. Zhang, Y. Zhou, M. Ezawa, and C. I. L. de Araujo, J. Magn. Magn. Mater. 455, 25–31 (2018). 28M. Cubukcu, O. Boulle, N. Mikuszeit, C. Hamelin, T. Br €acher, N. Lamard, M.- C. Cyrille, L. Buda-Prejbeanu, K. Garello, I. M. Miron et al. ,IEEE Trans. Magn. 54, 9300204 (2018). FIG. 5. (a) Schematic view in perspective of the possible states of layers magneti- zation with state in which the TDW is at the notch far from the reference layer 0(before the application of the ﬁrst current pulse) and in the state in which the TDW is at the notch near the reference layer 1 (shortly after the application of the ﬁrst current pulse). (b) and (c) Time evolution of the TDW velocity and TMG, respec-tively. Multimedia view: https://doi.org/10.1063/1.5089949.2Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 114, 212403 (2019); doi: 10.1063/1.5089949 114, 212403-5 Published under license by AIP Publishing
1.4979032.pdf	Switching field reduction of a perpendicular magnetic nanodot in a microwave magnetic field emitted from a spin-torque oscillator Hirofumi Suto , Taro Kanao , Tazumi Nagasawa , Kiwamu Kudo , Koichi Mizushima , and Rie Sato Citation: Appl. Phys. Lett. 110, 132403 (2017); doi: 10.1063/1.4979032 View online: http://dx.doi.org/10.1063/1.4979032 View Table of Contents: http://aip.scitation.org/toc/apl/110/13 Published by the American Institute of Physics Articles you may be interested in Nanosecond magnetization dynamics during spin Hall switching of in-plane magnetic tunnel junctions Applied Physics Letters 110, 122402 (2017); 10.1063/1.4978661 Effect of inserting a non-metal C layer on the spin-orbit torque induced magnetization switching in Pt/Co/Ta structures with perpendicular magnetic anisotropy Applied Physics Letters 110, 132407 (2017); 10.1063/1.4979468 High frequency out-of-plane oscillation with large cone angle in mag-flip spin torque oscillators for microwave assisted magnetic recording Applied Physics Letters 110, 142403 (2017); 10.1063/1.4979324 Electrical detection of magnetic states in crossed nanowires using the topological Hall effect Applied Physics Letters 110, 132405 (2017); 10.1063/1.4979322 Current-driven skyrmion dynamics in disordered films Applied Physics Letters 110, 132404 (2017); 10.1063/1.4979316 Chiral magnetoresistance in Pt/Co/Pt zigzag wires Applied Physics Letters 110, 122401 (2017); 10.1063/1.4979031Switching field reduction of a perpendicular magnetic nanodot in a microwave magnetic field emitted from a spin-torque oscillator Hirofumi Suto,a)Taro Kanao, Tazumi Nagasawa, Kiwamu Kudo, Koichi Mizushima, and Rie Sato Corporate Research and Development Center, Toshiba Corporation, Komukai-Toshiba-cho 1, Saiwai-ku, Kawasaki 212-8582, Japan (Received 27 November 2016; accepted 5 March 2017; published online 29 March 2017) We demonstrate microwave-assisted magnetization switching of a perpendicular magnetic nanodot in a microwave stray ﬁeld from a spin-torque oscillator (STO). The switching ﬁeld decreases when the STO is operated by applying a current. The switching ﬁeld reduction is almost the same as that in a microwave magnetic ﬁeld generated by a signal generator despite the ﬂuctuations of the STO oscilla-tion. The switching ﬁeld distribution, however, is broader when the STO is used. We also examine the magnetization switching process in the nanosecond region by applying a nanosecond-order pulse current to the STO and measuring the STO signal waveform. The onset of the STO oscillation andsubsequent assisted switching occur within a few nanoseconds. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4979032 ] Microwave-assisted magnetization switching (MAS) 1 has attracted attention recently because of its application to next-generation hard disk drive (HDD) technologies, such as microwave-assisted magnetic recording2,3and three-dimensional magnetic recording.3–5MAS can also be employed in magnetoresistive random-access memory(MRAM). 6So far, MAS has been experimentally studied in many magnetic systems, including in-plane and perpendicu-lar magnetic nanodots, 7–9granular media materials,10anti- ferromagnetically coupled media materials,11magnetic tunnel junctions,12,13and double-layer magnetic nanodots.14 These studies have demonstrated the practicality of MAS and deepened the understanding of the principles of MAS.Technically, microwave magnetic ﬁelds are generated byintroducing a microwave signal from a signal generator (SG) to a waveguide fabricated near magnets. In actual magnetic recording devices, however, spin-torque oscillators (STOs) 15 are proposed as the microwave-ﬁeld source. STOs generatean oscillating stray ﬁeld when the magnetization oscillatesby applying a dc current, and STOs are small enough toallow integration into the head elements in HDDs or memory cells in MRAM. The use of STOs may result in switching behavior different from that when SGs are used because theamplitude and frequency of the microwave ﬁeld from STOsﬂuctuate whereas the microwave ﬁeld generated by SGs isstable. 16,17In addition, the magnetization of an STO interacts with the magnet to assist through the mutual stray ﬁelds.6 Therefore, MAS based on STOs needs to be studied for theimplementation of magnetic recording using STOs. In this study, we fabricate a nanoscale stack consisting of an STO and a perpendicular magnetic layer (PL) andinvestigate the switching of the PL in a microwave stray ﬁeld from the STO (STO-MAS). We employ an in-plane magne- tized STO because it emits a large-amplitude electrical signal(STO signal) and is advantageous for the analysis of the mag-netization dynamics. Note that, for the magnetic recordingapplication, an STO with a perpendicularly magnetizedpinned layer is proposed because its out-of-plane magnetiza- tion trajectory emits a large-amplitude microwave ﬁeld. 18,19 The sample also has a microwave waveguide, which enables the comparison of STO-MAS and MAS in a microwave mag- netic ﬁeld generated by an SG (SG-MAS). We show that a nearly identical switching ﬁeld reduction is achieved for the two different microwave-ﬁeld sources. The switching ﬁeld distribution, however, is broader for STO-MAS probably owing to the ﬂuctuations of the STO oscillation. The magne- tization dynamics of the STO and PL during the switching process is studied by using micromagnetic simulations. Furthermore, we examine the switching process in the nano- second region by applying a nanosecond-order pulse current to the STO. The onset of the STO oscillation and subsequent MAS occur within a few nanoseconds. Figure 1(a) shows the sample structure and measure- ment setup. The sample is a stack consisting of an STO and a PL. The STO consists of antiferromagnetically coupled in- plane magnetic layers with pinned magnetizations and an in- plane magnetic free layer (FL). The PL consists of a Co/Pt multilayer with a ferromagnetic resonance (FMR) frequency of 15 GHz for the unpatterned ﬁlm [Fig. 1(b)]. Switching of the PL is examined by applying an external magnetic ﬁeld Hexttilted 10/C14from the /C0zdirection. In the absence of the microwave ﬁeld, the PL reverses at Hext¼þ4 kOe, as shown in Fig. 1(c). To analyze MAS of the PL, we conduct the fol- lowing four measurements [Fig. 1(a)shows the setup of the fourth measurement]. (1) To measure the spectrum of the STO oscillation, a dc current ( Idc) is applied to the STO through the dcport of the bias tee and the STO signal result- ing from the magnetization excitation is measured by a spec- trum analyzer connected to the acport of the bias tee. (2) To apply a microwave magnetic ﬁeld by using an SG, a micro-wave signal is introduced to a waveguide fabricated beneath the sample, which generates a microwave ﬁeld along the x direction. Because of the size of the waveguide and fre- quency range, a microwave current is expected to ﬂow almost uniformly in the waveguide. Therefore, the micro- wave ﬁeld amplitude ( H rf) can be estimated from thea)E-mail: hirofumi.suto@toshiba.co.jp 0003-6951/2017/110(13)/132403/5/$30.00 Published by AIP Publishing. 110, 132403-1APPLIED PHYSICS LETTERS 110, 132403 (2017) magnetic ﬁeld generated from a dc current in the waveguide, which is measured from the shift of the sample resistance versus external magnetic ﬁeld curve along the xdirection when a dc current of 650 mA is introduced to the wave- guide. (3) To apply a pulse current ( Ipulse) to the STO, a pulse generator is connected to the acport of the bias tee. (4) To measure the waveform of the STO signal during MAS, apulse current from the pulse generator is processed by a low-pass ﬁlter (LPF) with a cutoff frequency of 500 MHz and divided into two signals by a resistive divider. One signal enters a band-pass ﬁlter (BPF) with a pass frequency of2.5–18 GHz and is reﬂected. The other signal enters the STOand induces magnetization excitation. The STO signal result- ing from the magnetization excitation passes the BPF and is measured by an oscilloscope with an 80 GHz sampling rate.All measurements are carried out at room temperature. Figure 2(a)shows the dependence of the STO spectra on H extobtained by applying Idc¼þ0:65 mA. An STO oscilla- tion peak appears and its frequency increases from 3 GHz to 6 GHz with increasing Hext. The abrupt frequency change at Hext¼þ2:5 kOe corresponds to PL switching. This switching ﬁeld ( Hsw) is reduced from the intrinsic Hswofþ4k O e . T h e inset shows the STO spectrum slightly below the Hsw.T h e power, center frequency, and full-width at half-maximum(FWHM) are 30 nW, 4.28 GHz, and 140 MHz, respectively.The H swreduction is caused not only by the microwave stray ﬁeld from the STO but also by the Oersted ﬁeld and tempera- ture rise induced by Idc.W h e nt h e Idcis approximated by a uniform current ﬂowing in a circular pillar with a diameter of50 nm, the Oersted ﬁeld is along the circumferential direction and its amplitude is at most 52 Oe at the outer rim of the pillar. Considering that the in-plane component of H extisapproximately 435 Oe at the Hswofþ2:5 kOe, the Oersted ﬁeld is negligible. To evaluate the temperature effect, the cur- rent direction is reversed. The temperature rise is expected tobe almost the same for positive and negative currents becausethe sample resistance is almost the same regardless of the cur-rent direction (data not shown). On the other hand, the nega- tive current does not efﬁciently excite the STO. As shown in Fig.2(b), the STO signal becomes weak, and the abrupt fre- quency change corresponding to switching of the PL appearsatH ext¼þ3:2 kOe. This Hswis still smaller than the intrinsic Hswprimarily because of the temperature effect but is larger than that for the positive current. The different Hswobtained for positive and negative currents evidences that the micro-wave stray ﬁeld from the STO assists PL switching when theSTO is operated by applying a positive current. Figures 2(c)and2(d) show the dependence of the STO spectra on I dcforHext¼þ2:5 kOe and the corresponding FIG. 1. (a) Sample structure and experimental setup. The lateral size of the sample is 55 /C245 nm with the major axis along the xdirection. The ﬁlm structure consists of the following layers from bottom to top: Ir 18Mn8270/ Co70Fe3025/Ru 8.5/Co 60Fe20B2012/Ta 2/Co 60Fe20B2012/Co 50Fe506/MgO/ FL [Co 40Fe40B2022]/Ta 84/Ru 10/PL[Pt 6/(Co 12.1/Pt 6) /C23]. Thicknesses are given in angstroms. The MgO layer thickness is adjusted to yield a resis- tance area product of 1 X/C1lm2. The solid arrow denotes the direction of Hext, and the dotted arrow denotes the current direction. (b) Vector-network-ana- lyzer FMR spectra versus the z-direction magnetic ﬁeld obtained for the ﬁlm sample. (c) Sample resistance versus Hextobtained by applying Idc¼þ5 lA. Abrupt resistance changes at Hext¼0 Oe and þ4 kOe correspond to magnetization switching of the FL and PL, respectively. FIG. 2. (a) Power spectral density (PSD) of STO signal versus Hextobtained by applying Idc¼þ0:65 mA. The abrupt frequency change at Hext¼þ2:5 kOe corresponds to PL switching and is caused by reversal of the stray ﬁeld from the PL. Inset shows the PSD at Hext¼þ2.4 kOe. (b) PSD of STO signal versus Hextobtained by applying Idc¼/C00:65 mA. (c) PSD of STO signal versus Idcobtained at Hext¼þ2.5 kOe. (d) Corresponding power and FWHM. Solid and open circles are the values before and after PL switching. Dashed lines in (a)–(c) are eye-guides to show the abrupt frequency change corresponding to PL switching. (e) Hsw versus frffor several Hrfvalues. Circles and error bars respectively represent the average, maximum, and minimum values among 10 repeated measure- ments. Dotted and solid lines are obtained by applying Idc¼þ5lA and /C00:65 mA. The microwave signal is pulse-modulated with a duration of 20 ns and a repetition frequency of 1 kHz to avoid additional temperature rise. (f) Average Hswin (c) for Hrf¼232 Oe and Idc¼/C00:65 mA plotted on the background of the data from (a).132403-2 Suto et al. Appl. Phys. Lett. 110, 132403 (2017)power and FWHM. The STO signal shows a red shift with respect to Idc. The threshold Idcis estimated to be about þ0:35 mA where the FWHM broadens and then narrows. Switching of the PL occurs at þ0:63 mA. We next compare STO-MAS with SG-MAS. Figure 2(e) shows the dependence of Hswon microwave ﬁeld frequency (frf) for several Hrfvalues. For this measurement, we use two Idcvalues: Idc¼þ5lA and /C00:65 mA. The result for Idc¼þ5lA shows MAS at room temperature. The result for Idc¼/C00:65 mA includes the temperature effect and is com- parable with the STO-assisted Hsw obtained for Idc¼þ0:65 mA. For both Idc¼þ5lA and /C00:65 mA, Hsw linearly decreases with increasing frfand suddenly increases at the critical frequency. Such triangular switching regionsare typical of MAS. 3The STO-assisted Hswofþ2:5k O e realized by the STO oscillation frequency of 4.28 GHz [Fig. 2(a)] does not agree with this triangular switching region. We discuss this disagreement later. The Hswcurve for Hrf¼232 Oe and Idc¼/C00:65 mA exhibits a dip at frf¼4.5 GHz. Under this condition, the microwave ﬁeld does not directly excite the PL magnetiza-tion but excites the STO, and this STO excitation then assists PL switching through the microwave stray ﬁeld as in the case of the I dc-induced STO-MAS. Figure 2(f)plots the Hsw of SG-MAS over the background of the STO spectra, which clearly shows agreement between the dip and the STO-MAS. This dip does not appear in the Hswcurves for smaller Hrf values (116 Oe and 174 Oe) because these microwave ﬁelds do not excite the STO sufﬁciently to assist PL switching. The disagreement between the STO-MAS and the trian- gular SG-MAS regions can be explained as follows. Themagnetization trajectory of in-plane STO oscillation gener-ates not only a microwave stray ﬁeld with the oscillation fre- quency but also one with double the oscillation frequency, and the double-frequency component assists PL switching.Therefore, the H swof STO-MAS is compared with the Hsw of SG-MAS for frf¼8.5 GHz, which is approximately double the STO oscillation frequency, and the two Hswvalues coin- cide. This coincidence indicates that the same order of Hsw reduction is achieved for STO-MAS and SG-MAS despite the large linewidth of the STO oscillation. The STO assistance originating from the double- frequency stray ﬁeld is veriﬁed by zero-temperature micro-magnetic simulations based on the Landau-Lifshitz-Gilbert equation. A simulation model is constructed by discretizing the sample structure into 2 /C22/C21 nm cells. Simulation parameters are shown in Table I. The perpendicular anisot- ropy of the PL is derived from the measured FMR frequencyfor the ﬁlm sample [Fig. 1(b)]. In the simulations, a magneticﬁeld H PLtilted 10/C14from the –zdirection to the þxdirection is applied to the PL. An x-direction magnetic ﬁeld of þ200 Oe is applied to the FL, so that the oscillation fre- quency is approximately 4 GHz and reproduces the STO fre-quency in the experiment. Figure 3(a) shows the time evolution of the average magnetizations of FL and PL forH PL¼þ5.4 kOe. After a current of þ0.3 mA is applied from 5 ns, the magnetization excitation of the FL gradually grows.Simultaneously, the stray ﬁeld from the STO excites the PLmagnetization until PL switching occurs at approximately12 ns. After that, the PL magnetization relaxes to theswitched state. During this relaxation, the STO oscillation isdisturbed by the PL, and the amplitude becomes smaller. Asseen in the enlargement, before the PL switching [Fig. 3(b)], they-component magnetization oscillates at approximately 4 GHz and the x-component magnetization oscillates at the double frequency because of the magnetization trajectory of the in-plane oscillation. It is also seen that the PL magnetiza-tion synchronizes with the x-component of the FL magneti- zation and oscillates at approximately 8 GHz before theswitching, showing that the double-frequency stray ﬁeldassists PL switching. We also conduct the simulation withoutcurrent applied to the STO. In this case, PL switchingrequires H PL¼þ5.9 kOe (data not shown). These simulation results qualitatively explain the STO-MAS in the experi-ments, although the H swvalues are different because the sim- ulations do not take account of thermal ﬂuctuation. We next experimentally apply a nanosecond-order pulse current to the STO. Figure 4(a)shows the dependence of Hsw onIpulse.H e r e , Ipulseis calculated from the output voltage of the pulse generator with the characteristic impedance of 50 X and the sample resistance. The current dependence of the TABLE I. Simulation parameters. PL FL IL 1 IL2 Saturation magnetization [emu/cm3]900 1200 1200 1200 Damping constant 0.02 0.02 Remarks Perpendicular magnetic anisotropy 7.5/C2106erg/cm3FL/MgO interfacial perpendicular magnetic anisotropy 1.0 erg/cm2Fixed in /C0xdirection Fixed in þxdirectionFIG. 3. (a) Calculated time evolution of the average magnetization of the FL and PL. Current is applied from 5 ns with a rise time of 0.1 ns. (b) Enlargement of (a).132403-3 Suto et al. Appl. Phys. Lett. 110, 132403 (2017)sample resistance is taken into account. The inset shows an example of the Ipulsewaveform. As jIpulsejincreases, Hswgrad- ually decreases for both negative and positive currentsbecause of the temperature effect, and from approximately jI pulsej¼0.7 mA, Hswfor the positive current shows a steep decrease. This decrease indicates that the magnetization exci-tation of the STO becomes sufﬁcient to assist PL switching. When I dc¼þ0:65 mA was applied, Hswis reduced to þ2.5 kOe [Fig. 2(a)], but the same Hswis achieved at Ipulse¼þ0.75 mA. This disagreement in dc and pulse current amplitude originates from the duty ratio of Ipulse(2/C210/C05) and errors in calculating Ipulse. When Hswis reduced by the STO, the Hswdistribution becomes broader. This is in contrast to the results for SG-MAS, which indicated narrowing of the Hswdistribution.9The broad Hswdistribution may reﬂect the ﬂuctuations of the STO oscillation. Figure 4(b)shows the dependence of Hswon pulse dura- tion. At jIpulsej¼0.4 mA where only the temperature effectoccurs, the Hswcurves for positive and negative currents almost overlap and gradually decrease until they become constant at around 4 ns. At jIpulsej¼0.84 mA, the Hswcurve for positive current becomes smaller than that for negativecurrent from around 2 ns because of the STO assistance and becomes constant at around 5 ns, showing that the onset of the STO oscillation and subsequent MAS of the PL occur onthis time scale. We ﬁnally present a waveform of the STO signal during MAS of the PL. Figure 5(a) shows an example of the I pulse waveform measured at the acþdcport of the bias tee. The amplitude is reduced to approximately 75% of the initialamplitude because it is the sum of the signal from one port of the divider that directly enters the STO (50%) and the sig- nal from the other port of the divider that is reﬂected by theBPF and then enters the STO after passing the divider again (25%). Because the divider and the BPF are directly con- nected, the time lag between these two signals is negligible.A ripple appears because the LPF modiﬁes the pulse wave- form. The STO resistance does not match the characteristic impedance and multiple reﬂection of I pulseoccurs between the STO and the BPF. To prevent these reﬂections from overlapping, the transmission time of microwave signal between the STO and the BPF is set to be sufﬁciently long.Both I dc¼þ0:4 mA and Ipulse¼þ0:45 mA are simulta- neously applied to reduce the effect of the ripple. Here, Ipulse is deﬁned by the current amplitude of the pulse plateau after the ripple. The external ﬁeld is set to Hext¼þ2:2 kOe. Note that Idc¼þ0:4 mA alone cannot induce PL switching at this Hext. Figure 5(b) shows a single-shot waveform of the STO signal. The amplitude of the waveform increases during the application of Ipulse. Figure 5(c) shows the corresponding instantaneous frequency estimated from the zero-cross inter-vals of ﬁve wave cycles. The frequency changes from 4.5 GHz to 3.5 GHz during the I pulseapplication because of the red shift characteristic of the STO, and after that, the fre-quency increases to 5.3 GHz. Figure 5(d) shows the STO spectra obtained for I dc¼þ0:4 mA with the PL magnetiza- tion in the þzand/C0zdirections. The frequencies of these spectral peaks agree with the instantaneous frequencies before and after the Ipulseapplication, showing that switching of the PL occurs during the Ipulseapplication. Regarding the exact time of PL switching, it can be detected by a frequency change as already indicated by the spectral measurements. In addition, according to the simulation results in Fig. 3(a), the PL switching disturbs the STO oscillation and decreases the oscillation amplitude for a short time. A similar frequency change and an amplitude decrease occur at 12 ns in Figs.5(b) and 5(c), which might be caused by PL switching. However, such changes in the STO waveform are not always evident in repeated measurements because of the amplitudeand frequency ﬂuctuations of the STO signal. To determine the exact time of PL switching reliably from the STO wave- form, more stable STO oscillation is needed. In summary, we demonstrated MAS of a perpendicular magnetic nanodot caused by a microwave stray ﬁeld from an STO. Because of the magnetization trajectory of the in-planeoscillation, the STO emits a double-frequency microwave stray ﬁeld, which assists PL switching. The H swreduction of STO-MAS is almost the same as that of SG-MAS despite theFIG. 4. (a) Hswversus jIpulsejfor positive and negative currents. The duration and repetition frequency of Ipulseare 5 ns and 4 kHz, respectively, making the duty ratio equivalent to that of the pulse-modulated microwave signal in Fig.2(e). The inset shows the waveform of Ipulse. (b) Hswversus duration time of Ipulse. The repetition frequency is kept at 4 kHz. Circles and error bars in (a) and (b), respectively, represent the average, maximum, and mini- mum values among 10 repeated measurements. FIG. 5. (a) Waveform of Ipulsemeasured at the acþdcport of the bias tee and calculated waveform applied to the STO that includes multiple reﬂections. (b) Single-shot waveform of the STO signal obtained for Idc¼þ0:4 mA, Ipulse¼þ0:45 mA, and Hext¼þ2.2 kOe. The waveform is shifted horizontally such that the time axis aligns with that of (a). (c) Instantaneous frequency estimated from the data in (b). (d) PSD of the STOsignal obtained for I dc¼þ0:4 mA with the PL magnetization in the þzand /C0zdirections.132403-4 Suto et al. Appl. Phys. Lett. 110, 132403 (2017)ﬂuctuations of the STO oscillation. The Hswdistribution, however, is broader for STO-MAS. We also investigated STO-MAS in the nanosecond region and showed that the onset of the STO oscillation and subsequent MAS occurwithin a few nanoseconds. These ﬁndings present the appli- cability of the STO-based writing method for next- generation magnetic recording devices. This work was supported by Strategic Promotion of Innovative Research and Development from Japan Science and Technology Agency, JST. 1C. Thirion, W. Wernsdorfer, and D. Mailly, Nat. Mater. 2, 524 (2003). 2J.-G. Zhu, X. Zhu, and Y. Tang, IEEE. Trans. Magn. 44, 125 (2008). 3S. Okamoto, N. Kikuchi, M. Furuta, O. Kitakami, and T. Shimatsu, J. Phys. D: Appl. Phys. 48, 353001 (2015). 4G. Winkler, D. Suess, J. Lee, J. Fidler ,M .A .B a s h i r ,J .D e a n ,A .G o n c h a r o v , G. Hrkac, S. Bance, and T. Schreﬂ, Appl. Phys. Lett. 94, 232501 (2009). 5H. Suto, K. Kudo, T. Nagasawa, T. Kanao, K. Mizushima, and R. Sato, Jpn. J. Appl. Phys. 55, 07MA01 (2016). 6K. Kudo, H. Suto, T. Nagasawa, K. Mizushima, and R. Sato, Appl. Phys. Express 8, 103001 (2015). 7Y. Nozaki, M. Ohta, S. Taharazako, K. Tateishi, S. Yoshimura, and K. Matsuyama, Appl. Phys. Lett. 91, 082510 (2007).8S. Okamoto, N. Kikuchi, M. Furuta, O. Kitakami, and T. Shimatsu, Appl. Phys. Express 5, 093005 (2012). 9S. Okamoto, N. Kikuchi, M. Furuta, O. Kitakami, and T. Shimatsu, Phys. Rev. Lett. 109, 237209 (2012). 10S. Okamoto, N. Kikuchi, A. Hotta, M. Furuta, O. Kitakami, and T. Shimatsu, Appl. Phys. Lett. 103, 202405 (2013). 11Y. Nakayama, Y. Kusanagi, T. Shimatsu, N. Kikuchi, S. Okamoto, and O. Kitakami, IEEE Trans. Magn. 52, 1 (2016). 12T. Moriyama, R. Cao, J. Q. Xiao, J. Lu, X. R. Wang, Q. Wen, and H. W. Zhang, Appl. Phys. Lett. 90, 152503 (2007). 13H. Suto, T. Nagasawa, K. Kudo, K. Mizushima, and R. Sato, Appl. Phys. Express 8, 023001 (2015). 14H. Suto, T. Nagasawa, K. Kudo, T. Kanao, K. Mizushima, and R. Sato, Phys. Rev. Appl. 5, 014003 (2016). 15S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph, Nature 425, 380 (2003). 16M. Quinsat, D. Gusakova, J. F. Sierra, J. P. Michel, D. Houssameddine, B. Delaet, M.-C. Cyrille, U. Ebels, B. Dieny, L. D. Buda-Prejbeanu, J. A. Katine, D. Mauri, A. Zeltser, M. Prigent, J.-C. Nallatamby, and R.Sommet, Appl. Phys. Lett. 97, 182507 (2010). 17T. Nagasawa, H. Suto, K. Kudo, K. Mizushima, and R. Sato, J. Appl. Phys. 109, 07C907 (2011). 18D. Houssameddine, U. Ebels, B. Delaet, B. Rodmacq, I. Firastrau, F. Ponthenier, M. Brunet, C. Thirion, J.-P. Michel, L. Perjbeanu-Buda, M.-C. Cyrille, O. Redon, and B. Dieny, Nat. Mater. 6, 447 (2007). 19H. Suto, T. Yang, T. Nagasawa, K. Kudo, K. Mizushima, and R. Sato, J. Appl. Phys. 112, 083907 (2012).132403-5 Suto et al. Appl. Phys. Lett. 110, 132403 (2017)
1.403969.pdf	Boundary integral equation method for source localization with a continuous wave sonar Yongzhi Xu Institute for Mathematics and its Applications, University of Minnesota, 514 Vincent Hall, 206 Church Street SE, Minneapolis, Minnesota 55455 Yi Yan Department of Mathernatics, Unioersity of Kentucky, Lexington, Kentucky 40506 (Received 17 January 1992; accepted for publication 22 April 1992 ) In this paper, matched-field processing is combined with the boundary integral equation method (BIEM) of scattering theory to study a sound source localization problem in a perturbed shallow ocean. It is assumed that there is a known inclusion embedded in a shallow water waveguide. Continuous waves (cw), produced by a sound source, are scattered by the inclusion and then received by a hydrophone array. Because the symmetry of the waveguide has been destroyed by the existence of the inclusion, a proper procedure is required to avoid the mismatching. A numerical scheme is presented that makes use of the separation of the source and the detection array, and greatly reduces the computation. A numerical simulation using this method is presented. PACS numbers: 43.30.Wi, 43.30.Bp, 43.20.Mv INTRODUCTION Localization of an acoustic source in waveguides has been studied by many authors in recent years. -5 One of the most significant advances is probably the "matched-field processing" method that was proposed by Bucker 2 in 1976. The main idea of the matched-field processing method is outlined by the title of Bucker's paper, "Use of calculated wave field and matched-field detection to locate sound source." The matched-field processing is usually performed in either "phone space" (matching the total field received by each hydrophone) or in "mode space" (matching the re- solved modes). On the other hand, the classical inverse scattering theo- ries that usually involve more mathematics have been rapid- ly developed in about the same time. 6-8 The basic idea of the inverse scattering theory is based on the physical idea of scattering one or more "plane waves" off the unidentified inclusion and then trying to identify the shape of the inclu- sion or its location from its far-field patterns. Recently, Gil- bert and Xu have generalized this idea to the direct and in- verse scattering problems in a shallow ocean. 9-2 The direct scattering problems in waveguides have been studied by us- ing boundary integral equation methods. 3-5 However, there is a concern remaining, in particular from the engineering point of view, in the inverse scattering theory in a shallow ocean. That is, a "complete set of data" is required in order to find a reasonable solution. Unfortunate- ly, these "complete data" are not always available in prac- tice. It raises a question' that is, if we can find a complement between inverse scattering theory and matched-field signal processing we can use less detected information to estimate the unknown object, or localize the sound source in a more complicated environment. In this paper, we combine matched-field processing with the boundary integral equation method of scattering theory to study a sound source localization problem in a perturbed shallow ocean. We assume that there is a known inclusion embedded in a shallow water waveguide. Contin- uous wave (cw), produced by a sound source, is scattered by the inclusion and then received by a hydrophone array (Fig. 1). We present a numerical scheme for the sound source localization in Sec. I, where the source location and the de- tection array are separated, which leads to the reduction of computation load. Some numerical experiments are demon- strated in Sec. II. A numerical method for the boundary integral equation is essential in our computation, and is in- cluded in the appendix. . I. MODELING AND METHODOLOGY A. Modeling The synthetic modeling of the perturbed waveguide is depicted in Fig. 1. We denote lhe waveguide with depth d as R -- {(x,x2) I -- o < x < o,0<x2<d}. An inclusion, which is a bounded region located in the waveguide, is denoted as 1. For the sake of illustrating our method, we shall assume that the inclusion has a sound-soft boundary o91. A time- harmonic acoustic source locates at x s-- (x ,x} ). The hy- drophone array consists of L hydrophones at x= (x ,x2 ), l - 1,2,...,L. The time-harmonic waves of the form P(x,t;x s) _ p(x;xS)e- iot radiated from x s and scattered by 1, propa- gates outward to xl- o. Here P(x,t;x s) is the acoustic pressure at x - (x,x2), emitted from the acoustic source at x s, and k -- co/c is the wave number, where co is the circular 995 J. Acoust. Soc. Am. 92 (2), Pt. 1, August 1992 0001-4966/92/080995-08500.80 © 1992 Acoustical Society of America 995 Downloaded 01 Dec 2012 to 150.135.135.70. Redistribution subject to ASA license or copyright; see http://asadl.org/termsWaveguide surface (p=O) Acoustic source Inclusion (p=O) Perturbed Waveguide p+k2p=O) Waveguide bottom (Px2=O) FIG. 1. Acoustic source in a perturbed waveguide. frequency and c is the speed of the time-harmonic acoustic wave. If the waveguide has a pressure released surface x2 -- 0 and a rigid bottom x2 = d, thenœ(x;xS), the total field of the outgoing wave is governed by the following system: Ap(x;x s) + k 2p(x;x s) = - (x, - x )(x2 - x ), x = (x,,x:)R f, ( 1 ) p(x,O;x s) -- O, _c2P (x,d;xS) -- O, (2) p(x;x ) = 0, for x8. (3) Moreover, p(x;x s) satisfies an outgoing radiation condition, i.e., for Ix, I - oo, p (x;x s) has an expansion œ(x,,x:) = œ,b, (x:)e ik"lx'l , (4) n----1 where k, = [ k: - ( n - «): ( rr/d :) ] '/: is the horizontal wave number, and the coefficients p, depend on x s and the sign of x,, and b, (x:) = sin[(n ---) rx: ] . (5) d Now we can state our source localization problem as follows: given the acoustic pressure at points x , 1 -- 1,2,...,L in the aforementioned perturbed waveguide, estimate the lo- cation of the sound source x s. B. Construction of the propagator The propagating acoustic wave emitting from a point source at x s (which is called propagator) in a perturbed waveguide can be constructed in the following way. Let po(x;x s) be the Green's function in a waveguide without inclusion, i.e., po(x;x s) satisfies Apo(X;X s) + k po(X;X s) = - (x, - x )(x - x ), x = (x,,x:)R, (6) po(x,,O;x s) = O, Pø (x,,d;x s) = 0, (7) and po(x;x s) is outgoing. By separation of variables, we can represent po(x;x s) as Pø(X;XS) = dk, qb, (x,)b, (x) . (8) n--1 We write the propagator in the waveguide with an inclu- sion as p ( x;x s) = po ( x;x s) -3- p, ( x;xS) . Then p = p -- Po is a solution of the problem Ap, (x;x s) + k :zp, (x;x s) = O, xR, p, (x,,O;x s) = O, _P' (x,,d;x s) = O, p, (x;x s) = -- po(x;xS), for (9) (lO) (11) (12) and p, (x;x s) is outgoing as Ix, I - oo. The physical meaning of this problem is that an incident wavepo upon the inclusion fl produces the scattered wave p. The propagator p is the composition of the incident wave Po and the scattered wave P. The scattered wavep can be constructed by the bound- ary integral equation method. We represent it by a double layer potential p,(x;x s) =fan c9pø(x;Y) ½(y;xS)dø"v' for xR, (13) where ½ is the solution of the boundary integral equation ½(x;x s) + 2 fo, O? (x;Y)½(y;xS)clø"' = - 2œo(X,X), for xSfl. (14) If k is not an eigenvalue of the interior Neumann problem in , then Eq. (14) has a unique solution. Symbolically we denote the boundary integral equation (14) as b + Kb = - 2œ0, (15) where K is the integral operator K½Cx;xs) ' -- 2 ;,m ?Pø cx;y)t/'Cy;xs)aø"v' for x9fl. (16) If k is not an eigenvalue of the interior Neumann problem in , then I + K is invertible. We can write and (x;x s) = -- 2(1 + K) - 'po(X;X s) (17) p(x;x s) =po(x;x s) -- 2 f cgpø(x;Y) (I + K)- 1 on ,9% XPo (y,x) day, for xR3 fl. (18) By the assumption of the boundedness of the inclusion , we know that for Ix, I large enough (say, Ix, I Xo for some constant Xo), p(x;x s) is expressed by a summation of normal modes: P(x;xs) = E '/In(xs)qn (x2)eik"lx'l ' (19) nl 996 J. Acoust. Soc. Am., Vol. 92, No. 2, Pt. 1, August 1992 Y. Xu and Y. Yan: Shallow water source localization 996 Downloaded 01 Dec 2012 to 150.135.135.70. Redistribution subject to ASA license or copyright; see http://asadl.org/termswhere A n (X s) is the modal amplitude given by i (bn(x[)e An (x s) -- dkn -- iknx sgn (x,) -- 2 a [b n (y2)e -ik'½"sgn(x') ] cvy _ .s ) X(I+K) 'po(y,x )d% , (2O) An approximate boundary integral equation method for the numerical solution of (15) is outlined in the Appendix. For more detailed discussion of this method, readers could refer to Ref. 16. C. Construction of estimators Using the representations for the propagator and its modal amplitude, we now construct the estimators in both phone space and mode space. 1. Estimator in phone space Let {Pmt} be the detected data set consisting of the acoustic pressure field Pmt sampled on each hydrophone lo- rn l cated at (x, ,x2 ), m = 1,2,...,M; ! = 1,2,...,L. The estimator in phone space is defined as follows: s m l s 2 - Ip(Xl ,x2' s , Fp (x ,x2 ) -- ,X 1 ,X2 ) * œmzl I lm=l (21) m I . s s where p (x ,x2 ,x ,x2 ) is the calculated acoustic pressure field at ( m 1 X ,X2 ). It can be computed by (18) for given x s using the method outlined in the Appendix. 2. Estimator in mode space Let {p'} be the data set consisting of the acoustic pres- sure field p' sampled on each hydrophone located at x2, l = 1,2,...,L of a vertical array. Using a mode filtering ap- proach (for example, by least-squares best fitting, damped least-squares best fitting, or singular value decomposition), we obtain a set of complex modal amplitudes A * n = 1,...,N. The estimator in the mode space is defined as follows: Fm (x ,x ): lag <x,x > - A :l (22) 1 where A n (x ,x ) is the calculated complex modal ampli- tudes. It can be computed by (20) for each given x s. N and N2 are the grid numbers of range and depth, respec- tively. We know that in a stratified ocean, some effective source localization processing methods have been discov- ered. For example, in Ref. 5, Shang presented a high-resolu- tion method of source localization processing in mode space, which requires only N q-N2 searching number. But in a waveguide with an inclusion, it is no longer proper to sepa- rate the depth search and the range search, because the sepa- ration of variables is no longer valid in the whole waveguide. Fortunately, the representation (18) can be used to sep- arate the source location and the detecting locations. This will greatly reduce the computation load. In view of ( 7 ), we can rewrite ( 18 ) as (for x < y < x ) s -- ikanx] ( eikn x, oo i bn (X2 )e n (X2) P(X;xs): Z dk n n=l -- 2 fo 3po(x;y) (I + K) -' X [ qn (Y2 ) eik"V' ] drr. v ) . (23) For other cases of x ,y, and x, we can get a similar repre- sentation with a proper change of the signs of x ,y, and x. Hence, we can approximate p(x;x s) by v i -ikn'l (24) pN(X;X s) = n B n (x) n (x )e , = dkn where B n (x) -- n (X2) eiknx'-- 2 I 3pO(x;Y) X (I + K) - [ bn (Y2) eik"v' ] drry, (25) and Nis a properly chosen positive number. Note that B n (x) does not depend on x s. Therefore, we can compute Pv (x; xs) in two separated steps. ( 1 ) Compute B n (x ) for given x t, l = 1,2 ..... L. First we solve the integral equation (15) where the right-hand side is ik x changed to -- 2n (X2) e n , with n = 1,2,...,N. Then substi- tuting the solution (I q- K) - [ -- 2n (x2)e iknx' ] into (25), we obtain the B n (x ) for n = 1,2,...,N. This calculation re- quires us to solve the integral equation for N times, and re- quires L times potential evaluation for each solution of the integral equations. (2) Compute PN ( x; xs) for given x s. After B n ( x t) are obtained, pN (xt;x s) can be calculated using (24) where no integral equations are involved. It is clear that the computation ofps (x;x s) involves so- D. Approximation of the estimators The estimators presented in the last section provide a tool to localize an acoustic source in a shallow ocean with a known inclusion. This inclusion can be arbitrarily large. To scan an area, we may compute the estimator for each chosen point Xs in the area. If the point Xs is close to the real location of the acoustic source, the estimator may appear as a large number. For uniform searching in a rectangular area, this scheme requires a source searching number N X N2, where Waveaulde bottom (x2=100) ½x;,x9 Acoustxc Source Waveguide surface (x2=O) FIG. 2. Detection of acoustic source by vertical hydrophone array. 997 d. Acoust. Soc. Am., Vol. 92, No. 2, Pt. 1, August 1992 Y. Xu and Y. Yan: Shallow water source localization 997 Downloaded 01 Dec 2012 to 150.135.135.70. Redistribution subject to ASA license or copyright; see http://asadl.org/termslution of the integral equation for only N times, which is independent of the source searching number. Hence, this computation becomes economical when a large source searching number is required. II. COMPUTER SIMULATION Computer simulations using the aforementioned meth- od are carried out on Cray2 of Minnesota Supercomputer Center. In this section we present two examples from our computations. A. Example 1- Vertical hydrophone array The configuration for the computer simulation is de- picted in Fig. 2. We assume the waveguide has' depth of 100 m. The sound speed is assumed to be 1500 m/s. An acoustic source S located at ( -- 500/rr, 100/rr) emits a time-harmonic wave at the frequency f= 30 Hz. The hydrophone array is arranged vertically at (600/rr,2.5j), j = 0,1,...,40. There is an inclu- sion 1 with a pressure release surface that occupies the re- gion ( l + - If the waveguide is normalized to depth rr, then the normal- ized wave number k -- 4, which means there are four propa- gating modes for the acoustic wave at the given frequency. We first generate the propagating wave by our approxi- mate boundary integral equation method. More precisely, we solve the integral equation (14) for ½(x;x), where po(x;x ) is given by (8) with truncation at n -- 30 and x = ( -- 500/rr, 100/rr), and substitute the ½(x;x ) into (18) to get the propagating field p(x;x). [A contour plotting of the propagating wave with the source at x = ( -- 350/rr, 100/rr) is plotted in Fig. 3. ] In particular, we obtain Pm = P (600/rr,2.Sm; x),m = 0,1,...,40. To make these data closer to reality, we add some Gaussian noise (generated by the NAG subroutine gO5ddfin our computa- tion) to these data and use them as our detected data. The second step is to compute the estimator. Since there are only four propagating modes, we choose N = 10 and compute B n (x) ,It = 1,2,..., 10. Using these B n (x), we search -!.0 g, (100/) FIG. 3. Propagating wave in perturbed waveguide. FIG. 4. Theoretical estimator (3-D plotting). -9.0 -8.'l -7.8 -7.2 -6.6 -6.0 -S.'l -'l.8 -'l.2 -3.6 ß , (100/r) FIG. 5. Theoretical estimator (contour plotting). FIG. 6. Estimator when detected data with Gaussian noise (3-D plotting). 998 J. Acoust. Soc. Am., Vol. 92, No. 2, Pt. 1, August 1992 Y. Xu and Y. Yan: Shallow water source localization 998 Downloaded 01 Dec 2012 to 150.135.135.70. Redistribution subject to ASA license or copyright; see http://asadl.org/terms-9.o -.'., ' -i.e ¾.2 .-6'.6 FIG. 7. Estimator when detected data with Gaussian noise (contour plot- ting). FIG. 10. Estimator Fp (xS), a filter with threshold value 0 55 is used (con- tour plotting). ß _. .. .... (x =lOO) Hydrophone array (x ,x ) Inclusion (1 x 7) Acoustic Source Haveguide surface (x2=O) FIG. 11. Detection of acoustic source by horizontal hydrophone array. o c' .. o .. -e'., -;% -;'.2 -œ.6 .o -s'., ß , (:] oo/-) FIG. 9. E,'timator Fp (x9, a filter with threshold value 0.5 is used (contour plotting). FIG. 12. Theoretical estimator (3-D plotting). 999 J. Acoust. Soc. Am., Vol. 92, No. 2, Pt. 1, August 1992 Y. Xu and Y. Yan: Shallow water source localization 999 Downloaded 01 Dec 2012 to 150.135.135.70. Redistribution subject to ASA license or copyright; see http://asadl.org/terms-8.4 -7.8 -7.2 -6.6 -6.0 x (100/r) -. .... -4.2 -3.6 -3.n FIG. 13. Theoretical estimator (contour plotting). FIG. 14. Estimator when detected data with Gaussian noise (3-D plotting). the area of [ -- 900/rr, -- 300/rr] X [0,100], and plot the es- timator F, (x s) for xS [ -- 900/rr, -- 300/rr ] X [ 0,100 ]. ( See Figs. 4-10.) Figures 4 and 5 show the estimator F, (x s) for the de- tected data Pm = p(600/rr,2.Sm; xs),m = 0,1,...,40 without adding Gaussian noise. Though these beautiful plots have not too much sense in practice, we present them here as theo- retical expectancies and use them for comparison. Figures 6 and 7 show the estimator F, (x s) for the de- tected data Pm = p(600/rr,2.Sm;x s),m = 0,1,...,40 which contain Gaussian noise with signal-to-noise ratio S/N = 10 dB. Figures 8-10 show the estimator F, (x s) for the detected data Pm = p(600/rr,2.5m;x s),m = 0,1,...,40 which contain Gaussian noise with signal-to-noise ratio S/N = 0 dB. In Fig. 8, we plot the contour of Fp (xS), which shows that the signal is buried by the noise. In Fig. 9, a filter with the thresh- old value F, (x s) = 0.5 is used, i.e., we set F, (x s) = 0 if F, (x s) < 0.5. In Fig. 10, the threshold value is increased to F, (x s) = 0.55 and the source is clearly identified. B. Example 2: Horizontal hydrophone array The configuration for the computer simulation is de- picted in Fig. 11. We assume the waveguide,the inclusion and the other acoustic parameters are the same as that in example 1 except that the hydrophone array is arranged horizontally at [ ( 100j + 3000)/6rr,25/rr], j ---- 0,1,...,6. In the same way as in example 1, we compute the estimator F, (x s) and plot it in Figs. 12-15. Figures 12 and 13 show the estimator F, (x s) for the detected data Pm =P[ ( 100j + 3000)/6rr,25/rr], j = 0,1,...,6 without adding Gaussian noise. Figures 14 and 15 show the estimator F, (x s) for the detected data Pm =P[ ( 100j + 3000)/6rr, 25/rr], j = 0,1,...,6 which contain Gaussian noise with signal-to- noise ratio S/N = 10 dB. FIG. 15. Estimator when detected data with Gaussian noise (contour plot- ting). III. CONCLUSION (1) Matched-field signal processing in complex envi- ronments are very interesting problems. One of the essential parts of these problems is to find an efficient and accurate algorithm to solve the propagating field. The scheme used here makes use of the separation of the source and the detec- tion array, and greatly reduces the computation. (2) The signal processing method used here is a high resolution method. It localizes the source nicely even when a substantial amount of noise exists. ACKNOWLEDGMENTS The research of Yongzhi Xu was supported in part by the Institute for Mathematics and its Applications with funds provided by the National Science Foundation, the Minnesota Supercomputer Institute, and the Alliant Tech- system Inc. The work ofYi Yan was supported in part by the National Science Foundation Grant RII-8610671 and the Commonwealth of Kentucky through the University of 1000 J. Acoust. Soc. Am., Vol. 92, No. 2, Pt. 1, August 1992 Y. Xu and Y. Yan: Shallow water source localization 1000 Downloaded 01 Dec 2012 to 150.135.135.70. Redistribution subject to ASA license or copyright; see http://asadl.org/termsKentucky's Center for Computational Sciences. The au- thors would like to thank the referees for helpful and con- structive comments of the original manuscript. APPENDIX An approximate boundary integral equation method la is included in this Appendix for solving the boundary inte- gral equation ½(x) + 2 vy(x;y)½(y)&ry - 2f (x), for (A1) where we assume without loss of generality the depth of the waveguide d = rr. Let Go ( x;y ) : = 67o ( X1,x:;y1,y: ) =1 77'(1/ -- ) n (X2)n (y2)e-(n-- 1/2)Ix , and M(xv)' = po(x;y) -- Go(x;y) = 1 n (X2)n (Y2) I i ikanlx, -- Y, I X k;n e where a n = [ 1 -- ( 2n -- 1 ) 2/4k 211/2. We can rewrite (A 1 ) in the form (A2) -- (n- 1/2)lx--yl / , (A3) p(x) +2fo 8Go fon 8M 8vy (x;y)p(y)&ry + 2 (x;y) X ½(y)&ry = 2f (x), for (A4) We assume that the boundary 1 is given by a 2rr-periodic parametric representation with I Y' (s) 1-7 = 0 for all s. Furthermore, we assume that y is a C © function. Denote the kernel of the integral equation (A1) by 8 8 Ko(x;y) = 2 8vy G(x;y), Kl(X;y) = 2 8% M(x;y), and set w(s = g(s) = f(r(s)), Lo(s,) = I, Ll(S,a) = K,½r(s);r(a))lr'(a)l. Thus Eq. (A 1 ) reduces to w(s) + w(a)Lo(s,a)da + w(a)L1 (s,a)da = g(s), (A5) s[ -- rr, rr]. (A6) It is shown 16 that Lo(s,a) is continuous for (s, rr)[ -- rr, rr] X [ -- rr, rr], and that L 1 (S,O') can be written as $10- L l(s,O') = -- a(s,a)log 2 sin + b(s,a) 2 X (arctan cot S+a+sgn(s 2-- 2 q- L2 (s,o'), (A7) where a(s,a), b(s,a), and L2 (s, rr) are continuous and differ- entiable for (s,a)[ -- rr, rr] X [ -- rr, rr]. We use the ordi- nary rectangular formula rr N /2 v()d=h v(ti ), (AS) --rr k= --N/2+ 1 the weighted quadrature formula -- v(tr) log s sin s -- tr N/2 h R 1(s- tk )o(tk ), (AS) k= --N/2+ 1 and the weighted quadrature formula ( v (a) arctan cot s + a - ,r 2 + sgn(s2 -- 0'2) da N/2 h R 2(S,t )v(t ), (A10) k= --N/2+ 1 where t = kh with h = 2r/N and N an even integer are the equidistant quadrature knots and the weights are given by N/-- 1 1 2 ei(N/2)s R 1 (S) -- COS IS + -- 2_-1 -]- N and N /2 ( ie- ils -- ilt k 7't' R ) - ,s.i.n/Isl + + Isl ß l=-N/2+1 I 21 ! 2 l,aO Applying the quadrature formula (A8), (AS), and (A 10) to the integrals in (A6), we replace the integral equa- tion (A6) by the linear system N/2 w q- h [Rl(tj_)a(t,t ) q- R2(ti,t)b(tj,t ) k= --N/2+ 1 q- Lo(ti,t ) q- L2(ts,t ) ]wk = g, j= --N/2 + 1,...,N/2, (All) for the approximate values ws to w(ts ), where gs = g(ts ). For more detailed discussions, the reader could refer to Ref. 16. A. B. Baggeroer, W. A. Kuperman, and H. Schmidt, "Matched field pro- cessing: Source localization in correlated noise as an optimum parameter estimation problem," J. Acoust. Soc. Am. 83, 571-587 (1988). 2H. P. Bucker, "Use of calculated wave field and matched field detection to locate sound source in shallow water," J. Acoust. Soc. Am. 59, 368-373 (1976). 3M. B. Porter, R. L. Dicus, and R. G. Fizell, "Simulation of matched-field processing in a deep-water Pacific environment," IEEE Trans. Ocean Eng. OE-12, 173-181 (1987). 4E. C. Shang, C. S. Clay, and Y. Y. Wang, "Passive harmonic source rang- 1001 J. A½oust. Soc. Am., Vol. 92, No. 2, Pt. 1, August 1992 Y. Xu and Y. Yan: Shallow water source localization 1001 Downloaded 01 Dec 2012 to 150.135.135.70. Redistribution subject to ASA license or copyright; see http://asadl.org/termsing in waveguides by using mode filter," J. Acoust. Soc. Am. 78, 172-175 (1985). 5E. C. Shang, "An efficient high-resolution method of source localization processing in mode space," J. Acoust. Soc. Am. 86, 1960-1964 (1989). 69. Colton, "The inverse scattering problem for time-harmonic acoustic waves," SIAM Rev. 26, 323-350 (1984). ?B. D. Sleeman, "The inverse problem of acoustic scattering," IMA J. Appl. Math. 29, 113-142 (1982). 8D. Colton, R. Ewing, and W. Rundell, Inverse Problem in Partial Differen- tial Equations (SIAM, Philadelphia, 1990). 9R. P. Gilbert and Y. Xu, "Starting fields and far fields in ocean acoustics," Wave Motion 11, 507-524 (1989). øR. P. Gilbert and Y. Xu, "Dense sets and the projection theorem for acoustic harmonic waves in homogeneous finite depth oceans," Math. Methods Appl. Sci. 12, 69-76 (1989). Y. Xu, "An injective far-field pattern operator and inverse scattering problem in a finite depth ocean," Proc. Edinburgh Math. Soc. 34, 295-311 (1991). 2y. Xu, T. C. Poling, and T. Brundage, "Direct and inverse scattering of time harmonic acoustic waves in inhomogeneous shallow ocean," IMA Preprint 821 ( 1991 ), to appear in Proceedings of Third IMACS Sympo- sium on Computational Acoustics, Harvard University, 1991. 3T. W. Dawson and J. A. Fawcett, "A boundary integral equation method for acoustic scattering in a waveguide with nonplanar surfaces," J. Acoust. Soc. Am. 87, 1110-1125 (1990). 4G. V. Norton and M. F. Werby, "Some numerical approaches to describe acoustical scattering from objects in a waveguide," Math. Comput. Mod- eling 11, 81-86 (1988). SM. F. Werby, Comput. Acoust. 2, 93-112 (1990). 6y. Xu and Y. Yan, "An approximate boundary integral equation method for acoustic scattering in shallow oceans," to appear in J. Comput. Acoust. (1992). 1002 J. A½oust. So½. Am., Vol. 92, No. 2, Pt. 1, August 1992 Y. Xu and Y. Yan: Shallow water source localization 1002 Downloaded 01 Dec 2012 to 150.135.135.70. Redistribution subject to ASA license or copyright; see http://asadl.org/terms
1.1469201.pdf	Domain wall motion effect on the anelastic behavior in lead zirconate titanate piezoelectric ceramics El Mostafa Bourim, Hidehiko Tanaka, Maurice Gabbay, Gilbert Fantozzi, and Bo Lin Cheng Citation: Journal of Applied Physics 91, 6662 (2002); doi: 10.1063/1.1469201 View online: http://dx.doi.org/10.1063/1.1469201 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 An in situ diffraction study of domain wall motion contributions to the frequency dispersion of the piezoelectric coefficient in lead zirconate titanate Appl. Phys. Lett. 102, 042911 (2013); 10.1063/1.4789903 Evaluation of domain wall motion in lead zirconate titanate ceramics by nonlinear response measurements J. Appl. Phys. 103, 054108 (2008); 10.1063/1.2894595 Internal friction study on low-temperature phase transitions in lead zirconate titanate ferroelectric ceramics Appl. Phys. Lett. 82, 109 (2003); 10.1063/1.1534610 Dielectric and piezoelectric response of lead zirconate–lead titanate at high electric and mechanical loads in terms of non-180° domain wall motion J. Appl. Phys. 90, 5278 (2001); 10.1063/1.1410330 Effects of fluorine–oxygen substitution on the dielectric and electromechanical properties of lead zirconate titanate ceramics J. Appl. Phys. 86, 5747 (1999); 10.1063/1.371588 [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: 170.140.105.10 On: Mon, 24 Nov 2014 16:04:49Domain wall motion effect on the anelastic behavior in lead zirconate titanate piezoelectric ceramics El Mostafa Bourim and Hidehiko Tanakaa) National Institute for Materials Science (NIMS), Namiki 1-1, Tsukuba, Ibaraki 305-0044, Japan Maurice Gabbay and Gilbert Fantozzi Institut National des Sciences Applique ´es (INSA) de Lyon - GEMPPM, UMR 5510 CNRS, Ba ˆtiment B. Pascal, 69621 Villeurbanne Cedex, France Bo Lin Cheng IRC in Materials, University of Birmingham, Edgbaston, Birmingham B15 2TT, United Kingdom ~Received 11 December 2000; accepted for publication 22 February 2002 ! Three undoped lead zirconate titanate ~PZT!ceramics were prepared with compositions close to the morphotropicphaseboundary:Pb ~Zr0.50Ti0.50!O3,Pb~Zr0.52Ti0.48!O3,andPb ~Zr0.54Ti0.46!O3.Internal frictionQ21and shear modulus Gwere measured versus temperature from 20°C to 500°C. Experiments were performed on an inverted torsional pendulum at low frequencies ~0.1, 0.3, and 1 Hz!. The ferroelectric–paraelectric phase transition results in a peak ~P1!ofQ21correlated with a sharpminimum M1ofG.Moreoverthe Q21(T) curvesshowtworelaxationpeakscalledR 1andR2 respectively, correlated with two shear modulus anomalies called A 1and A2on theG(T) curves. The main features of the transition P1peak are studied, they suggest that its behavior is similar to the internal friction peaks associated with martensitic transformation. The relaxation peak, R 1and R2are both attributed to motion of domain walls ~DWs!, and can be analyzed by thermal activated process described by Arrhenius law. The R 2peak is demonstrated to be due to the interaction of domain walls and oxygen vacancies because it depends on oxygen vacancy concentration andelectrical polarization. However, the R 1peak is more complex; its height is found to be increased as stress amplitude and heating rate increase. It seems that the R 1peak is inﬂuenced by three mechanisms: ~i!relaxation due to DW–point defects interaction, ~ii!variation of domain wall density, and ~iii!domain wall depinning from point defect clusters. © 2002 American Institute of Physics. @DOI: 10.1063/1.1469201 # I. INTRODUCTION Lead zirconate titanate Pb ~Zr12xTix)O3~usually called PZT!ceramics are used as the active material for various sensors and actuators. These devices require high electrome-chanical coupling constants as well as low dielectric and me-chanical losses. The energy dissipation from dielectric andmechanical losses causes a change of physical properties inpiezoelectric materials. It is important, therefore, to investi-gate the mechanisms that control these losses in order toreduce them. The losses are mainly associated with domainwall motion, 1but also with point defects.2–4Postnikov et al.5 have shown that the peaks of mechanical losses observed in PZT ceramics are linked to the interaction between DWs andmobile point defects. In this paper, the anelastic behavior of PZT ceramics is studied through the analysis of internal friction peaks due onthe one hand to the ferroelectric–paraelectric phase transi-tion, and on the other hand to the relaxation linked to themotion of DWs. This is in order to reveal the physicalmechanisms responsible for mechanical losses in the ferro-electric PZT ceramics.II. MATERIALS AND EXPERIMENTAL PROCEDURE A. Studied chemical compositions Undoped PZT ceramics were prepared by conventional sintering ~solid solution !from the starting powders of PbO, ZrO2, and TiO 2of 99.9% pure reagent grade. Mixtures of desired molar ratio were chosen near the morphotropic phaseboundary with the following Zr/Ti ratios: Pb ~Zr 0.50 Ti0.50)O3,P b~Zr0.52Ti0.48!O3, and Pb ~Zr0.54Ti0.46!O3: so- called PZT 50/50, PZT 52/48, and PZT 54/46, respectively.X-ray diffraction showed that the structures of the PZT50/50and the PZT 52/48 compositions were tetragonal, as identi-ﬁed by the associated (002) T/(200) Tdoublet lines ~Fig. 1 !. In the PZT 54/46 composition, the x-ray diagram indicatedthe presence of two phases, a tetragonal one related to the (002) T/(200) Tdoublet lines, and a rhombohedral one related to the (200) Rline~Fig. 1 !. The coexistence of this double structure is speciﬁc to the morphotropic phase boundaries.6,7 The plateau shape, which corresponds to the additional dif-fracted intensity between the doublet lines, is a ﬁngerprint ofthe ferroelectric microstructure related to the 90° DWs. 8 Thus, for two associated 90° domains ~indicated, for ex- ample, by DI and DII respectively !, the corresponding x-ray diffraction is a doublet, in which the ﬁrst line would resultfrom DI domain diffraction, the second line would result a!Electronic mail: tanaka.hidehiko@nims.go.jpJOURNAL OF APPLIED PHYSICS VOLUME 91, NUMBER 10 15 MAY 2002 6662 0021-8979/2002/91(10)/6662/8/$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: 170.140.105.10 On: Mon, 24 Nov 2014 16:04:49from DII domain diffraction, and the plateau would arise from the connection—a zone frontier existing between thetwo domains. Consequently, the 90° domain wall corre-sponds to a distorted zone with continuous variation of theinterplanar distance from DI domain to its associated DIIdomain. B. Internal friction and shear modulus measurements The specimens were rectangular bars ~dimensions: 40 3531m m3!. After machining, the specimens were an-nealed ~500°C in air for 1 h !in order to remove residual stresses. An inverted torsion pendulum is used to measurethe internal friction Q 21and the shear modulus Gas a func- tion of temperature from 20 to 500°C.9 Asample was ﬁxed at its lower end to a ﬁxed clamp and at its higher end to a mobile clamp interdependent to a rigidtransmission stem via which the sample was excited by tor-sional oscillations. In an anelastic solid, the application to thesample of a sinusoidal torsional stress s(t)5s0eivtinduces a sinusoidal strain ewith the same pulsation vbut with a phase lag fbehind the stress: e(t)5e0ei(vt2f). The lag angle fis typical of an anelastic behavior of the material and allows to compute the mechanical loss ~Q21!. The dynamic elastic shear modulus is given by the ratio between the applied stress and the measured strain; it is acomplex number G expressed by the following relation: G5s~iv! e~iv!5Gexp~if!5G81iG9. Thus, the internal friction ~Q215tanf!is calculated by the ratio ofG9/G8, and the shear modulus Gis estimated to be equal toG8in the case of weak mechanical losses. Three sinusoidal vibration frequencies were used: 0.1, 0.3, and 1.0 Hz with a strain amplitude lower than 1025.T o neglect the air damping to the oscillation movement ofsample, all measurements were performed under vacuum(’10 23Torr!. III. RESULTS AND DISCUSSION Figure 2 shows Q21(T) andG(T) curves recorded dur- ing the ﬁrst heating run for the three different vibration fre-quencies ~0.1, 0.3, and 1 Hz !on the PZT 52/48 and PZT 54/46 ceramics. The following anelastic events can be iden-tiﬁed: ~i!TheP 1peak ofQ21correlated with a sharp mini- mumM1of shear modulus Gare due to the tetragonal–cubic phase transition. ~ii!TheP2peak ofQ21and theM2mini- mum ofGare associated with the rhombohedral–tetragonal phase transition ~in the case of the PZT52/48 and PZT50/50 ceramics,therhombohedral–tetragonalphasetransitionchar-acteristics are not visible because their temperatures areabout 269 and 2139°C, respectively !. 9–11~iii!Two relax- ation peaks R 1and R2correlated with two shear modulus anomalies A 1and A 2, respectively, on the G(T) curves. Similar results were observed for the PZT50/50 compositionceramic.The analysis of the three main peaks P 1,R2, and R 1 is presented below. A.P1peak analysis The height of the P1peak depends on the following parameters: vibration frequency f, temperature rate T˙~Fig. 7 ! and stress amplitude s~Fig. 8 !. All these characteristics are very similar to those of materials that undergo martensitictransformations, suggesting that its behavior is similar to theinternal friction peaks associated with martensitic transfor-mation. Among the various theoretical models reviewed byVan Humbeeck, 12there is an interesting qualitative agree- ment with the phenomenological models presented by Belkoet al., 13Delorme,14and De Jonghe et al.15 FIG. 1. X-ray diffraction 002/200 doublet lines evolution with phase structure.6663 J. Appl. Phys., Vol. 91, No. 10, 15 May 2002 Bourimet 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: 170.140.105.10 On: Mon, 24 Nov 2014 16:04:49Internal friction Q21calculated by Belko et al.is given by the relation Q215Gba2 kTm˙ v whereGis the shear modulus, bis the volume of critical nucleus,ais the non-elastic deformation at Curie tempera- ture,m˙is the relative volume of the transformed phase per unit of time, vis the angular frequency, Tis the temperature, andkis Boltzmann’s constant. However, this relation does not take into account the applied stress, which is also thecase with Delorme’s model. De Jonghe et al.propose a model in which the amount of transformed material per unit of time takes into account not only the temperature rate T˙but also the applied stress s, thus the amount of transformed phase per unit of time m˙5f(T,s) is expressed according the following relation: dm dt5]m ]T]T ]t1]m ]s]s ]t. Nevertheless, the applied stress only contributes to phase transformation over a critical stress sC, and thus De Jonghe et al.propose the calculation of internal friction Q21using the following relation: Q215A 2p.H]m ]T]T ]t1 f14 3s0]m ]s.F12SsC s0D3GJ With it, De Jonghe et al.have proposed various peak shapes for martensitic transformation according to the mobility ofthe interfaces and the applied stress value. This model pre- dicts a peak shape for the phase transition similar to thetransition peak shape obtained by Gridnev et al. 16with niobium-doped PZT ceramics. Such a doping allows to sup-press the relaxation peaks R 1and R2and to make the low temperature side of the P1peak clearly observable. Thus, it is possible to improve our understanding of phase transitionmechanisms. Moreover this also facilitates the decomposi-tion ofQ 21(T) curves to elucidate the R 1peak. Our results show that the P1peak is more visible when the vibration frequency is lower ~Fig. 2 !: the peak height (Qmax21)P1increases as the vibration frequency decreases @Fig. 3~a!#, and the peak temperature is insensitive to the fre- quency and to the thermal cycle ~heating and subsequent cooling !, which have an effect only on the peak height @Fig. 3~a!#. TheP1peak height increases with increasing tempera- ture rate T˙~Fig. 7 !with a nonlinear evolution as shown in Fig. 3 ~b!from the plot ~Qmax21)P1versusT˙/f, and the peak temperature remains constant. Furthermore the P1peak height is sensitive to the stress amplitude of measurement~Fig. 8 !: the increase is quasilinear @Fig. 3 ~c!#. These results are roughly described by the preceding models, but for the recently observed nonlinear relations be- tweenQ 21and the temperature rate T˙as well as Q21and the vibration frequency, Zhang et al.17suggested that the P1 peak observed at the transition temperature arises from the motion of phase interfaces during the ﬁrst-order phase tran-sition and obtained a relation which is well veriﬁed by themeasurements. Similar relation was obtained by Wang FIG. 2. Curves Q21(T) andG(T) curves recorded at low frequencies during the ﬁrst heating run.6664 J. Appl. Phys., Vol. 91, No. 10, 15 May 2002 Bourimet 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: 170.140.105.10 On: Mon, 24 Nov 2014 16:04:49et al.18by using models developed for ﬁrst-order transition. The effect of stress can be introduced by considering anequivalent depinning process. B. R2peak analysis The R 2peak is controlled by a relaxation mechanism because it is frequency dependent. The temperature of thepeak depends on the frequency according to an Arrheniusequation and the relaxation time can be written as: t 5t0exp(H/kT), whereHis the activation energy, t0the pre- exponential factor, Tis the absolute temperature, and kis Boltzmann’s constant. For a Debye peak, the peak maximumoccurs when vt51, that is to say: ln~vt0!1H kTP50, whereTPis the peak temperature and v52pf, withfthe vibration frequency. TheArrhenius plots of ln( v) versus 1/ TP~Fig. 4 !lead to the following activation parameters: activation energyH (R2)5160.1 eV and pre-exponential factor t0(R2) 5102(1361)s. The magnitude of the pre-exponential factor t0(R2)is coherent with a point defect relaxation, and the activation energy H(R2)of about 1 eV is the typical value for the migration enthalpy of oxygen vacancies in perovskiteoxides. 2–19So, what is the physical mechanism controlling the relaxation process linked to the R 2peak? As observed in BaTiO 3,20the R2peak could be linked to the domain wall. As proposed by Postnikov et al.,5the R2 peak can be related to a relaxation mechanism which in- volves an interaction between the DWs and mobile pointdefects. Postnikov et al.have proposed a model of the relax- ation in ferroelectric materials based on an interaction be-tween mobile point defects and immobile DWs. In thismodel, under external mechanical stress, there is an increasein the number of bound electric charges on the DWs via thepiezoeffect. Electrical compensation of these charges by themigration of mobile charged point defects present in the lat-tice leads to a change in the electric ﬁeld within the domainwith time. Thus, by the inverse piezoeffect, a mechanicalrelaxation takes place. The Postnikov model predicts, for small concentration of mobile point defects, that the relaxation peak height ~Q max21! and the relaxation time tare given by the following rela- tions: FIG. 3. Variation of P1peak height ~Qmax21)P1as function of vibration fre- quencyfand 10/f~a!,T˙/f~b!, and stress amplitude oscillation s~c!. FIG. 4. Arrhenius plots for the three different ceramic compositions PZT 50/50, PZT 52/48, and PZT 54/46.6665 J. Appl. Phys., Vol. 91, No. 10, 15 May 2002 Bourimet 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: 170.140.105.10 On: Mon, 24 Nov 2014 16:04:49Qmax2152~d332d31!2c0L2q2 p4er2e0kTs, ~3a! t5L2 p2D5L2 p2D0expSH kTD, ~3b! where,c0is the equilibrium concentration of point defects, L is the 90°-domain width, dijis the piezoelectric constant, qis the point defect charge ~1.6310219C!,eris the relative permittivity, e0is the vacuum permittivity, sis the unrelaxed elastic compliance, kis Boltzmann’s constant, His the acti- vation energy, and Dis the diffusion coefﬁcient of the point defects. In these Eqs. ~3a!and~3b!the height of internal friction peak depends on point defect concentration c0and domain width L, and the relaxation time varies as L2. Figures 5 and 6, respectively, show the inﬂuences of thermal treatment and electrical polarization on the R 2peak. High temperature vacuum annealing can introduce excess ofoxygen vacancies, the R 2peak height is clearly increased by such thermal treatment, but an annealing in air at the samehigh temperature reduces it. And regarding the polarization,which has a direct effect on the variation of domain size or domain wall density, the peak height is decreased by suchpoling under an electrical ﬁeld of 3 kV/mm at 130°C during30 min. These results suggest that the R 2peak is related to a relaxation mechanism involving both, point defects ~oxygen vacancies !and DWs. A good agreement was observed between the R 2peak features and Postnikov’s model. The inﬂuence of oxygen va-cancy concentration and domain size variations on the R 2 peak height veriﬁes the Eq. ~3a!. Furthermore, the measured activation energy corresponds to the activation energy fordiffusion of oxygen vacancies. This hypothesis is conﬁrmedby the results of Postnikov et al.and Gridnev et al., who observed that the R 2peak is suppressed by niobium doping which decreases the oxygen vacancy content as foreseen byEq.~3a!. In addition to the sensitivity of the R 2peak height to thermal treatment and effect of poling, we notice also a shiftin its temperature position T P. Analysis of the peak shift from the thermal treatment results by theArrhenius equationhas showed that the activation energy H (R2)remains approxi- mately about 1 eV, independent on the changes in oxygenvacancy concentration. This means it is subject to the sameenergy barrier for the relaxation process of the R 2peak, al- though the pre-exponential factor t0(R2)increases from >10215s after annealing in air up to >10212s after anneal- ing under vacuum. So, according to the Eq. ~3b!relating the relaxation time to the domain size, we can say that the oxy-gen vacancy concentration has also an effect on the domaindimension, which inﬂuences the peak temperature position. In the Eq. ~3b!the pre-exponential factor t0is expressed byL2/p2D0. Therefore, at peak temperature condition vt 51, with constant vibration frequency and constant activa- tion energy as was checked on the two kinds of annealing,we ﬁnd that the peak temperature T Pis proportional to pre- exponential factor t0, and consequently to the domain width L, as shown in the following relationship: vL2 p2D0expSH kTpD51. From the above analysis, the hypothesis that the domain di- mension is dependent to the oxygen vacancy concentration isfeasible when we consider oxygen vacancies as an obstacleto the movement of DWs during their apparitions to form the90° domain frontiers in the ferroelectric phase. So, at theparaelectric-ferroelectric phase transition during the subse-quent cooling after air annealing at 600°C for 6 h, the weakoxygen vacancy concentration will not be high enough toblock domain wall motion, and a ﬁne domain microstructurewill be established.And thus the small Lsize implies that the weak t0(R2)results in low TR2. On the other hand, during subsequent cooling under vacuum annealing 600°C for 6 h,the high concentration of oxygen vacancies slows down do-main wall motion, excess oxygen vacancies are segregatedpreferably at the DWs as observed directly in high resolutiontransmission electron microscopy ~TEM !by Tanet al.. 21 This kind of pinning to the motion of DWs induces the for- mation of a large domain microstructure, and thus the large L FIG. 5. Inﬂuence of thermal treatment on the Q21(T) curve. FIG. 6. Inﬂuence of electrical polarization on the Q21(T) curve.6666 J. Appl. Phys., Vol. 91, No. 10, 15 May 2002 Bourimet 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: 170.140.105.10 On: Mon, 24 Nov 2014 16:04:49size implies that high t0(R2)leads to high TR2. This is pre- cisely what we observed in the analysis of the R 2peak be- havior with the thermal treatment effect. The scenario of an oxygen vacancy concentration effect on domain size is further supported by the electrical polingwhich has a direct effect on changes in domain size. It isknown that polarization increases domain size. According toPostnikov’s model, an increase in domain size will cause thepeak to shift toward high temperatures with an increase in itslevel, but the poling effect has resulted in the opposite be-havior for the R 2peak ~Fig. 6 !. In fact, during the poling operation of a ferroelectric ceramic, the domains ﬁrst un-dergo a reduction in their size 22and, when the electric ﬁeld reaches a sufﬁcient value, an increase in the domain sizestarts to take place. Considering the fact that the appliedelectric ﬁeld to our ceramic was weak compared to a coerci-tive ﬁeld, it is likely that only a decrease in domain size tookplace. Consequently, the decrease in domain width Lin- volves a decrease in the relaxation time as foreseen by Eq.~3b!, which makes the peak R 2move toward low tempera- tures, and also leads to a reduction in its height according toEq.~3a!. This is again precisely what we observed in the analysis of the R 2peak behavior for the polarization effect. In conclusion, our results suggest that the R 2peak relax- ation could be related to the interaction between 90° DWsand mobile oxygen vacancies. Furthermore, oxygen vacan-cies have a direct inﬂuence on the conﬁguration of ferroelec-tric domains at time of their formation. C. R1peak analysis The R1peak behavior is very complex because it is in- ﬂuenced on both sides by the R 2andP1peaks ~as for its temperature is near the Curie temperature TC). Nevertheless the following three main features were observed. 1. Frequency dependence The R1peak is controlled by a relaxation mechanism as the R2peak. The activation parameters deduced from Arrhenius plots are H(R1)51.860.2 eV and t0(R1) 5102(1861)s for all the PZT 50/50, PZT 52/48, and PZT 54/46 ceramics. The activation energy H(R1)is much higher than one for the diffusion of oxygen vacancies. Its value letsus suppose that the relaxation process is due to an interactionof the DWs with the point defects with signiﬁcant diffusionactivation energy. Thus, the R 1peak could be attributed to the interaction of 90° DWs with Ti or Zr vacancies. Thishypothesis is conﬁrmed by the decrease of the R 1peak due to a reduction of the Ti or Zr vacancies by introducing nio-bium oxide Nb 2O5in PZT ~Nb replaces Ti or Zr atoms !as observed by Postnikov et al.However, the pre-exponential factor t0(R1)of relaxation time is very short and it is difﬁcult to give it a physical meaning. The shortness of t0(R1)could be due to the reduction in the domain wall inertia generatedby the thinning of the wall thickness with the temperature. In general, the determined activation parameter values are only the apparent ones, and suggest that the R 1peak is related to a combination process. As shown below, the R 1 peak is dependent on other parameters.2. Temperature rate T ˙dependence Figure 7 shows an increase of the R 1peak height as a function of T˙as for the P1peak. Such T˙effect on a relax- ation peak associated with a very low pre-exponential factor t0have been attributed to microstructure changes due to an evolution of the material.23In fact, the appearance of the R 1 peak near the ferroelectric-paraelectric phase transition ( P1 peak!suggests an instability in domain arrangement. This state is identiﬁed in TEM observations:9the density of DWs increases with temperature, then domains disappear at Curietransition T C; inasmuch as the domain interface energy evolves to approaching zero when the temperature tends to-wardT C. According to the model of Wang et al.24based on the viscous motion of the DWs,25the number of domains N is proportional to ( TC-T)21, hence, N increases with tem- perature, which results in an increase of internal friction, andwhen the domain density reaches a critical value such thatthe interaction between walls tends to reduce their mobility,the internal friction decreases, thus the formation of the peaktakes place. Thus, the R 1peak could be linked to domain wall motion controlled by the nucleation of new domainsover a large temperature range before the Curie temperature. 3. Stress amplitude dependence Figure 8 shows an increase in R 1peak height with the oscillating stress amplitude. This increase could be due to‘‘dragging-depinning’’ process similar to the dislocation de-pinning from point defects. According to this mechanism,mechanical loss in the R 1peak could be activated by a drag- ging force produced by point defects opposing the domainwall motion. Thus, for a signiﬁcant stress or deformationamplitude, the displacement of the domain wall would alsobe signiﬁcant, and consequently a dependence of peak heighton the applied stress amplitude would be obtained. Indeed, the stress amplitude dependence can be given by the classical Granato and Lu ¨cke model 26which is relative to the dislocation depinning and can be expressed by the fol-lowing relationship: FIG. 7. Inﬂuence of temperature rate on Q21(T) curve.6667 J. Appl. Phys., Vol. 91, No. 10, 15 May 2002 Bourimet 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: 170.140.105.10 On: Mon, 24 Nov 2014 16:04:49Q21~e!5C1 eexpS2C2 eD, whereC1andC2are constants without dimensions, C2is proportional to the point defect concentration along the dis-location line, and eis the deformation. The Granato and Lu ¨cke plot is expected to be a straight line and predicts that its slope ~C2!is inversely proportional to the dislocation loop length between soft pinning points,and its intercept ~C 1!at the axis 1/ e50 is proportional to the dislocation density ~N!. Figure 9 shows the Granato and Lu ¨cke plot ~logQ21e versus 1/ e!in the temperature range between the R 1andP1 peaks. Such a plot exhibits a positive curvature characteristic of an interaction of dislocations with immobile point defectsdistributed in a glide plane, and accompanied at same timewith an increase of the dislocation density which appears bythe monotonous raising of the curve slope with deformation amplitude. By analogy, this simply means an interaction of adomain wall with point defects with a parallel increase in thedomain wall density under the stress amplitude effect. This inﬂuence of mechanical stress on the nucleation of new DWs has been highlighted also by Sarrazin et al. 27by optical observations in a BaTiO 3crystal subjected to local pressure stresses. They showed that the 90° domain densityincreases during the progressive application of the stress. It can be noticed that the R 1peak temperature does not depend on stress amplitude, contrary to a depinning process.The peak temperature T Pshould decrease as the applied stress sincreases.The lack of temperature shift could be due to simultaneous variation in the domain wall density, whichevolves with temperature and stress amplitude too. In theWanget al.model, 24which predicts that the peak tempera- ture is controlled by a self-locking effect when the DWsreach a critical density, this mechanism should shift the R 1 peak to high temperatures as the stress amplitude increases.However, an increase in stress also helps to increase the do-main wall density; hence, the additional density increasingunder stress plays a counterbalancing role in retaining thepeak at a similar temperature level. Thus, the R 1peak could be related to a viscous motion of DWs, and the peak height increasing could be due to adragging-depinning process involving the interaction ofpoint defects with DWs whose microstructure evolves withtemperature and stress amplitude. Moreover, the dragging-depinning process of DWs from oxygen vacancies can befurther supported, since the R 1peak shape underwent some variation when the oxygen vacancy concentration was de-creased by annealing in air ~Fig. 5 !. These different hypotheses must be veriﬁed by further experimental and theoretical studies. IV. CONCLUSION In this study of anelastic behavior in PZT ceramics by internal friction, three important peaks were observed: P1, R2, and R 1. TheP1peak is due to ferroelectric–paraelectric phase transition.The main features of this peak are similar tointernal friction peaks associated with a ﬁrst-order phasetransition. The R 2peak is controlled by a relaxation mecha- nism with activation energy H(R2)>1 eV and pre- exponential relaxation time t0(R2)>10213s. The R 2peak relaxation process involves interaction between DWs andpoint defects, such as oxygen vacancies. The R 1is a more complex peak with activation energy H(R1)>1.8 eV and t0(R1)>10218s. In addition to its relaxational behavior, the height of R 1peak depends on heating rate and stress ampli- tude. The R 1peak is probably controlled by at least three mechanisms: ~i!a relaxation mechanism involving interac- tion of DWs with point defects ~as Ti or Zr or oxygen va- cancies !,~ii!a domain density variation mechanism in a tem- perature range approaching the Curie transition, and ~iii!a hysteresis mechanism of domain wall depinning from aggre-gates of point defects with a stress amplitude dependence. Finally, it is of interest to note that the existence of struc- tural defects in the ferroelectric materials plays an important FIG. 8. Inﬂuence of stress amplitude vibration on Q21(T) curve. FIG. 9. Granato and Lu ¨cke plot at different temperatures around the R1and P1peaks.6668 J. Appl. Phys., Vol. 91, No. 10, 15 May 2002 Bourimet 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: 170.140.105.10 On: Mon, 24 Nov 2014 16:04:49role on the conﬁguration of domain size as well as on the motion of DWs. 1P. Gerthsen, K. H. Ha ¨rdtl, and N. A. Schmidt, J. Appl. Phys. 51, 1131 ~1980!. 2K. Carl and K. Ha ¨rdlt, Ferroelectrics 17, 473 ~1978!. 3G. H. Jonker, J. Am. Ceram. Soc. 55,5 7~1972!. 4U. Roblels and G. Arlt, J. Appl. Phys. 73, 3454 ~1993!. 5V. S. Postnikov, V. S. Pavlov, S.A. Gridnev, and S. K. Turkov, Sov. Phys. Solid State 10, 1267 ~1968!. 6P. Ari-Gur and L. Benguigui, J. Phys. D 8, 1856 ~1975!. 7F. Vasiliu, P. Gr. Lucuta, and F. Constantinescu, Phys. Status Solidi A 80, 637~1983!. 8C. M. Valot, N. Floquet, M. Mesnier, and J. C. Niepce, Fourth European Powder Diffraction Conference EPDICIV, Mater. Sci. Forum 228-231,5 9 ~1996!. 9E. M. Bourim, Ph.D. Thesis, INSA-Lyon, France ~1998!. 10E. M. Bourim, H. Idrissi, B. L. Cheng, M. Gabbay, and G. Fantozzi, J. Phys. IV 6, C8, 633 ~1996!. 11E. M. Bourim, H.Tanaka, M. Gabbay, and G. Fantozzi, Jpn. J.Appl. Phys. 39, 5542 ~2000!. 12J. V. Humbeeck, in Proceedings of the Summer School on IFS, Cracow, Poland, 14–17 June 1984, p. 131.13V. N. Belko, B. M. Darinskii,V. S. Postnikov, and I. M. Sharshakov, Phys. Met. Metallogr. 27, 140 ~1969!. 14J. F. Delorme, Ph.D. Thesis, Universite ´Claude Bernard-Lyon I, France ~1971!. 15W. DeJonghe, R. DeBatist, and L. Delaey, Scr. Metall. 10,1 1 2 5 ~1976!. 16S. A. Gridnev, B. M. Darinskii, and V. S. Postnikov, Bull. Acad. Sci. USSR, Phys. Ser. ~Engl. Transl. !33, 1106 ~1969!. 17J. X. Zhang, W. Zheng, P. C. W. Fung, and K. F. Liang, J.Alloys Compd. 211Õ212, 378 ~1994!. 18Y.Wang, X. Chen, and H. Shen, Chin. J. Met. Sci.Technol. 7,1 5 7 ~1991!. 19T. Ishigaki, S. Yamauchi, K. Kishio, J. Mizusaki, and K. Fueki, J. Solid State Chem. 73, 179 ~1988!. 20B. L. Cheng, M. Gabbay, and G. Fantozzi, J. Mater. Sci. 31, 4141 ~1996!. 21Q.Tan,Z.Xu,J.F.Li,andD.Viehland,Appl.Phys.Lett. 71,1062 ~1997!. 22A. R. Von Hippel, Molecular Science and Molecular Engineering ~Tech- nology Press of M.I.T, New York, 1959 !, pp. 237–276. 23J. J. Amman and R. Schaller, J. Alloys Compd. 211Õ212,3 9 7 ~1994!. 24Y.Wang,W. Sun, X. Chen, H. Shen, and B. Lu, Phys. Status SolidiA 102, 279~1987!. 25Y. N. Huang,Y. N. Wang, and H. M. Shen, Phys. Rev. B 46, 3290 ~1992!. 26A. V. Granato and K. Lu ¨cke, J. Appl. Phys. 27,5 8 3 ~1956!. 27P. Sarrazin, B. Thierry, and J. C. Niepce, J. Eur. Ceram. Soc. 15, 623~1995!.6669 J. Appl. Phys., Vol. 91, No. 10, 15 May 2002 Bourimet 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: 170.140.105.10 On: Mon, 24 Nov 2014 16:04:49
5.0039923.pdf	J. Chem. Phys. 154, 054313 (2021); https://doi.org/10.1063/5.0039923 154, 054313 © 2021 Author(s).Vibronic coupling in the first six electronic states of pentafluorobenzene radical cation: Radiative emission and nonradiative decay Cite as: J. Chem. Phys. 154, 054313 (2021); https://doi.org/10.1063/5.0039923 Submitted: 09 December 2020 . Accepted: 13 January 2021 . Published Online: 05 February 2021 Arun Kumar Kanakati , and S. Mahapatra COLLECTIONS Paper published as part of the special topic on Quantum Dynamics with ab Initio Potentials ARTICLES YOU MAY BE INTERESTED IN Relaxation dynamics through a conical intersection: Quantum and quantum–classical studies The Journal of Chemical Physics 154, 034104 (2021); https://doi.org/10.1063/5.0036726 Electronic structure software The Journal of Chemical Physics 153, 070401 (2020); https://doi.org/10.1063/5.0023185 Vibronic coupling in the ground and excited states of the pyridine radical cation The Journal of Chemical Physics 153, 164307 (2020); https://doi.org/10.1063/5.0024446The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Vibronic coupling in the first six electronic states of pentafluorobenzene radical cation: Radiative emission and nonradiative decay Cite as: J. Chem. Phys. 154, 054313 (2021); doi: 10.1063/5.0039923 Submitted: 9 December 2020 •Accepted: 13 January 2021 • Published Online: 5 February 2021 Arun Kumar Kanakati and S. Mahapatraa) AFFILIATIONS School of Chemistry, University of Hyderabad, Hyderabad 500 046, India Note: This paper is part of the JCP Special Topic on Quantum Dynamics with Ab Initio Potentials. a)Author to whom correspondence should be addressed: susanta.mahapatra@uohyd.ac.in ABSTRACT Nuclear dynamics in the first six vibronically coupled electronic states of pentafluorobenzene radical cation is studied with the aid of the stan- dard vibronic coupling theory and quantum dynamical methods. A model 6 ×6 vibronic Hamiltonian is constructed in a diabatic electronic basis using symmetry selection rules and a Taylor expansion of the elements of the electronic Hamiltonian in terms of the normal coordinate of vibrational modes. Extensive ab initio quantum chemistry calculations are carried out for the adiabatic electronic energies to establish the diabatic potential energy surfaces and their coupling surfaces. Both time-independent and time-dependent quantum mechanical methods are employed to perform nuclear dynamics calculations. The vibronic spectrum of the electronic states is calculated, assigned, and compared with the available experimental results. Internal conversion dynamics of electronic states is examined to assess the impact of various couplings on the nuclear dynamics. The impact of increasing fluorination of the parent benzene radical cation on its radiative emission is examined and discussed. Published under license by AIP Publishing. https://doi.org/10.1063/5.0039923 .,s I. INTRODUCTION Vibronic coupling, i.e., coupling between electronic and nuclear motion, is ubiquitous in polyatomic molecules. Such cou- pling causes a breakdown of the Born–Oppenheimer (BO) approx- imation,1and electronic transition takes place during nuclear motion. A generic feature of vibronic coupling is the occurrence of conical intersections (CIs) of potential energy surfaces (PESs),2–5 signature of which is often imprinted in the molecular electronic spectrum and absence of radiative emission of excited molec- ular states. The standard vibronic coupling theory2,6,7has been overwhelmingly successful in treating molecular processes in the non-BO situation. It relies on the concept of a diabatic electronic basis, symmetry selection rule, and a Taylor series expansion of the electronic Hamiltonian.2 Electron spectroscopy of benzene (Bz) and benzene radical cation (Bz+) and their fluoro derivatives have been extensively stud- ied experimentally8–29and also to a large extent theoretically30–39in the past decades. Fluorination of the benzene ring stabilizes the states arising out of σ-type orbitals of Bz in its fluoro derivatives. The extent of stabilization increases with increasing fluorination. This is a consequence of the electronic effect of fluorine atom and termed as the perfluoro effect.40Stabilization of electronic states in the fluoro Bz and Bz+causes their energetic re-ordering. As a result, vibronic coupling becomes an important mechanism and largely governs the mechanistic details of the spectroscopy and dynamics of electronically excited states of these molecules. The observation of radiative emission and structureless elec- tronic bands in fluoro Bz+, in particular, motivated detailed theoret- ical studies of the structure and dynamics of their electronic excited states. It was found that less than threefold fluorination of Bz+does not give rise to fluorescence emission.26,27Köppel and co-workers carried out benchmark theoretical studies for the first time on Bz+ and its mono- and di-fluoro derivatives.30–34They devised multi- state and multi-mode vibronic coupling models through extensive electronic structure calculations and carried out detailed nuclear J. Chem. Phys. 154, 054313 (2021); doi: 10.1063/5.0039923 154, 054313-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp dynamics studies. It turned out from these studies that energeti- cally accessible CIs among states drive the nonradiative decay of excited states in Bz+, F-Bz+, and F 2-Bz+. Among the three isomers [ortho (o),meta (m), and para (p)] of the latter, the m-isomer is weakly emissive.34It was found that the energetic minimum of its ̃Cstate occurs at higher energy as compared to the two other iso- mers, and therefore, the nonradiative decay of this state becomes partially feasible. In a later study on 1,3,5-trifluoro Bz+, we found that its excited electronic state of ̃A2A′′ 2symmetry is energetically well separated from the other states, and therefore, the observed radiative emission of this radical cation is explained to be arising out of this state.38The radiative emission and nonradiative decay were also studied recently both experimentally and theoretically for phenol and pentafluoro phenol.41,42In this case, the coupling between optically bright1ππ∗and optically dark1πσ∗states gov- erns the radiative emission. In phenol, these states are energetically apart, whereas in pentafluoro phenol, they are energetically close. Therefore, while radiative emission of the1ππ∗state dominates in phenol, it is significantly quenched in pentafluoro phenol owing to a large nonradiative population transfer to the optically dark1πσ∗ state.42 The photoelectron spectrum of pentafluorobenzene (PFBz) has been recorded by Bieri et al.8using He II radiation as the ioniza- tion source. Four distinct bands observed in the ∼9 eV–16 eV energy range were attributed to result from an ionization from the six valence molecular orbitals (MOs) of neutral PFBz. Among the four electronic bands of the pentafluorobenzene radical cation (PFBz+), the first, third, and fourth bands revealed the overlapping vibronic structure and therefore carried the signature of vibronic coupling in the energetically low-lying electronic states of PFBz+. Radiative emission and the highly overlapping electronic band structure motivated us to investigate vibronic coupling and nuclear dynamics in the energetically low lying electronic states of PFBz+. In the following, vibronic interactions in the ener- getically lowest six electronic states of PFBz+have been inves- tigated. These states result from ionization from the occupied valence MOs of PFBz. The MO configuration of the latter is (core)(13b2)2(19a1)2(14b2)2(4b1)2(5b1)2(3a2)2. Ionization from the highest occupied MO and the inner ones gives rise to ̃X2A2,̃A2B1, ̃B2B1,̃C2B2,̃D2A1, and̃E2B2electronic states of PFBz+in the order of increasing energy. Hereafter, these states will be identified as ̃X,̃A, ̃B,̃C,̃D, and̃Ein the rest of this paper. A vibronic coupling model is developed here to investigate the nuclear dynamics in the mentioned six electronic states. The electronic PESs are calculated ab initio by both complete active space self-consistent field (CASSCF)43,44-multi reference configu- ration interaction (MRCI)45,46and equation of motion ionization potential coupled cluster singles and doubles (EOMIP-CCSD)47,48 methods. The coupling strength of all vibrational modes on six elec- tronic states is calculated, and the relevant vibrational modes are included in the study based on the coupling strength. A first princi- ples nuclear dynamics study is carried out by both time-independent and time-dependent quantum mechanical methods. The vibronic coupling model is developed here with the aid of the standard vibronic coupling theory.2The latter relies on the concept of the diabatic electronic state, Taylor expansion of the elements of the diabatic electronic matrix in terms of the normal coordinate of vibrational mode, and elementary symmetry rules. Thedynamics study is carried out by a matrix diagonalization method in the time-independent framework.2This enables to determine the precise location of the vibronic energy levels and aids in their assignments. The time-dependent calculations are carried out by propagating wave packets (WPs) with the aid of the multi- configuration time-dependent Hartree (MCTDH) method devel- oped at Heidelberg.49–52This exercise enables us to calculate the broad band electronic spectra and to study the mechanistic details of radiative and nonradiative decay of excited electronic states. The results of this study are shown to be in good accord with the available experimental results. The rest of this paper is organized in the following way. In Sec. II, the quantum chemistry calculations are discussed. The vibronic model is established in Sec. III. The salient features and topography of adiabatic and diabatic electronic states are presented in Sec. IV. Theoretical methodologies to treat nuclear dynamics and to calculate dynamical observables are presented in Secs. V and VI, respectively. Finally, the summarizing remarks are presented in Sec. VII. II. QUANTUM CHEMISTRY CALCULATIONS The optimized equilibrium geometry of the electronic ground state of the PFBz molecule is calculated at the second-order Møller–Plesset perturbation theory (MP2) level employing both the augmented correlation-consistent polarized valence double zeta (aug-cc-pVDZ) basis set of Dunning53and def2-TZVPPD54–56basis set. Gaussian-0957suite of program is used for the calculations. The electronic ground term of PFBz is1A1, and the equilibrium geome- try possesses C 2vpoint group symmetry. This is the reference state in this study, and the vibrational motions in this state are treated as harmonic. The frequency ( ωi) of the thirty vibrational modes at the optimized equilibrium geometry is calculated by diagonalizing the kinematic ( G) and ab initio force constant ( F) matrix at the same level of theory. The mass-weighted normal displacement co- ordinates are derived from the eigenvectors of the GFmatrix and are transformed to the dimensionless form ( Q) by multiplying with√ωi(in a.u.).58 The optimized equilibrium geometry of PFBz is shown in Fig. 1 with atom numbering, and the equilibrium geometry parameters are given in Table I. The harmonic frequency of the vibrational modes and their symmetry are given in Table II along with the literature data59,60for comparison. It can be seen from Table II that the present data compare well with the experimental as well as the theoretical results available in the literature. In order to study the nuclear dynamics, the PESs of the six elec- tronic states of PFBz+mentioned in the introduction are calculated along the dimensionless normal displacement coordinates of the ref- erence electronic ground state of PFBz. The adiabatic potential ener- gies are calculated both by the CASSCF-MRCI and EOMIP-CCSD methods employing the aug-cc-pVDZ basis set. The CASSCF-MRCI and EOMIP-CCSD calculations are carried out using MOLPRO61 and CFOUR62suite of programs, respectively. The vertical ioniza- tion energies (VIEs) are calculated along the dimensionless normal displacement coordinates of each vibrational mode. The CASSCF- MRCI calculations are carried out with a (12,10) active space, which includes six valence occupied orbitals and four virtual orbitals with J. Chem. Phys. 154, 054313 (2021); doi: 10.1063/5.0039923 154, 054313-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1 . Schematic representation of the equilibrium minimum structure of the electronic ground state of PFBz. twelve electrons for PFBz. The electronic states of PFBz+have an open shell configuration, and a (11,10) active space is used. We note that many test calculations are carried out with varying active space, and the chosen ones yield the best result with an affordable computational cost. TABLE I . Optimized equilibrium geometry of the electronic ground state of PFBz. The bond length (R) and bond angle ( ∠) are given in Å and degrees, respectively. Parameters aug-cc-pVDZ def2-TZVPPD R(C1–C2,C1–C6) 1.40 1.38 R(C2–C3,C5–C6) 1.40 1.39 R(C3–C4,C4–C5) 1.40 1.39 R(C1–H7) 1.09 1.08 R(C2–F10,C6–F12) 1.35 1.33 R(C3–F9,C5–F11) 1.35 1.33 R(C4–F8) 1.35 1.33 ∠(C1–C2–C3,C1–C6–C5) 121.48 121.28 ∠(C2–C3–C4,C4–C5–C6) 119.15 119.20 ∠(C3–C4–C5) 120.39 120.37 ∠(C2–C1–C6) 118.34 118.68 ∠(H7–C1–C2,H7–C1–C6) 120.83 120.66 ∠(F10–C2–C1,F12–C6–C1) 119.97 120.11 ∠(F10–C2–C3,F12–C6–C5) 118.55 118.61 ∠(F9–C3–C2,F11–C5–C6) 120.94 120.94 ∠(F9–C3–C4,F11–C5–C4) 119.90 119.86 ∠(F8–C4–C3,F8–C4–C5) 119.81 119.81The VIEs calculated at the equilibrium geometry of the refer- ence state are given in Table III along with the literature data. In addition to the CASSCF-MRCI and EOMIP-CCSD results, the VIEs calculated by the outer valence Green’s function (OVGF) method are also given in Table III. It can be seen from Table III that both the OVGF and EOMIP-CCSD results are closer to the experimental data as compared to the CASSCF-MRCI results. The EOMIP-CCSD results appear to be closest to the experimental data. A close look at the data given in Table III reveals that the ̃Xand̃Astates are ener- getically close at the vertical configuration. A similar observation can also be made for the ̃B–̃C–̃D–̃Eelectronic states. Therefore, vibronic coupling appears to be an important mechanism to govern nuclear dynamics in these states. III. THE VIBRONIC MODEL In this section, a vibronic coupling model of the six ener- getically lowest electronic states ̃X,̃A,̃B,̃C,̃D, and̃Eof PFBz+ is developed. The model is based on the framework of standard vibronic coupling theory, symmetry selection rules, a diabatic elec- tronic basis, and dimensionless normal displacement coordinates of the vibrational modes.2The thirty vibrational modes of the elec- tronic ground state of PFBz transform to the following irreducible representations (IREPs) of the C 2vsymmetry point group: Γvib=11a1⊕6b1⊕10b2⊕3a2. (1) Using symmetry selection rules and standard vibronic coupling the- ory, the Hamiltonian can be written in a diabatic electronic basis as2 H=H016+ΔH, (2) with H0=−1 230 ∑ i=1ωi(∂2 ∂Q2 i)+1 230 ∑ i=1ωiQ2 i (3) and ΔH=⎛ ⎜⎜⎜⎜⎜⎜⎜ ⎝WXXWXAWXBWXCWXDWXE WAAWABWACWADWAE WBBWBCWBDWBE WCCWCDWCE h.c. WDDWDE WEE⎞ ⎟⎟⎟⎟⎟⎟⎟ ⎠. (4) In Eq. (2), the quantity 1represents a (6 ×6) unit matrix. The Hamil- tonian of the harmonic reference electronic ground state of PFBz is denoted by H0and is defined in Eq. (3). The quantity ΔHdefines the change in electronic energy upon ionization to PFBz+. The elements of the matrix Hamiltonian ΔHare expanded in a Taylor series around the equilibrium geometry of the reference state atQ= 0 as J. Chem. Phys. 154, 054313 (2021); doi: 10.1063/5.0039923 154, 054313-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE II . Symmetry designation and harmonic frequency (in cm−1) of vibrational modes of the electronic ground state of PFBz calculated at the MP2 level of theory. This work Expt. Sym. Mode aug-cc-pVDZ def2-TZVPPD Reference 59 Reference 60 Description of the modes a1ν1 3257 3262 3103 3105 C–H stretching in plane ν2 1682 1683 1648 1648 C–C–C bending ν3 1533 1552 1516 1514 C–C and C–F stretching ν4 1422 1444 1413 1410 C–C stretching ν5 1268 1302 1291 1286 C–C stretching (Kekule) ν6 1063 1092 1078 1082 C–F stretching ν7 716 729 719 718 C–C–C trigonal bending ν8 574 584 577 580 Ring breathing ν9 467 474 474 469 C–C–C in plane bending ν10 324 329 327 325 C–F in plane bending ν11 267 269 272 272 C–F in plane bending a2ν12 632 671 661 . . . C–C–C out-of-plane ν13 384 398 387 391 C–F out-of-plane bending ν14 132 132 142 171 C–F out-of-plane b1ν15 841 840 837 838 C–H out-of-plane bending ν16 591 636 715 689 C–H and C–F out of plane trigonal ν17 543 560 556 556 C–H and C–C–C out of plane ν18 317 323 321 . . . C–F out-of-plane bending ν19 204 208 206 . . . C–F out-of-plane bending, in phase ν20 158 155 158 . . . C–F out-of-plane bending b2ν21 1679 1685 1648 1648 C–C stretching ν22 1552 1570 1540 1535 C–C stretching ν23 1478 1455 1269 1268 C–C stretching ν24 1185 1207 1182 1182 C–H bending, in plane ν25 1129 1163 1143 1138 C–F stretching, in plane ν26 947 969 958 953 C–F stretching and C–H bending, in plane ν27 684 694 692 662 C–F in plane bending ν28 429 436 433 436 C–C–C in plane bending ν29 300 303 303 300 C–F in plane bending ν30 272 274 256 . . . C–F in plane bending TABLE III . Vertical ionization energy (in eV) of the energetically lowest six electronic states of PFBz+calculated at the equilibrium geometry of the electronic ground state of PFBz (reference). State OVGF CASSCF-MRCI EOMIP-CCSD RI-SCS-CC2aExpt. ̃X2A2 9.63 10.42 9.91 9.86 9.9b ̃A2B1 9.94 10.69 10.27 10.49 10.1c/10.06a ̃B2B1 12.89 13.54 13.07 12.63 12.7c/12.74d ̃C2B2 14.26 15.72 13.98 . . . 13.9c ̃D2A1 14.53 16.08 14.39 . . . . . . ̃E2B2 15.20 17.00 14.93 . . . 14.9c aReference 71. bReference 23. cReference 8. dReference 73. J. Chem. Phys. 154, 054313 (2021); doi: 10.1063/5.0039923 154, 054313-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Wαα=E0 α+∑ i∈a1κ(α) iQi+1 2!∑ i,j∈a1,a2,b1,b2γ(α) ijQiQj +1 3!∑ i∈a1η(α) iQ3 i+1 4!∑ i∈a1,a2,b1,b2ζ(α) iQ4 i (5) and Wαα′=W∗ α′α=∑ iλαα′ iQi. (6) In the above equations, αandα′are the electronic state indices and i andjare the indices representing vibrational modes. The VIE of the αth electronic state is denoted by E0 α. The quantities κα i,γα ij,ηα i, andζα i represent the linear, quadratic, cubic, and quartic coupling param- eters, respectively, within the αth electronic state. The quantity λαα′ i denotes the linear inter-state coupling parameter between the states αandα′, coupled through ith vibrational mode. The numerical val- ues of the above parameters are derived by fitting the adiabatic elec- tronic energies calculated ab initio to the diabatic electronic Hamil- tonian introduced above. The Hamiltonian parameters of all six electronic states calculated in that way are given in Tables S1–S4 of the supplementary material. We note that while a second-order Taylor expansion resulted a good fit (along the totally symmetric modes) of the electronic energies calculated by the EOMIP-CCSD method, the CASSCF-MRCI energies required a higher order fit. All the higher order fit parameters are given in Tables S5–S7 ofthe supplementary material. Along with this, we have estimated the diagonal bilinear coupling parameters along the five ( ν2,ν3,ν4,ν9, andν11) totally symmetric vibrational modes by a two-dimensional fit (using the Levenberg Marquardt algorithm as implemented in MATLAB63). The diagonal bilinear parameters are given in Table S8 of the supplementary material. We also estimated the third-order coupling parameters along the coupling modes. The magnitude of these parameters is of the order of 10−3eV or less. Therefore, a linear expansion of the coupling elements is retained in Eq. (6). IV. POTENTIAL ENERGY SURFACES AND CONICAL INTERSECTIONS The topography of the adiabatic potential energy surfaces of thẽX,̃A,̃B,̃C,̃D, and̃Eelectronic states of PFBz+is discussed in this section. One dimensional cuts of the multidimensional potential energy hypersurface of the electronic states are presented. These are plotted along the normal displacement coordinates of some selected totally symmetric vibrational modes ( ν2–ν4,ν9, andν11) in Fig. 2. The points in the figure represent the adiabatic electronic ener- gies calculated by the CASSCF-MRCI [Fig. 2(a)] and EOMIP-CCSD [Fig. 2(b)] methods. The superimposed solid curves represent the analytic fit of the corresponding points. The parameters derived from the fits are reported in Tables S1–S4 of the supplementary material, respectively. FIG. 2 . One dimensional cuts of the adiabatic potential energy surface of thẽX,̃A,̃B,̃C,̃D, and ̃Eelectronic states of PFBz+along the dimension- less normal displacement coordinate of the totally symmetric vibrational modes mentioned in the panel. The poten- tial energies obtained from the present theoretical model and calculated ab ini- tio[column (a): CASSCF-MRCI and col- umn (b): EOMIP-CCSD] are shown by the solid lines and points, respectively. J. Chem. Phys. 154, 054313 (2021); doi: 10.1063/5.0039923 154, 054313-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp It can be seen from Fig. 2 that the ̃Xand̃Astates are energeti- cally very close in the entire range of nuclear coordinates in both sets of data. The crossing of these states can be clearly seen along ν2and ν9vibrational modes. Such curve crossings acquire the topography of CIs in multi-dimensional space. The location of the ̃Bstate is ener- getically closer to the ̃C–̃D–̃Eelectronic states in the EOMIP-CCSD energy data (cf., Table III). In both (CASSCF-MRCI and EOMIP- CCSD) energy data, the entanglement of ̃C–̃D–̃Estates can be seen (cf., Fig. 2). Multiple crossings of these states lead to multiple multi- dimensional CIs. The greater anharmonicity of the CASSCF-MRCI energies is also revealed by the data plotted in Fig. 2. Various station- ary points, viz., the energy of the minimum of the seam of CIs and the minimum of the states, are calculated with the EOMIP-CCSD potential energy curves using a minimization algorithm employing Lagrange multipliers. The numerical tools available in MATHE- MATICA64are used for this purpose. The results are tabulated in a matrix array in Table IV. In the latter, the energies in the diagonal represent the minimum of a state, and those in the off-diagonal are the minimum of the intersection seam. The following remarks can be readily made by examining the data given in Table IV. The energetic minimum of the ̃Bstate occurs well above its minimum of intersections with the other states. The energetically closest one is the ̃B–̃Cintersection minimum occurring ∼2.25 eV above the minimum of the ̃Bstate. The minimum of the ̃C state is, however, closer, ∼1.33 eV lower than the ̃B–̃Cintersection minimum. The ̃C–̃D–̃Eelectronic states of PFBz+are energetically close. The ̃C–̃Dintersection minimum is closer to their respective equilibrium minimum. This is also true for the ̃D–̃Eintersection minimum. The latter is almost quasi-degenerate with the minimum of thẽEstate. It emerges from the above results and also from the potential energy curves of Fig. 2 that ̃X–̃Astates of PFBz+form an isolated pair and are energetically well separated from the rest of their neigh- bors. The excitation strength of the vibrational modes is also simi- lar in both these states (cf., Tables S1 and S3 of the supplementary material), except that the vibrational mode ν8has somewhat larger coupling strength in the ˜Astate. ThẽX–̃Acoupling is fairly strong along the ν28mode of b2sym- metry, and the coupling is moderate along the vibrational modes ν21 andν30ofb2symmetry (cf., Tables S9 and S10 of the supplementary material). Although the ̃C–̃D–̃Estates are energetically close and TABLE IV . Energy (in eV) of the equilibrium minimum of the state (diagonal entries) and the minimum of its intersection seam with its neighbors (off-diagonal entries) of PFBz+calculated within a second-order coupling model and the EOMIP-CCSD electronic energy data. ̃X2A2̃A2B1̃B2B1̃C2B2̃D2A1̃E2B2 ̃X2A2 9.74 10.13 32.90 27.42 . . . 21.29 ̃A2B1 . . . 10.12 . . . 26.71 22.60 23.25 ̃B2B1 . . . . . . 12.97 15.22 15.62 17.00 ̃C2B2 . . . . . . . . . 13.89 14.37 . . . ̃D2A1 . . . . . . . . . . . . 14.26 14.89 ̃E2B2 . . . . . . . . . . . . . . . 14.84their respective equilibrium minimum is closer to various intersec- tion minima (cf., Table IV), the coupling of ̃C–̃Dand̃D–̃Estates is not very strong. As can be seen from Tables S9 and S10 of the supplementary material, ̃C–̃Dstates are moderately coupled through vibrational modes ν22andν29and weakly coupled through ν26of b2symmetry. Likewise, ̃D–̃Estates are moderately coupled through ν21andν28and weakly coupled through ν24,ν27, andν30vibrational modes of b2symmetry. The impact of these couplings on the nuclear dynamics is examined below. V. NUCLEAR DYNAMICS The nuclear dynamics study is carried out from first principles by both time-independent and time-dependent quantum mechan- ical methods. In the time-independent method, the vibronic spec- trum is calculated by the golden rule equation of the spectral intensity,2,65 P(E)=∑ n∣⟨Ψf n∣ˆT∣Ψi 0⟩∣2δ(E−Ef n+Ei 0). (7) In the above equation, ∣Ψi 0⟩and∣Ψf n⟩are the initial and final vibronic states with energy Ei 0and Ef n, respectively. The quantity ˆTis the transition dipole operator. The reference electronic ground state of PFBz,∣Ψi 0⟩, is assumed to be vibronically decoupled from the excited electronic states and is given by ∣Ψi 0⟩=∣Φi 0⟩∣χi 0⟩, (8) where∣Φi 0⟩is the diabatic electronic part and ∣χi 0⟩is the nuclear part of this state. The nuclear component of the wave function in Eq. (8) is given by the product of eigenfunction of the reference harmonic Hamiltonian H0, as a function of the normal coordinates of the vibrational modes. The final vibronic state of PFBz+can be expressed as ∣Ψf n⟩=∑ m∣Φm⟩∣χm n⟩. (9) In the above equation, mandnare the electronic and vibrational index, respectively. With the above definitions, the spectral intensity of Eq. (7) assumes the form2 P(E)=∑ n,m∣τm⟨χm n∣χ0⟩∣2δ(E−Ef n+Ei 0), (10) where τm=⟨Φm∣ˆT∣Φ0⟩ (11) represents the transition dipole matrix elements in the diabatic elec- tronic basis. These are treated as constants assuming the validity of the Condon approximation in this basis.66 J. Chem. Phys. 154, 054313 (2021); doi: 10.1063/5.0039923 154, 054313-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp The time-independent Schrödinger equation of the vibronically coupled states is solved by representing the Hamiltonian [cf., Eq. (2)] in the direct product harmonic oscillator (HO) basis of the reference state. The vibrational wave function, ∣χm n⟩, in this basis is given by ∣χm n⟩=∑ n1,n2,...,nkam n1,n2...,nk∣n1⟩∣n2⟩. . .∣nk⟩. (12) In the above equation, nlis the quantum number associated with thelth vibrational mode, and kis the total number of such modes. The summation runs over all possible combinations of quantum numbers. The Hamiltonian matrix represented in the above basis is diagonalized by the Lanczos algorithm.67The eigenvalues of this matrix yield the location of the vibronic energy levels, and the inten- sity is calculated by squaring the first component of the eigenvector matrix.67 In a time-dependent picture, the spectral intensity defined in Eq. (7) translates to a Fourier transform of the time autocorrelation function of the WP evolving on the final electronic state,2,65 P(E)≈6 ∑ m=12Re∫0∞ eiEt/̵h⟨χ0∣τ†e−iHt/̵hτ∣χ0⟩dt, (13) ≈6 ∑ m=12Re∫0∞ eiEt/̵hCm(t)dt, (14) where Cm=⟨Ψm(0)\|Ψm(t)⟩represents the time autocorrelation function of the WP initially prepared on the mth electronic state. While the matrix diagonalization method (discussed above) yields the precise location of the vibronic eigenstates, the applicability of the method is limited to a smaller number of electronic and nuclear degrees of freedom (DOF) because of a huge increase of compu- tational overheads. For large systems, the MCTDH method49–52 emerged as a state-of-the-art tool to obtain a numerically exact solu- tion of the time-dependent Schrödinger equation. In this method, the time-dependent Schrödinger equation is numerically solved by propagating WPs with a discrete variable representation (DVR) based scheme and a variational ansatz. The dimensionality of the system is effectively reduced by its multi-set formalism. For the details of this method and algorithm, the readers are referred to the original research papers.49–52 VI. RESULTS AND DISCUSSION The vibronic band structure of the ̃X–̃A–̃B–̃C–̃D–̃Ecoupled electronic manifold of PFBz+is calculated and compared with the experimental photoionization spectroscopy results of Ref. 8. In order to develop a systematic understanding of the details, we in the fol- lowing examine the vibronic energy level structure of the uncoupled electronic states first and include the coupling between states sub- sequently to reveal its impact on the energy level structure. A time- independent matrix diagonalization method is used to calculate the precise location of the energy levels of the uncoupled electronic states and coupled two electronic states. Because of the dimension- ality problem (as mentioned above), this method could not be usedin the complete coupled state situation. The final spectral envelope for the entire coupled state situation is therefore calculated by a time-dependent WP propagation method employing the Heidelberg MCTDH49program modules. A. Vibrational energy level spectrum of the uncoupled ̃X,̃A,̃B,̃C,̃D, and ̃Eelectronic states of PFBz+ The vibrational energy level spectrum of the uncoupled ̃X,̃A,̃B, ̃C,̃D, and̃Eelectronic states of PFBz+is calculated by a matrix diag- onalization approach2using the Lanczos algorithm. The theoretical calculations are carried out with ten totally symmetric vibrational modes (ν2–ν11) and the vibronic Hamiltonian of Sec. III and the parameters of Tables S1–S4 of the supplementary material. Both set of parameters derived from the CASSCF-MRCI and EOMIP- CCSD electronic energies are used for these calculations, and the corresponding results are shown in panels (a) and (b) of Fig. 3, respectively. The HO basis functions used along each mode in these calculations are given in Table S11 of the supplementary material. The Hamiltonian of each state represented in the HO basis is diagonalized using 10 000 Lanczos iterations. The theoretical stick spectrum obtained from the diagonalization of the Hamiltonian matrix is convoluted with the Lorentzian line shape function of 40 meV full width at the half maximum (FWHM) to generate the spectral envelopes shown in Fig. 3. The excitation of the fundamental of vibrational modes ν8,ν9, andν11is found in the ˜Xstate of PFBz+calculated with both the CASSCF-MRCI and EOMIP-CCSD Hamiltonian parameters. The FIG. 3 . The stick vibrational spectrum and the convoluted envelope of the uncou- pled̃X,̃A,̃B,̃C,̃D, and̃Eelectronic states of PFBz+, calculated with totally sym- metric vibrational modes using the CASSCF-MRCI [panel (a)] and EOMIP-CCSD [panel (b)] Hamiltonian parameters. J. Chem. Phys. 154, 054313 (2021); doi: 10.1063/5.0039923 154, 054313-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp peaks are ∼578 cm−1,∼465 cm−1, and ∼303 cm−1(CASSCF-MRCI) and∼577 cm−1,∼488 cm−1,∼298 cm−1(EOMIP-CCSD) spaced in energy and correspond to the frequency of the vibrational modes ν8, ν9andν11, respectively. Peak spacings of ∼572 cm−1,∼458 cm−1, ∼285 cm−1(CASSCF-MRCI) and ∼572 cm−1,∼460 cm−1, and ∼285 cm−1(EOMIP-CCSD) corresponding to the excitation of ν8, ν9, andν11vibrational modes, respectively, are found in the ̃Astate. The extended progression of all the modes excited in both the ̃X and̃Astates is assigned and given in Tables S12 and S13 of the supplementary material, respectively. In addition to the energetic location and excitation strength analysis, the assignment of the peaks is also confirmed by examin- ing the nodal pattern of the vibrational wave functions. These wave functions are calculated by a block improved-relaxation method as implemented in the MCTDH program module.68–70In Figs. S1–S4 of the supplementary material, we present a few vibrational eigen- functions of the ̃Xand̃Astates. In these figures, the wave function probability density is plotted in a suitable reduced dimensional space of normal coordinates. In panels (a)–(c), the wave function of the fundamental of ν8,ν9, andν11is shown, respectively. It can be seen from these plots that the wave function develops a node along the respective normal coordinate. The wave function for the overtone peaks of the excited vibrational modes is shown in panels (d)–(f). Two, three, and four quantum excitations along the first, second, and third overtones, respectively, can be seen from the plots. Some combination peaks are shown in panels (g)–(l) of Figs. S1–S4 of the supplementary material.B. Coupled two-states results In order to assess the impact of nonadiabatic coupling on the vibronic structure of an individual state, we performed several cou- pled two states calculations. The overall structure of the spectrum of thẽXstate does not change upon inclusion of its coupling with the other states (i.e., ̃A,̃B,̃C, and̃E) even though the coupling strength is moderate (cf., Table S10 of the supplementary material). This is because except ̃A, the other states are energetically (vertically) well separated from the ̃Xstate (cf., Table III), and the energetic min- imum of the seam of its intersection with them lies well above its equilibrium minimum (cf., Table IV). The WP initially prepared on the ̃Xstate does reaches the ̃X–̃A crossing seam, and some population flows to the ̃Astate [cf., panel (a) of Fig. 4]. In this case, the energetic minimum of the intersec- tion seam occurs ∼0.39 eV and ∼0.01 eV above the minimum of the ̃Xand̃Aelectronic states, respectively (cf., Table IV). As a result, the impact of the coupling is significant on the ̃Astate. The vibronic structure of the ̃Astate and its electronic population dynamics bears the signature of this coupling effect. The WP initially prepared on thẽAstate accesses the ̃X–̃Aintersection seam, and more than ∼80% electronic population flows to the ̃Xstate within ∼22 fs [cf., panel (b) of Fig. 4]. Such a huge population exchange causes a large increase in the spectral line density and broadening of the vibronic spectrum of the ̃Astate. The ̃Xand̃Aelectronic bands resulting from these coupled ̃X–̃Astates calculations are shown in panels (a) and (b) of Fig. 5, respectively. FIG. 4 . Time-dependence of the diabatic electronic populations in the coupled ̃X–̃A,̃B–̃C,̃C–̃D, and̃D–̃Estate dynam- ics obtained by locating an initial WP on each electronic state separately is shown in the panels (a)–(h), respectively. EOMIP-CCSD Hamiltonian parameters are used for these calculations (see text for details). J. Chem. Phys. 154, 054313 (2021); doi: 10.1063/5.0039923 154, 054313-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 5 . Composite vibronic band struc- ture of the coupled ̃X–̃A,̃B–̃C,̃C–̃D, and̃D–̃Estates of PFBz+are shown in the panels (a)–(h), respectively. The band structures are calculated using the Hamiltonian parameters derived from the EOMIP-CCSD energy data. In addition to this, we examined the ̃X–̃Acoupled state results obtained by the matrix diagonalization method. Based on the excita- tion strength (cf., Tables S1, S3, S9, and S10 of the supplementary material), five totally symmetric vibrational modes ( ν2,ν3,ν4,ν9, andν11) and five coupling vibrational modes ( ν21,ν24,ν28,ν29, and ν30) ofb2symmetry are included in the calculation. The compos- ite vibronic spectra of the ̃X–̃Acoupled states of PFBz+are shown in panels (b) and (c) of Fig. 6 and compared with the experimen- tal band plotted in panel (a). The results of the panels (b) and (c) are obtained with CASSCF-MRCI and EOMIP-CCSD Hamilto- nian parameters, respectively. The theoretical spectral envelope is obtained by convoluting the vibronic stick lines with a Lorentzian line shape function of 40 meV FWHM. The theoretical spectrum of thẽXstate given in panels (b) and (c) is shifted by ∼0.9 eV and ∼0.6 eV, respectively, along the abscissa to reproduce the experimen- tal8adiabatic ionization energy. Because of reduced dimensional calculations, such shifts were necessary to account for the zero- point energy. We also calculated the adiabatic ionization energy of the ̃Xstate by the CCSD method. We obtained a value of ∼9.56 eV as compared to its experimental value of ∼9.64 eV.8It can be seen from Fig. 6 that the theoretical results are in very good accord with the experimental band structure of the ̃X–̃Acoupled states. The vibronic energy levels of the ̃X–̃Acoupled states are assigned by examining the WP density plots in an analogous way, as described in Sec. VI A. The most probable assignments ofvibronic energy lines are presented in Tables S14 and S15 of the supplementary material, and the WP density plots of some of these assignments are shown in Figs. S5–S8 of the supplementary material. The comparison with the data presented in Tables S12 and S13 of the supplementary material reveals a slight change of the energetic loca- tion of the fundamentals of the totally symmetric vibrational modes. In contrast to the uncoupled state spectrum, the combination peaks of the totally symmetric vibrational modes are not found in the cou- pled states spectrum of the ̃Xstate. However, they are found in the ̃Astate both in the uncoupled state and coupled state situations. In the ̃X–̃Acoupled states, spectrum excitation of the vibrational modes of b2symmetry is found. These vibrational modes also form combination peaks between them and also with the totally symmet- ric modes. A small number of combination peaks are found with the EOMIP-CCSD parameters as compared to the CASSCF-MRCI parameters. It can be seen from the results presented above that the funda- mental of the totally symmetric ν11andν9vibrational modes appears at∼300 cm−1and∼460 cm−1, respectively (cf., Tables S14 and S15 of the supplementary material). The latter is reported at ∼474 cm−1 in the experiment71and is reasonably in good agreement with the present result. Inclusion of ̃X–̃Acoupling increases the vibronic line density and causes a broadening of the spectral envelope. Because of large energy separation, the vibronic spectrum of the ̃B,̃C,̃D, and ̃Estates is not affected by their coupling with the ̃Xstate, as signifi- cantly as the ̃Astate. According to the symmetry rules, ̃Aand̃Bstates J. Chem. Phys. 154, 054313 (2021); doi: 10.1063/5.0039923 154, 054313-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 6 . Stick vibronic spectrum and convoluted envelope of the ̃X-̃Acoupled electronic states of PFBz+. Panels (b) and (c) are obtained with the Hamiltonian parameters derived from the CASSCF-MRCI and EOMIP-CCSD, respectively. The experimental ̃X–̃Aband is reproduced from Ref. 8 and shown in panel (a). are not coupled with each other. The coupling of ̃Band̃Cstates does not have impact on their respective vibronic structure, as indicated by very little population exchange between them [cf., panels (c) and (d) of Fig. 4]. In contrast to the above, the coupling between ̃C–̃Dand ̃D–̃Estates have strong impact on their respective vibronic struc- ture. In the case of ̃C–̃Dcoupled states, the energetic minimum of the intersection seam occurs ∼0.48 eV and ∼0.11 eV above the estimated equilibrium minimum of the ̃Cand̃Dstates, respectively (cf., Table IV). The coupling between these states is also fairly strong. As a result, large population exchange occurs between them. In order to illustrate, the time-dependence of the diabatic electronic popu- lation for an initial transition to the ̃Cand̃Dstates in the ̃C–̃D coupled state situation is shown in panels (e) and (f) of Fig. 4, respec- tively. It can be seen from panel (e) of Fig. 4 that the population of thẽCstate monotonically decays to ∼0.37 and that of the ̃Dstate grows to ∼0.67 in about 200 fs. As can be seen from panel (f) of Fig. 4, a large fraction of population flows to both the electronic states in this case. This is because the equilibrium minimum of the ̃Dstate is energetically very close to the minimum of the ̃C–̃DCIs(cf., Table IV). A sharp decay of population occurs within ∼20 fs followed by quasi-periodic recurrences at longer times. In the case of coupled ̃D–̃Estates, the energetic minimum of the intersection seam occurs ∼0.63 eV and ∼0.05 eV above the estimated equilibrium minimum of ̃Dand̃Estates, respectively (cf., Table IV). This leads to a very small amount of population transfer to the ̃Estate when the WP is initially launched on the ̃Dstate [cf., panel (g) of Fig. 4]. Because of fairly strong coupling between the ̃Dand̃Estates and the fact that the minimum of their intersection seam is energet- ically very close to the minimum of the ̃Estate (cf., Table IV), the coupling has strong impact on the dynamics of the ̃Estate. The pop- ulation of the ̃Estate sharply decays to ∼0.47 and that of the ̃Dstate grows to ∼0.53 within a short time of ∼23 fs [cf., panel (h) of Fig. 4] when the WP is initially located on the ̃Estate. At longer times, ̃E state population decreases monotonically. The vibronic band struc- tures resulting from the above coupled-states calculations are shown in Fig. 5. C. Vibronic spectrum of coupled ̃X–̃A–̃B–̃C–̃D–̃E electronic states The vibronic spectrum of the coupled ̃X–̃A–̃B–̃C–̃D–̃Eis calcu- lated and presented in this section. Because of large vertical energy separation of the ˜Bstate from all other states, we have performed nuclear dynamics calculations with two separate group of states, viz., ̃X–̃A–̃Band̃B–̃C–̃D–̃E. Both the CASSCF-MRCI and EOMIP-CCSD Hamiltonian parameters are employed in the calculations. With the CASSCF-MRCI parameters, 16 vibrational modes were necessary for both group of states, and with the EOMIP-CCSD parameters, 16 and 24 vibrational modes, respectively, were necessary for the two group of states noted above. A different coupling mechanism revealed by the CASSCF-MRCI and EOMIP-CCSD parameters is reflected in the electronic population dynamics discussed below. The different number of vibrational DOFs required for ̃B–̃C–̃D–̃E coupled states dynamics is assessed from the interstate coupling parameters obtained from the two sets of electronic energy data. The coupling between ̃B–̃Dstates is absent in the case of CASSCF- MRCI (cf., Table S9 of the supplementary material) data. It can be seen from panel (d) of Figs. 7 and 8 that the electronic population dynamics calculated with two sets of data differs significantly when thẽCstate is initially populated. The dynamics calculations are carried out by propagating WPs on the coupled electronic states using the Heidelberg MCTDH suite of program modules.49Six WP calculations are performed by launching the initial WP on each of the six electronic states sepa- rately. The details of the mode combination and the sizes of the basis sets are given in Table S16 of supplementary material. In each calcu- lation, WP is propagated for 200 fs. The time autocorrelation func- tion is damped with an exponential function of relaxation time 33 fs and then Fourier transformed to obtain the spectrum. The results from six different calculations are combined with equal weightage to generate the composite theoretical band. The results obtained with the CASSCF-MRCI and EOMIP-CCSD Hamiltonian parameters are shown in Fig. 9 along with the experimental results reproduced from Ref. 8. It can be seen from the panels (a)–(d) of Fig. 9 that the theoretical results are in good accord with the experimental band structures. While the first band originates from highly overlapping ̃Xand̃Aelectronic states, the third and fourth bands are formed by J. Chem. Phys. 154, 054313 (2021); doi: 10.1063/5.0039923 154, 054313-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 7 . Time evolution of the diabatic electronic populations obtained in the coupled ˜X-˜A-˜B-˜C-˜D-˜Estate situation (using the parameter set derived from the CASSCF-MRCI energy data) by locating an initial WP on each of the ˜X, ˜A,˜B,˜C,˜D, and ˜Eelectronic states sep- arately is shown in the panels (a)–(f), respectively. highly overlapping ̃C,̃D, and̃Eelectronic states. To this end, we note that the vibronic band structures remain unchanged [cf., panel (d) of Fig. 9] upon inclusion of the bilinear coupling parameters given in Table S8 of the supplementary material.D. Internal conversion dynamics The time-dependent populations of the six diabatic electronic states of PFBz+in the coupled (i.e., ̃X–̃A–̃Band̃B–̃C–̃D–̃E) state FIG. 8 . Time evolution of the diabatic electronic populations obtained in the coupled ˜X-˜A-˜B-˜C-˜D-˜Estate situation (using the parameter set derived from the EOMIP-CCSD energy data) by locat- ing an initial WP on each of the ˜X,˜A, ˜B,˜C,˜D, and ˜Eelectronic states sep- arately is shown in the panels (a)–(f), respectively. J. Chem. Phys. 154, 054313 (2021); doi: 10.1063/5.0039923 154, 054313-11 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 9 . Composite vibronic band structure of the coupled ̃X-̃A-̃B-̃C-̃D-̃Eelectronic states of PFBz+. The band structures calculated using the Hamiltonian parameters derived from the CASSCF-MRCI and EOMIP-CCSD energy data are, respec- tively, shown in panels (b) and (c). The band structures obtained by including the bilinear coupling parameters of Table S8 of the supplementary material are plotted in panel (d). The experimental result reproduced from Ref. 8 is shown in panel (a). The intensity in arbitrary units is plotted as a function of the energy of the cationic vibronic states. The zero of the energy scale corresponds to the energy of the equilibrium minimum of the electronic ground state of neutral PFBz. situation are recorded and examined in this section. This is to unravel and understand the impact of various couplings on the dynamics of a given state. The results obtained by initially pop- ulating the ̃X,̃A,̃B,̃C,̃D, and̃Eelectronic states are shown in panels (a)–(f) of Figs. 7 and 8 calculated with CASSCF-MRCI and EOMIP-CCSD parameters, respectively. The electronic populations are color-coded (online version) in the same way in all panels of the respective figures. The electronic populations for an initial location of the WP on the ̃Xstate shown in panel (a) of both Figs. 7 and 8 reveal a very little amount of population transfer to the ̃Aand̃B states. The minimum of the ̃X–̃ACIs located ∼0.39 eV above the minimum of the ̃Xstate, and therefore, the population transfer to thẽAstate is not significant. Because of the large energy separation between the ̃Xand̃Bstates, the population transfer to the ̃Bstate is also negligible. On the other hand, a large amount of population flows to the ̃Xstate when the WP initially placed on the ̃Astate [cf., panel (b) of Figs. 7 and 8]. A decay rate of ∼15 fs can be estimated from the initial fast decay of the population of the ̃Astate. The ener- getic minimum of the ̃X–̃ACIs occurs ∼0.01 eV above the minimum of thẽAstate and therefore causes such a rapid decay of the ̃Astate population.The population for an initial excitation of the WP to the ̃Bstate is shown in panel (c) of Figs. 7 and 8. It can be seen that practically no population flows to all other states when the WP is initially prepared on the ̃Bstate. This is due to the fact that the ̃Bstate is vertically well separated from all other states, and the CIs of the ̃Bstate with all other states are located at high energies and are not accessible to the WP during its evolution on this state. This results into the observed sharp vibrational level structure of the ̃Bband [cf., panels (a) and (b) of Fig. 3]. This implies a long-lived nature of the ̃Bstate and gives rise to the observed emission of PFBz+. We will return to this point again later in the text. Time-dependence of electronic populations for an initial loca- tion of the WP on the ̃Cstate is shown in panel (d) of Figs. 7 and 8. In this case, the internal conversion takes place to both ̃Dand̃Bstates via the low-lying ̃C–̃Dand̃B–̃DCIs, respectively. At longer times, the WP from the ̃Dstate moves to the ̃Bstate via ̃B–̃DCIs, minimum of which occurs ∼1.36 eV above the minimum of the ̃Dstate. Although the overall picture remains similar, the extent of population transfer obtained with the EOMIP-CCSD parameters is far greater [cf., panel (d) of Fig. 8]. The WP initially prepared on the ̃Dstate quickly flows to the ̃C state [shown in panel (e) of Figs. 7 and 8] via the energetically low- lying̃C–̃DCIs. The minimum of the ̃Dstate is only ∼0.11 eV below the minimum of ̃C–̃Dintersections. The internal conversion to the ̃B state appears to occur through the ̃Cstate as these states are strongly coupled via the ν13mode of a2symmetry (cf., Table S9 and S10 of the supplementary material). A nonradiative decay rate of ∼16 fs can be estimated from the population curve of the ̃Dstate given in panel (e) of Figs. 7 and 8. The electron population dynamics becomes more complex and involved when the WP is initially prepared on the ̃Eelectronic state [shown in panel (f) of Figs. 7 and 8]. In this case, most of the population flows to the ̃Cand̃Delectronic states. This is because of strong nonadiabatic coupling among ̃C,̃D, and̃Eelec- tronic states. In addition, a large population transfer is facilitated by the energetic proximity of the equilibrium minimum and the mini- mum of various intersection seams in the ̃C–̃D–̃Estates. The initial fast decay of the population relates to a life-time of ∼64 fs of the ̃Estate. In summary, the results presented above show that the observed broad band photoionization spectrum of PFBz+is better repro- duced with the Hamiltonian parameters extracted from the EOMIP- CCSD electronic structure data, as compared to the same with the CASSCF-MRCI data. The overall dynamical mechanism is qual- itatively same in both the cases as discussed in relation to the population dynamics. The superiority of the EOMIP-CCSD data cannot be judged in the present work; it requires more resolved experimental data to be available in order to make a conclusive remark. E. Radiative emission The radiative emission of Bz+and its fluoro derivatives was studied both experimentally8–29and theoretically.30–34,38A clear radiative emission was observed for threefold fluorination or more of Bz+. It was found that Bz+and its monofluoro and diflu- oro (abbreviated as MFBz+and DFBz+, respectively) derivatives are non-emissive, except the m-DFBz+(the meta isomer), which J. Chem. Phys. 154, 054313 (2021); doi: 10.1063/5.0039923 154, 054313-12 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp emits weakly.22,34Fluorescence emission was observed for the 1,3, 5-trifluorobenzene radical cation (TFBz+). In the recent past, some of the experimental observations were explained in several exten- sive theoretical studies on the electronically excited fluorobenzene radical cations.30–39Vibronic coupling among electronic states was established to be the crucial mechanism that governs the nonradia- tive decay and radiative emission in fluorinated Bz+. Fluorination of Bz causes a stabilization of its σ-type of MOs. The stabilization increases with increasing fluorination and causes an energetic re-ordering of the cationic states. To understand the energetic ordering of electronic states of the fluorobenzene cations more clearly, we have calculated the six lowest valence MOs of Bz, monofluorobenzene (MFBz), difluorobenzene (DFBz) ( o,m, and p), trifluorobenzene (TFBz), PFBz, and Hexafluorobenzene (HFBz) and plotted their energies in Fig. S9 of the supplementary material. It can be seen from the figure that the HOMO of all molecules is of πtype. Fluorination causes a reduction in symmetry of Bz (D 6h), which is restored again in HFBz. Due to this symmetry reduction, the degen- erate E1gMO transforms to two non-degenerate MOs in MFBz and DFBz. Because of high symmetry of 1,3,5-TFBz (D 3h), the degener- acy of MOs is restored. The degeneracy is again split in PFBz (C 2v). It can be seen that the σ-type E2gMO of Bz (HOMO-1) undergoes con- siderable energy shift upon fluorination, as compared to the E1g(π) MO. The electronic states of the radical cations originating from ion- ization of an electron from the above MOs are portrayed in Fig. 10. In this figure, the VIEs of the cationic states are calculated with the EOMIP-CCSD/aug-cc-pVDZ level of theory and plotted. First of all, it can be seen that the cationic states form two groups, ̃X–̃A and̃B–̃C–̃D–̃E. These two groups are fairly well separated in energy. The nonradiative decay is governed by the interactions within and between the two groups. A second observation that can be clearly made from the plot is that the states arising from the2E2g(σ) MO of Bz are all shifted to higher energies in the fluorinated Bz+. Thisis due to a stabilization of the corresponding orbitals in the neu- tral molecules (cf., Fig. S9). A third observation that can be made from Fig. 10 that the states arising out of a2u(π) MO of Bz remain energetically unaffected for all fluorobenzene cations. In Bz+, the Jahn–Teller split components of the ̃Xand̃B states form low energy CIs, which facilitates nonradiative decay and quenching of fluorescence.72The interaction between the ̃X–̃Aand ̃B–̃C–̃Dgroup of states gives rise to energetically accessible CIs for nonradiative decay in MFBz+and DFBz+. Among the three DFBz+ (o,mand p), the energetic minimum of the relevant CIs occurs relatively at higher energy in the m-isomer and gives rise to weak radiative emission of its ̃Cstate. The degenerate ̃X2E“and excited ̃B2E′electronic states of 1,3,5-TFBz+are energetically well separated, and the intersections of these states with its ̃A2A′′ 2state occurring in between occur at higher energies relative to the minimum of the latter state. As a result, minimal electronic population flows to the ̃A2A′′ 2state when the WP initially prepared on any of the remaining states. Furthermore, the electron population dynamics of this state is not affected at all by its coupling with the other states. The popu- lation of this state remains at ∼100% for a long time and gives rise to radiative emission in 1,3,5-TFBz+. An analogous situation (as in case of 1,3,5-TFBz+) can be sketched in the case of PFBz+by examining the results presented in Sec. VI D. The data presented in Table IV reveal that the ̃Bstate of PFBz+is∼2.85 eV above the ̃Astate and ∼1.0 eV below the ̃Cstate. ThẽBstate is not coupled with the ̃Astate on symmetry ground. However, it is coupled to the ̃Cstate, and the energetic minimum of the ̃B–̃CCIs occurs at ∼15.22 eV, which is ∼2.25 eV above the ̃Bstate minimum. Despite strong ̃B–̃Ccoupling through ν13and ν14vibrational modes of a2symmetry (cf., Table 9), the coupling effect on the population dynamics of the ̃Bstate is weak because of large energy gap. In fact, the electron population does not flow to the other states when the WP is initially prepared on the ̃Bstate FIG. 10 . VIEs of Bz and its fluoro derivatives. J. Chem. Phys. 154, 054313 (2021); doi: 10.1063/5.0039923 154, 054313-13 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp (cf., Figs. 7 and 8). The population curve of the ̃Bstate remains par- allel to the time axis. This indicates a long-lived nature of the ̃Bstate, which gives rise to radiative emission in PFBz+. VII. SUMMARY Vibronic coupling and quantum nuclear dynamics in the ener- getically lowest six electronic states of PFBz+is studied in this arti- cle. Detailed electronic structure calculations are carried out by dif- ferent ab initio quantum chemistry methods. With the aid of the electronic structure results, a model vibronic Hamiltonian is con- structed in a diabatic electronic basis in terms of the dimensionless normal displacement coordinates of the vibrational modes. The cou- pling among different electronic states is evaluated by the standard vibronic coupling theory and elementary symmetry selection rules. The nuclear dynamics calculations are carried out from first princi- ples by both time-independent and time-dependent methods. For the latter calculations, the Heidelberg MCTDH suite of program modules is utilized. It appears from the electronic structure data and subsequent dynamics results that the EOMIP-CCSD method does a superior job in this case. It is by no means a conclusive remark in the absence of high resolution spectroscopy data. It is established that the energetically lowest six electronic states separates into two groups, viz., ̃X–̃Aand̃B–̃C–̃D–̃E. ThẽXand̃Astates form energet- ically accessible CIs. The effect of the latter on the dynamics of the ̃Xstate is not as much as on the same on the ̃Astate. This is because the minimum of the ̃Xstate is energetically well separated from the minimum of the ̃X–̃ACIs. The minimum of the ̃Astate on the other hand is energetically very close to the minimum of the ̃X–̃Ainter- sections. Therefore, the ̃X–̃Acoupling has a significant effect on the vibronic structure of the ̃Astate. It is found that the ̃Bstate is energetically well separated from the rest of the states. The coupling of ̃Bstate with others therefore has no significant effect on the vibronic structure of the ̃Bstate. The population of this state remains ∼100% for a long time when the dynamics started on it. The radiative emission in PFBz+therefore originates from this state. The ̃C–̃D–̃Eelectronic states are energet- ically close and therefore give rise to highly overlapping vibronic bands. The theoretical results are shown to be in good accord with available experimental results. SUPPLEMENTARY MATERIAL See the supplementary material for the coupling parameters of the electronic Hamiltonian [cf., Eqs. (5) and (6)] of PFBz+ (Tables S1–S10) and for the numerical details of the calculations (Tables S11 and S16), vibrational energy levels and their assign- ments (Tables S12–S15), vibronic wave functions of ̃Xand̃Astates of PFBz+(Figs. S1–S8), and the six lowest valence MOs of Bz and fluorobenzenes (Fig. S9). ACKNOWLEDGMENTS A.K.K. acknowledges the Council of Scientific Industrial Research, New Delhi, for a doctoral fellowship. This study is supported, in part, by a research grant (No. EMR/2017/004592)from the Department of Science and Technology, New Delhi. 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Phys. 154, 054313 (2021); doi: 10.1063/5.0039923 154, 054313-15 Published under license by AIP Publishing
1.4994219.pdf	Using magnetic charge to understand soft-magnetic materials Anthony S. Arrott , and Terry L. Templeton Citation: AIP Advances 8, 047301 (2018); View online: https://doi.org/10.1063/1.4994219 View Table of Contents: http://aip.scitation.org/toc/adv/8/4 Published by the American Institute of Physics Articles you may be interested in Very low frequency noise reduction in orthogonal fluxgate AIP Advances 8, 047203 (2017); 10.1063/1.4994208 Decomposing the permeability spectra of nanocrystalline finemet core AIP Advances 8, 047205 (2017); 10.1063/1.4991941 Critical exponent analysis of lightly germanium-doped La 0.7Ca0.3Mn1-xGexO3 (x = 0.05 and x = 0.07) AIP Advances 8, 047204 (2017); 10.1063/1.4993412 Investigation of measurement method of saturation magnetization of iron core material using electromagnet AIP Advances 8, 047202 (2017); 10.1063/1.4993998 Dependence of magnetic permeability on residual stresses in alloyed steels AIP Advances 8, 047201 (2017); 10.1063/1.4994202 Current-driven domain wall motion based memory devices: Application to a ratchet ferromagnetic strip AIP Advances 8, 047302 (2017); 10.1063/1.4993750AIP ADV ANCES 8, 047301 (2018) Using magnetic charge to understand soft-magnetic materials Anthony S. Arrottaand Terry L. Templeton Physics Department, Simon Fraser University, Burnaby, BC, Canada, V5A 1S6 (Received 4 July 2017; accepted 26 August 2017; published online 17 October 2017) This is an overview of what the Landau-Lifshitz-Gilbert equations are doing in soft- magnetic materials with dimensions large compared to the exchange length. The surface magnetic charges try to cancel applied magnetic ﬁelds inside the soft mag- netic material. The exchange energy tries to reach a minimum while meeting the boundary conditions set by the magnetic charges by using magnetization patterns that have a curl but no divergence. It can almost do this, but it still pays to add some divergence to further lower the exchange energy. There are then both positively and negatively charged regions in the bulk. The unlike charges attract one another, but do not annihilate because they are paid for by the reduction in exchange energy. The micromagnetics of soft magnetic materials is about how those charges rearrange themselves. The topology of magnetic charge distributions presents challenges for mathematicians. No one guessed that they like to form helical patterns of extended multiples of charge density. © 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/). https://doi.org/10.1063/1.4994219 This article is an extension of the ideas introduced in “Visualization and Interpretation of Mag- netic Conﬁgurations Using Magnetic Charge” by A. S. Arrott.1Theexchange energy density, "x, responsible for ferromagnetism, is expressed as "x=A=a2A[mr(rm)mrr m], (1) where Ais the exchange constant, an energy per unit length, ais an atomic distance, and mis a unit vector in the direction of the magnetization, M=Msm. The ﬁrst term in the [. . . ] vanishes if mhas no divergence. The second term vanishes if mhas no curl. There are two types of curls: bends and twists (think of vortices and Bloch domain walls). This is about the physics of unit vector ﬁelds. It is thatmis a unit vector that allows one to express the exchange energy, which is the sum of the squares of ﬁrst derivatives, by second derivatives using the vector Laplacian. Thinking in terms of curls and divergences has been helpful in visualizing three-dimensional magnetism. Magnetic charges aid in the understanding of the solutions to the Landau–Lifshitz–Gilbert (LLG) equations of micromagnetism. These are simultaneous, non-linear, integro-differential time- dependent equations (often millions) with damping. The non-linearity arises from imposition of the constraint that the magnetization Mis a vector of constant magnitude, that is M=Msm. (2) Magnetic charge appears because rB=r0(H+M)=0. (3) His a mixed vector that has real current density jas sources for its curl and the magnetic charge density m-rM (4) aCorresponding author: arrott@sfu.ca 2158-3226/2018/8(4)/047301/10 8, 047301-1 ©Author(s) 2017 047301-2 A. S. Arrott and T. L. Templeton AIP Advances 8, 047301 (2018) as sources for its divergence, that is rH=m. (5) IfMis discontinuous at a boundary, the divergence is a surface charge density, mnM, (6) where nis the outward surface normal. There are, generally, four terms in the micromagnetic energy: (1) It is fundamental that the Zeeman energy for an entity with magnetic moment isEZ= -B, where the ﬁeld Bis the inductance which can be expressed in terms of the vector potential A; B=OxA.In micromagnetics, the quantum importance of the vector potential Ais absent. The Zeeman energy density "Z= -M0Ha, where Haarises from current sources; this energy produces surface charges that try to exclude Ha; (2) The exchange energy E x, which is minimized when all the magnetic moments align parallel to one another, is responsible for ferromagnetism. The exchange energy increases with spatial changes in M(hence the differentials in LLG) and must increase to meet the boundary condition imposed by the surface charges; (3) The magnetostatic energy Edis the sum of all the charge–charge interactions; (accounting for the integrals in LLG). The exchange produces as little volume charge as possible; the +and - volume charges interact, but do not annihilate; (LLG is necessary to calculate the interactions of the magnetic charges); and, (4) The anisotropy energy EK, depends on the orientation of mwith respect to the local crystal axes. (In a single crystal, there are preferred directions and the anisotropy increases the magnetic charges. In a polycrystalline, charges appear near the grain boundaries and they can suppress much of the inﬂuence of the anisotropy.) There are also magneto-elastic energies that are often neglected. ZEEMAN ENERGY Just as an electrical conductor acts against an external electric ﬁeld to have the electrical ﬁeld vanish in its interior, in the presence of an externally applied magnetic ﬁeld, Ha, theMpattern of a magnetically soft ferromagnetic material produces a surface magnetic charge density mthat creates a magnetic ﬁeld almost equal and opposite to Ha. The cancellation is not as complete as in the electric ﬁeld case, for there will be some Omin the interior of an iron bar. In zero ﬁeld, there will be magnetic charge in the necessary swirls. In a high ﬁeld, Mno longer produces enough mto cancel the applied ﬁeld. In the electrical conductor, the ﬁeld in the interior is canceled by a small fraction of the electrical charge of the surface atoms. The uniform Eﬁeld in a conducting wire arises from a linear variation ofealong the wire. The deﬁnition of the ampere fails to include the word “shielded” in front of the word wire, indicating the collective ignorance of the electrical engineers who gave us the SI units by a one vote margin 1931. Heaviside knew this in the 19thCentury. In an iron bar in moderate ﬁelds, mis close to linear along the sides of the bar. Magnetization processes in ideally soft materials In the 1970’s B. Heinrich and A.S. Arrott studied ideally soft magnetic materials, which were deﬁned as those with zero anisotropy and with div M= 0. This work, centered on iron whiskers, was in the time of the “lost years of magnetism” in which the publications of the annual Magnetism and Magnetic Materials conferences were buried in the proceedings of the American Physical Society. An example of an ideally soft magnetic material was the analytical solution for the magnetization of a thin toroidal shell.2The analytical result was for a toroid with dimensions greatly larger than the exchange length. Solving the equation div M= 0, for a unit vector magnetization in toroidal coordinates, provided a result that was veriﬁed 30 years later by micromagnetic calculations in smaller systems where the exchange mattered,3but it did not alter the main concept of the ideally soft magnetic material. The solutions of the LLG equations are quite close to being divergence free047301-3 A. S. Arrott and T. L. Templeton AIP Advances 8, 047301 (2018) when the anisotropy is small. The magnetization lies everywhere in the toroidal shell. The surface charge necessary for this to happen is extremely small for a large toroidal shell. The magnetization pattern is determined by geometry alone. J. A. Ewing knew this in the 19th Century.4 The micromagnetics of soft-magnetic materials starts with Ewing, but it is the work of L. Landau and E. Lifshitz that gave the ﬁrst three-dimensional vision of magnetic materials.5They drew the structure of Fig. 1A. If they had had PowerPoint in 1935, they would have colored the two elongated domains red and green while using blue and yellow for the closure domains; see Fig. 1B. They would then have wondered about what color to use for the domain walls. N ´eel partially answered that question more than a decade later,6saying that the magnetization should lie in the surface and the wall between the green and the blue should be blue-green, but the wall between the green and the red could be either blue or yellow; see Fig. 1C and 1D. In 1975, Arrott pointed out that N ´eel’s answer led to an additional symmetry breaking.7Landau and Lifshitz already had broken the inversion symmetry on choosing the direction of rotation of the magnetization about the Bloch wall between the elongated domains as well as introducing the handedness of the Bloch wall. The blue or yellow domain wall produces a singularity at one end of the wall but not at the other; see Fig. 1F and 1G. Alex Hubert,8properly, gave the name “swirls” to these singularities. At least two of these are required on any singly-connected magnetic body, if there is a component of the magnetization that lies in the surfaces. The swirls are topological entities with properties and each has a life of its own. They can develop a handedness. They can occupy positions of symmetry, but with changes in ﬁeld they can move off the symmetry position. They are connected by tubes of energy density when both swirls have the same handedness. Just as there are vortices and anti-vortices, there are swirls and anti-swirls. A six-sided ferromagnetic brick can support six swirls, one on each face with two of these being anti-swirls.7 When a swirl moves off center, breaking symmetries, the tube of energy density can spiral from the offset swirl on the surface to the original line of symmetry well below the surface. The tubes of magnetic charge that accompany the swirl can form a helical pattern. None of this was envisioned in the forty years between the original work of Landau and Lifshitz and when Arrott took the knowledge he gained from studying liquid crystals, back to magnetism. The classical “Landau structure” evolves from the illustration in the 1935 paper and is shown in Fig. 2 along with a modern version of that structure using isosurfaces of the components of the magnetization. The idea of Landau and Lifshitz was to create a charge free structure. The Bloch wall FIG. 1. Landau and Lifshitz original drawing is indicated in A. The Landau structure was extracted from this and portrayed as in E, for the next 40 years. Had Lifshitz had PowerPoint, he would have tried to color the domains. He would have anticipated the modern convention of coloring the long domains as Red to the Right and green to the left, naturally, with blUe as Up and yellOW as down (on the page) for the closure domains. Then he would have noted some problems. The ﬁrst one is that the Bloch wall between the red and green domains has the magnetization out of the surface; shown in white in B. N ´eel partially answered that question more two decades later, saying that the magnetization should lie in the surface in what are now called N´eel caps. There are two ways for this to happen as shown in C, turning the white area yellow as shown for the Landau structure in F, or in D, turning the white area blue as shown in G. Either way, swirls appear at one end of the N ´eel cap and not at the other. The second problem is that where the yellow and blue domains meet on the side surface, the magnetization is either in the +z-direction or the -z-direction, leading to troubles at the surfaces, marked by black partial circles. These are partial anti-swirls.047301-4 A. S. Arrott and T. L. Templeton AIP Advances 8, 047301 (2018) FIG. 2. Three views of the Landau structure, showing the isosurface of mz = 0.8 for the brick and the ellipsoid. is divergentless, but produces surface charges mwhere it intersects the top and bottom surfaces. N´eel’s solution was to keep min the z-surface, but in doing so he created some bulk charges because myis changing in the y-direction, ﬁrst increasing and then decreasing. The divergence also has a term from the change of mzin the z-direction. These two terms tend to cancel one another by shifting the position of the N ´eel cap with respect to the Bloch wall. The resulting structure is not quite charge free. Again, it is geometry that is dictating the structure. The Bloch wall with its Neel caps and the closure domains are there to suppress magnetic charge in the bulk. The equivalent of these are seen the calculations for the iron ellipsoid; see Fig. 2. Magnetic charges get into the act during the processes that lead to the formation of the Landau structure, starting from the saturated state in a large ﬁeld. We do not know how this happens in the laboratory despite many years of measurements on iron whiskers.047301-5 A. S. Arrott and T. L. Templeton AIP Advances 8, 047301 (2018) What appears from susceptibility measurements to be the Landau state, appears only after a whisker is cycled back in forth between ﬁelds sufﬁciently strong to cause the center of the whisker to be close to uniformly magnetized, ﬁrst in one direction and then the other. On the computer, one must break the inversion symmetry. If this is not done, the two ends have the opposite hand- edness for the two swirls that ﬁrst form on the ends when lowering the applied ﬁeld. Inversion symmetry is broken when the swirls move off center, but the handedness of the two ends remains opposite. While all this has been long known, it is only recently that emphasis has been placed on the role of magnetic charge in the ferromagnetic parallelepiped (brick). Two iron bricks are compared for dimensions 240 nm x 240 nm x 1440 nm with and without inversion symmetry. A brick has inversion symmetry, as do the LLG equations. One way to break the symmetry is to pass a current along the long axis (z-axis). This is done for the brick in a ﬁeld 0Ha greater than 0.8 T. The ﬁeld is then lowered through 0.8 T as swirls appear on the end surfaces of the long brick. The current is sufﬁcient to force the swirls to have the same handedness as the magnetization circulates about the z-axis in the sense dictated by the current. When the ﬁeld is sufﬁciently below 0.8 T, the current can be removed and the sense of rotation will remain, unless a large sudden change in ﬁeld is made that restores the inversion symmetry through the dynamic response. The difference, between the inversion symmetry state and that prepared by passing the current and then removing it, is most striking below 200 mT. Both the magnetization patterns and the accom- panying magnetic charge densities are shown using isosurfaces of constant scalar values, m zandm in Fig. 3. To get the Landau structure from this vision of the inversion symmetry state just by reversing the applied ﬁeld is like trying to untie the Gordian knot. This is partially achieved by sequential nucleation of pairs of solitons on each of two adjacent edges at the central cross section. The details concerning the breaking of inversion symmetry will be presented elsewhere.9Once the Landau structure is obtained, it is not so difﬁcult to explain how it reappears again and again on cycling an applied ﬁeld, provided the ﬁeld is never sufﬁcient to restore the inversion symmetry. To get the Landau structure, it is necessary to reverse the magnetization along two adjacent edges of the long whisker. This necessitates the formation of edge solitons which propagate and reverse the adjacent edges. Charge density plays an important role in the nucleation of the edge solitons on adjacent edges. The symmetry breaking necessary to select out two adjacent edges is aided by using a whisker with rectangular cross section. With a square cross section, the solitons can sometimes form with two edges selected differently on the two ends of the whisker. FIG. 3. Lines of constant m zare shown on the left for the mid-section of an iron brick, 240 nm x 240 nm x 1440 nm in which the inversion symmetry has been suppressed. The contours of m z= 0 are shown as a heavy dark line. There are ﬁve tubes of magnetization for a given m z. The center tube is equivalent to the four tubes in the corners, which can be called quarter-tubes. These tubes are shown again in on the right, but now viewed looking down in slice that extends from z = 360 nm to 1080 nm. What is new about this diagram is a view of the tubes of magnetic charge with blue for positive charge and red for negative charge. These charges are further explored in Figs. 4, 5, 6, and 7.047301-6 A. S. Arrott and T. L. Templeton AIP Advances 8, 047301 (2018) FIG. 4. A section of long iron brick from 360 nm to 1080 nm showing the isosurfaces for m z= 0.9 in black and isosurfaces for both plus and minus magnetic charge in blue and red. The view in Fig. 3b is looking down on this section. The tubes of blue charge grow out the top section of the long brick (not shown) while the tubes of red charge grow out of the bottom section from 0 to 360 nm shown in Fig. 5. The magnetic charges form when there is a central core of magnetization in the direction of the applied ﬁeld that is surrounded by a region where the magnetization is reversing when the demagnetizing ﬁeld is more negative than the applied ﬁeld is positive. While the magnetization in the planes perpendicular to the long z-axis is primarily circulating about the z-axis, the exchange energy develops some divergence by producing a radial component to the magnetization. This is encouraged by crystalline anisotropy and by the effects of the geometry of the cross-section not being circular. In a square cross section [1 0 0] oriented whisker, the charge density forms an elongated octupole for much of the length of the whisker. There are eight tubes of charge density in a square pattern with adjacent tubes of opposite charge. In a [1 1 1] oriented whisker, the anisotropy and the hexagonal shape cause the charge density to form an elongated hexapole or dodecapole. There are then six tubes of charge on the corners of a hexagon alternating in charge from one tube to the next. The symmetric arrangement of the cylindrical tubes is not always the stable state. Each tube has magnetostatic self-energy, which can be reduced by distortions of the tubes. The tube conﬁguration can move off center. Each tube can lower itself energy by breaking into segments. If the tubes could stretch out, the self-energy would be lowered. Stretching out is achieved by making a helical pattern. In the hexagonal whisker, the tubes can move off center ﬁrst, decreasing the charge in two of the tubes while the other four tubes create a quadrupole. After that, helical distortions further lower the magnetic self-energy by decreasing the self-energy in each tube while bringing the tubes closer together to lower the energy further by the attractive interaction of the opposite charges. In the [1 0 0] whisker, a quadrupole moves off center bringing the reversed magnetization closer to two of the edges, resulting in nucleation of pairs of solitons on each of those two edges, followed047301-7 A. S. Arrott and T. L. Templeton AIP Advances 8, 047301 (2018) FIG. 5. The continuation of Fig. 4 to the bottom of the long iron brick. The bottom 0 to 360 nm shown here is almost indistinguishable from the same section of the brick when the inversion symmetry is maintained. In both cases the top sections are the same with red and blue interchanged. But the middle section for inversion symmetry is quite different as shown in Fig. 6. FIG. 6. The isosurfaces of magnetic charge and magnetization near the center of the iron brick with inversion symmetry. As this is a bit much for the mortal mind, it is simpliﬁed in Fig. 7 by making the magnetic charge quite transparent.047301-8 A. S. Arrott and T. L. Templeton AIP Advances 8, 047301 (2018) FIG. 7. The magnetic charge in Fig. 6 is shown over the full sample length and has been made transparent to show the isosurfaces of m z= 0.9 from a slightly different angle. There is a ring of magnetization in the midplane which connects to four quarter-tubes of magnetization in both the upper and lower portions of the magnetic structure. The central tubes cannot join in the midplane because they have opposite handedness. As they approach each other they veer away from the central axis and join the ring. Magnetism is complex. by the reversal of two adjacent edges necessary for the formation of the Landau structure with its two domains separated by the Bloch wall with its N ´eel caps. The importance of thinking about magnetic charge was reinforced recently by the discovery that measurements of Scott D. Hanham,1040 years ago, on hexagonal whiskers can now be explained. There were six peaks in the ac susceptibility separated in ﬁeld by 0.2 mT. With the recognition of the role of edge solitons in explaining behaviors of thin ﬁlm elements for magnetic random-access memories, it seemed possible that the six peaks could be explained by soliton propagation down the six edges of the hexagonal cross-section whiskers. What was not expected was an explanation of047301-9 A. S. Arrott and T. L. Templeton AIP Advances 8, 047301 (2018) FIG. 8. Isosurfaces of magnetic charge and of m z= 0.9 (in black) for a circular cross-section iron cylinder with the hexagonal hard axis of magnetization along the axis. The cylinder is 1000 nm long with diameter of 163 nm. The top 50 nm of the cylinder is not shown. The helical state on the left changes to the nano-helical state on the right in a ﬁrst order change with ﬁeld from -32.0 mT to -32.2 mT. The nano-helical state has a helical distortion of at most a few nanometers and generally less. It is the response of the central part of the magnetization to the swirls on the top and bottom surfaces. It is distinct from the helices on the left that are spontaneously formed in the ﬁrst order change in conﬁguration. Hanham’s observation of stripe domain patterns in the hexagonal whiskers, which are many orders of magnitude bigger than can be calculated by micromagnetics. Micromagnetic calculations for hexagonal cross-section cylinders with 75 nm edges accounted for the six peaks and showed that the magnetization took a helical conﬁguration during the process of magnetization reversal with the helical patterns accounting for the observations of the stripe domains. Fig. 8 shows the collapse of the helical pattern in a circular cross-section iron cylinder with a hard axis of cubic anisotropy along the axis of the cylinder. The abstract submitted for this conference also pointed to the importance of magnetic charge in suppressing the adverse effects of anisotropy in polycrystalline iron, but that will wait for another time1,11as this has already exceeded the bounds of space and time. 1A. S. Arrott, “Visualization and interpretation of magnetic conﬁgurations using magnetic charge,” IEEE Magnetics Letters 7, 1108505A (2016). 2A. S. Arrott, B. Heinrich, and D. S. Bloomberg, “Micromagnetics of magnetization processes in toroidal geometries,” IEEE Transactions on Magnetics 10, 950–953 (1974). 3A. S. Arrott and R. Hertel, “Mode anticipation ﬁelds for symmetry breaking,” IEEE Transactions on Magnetics 43, 2911 (2007). 4A. S. Arrott, “The past, the present and the future of soft magnetic materials,” Journal of Magnetism and Magnetic Materials 215/216 , 6–10 (2000). 5L. Landau and E. Lifshitz, “On the theory of the dispersion of magnetic permeability in ferromagnetic bodies,” Phys. Zeitsch. der Sow. 8, 153–169 (1935), online at http://www.ujp.bitp.kiev.ua/ﬁles/journals/53/si/53SI06p.pdf. 6L. N´eel, “ ´Energie des parois de Bloch dans les couches minces,” C. R. Acad. Sci. 241, 533–536 (1955), (Bloch wall energy in thin layers). 7A. S. Arrott, “Dipole-dipole interactions in the computational micromagnetism of iron (1955–2010),” J. Appl. Phys. 109, 07E135 (2011). 8A. Hubert, “The role of ‘magnetization swirls’ in soft magnetic materials,” J. Phys. Colloq 49, C8-1859–C8-1864 (1988).047301-10 A. S. Arrott and T. L. Templeton AIP Advances 8, 047301 (2018) 9T. L. Templeton, S. D. Hanham, and A. S. Arrott, “Helical patterns of magnetization and magnetic charge density in iron whiskers,” abstract MMM 2017. 10S. D. Hanham, (1980), Thesis for the Ph.D., Simon Fraser Univ.Burnaby, BC, Canada, “The magnetic behaviour of the (111)-oriented iron whisker,” Fig. 14, sfu.ca/system/ﬁles/iritems1/3926/b12516338.pdf. 11A. S. Arrott, C. M. Williams, and E. Negusse, “Magnetic charges suppress effects of anisotropy in polycrystalline, soft, ferromagnetic materials,” abstract MMM 2017.
1.3453683.pdf	A time-dependent semiempirical approach to determining excited states Lizette A. Bartell, Michael R. Wall, and Daniel Neuhauser Citation: J. Chem. Phys. 132, 234106 (2010); doi: 10.1063/1.3453683 View online: http://dx.doi.org/10.1063/1.3453683 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v132/i23 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 13 Jan 2013 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsA time-dependent semiempirical approach to determining excited states Lizette A. Bartell, Michael R. Wall, and Daniel Neuhausera/H20850 Department of Chemistry and Biochemistry, University of California, Los Angeles, California 90095-1569, USA /H20849Received 28 December 2009; accepted 24 May 2010; published online 16 June 2010 /H20850 We study a time-dependent semiempirical method to determine excitation energies, TD-PM3. This semiempirical method allows large molecules to be treated. A Linear-response Chebyshev approachyields the TD-PM3 spectrum very efﬁciently. Spectra and excitation energies were tested bycomparing it with the results obtained using TD-DFT /H20849Time Dependent-Density Functional Theory /H20850, using both small and large basis sets. They were also compared to PM3-CI, TimeDependent-Hartree Fock using the STO-3G basis set, and to experiment. TD-PM3 results generallymatch better the large-basis set calculations than the small-basis TD-DFT do; excitation energies arealmost always accurate to within about 20% or less, except for a few small molecules. Accuracyimproves as the molecules get larger. © 2010 American Institute of Physics . /H20851doi:10.1063/1.3453683 /H20852 I. INTRODUCTION Semiempirical methods have an important role in large scale simulations, allowing treatment of very largesystems. 1,2Traditionally, semiempirical methods have been mostly used for time-independent ground-state simulations.However, with the rising interest in excited state dynamics,and the advent of large scale iterative computational meth-ods, a natural question arises whether semiempirical time-dependent and/or iterative methods can be as useful for dy-namics and for excited states. At present there are severalsemiempirical methods which have been used for excited-state dynamics. One is time-dependent tight-binding DFT, 4,5 a method which bridges DFT /H20849Density Functional Theory /H20850 and semiempirical tight binding6,7in order to also treat sys- tems such as organic molecules and biological moleculeswith atoms of different electronegativities, rather than thesolid-state systems that tight binding has been generally usedfor. The other approach is to use a semiempirical MO-CI/H20849molecular orbital-conﬁguration interaction /H20850method. 8–10 This involves using a semiempirical program such as PM3 or MNDO along with a CI /H20849conﬁguration interaction /H20850calcula- tion such as CI singles or CI doubles in order to get single ordouble excited states. A decade ago some interest has also risen in using time- dependent methods for polymeric systems. It was also real-ized that a Krylov subspace approach could be used to turnthe time-dependent equation into a linear-response equationwith an easily calculated action of a time-independent Liou-ville operator. 12Here, we systematically investigate the PM3 approach using a Chebyshev framework of polynomialexpansion of linear-response time-dependent densitymethods. 14,15We examine such an approach for a collection of small molecules, showing that even for small systems theexcitation energies are quite accurate. The resulting approachis therefore very efﬁcient numerically compared with directreal time propagation since no time-dependent propagation is needed and the results are calculated directly in frequencyspace, iteratively, without any matrix diagonalization. Conceptually, PM3 may seem unnatural as a time- dependent method, as it has been parametrized for groundstates. However, it is known that linear-response time-dependent methods tend to have surprisingly goodaccuracies. 14–17The formal reason is that much of the effects which are missing in time-independent descriptions are justthe polarization of the electron cloud /H20849due to the Hartree terms /H20850and a time-dependent treatment automatically takes those into account. /H20851Formally, a time-dependent treatment takes the RPA /H20849random phase approximation /H20850diagrams into account. /H20852We therefore examine here a straightforward appli- cation of PM3 to excited-state studies, and, indeed, ﬁnd thatthe method is surprisingly accurate. Speciﬁcally, we showbelow that even without any parameter tweaking, a linear-response PM3 approach yields excitation energies which areoften ﬁve times more accurate than the time-independenthighest occupied molecular orbital-lowest unoccupied mo-lecular orbital /H20849HOMO-LUMO /H20850gap, and are generally accu- rate to within 10%–20%. The paper is arranged as follows. Section II reviews PM3 and linear response. Section III shows results, and con-clusions follow in Sec. IV. II. THEORY A. PM3 equations PM3 is an acronym for the modiﬁed neglect of diatomic orbital method-parametrized model 3 or MNDO-PM3. Thismodel makes several assumptions in order to make its calcu-lations feasible and efﬁcient. It only treats the valence elec-trons of an atom, in a minimal basis, and approximates itsinner shell electrons and the rest of its nucleus as a ﬁxedcore. The electronic energy is calculated using a/H20850Electronic mail: dxn@chem.ucla.edu.THE JOURNAL OF CHEMICAL PHYSICS 132, 234106 /H208492010 /H20850 0021-9606/2010/132 /H2084923/H20850/234106/6/$30.00 © 2010 American Institute of Physics 132, 234106-1 Downloaded 13 Jan 2013 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsE=1 2/H20858 ijPij/H20849Hij+Fij/H20850. /H208491/H20850 Here, Pis the density matrix; His the core Hamiltonian matrix, which includes the usual core Hamiltonian togetherwith a dipole term H ij=H0,ij+E·Dij, /H208492/H20850 due to any external electric ﬁelds, E/H20849as will be important for the linear response evolution below /H20850, D is the electric dipole matrix, and Fis the Fock matrix. The basis set for the ma- trices is composed of the valence shell atomic orbitals ofeach atom in the molecule. The equations used to calculate the core Hamiltonian matrix and the Fock matrix are detailed by Stewart, Dewar,and Thiel 19–21and are brieﬂy described below. We use the close-shell version, since if the ground state is made from closed-shell orbitals, it is sufﬁcient to consider only theclosed-shell equations for singlet-singlet transitions. The core Hamiltonian matrix is composed of one- electron terms and is calculated by H /H9262/H9262=U/H9262/H9262+/H20858 BV/H9262/H9262,B, /H208493/H20850 H/H9262/H9263=/H20858 BV/H9262/H9263,B, /H208494/H20850 H/H9262/H9261=1 2/H20849/H9252/H9262A+/H9252/H9261B/H20850S/H9262/H9261, /H208495/H20850 where the symbols are deﬁned as follows. First, here and in the following, /H9262and/H9263correspond to matrix elements pertaining to atomic orbital in atom A. The subscripts /H9261and/H9268denote atomic orbitals in an atom Bwhich is different from atom A. U/H9262/H9262is the sum of the kinetic energy of the electron in orbital /H9262and the potential energy of the attraction between the electron and the core of the atom in which this orbitalresides. This parametrized term is determined by ﬁtting sev-eral of its theoretical valence energies against the corre-sponding spectroscopic results. V /H9262/H9263,Bis the potential energy of the attraction between the electron in atom Aand the core of atom Band is calculated by evaluating the two center integral representing the repul-sion interactions between the charge distribution of theatomic orbitals represented by /H9262and/H9263in atom Aand a purely spherical /H20849s-type /H20850charge distribution in atom Bwhich approximates the core of atom B. The/H9252’s are parameters speciﬁc to the atom and the type of atomic orbital, i.e., whether it is s or p /H20849there have been extensions to d- and higher order orbitals, but for most ap-plications s and p orbitals sufﬁce /H20850. Finally, S /H9262/H9263is an element from the overlap matrix cal- culated from the overlap integrals of the individual minimalbasis Slater orbitals. Next, we turn to the Fock matrix in astatic ﬁeld. It is composed of the core Hamiltonian and two-electron terms, the open shell equations for the alpha/H20849spinup /H20850Fock matrix areF /H9262/H9262/H9251=H/H9262/H9262+/H20858 /H9263A /H20851P/H9263/H9263/H9251+/H9252/H20849/H9278/H9262A/H9278/H9262A,/H9278/H9263A/H9278/H9263A/H20850 −P/H9263/H9263/H9251/H20849/H9278/H9262A/H9278/H9263A,/H9278/H9262A/H9278/H9263A/H20850/H20852+/H20858 B/H20858 /H9261,/H9268B P/H9261/H9268/H9251+/H9252/H20849/H9278/H9262A/H9278/H9262A,/H9278/H9261B/H9278/H9268B/H20850, /H208496/H20850 F/H9262/H9263/H9251=H/H9262/H9263+2P/H9262/H9263/H9251+/H9252/H20849/H9278/H9262A/H9278/H9263A,/H9278/H9262A/H9278/H9263A/H20850−P/H9262/H9263/H9251/H20851/H20849/H9278/H9262A/H9278/H9263A,/H9278/H9262A/H9278/H9263A/H20850 +/H20849/H9278/H9262A/H9278/H9262A,/H9278/H9263A/H9278/H9263A/H20850/H20852+/H20858 B/H20858 /H9261,/H9268B P/H9261/H9268/H9251+/H9252/H20849/H9278/H9262A/H9278/H9263A,/H9278/H9261B/H9278/H9268B/H20850,/H208497/H20850 F/H9262/H9261/H9251=H/H9262/H9261−/H20858 /H9263A /H20858 /H9268B P/H9263/H9268/H9251/H20849/H9278/H9262A/H9278/H9263A,/H9278/H9261B/H9278/H9268B/H20850. /H208498/H20850 Here, /H9272represents the atomic orbitals of the speciﬁed atom. The terms in parentheses composed of atomic orbitals in thesame atom Aare the one-center two-electron repulsion inte- grals due to exchange and Coulomb forces between two elec-trons in different atomic orbitals but in the same atom. Theseare parametrized speciﬁcally to each atom using experimen-tal data. The terms in parentheses composed of atomic orbitals from two different atoms, AandB, are the two-center two- electron repulsion integrals due to the repulsion forces be-tween two electrons in two different atoms. /H20849All three- and four-center integrals are neglected. /H20850The two-center integrals were calculated using the method and equations by Dewarand Thiel, 21where the interaction between orbitals on differ - ent atoms is approximated from electrostatic moments. As the equations above show, a trait of semiempirical methods is the simpliﬁcation of the Hamiltonian by replacingsome of the terms with parameters and equations obtained byderiving them from and ﬁtting them against experimentalresults and data. 1,2,23–26 The ﬁrst stage in the simulation is completed by itera- tively preparing the ground-state Fock and density matrices P0,F0fulﬁlling P0=/H9258/H20849/H9262−F0/H20849P0/H20850/H20850, /H208499/H20850 where we introduced the chemical potential and step func- tion. Standard sparse-matrix methodologies can be used toefﬁciently do the Hartree–Fock /H20849HF/H20850iterations for large sys- tems. The ground-state density matrix is then used as aninput to the time-dependent stage. B. Time-dependent PM3 After the electronic energy converges, the time- dependent response is mostly simply calculated in real timefrom evolving the time-dependent equation i/H11509P /H11509t=/H20851F/H20849P/H20849t/H20850/H20850,P/H20849t/H20850/H20852. /H2084910/H20850 The time dependence is induced by a simple addition of an electric ﬁeld delta-function /H20849in time /H20850perturbation to the initial Fock matrix, i.e., using234106-2 Bartell, Wall, and Neuhauser J. Chem. Phys. 132, 234106 /H208492010 /H20850 Downloaded 13 Jan 2013 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsE/H20849t/H20850=/H9254/H20849t/H20850E0. /H2084911/H20850 After the delta-function perturbation ends, the density matrix /H20849denoted as Pstartto distinguish it from the original density matrix, P0/H20850takes the form Pstart=P0−i/H20851E·D,P0/H20852. /H2084912/H20850 Pstartis used as the initial condition to the time-dependent propagation. A full time-dependent linear propagation is often expen- sive, although there has been considerable progress in directreal time propagation /H20849see, e.g., Ref. 27and references therein /H20850. Since the vast majority of applications will be con- cerned with linear response, we used here a more efﬁcientiterative Chebyshev approach /H20849see Refs. 14and15/H20850. This approach is different from the more commonly usedfrequency-based linear-response approach used byCasida 28,29and others16,17,30in that it processes all frequen - cies at once without matrix diagonalization. Using this ap-proach allows for the excitation energies to be found simplyfrom the Fourier transform of the propagation of the densitymatrix over time. For greater efﬁciency, a similar iterativeapproach to that developed in Refs. 14and15but which extracts the spectrum directly in frequency space was usedhere. This approach will be explained below. C. Manifestly linear evolution equations The linear response approach to PM3 and density matri- ces in general is even simpler than the previously introducedwave function approach. 14,15 We deﬁne the deviation between the actual and ground- state density matrix as W/H20849t/H20850=P/H20849t/H20850−P0. /H2084913/H20850 Deﬁning F0=F/H20849P0/H20850and using /H20851F0,P0/H20852=0, we expand the evolution equation /H20851Eq. /H2084910/H20850/H20852for the density matrix ignoring terms of order W2, i/H11509 /H11509tW/H20849t/H20850=/H20851F/H20849W/H20849t/H20850+P0/H20850−F0,P0/H20852+/H20851F0,W/H20849t/H20850/H20852, /H2084914/H20850 a form which is linear in Wfor small enough deviations. The-F0term in Eq. /H2084914/H20850is important to impose linearity in the typical case where numerically /H20851F0,P0/H20852is small but non- vanishing. The linearity can be further imposed by scaling through a small constant, denoted by g, resulting at /H11509 /H11509tW/H20849t/H20850=LW, /H2084915/H20850 where the Liouville superoperator is deﬁned as LW=−i g/H20851F/H20849gW/H20849t/H20850+P0/H20850−F0,P0/H20852−i/H20851F0,W/H20852/H20849 16/H20850 /H20851in practice we found that a variable g, equal to a small number /H20849e.g., 10−5/H20850times the norm of W, leads to uniformly stable results /H20852. The initial density matrix is then obtained by applying a delta-function electric ﬁeld perturbation, which results in astarting density ofW start=Pstart−P0=−i/H20851E·D,P0/H20852. /H2084917/H20850 The linear evolution equation is then solved by the itera- tive Chebyshev algorithm. Formally, the time-dependentpropagation is represented as W/H20849t/H20850=e LtWstart=/H20858 n/H208492−/H9254n0/H20850Jn/H20849t/H9004/H20850Tn/H20873L /H9004/H20874Wstart =/H20858 n/H208492−/H9254n0/H20850Jn/H20849t/H9004/H20850/H9256n, /H2084918/H20850 where /H9004is a parameter essentially equaling to /H20849or somewhat larger than /H20850the typical energy range in the Fock operator, and we introduced the Bessel function and the modiﬁedChebyshev series, formally deﬁned as /H9256n=Tn/H20873L /H9004/H20874Wstart, /H2084919/H20850 where Tnare modiﬁed Chebyshev operators deﬁned as Tn/H20849x/H20850=i−nacos /H20849ix/H20850. /H2084920/H20850 In practice, the series is evaluated as /H92560=Wstart, /H92561=LWstart /H9004, /H2084921/H20850 /H9256n=2L/H9256n−1 /H9004+/H9256n−2. Note that each element /H9256nis itself a density matrix of the same dimensions as P0. In practice we are typically interested in the absorption spectrum. For that, we need the time-dependent dipole, d/H20849t/H20850=T r /H20849DW/H20849t/H20850/H20850. /H2084922/H20850 The dipole will yield the absorption cross section deﬁned as B/H20849/H9275/H20850=/H9275Im/H20849E0·d/H20849/H9275/H20850/H20850, /H2084923/H20850 where d/H20849/H9275/H20850=1 2/H9266/H20885 0/H11009 e−t2a2/2ei/H9275td/H20849t/H20850dt. Noticing that Wstartis purely imaginary, we get Imd/H20849/H9275/H20850=1 2/H9266Im/H20885 0/H11009 Tr/H20849D·ei/H9275teLtWstart/H20850dt =T r /H20849D·/H9254/H20849iL−/H9275/H20850Wstart/H20850dt, /H2084924/H20850 where the delta function is evaluated by a Chebyshev itera- tive algorithm. In practice, it is Gaussian broadened and de-ﬁned as /H9254/H20849iL−/H9275/H20850=1 /H208812/H9266aexp/H20873−/H20849iL−/H9275/H208502 2a2/H20874, /H2084925/H20850 where ais a frequency-width parameter; we then follow with the expression234106-3 Allows efﬁcient treatment of large molecules J. Chem. Phys. 132, 234106 /H208492010 /H20850 Downloaded 13 Jan 2013 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions1 /H20881/H9266aexp/H20873−/H20849iL−/H9275/H208502 a2/H20874=/H20858 nTn/H20873L /H9004/H20874fn/H20849/H9275/H20850, /H2084926/H20850 where fn/H20849/H9275/H20850are frequency dependent coefﬁcients /H20849evaluated in the Appendix /H20850. The equations above result in Gaussian broadened density matrices, W/H20849/H9275/H20850/H110131 /H20881/H9266aexp/H20873−/H20849iL−w/H208502 a2/H20874Wstart=/H20858 nfn/H20849/H9275/H20850/H9256n,/H2084927/H20850 so that ﬁnally, B/H20849/H9275/H20850=/H9275/H20858 nfn/H20849/H9275/H20850D·Rn, /H2084928/H20850 where the residues Rn=T r/H20873DTn/H20873L /H9004/H20874Wstart/H20874=T r /H20849D/H9256n/H20850/H20849 29/H20850 are each a length-three vector. The ﬁnal result is therefore very simple, as in practice the calculation of the absorption spectrum amounts to thecalculation of Chebyshev series, and while it is evaluated theresidues need to be collected. Then, for each desired damp-ing parameter, the spectrum is calculated from Eq. /H2084928/H20850. The number of Chebyshev terms in our simulations is about 4 /H9004/a, which for typical spectra yields several thou- sand terms. There are several methods for expediting conver-gence of a Chebyshev series and these should yield up to anorder of magnitude improvement in the number of terms/H20849and even more for isolated spectra /H20850as will be investigated in future studies. III. SIMULATIONS A. Overview For this test study, several small representative mol- ecules were ﬁrst used: dihydrogen, hydrogen ﬂuoride, diﬂuo-ride, carbon monoxide, methane, ethene, and formaldehyde.These molecules are well represented by established methodssuch as DFT and TD-DFT /H20849Time Dependent-Density Func- tional Theory /H20850with standard functionals. In addition, the pro- gram was tested on the following aromatics, benzene, naph-thalene, anthracene, tetracene, and pentacene, in order todetermine its accuracy as the molecules increase in size.The ﬁrst step in the simulations is the construction of the ground-state density and Fock matrices by established meth-ods. The electric dipole perturbation is then added and thedensity matrix is iterated over time using Eq. /H2084914/H20850. Next, Eq. /H2084928/H20850is applied; typically we use a few thou- sand terms for convergence /H20849indicated by a totally positive spectrum without any negative parts which will be artifactsof lack of convergence /H20850. The width parameters taken were a= 0.05 eV, /H9004=5 0 e V , /H2084930/H20850 where awas chosen to yield well-isolated peaks in the spec- tra, while /H9004was chosen to ensure convergence of the Cheby- shev expansion /H20849the only requirement on /H9004is that it needs to be higher than the half width of the spectrum of L; the sim- plest way to ensure this requirement is by empirically choos-ing a low enough where the expansion still converges /H20850. From the spectrum we extract the lowest excitation en- ergies. For the test calculations, we ﬁrst checked our time-independent PM3 results against that of established PM3routine in Gaussian, obtaining essentially identical results.The HOMO-LUMO gap was then reproduced by direct di-agonalization of the time-independent Fock. The PM3 program in the molecular package MOPAC /H20849Ref.20/H20850was ﬁrst used to optimize the geometry of the small molecules. The DFT program in the molecular package Q-CHEM /H20849Ref. 31/H20850, was ultimately used to optimize the ge- ometry using the B3LYP functional with the 6-311G/H11569/H11569basis set. The excitation energies were then found for the mol-ecules in their optimized geometry using the resulting linear-response time-dependent PM3 and also the TD-DFT programin Q-CHEM . B. Results Table Ishows the lowest allowed singlet vertical excita- tions with signiﬁcant oscillation strengths /H20849generally above 10−4a.u./H20850calculated using the time-dependent PM3 pro- gram by the TDDFT module in Q-CHEM using two basis sets, large /H20849aug-cc-pvtz /H20850and small /H208493-21G /H20850, the PM3-CI program inMOPAC with ﬁve states, and experimental values. In all cases we compared closed-shell simulations, where theground-state density matrix is equal for both spins. There aresome weak transitions which have very little overlap withsymmetric closed-shell transitions which are therefore notTABLE I. Lowest excitation energies of small molecules obtained using TD-PM3 and various methods /H20849energies in eV; TD-DFT using B3LYP /H20850. Molecule Transition TD-PM3 TD-DFT aug-cc-pvtz TD-DFT 3-21G TD-HF STO-3G PM3-CI Band gap Expt. H2 /H9018u:/H9268→/H9268/H115699.96 11.74 15.76 15.07 10.25 20.86 11.19a HF /H9018g:n→/H9268/H115698.66 9.33 9.57 13.11 8.59 19.75 10.35b F2 /H9016u:/H9266/H11569→/H9268/H115694.85 5.26 5.29 7.16 4.52 15.53 4.4c CO /H9016:/H9268→/H9266/H115697.56 8.60 8.43 8.59 7.01 14.03 8.55d CH4 T2:/H9268→/H9268/H115698.76 9.63 13.43 24.03 8.59 17.88 9.7d C2H4 B3u:/H9266→3s 8.26 6.69 10.11 15.37 8.43 7.11d B1u:/H9266→/H9266/H115695.76 7.47 8.76 11.41 6.65 11.72 7.60d CH2OB2:n→3s 5.56 6.48 9.15 18.65 5.57 11.37 7.11d aReference 3. bReference 13. cReference 18. dReference 22.234106-4 Bartell, Wall, and Neuhauser J. Chem. Phys. 132, 234106 /H208492010 /H20850 Downloaded 13 Jan 2013 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsshown, such as a weak A 2transition in TDDFT and experi- ment around 4.0 eV for CH 2O. The TDDFT calculations used the hybrid exchange and correlation functional B3LYP. Generally, the larger the basisset the more accurate is the result obtained by a TDDFTcalculation. Accuracy can also be improved by choosing abasis set that includes diffusive and valence polarizationfunctions. While the large basis set will be more accurate, itis limited in practice for large systems due to the numericalcost; this was the reason we compared PM3 to smaller basissets. In practice, the lowest excitation energies were obtained using the time-dependent iterations and the graphs producedand taking the lowest peak /H20849and as mentioned, more efﬁcient approaches can be envisioned for extracting the gap from theChebyshev series if only the low energy spectrum is desired /H20850. The peaks give energies that are allowed and have signiﬁcantoscillator strengths. The transitions were determined by ob-serving the direction of the dipole applied on the molecule. Table Ishows mixed quality results. The TD-PM3 re- sults were much better than the band gap and in many casesbetter than TD-DFT with the 3-21G basis set. They weregenerally about 20% lower than the more accurate large-basis /H20849aug-cc-pvtz /H20850TD-DFT results as well as the experi- mental results. The main exception is C 2H4, where the true lowest state with signiﬁcant dipole strength is a transition to /H9266→3s transition, while TD-PM3 predicts a /H9266→/H9266/H11569as lower. Table IIcompares TD-PM3, TD-DFT, and PM3-CI for a series of aromatic rings. PM3-CI is accurate for the smalleraromatics, but for the ﬁxed number n=5 of conﬁgurations used here, it deteriorates, indicating that more conﬁgurationsare needed for larger systems. The TD-PM3 results improved/H20849relatively and absolutely /H20850as the molecules got bigger. IV. DISCUSSION AND CONCLUSIONS Our results indicate that a time-dependent application of a semiempirical method /H20849in this case PM3 but the results should be of general validity /H20850is useful for large molecules. The results, especially for larger molecules, were surpris-ingly accurate, especially considering that we did not reopti-mize the parameters. The PM3 parameters that were em-ployed have been optimized previously to yield accurateground-state properties, and this work shows that these same parameters lead to surprisingly accurate excitation energieswhen used in a TD-PM3 scheme. The timings on the method are interesting. The nonopti- mized TDPM3 code was about 200–150 times faster than thelarge-basis TDDFT code. PM3-CI with n=5 conﬁgurations was faster signiﬁcantly /H20849more than an order of magnitude /H20850 than TD-PM3, especially since at present TD-PM3 scaleslike the cube of the number of orbitals because of the matrixmultiplication /H20849/H20851F,H/H20852/H20850in the time evolution. TD-PM3 is clearly not a method for small molecules, but rather for large systems for two purposes: real time dynamics, or for spectralapplications, once the method is made numerically more ef-ﬁcient, especially by accounting for the sparsity in the appli-cation of FonH. Numerical efﬁciency and its improvements will be discussed in more details in future publications. Further improvements can still be made. PM3 equations and parameters for d-orbital atoms have already beendeveloped 32,33so this program can be revised to include d-orbital atoms. Another improvement is to modify the pa- rameters used in the PM3 program. Since the parametersused are based on the ground state of the molecule, in prin-ciple they could be modiﬁed to yield better spectra whileretaining reasonable accuracy for ground-state properties. Ina future publication we discuss the application of these con-cepts to more general quantities than absorption, as well asmore rapid extraction of the frequency information. In addition, the same concepts and methods implied here can be applied directly to other semiempirical methods suchas INDO/S /H20849Intermediate Neglect of Differential Overlap/ Screened Approximation /H20850, which has been popular for com- puting vertical excitation energies; future publications willexamine where TD-INDO/S will outperform TD-PM3. To conclude, our results show that a time-dependent ap- plication of a semiempirical method should be useful forlarge systems, where highly quantitative results are notneeded but accuracies of /H1101120% are desired. Further numeri- cal developments to improve the scaling should make themethod applicable for a range of large scale problems. ACKNOWLEDGMENTS We are grateful to Roi Baer for helpful conversations and for the referees for their useful comments. This materialTABLE II. Lowest excitation energies of aromatics obtained using TD-PM3 and various methods /H20849energies in eV; TD-DFT using B3LYP /H20850. Molecule Transition TD-PM3 TD-DFT 6-311G/H11569/H11569PM3-CI Band gap Expt.a Benzene 1E1u:/H9266→/H9266/H115695.56 7.34 6.25 10.12 6.9 Naphthalene 1B2u:/H9266→/H9266/H115693.45 4.41 4.09 8.34 4.45 2B3u:/H9266→/H9266/H115694.85 6.03 5.42 5.89 Anthracene 1B2u:/H9266→/H9266/H115692.95 3.24 3.51 7.18 3.31 1B3u:/H9266→/H9266/H115694.35 5.27 4.92 4.92 Tetracene 1B2u:/H9266→/H9266/H115692.55 2.47 3.10 6.39 2.63 2B3u:/H9266→/H9266/H115693.95 4.71 4.58 4.51 Pentacene 1B2u:/H9266→/H9266/H115692.25 2.17 2.95 5.82 2.12 2B3u:/H9266→/H9266/H115693.75 4.63 3.99 4.10 aReference 11.234106-5 Allows efﬁcient treatment of large molecules J. Chem. Phys. 132, 234106 /H208492010 /H20850 Downloaded 13 Jan 2013 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsis based upon work supported as part of the Molecularly Engineered Energy Materials, an Energy Frontier ResearchCenter funded by the U.S. Department of Energy, Ofﬁce ofScience, Ofﬁce of Basic Energy Sciences under Award No.DE-SC0001342. APPENDIX: FREQUENCY DEPENDENT EXPANSION COEFFICIENTS The calculation of the coefﬁcients in Eq. /H2084926/H20850is straight- forward and well known, and is presented here for complete-ness, since we use the modiﬁed Chebyshev polynomialsrather than the regular ones /H20851deﬁned as in Eq. /H2084920/H20850without thei’s/H20852. A general function of iL− /H9275is written as f/H20849iL−/H9275/H20850=/H20858 nTn/H20873L /H9004/H20874fn/H20849/H9275/H20850. /H20849A1/H20850 Using the deﬁnition of the modiﬁed Chebyshev operator, we have /H20885 02/H9266 Tn/H20849i−1cos/H20849/H9258/H20850/H20850Tm/H20849i−1cos/H20849/H9258/H20850/H20850d/H9258=2/H9266 2−/H9254n0/H9254nm,/H20849A2/H20850 so that fn/H20849/H9275/H20850=/H208492−/H9254n0/H20850 2/H9266/H20885 02/H9266 cos/H20849n/H9258/H20850f/H20849/H9004· cos/H9258−/H9275/H20850d/H9258 =R e/H208492−/H9254n0/H20850 2/H9266/H20885 02/H9266 ein/H9258f/H20849/H9004· cos/H9258−/H9275/H20850d/H9258, /H20849A3/H20850 where the last step is valid for real functions. Therefore, the coefﬁcients are easily obtained by a simple Fourier trans-form. 1J. J. P. Stewart, J. Comput. Chem. 10, 209 /H208491989 /H20850. 2M. J. S. Dewar, E. G. Zoebisch, E. F. Healy, and J. J. P. Stewart, J. Am. Chem. Soc. 107,3 9 0 2 /H208491985 /H20850. 3R. F. da Costa, F. da Paixao, and M. A. P. 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Neuhauser and R. Baer, J. Chem. Phys. 123, 204105 /H208492005 /H20850. 15R. Baer and D. Neuhauser, J. Chem. Phys. 121, 9803 /H208492004 /H20850. 16R. E. Stratmann, G. E. Scuseria, and M. J. Frisch, J. Chem. Phys. 109, 8218 /H208491998 /H20850. 17S. Hirata and M. Head-Gordon, Chem. Phys. Lett. 302,3 7 5 /H208491999 /H20850. 18P. J. Hay and D. C. Cartwright, Chem. Phys. Lett. 41,8 0 /H208491976 /H20850. 19M. J. S. Dewar and W. Thiel, J. Am. Chem. Soc. 99, 4899 /H208491977 /H20850. 20J. J. P. Stewart, J. Comput.-Aided Mol. Des. 4,1/H208491990 /H20850. 21M. J. S. Dewar and W. Thiel, Theor. Chim. Acta 46,8 9 /H208491977 /H20850. 22S. Hirata and M. Head-Gordon, Chem. Phys. Lett. 314,2 9 1 /H208491999 /H20850. 23R. F. Stewart, J. Chem. Phys. 52,4 3 1 /H208491970 /H20850. 24R. C. Bingham, M. J. S. Dewar, and D. H. Lo, J. Am. Chem. Soc. 97, 1285 /H208491975 /H20850. 25J. J. P. Stewart, J. Comput. Chem. 10, 221 /H208491989 /H20850. 26J. J. P. Stewart, J. Comput. 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1.58237.pdf	Laser cooling and trapping of neutral atoms Luis A. Orozco Citation: AIP Conference Proceedings 464, 67 (1999); doi: 10.1063/1.58237 View online: http://dx.doi.org/10.1063/1.58237 View Table of Contents: http://aip.scitation.org/toc/apc/464/1 Published by the American Institute of PhysicsLaser Cooling and Trapping of Neutral Atoms Luis A. Orozco Department of Physics and Astronomy, State University of New York at Stony Brook, Stony Brook, NY 1794-3800, United States Abstract. The forces felt by atoms when illuminated with resonant radiation can re- duce their velocity dispersion and confine them in a region of space for further probing and experimentation. The forces can be dissipative or conservative and allow manipu- lations of the external degrees of freedom of atoms and small neutral particles. Laser cooling and trapping is now an important tool for man5" spectroscopic studies. It en- hances the density of atoms in phase space by many orders of magnitude reducing the need of large samples. These lecture notes review the fundamental principles of the field and show some of the applications to the study of the spectroscopy of radioactive atoms. I INTRODUCTION These notes are based on the lectures I gave at the Escuela Latinoamericana de Fisica in M4xico City during the summer of 1998. The purpose of the course was to familiarize the participants with the exciting new developments in atomic physics during the last decade. We have gained unprecedented abilities to control the positions and velocities of neutral atoms, that have opened new possibilities in the investigation of' their spectroscopy and collective behavior. There are excellent reviews and summer school proceedings in the literature [1-4]. In these notes I treat only very general aspects of laser cooling and trapping without the careful detail given in the above reviews. The covered material follows the presentation of Ref. [5]. The aim is to develop an intuitive understanding of the principles and the basic mechanisms for laser cooling and trapping of neutral atoms. Last century the electromagnetic theory of Maxwell gave a quantitative expla- nation to the pressure associated with light. This idea was not new, it had been proposed at least in the XVII century, to explain why comet tails point away from the sun. At the beginning of this century Einstein studied the thermodvnamics of emission and absorption of radiation in his paper on blackbody radiation [6]. He remarked on the transfer of momentum in spontaneous emission that 'the smallness CP464, Latin-A merican School of Physics XXXI ELAF edited by S. Hacyan, R. Jhuregui-Renaud, and R. L6pez-Pefia ~~ 1999 The American Institute of Physics 1-56396-856-8/99/$15.00 67 of the impulses transmitted by the radiation field implies that these can almost al- ways be neglected in practice'. At that time, given the available light sources, any mechanical effects were extremely difficult to detect. Frisch observed the deflection of an atomic beam of Na by resonant light from a Na lamp in 1933 [7]. Ideas about using light to manipulate atoms and particles continued to appear in the literature and the invention of the laser helped trigger some of them. H/insch and Schawlow [8] and Wineland and Dehmelt [9] realized that high brightness sources can exert a substantial force on atoms or ions, potentially cooling their velocity distributions. The advent of tunable lasers during the 1970s with very narrow linewidths made pi- oneering experiments possible. Since then a long list of people have contributed to advances in the development of laser cooling and trapping. Among the spectacular achievements facilitated by the new techniques is the Bose-Einstein condensation of a dilute gas of alkali atoms, (see the lectures of S. Rolston). Finally, the field of laser cooling and trapping received the 1997 Physics Nobel Prize in the persons of Steve Chu, Claude Cohen-Tannoudji and William Phillips [10]. It is possible to say that laser trapping and cooling is now part of the cannon of physics. In the course of this lectures we will try to understand how to cool and trap neutral atoms using forces derived from the interaction of light with atoms. Sec- tion II introduces the light forces. Section III shows the velocity dependent force and the associated cooling mechanisms. The position dependent force is discussed in section IV. Section V shows how the forces combine to form an optical trap. Finally, in section VII have included some examples drawn from the work with radioactive neutral atoms where I have been involved. II THE LIGHT FORCES The origin of the light force is the momentum transferred when an atom absorbs a photon from a laser beam. The momentum of the atom changes by hk, where k is the wave vector of the incoming photon. After the emission of a photon by an atom the atom recoils. The associated recoil velocity Vrec and recoil energy Erec for an atom of mass M are: hk Vrec M' (1) h21kl 2 J~rec ~- 2A//" (2) II. A Spontaneous emission force If the excitation is followed by spontaneous emission, the emission can be in any direction, but because the electromagnetic interaction preserves parity, the emission will be in a symmetric pattern with respect to the incoming photon. In this case the recoil momentum summed over many absorption and emission cycles will average 68 of the two-level transition (3A2/27). The rate of fluorescence (see Eq. (6)) depends on the detuning A between the atom and the laser. At low intensities the scattering rate is proportional to the saturation parameter, but as the intensity grows it shows power broadening and the rate saturates at 1/2F. The FWHM of the Lorentzian goes from F at low intensities S0 << 1 to Fv/1 + So for S0 > 1. Power broadening can be thought as arising from the absorption- stimulated emission cycles that do not contribute to the force because the emission is into the same laser beam. The on-resonance atoms are already saturated and it is only those off resonance that can contribute and broaden the width. The force is small but a two-level atom it returns to the ground state after emission of a photon and can be re-excited by the same laser beam. When such a transition exists in real atoms it is called a cycling transition. Alkali atoms with nuclear spin I and total angular momentum F have the transition from the S~/2 ground state with F = I + 1/2 to the Pa/2 state with F = I + 3/2 that satisfies the cycling condition. The excited state can not decay to the other hyperfine level (F = I - 1/2) of the ground state because of the AF = 0, ±1 selection rule. This transition in the D2 line is commonly used for trapping alkali atoms. The saturation intensities of their cycling transitions are in the range of 1 mW/em 2 < I~t < 10 mW/cm 2. II. B Stimulated emission force If the absorption of a photon is followed by stimulated emismon into the same laser beam, the outgoing photon will again carry away hk, so there is no momentum transferred. However, if the emission is into another laser beam, there is a redistri- bution of laser photons causing a force proportional to the difference between the two k vectors Ak = kl - k=. The absorption an emission are correlated events and they are coherent scattering of photons. This redistributon of momentum is what happens in an optical lens and a positive lens will be drawn towards regions of high intensity as a consequence of the third law of Newton. To calculate the index of refraction of an atom it is necessary to add the amplitude of the incident light field with the dipole field generated by the driven atomic electrons. An optical field E of the light induces a dipole moment d on the atom. Considering the electron as a harmonic oscillator, the induced dipole moment can be in phase or out of phase depending on the detuning of the driving frequency with respect to resonance. When it is in phase, the interaction energy between the dipole and the field U = -d-E is lower in high field regions. When it is out of phase U increases with E and a force will eject the atom out of the field. On resonance the oscillator is orthogonal to E and there is no force. If the atom is illuminated only with a plane wave the stimulated force will be zero as all the k vectors are the same. A force from stimulated emission needs a gradient in tile intensity of the light such that the k vectors point in different ways. This force is sometimes called the dipole or stimulated force. A force will act on an 69 to zero. The atom then gains momentum in the direction of the wave vector of the incoming laser beam. The resulting force is sometimes called Doppler, radiation pressure, scattering, or the spontaneous force. The variance of the momentum transferred does not vanish, and the atom performs a random walk in momentum space as it emits spontaneously. These fluctuations limit the lowest temperature achievable when the laser beam is present. F = Fabs + Fern, F = < Fabs > + < Fern > ~-5Fem <F> =/~phk, (3) (4) (5) where F is the force on the atom,/~sp is the rate of fluorescence scattering in cycles per second, and 5F represents the random fluctuations from the recoiling atoms. The repeated transfer of momentum from a light beam to the atom by absorption and spontaneous emission provides the spontaneous light force. The mean number of fluorescence cycles per second from a two level atom illu- minated by a laser beam near or at resonance with the transition is equal to the population in the excited state times the Einstein A coefficient F. To precisely cal- culate the population it is necessary to include off diagonal elements in the density matrix and solve in steady state the optical Bloch equations (see for example [11]). Here we present the result without deriving it. The fluorescence depends on the amount of power available for the excitation (governed by the saturation parameter So) and the full width at half maximum (FWHM) F of the Lorentzian lineshape. The radiative lifetime of the transition T ----- 1/ F is the inverse of the Einstein A-coefficient. The fluorescent rate is: r So R~(A) = -~ 1 + So + (2A/r) 2' (6) where A is the laser detuning from resonance, A ~ ~dlase r -- ~dato m, (7) and the on-resonance saturation parameter S0 = Iexp/Isat is the ratio between the available intensity Iexp and the saturation intensity Is~t. At So = 1 and on resonance the atom scatters at half of the maximum possible rate. There are different definitions of So in the literature depending on particular definitions of I~at and the reader has to pay attention to the particular one used. Here we follow the work of Citron et al. [12]. hTr cF Isat- 313 • (8) With this definition, an intensity of I~t corresponds to providing the energy of one photon (hw) every two lifetimes (2/F) over the area of the radiative cross section 70 induced dipole if there is a gradient in the intensity, it can be attractive or repulsive depending on the drive detuning with respect to resonance. Any material with an index of refraction feels a force in the presence of a gradient of the intensity,. The dipole force acts cells, organelles and even DNA, providing 'optical tweezers' for their manipulation. (See for example the Nobel lecture of S. Chu [10]). III VELOCITY DEPENDENT FORCE The spontaneous force Fspon t is a velocity dependent force because the resonance condition of an atom depends on its velocity v through the Doppler shift k • v. F So Fspon t = ttk ~ 1 + S0 + (2(/_.k - k. v)/F) ~" (9) This force saturates at hkF/2 and is limited by the spontaneous decay time of the atomic level. The force felt by an atom when the intensities are large (So ~ 1) are more complicated since stimulated emission is significant. We limit the discussion to the case where those processes are negligible. The velocity range of the force is significant, for atoms with velocity such that their Doppler detuning keeps them within one linewidth of the Lorentzian of Eq. (9). See Fig. 1. This condition states that: ]A - k. vl _< F,~7+ S0. (10) III. A Deceleration of an atomic beam The maxmmm acceleration of a sodium atom interacting with resonant laser light in the D2 cycling transition shows that light can decelerate an atom in a very short time. hk 1 amax -- .~-f 27' (11) Vrec z 57- 3 × lO-2m/s 2 2x 16x 10-9s ' 106rn/s 2, 1059. The thermal velocities of atomic beams are in the order of a thousand meters per second, so the stopping time is about one millisecond at a .... ~x, stopping in about one meter. However, these estimates do not consider that the force will be different for atoms with changing velocities through the Doppler effect. The spontaneous 71 S t Absorption / ',. / \ / \ ......... ¸Ir/j 1 w n -kv w I wj +kv ' ' ~ I .... F -I 0 ' ' ' ' I .... 1 D /G w-kv w+kv + Spontaneous emission ~.~i i:.~ 1,- :4 II ql N~ fore = -J FIGURE 1. Velocity dependent force force can act on atoms that have a velocity range where the force is significant: A Doppler shift of the order of the linewidth of the transition. F VDop k' (12) where k = Ik], and for Na VDop ~ 6 m/s which is two orders of magnitude smaller than the thermal velocity and three orders of magnitude higher than the recoil velocity. A laser beam red detuned with respect of the resonant transition and counter- propagating with a beam of atoms at velocity v can decelerate a velocity class of atoms with a width of VDop and pile them at a lower velocity. To compensate for the resonant changing transition it is necessary to either tune the energy level of the atom in space or to change the frequency of the laser in time to keep it resonant with a group of atoms while they decelerate. Real atoms have more complications, cycling transitions are not perfect. For example, there is hyperfine structure in alkali atoms and some of the off resonant excitation can optically pump the atom into the non-cycling ground state (F=I for Na). Then the atom no longer feels the force. The methods developed for deceleration maintain the atom in a cycling tran- sition. They use the selection rules from the polarization of the light in the presence of a magnetic field, take advantage of the Clebsh-Gordan coefficients between the levels, and sometimes require excitation at other frequencies. 72 III. B Zeeman slowing One approach to slowing atoms uses the Zeeman effect in a spatially varying magnetic field to tune the atomic energy levels with the changing velocity. The magnetic field is shaped to optimize the match between velocity and Zeeman de- tuning and keep a strong scattering of photons along the solenoid [13]. The method works if the g-factors of the levels that scale the Zeeman shifts of the ground and excited states are different so that their resonant frequency shifts. The largest ground state m sub-level in the D2 line of Na shifts 1.4 MHz/G while the excited state shifts by 2.8 MHz/G. As a result of their difference the magnetic field can shift the transition energy and can compensate for the Doppler shift along the path of a moving atom. Assuming that the atoms decelerate with a constant acceleration a from an initial velocity v0, the position dependent velocity v(z) is: v(z) = V/~0 z - 2az. (13) We take the changing Doppler shift k. v(z) equal it to the Zeeman shift fi- B(z), where fi is the magnetic moment of the transition, to find the shape of the com- pensating magnetic field. B(z) = Bo~/1 z, (14) V Zo ICVo B0 - , (15) (16) z0 -- 2a the field B0 induces a Zeeman shift equal to the Doppler shift of an atom having velocity v0. A tapered solenoid produces a field of such spatial dependence. In certain applications it may be necessary to add a uniform bias field Bb to keep the field high enough to avoid optical pumping [13]. The atomic beam comes from a thermal source with a dispersion of velocities comparable to its mean velocity. ]t enters a tapered solenoid where the field is higher at the oven side. The laser is resonant with atoms of a given velocity v0, usually around the mean of the thermal distribution, but this transition is modified by the Zeeman shift at the entrance and by the Doppler shift. These atoms at v0 decelerate. As their velocity changes, their Doppler shift changes but it is compensated by a different Zeeman shift. The initially fast atoms continue to be on resonance. As they decelerate and move downstream in the magnet more atoms come on resonance and start feeling the light force of the opposing laser beam. At the end of the tapered solenoid all the atoms with velocities smaller than v0 are decelerated to a final velocity that depends on the details of the solenoid and the laser detuning. The result is a significant enhancement of density in phase space; 73 TABLE 1. Trapping and cooling parameters for alkali atoms from a source at 1000 K. Atom A AD2 TD2 TDop. lzeeman [rim] [~ecl [,KI [cm] Na 23 589 16.2 235 40 K 39 766 26.3 145 84 Rb 87 780 26.2 145 85 Cs 133 852 30.4 125 108 Fr 210 718 21.0 181 63 despite the fact that the diffusion process associated with the cooling increases significantly the divergence in the transverse direction. Table 1 gives lengths for Zeeman slowers, required to bring different alkali atom with velocities Vtherma I = ~/2kBT/M to halt by driving it fully saturated transition. a Oil a III. C Frequency chirping Another method to slow atoms in a beam is to chirp the frequency of the laser maintaining the resonant interaction with a group of atoms and leaving the oth- ers without deceleration [14]. The instantaneous acceleration is negative and the varying laser detuning compensates for the changing Doppler shift. = + (17) where A'(t) is the time varying laser detuning of the laser frequency. In the decel- eration frame the force on an atom at velocity v is: hkr [ -So So] F(v) = ~ 1 + So + 2(a+kv)~r + 1 + So + (_~)2 ' (18) expanding near v = 0 2 /r ] F(v) = 2hk2So [1 + So + (2A/F)2] 2j v. (19) The force is proportional to the velocity and the proportionality constant is a friction coefficient. The method is self correcting and works in batches of atoms. All velocities near v(0) damp towards v(t). Any velocities not initially near v(0) become close to v(t) at a later time. Changes in the saturation parameter from the attenuation of the laser beam as it propagates through the beam can be com- pensated. The chirp rate of the laser frequency to obtain deceleration (A < 0) is 74 dA r - ka, (20) dt F(v) a - (21) M A chirp rate of 780 MHz/ms can stop an initially thermal Na atom. III. D Optical molasses ¥'elocity dependent forces are necessary to cool an atom and reduce its velocity. They do not confine the atom, but they provide what has been termed 'optical molasses'. The damping felt by the atoms is substantial and the study of the cooling mechanisms has been discussed in the literature. (See for example the review paper of Metcalf and van der Straten [1]). An atom subject to two laser beams in opposite directions will feel a force F(v) coming from its interaction with both beams. If the counterpropagating laser beams are detuned to the red of the zero velocity atomic resonance, a moving atom will see the light of the opposing beam blue shifted in its rest frame (See Fig. 1). The beam in the same direction as the atom will be further red shifted in its rest frame. Considering only one dimension and So << 1, the force opposing the motion will always be larger than the force in the direction of the motion, and this leads to Doppler cooling. F(v) = Fspont(k) + Fspont(-k). (22) The sum of the two forces, with the semiclassical assumption that the recoil shift is negligible kVrec <~ I ~ gives in the limit where v 4 << (F/k) 4, 8hk2 SoA F(v) ~ r(1 + x0 + (2A/17)2) 2v' (23) S(v) ~ ~v. (_94) The force is proportional to the velocity of the atom through the friction coef- ficient a and depends on the sign of the laser detuning A. Figure 2 shows the Doppler cooling force in one dimension as a function of velocity and detuning for the D2 line of francium. This force is limited by the spontaneous decay time of the atomic level. An estimate for the maximum velocity an atom can have and still feel the light force is when the Doppler shift is equal to the laser detuning from the transition: vma~ ~ A/k. Only a very small fraction of the thermal distribution of atoms at room temperature can be cooled in optical molasses. III. E Cooling Limits Optical molasses provides a velocity dependent or viscous force. In the three- dimensional configuration atoms get slowed wherever they are in the region defined 75 0 4J ~J o o < 80000 D=-G - D=-2~ D=-G/2 i oooo i [ D=~/!0 I i -40004 -80000 ' J ..... :_ , , i _ ..... ~ ..... -25 -20 -15 -10 -5 0 5 10 15 20 25 Velocity [m/s] FIGURE 2. Doppler cooling in one-dimensional optical molasses. The numerical values are for the francium D2 line at So = 1. (From Ref. [21]). by the overlap of the six orthogonal beams. Large laser beams will increase the total number of cooled atoms, but the atomic density remains constant. Because of the variance of the momentum coming from the repeated random spontaneous emission, atoms can diffuse out of the molasses region because this is not a trap. The competition between the cooling process and the diffusion of the momentum reaches an equilibrium that determines the lowest temperature of the atoms [1]. III. E. i The Doppler Cooling Limit The atomic momentum and energy change by hk and Ere¢ after each interaction with the laser beam. Following the one dimensional treatment of the force above, the change of the energy has an associated change in the frequency of the transition such that Erec -- h~vrec. Then the average frequencies of absorption and emission are: (X)ab s = (Matom -- (.drec, (25) ~.dem = 03atom -~ (,Ore c. (26) The light field losses every cycle an average energy of: ~(aJabs -- 02em) = 2hCdrec, and the power lost by the laser field becomes atomic kinetic energy. heating should equal the rate of cooling in thermal equilibrium and ~Jrec F.v - (28) 1/ Rsc' Oz~) 2 _-- ~Orec 1/Rs (29) (27) The rate of 76 The cooling force in the optical molasses is proportional to the velocity through the friction coefficient c~. The temperature associated with the kinetic energy is: 2] 2A/F " (30) This expression becomes independent of So in the limit of low intensity and has a minimum for A = -I2/2. This temperature is called the Doppler cooling limit ZDoppler. hF '~Doppler -- 2]~B" (31) The lowest temperature in optical molasses is independent of the optical wave- length, atomic mass, and, in the limit of low intensity, also of taser intensity. The only atomic parameter that enters is the rate of spontaneous emission F. The value for Na is 240 pK which corresponds to an average velocity of 30 cm/s four orders or magnitude smaller than the typical thermal velocities produced out of an effusive oven (See Table 1). III. E. 2 Beyond Doppler cooling In 1988 the NIST group [15] discovered that the temperature of sodium atoms in optical molasses was a factor of six lower than the Doppler cooling limit. The quantitative understanding of this result requires the inclusion of all the energy levels that are present in an atom, the effects of the polarization of the different laser beams, and the non-adiabatic response of a moving atom to the light field [3,16,17]. The atom has a finite response time r~t to adjust its internal state cT to a new environment, a depends oil the position z and velocity v of the atom and in general lags behind the steady state of an atom which would be at rest in z d (32) The non-adiabaticity parameter in the problem is: v~,~ (33) =kv~t. (34) The frictional force is going to be linear in t, as long as e < 1. The equilibrium temperature of the system is: h, ksT ~ --. (35) Ti~t 77 For a two level atom there is a single internal time ~-i~t = l/F, the radiative lifetime of the excited state. The non-adiabaticity parameter is the ratio of the Doppler shift divided by the natural width of the transition. The temperature reachable is of the order of hF. This result is in agreement with the TDoppler cal- culated based on the change in the energy of the laser field from Eq. (31). The Doppler limit is independent of the intensity. However; a multilevel atom, for example an alkali, has hyperfine splitting and Zeeman sublevels. There is a new internal time: The optical pumping time between ground state sublevels. Let F' be the absorption rate from Ig >, this number depends on the intensity and will give a different value for the lowest temperature than Doppler cooling. At low intensities So <:< 1 and F ~ << F. The associated d, which is the ratio of the Doppler shift to the optical pumping rate, will be very large. III. E. 3 Sisyphus cooling When the intensity and detuning of the laser beams are significant, a different mechanism can cool an atom. It requires an AC Stark Shift of the atomic ground state. The dressed atom formalism of the atom + photon interaction shows (see for example the contribution of Cohen-Tannoudji in Ref. [3]) that the light shift for the ground state ~ in the presence of a field with Rabi frequency ~ much smaller than the absolute value of the detuning between the laser and the atomic transition A is: ~2 5' (36) =~-~- The light shifts are proportional to the intensity (~2), the sign depends on the detuning A of the laser with respect to the atomic transition. If an atom is illu- minated by two detuned laser beams counterpropagating but one with horizontal polarization and the other with vertical polarization, the atom will feel a very different force from the spontaneous force. The resulting field has polarization gra- dients. The field has negative circular helicity in one point in space, a distance A/4 away has positive helicity, and is elliptically polarized in between with linearly polarized light exactly at A/8 of the point with purely circular light. (See the No- bel lecture of Cohen-Tannoudji [10]). For a case where the ground state has two sublevels Jg = 1/2 and the excited state four J~ = 3/2 the optical pumping rates are the largest from the highest sublevel of the ground state to the lowest sublevel of the ground state. If w-i~t ~ ~/2~ the atom can climb a potential hill and reach the top before being pumped back to a valley. The atom is always climbing in analogy to the Greek Sisyphus. There is a decrease of the kinetic energy and the dissipation of potential energy is by spontaneous anti-Stokes Raman photons. The equilibrium temperature comes when the atoms gets trapped in one of the poten- tial wells formed by the position dependent AC Stark Shift, then the equilibrium temperature is of the order of the well depth: 78 h~ 2 kBT ~ -IA I , (37) further cooling in the well is possible using adiabatic expansion by lowering the laser intensity at a rate slow compared to the frequency of oscillation of the trapped atom in the potential well. Another way to understand Sisyphus cooling is the following (See Ref. [4] and the contribution of S. Chu in [2]). The induced electric dipole d of an atom in the presence of an off-resonant field minimizes its energy when it aligns with the optical electric field E. If an atom at a point of linearly polarized light moves a distance ,~/8 the polarization is now circular because of the way the opposite polarizations add at each point in space. The atom can only follow a change in field alignment with a finite time delay characteristic of the damping process. This process changes kinetic energy into potential energy which is lost from damping as the dipole relaxes to the new state of polarization. There are other configurations that produce Sisyphus cooling, for example two counterpropagating beams with cr + and or- polarizations. The polarization of the field is always linear but it changes directions continuously over one wavelength. The atomic dipole sees a change in tile direction it should oscillate. All the mechanisms described before rely on absorption and spontaneous emission of photons. A natural limit, to the lowest achievable temperature is given by the recoil energy kbTrec/2 = Erec. Finding a way to ~protect' the atoms from light can bypass this limit. Two laser cooling methods are known to reduce the temperature of the atoms beyond Tree: velocity selective coherent population trapping (VSCPT) and Raman cooling. VSCPT prepares the atom in a 'dark state' that does not absorb any light eliminating the possibility of recoil. This state is stationary and an atom that diffuses into it will be trapped (See [18] and the Nobel lecture of Cohen-Tannoudji [10]). In Raman cooling, a series of light pulses, with well defined frequency and du- ration, produces an excitation profile that constitutes a 'trap' in velocity space for the atom. (See S. Chu in [2]). IV POSITION DEPENDENT FORCE The position dependent force is necessary to construct a trap but is more subtle than the velocity dependent force. A series of no trapping theorems constrain the distribution of electric and magnetic fields for capturing neutral atoms. (See the contribution by S. Chu in Ref. [2]). Earnshaw theorem states that is is impossible to arrange any set of static charges to generate a point of stable equilibrium in a charge-free region. The electrostatic potential 0 satisfies V24~ = 0, then 6(z, y, z) at any point is the average of 0 on the surface of the sphere centered at (~', g, z). There can not be an extremum of 0 and since the electrostatic energy is proportional to the potential, there is no minimum 79 of the energy. Similarly: V. E = 0 and all the lines of force that go in are balanced by lines that go out of it. The optical Earnshaw theorem uses the Poynting vector of the field S and it applies to the scattering force. The light flux can not point inward everywhere, so a light trap is unstable (V - S = 0). A solution is not to use static light beams, but alternate them in time to gen- erate a trap following the ideas of the Paul trap. Another way to circumvent the optical Earnshaw theorem is to exploit the internal structure of the atoms. The effective atomic polarizability P can be position dependent through the presence of an external magnetic field B resulting in a negative divergence of the spontaneous light force, since the force is proportional to P. J. Dalibard proposed a solution to the neutral atom trapping using the sponta- neous light force. His idea became the basis of the Magneto-Optical Trap (MOT). The solution of Dalibard was to add a spatially varying magnetic field, so that the shifts in the energy levels make the light force dependent on the position. Soon afterwards this scheme was generalized to three dimensions and it was successfully demonstrated with Na atoms by Raab et al. [19]. Despite many new developments the MOT remains the workhorse of laser trapping due to its robustness, large vol- ume and capture range. The next section discusses this trap in more detail since this type has been used in the successful trapping of radioactive atoms [5]. V OPTICAL TRAPS V. A The Magneto-Optical Trap This section presents a simplified one-dimensional model to explain the trapping scheme in a J = 0 -+ J = 1 transition. Figure 3 shows a configuration similar to optical molasses. Two counterpropa- gating, circularly polarized beams of equal helicity are detuned by A to the red of the transition. In addition there is a magnetic field gradient, splitting the J = 1 excited state into three magnetic sublevels. If an atom is located to the left of the center, defined by the zero of the magnetic field, its J = 0 -+ J = 1, m = 1 transition is closer to the laser frequency than the transitions to the other m-levels. However, Am = +1 transitions are driven by a + light. Atoms on the left are more in resonance with the beam coming from the left, pushing them towards the center. The same argument holds for atoms on the right side. This provides a position dependent force. The Doppler-cooling mechanism is also still valid, providing the velocity dependent force. Writing the Zeeman shift as Sx, where x is the coordinate with respect to the center, the total force is: hkF [ So FMOT=~ l+So+(2(A-~)/r) 2 So ] (38) 1 + So + (2(A + ~)/F) 2) where 80 1 - JJ 0 1 -1 v v B i, 0 x FIGURE 3. Simple 1-D model of the b1OT. (From Ref [21]). = kv + flz. (39) For small detunings, expansion of the fractions in the same way as in Eq. (24), shows the fbrce proportional to ( (see W.D. Phillips in, [2]). In the small-field, low-velocity limit the system behaves as a damped harmonic oscillator subject to the force: and with 4hkSo(2A/F)(kv + fix) F(v, z) = [1 + (2A/F)2] 2 ' (40) "2 [1: q- 7~ -F ~trapX = O, (41) 4hk2&(2A/r) (42) ' = M[1 + (2~,/r)212' 4hkgSo(2~/r) (43) 2 wt'~P = M[1 + (2.:X/F)2] 2' The motion of the atom in the harmonic region of the trap is overdamped since 72/4~Z~,a; > 1. This same ratio in terms of the recoil energy and the Zeeman shift over one waveleght is: ~2 /rErec 4~'~,; 4Ah3 (44) 81 (~+l I FIGURE 4. Laser beams and coils for a MOT. (From Ref. [22]). A trap with a magnetic field gradient that produces a Zeeman shift of ~ = 14 MHz/cm has a trapping frequency of a few kilohertz and an Eq. (44) of the order of 10. The real world requires three-dimensional trapping, and in alkalis a J = 0 -~ J = 1 transition is hard to find. For an alkali atom with non-vanishing nuclear spin the ground state (nS1/2) splits into two levels. The transition to the first P3/2 excited state has four levels (for J < I), yet the trap works quite well under these conditions. Ideally, the transition from the upper ground state to the highest excited state F- level is cycling, and one can almost ignore the other states. Due to finite linewidths, off resonance excitation, and other energy levels the cycling is not perfect. An atom can get out of the cycling transition and an extra beam, a weak 'repump' laser, can transfer atoms from the 'dark' lower ground state to the upper one. A magnetic quadrupole field, as produced by circular coils in the anti-Helmholtz configuration, provides a suitable field gradient in all three dimensions. The exact shape of the field is not very critical, and the separation between the two coils does not have to be equal to the radius. Typical gradients are 10 G/cm. A large variety of optical configurations are available for the MOT. The main condition is to cover a closed volume with areas normal to the k vectors of the laser beams with the appropriate polarized light. (See Fig. 4). The realization with three retro-reflected beams in orthogonal directions requires quarter-wave plates before 82 7000 t "~ 6000 *~ v s00o t 4000" 3000 e000 o~~ ~+~gl .J Position (ram) 0 0 FIGURE 5. Two-dimensional CCD image of the fluorescence from francium atoms trapped in a MOT. (From Ref. [23]). entering the interaction region. In order to have tile appropriate polarization on the retro-reflected beam the phase has to advance half a wavelength. The usual arrangement is to place a quarter wave plate in front of a plane mirror, but two reflections can also provide the same phase shift [20]. The intensity of the laser beams should provide a saturation parameter So ~ 1. The MOT can work with significantly less intensity but it becomes more sensitive to alignment. In general the MOT is averv forgiving trap as far as polarization and intensities. The retro-reflecting technique for traps, despite the scatter losses in the windows and the beam divergence as it propagates, works very well. The well depth of a MOT is set bv the maximum capture velocity Vm~. For alkali atoms and & ~ 2F it is close to 1 K. The background pressure around the MOT limits its lifetime and consequently the maximum number of atoms in steady state. A pressure of I x 10 .8 Torr produces a trap lifetime of the order of 1 s. The characteristic size x0 of the trapping volume is set by tile gradient and the detuning of the specific realization of a MOT: x0 = &/~q. Xo is about 1 cm and to obtain larger volumes larger laser beams are required. The captured atoms concentrate in a region much smaller than the trapping volume. The size of the fluorescing ball of less than 10 6 captured atoms is smaller than 1 mm in diameter. It depends on the temperature and is related to the laser beams shape, magnetic environment, and polarization. The shape of tile fluorescence when integrated in a charge couple device (CCD) camera is usually Gaussian (see Fig. 5). If the alignment of tile laser beams is not good there can be a torque impressed into the trap and satellites can form. Fringes in the beams can also generate 83 satellites. As the number of atoms increases there is a limit to the size of the trap. A similar effect to space charge appears. The optical density is thick enough to create an imbalance in the two counterpropagating beams; also the atoms can absorb spontaneously emitted light that is not red-detuned from neighboring atoms. The trap is no longer optically transparent with an extra internal radiation pressure that may eject the atoms out of the trap. To increase the density and the number of atoms beyond the point where the repelling force turns on, Ketterle et al. [24] developed the dark MOT. The repump- ing beams are blocked from the central region of the trap. The trap maintains the atoms in a non-cycling state and only repumps them to the cycling transition when they stray to the edge of the trapping volume. This approach works with alkali atoms since the ground state hyperfine splitting already requires a repumping laser. The first experiments with a MOT by Raab et al. [19] reported the capture of atoms from the residual background gas in the vacuum chamber without need of deceleration. In 1990 Monroe et al. [25] showed trapping in a glass cell from the residual vapor pressure of a Cs metal reservoir. If the vapor pressure of an element is sufficiently high, a MOT inside a cell filled with a vapor continuously captures atoms from the low-velocity tail of the Maxwell-Boltzmann distribution. The remaining atoms thermalize during wall collisions and form a new Maxwell- Boltzmann distribution. From this the MOT can again capture the low velocity atoms. The trapping efficiency depends on the number of wall contacts that an atom can make before leaving the system. Since alkali atoms tend to chemisorb in the glass walls, special coatings can prevent the loss of an atom [26]. If the wall is coated, the atom physi-sorbs for a short time, thermalizes and then is free to again cross the capture region and fall into the trap. The capture range of the MOT is enhanced with the help of large and intense laser beams. Gibble et aI. [27] reported that for their large trap they captured atoms with initial velocities below about 18 % of the average thermal velocity at room temperature. However, the fraction of the Maxwell-Boltzmann distribution of atom velocities below the capture velocity of the trap is too small to capture a significant fraction of scarce radioactive atoms on a single pass through the cell. Wall collisions are critical to provide multiple opportunities for capture in the vapor cell technique. On the one hand they provide the thermalization process, but they also increase the possibility of losing the atom by chemical adsorption onto the wall. No significant vapor pressure of stable alkali atoms normally builds up unless the walls of the glass cell are coated by a mono-layer of the atom to be trapped. For most radioactive samples this is impossible, and also not desired since that will create a source of background for the study of the decay products. An alternative is to coat the cell with a special non-stick coating. The coatings are in general silanes and have been extensively studied for optical pumping applications of alkali atoms. Collisions with the bare glass walls destroy the atomic polarization and the coatings can provide a 'soft surface' for reflection. The Stony Brook group uses one commercially identified by the name of Dryfilm (a mixture of dichlorodimethylsi- 84 lane and methyltrichlorosilane). The coating procedure follows the techniques of Swenson et al. [26]. The choice of a particular coating depends on many issues. For example: The difficulties in the application of the coating to the surface, how well the coating withstands high temperatures present nearby in the experimental apparatus. The coating of choice constrains the attainable background pressure in the cell and the geometry of the vacuum container. Nevertheless the vapor cell is appealingly simple. As long as a coating is known to work for a stable alkali it seems to work for the radioactive ones. The Colorado group has studied different coatings extensively [28], and have developed curing procedures to optimize the performance of the coatings. The glass cell method relies on the non-stick coatings and works well for ra- dioactive alkali atoms, but tbr other radioactive elements it may not be so easily implemented and the Zeeman slower could prove more effective to load atoms into a MOT. The group of the University of Colorado has published a resource letter on laser trapping and cooling [29]. They also published a detailed explanation, including electronic diagrams, on how to build a glass cell MOT for Rb or Cs using laser diodes [30]. V. B The dipole force trap An electric or magnetic dipole in an inhomogeneous electric or magnetic field feels an attractive or repulsive force depending on the specific conditions. A strong laser field can induce an electric dipole in an atom. In 1968 Letokhov [31]proposed laser traps based on the interaction of this induced electric dipole moment with the laser field. Later, Ashkin [32] proposed a trap that combined this dipole force and the scattering force. The first laser trap for neutral atoms was of this type [33]. The trap depth is proportional to the laser intensity divided by the detuning hft2/tAt. In order to minimize heating from spontaneous emission, the frequency of the in- tense laser is tuned hundreds of thousands of linewidths away from resonance. The heating is greatly reduced since the emission rate is proportional to the laser inten- sity divided by the square of the laser detuning. The off-resonance nature of the trap requires very intense beams with an extremely tight focus, and is often referred to as a Far Off Resonance Trap (FORT). A single laser red-detuned tightly focused has a gradient large enough to capture atoms from a MOT. The well depth is very small, fractions of a milliKelvin, depending oil precooled atoms and very good vac- uum for an extended residence in the trap. The atoms reside in a conservative trap and can cool down further by other mechanisms like evaporative cooling [34]. This kind of trap has found applications in the manipulation of extended objects as a form of optical tweezers. 85 V. C Other traps and further manipulation Although the MOT is a proven trap for radioactive atoms, it may not be the ideal environment for some of the experiments now planned. The atoms are not polarized because there are all helicities present in the laser field, and the magnetic field is inhomogeneous. There have been a series of traps developed in conjunction with the pursuit of Bose Einstein condensation (BEC) [35-38]. that may have application in the field of radioactive atom trapping. In this quest for even higher phase space densities, new techniques for transport and manipulation of cold atoms have also appeared. V. C. 1 Cold atom manipulation To move the accumulated atoms in a MOT to a different environment requires some care. Simply turning the trapping and cooling fields off will cause the atoms to fall ballistically. The trajectories out of the trap will map out the original velocity distribution of the captured atoms, dispersing the atoms significantly as they fall. An auxiliary laser beam can push the atoms in one direction, but it has a limited interaction range since the atoms accelerate until they are shifted out of resonance by their Doppler shift. The acceleration is in only one direction and there is still ballistic expansion of the cold atoms. Gibble et al. [39] created a moving molasses with the six beams of the MOT. By appropriate shifting of the frequencies of the beams, the atoms accelerate in the 111 direction (along the diagonal of the cube formed by the beams), but they are kept cold by the continuous interaction with the six beams. VI COOLING AND TRAPPING OF FR Francium is the heaviest of the alkali atoms and has no stable isotopes. It occurs naturally from the c~ decay of actinium or artificially from fusion or spallation nuclear reactions in an accelerator. Its longest lived isotope has a half-life of 22 minutes. Previously, experiments to study the atomic structure of francium were possible only with the very high fluxes available at a few facilities in the world [40], or by use of natural sources [41]. Because of its large number of constituent particles, electron correlations and relativistic effects are important, but its structure is calculable with many-body perturbation theory (MBPT). Its more than two hundred nucleons and simple atomic structure make it an attractive candidate for a future atomic parity non- conservation (PNC) experiment. (See Ref. [42] for the most recent results in Cs). The PNC effect is predicted to be 18 times larger in Fr than Cs [43]. The present francium spectroscopy serves to test the theoretical calculations in a heavier alkali. This ensures that the Cs structure, calculated with the same techniques, is well understood. 85 l.tmcr Dryfdm Coted Cell ~- ? FIGURE 6. Schematic view of target, ~on transport system, and magneto optical trap. (From Ref. [23]) Heavy-ion fusion reactions can, by proper choice of projectile, target and beam energy, provide selective production of the neutron deficient francium isotopes. Gold is an ideal target because it is chemically inert, has clean surfaces, and a low vapor pressure. The i97Au(iSO,xn) reaction at 100 MeV produces predominantly 21°Fr, which has a 3.2 min half-life. Changing the energy and the isotope of the oxygen beam maximizes the production of isotopes 208, 209 or 211. The reaction 198pt(19F,5~) produces 212Fr. Fig. 6 shows the apparatus to trap and produce Fr at Stony Brook 1012 lSO ions/s on Au produce 21°Fr in the target, with less than 10% of other isotopes. The target is heated to ~ 1200 K bv the beam power and by an auxiliary resistance heater. The elevated temperature is necessary for the alkali elements to rapidly diffuse to the surface and be surface ionized. Separation of the production and the trapping regions is critical in order to operate the trap in a UHV environment. Extracted at 800 V, the ~ 1 x 106/s 21°Fr ions travel about one meter where they are deposited on the inner surface of a cylinder coated with yttrium which is heated to 1000 K and located 0.3 em away from the entrance of the cell. Neutral Fr atoms evaporate frorn the Y surface and form an atomic beam directed towards an aperture into the vapor cell MOT. The physical trap consists of a 10 cm diameter Pyrex bulb with six 5 cm diameter windows and two viewing windows 3 cm in diameter. The MOT is formed by six intersecting laser beams each with 1/e a (power) diameter of 4 cm and power of 150 mW, with a magnetic field gradient of 6 G/era. The glass cell is coated with a non- stick Dry-film coating [26] to allow the atoms multiple passes through the trapping region after thermalization with the walls [25]. The trapping laser operates in the D2 line of francium, while the repumper may operate in the Di or in the D2 lines depending on the measurement. The ground state hyperfine splitting of 2~°Fr is 46.7 GHz. ~\~ have recently captured francium atoms [44] in a magneto optical trap (MOT). opening the possibility for extensive studies of its atomic properties. (See Fig. 4 for an image of the fluorescence of Fr atoms in a MOT). 87 We have been studying the spectroscopy of francium in a magneto optical trap on-line with an accelerator. The captured atoms are confined for long periods of time moving at low velocity in a small volume, an ideal environment for precision spectroscopy. Our investigations have included the location of the 8S and 9S energy levels [45,48]. We have also made the first measurements of any radiative lifetime in Fr. The precision of our lifetime measurements of the D1 and D2 lines are comparable to those achieved in stable atoms [46,47]. They test atomic theory in a heavy atom where relativistic and correlation effects are large. ACKNOWLEDGMENTS During the years that I have been working in cooling and trapping of atoms I have benefited from the interaction with many people. Among them I would like to mention Jesse Simsarian, Jeff Ng, Gerald Gwinner, Jiirgen Gripp, Steve Mielke, Greg Foster, Joshua Grossman, Gene Sprouse, Hal Metcalf, Tom Bergeman, Simone Kulin, and Steven Rolston. I would like to thank the organizers of the 1998 Escuela Latinoamericana de Fisica for their invitation and hospitality. Support for the experiments with radioactive atoms has come from the National Science Foundation and the National Institute of Standards and Technology. REFERENCES 1. Metcalf, H., and van der Staten, P., Phys. Rep. 244, 203 (1994). 2. Arimondo, E., Phillips, W. D., and Strumia, F., eds., Laser Manipulation of Atoms and Ions, Amsterdam: North Holland, 1992. 3. Dalibard, J., Raimond, J.-M., and Zinn-Justin, J., eds., Fundamental Systems in Quantum Optics, Les Houches 1990, Session LIII, Amsterdam: North Holland, 1992. 4. Adams, C. S., and Riis, E.,Progress in Quantum Electronics (1996). 5. Sprouse, G. D., and Orozco, L. A., Annu. Rev. Nucl. Part. Sci. 47, 429 (1997). 6. Einstein, A., Physikalishe Zeit. 18, 121 (1917); translation in, Sources of Quantum Mechanics, van der Waerden, B., ed. Amsterdam: North-Holland, 1967. 7. Frisch, O., Zeit. f. Phys. 86, 42 (1933). 8. 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1.3488618.pdf	Magnetization pinning at a Py/Co interface measured using broadband inductive magnetometry K. J. Kennewell, M. Kostylev, N. Ross, R. Magaraggia, R. L. Stamps, M. Ali, A. A. Stashkevich, D. Greig, and B. J. Hickey Citation: Journal of Applied Physics 108, 073917 (2010); doi: 10.1063/1.3488618 View online: http://dx.doi.org/10.1063/1.3488618 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/108/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Precessional magnetization induced spin current from CoFeB into Ta Appl. Phys. Lett. 103, 252409 (2013); 10.1063/1.4853195 Effects of radio-frequency noise suppression on the microstrip line using FeCoNiB soft magnetic thin films J. Appl. Phys. 113, 043922 (2013); 10.1063/1.4789607 Spin wave resonance excitation in ferromagnetic films using planar waveguide structures J. Appl. Phys. 108, 023907 (2010); 10.1063/1.3435318 Optic and acoustic modes measured in a cobalt/Permalloy exchange spring bilayer using inductive magnetometry J. Appl. Phys. 97, 10A707 (2005); 10.1063/1.1849551 A coplanar waveguide permeameter for studying high-frequency properties of soft magnetic materials J. Appl. Phys. 96, 2969 (2004); 10.1063/1.1774242 [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: 128.171.57.189 On: Thu, 21 Aug 2014 08:43:21Magnetization pinning at a Py/Co interface measured using broadband inductive magnetometry K. J. Kennewell,1M. Kostylev,1,a/H20850N. Ross,1R. Magaraggia,1R. L. Stamps,1M. Ali,2 A. A. Stashkevich,3D. Greig,2and B. J. Hickey2 1School of Physics, The University of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia 2School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom 3LPMTM CNRS (UPR 9001), Université Paris 13, 93430 Villetaneuse, France /H20849Received 14 February 2010; accepted 6 August 2010; published online 8 October 2010 /H20850 Broadband ferromagnetic resonance responses for metallic single-layer and bilayer magnetic ﬁlms with total thicknesses smaller than the microwave magnetic skin depth have been studied. Twodifferent types of microwave stripline transducers were used to excite and detect magnetizationprecession: a coplanar waveguide and a microstrip line both with characteristic width larger than thefree propagation path for traveling spin waves along the ﬁlm. Both transducers show efﬁcientexcitation of higher-order standing spin wave modes across the ﬁlm thickness in samples 30–91 nmthick. The ratio of amplitudes of the ﬁrst standing spin wave to the fundamental resonant mode isindependent of frequency for single-layer permalloy ﬁlms. In contrast, we ﬁnd a strong variation inthe amplitudes with frequency for cobalt–Permalloy bilayers and the ratio is strongly dependent onthe ordering of layers with respect to a stripline transducer. Most importantly, cavity ferromagneticresonance measurements on the same samples show considerably weaker amplitudes for thestanding spin waves. All experimental data are consistent with expected effects of eddy currents inﬁlms with thicknesses below the microwave magnetic skin depth. Finally, conditions for observingeddy current effects in different types of experiments are critically examined. © 2010 American Institute of Physics ./H20851doi:10.1063/1.3488618 /H20852 I. INTRODUCTION Broadband ferromagnetic resonance /H20849FMR /H20850microwave spectrometers1–8have become a common experimental tool with which to study dynamic properties of magnetic thinﬁlms and nanostructures /H20849see, e.g., Refs. 9–14/H20850. In this paper we demonstrate a unique ability of this technique for study-ing exchange effects in magnetic ﬁlms and at buried inter-faces in multilayer geometries. Resonance and standing spinwaves are measured for permalloy /H20849Py, Ni 80Fe20/H20850ﬁlms and permalloy/cobalt bilayers, and we show how frequencies andamplitudes can be completely understood in terms of con-ductive layer microwave response. Standing spin wave modes /H20849SSWMs /H20850are excitations conﬁned by the thickness of the ﬁlm. The wavelengths ofSSWMs are determined by the ﬁlm thickness and pinning atthe surfaces and interfaces. It is well known that the homo-geneous microwave magnetic ﬁeld typically used for FMRcavity experiments does not allow SSWM observation unlesspinning 15–17of magnetization is present at the ﬁlm surfaces. Driving using a nonhomogeneous ﬁeld, e.g., by placing it thesample over a hole in a wall of a microwave cavity, 18can be used instead to observe the SSWM. Recently it was showntheoretically that a microwave microstrip transducer can be used to couple efﬁciently to the SSWM. 19In this scheme, resonant absorption by higher-order SSWM modes of anyparity is predicted due to effects of eddy currents excited bythe microwave ﬁeld of stripline transducers. This in fact hasallowed us to experimentally study the efﬁciency of coupling to these modes for an in-plane geometry using a broadbandstripline FMR technique. We show experimentally that for the broad stripline transducers considered in the present paper, the homogeneityof the microwave ﬁeld in the ﬁlm plane form conditions forobservation of pronounced eddy current effects for ﬁlmthicknesses small compared to the microwave skin depth. Wepresent below experimental evidence for eddy current effectsin what follows, and show that these effects provide quanti-tative descriptions for observed SSWM intensities. Unlike what is observed for single ﬁlms, we ﬁnd a strong frequency dependence of the amplitude of the ﬁrstSSWM relative to the fundamental mode amplitude in strip-line response for bilayers. Most signiﬁcantly, the responseamplitude is strongly dependent on the ordering of layerswith respect to a stripline transducer. As discussed previ-ously, quantitative analysis of the observed mode amplitudescan be made in terms of screening by eddy currents existingin the metallic ﬁlms with thicknesses below the microwavemagnetic skin depth. This effect, as illustrated by our ex-amples of exchange coupled Co/permalloy bilayers, is usefulfor studying buried magnetic interfaces and exchange effectsin conducting structures. The plan of the paper is as follows. Results from experi- ments on single and bilayer structures with total thicknessessmaller than the microwave magnetic skin depth are pre-sented and discussed in the following three sections. Thepaper concludes with a short theoretical discussion of cir-cumstances under which conductivity can be expected tohave signiﬁcant effects in spin wave experiments. In particu- a/H20850Electronic mail: kostylev@cyllene.uwa.edu.au.JOURNAL OF APPLIED PHYSICS 108, 073917 /H208492010 /H20850 0021-8979/2010/108 /H208497/H20850/073917/12/$30.00 © 2010 American Institute of Physics 108 , 073917-1 [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: 128.171.57.189 On: Thu, 21 Aug 2014 08:43:21lar, we discuss the reasons why efﬁcient excitation of high order SSWMs has been observed with broad coplanar andstripline transducers used in the present paper, but not withcavity FMR or with micron-wide transducers in experimentson excitation of traveling spin waves in permalloy ﬁlms. II. EXPERIMENT Measurements are made with the sample placed on a section of the microwave stripline transducer carrying a mi-crowave current as illustrated in Fig. 1. The magnetization is aligned along the axis of the transducer /H20849along z/H20850by an ap- plied static dc bias ﬁeld. Current through the transducer gen-erates an oscillating magnetic ﬁeld in the sample perpendicu-lar to the equilibrium magnetization. Resonant absorption isdetected by measuring microwave transmission loss throughthe transducer. Transmission measurements were made withvery little attenuation inserted by the cables and the trans-ducer. The strength of the bias ﬁeld is swept from 0 to 0.6 T,at a ﬁxed frequency of the ac current. This is repeated acrossthe available range of frequencies /H20849100 MHz to 20 GHz /H20850.I n this way frequencies are chosen which avoid any transmis-sion resonances of nonmagnetic nature. The magnetic contri-bution of the signal is extracted by measuring a referencesignal at a ﬁeld large enough /H208491T/H20850to suppress any of the resonances. This range allows an optimal compromise be-tween choosing thicker ﬁlms in order to detect low frequencySSWM modes and measuring surface effects that fall offaccording to 1 /L. The samples were deposited by magnetron sputtering at an argon working pressure of 2.5 mTorr. Two series ofsamples were grown, each containing bilayer and single-layer ﬁlms. Details of each series are listed in Table I. All samples in a given series were grown during the samevacuum cycle in order to ensure consistency. The basepressure prior to the deposition was of the order of 1 /H1100310 −8Torr. The ﬁlm structures Ta /H208495n m /H20850/Py/H20849Xnm/H20850/ Co/H20849Ynm/H20850/Ta/H208512.5 nm /H20849Series 1 /H20850o r5n m /H20849Series 2 /H20850/H20852were deposited onto silicon /H20849100/H20850substrates in an in-plane form- ing ﬁeld of magnitude 200 Oe at ambient temperature. Depo-sition rates were determined by measuring the thickness ofcalibration ﬁlms by low angle x-ray reﬂectometry. Note that the Ta capping and the seed layer have the same thickness in Series 2, in contrast to the structuresgrown in Series 1. This gave Serie s 2 a symmetric combina- tion with respect to transducer coupling to the ﬁlm and sub-strate through the capping/seed layers. Comparison of theFMR data obtained on both series reveals no effect of changein the capping layer design. All bilayers of Series 1 consist of a cobalt /H20849Co/H20850layer grown on top of a thicker permalloy /H20849Py/H20850layer /H20849“Si/Py/Co” geometry /H20850. Thicknesses of each Co and Py layer are varied across the series /H20849see Table I/H20850. All bilayers of Series 2 contain a 10 nm thick Co layer and are divided into two subseries.The “Si/Py/Co” subseries has the cobalt layer grown on topof the Py layer as Si/Ta/Py/Co/Ta, similar to the structures inSeries 1. The “Si/Co/Py” subseries has reversed ordering ofPy and Co layers: Si/Ta/Co/Py/Ta. The total sample thicknessfor all samples in Series 1 and Series 2 is smaller than themicrowave magnetic skin depth 20over the frequency range 100 MHz–20 GHz. Note that the microwave magnetic skindepth varies from 102 nm at 7.5 GHz to 111 nm at 18 GHzas calculated from the material parameters derived from thesingle-layer reference Py ﬁlms, and assuming 0.008 for the Gilbert damping parameter and of 4.5 /H1100310 6S/m for the conductivity of Py.21Whereas the classical skin depth actu- ally decreases with increasing frequency, the magnetic skin depth instead increases slightly because of increased mag-netic losses /H9251/H9275/H20851see Eq. /H208492.9/H20850in Ref. 20/H20852. An Agilent N5230A PNA-L microwave vector network analyzer /H20849VNA /H20850was used to apply the microwave signal to the samples and to measure magnetic absorption. As a mea-sure of the absorption we use the microwave scattering pa-rameter S21. 3The sample sits on top of the transducer with the magnetic layers facing it. To avoid direct electric contactthe sample surface is separated from the transducer by a15 /H9262m thick Teﬂon layer. The microwave frequency is held constant and the static magnetic ﬁeld His slowly increased. The raw data then appears as resonance curves in the form ofS21/H20849H/H20850as a function of H. This process is repeated for a number of frequencies. We also measure S21 for the trans- ducer with no sample /H20851S21 0/H20849H/H20850/H20852to eliminate any ﬁeld- FIG. 1. /H20849Color online /H20850Cross-sections of the coplanar /H20849a/H20850and the microstrip /H20849b/H20850broadband FMR transducers with a sample on top. TABLE I. List of the samples studied in this work. Py denotes a permalloy layer; Co: a cobalt layer, Ta: a tantalum capping or seed layer; Si: silicon subs trate on which the ﬁlm was grown. Data in square brackets are thicknesses of respective layers. Series 1, single-layers /H20849Si/Ta /H208515n m /H20852/ Py/H20851Xn m /H20852/Ta/H208512.5 nm /H20852/H20850Series 1, bilayers /H20849Si/Ta /H208515n m /H20852/Py/H20851Xn m /H20852/ Co/H20851Yn m /H20852/Ta/H208512.5 nm /H20852/H20850Series 2, single-layers /H20849Si/Ta /H208515n m /H20852/ Py/H20851Xn m /H20852/Ta/H208515n m /H20852/H20850Series 2, Si/Py/Co bilayers /H20849Si/Ta /H208515n m /H20852/Py/H20851Xn m /H20852/ Co/H2085110 nm /H20852/Ta/H208515n m /H20852/H20850Series 2, Si/Co/Py bilayers /H20849Si/Ta /H208515n m /H20852/Co/H2085110 nm /H20852/ Py/H20851Xn m /H20852/Ta/H208515n m /H20852/H20850 X=38 X=30 X=60.5 X=60.5, Y=0.2, 0.5, 1, 2, 3, 4, 5, and 10 X=40 X=40 X=40X=74 X=74, Y=0.2, 0.5, 1, 2, 3, 4, 5, and 10 X=60 X=60 X=60X=91 X=91, Y=0.2, 0.5, 1, 2, 3, 4, 5, and 10 X=80 X=80 X=80073917-2 Kennewell et al. J. Appl. Phys. 108 , 073917 /H208492010 /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: 128.171.57.189 On: Thu, 21 Aug 2014 08:43:21dependent background signal from the results. The results presented below are Re/H20851S21/H20849H/H20850/S210/H20849H/H20850/H20852. It is worth noting that the raw data /H20841S21/H20849H/H20850/H20841show the same qualitative behav- ior, so artifacts arising from the mathematical processing of data are not signiﬁcant. Two types of stripline transducers were utilized: copla- nar and microstrip. Detailed broadband FMR measurementson the samples from Series 1 were taken using a coplanartransducer and compared to additional results made with amicrostrip transducer. One measurement run at a single mi-crowave frequency requires about 15 min of VNA time, ascan across the entire frequency range together with a back-ground scan requires a day. These large time requirementslimited the number of cases examined and only two frequen-cies were studied across the entire range using the microstriptransducer. The microstrip results were in complete agree-ment with those from the coplanar waveguide. The measure-ments on Series 2 were made using the microstrip transduceronly, allowing also data to be taken for two sample orienta-tions: one with the ﬁlm facing the transducer and one withthe substrate facing the transducer. Moreover, the microstriptransducer turned out to be of a slightly better microwavequality than the coplanar one. Three additional measurements were made in order to make comparison with very different techniques: using aVarian 4 cavity, a section of a hollow waveguide in reﬂec-tion, and using a Brillouin light scattering /H20849BLS /H20850technique. III. SINGLE-LAYER FILMS Results from the single-layer reference Py ﬁlms in Series 2 are shown in Fig. 2for driving at 7.5 and 18 GHz. A fundamental resonance and a SSWM can be identiﬁed foreach ﬁlm except the thinnest 30 and 40 nm thick ﬁlms at 7.5GHz. For these ﬁlms this frequency is lower than the mini-mum frequency for observation of the ﬁrst SSWM. A comparison of the VNA FMR data with results from cavity FMR measurements taken at 9.47 GHz is shown inFig.3/H20851panels a /H20850–/H20849c/H20850/H20852. Three single-layer ﬁlms of Series 2 of different thicknesses /H2084940, 60, and 80 nm /H20850were used to pro- duce these data. The strongest absorption peak in the cavity FMR data is conventionally identiﬁed with the fundamental resonancemode. The smaller peak is then an SSWM. One notices thatthe SSWMs are clearly visible in the broadband FMR butappear very weak in the cavity FMR experiment. We expectthat the unpinned SSWM should not produce a strong re-sponse in the cavity due to the high antisymmetry of themode, resulting in a low overlap integral with the highlyhomogeneous driving ﬁeld. Nevertheless, some signature ofthis mode is apparent in all the ﬁlms from this series. Thissuggests that a weak surface pinning may be present in allsamples. The pinning must be asymmetric; i.e., larger at one ﬁlm surface than the other, since it is known that the type ofSSWM observed in the cavity FMR measurements dependson the symmetry of the surface pinning. 22The odd-symmetry modes are seen, provided pinning conditions are asymmetric,i.e., surface anisotropies at two ﬁlm surfaces differ from eachother. If the surface anisotropy strengths at both surfaces are equal but do not vanish, only the even-symmetry modes areobservable. /H20849If pinning vanishes completely only the funda- mental mode is observable. /H20850 A BLS technique was utilized in order to identify the SSWM seen in the cavity FMR data. In thermal BLS, SS-WM’s are excited through thermal ﬂuctuations and cantherefore appear regardless of mode symmetry. 23A BLS study of the 60 nm thick single-layer ﬁlm was performed. ABLS intensity spectrum measured at the angle of light inci-dence of 5° from the normal to the ﬁlm is shown in panel inFig. 3/H20849d/H20850. We found that the frequency position of the ﬁrst BLS peak above the fundamental /H20849F/H20850dipole mode for a given ﬁeld and a given magnon wave number is consistentwith the ﬁeld position of the lower ﬁeld peak detected in thebroadband FMR. This unambiguously identiﬁes the smallerpeak seen in both broadband and cavity FMR as the ﬁrstodd-symmetry SSWM /H20851SSWM1 in Fig. 3/H20849d/H20850/H20852. Recall, that the odd-symmetry mode is observable in a cavity FMR experiment, provided there is an asymmetry inpinning conditions for surface magnetization. The BLS dataindicate the presence of a dipole gap 24in the spin wave spec- trum, where the Damon–Eschbach mode repels the ﬁrst di-pole exchange branch. 25Without asymmetric pinning, the odd-symmetry modes are practically orthogonal to the fun-damental mode and the dipole gaps formed where the modesrepel are negligibly narrow. Therefore the existence of a gapalso evidences some small degree of asymmetric pinning atthe ﬁlm surfaces that modiﬁes the symmetry of the modes. FIG. 2. Microwave broadband FMR absorption data for single-layer permal- loy ﬁlms. Microstrip transducer is 1.5 mm in width. Left-hand panels /H20849a/H20850– /H20849d/H20850: driving frequency is 7.5 GHz. Right-hand panels /H20849e/H20850–/H20849h/H20850:1 8G H z .F i l m thicknesses: /H20849a/H20850and /H20849e/H20850:3 0n m ; /H20849b/H20850and /H20849f/H20850:4 0n m ; /H20849c/H20850and /H20849g/H20850: 60 nm; /H20849d/H20850 and /H20849h/H20850:8 0n m .073917-3 Kennewell et al. J. Appl. Phys. 108 , 073917 /H208492010 /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: 128.171.57.189 On: Thu, 21 Aug 2014 08:43:21From Fig. 2one sees that the ﬁeld position of the fun- damental mode does not depend on the ﬁlm thickness, butthe SSWM peak shifts to lower ﬁelds with decrease in L.I n accordance with Kittel’s formula, 15the frequency shift for a mode due to the exchange contribution to the mode energyscales as a square of its standing-wave wave number acrossthe ﬁlm thickness. If the resonance spectra are measuredﬁeld-resolved while keeping the driving frequency constant,one observes a downshift in the resonance ﬁeld instead of anincrease in the resonance frequency. The standing-wavewave number is inversely proportional to the ﬁlm thicknessand also depends on the surface anisotropy. The larger thesurface magnetization pinning, the larger the mode ﬁelddownshift. We have already discussed ﬁnding some degreeof asymmetric pinning of surface magnetization for the ﬁlms.For the fundamental mode in the absence of surface pinningthe standing-wave wave number is zero but becomes nonva-nishing if a surface pinning is present. For the same degreeof pinning the standing-wave wave number for the funda- mental mode scales as 1 /L. Therefore the observed indepen- dence of the ﬁeld position for the fundamental mode on thethickness suggests that its standing-wave wave number isvery close to zero, thus the surface anisotropy is indeedsmall. The ﬁeld downshift for SSWMs with respect to the fun- damental mode should also scale approximately as 1 /L 2. This is clearly seen from comparison of the ﬁeld positionsfor SSWMs in Figs. 3/H20849a/H20850and3/H20849c/H20850: the sample thickness in trace /H20849c/H20850is twice larger than in trace /H20849a/H20850, but the difference in resonance ﬁelds for the fundamental mode and SSWM isfour times larger in /H20849a/H20850than in /H20849c/H20850. Lastly, we comment on eddy current effects on the broadband FMR response. As noted earlier, efﬁcient excita-tion of the asymmetric modes is not possible unless a ﬁlmhas very different pinning conditions for magnetization attwo ﬁlm surfaces. Using the approach of Ref. 26which is valid for insulating ﬁlms we ﬁnd that the experimental am-plitudes in Fig. 2for the SSWM cannot be obtained unless one assumes a near complete pinning of magnetization at oneof the ﬁlm surfaces. This is inconsistent with our ﬁnding ofindependence of the ﬁeld position of the fundamental modein Fig. 2on the ﬁlm thickness: this value of pinning consid- erably shifts the fundamental and the other mode down-wards. The shift is proportional to the ﬁlm thickness and isroughly one ninth of the ﬁeld distance between the funda-mental mode and the ﬁrst SSWM. Therefore it should bewell-seen in Figs. 2/H20849a/H20850and2/H20849b/H20850compared with Fig. 2/H20849d/H20850. On the other hand, observing larger amplitudes for the SSWM in the broadband FMR /H20849Figs. 2and3/H20850than in the cavity FMR /H20849Fig. 3/H20850is in agreement with theory in. 19In particular, it is shown that large amplitude SSWM peaks canbe observed from metallic ﬁlms without surface magnetiza-tion pinning provided that an asymmetric eddy current con-tribution to the total microwave magnetic ﬁeld exists. Thiscan in fact be realized simply by stripline transducers andused to detect SSWM resonances that are weak or not visiblein conventional cavity FMR. This eddy current theory alsodescribes, as experimentally observed, that the SSWM am-plitudes in single-layer ﬁlms are practically independent of driving frequency. A detailed study of SSWM’s was carried out using the reference 74 nm thick single-layer Py ﬁlm from Series 1. Acoplanar transducer was chosen to drive and detect magneti-zation precession. Measurements were taken with the staticapplied ﬁeld aligned along the direction of the uniaxial an-isotropy axis, and ﬁeld sweeps were made for numerous fre-quencies in the range from 100 MHz up to 15GHz. Theobtained resonance spectra are very similar to ones shown inFig.2for the single layers from Series 2, therefore, are not presented here. The measured dependencies of resonanceﬁelds for the observed modes on the driving frequency areshown in Fig. 4/H20849a/H20850/H20851Fig.4/H20849b/H20850will be discussed in Sec. IV /H20852. For this ﬁlm a best ﬁt gives a saturation magnetization valueof 4 /H9266Ms=8120 /H1100660 G, g-factor of g=2.05, and an ex- change constant of A/H112290.41/H1100310−6erg /cm. These param- eters provide consistent results for all measured Py thick-nesses for Series 1. FIG. 3. /H20849Color online /H20850Cavity FMR /H20851panels /H20849a/H20850–/H20849c/H20850, dashed line /H20852and BLS data /H20851panel /H20849d/H20850/H20852in comparison with broadband FMR data /H20851panels /H20849a/H20850–/H20849c/H20850, solid lines /H20852for the single-layer ﬁlms with different thicknesses. /H20849a/H20850:4 0n m ; /H20849b/H20850:6 0n m ;a n d /H20849c/H20850: 80 nm. Driving frequency is 9.47 GHz. Broadband transducer: the same 1.5 mm wide microstrip. Cavity: Varian-4 ESR spec-trometer cavity. BLS data are taken for the 60 nm thick ﬁlm at an incidenceangle of 5° and in an applied ﬁeld of 500 Oe. “F” in panel /H20849d/H20850indicates the fundamental /H20849Damon–Eschbach /H20850BLS peak, “SSWM1” is the ﬁrst /H20849odd- symmetry /H20850SSWM, and “SSWM2” is the second /H20849even-symmetry /H20850SSWM.073917-4 Kennewell et al. J. Appl. Phys. 108 , 073917 /H208492010 /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: 128.171.57.189 On: Thu, 21 Aug 2014 08:43:21The value for 4 /H9266Msderived from the broadband FMR measurements is consistent with results obtained with super-conducting quantum interference device /H20849SQUID /H20850magne- tometry. A saturation magnetization for the Py given by4 /H9266Ms=8000 G was obtained from out-of-plane saturation and volume magnetization measurements. Note, that it isnot unusual that saturation magnetization for permalloysamples differs considerably from the standard value 4 /H9266Ms =10 800 G. Values in the range from 8000 G to the standardvalue were found by different authors /H20849see, e.g., Refs. 27–29/H20850. The extracted value for the exchange constant is also lower than the standard for permalloy /H20849A=1.3 /H1100310 6erg /cm/H20850. Structural and composition analysis of the ﬁlms is necessary to explain the decreased value of A. This is out of scope of this paper which has emphasis on electrody-namical properties of the material. The ﬁts were made as follows. Since it was found that conductivity and an eventual weak surface pinning of mag-netization have negligible effect on the resonant ﬁeld for thefundamental mode, the /H9275/H20849H/H20850dependence for this mode was used in the expression for in-plane FMR, /H92752=/H92532H/H20849H +4/H9266Ms/H20850, in order to determine 4 /H9266Ms. A best ﬁt was obtained using regression analysis. The values of 4 /H9266Msobtained this way were then used to ﬁt the experimental data for theSSWM frequency /H9275//H208492/H9266/H20850=15 GHz with our theory which includes the effect of conductivity on SSWM excitation,19leading to the above value of A. A value of 4.5 /H11003106S/m2 for conductivity of permalloy21was used to produce the ﬁts. As it was done for data from Series 2, we have assumed that the SSWM peak corresponds to the ﬁrst SSWM, sincethe ﬁlm has a similar composition and a similar thickness tothe 80 nm thick ﬁlm from Series 2 /H20851Figs. 2/H20849d/H20850and2/H20849h/H20850/H20852and shows absorption at the ﬁelds close to ones for that ﬁlm.Moreover, if one assumes that the SSWM is instead thesecond, symmetric, SSWM one then obtains an unrealisti-cally small value for the exchange constant: A/H112290.11 /H1100310 −6erg /cm. IV. BILAYERS Resonance curves for bilayers from Series 2 are shown in Fig. 5. Except for panels in Figs. 5/H20849a/H20850and5/H20849e/H20850, each plot contains a response from a Si/Py/Co bilayer and a responsefrom a Si/Co/Py structure. The Py layer thickness is the samefor both orderings in each panel. One sees that the responseof the Si/Co/Py structures is characterized by a single ab-sorption peak located at a ﬁeld slightly little smaller than thatof the fundamental mode for the corresponding Py single-layer. Therefore this peak is identiﬁed as the fundamentalmode of the bilayer. We will denote this single-peak responseas “Type A.” The ﬁeld downshift decreases as the Py layerthickness is increased, indicating a magnetization pinning.The reversed ordering of layers /H20849Si/Py/Co /H20850provides a quite different response, and we denote this as a “Type B” re-sponse. Two peaks with comparable intensities are seen at7.5 GHz in panels in Figs. 1/H20849b/H20850–1/H20849d/H20850. The high ﬁeld peak is located at the ﬁeld very close to the ﬁeld position for thefundamental mode for the respective single-layer permalloyﬁlm and is identiﬁed as the fundamental. The low ﬁeld peak/g19 /g24/g19/g19 /g20/g19/g19/g19 /g41/g85/g72/g84/g88/g72/g81/g70/g92/g11/g42/g43/g93/g12/g19 /g21 /g23 /g25 /g27 /g20/g19 /g20/g21 /g20/g23/g36/g83/g83/g79/g76/g72/g71 /g73/g76/g72/g79/g71 /g11/g50/g72/g12/g19/g24/g19/g19/g20/g19/g19/g19/g20/g24/g19/g19/g21/g19/g19/g19/g21/g24/g19/g19 /g41/g85/g72/g84/g88/g72/g81/g70/g92/g11/g42/g43/g93/g12/g19/g21/g23/g25/g27/g20 /g19 /g20 /g21 /g20 /g23/g53/g72/g79/g68/g87/g76/g89/g72/g44/g81/g87/g72/g81/g86/g76/g87/g92 /g85/g76/g18/g85/g19 /g19/g17/g19/g19/g17/g23/g19/g17/g27/g20/g17/g21/g20/g17/g25/g73/g20/g21/g22 /g73 /g20/g11/g68/g12 /g11/g69/g12 /g20/g21 /g21/g22 FIG. 4. /H20849Color online /H20850/H20849a/H20850Resonant ﬁelds as a function of driving frequency. /H20849b/H20850Relative mode intensities. Coplanar transducer with a central conductor 0.3 mm in width is used. Black triangles: single-layer 74 nm thick ﬁlm. Reddots: bilayer ﬁlm /H20849Py:74 nm, Co:10 nm /H20850. Black dashed lines: ﬁts for the single-layer ﬁlm. Blue solid lines: ﬁts for the bilayer ﬁlm. f: fundamentalmode. 1–3: SSWMs with respective numbers. Difference between the blackdashed and blue solid lines is smaller than the size of the symbols whichshow experimental points, therefore it is not well resolved in the graph. Insetin Fig. 4/H20849a/H20850: raw data used to construct the plots in Fig. 4. Red solid line: experimental absorption trace; blue dashed line: theoretical absorption trace.Frequency: 9.53 GHz. The dashed vertical line in Fig. 4/H20849a/H20850shows the fre- quency position for these data. Horizontal axis for the inset: applied ﬁeld inoersteds. Vertical axis: the same as in Fig. 3.FIG. 5. /H20849Color online /H20850Broadband FMR absorption for bilayers. Microstrip transducer is 1.5 mm in width. Left-hand panels /H20849a/H20850–/H20849d/H20850: driving frequency is 7.5 GHz. Right-hand panels /H20849e/H20850–/H20849h/H20850: 18 GHz. /H20849a/H20850and /H20849e/H20850: Single-layer permalloy ﬁlms /H20849Si/Ta/Py/Ta, given here for comparison /H20850; solid line: 80 nm thick; dashed line: 40 nm thick; /H20849b/H20850–/H20849d/H20850and /H20849f/H20850–/H20849h/H20850are bilayer responses. Red solid lines are for Si/Ta/Py/Co/Ta and dashed blue lines are for Si/Ta/Co/Py/Ta structures. In all ﬁgures cobalt layer is 10 nm thick. /H20849b/H20850and /H20849f/H20850: permalloy thickness is 40 nm; /H20849c/H20850and /H20849g/H20850:6 0n m ; /H20849d/H20850and /H20849h/H20850:8 0n m .073917-5 Kennewell et al. J. Appl. Phys. 108 , 073917 /H208492010 /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: 128.171.57.189 On: Thu, 21 Aug 2014 08:43:21is at a ﬁeld close to the ﬁrst SSWM observed in the single- layer ﬁlm, and should therefore be the ﬁrst SSWM mode forthe bilayer. A ﬁeld downshift is observed that decreases withincrease in Py thickness. Figure 6/H20849a/H20850shows cavity FMR data for the Si/Py/Co and the Si/Co/Py samples with 80 nm of Py,and 10 nm of Co along with respective broadband FMR data.The measurements were taken at 9.47 GHz. One sees thatthese two cavity traces are very close, and the resonanceﬁelds for all modes are practically the same for both bilayers.Furthermore, three distinct peaks are seen. The largest peak is the fundamental mode and the two other peaks are the ﬁrstand the second SSWM. From this panel two important factscan be determined. First, additional SSWMs are detected thatare not visible in the microstrip broadband FMR data fromthe Si/Co/Py structure. Second, the SSWM amplitudes areconsiderably smaller than corresponding ones found for theSi/Py/Co structure. Furthermore, the cavity FMR data clearlyshow that the difference in Type A and Type B responsescannot be explained in a conventional way, 15i.e., assuming that all samples from the Si/Py/Co subseries have stronglypinned surface spins at the one of the external bilayer sur-faces or at the layer interface, but the samples from the Si/Co/Py subseries do not. Note that for all Si/Py/Co structures in Fig. 5the experi- mental resonance ﬁelds for the fundamental peaks are alwaysslightly larger than for the Si/Co/Py bilayers. The minimumdifference is 5 Oe /H20851Fig.5/H20849b/H20850/H20852, and the largest one is 20 Oe /H20851Fig.5/H20849h/H20850/H20852. The cavity FMR data taken at 9.54 GHz show a difference of 10 Oe which is the same as for the broadbandFMR data for 7.5 GHz for the same samples /H20851Fig. 5/H20849d/H20850/H20852. About 5 Oe can be attributed to the effect of conductivity, asour simulations based on the theory in 19show. The remain-ing 5 Oe difference is possibly due to ﬁlm growth conditions: one may expect a slight difference in pinning conditions formagnetization at the bilayer external surfaces and the inter-face between the layers for different orderings of layer depo-sition. It can be a small difference in the values for the in-terlayer exchange constant A 12or difference in interface anisotropies.30In our simulations we use the value for A12 =2/H1100310−6erg /cm2for which the exchange coupling of lay- ers “saturates:” for this A12value the resonance ﬁeld down- shift for the fundamental mode reaches a plateau and doesnot increase noticeably with a further increase in A 12.I n reality the interlayer exchange may be slightly more “satu-rated” for the Si/Py/Co structures than for the Si/Co/Py one,as the data in Fig. 5suggest. Furthermore, for simplicity the theory we use to ﬁt the experimental data does not account for possible contributionto magnetization pinning at the layer interface by the inter-face anisotropy /H20849the factors K iin the interface boundary con- dition /H20851Eq./H2084916/H20850in Ref. 30/H20852were set to zero /H20850. Therefore, an alternative explanation is that deposition of the layers in dif-ferent orders may have produced slightly different interfaceanisotropies. However, it is clear that the effect of the inter-face anisotropy on the resonance ﬁeld should be of the sameorder of magnitude as the effect on the resonance ﬁeld of asurface anisotropy of the same magnitude in the case of asingle-layer ﬁlm of the same thickness. A simple calculationof the absorption amplitude for an insulating magnetic ﬁlmof the same thickness shows that an increase in the surfaceanisotropy /H20849i.e., increase in the surface magnetization pin- ning /H20850which produces a 20 Oe downshift in the resonance ﬁeld for the fundamental mode cannot result in a qualitativedifference in absorption amplitudes: it is not possible totransform a response of Type A into one of Type B due tothis slight increase in surface anisotropy. Therefore, in thefollowing we will neglect this small difference in the reso-nance ﬁelds. Furthermore, in Sec. V we will provide experi-mental evidence that the difference in Type A and Type Bresponses is not related to the difference in the ﬁlm growthconditions. Coplanar-transducer studies were carried out for Series 1. As with the Si/Py/Co subseries 2 of Series 2, only Type B responses were detected for Series 1, since it contains Si/Py/Co structures only /H20849Table I/H20850. This series contains samples which have cobalt layers of different thicknesses. With afamily of samples having permalloy layers of the same thick-ness it was found that amplitudes of SSWMs gradually in-crease with increase in Co thickness. In parallel, ﬁeld posi-tions for all modes, including the fundamental one, graduallyshift downwards. At 12 GHz the ﬁeld position for the ﬁrstSSWM shifts by 100 Oe when the cobalt thickness is in-creased from 1 to 10 nm, which constitutes 10% of the mag-nitude of the resonance ﬁeld for this mode. A detailed report on the experimental data obtained for different thicknesses of Cobalt is out of scope of the presentpaper and will be published elsewhere. Most importantly, noabrupt qualitative difference in behavior was found when thecobalt thickness increases from 1 to 10 nm. Therefore, in thefollowing we concentrate on the sample Si/Py /H2085174 nm /H20852/Co/H2085110 nm/H20852from Series 1 which has the thickest Cobalt layer andFIG. 6. /H20849Color online /H20850/H20849a/H20850Cavity FMR data for the bilayers from Fig. 5/H20849d/H20850. Red solid line: Si/Co/Py, blue dashed line: Si/Py/Co. /H20849b/H20850and /H20849c/H20850: broadband FMR traces for Si/Py/Co and Si/Co/Py, respectively. Red solid lines in /H20849b/H20850 and /H20849c/H20850: ﬁlm faces the transducer; blue dashed lines in /H20849b/H20850and /H20849c/H20850: substrate faces the transducer. Frequency is 9.47 GHz.073917-6 Kennewell et al. J. Appl. Phys. 108 , 073917 /H208492010 /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: 128.171.57.189 On: Thu, 21 Aug 2014 08:43:21therefore shows the most pronounced effect of addition of a Cobalt layer. A summary of VNA FMR results obtained onthis sample is shown in Fig. 4along the ﬁttings with our theory. The inset in the ﬁgure shows a typical resonancespectrum for this sample and a result of its ﬁtting with ourtheory. From the inset one sees that the response is of thesame Type B as for the Si/Py/Co samples from Series 1. The ﬁelds and frequencies for the bilayer in Fig. 4/H20849a/H20850can be accounted for with 4 /H9266Ms=15 080 G for cobalt. This ex- tracted value is not far from that obtained by a SQUID de-termined value of 4 /H9266Ms=17 800 G for a reference 10 nm thick Co ﬁlm made in the same production run. Furthermore,from this SQUID measurement on the reference ﬁlm we es-timated the anisotropy for the Co layer. From the saturationpoint on the hard axis an in-plane anisotropy of approxi-mately 50 Oe was found. This small anisotropy was ne-glected when ﬁtting the experimental results with our theory. It is worth noting that cobalt is a highly anisotropic ma- terial and in a polycrystalline ﬁlm form often shows a stronginduced uniaxial anisotropy. The in-plane component of theeffective anisotropy ﬁelds is small, as already discussed. Theout-of-plane anisotropy cannot be determined from the in-plane FMR data, as its contribution cannot be separated fromsaturation magnetization. What one actually extracts fromthe raw FMR data is an effective saturation magnetizationfor cobalt which is the difference between 4 /H9266Msand the effective ﬁeld of the normal uniaxial anisotropy. For the pur-pose of our study extracting an effective 4 /H9266Msfor cobalt is sufﬁcient. Most interesting in Fig. 4/H20849b/H20850is not only the shift in reso- nant ﬁelds, but also the signiﬁcant change in the relativeamplitudes of the resonant modes. As stated in the previoussection, with a single layer of Py, the amplitude of the ﬁrstSSWM is relatively constant with respect to the fundamentalmode at a constant applied ﬁeld. However, as shown in Fig.4/H20849b/H20850, the bilayer ﬁlm with just 10 nm of Co on the single Py layer has a signiﬁcantly different distribution of relative am-plitudes. The relative amplitude r ifor a ith mode is calcu- lated as a ratio of its amplitude to the amplitude of the fun-damental mode r 0. This removes the effect of a decreasing precessional angle with a larger applied ﬁeld. The ﬁrst SSWM increases in amplitude as the applied ﬁeld increases,so much so in fact that it has a larger intensity than thefundamental mode. This effect is clearly seen for all sampleswith Co thicknesses greater than 5 nm, and for the wholerange of Py thicknesses studied /H2084940–91 nm /H20850. The theoretical intensities in Fig. 4/H20849b/H20850calculated using the theory in Ref. 19are in good qualitative agreement with experiment. Our theory treats the microwave transducer ﬁeldas absolutely homogeneous in the ﬁlm plane. Due to this andother simpliﬁcations used in the theory, a better quantitativeagreement is not to be expected. Furthermore, the theoreticalintensities strongly depend on a number of material param-eters, in particular on layer conductivities and the values theGilbert magnetic damping parameters for the layers. Thismakes the task of optimal ﬁtting somewhat complicated, asone has to ﬁt all curves for intensities /H20851Fig.4/H20849b/H20850/H20852and all for resonant ﬁelds /H20851Fig. 4/H20849a/H20850/H20852with the same set of parameters simultaneously. No attempt was made to obtain the optimalﬁt, as the most important task in the calculation in Fig. 4was to show that the theoretical curves exhibit the same behavioras in the experiment. The calculated relative intensities forSSWMs increase with frequency and, like the experimentaldata, reach a maximum at higher frequencies /H20849not shown in the graph, as the theoretical maximum for the ﬁrst SSWM isat about 18 GHz /H20850. Moreover, the theory explains the differ- ence in responses for Si/Py/Co and Si/Co/Py systems. Shortdetails of the theory are given in the next section. Thus weconclude that the eddy currents induced in the bilayer ﬁlmsby incident microwave ﬁelds give a major contribution to thebroadband FMR response. Additional measurements were carried out using the wide microstrip transducer and a hollow waveguide. As fol-lows from Ref. 31, the microwave ﬁeld of a 1.5 mm wide microstrip transducer should be homogeneous above thetransducer at distances less than 1.5 mm from the surface.This allows taking measurements with a sample placed onthe transducer with its Si-substrate /H208490.5 mm thick /H20850facing the transducer. Representative results of such a measurementwith a Si/Py/Co sample are shown in Fig. 6. For the Si/Co/Py structure with the ﬁlm facing the transducer, the response isType A. The response changes to Type B when the sample isplaced with the substrate facing the transducer. The Si/Py/Cobilayers behave in the opposite way. This is consistent withthe theoretical predictions given in Ref. 19. Following Appendix B, Case A in Ref. 19the same ef- fect of swapping response types should be seen for propa- gating plane electromagnetic waves incident normally onto the ﬁlm surface. We tested this using a hollow metallic wave-guide of rectangular cross-section to form conditions for thenormal incidence. The ﬁlms are placed in the cross sectionalplane of the waveguide. The measurements are made in re-ﬂection, so that the parameter S11 /H20849Ref. 3/H20850is obtained. The samples ﬁll about one half of the waveguide cross-section.For this reason one has to expect a microwave ﬁeld incidenton the far sample surface that includes contributions fromdiffraction around edges of the sample. The presence of thisdiffracted ﬁeld is conﬁrmed by VNA measurements. When asample is inserted into the waveguide, the transmission char-acteristic S21/H20849f/H20850acquires a nonmonotonic dependence on frequency fbecause of partial standing-wave resonances formed in the waveguide. Our ﬁeld-resolved measurementsare carried out in a local minimum of S21/H20849f/H20850in order to reduce effects of the diffracted microwave ﬁeld at the far sample surface. Representative data are displayed in Fig. 7. We ﬁnd the same tendency as in the microstrip experiments: responses ofType A are obtained when the Co layer of any structure facesthe incident ﬂux, and responses of Type B are obtained whenthe Py layer faces the incident ﬂux. V. DISCUSSION Here we give basic elements of the theory in a broader context of previous studies for eddy current effects for con-ducting ﬁlms. Details of the theory can be found in Ref. 19. The essential result of the theory is that conditions for ahighly inhomogeneous microwave magnetic ﬁeld are formed073917-7 Kennewell et al. J. Appl. Phys. 108 , 073917 /H208492010 /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: 128.171.57.189 On: Thu, 21 Aug 2014 08:43:21in the microstrip broadband FMR geometry due to micro- wave eddy currents in conducting samples. The eddy cur-rents in conducting ﬁlms thinner than the microwave mag-netic skin depth, 20hereby referred to as subskin depth ﬁlms “SSDF,” may strongly affect broadband FMR measurementresults. In summary, manifestations of this effect for SSDFare as follows: /H20849i/H20850the response of conductive multilayers may strongly depend on layer ordering with respect to themicrowave transducer location; /H20849ii/H20850extremely large ampli- tudes of high order SSWMs can be observed in some multi-layers; and /H20849iii/H20850the response of these systems can be strongly frequency dependent. Experimental data presented here are in the full agree- ment with these predictions. We note that to some extent thedriving of SSWM discussed here is similar to efﬁcient exci-tation of high order FMR modes by the microwave electricﬁeld observed by Wolf. 18In both cases, the excitation of SSWM depends upon inhomogeneous ﬁelds. In Wolf’s ex- periment, a conducting ﬁlm was placed on a hole in a cavitywall. Efﬁcient excitation of the inhomogeneous SSWM reso-nances was observed and can be understood as follows. Themicrowave electric ﬁeld across the hole drives a current inthe sample. The current creates a microwave Oersted ﬁeldwhich is antisymmetric across the sample thickness. Thisﬁeld can be approximated by h x/H20849y/H20850/H11008y/L−0.5 with y=0 at one of the ﬁlm surfaces /H20849see Fig. 1for the used frame of reference /H20850. This highly inhomogeneous magnetic ﬁeld efﬁ- ciently excites the SSWM resonances. Importantly, this ex-periment clearly demonstrates that magnetization precessionis driven by the total microwave magnetic ﬁeld to whichmicrowave currents in the sample contribute. In Ref. 19it was noted that an external in-plane micro- wave magnetic ﬁeld h eapplied to a ﬁlm medium is neces-sarily accompanied by an in-plane curling electric ﬁeld. This electric ﬁeld should induce a microwave current in thesample whose Oersted ﬁeld h Oeadds to the external micro- wave magnetic ﬁeld. For thick samples one recovers themagnetic skin depth effect and the amplitude of the total ﬁeldh t=he+hOefalls off exponentially with the distance from the sample surface facing the incident ﬁeld ﬂux. Such a thickﬁlm geometry has been discussed in the past. 20,32 It turns out that effects are striking in the case of thin ﬁlms also. Indeed, for a thin ﬁlm with thickness less than theSSDF, the total microwave magnetic ﬁeld decays morestrongly than exponentially. For Py of a thickness larger than30 nm, in contact with an adjoining media with a high char-acteristic impedance z 0/H1135050 Ohm, the ﬁeld is negligible at the far ﬁlm surface /H20851see Eq. /H2084944/H20850and Fig. 7 in Ref. 19/H20852. The derivation of this result is not trivial, and details can befound in Ref. 19./H20849See also Eqs. /H208493/H20850and /H208494/H20850in Sec. VI. /H20850 It is important to note that this effect is not seen in the cavity FMR measurements, as the microwave magnetic ﬁeldof the cavity is incident on both SSDF surfaces /H20851see Eq. /H208494.1/H20850 in Ref. 33/H20852. As a result the total microwave magnetic ﬁeld inside an SSDF sample is close to homogeneous. If there isno magnetization pinning at the ﬁlm surfaces, the fundamen-tal mode only displays an FMR response. Intensities forhigher-order odd-symmetry SSWMs are observable in thecavity FMR only for samples with a very high level of pin-ning asymmetry; e.g., with spins almost pinned at one of theﬁlm surfaces. The enhanced inhomogeneity described in Ref. 19origi- nates from an extension of the classical skin depth effect to samples of ﬁnite thicknesses /H20849see Refs. 34and35and refer- ences in Ref. 35/H20850. The phase of the back-reﬂection of the total microwave magnetic ﬁeld from the boundary betweentwo media with a large difference in values of electric con-ductivity is important for such samples. The total microwavemagnetic ﬁeld incident on the far surface interferes destruc-tively with the back reﬂected ﬁeld. As a result, the usual skin depth law /H20841h t/H20849y/H20850/H20841/H11008exp/H20851−/H20881/H20849i/H9268/H9275/H20850y/H20852which is valid for the half- space is modiﬁed. For a single layer of thickness Lthe ex- pression is htx/H20849y/H20850/H11008sinh/H20851/H20881/H20849i/H9268/H9275/H20850/H20849y−L/H20850/H20852, where y=0 is the co- ordinate for the ﬁlm surface facing the transducer and y=Lis for the far ﬁlm surface. For ﬁlms which are much thinnerthan the classical microwave skin depth this expression re-duces to a linear function h tx/H20849y/H20850/H11008/H20849y−L/H20850/L. From this for- mula one sees that the total microwave magnetic ﬁeld inside the samples is indeed highly inhomogeneous and stronglyasymmetric. Since magnetization precession is driven by thetotal ﬁeld, conditions are thereby formed for efﬁcient excita-tion of nonuniform eigenmodes of precession. If eigenmodesof the system lack inversion symmetry the SSDF broadbandFMR response will depend on layer ordering with respect tothe direction of the incident microwave ﬂux. Calculated mode proﬁles for the dynamic magnetization and the total ﬁeld are shown in Fig. 8for the bilayers. The left panels of this ﬁgure are for the Si/Py/Co structures withthe Co layer facing the transducer. The right panels are forSi/Co/Py with the Py layer facing the transducer. The fundamental mode displays a highly inhomogeneous dynamic magnetization across the Py, and has a minimum atFIG. 7. /H20849Color online /H20850Hollow waveguide data. Frequency is 8.99 GHz. /H20849a/H20850: response of Si/Ta/Py /H2085180nm /H20852/Co/H2085110 nm /H20852/Ta structure. /H20849b/H20850: response of Si/Ta/ Co/H2085110 nm /H20852/Py/H2085180 nm /H20852/Ta structure. Red solid line: ﬁlm facing the incident ﬂux. Blue dashed line: substrate facing the incident ﬂux. All measurementswere done in reﬂection.073917-8 Kennewell et al. J. Appl. Phys. 108 , 073917 /H208492010 /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: 128.171.57.189 On: Thu, 21 Aug 2014 08:43:21the interface /H20851see Fig. 8/H20849a/H20850and8/H20849c/H20850/H20852. This minimum is due to partial pinning of magnetization at the Py/Co interface be-cause of exchange coupling /H20849A 12/H20850to the cobalt layer. The ﬁrst higher-order SSWM is also strongly affected by pinning, and can be decomposed into a combination of the Co fundamen-tal mode and the ﬁrst Py SSWM. The broadband FMR re-sponse of a resonant mode can be approximated by an over-lap integral composed of the SSWM amplitude and h tx/H20849y/H20850. The total microwave magnetic ﬁeld in a bilayer is de- scribed by a linear function with a discontinuity at the mag-netic interface where the slope scales with layer conductivity.The effect is seen in the proﬁles shown in the panels /H20849b/H20850and /H20849d/H20850of Fig. 8. Comparing the left and the right panels, one sees that the overlap integral of the fundamental mode proﬁlewith the total ﬁeld is clearly dependent on the layer ordering.Note however that the overlap integral for the ﬁrst SSWM isonly weakly dependent on the layer ordering. We concludefrom this that the coupling of the fundamental mode to thetotal ﬁeld is efﬁcient for Si/Co/Py structures, and it should bethe dominant feature in an absorption spectrum. The re-sponse of the fundamental mode in the Si/Py/Co bilayers isweakened and becomes comparable with the response of the1st exchange mode /H20849whose response is much less dependent on the layer ordering /H20850. It is also worth noticing that in Figs. 2,3, and 5we do not convert the values of the measured sample responseS21/H20849H/H20850/S21 0/H20849H/H20850into the scalar magnetic permeability as is often done.5The reason for this is as follows. First, the per- meability one obtains in this way is an effective scalar per-meability /H9262but not the tensor of the microwave magnetic permeability.36Second, the effective permeability value which is extracted from the experimental data is found up toa constant, 4,37i.e., only functional dependence of /H9262on the applied ﬁeld or frequency and on the other relevant experi-ment parameters is extracted, but not the absolute value of /H9262. This constant depends on the geometry of the stripline wave-guide and should be obtained from theory which should beconstructed separately for each type of transducer. On thecontrary, the values of the measured S21/H20849H/H20850/S21 0/H20849H/H20850allowextraction of a physically meaningful parameter of the trans- ducer complex radiation resistance Zr=2z0lln/H20851/H20849S21 /S210/H20850/H20852, where z0is the characteristic impedance of the transducer andlis the sample length along the transducer. The imagi- nary part of the effective permeability scales linearly withRe/H20849Z r/H20850and its real part scales with Im/H20849Zr/H20850. We found that /H20841Im/H20849Zr/H20850/H20841/H11270Re/H20849Zr/H20850for all our experimental data. This is in good agreement with the theoretical result shown in Fig. 1 of Ref. 19. Thus S21/H20849H/H20850/S210/H20849H/H20850/H11015Re/H20851S21/H20849H/H20850/S210/H20849H/H20850/H20852and Re/H20849Zr/H20850/H110152z0lln/H20851Re/H20849S21 /S210/H20850/H20852. Note that Zrvalues extracted from the experiment are absolute, and therefore can be used for extraction of ﬁlmmaterial parameters using the existing theory in Ref. 19or similar. Furthermore, the Re/H20851S21/H20849H/H20850/S21 0/H20849H/H20850/H20852values off- resonance represent contribution to Zrfrom eddy current losses. For metallic samples this nonmagnetic contribution ismuch larger than the precessional magnetic one as seen fromFigs. 2,3, and 5. For different samples the off-resonance transmission Re/H20851S21/H20849H/H20850/S21 0/H20849H/H20850/H20852varies from 0.2 to 0.5 /H20849i.e., from /H1100214 to/H110026d B /H20850but the resonance contribution is less than 1% /H208490.1 dB /H20850of the off-resonance value in all panels. Finally, this effect should be observed for a number of different excitation geometries provided that the microwaveﬁeld ﬂux is incident on the bilayer structure from one surfaceonly. The present experiment with the hollow waveguide/H20849Fig.7/H20850is in full agreement with this prediction. VI. COMPARISON OF DIFFERENT MEASUREMENT TECHNIQUES The magnetic dynamics of thin magnetic ﬁlms driven by the microwave ﬁeld produced by the stripline transducerscan be understood by examining the quasistatic form of theMaxwell equations. We consider the geometry shown in Fig.1in which the static ﬁeld and transducer axes are in the z direction, and the normal to the ﬁlm is in the ydirection. Written in terms of the Fourier-components of the micro-wave ﬁeld, Maxwell’s quasistatic equations are: ikh ky−/H11509hkx//H11509y=/H9268ekz, kekz=−/H9275/H92620/H20849hky+mky/H20850, /H11509hky//H11509y−ikhkx=−/H11509mky//H11509y+ikm kx. /H208491/H20850 /H9275//H208492/H9266/H20850is the driving frequency, /H9268is the sample conductiv- ity, and all components of the microwave ﬁeld are presented as Fourier expansions in the in-plane direction xperpendicu- lar to the transducer longitudinal axis z:m,h,e /H11008/H20848−/H11009/H11009mk,hk,ekexp/H20849i/H9275t/H20850exp/H20849−ikx/H20850dx. These expansions are needed in order to describe the in-plane inhomogeneous mi- crowave ﬁeld of the stripline transducers. Equations /H208491/H20850can be reduced to a single second-order differential equation:38 /H115092hkx//H11509y2−/H20851i/H92620/H9268/H9275+k2/H20852hkx=/H20851i/H92620/H9268/H9275+k2/H20852mkx+ikm ky. /H208492/H20850 An analytical solution to this equation in the form of a Green’s function was obtained in Ref. 38. Here we brieﬂy FIG. 8. /H20849Color online /H20850Calculated proﬁles of the dynamic magnetization and of the total microwave magnetic ﬁeld for the bilayer Py /H2085174nm /H20852/Co/H2085110nm /H20852. /H20849a/H20850and/H20849c/H20850in-plane component of dynamic magnetization mxat 7.5 GHz. /H20849b/H20850 and /H20849d/H20850: total microwave magnetic ﬁeld. Left panels: permalloy layer faces the transducer. Right panels: cobalt layer faces the transducer. In /H20849a/H20850and/H20849c/H20850: thick lines: fundamental mode of the stack; thin lines without arrows: am-plitude /H20841m x/H20841of the ﬁrst higher-order SSWM of the stack; thin lines with arrows: phase the ﬁrst higher-order SSWM /H20849right axes /H20850.073917-9 Kennewell et al. J. Appl. Phys. 108 , 073917 /H208492010 /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: 128.171.57.189 On: Thu, 21 Aug 2014 08:43:21discuss essential peculiarities which follow from Eq. /H208492/H20850for conductive ﬁlms. Waves with nonzero kin the ﬁlm plane are traveling spin waves excited by the inhomogeneous ﬁeld created by thetransducers. 3The total microwave magnetic ﬁeld in the samples hkin Eqs. /H208491/H20850and /H208492/H20850consists of several contribu- tions. The ﬁrst contribution is the ﬁeld of the stripline trans-ducer h ek. Another is the ﬁeld induced by the eddy currents hOekwhich is also created in any nonmagnetic conducting body on whose surface a microwave magnetic ﬁeld hekis incident. The rest are magnetic contributions: the dipole hdk and the effective exchange hexckﬁelds of the precessing mag- netization, and the ﬁeld hprkwhich is created by the eddy currents which are induced by magnetization precession. Thecontributions h ekandhOektohkremain present in Eq. /H208492/H20850 even when mkxis set to zero, thus they represent the excita- tion ﬁeld which drives magnetization precession. In transducer experiments conductivity effects can be signiﬁcant for ﬁlms thinner than a magnetic skin depth, lsm and can considerably modify efﬁciency of excitation of mag- netization precession by an external microwave ﬁeld.19As we discussed in the previous sections, strong inhomogeneityof the internal microwave ﬁeld in conducting samples existsfor thin ﬁlms which is related to the phase of reﬂection of themicrowave ﬁeld from the ﬁlm surface. The theory 19was con- structed for a particular case of FMR driven by very widemicrostrip transducers /H20849k max=0/H20850. One may suppose that simi- lar effects should have been noticed in the traveling wave experiments as well. In the following we show that it isactually not the case, and the broadband microstrip FMRrepresents a unique tool to observe these effects. The microwave magnetic ﬁeld outside the ﬁlm is de- scribed by Eq. /H208492/H20850with a vanishing right-hand part and /H9268 =0. This ﬁeld is a combination of the microwave ﬁeld from the transducer and the dynamic magnetic ﬁeld from the ﬁlm.Solving this equation outside the ﬁlm, and applying the usualelectromagnetic boundary conditions, one arrives at condi-tions for the ﬁelds at the ﬁlm surfaces involving dynamicquantities inside the ﬁlm only. The conditions at the far ﬁlm surface y=L/H20849i.e., at the surface not facing the microwave ﬂux from the transducer /H20850are: ke zk+i/H9275/H20841k/H20841 khxk=0 . /H208493/H20850 For the ﬁlm surface facing the microwave ﬂux y=0, the con- dition is: kezk−i/H9275/H20841k/H20841 khxk=−i/H9275hk/H208490/H20850, /H208494/H20850 where hk/H208490/H20850is some quantity proportional to the stripline transducer ﬁeld hek. From /H208493/H20850and /H208494/H20850it follows that for k=0 the ﬁeld hxk/H20849y =0/H20850=const /HS110050, but hxk/H20849y=L/H20850=0. This is in contrast to the cavity FMR experiments in which the microwave ﬁeld is incident from both sides of the ﬁlm and the boundary condi- tion which is the limiting case of Eq. /H208494/H20850fork=0:hxk=hk/H208490/H20850is satisﬁed at both ﬁlm surfaces y=0 and y=L/H20851see Eq. /H208494.1/H20850in Ref. 33/H20852.Fork/HS110050 one can expect a signiﬁcant microwave mag- netic ﬁeld at the far ﬁlm surface. In the following we showthis with a simple calculation. The effects of conductivityhave been previously considered also by Almeida and Mills 20 in the limit of exchange free, dipolar spin waves and L /H11271lsm. Based on Eqs. /H208493/H20850and /H208494/H20850we extended this theory on smaller L-values. We begin the discussion by noting that according to Eq. /H208492/H20850, it would seem that conductivity will affect the microwave response for /H92620/H9268/H9275values comparable with k2, i.e., for spin waves with wavelengths 2 /H9266/kcompa- rable to the microwave skin depth in the material/H208812//H20849/H92620/H9268/H9275/H20850. However, as shown by Almeida and Mills,20the range of affected in-plane wave numbers is considerably larger. The important parameter turns out to be the micro-wave magnetic skin depth l smwhich can be considerably smaller than the classical skin depth lsc=/H208812//H20849/H92620/H9268/H9275/H20850, espe- cially at frequencies and applied ﬁelds close to those for the in-plane FMR. The dynamic ﬁelds are related by the microwave mag- netic susceptibility tensor /H9273ˆ, deﬁned by mk=/H9273ˆhk. The sus- ceptibility can be found from the linearized Landau–Lifschitz–Gilbert equation. 36This, together with Eqs. /H208491/H20850and /H208492/H20850form a system of equations with solution mk,hk /H11008exp/H20849/H11006Qy/H20850. The out-of-plane wave number Qis larger than the corresponding insulating value /H20849k/H20850, and is also larger than the quantity /H20881k2i/H92620+/H9268/H9275appearing in Eq. /H208492/H20850. The imaginary component of Q2, and the actual magnetic skin depth, are “ampliﬁed” by the off diagonal susceptibility /H20849or magnetic gyrotropy /H20850/H9273a:lsm=lsc//H20881/H9262V, and Q=/H20881k2/H92620−i/H9262V/H9268/H9275, where /H9262ˆ=1ˆ+4/H9266/H9273ˆand/H9262V=/H92622+/H9273a2//H9262.20 FMR in our geometry represents homogeneous preces- sion k=0. In the absence of magnetic losses at the resonance frequency, the diagonal component of the permeability tensorvanishes /H20849 /H9262=0/H20850. However the off diagonal component /H9262a responsible for gyrotropy does not vanish at resonance. This results in divergence of /H9262Vat resonance. In real materials /H9262Vis bounded due to magnetic losses, but nevertheless lsmis nearly an order of magnitude less than lscfor permalloy ﬁlms. For k/H110220 the conductivity contribu- tion to Qbecomes less important, but still large. This effect is shown in Fig. 9/H20849a/H20850. From this ﬁgure one sees that spin waves with in-plane wave numbers up to 40 000 rad /cm are affected by the conductivity. For larger wave numbers theout-plane wave number has the same asymptotic behavior asfor a respective insulating ﬁlm of the same thickness /H20851dotted line in Fig. 9/H20849a/H20850/H20852. The consequences for resonant absorption due to this enhanced skin depth appear when one considers the striplinegeometry. Spin waves propagating in the ﬁlm plane are ex-cited within an area in which the transducer’s magnetic ﬁeldis largest. They travel out of this region, mostly in directionsperpendicular to the transducer axis. It is known that spinwaves are excited by stripline transducers resonantly, so thata microwave ﬁeld of the frequency /H9275excites a spin wave with the same frequency /H20849see, e.g., Refs. 26and39–41/H20850with in-plane wave number kdetermined by the spin wave disper- sion/H9275/H20849k/H20850. The amplitude of an excited spin wave with wave073917-10 Kennewell et al. J. Appl. Phys. 108 , 073917 /H208492010 /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: 128.171.57.189 On: Thu, 21 Aug 2014 08:43:21vector kis proportional to the amplitude of the corresponding spatial Fourier-component of the transducer’s microwavemagnetic ﬁeld h ek. The spectrum of Fourier-components for the microwave ﬁeld from the microstrip antenna /H20851Fig. 1/H20849b/H20850/H20852is/H20841hek/H20841 /H11008sin/H20849kw /2/H20850//H20849kw /2/H20850/H20851see Eq. /H2084938/H20850in Ref. 42/H20852. The ﬁrst zero of this function is located at k=2/H9266/w. The Fourier spectrum of the microwave ﬁeld of coplanar transducers /H20851Fig.1/H20849a/H20850/H20852has a more complicated shape with several minor lobes groupedtogether /H20849see, e.g., Fig. 3 in Ref. 38/H20850. It is usually assumed that the wave number bandwidth excited by a coplanar trans-ducer is given by the width of the ﬁrst major lobe of theFourier spectrum. The peak with two maxima of differentamplitudes located between k=0 and 200 rad/cm in Fig. 1 in Ref. 38is an example of such a major lobe. From this we conclude that for both types of transducers the wave numberrange for spin wave excitation is from k min=0 to kmax =2/H9266/wchar, where wcharis the characteristic width of the transducer. For the microstrip transducers, wcharcoincides with the width of the microstrip win the xdirection. For coplanar transducers wchar=w+2/H9004, where wis the width of the center conductor and /H9004is the separation of the central conductor from a ground half-plane.38,40 Transducers having a characteristic width of 2–5 /H9262m were used in experiments on traveling spin waves reported inRefs. 28and43–47. These correspond to wave numbers k max=104–3/H11003104rad /cm. Typically, a single-layer 30–40 nm thick permalloy ﬁlm was utilized /H20849except for Ref. 28 where the ﬁlm was 200 nm thick /H20850. The calculated results shown in Fig. 9are for a 40 nm thick ﬁlm for comparison. One sees that conductivity effects in Q/H20849k/H20850/H20851Fig.9/H20849a/H20850/H20852appearthroughout the entire transducer wave number range. Fur- thermore, in BLS experiments the accessible wave numberrange extends to 2.5 /H1100310 5rad /cm/H20849see e.g.48/H20850, hence a large part of the accessible wave number range is affected also. Nevertheless, zero conductivity models the frequencies work exceptionally well for traveling wave experiments,such as BLS, and for coplanar-transducer driven travelingspin waves experiments on 30–40 nm thick conductingﬁlms. 45The reasons are very illuminating, and provide in- sight into the fundamental nature of conductivity effects. Observable quantities in traveling wave experiments are spin wave dispersion /H20849usually from BLS /H20850and spin wave am- plitudes /H20849most easily measured using transducer techniques /H20850. As pointed out in Ref. 20conductivity affects strongly the out-of-plane wave number. However this produces a rela-tively weak modiﬁcation of the spin wave dispersion. In-deed, one sees in Fig. 9/H20849a/H20850the dispersion calculated with nonzero conductivity and using the boundary conditions /H208493/H20850 and /H208494/H20850agrees very well with the dispersion calculated with zero conductivity /H20849thick solid line in this ﬁgure /H20850. From the values of the out-of-plane wave numbers in Fig.9/H20849a/H20850and the boundary conditions /H20851Eqs. /H208493/H20850and /H208494/H20850/H20852one can calculate the amplitude of the microwave magnetic ﬁeldat the far ﬁlm surface. This result is shown in Fig. 9/H20849b/H20850as h xk/H20849y=L/H20850/hxk/H20849y=0/H20850versus k/H20849red solid line /H20850. From this ﬁgure one ﬁnds that the strong ﬁeld inhomogeneity disappears at a k-value ksabout 200 rad/cm /H20849seen as hxk/H20849y=L/H20850/hxk/H20849y=0/H20850 =0.65 for this k-value /H20850. At yet larger ks-values this depen- dence approaches the one for an insulating ﬁlm /H20851blue dashed line in Fig. 9/H20849b/H20850/H20852. One can relate this to disappearance of the contribution of the eddy current ﬁeld hOekto the total dy- namic magnetic ﬁeld of the ﬁlm. In the case of transducer studies of traveling spin waves,43–47the wave number range from 0 to 200 rad/cm represents a small portion of the full transmission character-istic accessible. The remainder of the range is expected to bethat of a magnetic insulator. The smallness of the affected wave number range is the main reason why eddy currents arenot important in the traveling wave experiments with SSDFﬁlms. We can now understand the essential differences for typical broadband FMR experiments. 2,37The free spin wave propagation path depends on magnetic losses in the materialand on the sample thickness. Film thicknesses are typicallybelow 100 nm, and the transducers are orders of magnitudelarger /H20849in our case the width of the microstrip is w =1.5 mm /H208492 /H9266/w=40 rad /cm/H20850, and the width of the copla- nar waveguide at w=0.35 mm, with /H9004=0.6 mm /H208492/H9266//H20849w +2/H9004/H20850=41 rad /cm/H20850. These systems therefore fall in the lim- iting broad case kmax/H11270ks. Typically the transducer width is chosen such that it is larger than the free spin wave propa-gation path l SW=/H9251/H9275/Vgin order to ensure a quasihomoge- neous microwave ﬁeld in the ﬁlm plane /H20849here/H9251is the Gilbert damping constant and Vgis the spin wave group velocity /H20850.I n this way the traveling spin wave contribution3to the absorp- tion linewidth is minimized. However, for this geometry kmax is virtually zero, so that eddy currents should have a major impact on the magnetization precession, as seen from Fig. 9. Note that the eddy current effects cannot be observed di-/g44/g81/g16/g83/g79/g68/g81/g72/g90/g68/g89/g72/g81/g88/g80/g69/g72/g85 /g78/g11/g85/g68/g71/g18/g70/g80/g12/g19 /g21/g19/g19/g19 /g23/g19/g19/g19 /g25/g19/g19/g19/g41/g76/g72/g79/g71 /g68/g80/g83/g79/g76/g87/g88/g71/g72 /g75/g91/g78/g11/g47/g12/g18/g75/g91/g78/g11/g19/g12 /g19/g17/g19/g19/g17/g23/g19/g17/g27 /g41/g85/g72/g84/g88/g72/g81/g70/g92/g11/g42/g43/g93/g12 /g20/g22/g17/g27/g20/g22/g17/g28/g20/g23/g17/g19/g20/g23/g17/g20/g20/g23/g17/g21/g44/g81/g16/g83/g79/g68/g81/g72/g90/g68/g89/g72/g81/g88/g80/g69/g72/g85 /g78/g11/g20/g19/g24/g85/g68/g71/g18/g70/g80/g12/g19/g17/g19 /g19/g17/g23 /g19/g17/g27/g58/g68/g89/g72/g81/g88/g80 /g69/g72/g85/g52 /g11/g20/g19/g24/g85/g68/g71/g18/g70/g80/g12 /g19/g17/g19/g19/g17/g23/g19/g17/g27 /g41/g85/g72/g84/g88/g72/g81/g70/g92/g11/g42/g43/g93 /g12 /g20/g23/g20/g24/g20/g25/g20/g26 /g11/g68/g12 /g11/g69/g12 FIG. 9. /H20849Color online /H20850Solution of the exchange-free equations. /H20849a/H20850: red solid line: real part of the out-of-plane wave number Q. Blue dashed line: its imaginary part. Green dotted line: asymptotics Q=kvalid for large wave numbers k./H20849b/H20850: Thin red solid line: relative amplitude of the microwave magnetic ﬁeld hxk/H20849y=L/H20850/hxk/H20849y=0/H20850at the far ﬁlm surface y=Lfor a conduct- ing ﬁlm. Thin blue dashed line: the same but for an insulating ﬁlm. Greendotted line: exponential asymptotics exp /H20849−ky/H20850valid for both for large wave numbers k. Thick black solid line in both panels: Dispersion of the Damon– Eshbach wave in an insulating ﬁlm. It coincides with graphical accuracywith the result obtained including the electric conductivity. Parameters ofcalculation: ﬁlm thickness: 40 nm, ﬁlm saturation magnetization 4 /H9266Ms =10 000 G, applied ﬁeld is 2000 Oe, ﬁlm conductivity is 4.6 /H11003106S/m.073917-11 Kennewell et al. J. Appl. Phys. 108 , 073917 /H208492010 /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: 128.171.57.189 On: Thu, 21 Aug 2014 08:43:21rectly, however as shown in Ref. 19looking at exchange effects with broadband FMR allows one to determine theirpresence. Furthermore, neither traveling wave experiments nor cavity FMR results are noticeably impacted by the eddy cur-rents in the sample. For the traveling wave experiments h Oek andhprkare negligible compared with hdk. For the cavity FMR, although hOekandhprkare present and large, they are quasihomogeneous across Land thus simply renormalize the homogeneous driving ﬁeld. Furthermore, experiments onBLS on thermal magnons /H20849see, e.g., Ref. 48/H20850do not reveal any noticeable impact of the eddy currents either, as the onlypossible eddy current contribution to the total ﬁeld of ther-mal magnons is h prkwhich has a negligible effect on magnon dispersion. VII. CONCLUSION In this work we studied experimentally the broadband FMR responses for metallic single-layer and bilayer mag-netic ﬁlms with total thicknesses smaller than the microwavemagnetic skin depth. We found that stripline transducers withcharacteristic width larger than the free propagation path fortraveling spin waves along the ﬁlm efﬁciently excite higher-order SSWMs across the ﬁlm thickness in samples 30–90 nmthick. We ﬁnd a strong variation in the amplitudes with fre-quency for cobalt–permalloy bilayers. The ratio is stronglydependent on the ordering of layers with respect to a striplinetransducer. Most importantly, cavity FMR measurements onthe same samples show considerably weaker amplitudes forthe standing spin waves. All experimental data are consistentwith expected effects of eddy currents in ﬁlms with thick-nesses below the microwave magnetic skin depth. The observed microwave magnetic dynamics in conduct- ing ﬁlms driven by wide stripline transducers is thus quitedifferent from that observed in the traveling wave experi-ments and cavity FMR experiments on the submagnetic skindepth ﬁlms. Our theory provides an explanation of how thisdifference arises. ACKNOWLEDGMENTS Australian Research Council support through Discovery Projects and Postgraduate Awards is acknowledged. We alsoacknowledge support from the University of Leeds, theEPSRC, and from the University of Western Australia. 1A. Fessant, J. Geraltowski, J. Loaec, and H. Legall, J. Magn. Magn. Mater. 83,5 5 7 /H208491990 /H20850. 2T. J. Silva, C. S. Lee, T. M. Crawford, and C. T. Rogers, J. Appl. Phys. 85, 7849 /H208491999 /H20850. 3G. Counil, J.-V. Kim, T. Devolder, C. Chappert, K. Shigeto, and Y. Otani, J. Appl. Phys. 95, 5646 /H208492004 /H20850. 4G. Counil, P. Crozat, T. Devolder, C. Chapper, S. Zoll, and R. Fournel, IEEE Trans. Magn. 42, 3321 /H208492006 /H20850. 5C. 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1.4864793.pdf	A study of infrasound propagation based on high-order finite difference solutions of the Navier-Stokes equations O. Marsden,a)C. Bogey, and C. Bailly Laboratoire de M /C19ecanique des Fluides et d’Acoustique, UMR CNRS 5509, Ecole Centrale de Lyon, Universit /C19e de Lyon, 69134 Ecully cedex, France (Received 30 September 2013; revised 14 January 2014; accepted 28 January 2014) The feasibility of using numerical simulation of ﬂuid dynamics equations for the detailed description of long-range infrasound propagation in the atmosphere is investigated. The two dimensional (2D) Navier Stokes equations are solved via high ﬁdelity spatial ﬁ nite differences and Runge-Kutta time integration, coupled with a shock-capturing ﬁlter procedure all owing large amplitudes to be studied. The accuracy of acoustic prediction over long distances with this approach is ﬁrst assessed in the linear regime thanks to two test cases featuring an acoustic source placed above a reﬂective ground in a homogeneous and weakly inhomogeneous medium, solved for a range of grid resolutions. An atmospheric model which can account for realistic features affecting acoustic p ropagation is then described. A 2D study of the effect of source amplitude on signals recorded at gr ound level at varying distances from the source is carried out. Modiﬁcations both in terms of wave forms and arrival times are described. VC2014 Acoustical Society of America .[http://dx.doi.org/10.1121/1.4864793 ] PACS number(s): 43.28.Dm, 43.28.Js, 43.28.Fp [RMW] Pages: 1083–1095 I. INTRODUCTION It has been known since the early modern period and the rapid development of cannons in warfare that low frequencysounds produced on battle ﬁelds can be heard at great dis- tance, and also that sound amplitude is a non-monotonous function of distance from the source. A detailed history ofsuch observations as well as the chronology of work aiming to explain propagation phenomena can be found in the review article by Delany. 1 Although most of the theory regarding long-distance acoustic propagation in the atmosphere is now agreed upon, the prediction of the time signature of a given sourceat a given distance remains a complicated task, due to the variety of phenomena which affect propagation. A brief, non-exhaustive list includes convection due to wind, refrac-tion due to both temperature and wind speed gradients, scattering on smaller scale meteorological inhomogeneities, non-linear waveform distortion, caustics, atmosphericabsorption, ground and terrain effects. 2–5Simpliﬁed model- ing approaches easily amenable to propagation over long distances are not able to account for all of these physicaleffects. This is the case for approaches based on the fast ﬁeld program (FPP) and parabolic equations (PE), which have been used extensively to study acoustic propagation. 6–8Such approaches provide approximate solutions to the wave equa- tion, and suffer from limitations due both to the resolution technique, e.g., FFP is limited to horizontally homogeneousproblems while PE methods have angular limitations, and to the wave equation itself, in particular its linearity. It should be noted that recently developed non-linear parabolic equa-tions 9alleviate the latter problem. Ray tracing, based on the geometrical acoustics approximation, has also been widelyemployed for propagation problems since the early work of Blokhintsev.10,11State of the art ray tracing developments allow ﬁnite amplitude signals to be modeled along ray trajec- tories.12However ray tracing does not in itself predict scat- tering and diffraction due to atmospheric inhomogeneitiesand caustics, both of which can have a signiﬁcant impact on pressure signals received far from the source. 13,14Recently, efforts have been made toward long-range propagation stud-ies based directly on the full Navier-Stokes equations, 15–17 or on a set of linearized ﬂuid dynamic equations.18–20The full set of equations should allow a correct description of thewhole gamut of propagation effects mentioned previously, but the resolution of these equations in a computationally affordable way requires the use of well suited numericaltechniques. Accurate predictions of long distance propaga- tion in realistic conditions is of use to a range of ﬁelds, including international military monitoring 21and atmos- pheric studies.22 In this work, the full two-dimensional Navier-Stokes equations are solved to model the propagation of low-frequency sound waves through the atmosphere. The atmos- phere is modeled from ground level to an altitude of 160 km. It is stratiﬁed due to gravity, and has a mean temperatureproﬁle which mimics the large-scale variations observed in experimental proﬁles. 14,23Explicit ﬁnite differences based on 11-point stencils are used to compute the spatial deriva-tives involved in the Navier-Stokes equations. Time integra- tion is performed with a six-stage optimized Runge-Kutta scheme. Additionally, a shock-capturing ﬁltering techniqueis employed in order to handle the discontinuities that appear in the vicinity of shock waves. The full numerical algorithm has been implemented in a functionally equivalent mannerin FORTRAN 90 and in OpenCL allowing performance to be ascertained on a variety of hardware. Benchmark problems in both homogenous and inhomogeneous atmospheric condi-tions are used to ascertain the effect of grid resolution ona)Author to whom correspondence should be addressed. Electronic mail: olivier.marsden@ec-lyon.fr J. Acoust. Soc. Am. 135(3), March 2014 VC2014 Acoustical Society of America 1083 0001-4966/2014/135(3)/1083/13/$30.00 long range acoustic predictions. The effect of source amplitude on long-distance 2D propagated time signals is examined. The paper is organized as follows. After a general intro- duction, the set of equations and numerical algorithm are described in Sec. II. Section IIIis devoted to an examination of the algorithm’s ﬁdelity as a function of grid resolution, for two propagation test problems. In Sec. IVa description of time signals resulting from long range propagation isgiven, along with a study of the modiﬁcations of time signa- tures due to variations in source amplitude. II. NUMERICAL ALGORITHM Propagation over long distances of large-amplitude sounds is not correctly described by a linear model, as willbe illustrated later in this paper. Therefore, ﬂuid motion is modeled in this work in two dimensions with the standard non-linearized ﬂuid dynamics equations, namely that ofmass conservation, the Navier-Stokes equation and an energy equation, completed by the perfect gas law. In Cartesian coordinates, this set of equations governing theﬂow variables U¼ðq;qu 1;qu2;qetÞT, where etis the spe- ciﬁc total energy given for a perfect gas by qet¼p=ðc/C01Þþ1=2qu2 i, can be written as @U @tþ@E1 @x1þ@E2 @x2/C0@V1 @x1/C0@V2 @x2þ@Q1 @x1þ@Q2 @x2þC¼0; (1) where the Eulerian, viscous and thermal ﬂuxes are deﬁned by E1¼½qu1;pþqu2 1;qu1u2;ðqetþpÞu1/C138T; E2¼½qu2;qu1u2;pþqu2 2;ðqetþpÞu2/C138T; V1¼ð0;s11;s12;u1s11þu2s12ÞT; V2¼ð0;s21;s22;u1s21þu2s22ÞT; Qi¼½0;0;0;/C0ðlcp=rÞ@T=@xi/C138T; with sij¼lð@ui=@xjþ@uj=@xi/C02=3dij@uk=@xkÞ; and C¼ð0;0;qg;qgu2ÞT; (2) where lis the dynamic viscosity coefﬁcient, ris the Prandtl number of the ﬂuid, cpis the speciﬁc heat at constant pres- sure, and cis the ﬂuid’s equilibrium speciﬁc heat ratio. The previous set of Eqs. (1)and(2)does not describe molecular relaxation effects, which contribute to acousticabsorption and dispersion during propagation. Although the set of equations can be modiﬁed to account for these dissipa- tive and dispersive effects, 17,24they have been shown to be small when compared to other phenomena involved in long- range propagation.24Accordingly, these effects are not mod- eled in this work. This simpliﬁcation avoids the need to trackthe individual gaseous components of the atmosphere, allow- ing the mixture to be modeled as an equivalent perfect gas. Mean properties of the atmosphere are very strongly altitude-dependent, due in large part to the gravity-driven density stratiﬁcation, and to the temperature proﬁle. Indeed, over the altitude range relevant to long-distance propagation,from 0 to around 160 km, both average pressure and average density diminish by a factor of almost 10 orders of magni-tude. This poses signiﬁcant numerical difﬁculties, for exam- ple simply to ensure stability of the mean proﬁles, 25which stems from the hydrostatic equilibrium condition d/C22p=dx2¼ /C0/C22qgwhose ﬁnite difference approximation must be veriﬁed numerically to a high degree of accuracy. Rewriting Eq. (1) as @U @tþ@E1 @x1þ@E02 @x2/C0@V1 @x1/C0@V2 @x2þ@Q1 @x1þ@Q2 @x2þC0¼0; (3) with C0¼½0;0;ðq/C0/C22qÞg;ðq/C0/C22qÞgu2þ/C22p@u2=@x2/C138TandE02 ¼fqu2;qu1u2;ðp/C0/C22pÞþqu2 2;½qetþðp/C0/C22pÞ/C138u2gTis mathe- matically equivalent as long as mean ﬁelds are invariant in thex1direction, but numerically far more favorable because the aforementioned average hydrostatic stability condition isnot computed numerically at each time step. These equations are solved on a regular Cartesian grid with an optimized high-ﬁdelity numerical procedure based on explicit spatial ﬁ-nite differences and Runge-Kutta time integration. Where possible, spatial discretization is performed with explicit fourth-order 11-point centered ﬁnite differences optimized tominimize dispersion for wavenumbers discretized by between four and 32 grid points. 26Close to boundaries, be they the ground or radiation conditions, optimized explicitnon-centered differencing schemes are used. 27The non- centered differencing schemes are all based on 11-point sten- cils, including the one-sided stencil used for wall points. Timeintegration is performed with a six-step second-order opti- mized low-storage Runge-Kutta algorithm. 26Characteristics regarding dispersion and dissipation for the spatial differenc-ing schemes and the time integration scheme can be found in previous papers. 26,27The schemes’ properties mean that the behavior of waves discretized by at least four points per wave-length is accurately reproduced, with very low levels of dis- persion and dissipation, and is stable up to frequencies such thatxDt/C201:25/C2p. The determination of the computational time step Dtis based on a CFL (Courant-Friedrichs-Lewy) condition, CFL ¼c maxDt=Dxmin,w h e r e cmaxis the largest value of the speed of sound in the atmosphere modeled here,andDx minthe smallest grid spacing in the mesh. A value of CFL¼0:5 is used throughout this work. The ground is mod- eled as a non-slip boundary condition, except for the inviscidvalidation test cases, where the ground is modeled with a slip condition. The wall-point ﬂow variables are updated by solv- ing the governing equations with the aforementioned high-order non-centered differencing schemes. Spatial low-pass ﬁltering is carried out to ensure stable computations. An explicit sixth-order 11-point ﬁltering sten-cil is designed to remove ﬂuctuations discretized by less than four grid points per wavelength, while leaving larger wavelengths effectively untouched. 28As the differencing schemes used near boundaries are asymmetric, their effec- tive wavenumbers have an imaginary part which leads to them being unstable for very high frequencies.29It is there- fore, essential to use them in conjunction with appropriate highly selective ﬁlters, and to this end, the ﬁlters described 1084 J. Acoust. Soc. Am., Vol. 135, No. 3, March 2014 Marsden et al. : Simulation of infrasound propagationin Berland et al. ,27which also selectively damp ﬂuctuations with fewer than four points per wavelength have been imple-mented. Filters for grid points more than two points away from a boundary are built on 11-point stencils, while stencils for the wall point and for the ﬁrst point away from the wallare built on four and seven points, respectively. Thus at the ground, in the x 1direction, the centered 11-point ﬁlter is used, whereas in the x2direction the family of non-centered ﬁlters is applied. At the lateral radiation boundaries in the x1andx2direc- tions, Tam and Webb’s 2D far-ﬁeld radiation condition30is applied. The left and right radiation conditions are supple- mented by sponge zones combining grid stretching and low-order spatial ﬁltering, a technique commonly used incomputational aeroacoustics. 31A simple radiation boundary condition along the top boundary generates unsatisfactorily large reﬂected waves, which contaminate pressure signals atground level. Unfortunately, stratiﬁcation of the atmosphere due to gravity renders the top boundary less amenable than the side boundaries to a standard sponge layer approach.Indeed, for a given source strength the ratio of generated pressure ﬂuctuations to ambient pressure increases with alti- tude, 32and is proportional to ð1=rÞﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ c=ð/C22c2/C22pp Þin 3D, and ð1=ﬃﬃrpÞﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ c=ð/C22c2/C22pp Þin 2D as in the present work, where ris the propagation distance since the source. This means that extending the computational domain vertically with a spongezone will also increase the relative amplitude of the ﬂuctua- tions needing to be evacuated through the boundary condi- tion. The radiation condition being based on the linearizedEuler equations, 30the amplitude of spurious reﬂected waves generated by the boundary condition will increase linearly with the amplitude of physical outgoing waves. Thus, atground level, the spurious reﬂected wave amplitude does not decrease as could be expected when a simple sponge zone is applied. Instead, in this work, the useful computational do-main is extended vertically with a sponge zone in which the gravity proﬁle varies progressively from its expected value gðx 2Þto/C0gðx2Þfollowing a tanh ðx2=hÞvariation where his set to three wavelengths of the main source frequency at the altitude in question. Once negative gravity has been reached, the ratio of pressure ﬂuctuations to ambient pressure no lon-ger increases but decreases, allowing standard grid stretching techniques and low-order dissipation to operate as in a typi- cal sponge zone used in the computational aeroacousticscommunity. This approach allowed spurious reﬂections to be reduced to background error levels. Non-linear effects can be important in long-range atmospheric propagation, due both to the sizable propagation distances and to large relative amplitudes which are reached in the high atmosphere. For strong amplitude sources, acous-tic shocks are rapidly formed close to the source. It is also surmised that shock fronts may be formed in the thermo- sphere, regardless of the source amplitude. This aspect posespotential problems for standard ﬁnite-difference time-do- main acoustic solvers, which are not designed to cope with steep wave fronts and which can lead to unacceptably largeGibbs oscillations and divergent computations. The compu- tational ﬂuid dynamics (CFD) community has been dealing with shocked ﬂows for a long time, and has developed avariety of different techniques to avoid numerical problems associated with the presence of shocks. Standard shock-capturing schemes developed for CFD are however ill-suited to time-dependent problems, because they exhibit poor spec- tral accuracy, 33and tend to be excessively dissipative, par- ticularly in the context of long-distance propagation. Hence in this work we employ a non-linear ﬁltering method designed with acoustics and aeroacoustics in mind.28The methodology consists in applying a second-order conserva- tive ﬁlter only where necessary, i.e., only in the vicinity of shock fronts. Understandably, much of the method’s proper-ties come from the non-linear detection algorithm. Non- linear zones are identiﬁed thanks to a Jameson-like detector based on pressure ﬂuctuations. The ﬁrst step consists inextracting the high wavenumber components from pressure ﬂuctuations. This is done by applying a second-order ﬁlter- ing, as described in the following equation for grid point i,i n thex 1direction: Dpi¼ð /C0 piþ1þ2pi/C0pi/C01Þ=4 (4) and then deﬁning the high-pass ﬁltered squared pressure ﬂuctuation as D2 i¼1 2½ðDpi/C0Dpi/C01Þ2þðDpi/C0Dpiþ1Þ2/C138: (5) This squared pressure ﬂuctuation is used to deﬁne a sensor value as ri¼D2 i pi2þ/C15; (6) where /C15is a small parameter, typically 10/C016, whose role is to avoid numerical problems when dividing by ri, as will be seen subsequently, and is the averaged pressure at point i. The self-adjusting ﬁltering strength riat grid point iis com- puted according to ri¼1 21/C0rth riþ/C12/C12/C12/C121/C0rth ri/C12/C12/C12/C12 ! ; (7) where rthis a threshold constant whose value is rth¼10/C05. This ﬁltering strength has the desired properties of being equal to zero away from shocks, where ri<rth, and of increasing toward a value of 1 for increasing shock inten-sities. Conservative variables are ﬁltered conservatively, i.e., the ﬁltered term is computed as a difference of two ﬂuxes, as follows: U f i¼Ui/C0ar iþ1=2Fiþ1=2/C0ri/C01=2Fi/C01=2/C0/C1; (8) where riþ1=2is simply the average of previously calculated ﬁltering strengths riandriþ1, and Fiþ1=2¼Pn j¼1/C0ncjUiþj andFi/C01=2¼Pn j¼1/C0ncjUiþj/C01are the up-winded and down- winded ﬂuxes based on a dissipative second-order ﬁlter cj. The amplitude ais there to allow a ﬁne adjustment of the total ﬁltering magnitude, as will be seen subsequently. It is, in gen-eral, set to a¼1. A spectral analysis of the shock-capturing J. Acoust. Soc. Am., Vol. 135, No. 3, March 2014 Marsden et al. : Simulation of infrasound propagation 1085treatment is in Bogey et al. ,28and an example of the scheme’s behavior for an acoustic signal typical of longrange propagation scenarios is provided in Sec. IV C 1 . Examples of 2D acoustic diffraction and aeroacoustic ﬂows successfully simulated with the solver described in thiswork can be found in Marsden et al. 34and Berland et al.27 The same methodology has been applied to numerous aeroa- coustic studies in 3D,35–38and the generalization of the cur- rently described solver to three dimensions should pose no difﬁculties other than that of the computational cost of the resulting 3D computations. Finally, the entire code has been ported from FORTRAN 90 parallelized with OpenMP, to OpenCL, maintaining thecode functionally identical. This has allowed performanceto be tested on a variety of hardware, both CPU (Central Processing Unit) and GPU (Gr aphics Processing Unit). Results from this non-exhausti ve testing are summarized in Table I. The baseline is taken as the execution speed of the optimized Fortran code on a problem of size 5 M points in which IO has been removed, compiled with the IntelFortran compiler and run on a single Intel X5550 processor core (2.67 GHz). The same code, run in shared memory OpenMP mode on four cores, performs almost three timesfaster, providing reasonable strong scaling. Intel has rela- tively recently published an O penCL toolkit allowing code to be run on their CPUs. The OpenCL code running on fourcores slightly outperforms the OpenMP F90 version. Performance on the GPUs is vastly better. On a previous generation NVIDIA card destined to the HPC market, per-formance is 26 times higher than on a single CPU core, while on the current generation general public card from AMD, performance is an impressive 57 times higher. ThisGPU performance, combined with the CPU performance obtained with essentially the same code base, highlights the versatility and power of OpenCL as a cross-platform com-puting language. It should be noted that although very good performance is possible over a wide range of hardware, this performance is not obtained automatically. Code tuning foreach architecture was found to be highly beneﬁcial in this work, in particular to take into account large architectural differences between CPU and GPU, but also to accommo-date for smaller differences between GPUs, such as the size of local memory. III. APPLICATION TO BENCHMARK PROBLEMS The solver is applied to two test cases in order to study grid requirements for the correct representation of important physical phenomena relevant to long-distance atmospheric propagation.A. Acoustic source in a homogeneous atmosphere The ﬁrst problem is the inviscid linear prediction of the acoustic ﬁeld resulting from a harmonic monopole sourcelocated near a ﬂat rigid surface in a homogeneous atmos- phere. The analytical solution to this problem was published by Morse and Ingard, 39and results simply from the sum of the monopole’s direct radiated ﬁeld and of the ﬁeld radiatedby the monopole’s image with respect to the wall. For the present study, a monopolar source at a frequency of 100 Hz is placed at a height of 20 m above a wall, as in the work ofWilson and Liu 40and Ostashev et al.18The source shape is given by a Gaussian of half-width 0.8 m. Ambient pressure is set to 105Pa, and ambient density such that speed of sound is equal to c0¼340 m/s. In order to ensure negligible non- linear steepening, the monopolar source amplitude is set to a very low value of 10/C02Pa. Numerical solutions are com- pared to the analytical solution along a line parallel to the ground at the source altitude of 20 m, over a distance of 100 m. They are compared in terms of transmission loss,TLðxÞ¼20 log 10½pðxÞ=pref/C138where prefis the pressure ampli- tude at 1 m from the same monopole in a free ﬁeld, and of error, deﬁned here according to /C15¼ð1=99ÞÐ100 1jTLaðxÞ /C0TLsðxÞjdxwhere subscripts aandsrefer to the analytical and simulated transmission losses respectively. A grid con- vergence study is carried out, with discretizations ranging from just over four points per wavelength (ppw), corre-sponding to a Dxof 0.8 m, to 42 points per wavelength, i.e., Dx¼0:08 m. The computational domains cover the range /C0150/C20x 1/C20150 and x2/C20150, and grid sizes range from around 105grid points for the smallest mesh, to 107points for the ﬁnest mesh. Figure 1presents the results, with the transmission loss computed on the coarsest grid compared tothe analytical solution in Fig. 1(a), and error as a function of discretization in Fig. 1(b). Characteristic interference lobes are observed in the transmission loss, and the computed solu- tion with the coarsest discretization of only 4 ppw, despitesigniﬁcant attenuation, provides correct trends for these lobes, over the entire 100 m of the propagation domain. This distance corresponds to approximately 30 wavelengths, andthus the algorithm’s capacity to propagate acoustics with rel- atively low discretizations over long distances is illustrated. Figure 1(b) shows the convergence of the numerical so- lution toward the analytical one as the computational grid is reﬁned. This study is carried out with a constant value of CFL¼0.5, which means the time step varies linearly with Dx. In order for the cumulative effect of the high-order low- pass ﬁltering to be comparable for the different simulations, the low-pass ﬁltering strength r 27is chosen to be inversely proportional to the time step. For example, dividing the grid step by two also halves the time step, and the ﬁltering coefﬁ- cient ris therefore also halved. Mesh and simulation param- eters for this study are provided in Table II. The variation of error /C15with grid density is observed to be very close to third order, and thus the overall solver behavior here is of secondorder. This second-order behavior is due to the fact that for this relatively large CFL, time integration, performed with a second-order Runge-Kutta algorithm here, is the leadingsource of error. If a smaller CFL value were chosen, suchTABLE I. Normalized code performance, on one (1C) or four (4C) Intel X5550 CPU cores and on two different GPUs, for a 5 million point simulation. Intel X5550 (1C)X5550 (4C)X5550 (4C)NVIDIA M2050AMD HD7970 F90 F90-OpenMP OpenCL OpenCL OpenCL 1 2.8 3.2 26 57 1086 J. Acoust. Soc. Am., Vol. 135, No. 3, March 2014 Marsden et al. : Simulation of infrasound propagationthat the leading error term were due to the spatial differenc- ing, the ﬁnite difference scheme’s order would dictate the variation of /C15with grid density. B. Acoustic source in an upwardly refracting atmosphere A similar investigation is c arried out for an atmosphere with a vertically stratiﬁed spee d of sound, a condition typically found in a realistic atmosphere . Over long propagation distan- ces, acoustic refraction due to a gradient in sound speed canresult in the formation of wave guides or zones of silence, depending on the sign of the temperature gradient. 41Over the relatively small distance consid ered in the previous homogene- ous test case, and for a moderate celerity gradient, acoustic refraction will simply modify sli ghtly the transmission loss pro- ﬁle, in particular changing the d istance between consecutive TL extrema. This conﬁguration should both test the reﬂection con- dition in more realistic conditions , and demonstrate that refrac- tion effects are correctly captured by the numerical solution. Forthe present study, the temperature gradient is chosen negative in order to yield an upward refract ing atmosphere as often found at ground level in long-range propagation problems. As in the homogeneous case, a monopole at a frequency of 100 Hz is placed 20 m above a rigid ground. Ambient pressure is again set to 10 5Pa, but the speed of sound is now a function of altitude x2, and given by c0ðx2Þ¼c0=ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þ2x2=r0p , sometimes referred to as an n2-linear medium. Values of c0¼340 m/s and r0¼338:5 m are chosen to pro- vide a roughly linear variation of celerity close to the ground, of dc0=dx2’1=s. It can be noted that an analytical solution can be written if density variations are neglected,although it is rather more involved than for a homogeneous atmosphere. Developments for the axisymmetric case can be found in Chap. 9–5 of Pierce’s book 42and in Sec. 2.5.1 of Computational Ocean Acoustics .43In a gravitationally strati- ﬁed atmosphere, density decreases exponentially with alti- tude, but with the parameters of the present study theanalytical approximation is indistinguishable from the nu- merical solution obtained on the ﬁnest grid. The quality of the computational ﬁeld is assessed by comparing the numeri-cally obtained TL over a distance of 100 m to that obtained on the ﬁnest grid. Figure 2(a)provides a comparison of the reference trans- mission loss, computed on the ﬁnest grid with Dx¼0:06 m and that obtained on the coarsest mesh with Dx¼0:8m . I n Fig.2(b), the effect of grid resolution on computational error /C15is shown, for 4 :25/C20ppw/C2042:5 at ground level. It should be noted that due to the temperature proﬁle, the acoustic wavelength, and therefore also the grid resolution, diminishwith altitude. The numbers of points per wavelength are accordingly all given at ground level. As in the homogeneous case, even the coarsest grid resolution of 4.25 ppw provides areasonable match to the analytical solution, and error /C15is found to decrease again roughly with the third power of grid reﬁnement. Comparing Figs. 1(b) and2(b), it can be noted that for a given kDx, error is slightly higher for the temperature-stratiﬁed case than for the homogeneous case. This can be attributed to the reduction in acoustic wavelengthdue to thermal stratiﬁcation. These two propagation test cases show that the solver is capable of accurate predictions of both acoustic amplitudeand phase at long distances from a source, with grid discreti- zations greater than four points per wavelength. The solver is also shown to maintain its overall accuracy in the presenceof slowly varying atmospheric inhomogeneities. IV. SOURCE AMPLITUDE EFFECTS ON LONG DISTANCE PROPAGATION The numerical algorithm described in the previous sec- tions is used to perform an investigation of the effect of sourceamplitude on time signatures for long-distance infrasound propagation through a realistic atmosphere. After a presenta- tion of the problem’s atmospheric conditions and of the acous-tic source, a brief description of the resulting acoustic signals at long range is given. Aspects relating to the results’ accuracy are investigated, before discussing modiﬁcations to signalsrecorded at long range due to source amplitude. A. Atmospheric configuration The atmosphere used in this work is described below. Air is modeled as a perfect gas, with values of speciﬁc heat FIG. 1. (a) Transmission loss (TL) as a function of distance for a monopole above a rigid ground: — analytical solution, – – – – solution computed onDx¼0:8 m grid, i.e., 4.25 ppw. (b) Computational error /C15as a function of grid discretization in points per wavelength (ppw): – þcomputed error, –––– third order variation (/C15/C24ppw/C03). TABLE II. Grid and simulation parameters: Dx, time step Dtand ﬁltering strength rfor homogeneous atmosphere test case grid convergence study. Dx 0.08 0.125 0.25 0.5 0.8 Dt1:18/C210/C041:84/C210/C043:68/C210/C047:35/C210/C040.0012 r 0.064 0.1 0.2 0.4 0.64 J. Acoust. Soc. Am., Vol. 135, No. 3, March 2014 Marsden et al. : Simulation of infrasound propagation 1087ratio c¼1:4 and molecular weight moindependent of alti- tude. Values of important parameters are provided in Table III. The gravitational ﬁeld is assumed of constant strength, with g¼9:8ms/C02. A spline-based celerity proﬁle which follows the main trends of a realistic atmosphere is used to determine ambient pressure and density as a functionof altitude thanks to the hydrostatic equilibrium relation d/C22p=dx 2¼/C0/C22qg¼/C0g/C22p=RT, and a pressure of 105Pa at ground level. Spline knot locations and values are taken from,44with an additional knot location at 230 km altitude to allow a proﬁle to be deﬁned throughout the top sponge zone. The spline coefﬁcients are listed in Table IV, and used as follows: for x22½xi 2;xiþ1 2/C138,/C22cðx2Þ¼aciþbciþ1 þ1=6ða3/C0aÞc00iþðb3/C0bÞc00iþ1/C2/C3 ðxiþ1 2/C0xi 2Þ2, where a ¼ðxiþ1 2/C0x2Þ=ðxiþ1 2/C0xi 2Þandb¼1/C0a. The temperature proﬁle, and the corresponding speed of sound, are shown in Fig. 3along with the V €ais€al€a-Brunt fre- quency, deﬁned by N¼signðN2Þ/C2jN2j1=2, where N2¼/C0 g=/C22qd/C22q=dx2/C0g2=/C22c2. The proﬁles of temperature and celerity have two local minima, corresponding to the tro-popause and mesopause acoustic waveguides located at alti- tudes of around 18 and 90 km, respectively. The atmospheric proﬁle is statically stable, as indicated by the positive valuesofNðx 2Þ. All test cases in what follows are performed in an atmosphere at rest. The sound source in the computations, implemented as a forcing term to the energy equation, has a Gaussian spatial envelope and a simple time variation given by Sðx1;x2;tÞ¼1 2AsinðxstÞ1/C0cosxst 2/C18/C19/C20/C21 P /C22 Ts/C18 t/C0Ts 4/C19 ! e/C0ln 2 x2 1þx2 2ðÞ =b2; (9) with a frequency of fs¼1=Ts¼0:1 Hz and a half-width of b¼600 m. In the previous equation, PðxÞrepresents the standard box function. The parameter A, expressed in J m/C03, is used to adjust source strength. It is placed at ground levelat the origin of the domain, ðx1;x2Þ¼ð 0;0Þ. The signal resulting from this source is illustrated in Fig. 4, which presents pressure ﬂuctuations recorded at ground level at 1 km from the source, along with the corresponding energyspectral density. The physical part of the computational do- main spans 600 km in the xdirection and 160 km in the ver- tical direction. The reference grid has a spatial step ofDx 1¼Dx2¼200 m, and contains a total of 3.4 /C2106points. Additional details on grid resolution effects will be provided in Sec. IV C 2 . A CFL value of 0 :5 is used, which yields a time step of Dt¼0:5Dx=maxð/C22cÞ¼0:17 s based on the high- est frozen speed of sound, found in the thermosphere. Time signals are recorded at ground level every 50 km, and com-pared as a function of source amplitude A. In the equations used for this work, no account is taken of atmospheric absorption due to molecular relaxation.Relaxation effects are expected to be strongest in the high atmosphere, i.e., where non-linear propagation effects are also maximal. Hence quantitative aspects described in this workare likely to be slightly different once relaxational effects are included. 24The two-dimensional nature of this study also pre- cludes a direct comparison of quantitative aspects of this work to actual ground-recorded signals. Indeed, geometrical ampli- tude attenuation in two dimensions is proportional to 1 =ﬃﬃrp, where ris the propagation distance, while in three dimensions it is proportional to 1 =r. Different ground arrivals at a given location having propagated over different distances, thismeans that the relative amplitudes of acoustic phases in a 2D computation will not vary with distance as they would in three dimensions. As an additional consequence, non-linear effects,by essence amplitude-dependent, will not be quantitatively FIG. 2. (a) Transmission loss (TL) as a function of distance for a monopole above a rigid ground in a stratiﬁed atmosphere: — reference solution com- puted on Dx¼0:0 6mg r i d ,––––s o - lution computed on Dx¼0:8 m grid. (b) Computational error /C15as a function of grid discretization in points per wavelength (ppw) at ground level: – þ c o m p u t e de r r o r ,––––t h i r do r d e rv a r i - ation ( /C15/C24ppw/C03). TABLE III. Constant parameters for the atmospheric composition. c cp(J kg/C01K/C01)cv(J kg/C01K/C01)R(J kg/C01K/C01)m0(kg mol/C01) 1.4 1004.5 717.5 287 29 /C210/C03TABLE IV. Spline coefﬁcients for sound speed proﬁle used to generate the stratiﬁed atmospheric proﬁle. x2(km) cðx2Þ c00ðx2Þ 0 340 0 10 300 0.39300920 290 0.22796350 330 /C00.272237 70 290 0.0192425190 265 0.420267120 425 /C00.0970527 160 580 /C00.194266 230 450 0 1088 J. Acoust. Soc. Am., Vol. 135, No. 3, March 2014 Marsden et al. : Simulation of infrasound propagationcorrectly predicted. These effects are overestimated in all cases since amplitude decreases more slowly in 2D than 3D, and the overestimation will also be stronger for high traveling phases. Nevertheless, the qualitative variations that will bedescribed are expected to hold, and the computational approach followed in this work is readily applicable in 3D for future work. B. Overview of results Figure 5(a) shows a snapshot of the acoustic ﬁeld nor- malized by the square root of mean density,32att¼1320 s, which corresponds to the arrival time of the tropospheric phase Iwat the measurement location x¼400 km. The source amplitude in this case is A¼ 1Jm/C03. Acoustic rays are superposed in gray. Three separate arrivals are identiﬁa- ble in the acoustic ﬁeld. The ﬁrst wavefront to arrive is thetropospheric phase Iw, visible around x 1¼400 km. It is not captured by rays emitted from the source location. The two subsequent arrivals have traveled higher through the atmos-phere, both reaching the thermosphere. At distances larger than 330 km, the higher traveling It bphase is seen to arrive before the lower Itaphase, due to its larger trace velocity. A strong caustic is clearly visible in the thermosphere at an altitude of 110 km between x1¼180 and x1¼250 km, and a second one can be seen at the end of the shadow zone, de-scending down from the thermosphere to reach ground level around x 1¼270 km. The end of the shadow zone is also clearly visible in Fig. 5(b), where pressure signals recorded every 50 km along the ground are represented as a functionof time and distance from the source. This time-distance plot highlights the lengthening of the pressure traces as distance from the source increases, due to the multiple arrivals. Also clearly visible is the difference in trace velocity between dif-ferent phases, the higher traveling It bphase having a sub- stantially larger trace velocity than the other two. The brief description given above of the acoustic ﬂuctuations illus-trates well the fact that a simple initial source in a smooth atmospheric proﬁle can yield a complex acoustic pattern downstream from the ﬁrst shadow zone. C. Numerical accuracy and large-amplitude signals The overview of the acoustic ﬁeld far from the source suggests that care should be taken in the correct treatment ofnon-linear wave steepening. Indeed, thermospheric arrivals, which are seen to account for a substantial part of the signals recorded downstream of the shadow zone, will have spenttime during which the wave amplitude relative to ambient pressure is not small. In these conditions, signiﬁcant non- linear waveform modiﬁcation is expected. In this section, abrief assessment of the non-linear ﬁltering procedure described in Sec. IIis given, and grid discretization require- ments are revisited in light of the signals’ high frequencycontent. 1. Non-linear filtering for large-amplitude signals The effect of the non-linear ﬁltering technique described in Sec. IIis examined on a simple conﬁguration of FIG. 3. (a) Proﬁles of temperature (—) and of speed of sound (– – – –), and (b) the V €ais€al€a-Brunt frequency, as a function of altitude. FIG. 4. Source characteristics. (a) ﬂuc-tuating pressure signal at 1 km from the source for a source amplitude of A¼ 1Jm /C03. (b) Energy spectral den- sity (ESD) of the ﬂuctuating pressure signal. J. Acoust. Soc. Am., Vol. 135, No. 3, March 2014 Marsden et al. : Simulation of infrasound propagation 1089atmospheric acoustic propagation, designed to induce non-linear effects which occur in long-range propagation phenomena. This evaluation is carried out by studying the vertical propagation of a low-frequency high-amplitude acoustic sig- nal in the atmospheric model described in Sec. IV A . The source model used is the same as that of the main study given by Eq. (9), with an amplitude of A¼ 1000 J m/C03, located at ground level. The standard grid has a regular spac- ing of D0¼200 m in both directions. Three computations are performed. The ﬁrst is performed on the standard gridwith no non-linear ﬁltering, while the second is performed on the same grid but with the non-linear ﬁltering applied with a ﬁltering strength a, deﬁned in Eq. (8),o fa¼1. The ﬁnal computation, referred to as the reference computation, is performed on a ﬁner grid ðD 0¼D0=4Þand non-linear ﬁl- tering is applied with a magnitude of a¼0:01, to ensure nu- merical stability. Results from these three computations are compared at three different times in Figs. 6(a)–6(c). These plots show vertical cuts of p0=ﬃﬃﬃ/C22qpas a function of altitude above the source, at x1¼0. In Fig. 6(a) taken at t¼150 s, ﬁrst signs of wavefront steepening can be observed, due to the high source amplitude. There is no nota-ble difference between the three sets of results, as the signal is still properly resolved by the numerical scheme on the coarse grid. In Fig. 6(b), the solution obtained on the ﬁner grid, rep- resented by a solid black line, shows a well-deﬁned central N-wave, preceded by a smaller partly formed leadingN-wave. Weak Gibbs-type oscillations can be seen in the vi- cinity of the shocks, but their amplitude is a small fraction of the shock amplitudes. The coarse-grid solution obtained withthe non-linear ﬁltering (solid gray line) is in very goodoverall agreement with the reference solution. Gibbs oscilla- tions are effectively removed. The peak amplitude of the shocks is slightly lower than that of the reference peaks andthe shock fronts are slightly less steep, but this an unavoid- able consequence of any ﬁltering procedure. The coarse-grid unﬁltered solution, shown by a gray dashed line, is similar to the reference solution in overall shape, but exhibits strong oscillations around it. These oscillations can be interpretedas strong Gibbs oscillations, which are not properly resolved by the numerical scheme and which therefore are not propa- gated at the correct velocity. In the ﬁnal Fig. 6(c) taken at 375 s, two well deﬁned N-waves can be seen in the ﬁne-grid signal, the leading lower amplitude wave having had time to become fullyshocked. The coarse-grid solution obtained without non- linear ﬁltering shows a reasonable overall match with the ﬁne-grid signal, except in the vicinity of the leading shockfronts. Indeed, spurious peaks of amplitude greater than that of the reference signal are visible. The ﬁrst of these peaks is notably ahead of the reference signal. This can be explainedby noting the strong overshoots in Fig. 6(b)which will travel faster than the reference shock front. The analysis of such a signal obtained with no non-linear ﬁltering would yield,among other problems, erroneous arrival times. The signal obtained with the ﬁltering procedure does not exhibit this ar- tifact, on the contrary matching the reference signal well. In summary, a self-adjusting non-linear ﬁltering meth- odology has been brieﬂy tested for long-distance acoustic propagation computations with the Navier-Stokes equations.The non-linear ﬁltering technique successfully removes Gibbs oscillations, which can be a numerical requirement for computations dealing with strong shocks, and as such can beregarded as shock-capturing . Moreover it enables the proper FIG. 5. (Color online) (a) View of pressure ﬂuctuations p0normalized byﬃﬃﬃ/C22qpatt¼1320 s for a source ampli- tude of A¼ 1Jm/C03, with acoustic rays traced in gray; (b) pressure signalsrecorded at ground level as a function of retarded time and distance, for the same source amplitude: /H17034arrivals pre- dicted by ray-tracing, – – – – t/C0x 1=/C22cðx2¼0Þ¼0. FIG. 6. Computed signals p0=ﬃﬃﬃ/C22qpðx1¼0;x2Þat (a) t¼150 s, (b) t¼300 s, and (c) t¼375 s: — computation on ﬁner grid ( Dx=4), computation with non-linear ﬁltering and – – – – computation without non-linear ﬁltering, both on standard grid. 1090 J. Acoust. Soc. Am., Vol. 135, No. 3, March 2014 Marsden et al. : Simulation of infrasound propagationcomputation of shock-front velocities for relatively poorly discretized waves. This last point is essential if Navier-Stokes computations are to be used to study arrival times of acoustic signals over long propagation distances. 2. Grid resolution for realistic propagation Strong non-linear effects will be highlighted in the results described in Sec. IV D . These non-linear variations change the spectral content of time signals, and therefore their spatial and time discretization. A larger error will natu- rally be committed on smaller wavelengths generated bynon-linear processes than on the base wavelengths emitted by the source. A brief assessment of the impact of this is now provided, by comparing data obtained on the standardgrid with Dx 1¼Dx2¼D0to results from a second grid with twice the spatial resolution and a third grid which is three times ﬁner in both directions. The source described in Sec. IV A having a principal frequency of 0.1 Hz, the standard grid spacing of 200 m yields a discretization of 17 points per wavelength at groundlevel and of 13 points per wavelength at an altitude of 90 km where the sound speed proﬁle reaches its minimum. These values may seem large when compared with those from the test cases presented in Sec. III, but they are neces- sary to allow the non-linear wave steepening described previously. The minimum discretization of 13 points perwavelength for the source’s central frequency for example allows a reasonable representation of the second harmonic of this wavelength. For source strengths such that no non-linearsignal modiﬁcation is observed over the whole computa- tional domain, i.e., for A<1Jm /C03, results from a computa- tion performed on a grid which is twice as ﬁne shownegligible difference with those from the reference grid. As the source strength increases, differences appear, in particu-lar in the high frequencies, as can be seen in Fig. 7which traces the lower thermospheric arrival time signatures and energy spectral densities (ESD) for source amplitudes ofA¼ 10 2and 5 /C2103Jm/C03and for the three different grid resolutions, D0,D0=2, and D0=3. This arrival is as expected the most sensitive to grid step size. For the lower amplitudesource, the arrival is already strongly marked by non-linear effects, with a characteristic U-shape 45resulting from an N wave having traversed a caustic. With the larger source am-plitude, the period of the U-wave has increased from roughly 12 s to nearly 50 s, due to the increased non-linear lengthen- ing in this case. This shift to lower frequencies is also clearlyvisible in the ESD shown in Figs. 7(b) and7(d). For the source amplitude of 10 2Jm/C03, differences due to grid reso- lution are most visible in Fig. 7(a)between t/C0x1=/C22cðx2¼0Þ ¼335 and 345, where high-frequency oscillations are seen for the standard grid solution. These oscillations are far FIG. 7. Low thermospheric arrival recorded at x1¼400 km and x2¼0: (a) non-dimensional time signal and (b) ESD for A¼ 102Jm/C03; (c) non- dimensional time signal and (d) ESD for A¼ 5/C2103Jm/C03. Solutions obtained using – – – – standard grid withD0spacing, D0=2 grid spac- ing, and D0=3 grid spacing. FIG. 8. Transmission loss at ground level as a function of distance from the source, for source amplitudes: — A¼ 1 ,–––– A¼ 10, –/C1–/C1–/C1–A¼ 102, A¼ 103, A¼ 5/C2103Jm/C03, and : slope correspond- ing to 2D geometrical attenuation. J. Acoust. Soc. Am., Vol. 135, No. 3, March 2014 Marsden et al. : Simulation of infrasound propagation 1091smaller on the second grid with twice the spatial resolution, and on the third grid they have almost disappeared. Theoscillations are also visible in the ESD of this arrival shown in Fig. 7(b), where the ﬁrst and second harmonics of the base frequency are clearly overestimated on the coarse grid, whilethe third and fourth harmonics are overestimated on the sec- ond grid. Despite these differences, the main features of the signal are well preserved on the standard grid, in particularthe peak overpressure, the frequency shift due to shocked propagation, and the arrival time of this thermospheric phase. For the strongest source amplitude of 5 /C210 3Jm/C03, similar trends are observed in Figs. 7(c) and7(d), the fre- quency shift and arrival times appearing to be correctly described by the standard grid. In fact, the standard gridseems to provide a better match for the strongest source level than for the weaker one. This can be explained by noting from Fig. 7(d)that the energy for this source level is concen- trated at lower frequencies, and thus larger, better discre- tized, wavelengths. Overall, the standard grid used for the examination of source amplitude effects in this work appearssufﬁcient to justify the trends to be described in Sec. IV D . D. Effect of source strength The aim of this study is to look at the effect of the source amplitude on the individual elements of the time sig- nature. To this end, a parametric series of computations is performed for amplitudes ranging from A¼ 10/C03to 5/C2103 Jm/C03, corresponding to pressure signals of approximately 4/C210/C04to 2/C2103Pa at 1 km from the source.At a given distance from the source, the received signal amplitude is not simply proportional to the source amplitude,due to non-linear effects during propagation. This is illus- trated in Fig. 8, which plots the TL at ground level, based on a reference distance of 1 km, TL ðx 1Þ¼10 log10½Eðx1Þ= Eðx1¼1k mÞ/C138, where EðxÞ¼Ð p02ðxÞdt, for source ampli- tudes Aof 1, 10, 102,1 03;and 5 /C2103Jm/C03. In the zone of silence, between 150 and 200 km, transmission loss is seen todecrease slightly as source strength is increased. Inversely, downstream of the zone of silence, for distances greater than 300 km, transmission loss increases with source strength. Forsource levels below 10 3Jm/C03, transmission loss in this zone is in fact lower than that given by geometrical spreading, rep- resented as a thick dashed gray line in Fig. 8. The behavior in the shadow zone is not predictable by commonly used techniques based on geometrical acous- tics,46and is thus interesting to examine in more detail. Figure 9(a) shows the ﬁrst 200 s of the pressure signal recorded at ground level 150 km from the source, scaled by source amplitude A. The ﬁrst arrival, starting just after t /C0x1=/C22cðx2¼0Þ¼0 and thus corresponding to the tropospheric phase Iw, exhibits no effects of varying source strength. The second, however, arriving around t/C0x1=/C22cðx2¼0Þ¼100, shows distinct non-linear variat ion for large source amplitudes. This arrival is due to the acoustic trapping in the tropos- pheric waveguide and diffraction off the top of the tropo-spheric waveguide, visible in Fig. 10(a) around x 1¼130 km, x2¼40 km. Non-linear or self refraction is possibly also occurring due to the high acoustic intensity around thecaustic. For source amplitudes below A¼ 10 2Jm/C03, the FIG. 9. Acoustic signals in the shadow zone; (a) ﬁrst part of pressure signal at x1¼150 km and x2¼0, rendered non-dimensional by source strength A. —A¼ 1, A¼ 1 0 ,–––– A¼ 102, A¼ 103,–/C1–/C1–/C1– A¼ 5/C2103Jm/C03; (b) late part of non-dimensional pressure signal atsame location, same line styles as (a). FIG. 10. Acoustic ﬁeld in the shadow zone for a source amplitude of A¼ 10 J m/C03; (a) structure of the pressure ﬂuctuations p0=ﬃﬃﬃ/C22qparound the shadow zone att¼465 s, (b) view of the same ﬁeld at t¼930 s, with ray trajectories superimposed in black. 1092 J. Acoust. Soc. Am., Vol. 135, No. 3, March 2014 Marsden et al. : Simulation of infrasound propagationwaveform collapses almost perfectly to an antisymmetrical wavepacket of period roughly one sixth that of the source and of slightly lower amplitude than the direct arrival. For higher source levels, the waveform is ampliﬁed, reach-ing an amplitude larger than that of the direct arrival for A¼ 5/C210 3Jm/C03. Finally, at 150 km from the source, a strong non-linear ampliﬁcation in the pressure signal isobserved for delays of between 400 and 1400 s after the direct arrival, as can be seen in Fig. 9(b). The strongest source yields a scaled pressure signal of more than 10 timesthe amplitude of that resulting from the weakest source. The origin of this part of the signal is highlighted in Fig. 10(b) , which represents the scaled ﬂuctuating pressure ﬁeld duringthe late arrival. It comes from the thermospheric arrival, which, according to ray tracing does not touch ground before x 1¼280 km, whose wavefront diffracts downward into the shadow zone. It is this diffraction phenomenon which appears to be dependent on the amplitude of the acoustic signal. For the sound speed proﬁle speciﬁed in this study, the geometrical zone of silence ends at x1¼280 km. Downstream of this point, transmission loss increases withincreasing source amplitude, as previously noted from Fig.8. Time signals recorded at a distance of 300 km from the source are compared in Fig. 11for values of Aranging from 10 /C02to 5/C2103Jm/C03, to identify the causes of the TL increase. These signals have two distinct contributions, one from the stratospheric eigenray, and a second from thethermospheric eigenray. The stratospheric contribution, shown in Fig. 11(a) , the ﬁrst to arrive at 300 km from the source, is relatively independent of source amplitude, up toA¼ 10 2Jm/C03. As can be seen from Fig. 11(a) , it arrives with a delay of approximately 130 s compared to a signal propagating along the ground, corresponding to an effectivepropagation speed of 296 m/s which is very close to the mini- mum speed in the tropospheric waveguide. The thermospheric contribution, shown in Fig. 11(b) , is more sensitive to source amplitude, because it has traveled at a higher altitude and hence the ratio p 0=/C22phas reached larger values. Non-linear effects are visible for source amplitudes A/C21 1. A very large reduction in amplitude is observed as source strength is increased, with a signal 20 times less intense for A¼ 5/C2103 than for A/C20 1Jm/C03. The signal lengthening due to shockformation is better illustrated in Fig. 12in which pressure signals are scaled by the maximum amplitude of their ther- mospheric arrivals. A U-shape typical of an N-wave having traversed a caustic47is observed for source amplitudes larger than A¼ 1Jm/C03, and the stronger the source, the more stretched the U-shape, indicating shock formation occurring increasingly early along the propagation path.The arrival time of the peak overpressure for the largest amplitude source signal is brought forward by 42 s com- pared to the arrival times for very low source amplitudes.The signiﬁcantly shortened travel time is due to the high travel path of the thermospheric rays. Indeed, although the overpressure ratio p 0=/C22pobserved at ground level for the thermospheric arrivals is small, only 10/C04for the strongest source, this ratio is altitude -dependent during propagation, p0 /C22p¼p0 ﬃﬃﬃ/C22qp/C2ﬃﬃﬃ/C22qp /C22p¼p0 ﬃﬃﬃ/C22qp/C2ﬃﬃﬃﬃﬃﬃﬃc /C22c2/C22pr increasingly roughly exponentially with altitude. Assuming lossless propagation for simplicity and integrating backward along the ray trajectories, the overpressure ratio reaches a value of around one for the low thermospheric eigenray.This overpressure in turn induces signiﬁcantly faster pro- pagation, the shock wave traveling at a speed of v shock ¼ /C22c½1þðcþ1Þ=ð2cÞp0=/C22p/C1381=2for a perfect gas.48This simplis- tic estimation suggests that shock formation occurred as the ray was climbing toward the thermosphere before reaching FIG. 11. Pressure signals recorded atx1¼300 km and x2¼0: (a) ﬁrst arrival amplitude rendered non-dimensional by source strength, for —A¼ 10 /C02,–––– A¼ 1, –/C1–/C1–/C1– A¼ 10, A¼ 102, A¼ 103, A¼ 5/C2103J m/C03; (b) second arrival non dimen- sional amplitude at the same location, same line styles as (a). FIG. 12. Thermospheric arrivals of the pressure signals recorded at x1¼ 300 km and x2¼0, scaled by the maximum amplitude p/C3of these arrivals, for — A¼ 10/C02,–––– A¼ 1, –/C1–/C1–/C1–A¼ 10, A¼ 102, A¼ 103, A¼ 5/C2103Jm/C03. J. Acoust. Soc. Am., Vol. 135, No. 3, March 2014 Marsden et al. : Simulation of infrasound propagation 1093the caustic, which corresponds well with direct results from the simulation placing shock formation for this eigenrayaround x 1¼80 km and x2¼70 km. V. CONCLUSIONS A computational approach for the study of long-range atmospheric infrasound propagation, based on the resolutionof the full 2D Navier-Stokes equations with high-order space and time methods, is presented. This approach is used to examine the effect of source strength on pressure signalsrecorded at varying distance from the source. Time signa- tures recorded a long distance away from an acoustic source are shown to vary in a highly non-linear fashion as a functionof source amplitude. For very low source amplitudes, time signatures collapse cleanly. For higher source amplitudes, the relative level of arrivals due to the different eigenrays ishighly modiﬁed. Arrival times also vary strongly, with arriv- als occurring earlier for higher traveling rays. Signals recorded in the shadow zone are also observed to undergolarge modiﬁcations due to non-linearities. These non-linear waveform variations should be taken into account when attempting acoustic tomography of the atmosphere. ACKNOWLEDGMENTS This work was undertaken as part of a collaboration with the DASE department of the CEA/DAM/DIF. The authorswould like to thank Dr. Blanc-Benon, Dr. Sturm, and Dr. Gainville for stimulating discussions. This work was per- formed within the framework of the Labex CeLyA ofUniversit /C19e de Lyon, operated by the French National Research Agency (ANR-10-LABX-0060/ANR-11-IDEX-0007). 1M. E. 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1.2919734.pdf	Current-induced magnetization excitation in a pseudo-spin-valve with in-plane anisotropy Jie Guo, Mansoor Bin Abdul Jalil, and Seng Ghee Tan Citation: Applied Physics Letters 92, 182103 (2008); doi: 10.1063/1.2919734 View online: http://dx.doi.org/10.1063/1.2919734 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/92/18?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Current-induced magnetization reversal mechanisms of pseudospin valves with perpendicular anisotropy J. Appl. Phys. 102, 093902 (2007); 10.1063/1.2803720 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 Reduction in critical current of current-induced switching in exchange-biased spin valves J. Appl. Phys. 97, 10C712 (2005); 10.1063/1.1853279 In situ magnetoresistance measurements during nanopatterning of pseudo-spin-valve structures J. Appl. Phys. 97, 054302 (2005); 10.1063/1.1852067 Measurement of local magnetization in the buried layer of a pseudo-spin-valve submicron wire J. Appl. Phys. 95, 7028 (2004); 10.1063/1.1667798 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: 145.116.151.252 On: Mon, 24 Nov 2014 12:40:17Current-induced magnetization excitation in a pseudo-spin-valve with in-plane anisotropy Jie Guo,1,a/H20850Mansoor Bin Abdul Jalil,1and Seng Ghee T an2 1Information Storage Materials Laboratory, Electrical and Computer Engineering Department, National University of Singapore, 4 Engineering Drive 3, Singapore 117576, Singapore 2Data Storage Institute, DSI Building, 5 Engineering Drive 1, National University of Singapore, Singapore 117608, Singapore /H20849Received 5 February 2008; accepted 12 April 2008; published online 6 May 2008 /H20850 We study the magnetization dynamics of a pseudo-spin-valve structure with in-plane anisotropy, which is induced by the passage of a perpendicular-to-plane spin-polarized current. Themagnetization dynamics is described by a modiﬁed Landau–Lifshitz–Gilbert /H20849LLG /H20850equation, which incorporates two spin torque terms. The simulation results reveal two magnetization excitationmodes: /H20849a/H20850complete magnetization reversal and /H20849b/H20850persistent spin precession. The existence of these dual modes may be explained in terms of the competition between the four terms of themodiﬁed LLG equation. Our results give indications to the optimal operating conditions forcurrent-induced magnetization dynamics for possible device applications. © 2008 American Institute of Physics ./H20851DOI: 10.1063/1.2919734 /H20852 The magnetization of a ferromagnetic /H20849FM/H20850nanopillar can be manipulated by the transfer of spin angular momen-tum from a spin-polarized current. This concept, which wasproposed in 1996 by Berger 1and Slonczewski,2boasts the advantages of low power consumption and small risk ofcross writing compared with the conventional magnetic ﬁeldswitching method. The origin of spin-transfer torque is thes-dinteractions between the spin moments mof the conduc- tion electrons and the local dipole moments Mof the mag- netic layer, which consequently leads to a rotation or oscil-lation of M. The effects of spin torque on inducing magnetization switching have been conﬁrmed by experimen- tal and theoretical studies in a variety of nanostructures, 3–6 such as the pseudo-spin-valve /H20849PSV/H20850and magnetic tunnel junction multilayer structures. The threshold current densityfor complete magnetization switching has been observed tobe of the order of 10 7–108A/cm2.4–6However, besides in- ducing a magnetization rotation and reversal, the spin-transfer torque can also excite sustained precessional motionof the magnetization. Such spin-transfer oscillations in theradio frequency /H20849rf/H20850range have also been observed in PSV with either planar 3or perpendicular anisotropy,7as well as in the hybrid PSV which combines a perpendicular pinned layerwith a planar free layer. 8 Considering the existence of these two magnetization excitation modes, it is necessary to study the factors leadingto each particular mode, so that it can be optimally utilizedfor their respective practical application in magnetic randomaccess memory /H20849MRAM /H20850and rf devices. In this article, we analyze the effect of the spin-transfer torque on the magne-tization dynamics in a nanoscale PSV structure by incorpo-rating two spin torque terms into the Landau–Lifshitz–Gilbert /H20849LLG /H20850equation. Based on the modiﬁed LLG equation, we performed a micromagnetic simulation to studythe current-induced magnetization excitation mechanisms.Our analysis focuses on the conditions which lead to eithercomplete switching or sustained precessional motion, and therequired critical current density.The magnetic nanostructure under consideration is a PSV structure consisting of a Permalloy /H20849Py/H20850/Cu /Co trilayer with the cross-section area of 50 /H11003100 nm 2and the thickness of 3 nm for all three layers. The anisotropy direction of theFM layers is assumed to be in the in-plane direction, and thecurrent ﬂows in the perpendicular-to-plane direction. The ini-tia magnetization of the free /H20849Py/H20850and reference /H20849Co/H20850layers is given by M f=Mf/H20849cos/H92580ex+sin/H92580ey/H20850andMCo=MCoex, re- spectively with Mf/H20849Co/H20850being the saturation magnetization of the free /H20849reference /H20850layer and /H92580being the initial angular deviation between them. Theoretical studies have been undertaken to study the spin-transfer torque arising from a spin-polarized current indifferent transport regimes, e.g., purely diffusive, ballistic, ora combination of both. 9–11In this paper, we adopt the well- established spin drift-diffusive model of Zhang, Levy, andFert /H20849ZLF/H20850,12which is applicable for the PSV structure under consideration. The ZLF’s model includes an out-of-planetorque term /H20849also known as the effective ﬁeld term /H20850in addi- tion to the usual in-plane Slonczewski’s torque term, andpredicts the magnitude of the spin-transfer torque to be ofthe order of /H1101110 2–103Oe for a current density of j0=107A/cm2. After incorporating the two spin torque re- lated terms, the modiﬁed LLG equation governing the mag-netization dynamics of the free FM layer is given by dM f dt=−/H9253Mf/H11003Heff−/H9261Mf/H11003/H20849Mf/H11003Heff/H20850−aMf /H11003/H20849Mf/H11003MCo/H20850+b/H20849Mf/H11003MCo/H20850. /H208491/H20850 The/H9253and/H9261terms on the right-hand side of Eq. /H208491/H20850are the standard precessional and damping terms, respectively. Weapply the Landau–Lifshitz damping term instead of the alter-native Gilbert expression, since recent studies 13have shown that the former constitutes a more natural description of themagnetization dynamics. /H9253is the gyromagnetic constant, /H9261=/H9253/H9251, where /H9251is the dimensionless damping coefﬁcient, andHeff=−/H11509E//H11509Mis the effective ﬁeld arising from the ex- change, anisotropy, magnetostatic, and Zeeman contributionsto the free energy E. The ﬁnal two terms relate to the spin- transfer effect, with aand bdenoting the Slonczewski spin a/H20850Electronic mail: elegj@nus.edu.sg.APPLIED PHYSICS LETTERS 92, 182103 /H208492008 /H20850 0003-6951/2008/92 /H2084918/H20850/182103/3/$23.00 © 2008 American Institute of Physics 92, 182103-1 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: 145.116.151.252 On: Mon, 24 Nov 2014 12:40:17transfer and the spin torque effective ﬁeld coefﬁcients, re- spectively. The magnitude of these coefﬁcients is determinedby several critical parameters, such as the intrinsic spin po-larization of the free and reference layers, the angular devia-tion of their magnetization directions, and the saturationmagnetization and layer thickness of each individual layer. 2 They have also been experimentally measured for the case ofaC o /Cu /Co nanopillar structure.14The additional spin torque terms can be expressed as Mf/H11003HJ, where HJis ap- proximately given by HJ/H11015j/H20849bmCo,xex−am f,zmCo,xey+am f,ymCo,xez/H20850, /H208492/H20850 where mf,Co=/H20849Mf,Co /Mf,Co/H20850 and MCo=mCo/H208511−/H208492 /H1100310−6/H20850,10−3,10−3/H20852. The above approximation holds since mCo,x/H11271mCo,y,mCo,z. Note that a slight deviation of MCofrom thexaxis is assumed since a small component of MCoin the y-zplane is required to initiate the current-induced magneti- zation switching /H20849CIMS /H20850, and it is invariably present in prac- tice due to the presence of stray ﬁelds. The initial magneti-zation of the free layer is set to be in the antiparallelalignment to M Co, i.e., Mf=mf/H20849−1,0,0 /H20850and/H92580=/H9266. The fol- lowing material parameters are assumed for Co and Py: satu- ration magnetization MCo=1.4/H11003106A/m and Mf=8.6 /H11003105A/m and anisotropy constant Ku,Co=5.2/H11003105J/m3. In performing the micromagnetic simulations, the PSV stack is discretized into unit cells of dimensions 5 /H110035 /H110033n m3, with a uniform magnetic moment within each cell. The LLG equation is numerically solved by ﬁnitedifference methods based on the OOMMF micromagnetic code.15In our simulations, we apply a spin-polarized current with a step-wise increase in the current density /H20849starting from zero/H20850at different sweep rates. We consider a current sweep rate of 0.1 /H11003107A/cm2per ns. The spin torque coefﬁcient a is assumed to be either 200 Oe or 1 kOe per current densityofj 0=107A/cm2, while the value of bis varied from 400 Oe to 1 kOe per j0. The simulation results can be under- stood in terms of the competitions between the spin torqueand the damping torque,16,17which are respectively repre-sented by the aand the /H9261terms of Eq. /H208491/H20850. The relative orientation of the conduction electron spin mand the free layer magnetization Mfdetermines whether the spin torque favors or opposes the damping torque. When the spin-polarized current is passing through the free layer, the mag-netization M fwill be initially disturbed and will begin to precess about its static equilibrium state. The precession mo-tion will be rapidly damped if the spin torque is smaller thanthe damping torque, so that M fwill settle back to its original equilibrium orientation along HeffNumerically, it was found that this scenario occurs at the smaller value ofa=200 Oe /j 0for a current density up to 109A/cm2, i.e., up toj0a=20 kOe. On the other hand, when the spin torque is sufﬁciently large to overcome the damping torque, Mfwill deviate from its initial state to another equilibrium directionwhich is determined by the relative strength of the H effand the spin torque effective ﬁeld term b/H20849MCo/H20850. For example, for a=1 kOe /j0, the magnetization switching is observed for a current density in the range of 107–108A/cm2, i.e., j0a=1–10 kOe. Next, we investigate the effect of the bterm on the magnetization dynamics, by varying bvalue from 400 to 1000 Oe /j0, while keeping aﬁxed at 1 kOe /j0. The relative values of aand bare chosen to be in agreement with previous theoretical and experimental results,14,18which show that the magnitude of the bterm is generally smaller than the aterm. Figure 1plots the resulting trajectories of the magnetization of the free layer. For the largest bvalue con- sidered /H208491 kOe /j0/H20850, the net effect of the two torques will cause the magnetization to rapidly settle to its equilibrium orientation, as shown in Fig. 1/H20849a/H20850, i.e., an overall reversal of Mffrom its original orientation along the − xdirection to the direction of the ﬁxed layer magnetization MCoalong the + x direction. When the magnitude of bis decreased to 667 Oe /j0, the relative magnitude of the spin torque is still sufﬁciently large to facilitate eventual switching of Mfto its equilibrium state along the MCodirection. However, the magnetization Mfwill now undergo several cycles of preces- sion in its switching trajectory /H20851see Fig. 1/H20849b/H20850/H20852, rather than FIG. 1. Magnetization switching tra- jectories of the free layer in aPy /Cu /Co PSV structure with the initial point at /H20849−1,0,0 /H20850, correspond- ing to different magnitude of the b term of /H20849a/H208501k O e /j 0,/H20849b/H20850667 Oe /j0, /H20849c/H20850500 Oe /j0, and /H20849d/H20850400 Oe /j0. The aterm is ﬁxed at 1 kOe /j0, where j0=107A/cm2, while the cur- rent sweep rate is set at 0.1/H1100310 7A/cm2per ns.182103-2 Guo, Jalil, and T an Appl. Phys. Lett. 92, 182103 /H208492008 /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: 145.116.151.252 On: Mon, 24 Nov 2014 12:40:17assuming the smooth trajectory, as observed for the case of b=1 kOe /j0. If the magnitude of bis further decreased, the free layer magnetization Mfdoes not undergo complete switching to the + x/H20849MCo/H20850direction. Instead, Mfswitches from its initial − xdirection to some intermediate orientation. As shown in Figs. 1/H20849c/H20850and1/H20849d/H20850, the normalized Mfpre- cesses in a plane which lies normal to HeffandMCo.B y comparing Figs. 1/H20849c/H20850and1/H20849d/H20850, it can be seen that a further decrease in the magnitude of bfrom 500 to 400 Oe /j0will result in a more persistent precessional motion. The ﬁnalequilibrium state is also tilted more toward the initial − x direction, which is favored by the damping term H eff. The above micromagnetic results are consistent with the phasediagram of the switching and precession states calculatedbased on a macrospin approximation. 19Our results thus nu- merically conﬁrm the existence of the dual magnetizationexcitation modes and the relationship between these twomodes on factors, such as the damping parameter and thespin torque efﬁciency. Next, we investigate the dependence of the CIMS effect on the speed of the current sweep. Micromagnetic simula-tions are performed at different current sweep rates rangingfrom 0.05 /H1100310 7to 1.5/H11003107A/cm2per ns, and the corre- sponding critical current density for switching jcare plotted in Fig. 2.jcis deﬁned as the current density at which the magnetization switches to another equilibrium orientation orsettles down to a steady precessional motion. For the com-plete switching mode, we observe an initial trend of higher j c with increasing current sweep rate, which is in agreement with available theoretical and experimental results.20,21This suggests that it is energetically more favorable for the systemto undergo magnetization switching under quasistatic condi-tions in the presence of a low current sweep rate, rather thanin response to a fast-changing current which gives rise to arapidly varying effective ﬁeld both in terms of its magnitudeand direction. In practical terms, there is thus a downside intrying to achieve faster CIMS in PSV devices by increasingthe current sweep rate. Alternative methods, e.g., by optimiz-ing the current pulse proﬁle, 22may be required to accelerate the switching process without the undesired increase in jc.However, for sweep rates exceeding 1.0 /H11003107A/cm2per ns, the observed jcis signiﬁcantly reduced. This could be under- stood by the fact that for large sweep rate and effective ﬁeldbterm, the spin-transfer torque is sufﬁciently strong to cause the free layer magnetization M fto coherently reverse its di- rection from − xto + xdirections within a single current step, rather than undergoing a gradual switching process /H20849which is the case for lower sweep rates /H20850. In addition, it is also ob- served in Fig. 2that jcis lower for both low and high b values corresponding to the precession and stable switchingmodes, respectively. Regardless of the current sweep rates,the maximum j cvalues tend to occur at intermediate bvalues where both modes exist. This is a desirable trend in terms ofthe practical utilization of these magnetization excitationmodes in the data storage or rf oscillation devices since theirfunctioning is based on pure rather than mixed magnetizationmodes. In summary, the optimal condition for MRAM appli-cations can be achieved by having the effective ﬁeld termto be comparable to the Slonczewski’s torque term, i.e., an/H20849b/a/H20850ratio of /H110111. Such a /H20849b/a/H20850ratio would ensure complete magnetization switching at a low critical current density j c. For rf applications, a low /H20849b/a/H20850ratio of /H110110.4 is optimal as it excites a steady precessional mode at a critical low current density. For both applications, a fast magnetization excitationcan be achieved with a large current sweep rate /H20849in excess of 10 7A/cm2per ns /H20850, although this would result in some in- crease in jc. This work was supported by the National University of Singapore under Grant R-263-000-481-112. 1L. Berger, Phys. Rev. B 54, 9353 /H208491996 /H20850. 2J. C. Slonczewski, J. Magn. Magn. Mater. 159,L 1 /H208491996 /H20850. 3S. Urazhdin, N. O. Birge, W. P. Pratt, Jr., and J. Bass, Appl. Phys. Lett. 84, 1516 /H208492004 /H20850. 4G. D. Fuchs, N. C. Emley, I. N. Krivorotov, P. M. Braganca, E. M. Ryan, S. I. Kiselev, J. C. Sankey, D. C. Ralph, and R. A. Buhrman, Appl. Phys. Lett. 85,1 2 0 5 /H208492004 /H20850. 5J. Manschot, A. Brataas, and G. E. W. Bauer, Appl. Phys. Lett. 85,3 2 5 0 /H208492004 /H20850. 6Y. Jiang, T. Nozaki, S. Abe, T. Ochiai, A. Hirohata, N. Tezuka, and K. Inomata, Nat. Mater. 3, 361 /H208492004 /H20850. 7S. Mangin, D. Ravelosona, J. A. Katine, M. J. Carey, B. D. Terris, and E. E. Fullerton, Nat. Mater. 5, 210 /H208492006 /H20850. 8D. Houssameddine, U. Ebels, B. Delaët, B. Rodmacq, I. Firastrau, F. Ponthenier, M. Brunet, C. Thirion, J.-P. Michel, L. Prejbeanu-Buda, M.-C.Cyrille, O. Redon, and B. Dieny, Nat. Mater. 6, 447 /H208492007 /H20850. 9T. Seki, S. Mitani, K. Yakushiji, and K. Takanashi, Appl. Phys. Lett. 88, 172504 /H208492006 /H20850. 10J. Sort, F. Garcia, S. Auffret, B. Rodmacq, B. Dieny, V. Langlais, S. Suriñach, J. S. Muñoz, M. D. Baró, and J. Nogués, Appl. Phys. Lett. 87, 242504 /H208492005 /H20850. 11J. Guo and M. B. A. Jalil, Phys. Rev. B 71, 224408 /H208492005 /H20850. 12S. Zhang, P. M. Levy, and A. Fert, Phys. Rev. Lett. 88, 236601 /H208492002 /H20850. 13M. D. Stiles, W. M. Saslow, M. J. Donahue, and A. Zangwill, Phys. Rev. B75, 214423 /H208492007 /H20850. 14M. A. Zimmler, B. Özyilmaz, W. Chen, A. D. Kent, J. Z. Sun, M. J. Rooks, and R. H. Koch, Phys. Rev. B 70, 184438 /H208492004 /H20850. 15M. Donahue and D. Porter /H20849http://math.nist.gov/oommf/ /H20850. 16M. Covington, Science 307, 215 /H208492005 /H20850. 17J. Sun, Nature /H20849London /H20850425, 359 /H208492003 /H20850. 18H. Meng and J.-P. Wang, Appl. Phys. Lett. 88, 172506 /H208492006 /H20850. 19Spin Dynamics in Conﬁned Magnetic Structures III , edited by B. Hillebrands and A. Thiaville /H20849Springer, New York, 2006 /H20850, pp. 266–274. 20P. B. Visscher and D. M. Apalkov, J. Appl. Phys. 97, 10C704 /H208492005 /H20850. 21Y. F. Chen, M. Ziese, and P. Esquinazi, Appl. Phys. Lett. 88, 222513 /H208492006 /H20850. 22X. R. Wang and Z. Z. Sun, Phys. Rev. Lett. 98, 077201 /H208492007 /H20850. FIG. 2. /H20849Color online /H20850Critical current density for magnetization switching jcas a function of different magnitude of the bterm for different current sweep rates ranging from 0.05 to 1.5 /H11003107A/cm2per ns.182103-3 Guo, Jalil, and T an Appl. Phys. Lett. 92, 182103 /H208492008 /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: 145.116.151.252 On: Mon, 24 Nov 2014 12:40:17
1.4802687.pdf	Tuning of the nucleation field in nanowires with perpendicular magnetic anisotropy Judith Kimling, Theo Gerhardt, André Kobs, Andreas Vogel, Sebastian Wintz et al. Citation: J. Appl. Phys. 113, 163902 (2013); doi: 10.1063/1.4802687 View online: http://dx.doi.org/10.1063/1.4802687 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v113/i16 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 25 Apr 2013 to 152.2.176.242. 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_permissionsTuning of the nucleation field in nanowires with perpendicular magnetic anisotropy Judith Kimling,1Theo Gerhardt,1Andr /C19e Kobs,1Andreas Vogel,1Sebastian Wintz,2 Mi-Young Im,3Peter Fischer,3Hans Peter Oepen,1Ulrich Merkt,1and Guido Meier1 1Institut f €ur Angewandte Physik und Zentrum f €ur Mikrostrukturforschung Hamburg, Universit €at Hamburg, Jungiusstr. 11, 20355 Hamburg, Germany 2Institut f €ur Ionenstrahlphysik und Materialforschung, Helmholtz-Zentrum Dresden-Rossendorf, 01314 Dresden, Germany 3Center for X-ray Optics, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA (Received 28 February 2013; accepted 8 April 2013; published online 23 April 2013) We report on domain nucleation in nanowires consisting of Co/Pt multilayers with perpendicular magnetic anisotropy that are patterned by electron-beam lithography, sputter deposition, and lift-off processing. It is found that the nucleation ﬁeld can be tuned by changing the geometry ofthe wire ends. A reduction of the nucleation ﬁeld by up to 60% is achieved when the wire ends are designed as tips. This contrasts with the behavior of wires with in-plane anisotropy where the nucleation ﬁeld increases when triangular-pointed ends are used. In order to clarify the origin ofthe reduction of the nucleation ﬁeld, micromagnetic simulations are employed. The effect cannot be explained by the lateral geometrical variation but is attributable to a local reduction of the perpendicular anisotropy caused by shadowing effects due to the resist mask during sputterdeposition of the multilayer. VC2013 AIP Publishing LLC [http://dx.doi.org/10.1063/1.4802687 ] I. INTRODUCTION Magnetization reversal in ferromagnetic nanowires is governed by three processes: domain nucleation, domain- wall motion, and pinning. If the nucleation ﬁeld exceeds any propagation or depinning ﬁeld, no domain walls can be stati-cally observed. Thus, the nucleation ﬁeld determines the minimum potential depth of pinning sites to reliably trap do- main walls. Weak pinning sites and therefore small nuclea-tion ﬁelds are of interest in connection with current-driven depinning of domain walls, since high current densities, which are otherwise required, can modify or even destroy asample. 1It was experimentally found that the critical current density for magnetization reversal in systems with perpen- dicular magnetic anisotropy can be orders of magnitudesmaller than in soft-magnetic materials. 2The high efﬁciency of the underlying non-adiabatic spin-transfer torque arises from relatively narrow domain-wall widths.3,4In nanowires with perpendicular anisotropy, a reduced inversion symme- try can signiﬁcantly support the magnetization reversal via spin-orbit torque5and a recently discovered spin Hall effect based phenomenon.6Another advantage of systems with per- pendicular anisotropy is that generally Bloch walls appear,2 whereas in soft-magnetic wires complex two-dimensional spin structures with intrinsic degrees of freedom occur.7,8All in all, wires with perpendicular anisotropy, in particular Co/ Pt wires, are promising candidates for possible applicationsbased on the displacement of domain walls by spin-polarized currents like in the race-track memory 9as demonstrated in various studies (see, e.g., Refs. 10–12). Furthermore, Co/Pt layered structures are preferably used to investigate domain- wall resistance.13–15Recently, a new kind of effect, i.e., the anisotropic interface magnetoresistance, was found in these materials.16–18In the case of soft-magnetic wires, for example, patterned from permalloy (Ni 80Fe20), a common strategy to control domain nucleation for studying ﬁeld- and current- induced domain-wall dynamics is the usage of laterally extended pads of continuous ﬁlm attached to one of the wireends. The magnetization reversal commences in this so- called nucleation pad due to its smaller shape anisotropy compared to the wire region. 19A domain wall is injected from the pad into the wire and propagates until it pins at a defect or at an intentionally created pinning site, for exam- ple, a notch20or a magnetically softened region.21For wires with triangular-pointed ends, the formation of ﬂux-closure domains is suppressed. This leads to an increase of the switching ﬁelds compared to wires with ﬂat ends22,23or nucleation pads. In wires with perpendicular anisotropy, nucleation pads do not work as for in-plane magnetized samples since demag-netizing effects play only a minor role. In systems with few defects acting as nucleation sites, the size of the pad has to be in the order of several 100 lm to observe any effect. 24,25 Nevertheless, smaller nucleation pads were recently used for perpendicularly magnetized wires. Since no systematic stud- ies on the pad design were made, and the samples were usu-ally studied by methods based on transport measurements providing only little spatial resolution such as extraordinary Hall effect 10or giant magnetoresistance,26a proof of concept is still missing. A well known route to decrease the nucleation ﬁeld is lowering the perpendicular anisotropy or inducing defects acting as nucleation sites by means of local ion irradi-ation. 27For instance, Alvarez et al. demonstrated the func- tionality of Gaþ-irradiated nucleation pads by imaging the magnetization reversal of Co/Pt multilayer wires with a Kerrmicroscope. 28Another approach for the nucleation of domains in wires with perpendicular anisotropy is to exploit 0021-8979/2013/113(16)/163902/6/$30.00 VC2013 AIP Publishing LLC 113, 163902-1JOURNAL OF APPLIED PHYSICS 113, 163902 (2013) Downloaded 25 Apr 2013 to 152.2.176.242. 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 Oersted ﬁeld that accompanies a current pulse ﬂowing through a strip line attached to the wire.29 In this study, we demonstrate that in contrast to in-plane magnetized wires, where triangular-pointed ends suppress the nucleation of domains, the nucleation ﬁeld in Co/Pt wires with perpendicular anisotropy can be decreased by almost60% by designing tip-shaped wire ends. It is further shown that the nucleation ﬁeld can be intentionally tuned by varying the opening angle of the tip. Details about sample preparationand magnetic imaging are given in Section II.S e c t i o n III presents the experimental results and addresses aspects con- cerning the creation of pinning sites. The experimental resultsare further discussed in Section IValong with supporting micromagnetic simulations. Section Vconcludes our study. II. METHODS Samples were prepared by electron-beam lithography, sputter deposition of the Co/Pt multilayer, and lift-off proc- essing of the resist as illustrated in Fig. 1(a). As substrate a silicon-nitride membrane with a thickness of 150 nm wasused. 30The resist31was spin-coated with 8000 rpm, baked out at 170 8C for 30 min, and exposed with a scanning elec- tron microscope at 10 kV. After development, the samplewas treated by reactive-ion etching to remove residuals of the resist in the exposed areas. Then, the thin-ﬁlm stack Ptð5:0n m Þ=½Coð0:7n m Þ=Ptð2:0n m Þ/C138 /C24 was deposited at room temperature at a base pressure below 2 /C210 /C09mbar: First, a 4 nm thick Pt seed layer was grown via ion-beam sputtering utilizing an electron cyclotron resonance source.Subsequently, dc magnetron sputtering was employed to de- posit both a 1 nm thick Pt layer and right afterwards the Co/Pt multilayer. 32The lift-off was done by dissolving the resist in acetone. In a last step, the sample was rinsed with isopropanol and blow-dried with nitrogen. Magnetization reversal was imaged by transmission soft x-ray microscopy at the XM-1 full ﬁeld microscope at beam- line 6.1.2 of the Advanced Light Source in Berkeley, CA, USA.33The setup allows for the application of a magnetic ﬁeld perpendicular to the sample plane. Magnetic contrast is based on the x-ray magnetic circular dichroism (XMCD) effect.34It is proportional to the magnetization components along the pro-jection of the x-ray beam. The perpendicularly magnetized sample was inserted into the microscope setup such that the surface normal was in parallel to the beam direction.Micromagnetic simulations were performed with the MicroMagnum code. 35The external magnetic ﬁeld was applied at an angle of 3/C14to the surface normal. This symmetry breaking is required to enable domain nucleation since our simulations were performed for zero temperature and other ﬂuctuations are neglected. In case ﬂuctuations and imperfec-tions are not considered, there is usually an offset between the simulated values for the critical ﬁelds and the experimental results. This effect is known as Brown’s paradox. 36,37The thickness of the simulated wires was 0.7 nm according to the thickness of one Co layer in the actual sample, and the multi- layer structure of the Co/Pt ﬁlm was not taken into account.Discretization was done using ﬁnite differences. III. EXPERIMENTAL RESULTS Figure 2(a)depicts a transmission x-ray micrograph of a Co/Pt wire with a triangular-pointed end located at its top, astraight segment with a notch, and a nucleation pad at its bot- tom end which is not completely visible. The structure was saturated by applying a ﬁeld of about /C0200 mT before increasing the reverse ﬁeld in steps of þ0.1 mT. Figures 2(b) through 2(d) are differential images revealing the magnetiza- tion change between two ﬁeld steps. Magnetization reversaldoes not start in the nucleation pad but at the upper end of the wire at an applied ﬁeld of þ11.8 mT, i.e., the nucleation ﬁeld. More precisely, the reversal process commences withthe nucleation of an oppositely oriented domain within the triangular-pointed end. A domain wall propagating top-down is pinned twice before the entire magnetization is reversed atþ12.1 mT. The presence of the notch (see arrow in Fig. 2(a)) has no inﬂuence on the mobility of the domain wall. We attribute the observed nucleation behavior to a varia- tion of the local anisotropy constant due to shadowing effects during multilayer growth. The Co/Pt multilayer was sputter- deposited onto the resist mask ( /C25160 nm thick) created by FIG. 1. (a) Illustration of sample preparation. (b) Wire geometry: due to shadowing by the resist mask during sputter deposition of the metal ﬁlm, less material is deposited at the triangular-pointed wire end with tip-opening angle h. FIG. 2. (a) Transmission x-ray micrograph and (b) through (d) correspond- ing differential images (the respective previous image serves as reference) of a Pt ð5:0n m Þ=½Coð0:7n m Þ=Ptð2:0n m Þ/C138 /C24 wire with a nucleation pad (bottom, not completely visible), a notch marked by the arrow in (a), and a triangular-pointed end (top) recorded at the given ﬁeld values after satura- tion at /C0200 mT.163902-2 Kimling et al. J. Appl. Phys. 113, 163902 (2013) Downloaded 25 Apr 2013 to 152.2.176.242. 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_permissionselectron-beam lithography. Since for magnetron sputtering the diameter of the source is comparable to its distance to the sample, shadowing by the sample morphology has to be takeninto account. In case of the bottom-up approach used, the amount of deposited material locally varies and depends on the lateral distance to the edges of the resist mask. In particu-lar, at the triangular-pointed end, less material reaches the substrate compared to the straight wire segment. The same holds true for geometrical constrictions such as notches. Aschematic representation of a straight wire with a triangular- pointed end is shown in Fig. 1(b). The left hand side of the ﬁgure depicts the lateral geometry, while the wire’s cross-section is sketched on the right. Since Co and Pt are deposited at the same angle towards the sample surface and from the same distance under the same conditions, the ratio of Co andPt thicknesses in the multilayer stack is unaffected by the reduction of ﬁlm thickness towards the triangular-pointed wire end. It can be assumed that the saturation magnetizationM sremains basically constant down to very thin layer thick- nesses.38,39What is left as a changing property is the magne- tocrystalline anisotropy constant K1¼K1;effþl0M2 s=2w h i c h for the present system mainly originates from interface contri- butions. We consider a continuous reduction of the anisotropy constant K1towards the triangular-pointed end of a tip, as well as in other regions where less material is deposited, to be the origin for the reduction of the nucleation ﬁeld.40There are two reasons for the reduction of the anisotropy of theCo/Pt multilayer due to shadowing effects. The ﬁrst reason is the gradual reduction of the thickness of the Co layer as below t Co/H113510:5 nm the ﬁrst order anisotropy constant K1;eff decreases with decreasing tCo.41This behavior is in accord- ance with other studies, see, e.g., Refs. 42and43.T h es e c o n d reason for the gradual reduction of the anisotropy constant isconnected with the thickness of the Pt interlayer. When it falls below t Pt/C252 nm, we observe a decrease of the anisotropy for our multilayer system.32This is in agreement with previous results.44,45Consequently, a gradual reduction in perpendicu- lar magnetic anisotropy occurs in the regions where the Co- and Pt-layer thicknesses are gradually reduced due toshadowing effects. As discussed in Section IVit is conﬁrmed by micromagnetic simulations that the reduction of the nucle- ation ﬁeld for wires with triangular-pointed ends can indeedbe caused by such a reduced anisotropy. Shadowing effects not only change the anisotropy but also induce defects. Thus, they do not only affect domainnucleation but also the pinning of domain walls. We observe that in wires with periodic lateral modulations domain walls get pinned both at the transitions from wide to narrow andfrom narrow to wide regions (not shown). Here, the mecha- nism responsible for pinning is presumably not an increase in domain-wall energy when entering a wider segment, butconnected with defects and gradients of the anisotropy in the multilayer due to variations in the layer thicknesses caused by shadowing of the resist mask. Consequently, it has to becross-checked whether lateral geometrical variations, such as notches or anti-notches, or above mentioned effects caused by shadowing are responsible for pinning. In the latter case,the preparation has to be performed with particular care. The aspect of domain-wall pinning remains a major issue in theﬁeld of perpendicularly magnetized media, which shall not be addressed further in this paper, where the focus lies on the nucleation of domain walls. The dependence of the nucleation ﬁeld on the wire ge- ometry was studied in detail for three different widths of nanowires, namely, 320 nm, 430 nm, and 570 nm. Forincreasing tip-opening angles, the total length of a wire decreases from 28 lmt o2 0 lm. The reason for this is that the volume of the wires was kept constant for the differentgeometries in order to exclude a variation of the switching ﬁelds due to a change of the number of volume defects. 46,47 Figure 3(a) exemplarily depicts scanning electron micro- graphs of wire ends with different tip-opening angles. Switching ﬁelds were determined by transmission x-ray mi- croscopy. After saturation at an out-of-plane ﬁeld of about/C0200 mT, a reverse ﬁeld was applied and increased in steps ofþ1.1 mT. In the ﬁeld of view of the microscope, which has a diameter of 10 lm, about half of the total length of the wires could be imaged. Pinning of domain walls was never observed. Thus, it can be assumed that nucleation ﬁeld and switching ﬁeld are the same or close within the uncertaintyof a ﬁeld step of 1.1 mT. Figure 3(b)depicts the average switching ﬁeld as a func- tion of the tip-opening angle hdetermined from nine ﬁeld sweeps for each data point. For all three wire widths, the same behavior is observed. The switching ﬁelds of wires with a ﬂat end scatter around ðþ18:560:5ÞmT. For decreas- ing angle h, the switching ﬁeld decreases to a value of ðþ7:960:6ÞmT for h¼2:5 /C14. This corresponds to a reduc- tion of the nucleation ﬁeld by (57 65)%. The fact that there is no inﬂuence of the wire width on the switching ﬁeld FIG. 3. (a) Scanning electron micrographs of wires with various opening angles of the tip’s end. (b) Switching ﬁeld versus tip-opening angle determined by transmission x-ray microscopy for th ree different widths of nanowires.163902-3 Kimling et al. J. Appl. Phys. 113, 163902 (2013) Downloaded 25 Apr 2013 to 152.2.176.242. 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_permissionsindicates that nucleation takes place in a volume of the tip end that is provided independently of the tip’s length, which increases with increasing width of the wires. That the nuclea-tion takes place in the tip area is in complete agreement with the XMCD images presented in Fig. 2. As stated above, the reduction of the nucleation ﬁeld is assumed to originate froma decrease of the local anisotropy constant in the tip area due to shadowing during sputter deposition of the multilayer. Before presenting the results of micromagnetic simula- tions that support this hypothesis, the impact of the experi- mental results shall be discussed. The possibility to reliably control the nucleation ﬁeld shows that the bottom-up micro-fabrication approach used is worth being pursued. One fur- ther advantage of this preparation method is that it is based on a single lithography step. This distinguishes it from othermethods that require a second preparation step to diminish the nucleation ﬁeld, for example, local ion irradiation. A pos- sible disadvantage of the bottom-up approach used is thatshadowing effects do not only cause a modulation of the multilayer stack along the axis of a wire due to triangular- pointed ends. At all edges of a wire, the multilayer does notend abruptly but the thickness gradually decreases on the way to the edge according to the scheme of Fig. 1(a). The alternative patterning approach is to deposit the magneticﬁlm ﬁrst, to employ lithography with a negative resist, and to pattern the nanowires top-down by etching. Besides the dis- advantage of more steps of sample preparation in the top-down approach, also the problem of edge damage cannot be excluded with this method. It has been shown by Shaw et al. that even small variations in the edge properties can com-pletely change the reversal behavior of perpendicularly mag- netized nanostructures. 48Furthermore, possible damage of the entire multilayer system24can occur during etching if the ﬁlm is only protected by a resist mask and not, for example, by a titanium layer.25Anyway, with the bottom-up approach pursued in this work we can be sure that the multilayer stackis not signiﬁcantly modiﬁed in the center of the wires com- pared to the pristine ﬁlm, which can be concluded from the fact that rectangular wires with ﬂat wire ends of arbitrarywidth exhibit the same switching ﬁeld. IV. MICROMAGNETIC SIMULATIONS To study the inﬂuence of both the anisotropy variation and the tip geometry on the switching ﬁeld, micromagneticsimulations were performed. The width of the wires in the simulations was taken as 320 nm in accordance with one type of wires studied in the experiment. The local anisotropyconstant was reduced linearly in the tip area from the maxi- mum value K 1toK0¼a/C1K1at the very end as sketched on the right hand side of Fig. 4(a). The total reduction of the local anisotropy constant was varied between zero ( a¼1.0, K0 1;eff¼350 kJ =m3) and 20% ( a¼0.8, K0 1;eff¼34 kJ =m3). At an anisotropy reduction of a¼0.78, the spin-reorientation transition to in-plane anisotropy occurs. Figure 4(a) depicts the switching ﬁeld simulated for wires with tip-opening angles hbetween 10 8and 175 8. For a homogeneous aniso- tropy constant ( a¼1.0, black squares), the geometry does not inﬂuence the switching ﬁeld. Besides a reduction of theswitching ﬁeld, the same behavior is found for lower values of a homogeneous anisotropy constant down to 0 :8K1(not shown). This demonstrates that the reduction of the nuclea- tion ﬁeld reported above cannot be explained via the differ- ent opening angles of the tips. If we assume that theanisotropy constant is locally reduced in the range of the tips, we obtain a reduction of the switching ﬁeld with decreasing tip-opening angle h. This dependence is most pro- nounced for the highest reduction of K 0. It is thus the amount of material with reduced anisotropy that determines the switching ﬁeld. For an angle h¼10/C14, the maximum value of the switching ﬁeld (that is Hmax¼415 mT for ﬂat wire ends) is reduced by 12%, 29%, 48%, and 67% for anisotropy FIG. 4. (a) Dependences of the switching ﬁeld on the opening angle hof the tip simulated for a wire with homogeneous anisotropy (black squares) as well as for wires with a linear reduction of the local anisotropy constant in the tip area from K1toK0¼a/C1K1(other color) as sketched on the right hand side. (b) Dependence of the switching ﬁelds on the total reduction of the anisotropy constant K0along the decrease length dsimulated for rectangular-shaped wires. (c) Same data as in (b) plotted versus tip-opening angle hthat corresponds to a certain decrease length das illustrated on the right hand side.163902-4 Kimling et al. J. Appl. Phys. 113, 163902 (2013) Downloaded 25 Apr 2013 to 152.2.176.242. 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_permissionsreductions of 5%, 10%, 15%, and 20%, respectively. The simulated curves vary not only in the absolute reduction of the switching ﬁelds but also in the critical tip-opening angleat which the reduction sets in. While for a¼0.95 the switch- ing ﬁeld stays constant down to h¼130 /C14, it drops below the maximum value already at h¼155/C14fora¼0.9. The experi- mental values depicted in Fig. 3(b) stay constant for large angles as well and start to decrease somewhere between h¼85/C14andh¼75/C14. While this behavior corresponds to a reduction of the local anisotropy constant at the wire end of less than 5%, the relative decrease of the nucleation ﬁeld in our sample rather implies a reduction of about 18%. The experimental data show a nearly perfect linear de- pendence on the tip-opening angle hup to 90 8in accordance with the simulations. For higher angles, the experimentalnucleation ﬁeld remains constant, while a strengthened increase is found in the simulations. A possible explanation for this discrepancy is that shadowing effects during sputterdeposition lead to a continuous reduction of the ﬁlm thick- ness at all edges of the wires, while in the simulation abrupt edges are assumed. The gradual decay of the ﬁlm thicknessat the wire edges means that there is a gradual reduction of the anisotropy constant for all wires independent of the tip- opening angle. At sharp tips, the regions of both edges wherethe anisotropy decreases can overlap causing an anisotropy reduction in the whole tip area. For blunt tips, this effect is strongly reduced and the inﬂuence on the switching ﬁeld isfading away. The experimental results therefore indicate that the reduction of the anisotropy constant is localized at the edges in contrast to the simulations where the anisotropy istaken as reduced in the whole tip region. This probably leads to a “saturation” of the increase of the nucleation ﬁeld at h/C2590 /C14in the experiment, since the gradual reduction of the ﬁlm thickness at all edges has the same effect as the gradual reduction of the ﬁlm thickness in the triangular-pointed ends when the tip-opening angle exceeds a critical value. To further investigate the inﬂuence of the wire’s geome- try on the nucleation ﬁeld, additional simulations have been performed. Instead of having one triangular-pointed end,nanowires were designed with two ﬂat ends. Nevertheless, the anisotropy constant was reduced linearly at one end of the wire. Thereby the distance dover which the anisotropy decreases, corresponds to the length of a tip with a certain opening angle as sketched on the right hand side of Fig. 4(c). Figures 4(b) and4(c) reveal that the switching ﬁelds show the same dependences but are stronger reduced in compari- son to the switching ﬁelds of nanowires with triangular- pointed ends (compare Fig. 4(a)). This ﬁnding qualitatively shows that nucleation depends on the total area with reduced anisotropy constant, as in nanowires with ﬂat ends this area is much larger than in nanowires with triangular-pointedends. In particular, the region at the wire end, where the ani- sotropy constant is lowest, is signiﬁcantly reduced in nano- wires with triangular-pointed ends, compare sketch in Fig.4(c). Rectangular wires with a comparable linear reduction of the local anisotropy constant over the same decrease length d(and the corresponding tip-opening angle h) thus provide a larger area for a certain nucleation volume to reverse and consequently have lower switching ﬁelds. Withthe same arguments, it can be explained why the switching ﬁeld depends on the opening angle hin wires with triangular-pointed ends: the smaller h, the larger is the tip area available for nucleation. V. CONCLUSION We have shown that the critical ﬁeld for the nucleation and injection of domain walls in Co/Pt nanowires can be tuned and reduced by up to about 60% compared to theswitching ﬁeld of rectangular-shaped wires by designing tri- angularly pointed wire ends. The reasoning is based on the reduction of the perpendicular magnetic anisotropy withinthe tip region that is caused by shadowing effects during sputter deposition of the multilayer. As conﬁrmed by micro- magnetic simulations the reduction of the local anisotropyconstant accompanied by an increase of the nucleation area in sharper tips accounts for the effect observed. A low nucle- ation ﬁeld is a necessary prerequisite for the preparation ofdomain walls at comparably weak pinning sites as it is of in- terest for fundamental studies and applications. ACKNOWLEDGMENTS The authors gratefully acknowledge ﬁnancial support by the Deutsche Forschungsgemeinschaft via the Graduiertenkolleg 1286 and the Sonderforschungsbereich668. The operation of the x-ray microscope is supported by the Director, Ofﬁce of Science, Ofﬁce of Basic Energy Sciences, of the U.S. Department of Energy under ContractNo. DE-AC02-05-CH11231. 1P. S. Ho and T. Kwok, Rep. Prog. Phys. 52, 301 (1989). 2O. Boulle, G. Malinowski, and M. Kl €aui, Mater. Sci. Eng. R 72, 159 (2011). 3G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601 (2004). 4J. Xiao, A. Zangwill, and M. D. Stiles, Phys. Rev. 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1.3058623.pdf	Micromagnetic computation of interface conductance of spin-transfer driven ferromagnetic resonance in nanopillar spin valves M. Carpentieri1,a/H20850and L. T orres2 1Dipartimento di Elettronica, Informatica e Sistemistica, University of Calabria, Via P . Bucci, 42c 89100 Rende, Italy 2Departamento de Fisica Aplicada, University of Salamanca, Plaza de La Merced s/n, 37008 Salamanca, Spain /H20849Presented 11 November 2008; received 16 September 2008; accepted 20 October 2008; published online 4 February 2009 /H20850 Micromagnetic computations are used to describe spin-transfer driven ferromagnetic resonance in nanopillar spin valves with elliptical cross section. Analytical uniform magnetization modelsreproduce the resonance phenomenon adequately and these can be used to compute interfaceconductance. In this work, using the magnetic parameters extracted by ﬁtting staticmagnetoresistance measurements, mixing conductances are obtained; these values are 25% and 20%lower than the ones previously reported. Nonuniform magnetization resonance is found. © 2009 American Institute of Physics ./H20851DOI: 10.1063/1.3058623 /H20852 According to the predictions of Slonczewski 1and Berger,2the injection of an alternating spin polarized current through a nanomagnet should lead to a resonance at a fre-quency which depends on the effective ﬁeld experienced bythe nanostructure. Recently, Sankey et al. 3demonstrated a technique for measuring this spin-transfer driven ferromag-netic resonance /H20849STFMR /H20850in individual magnetic nanopillars. In our previous works, the normal modes excited by a directbias current in the mentioned experiment 4were identiﬁed by means of the micromagnetic spectral mapping technique.Frequencies, power, and linewidths were computed via mi-cromagnetic simulation and qualitative agreement with ex-perimental data was achieved. 3,4In this paper, our main ob- jective is forwarded to ﬁnd an easy method to computeinterface conductance values of nanopillar devices. 5,6 The nanopillar under study consists on a 20-nm-thick pinned layer /H20849PL/H20850of Permalloy /H20849Py/H20850, a 12 nm Cu spacer, and a free layer /H20849FL/H20850of 5.5 nm thick of Py 65Cu35alloy. The section is elliptical /H2084990/H1100330 nm2/H20850and the long and short axis are identiﬁed with xandydirections, respectively, whereas thezdirection perpendicular to the nanopillar /H20849see Fig. 1/H20850. Our own micromagnetic three-dimensional /H208493D/H20850dy- namic code has been used to perform the simulations.4It includes the Slonczewski term in the Gilbert equation whichgives rise to two terms in the equivalent Landau–Lifshitz–Gilbert equation. 4A polarization factor /H9257=0.3 and the spatial and angular dependences of Slonczewski’s polarizationfunction 1for each computational cell have been considered.7,8 In our model, the effective ﬁeld of the Landau–Lifshifz– Gilbert–Slonczewski equation includes not only all the clas-sical micromagnetic terms but also the magnetostatic cou- pling between the two magnetic layers and the Ampere ﬁeldfrom the electrical current. The effect of thermal activation isnot considered in this work. A ﬁnite difference scheme isused and the FL is discretized in computational cells of 2.5 /H110032.5/H110035.5 nm 3. The time step used is 32 fs and the current is considered positive for a ﬂow of electrons from the PLtoward the FL. 4The magnetic parameters used for the FL simulations have been obtained by ﬁtting static magnetore-sistance measurements. 3They are the saturation magnetiza- tion MS=27.85 /H11003104A/m and the exchange constant A =1.0/H1100310−11J/m. The magnetostatic coupling and the initial states of the ferromagnetic layers have also been computedby means of a 3D simulation of the whole structure. Theexternal ﬁeld is applied perpendicularly to the plane, whereasthe dynamic simulation of the FL is computed in two dimen-sions. The system is excited by applying I ac=I0cos/H208492/H9266ft/H20850, where I0=270/H9262A and the frequency franges from 1t o2 0G H z /H20849Idc=0/H20850. The normalized magnetization is cal- culated for different applied ﬁelds from 320 to 520 mT /H20849steps of 50 mT /H20850. At resonance, the ac and the spin valve resistance oscillate at the same frequency.9 Since the applied ﬁeld lies always in the zdirection /H20849per- pendicular to the plane of the sample, tilted /H110115° along the x direction /H20850larger ﬁelds give rise to more out of plane trajec- tories which lead to lower resistance variations. This fact canbe checked in the snapshots of the spatial distribution of the a/H20850Electronic mail: mcarpentieri@deis.unical.it. FIG. 1. Schematic drawing of the ferromagnet trilayer.JOURNAL OF APPLIED PHYSICS 105, 07D112 /H208492009 /H20850 0021-8979/2009/105 /H208497/H20850/07D112/3/$25.00 © 2009 American Institute of Physics 105 , 07D112-1magnetization presented in Fig. 2. In these snapshots it can be observed how slight nonuniformities are present whichcan only be analyzed from full micromagnetic simulations.These nonuniformities come from the self-demagnetizingﬁeld, magnetostatic coupling from the ﬁxed layer, classicalAmpere ﬁeld, and spatial distribution of the scalar polariza-tion function. The nonuniform effective ﬁeld produces a non uniform magnetization resonance state. For the largest applied ﬁeldsanalyzed, the resonance is more uniform as it was to expectand the magnetization oscillation is in the zdirection /H20849see panel I-L of Fig. 2for an applied ﬁeld of 520 mT /H20850. STFMR could be also used for quantitative studies of some physical properties of nanomagnets in a spin-transferdevice. One of these is the interface conductance between thespacer and the FL that could be calculated as shown in arecent theoretical analysis by Kupferschmidt et al. 5In this framework, our computed micromagnetic results are com-pared to the analytical ones computed by using the theory ofKupferschmidt et al. By means of this approach we demon- strate that it is possible to ﬁnd interface conductance valuejust from an individual computation. Figure 3shows the simulated normalized magnetization /H20849blue dotted line, no ﬁt- ting parameters are used /H20850compared to the data computed by using the equation of Kupferschmidt et al. /H20849black line /H20850for different applied ﬁelds. The asymmetrical nature of the lineshape depends on the anisotropy of the torque; so with increasing applied magneticﬁeld, the isotropic contribution increases and the lineshapebecomes more symmetric /H20849see Fig. 3/H20850. In order to compute the magnetization values following the analytical theory, thenext equation is used: 5m/H20849/H9275/H20850=m0/H20849J/e/H20850/H20851/H9275+−/H9275−cos/H9272+i/H9275/H20849/H9251++/H9251−/H20850/H20852 f/H20849/H9275/H20850, /H208491/H20850 where f/H20849/H9275/H20850=/H208491+/H9251+2−/H9251−2/H20850/H92752−2i/H9275/H20849/H9251+/H9275++/H9251−/H9275−cos/H9272/H20850+/H9275−2 −/H9275+2,m0=/H9253/H6036G−/2dM SG1+with G1+=G++/H20849G+2−G−2/H20850/g1,/H9251/H11006 is the dimensionless Gilbert damping /H9251+=/H9251 +/H20849g1/H9253/H60362/4de2MS/H20850/H208514−/H20849G+/G1+/H20850−/H20851g1//H20849g1+G1/H20850/H20852/H20852, and /H9251− =/H20849g1/H9253/H60362/4de2MS/H20850/H20851/H20849G+/G1+/H20850−/H20851g1//H20849g1+G1/H20850/H20852/H20852andJis the ac density. /H9275/H11006=/H20849/H92752/H11006/H92751/H20850/2 with /H92751and/H92752frequencies set by the energy cost for magnetization deviations along principal axes and /H9278/2 is the rotation angle between principal axes. G/H11006=/H20849G↑/H11006G↓/H20850/2 and G↑↓=G1+iG2are the interface con- ductance values for majority and minority electrons and mix- ing conductance for the interface between the ferromagneticsource and the nonmagnetic spacer, whereas g /H11006 =/H20849g↑/H11006g↓/H20850/2 and g↑↓=g1+ig2are the equivalent quantities for the interface between the spacer and the FL. Following Ref.5, in order to obtain expression /H208491/H20850,G2andg2have been set to zero since the imaginary parts of the mixing conduc-tances are numerically small for metallic junctions /H20849more de- tails in Ref. 5/H20850. The analytical theory of Kupferschmidt et al. for a uni- form magnetized nanomagnet shows excellent agreementwith our simulated data. In the comparison process, the con-ductance values were the only free ﬁtting parameters used inEq. /H208491/H20850. In order to obtain a good ﬁtting, in the preliminary steps, the values G +=0.4/H110031015and G−=0.2 /H110031015/H9024−1m−2, and G1=0.6/H110031015and g 1=0.55 /H110031015/H9024−1m−2were used /H20849taken from the literature, Table I FIG. 2. /H20849Color online /H20850Snapshots of the spatial distribution of the magneti- zation in the FL corresponding to a point of maximum /H20849left panel /H20850and minimum /H20849right panel /H20850of the xcomponent of the magnetization resonance with applied current Iac=270/H9262A/H20849varying the frequency from 1 to 20 GHz /H20850 and applied ﬁelds /H92620Happ=320 mT /H20849A-B /H20850, 370 mT /H20849C-D /H20850, 420 mT /H20849E-F/H20850, 470 mT /H20849G-H /H20850, and 520 mT /H20849I-L/H20850/H20849red and rightward: parallel state, blue and leftward: antiparallel state /H20850. FIG. 3. /H20849Color online /H20850Conductance calculation: simulated /H20849blue dotted line /H20850 and equation of Kupferschmidt et al. /H20849black line /H20850for applied current Iac =270/H9262A/H20849varying the frequency from 1 to 20 GHz /H20850and different applied ﬁelds: /H20849a/H20850320, /H20849b/H20850370, /H20849c/H20850420, /H20849d/H20850470, and /H20849e/H20850520 mT.07D112-2 M. Carpentieri and L. T orres J. Appl. Phys. 105 , 07D112 /H208492009 /H20850of Xia et al.10and Adam et al.11for Co /Cu /Co interfaces /H20850. With these values the agreement between simulated data andanalytical approach was not satisfactory. The ﬁtting proce-dure revealed an optimum ﬁtting using the same values ofG +=0.41/H110031015and G−=0.21/H110031015/H9024−1m−2, whereas G1 =g1=0.45/H110031015/H9024−1m−2for all the simulated external ﬁelds /H20849Fig.3/H20850. The use of a smaller conductance value for the interface PyCu /Cu than the tabulated value of Co /Cu interfaces is in agreement with the literature as explained in the paper ofBarnas et al. 12Therefore, the interface conductance values can be obtained from STFMR measurements or full micro-magnetic simulations using the uniform magnetization ana-lytical theory proposed by Kupferschmidt et al. A 25% de- crease in the interface conductance G 1and a 20% decrease in g1are obtained; we attribute this discrepancy to the use of Slonczewski’s expression for the torque computation and tothe effect of the nonuniformities present in the micromag-netic modeling /H20849and in the experiment /H20850. In fact, all experi- mental data are obtained on structures larger than the single-domain threshold so that effects of a noncollinearmagnetization state must be included to enable a quantitative/H20849and often even a qualitative /H20850comparison between theory and measurements. In our case, qualitative agreement withthe analytical theory of Kupferschmidt et al. is found; this fact is visible considering the magnetization oscillation forthe different applied ﬁelds. On the other hand the nonunifor-mities, as shown in the snapshots of Fig. 2, provide an ob- servable quantitative disagreement. The effect of these non-uniformities is quantiﬁed in the values of the interfaceconductance. It is also noteworthy how our computationsperformed using Slonczewski’s torque are reproduced in thewhole range of frequencies by the model of Kupferschmidtet al. /H20849see Fig. 3/H20850. It could be concluded that both models /H20849micromagnetic—Slonczewski and macrospin— Kupferschmidt et al. /H20850lead to the same physical results al- though different quantitative values of the conductivities areobtained. This work was partially supported by projects MAT2005-04827 and MAT2008-04706/NAN from Spanishgovernment, and SA063A05 and SA025A08 from Junta deCastilla y Leon. The authors would like to thank Professor S.Greco for his support with this research. 1J. C. Slonczewski, J. Magn. Magn. Mater. 159,L 1 /H208491996 /H20850. 2L. Berger, Phys. Rev. B 54,9353 /H208491996 /H20850. 3J. C. Sankey, P. M. Braganca, A. G. F. Garcia, I. N. Krivorotov, R. A. Buhrman, and D. C. Ralph, Phys. Rev. Lett. 96, 227601 /H208492006 /H20850. 4L. Torres, G. Finocchio, L. Lopez-Diaz, E. Martinez, M. Carpentieri, G. Consolo, and B. Azzerboni, J. Appl. Phys. 101, 09A502 /H208492007 /H20850;M .C a r - pentieri, L. Torres, G. Finocchio, and B. Azzerboni, J. Appl. Phys. 103, 07B107 /H208492008 /H20850. 5J. N. Kupferschmidt, S. Adam, and P. W. Brouwer, Phys. Rev. B 74, 134416 /H208492006 /H20850. 6A. A. Kovalev, G. E. W. Bauer, A. Brataas, Phys. Rev. B 75, 014430 /H208492007 /H20850. 7M. Carpentieri, L. Torres, B. Azzerboni, G. Finocchio, G. Consolo, and L. Lopez-Diaz, J. Magn. Magn. Mater. 316,4 8 8 /H208492007 /H20850. 8G. Finocchio, G. Consolo, M. Carpentieri, A. Romeo, B. Azzerboni, L. Torres, and L. Lopez-Diaz, Appl. Phys. Lett. 89, 262509 /H208492006 /H20850. 9A. A. Tulapurkar, Y. Suzuki, A. Fukushima, H. Kubota, H. Maehara, K. Tsunekawa, D. D. Djayaprawira, N. Watanabe and S. Yuasa, Nature /H20849Lon- don/H20850438,3 3 9 /H208492005 /H20850. 10K. Xia, P. J. Kelly, G. E. W. Bauer, A. Brataas, and I. Turek, Phys. Rev. B 65, 220401 /H20849R/H20850/H208492002 /H20850. 11S. Adam, M. L. Polianski, and P. W. Brouwer, Phys. Rev. B 73, 024425 /H208492006 /H20850. 12J. Barnas, A. Fert, M. Gmitra, I. Weymanna, and V. K. Dugaev, Mater. Sci. Eng., B 126, 271 /H208492006 /H20850.07D112-3 M. Carpentieri and L. T orres J. Appl. Phys. 105 , 07D112 /H208492009 /H20850
1.1839176.pdf	A 2 A 2 ←X 2 B 1 absorption and Raman spectra of the OClO molecule: A three- dimensional time-dependent wave packet study Zhigang Sun, Nanquan Lou, and Gunnar Nyman Citation: The Journal of Chemical Physics 122, 054316 (2005); doi: 10.1063/1.1839176 View online: http://dx.doi.org/10.1063/1.1839176 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/122/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Analysis and prediction of absorption band shapes, fluorescence band shapes, resonance Raman intensities, and excitation profiles using the time-dependent theory of electronic spectroscopy J. Chem. Phys. 127, 164319 (2007); 10.1063/1.2770706 Three-dimensional time-dependent wave-packet calculations of OBrO absorption spectra J. Chem. Phys. 123, 064316 (2005); 10.1063/1.2000259 Time-dependent quantum wave-packet description of the π 1 σ * photochemistry of phenol J. Chem. 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Downloaded to IP: 128.114.34.22 On: Tue, 02 Dec 2014 22:12:05A2A2]X2B1absorption and Raman spectra of the OClO molecule: A three-dimensional time-dependent wave packet study Zhigang Suna) State Key Laboratory of Molecular Reaction Dynamics, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Dalian 116023, People’s Republic of China and Graduate Schoolof Chinese Academy of Sciences, Dalian 116023, People’s Republic of China Nanquan Lou State Key Laboratory of Molecular Reaction Dynamics, Dalian Institute of Chemical Physics,Chinese Academy of Sciences, Dalian 116023, People’s Republic of China Gunnar Nyman Department of Chemistry, Physical Chemistry, Go ¨teborg University, SE-412 96 Go ¨teborg, Sweden ~Received 27 September 2004; accepted 2 November 2004; published online 21 January 2005 ! Time-dependent wave packet calculations of the ( A2A2ÃX2B1) absorption and Raman spectra of the OClO molecule are reported. The Fourier grid Hamiltonian method in three dimensions isemployed.The X 2B1ground state ab initiopotential energy surface reported by Peterson @J. Chem. Phys.109, 8864 ~1998!#is used together with his corresponding A2A2state surface or the revised surface of the A2A2state by Xie and Guo @Chem. Phys. Lett. 307, 109 ~1999!#. Radau coordinates are used to describe the vibrations of a nonrotating OClO molecule. The split-operator methodcombined with fast Fourier transform is applied to propagate the wave function. We ﬁnd that the ab initio A 2A2potential energy surface better reproduces the detailed structures of the absorption spectrum at long wavelength, while the revised surface of the A2A2state, consistent with the work of Xie and Guo, better reproduces the overall shape and the energies of the vibrational levels. Bothsurfaces of the A 2A2state can reasonably reproduce the experimental Raman spectra but neither does so in detail for the numerical model employed in the present work. © 2005 American Institute of Physics. @DOI: 10.1063/1.1839176 # I. INTRODUCTION The OClO molecule is of both experimental and theoret- ical interest, for instance, for its presumed role in polarstratosphericozonedepletion. 1Thereisawealthofstudiesof the dynamics of this molecule. Fragmentation into ClO(X 2P)1O(3P) and Cl(2P)1O2(3Sg,1Dg,1Sg), as well as the wavelength dependence of the branching ratio of thedissociation, have been studied with the molecular beamtechnique. 2–6 The allowed parallel transition from the ground X2B1 state to the A2A2state gives rise to a strong absorption band in the visible with a maximum at ;350 nm. As expected from the elongated bond lengths and a smaller bond angle ofthe excited A 2A2state, symmetric stretch and bend excita- tion on the upper surface were observed in the experimentalabsorption spectrum. 7Strong activity in the antisymmetric stretch excitation was also observed, which is not C2vsym- metry allowed. Therefore, a potential energy surface ~PES! which has a non- C2vequilibrium geometry and a double minimum along the antisymmetric stretch has been proposedto explain the absorption spectrum taken under jet-cooledconditions. 7–10The activity in the antisymmetric stretch has also been interpreted as due to strong coupling between thesymmetric and antisymmetric stretch modes, which suggestslarge anharmonicity in the antisymmetric stretch mode. 11,12 Recently, accurate ab initio based near-equilibrium po- tential energy surfaces of the X2B1ground and A2A2ex- cited states have been reported.13Theab initio PES of the A2A2state has C2vsymmetry at its equilibrium geometry and is characterized by strong coupling between the antisym-metric and the symmetric stretch modes. Variational calcula-tions of the vibrational energy levels 13,14using the ab initio PES approximately reproduce experimental observations.Therefore, a PES, where the antisymmetric stretch is highlyanharmonic and strongly coupled to the symmetric stretch,seems to be a good model for describing the potential of theexcitedA 2A2state of the OClO molecule. The concept of a PES is central in chemical physics and the ability to calculate the eigenstates it supports, and theprogression of its absorption spectrum, is helpful for the in-terpretation and understanding of the dynamics. Exact quan-tum dynamics calculations of polyatomic absorption spectra provide means for reﬁning multidimensional nuclear PESs.Especially for an ultrashort excitation process, where coher-ent dynamics of several vibrational states dominates, an ac-curate numerical quantum dynamics simulation based uponthe corresponding PES is necessary to quantitatively inter-pret the experimental observation. The OClO molecule hasbeen studied by several groups using ultrashort laserpulses. 15–17Unfortunately, limited by the available PES in- formation on the triatomic OClO molecule and our compu-a!Electronic mail: zsun@dicp.ac.cnTHE JOURNAL OF CHEMICAL PHYSICS 122, 054316 ~2005! 122, 054316-1 0021-9606/2005/122(5)/054316/7/$22.50 © 2005 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: 128.114.34.22 On: Tue, 02 Dec 2014 22:12:05tational resources, direct simulations of the pump-probe pro- cess have not appeared yet and the details underlying theexperimental results remain obscure. 15–18 Xie and Guo14recently revised the A2A2ab initio PES of Peterson13by ﬁtting the vibrational energies of the PES to those experimentally observed. Hereafter we refer to thisPES as XGPES. In this paper, XGPES as well as the ab initio PES of the excited state, are used together with the X 2B1ab initioground state PES of Peterson.13The latter is considered accurate and observed deviations between calculated and ex-perimental results are discussed in terms of inaccuracies inthe excited state surfaces ~or the numerical dynamics model used!. Xie and Guo found that XGPES reproduces the jet- cooled absorption spectrum better than the original ab initio PES. This conclusion was based upon an analysis of theFranck-Condon factors between the A 2A2and theX2B1 states, which they calculated by a variational time- independent method. We here report a thorough investigation of the two A2A2 PESs by comparing the calculated ( A2A2ÃX2B1) absorp- tion and Raman spectra with the related experimentalresults 6,7using time-dependent wave packet calculations in full dimensionality for a total angular momentum J50. We shall see that the ab initio PES better reproduces detailed vibrational structures in the experimental absorption spec-trum, while the XGPES correctly gives the vibrational peakpositions. However we note that these conclusions arereached by assuming that the Condon approximation is validand thatJ50 is a good approximation. The paper is arranged as follows: In Sec. II the numeri- cal method is described. The calculated absorption and Ra-man spectra for excitation wavelengths of 368.9 and 360.3nm are presented and compared with experimental spectra 6,7 in Sec. III. Section IV concludes by summarizing our ﬁnd-ings. II. COMPUTATIONAL ASPECTS The PESs of the two electronic states A2A2andX2B1 of the OClO molecule are involved in the calculations. For the ground X2B1state we have used the ab initiobased PES reported by Peterson.13For the excited state A2A2we have used the ab initio based PES of Peterson,13and also the XGPES in which some of the parameters have been revisedby Xie and Guo. 14The transition dipole moment has not been calculated yet.An arbitrary value of 1.0 a.u. is used andthe Condon approximation is invoked. It is known that thecoordinate dependence of the transition dipole function ismoderate near the ground state equilibrium geometry so theeffect of using the Condon approximation should bemoderate. 14We employ a transformed triatomic Hamiltonian in Radau coordinates, which allows the use of the split-operator method to propagate the wave packet. 19,20The ki- netic energy operators are evaluated by fast Fourier trans-form ~FFT!. A. Calculation of the initial wave packet The OClO molecule initially resides in its electronic and vibrational ground state. The corresponding wave function isfound by a variational method in Eckart coordinates for atotal angular momentum J50. 19,21The Hamiltonian used in the variational calculation can be written as22,23 Hˆ52\2 2F1 m1]2 ]r1211 m2]2 ]r221S1 2m1r1211 2m2r22 2cosu m3r1r2D]2 ]u2G2\2 m3Fsinu r1r2] ]u 2cosu 2S1 r1] ]r211 r2] ]r1D1cosu]2 ]r1]r2 2sinu] ]uS1 r1] ]r211 r2] ]r1DG1Vˆ~r1,r2,u!1DVˆ, ~1! where DVˆ5cos3u 4m3r2r2sin2u21 8S1 m1r1211 m2r22D~11csc2u! ~2! andVˆis the PES. The reduced masses are given by m1 5m25mClmO/(mCl1mO) and m35mCl.r1,r2, and uhave the usual meanings. In order to ﬁnd the eigenvalues and eigenfunctions of the Hamiltonian of the electronic ground state, the wave functionis expanded in a direct product basis of single-particle func-tions, C ~r1,r2,u!5( nr1,nr2,nuCnr1,nr2,nufnr1~r1!fnr2~r2!fnu~u! ~3! where fnri(ri) are Morse wave functions for the stretch modes and fnu(u) are harmonic oscillator wave functions for the bending mode. The expansion coefﬁcients Cnr1,nr2,nu are found by solving the secular equation for the vibrational Hamiltonian in Eq. ~1!using the direct product basis set in Eq.~3!. For constructing the secular equation, derivatives in the kinetic energy operators are obtained using ﬁnite-difference formulas. One- and three-dimensional integrationsover (r 1,r2,u) are performed using Gauss-Hermite quadra- ture. We note here that the procedure used for ﬁnding the secular equation makes the Hamiltonian matrix nonsymmet-ric, possibly due to the ﬁnite accuracy of the ﬁnite-differenceformulas.Although the eigenvalues thus obtained are slightlydifferent from those obtained by the discrete variable repre-sentation method, 13,20the eigenfunctions of low energy lev- els are accurate enough. We have checked the accuracy ofthe eigenfunction of the lowest energy level by the imaginarytime propagation method ~the relaxation method !. 24In the following, the number of single-particle wave functionsalong each dimension is chosen sufﬁciently large to make theaccuracy of the lowest eigenvalue 0.001 cm 21. B. The Hamiltonian in Radau coordinates The triatomic Hamiltonian ( J50) in Radau coordinates (R1,R2,w) is well known.25We use a transformed form of it in our calculations, viz.,20054316-2 Sun, Lou, and Nyman J. Chem. Phys. 122, 054316 (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: 128.114.34.22 On: Tue, 02 Dec 2014 22:12:05Hˆ52\2 2m1]2 ]R122\2 2m2]2 ]R222S\2 2m1R121\2 2m2R22D 3S]2 ]w211 4 sin2w11 4D1V~R1,R2,w,t!, ~4! R1,R2, and whave the usual meanings.20,25The volume element for Hˆin Eq. ~4!isa23dR1dR2dwwhere a2 5mCl/(2mO1mCl). The kinetic energy operators require only one forward-backward FFT in each dimension to evalu-ate the action of the transformed Hamiltonian on the wavefunction.Also, the Hamiltonian in Eq. ~4!does not mix local and nonlocal operators of the same coordinate, wherefore thesplit-operator method combined with FFT can be used topropagate the wave function. 19,20This simpliﬁes the calcula- tion and results in an efﬁcient computer code. C. Calculation of absorption and Raman spectra The absorption spectrum depends on the overlap of the wave functions for the intial ground state ~s!and the excited states. The absorption spectrum s~v!can be obtained by taking the Fourier transform F(v) of the time autocorrela- tion function C8(t) of the wave packet.26Here C8~t!5C~t!f~t!5^CA~t50!uCA~t!&f~t! ~5! and s~v!;vF~v!,F~v!5E 2‘1‘ C8~t!eiEtdt, ~6! where CA(t50)5CXmXAis the product of the lowest vibra- tional wave function of the ground electronic state CXwith the transition dipole function mXAandf(t) is an exponential damping function e2G0tused to approximately reproduce the experimental broadening of the spectral peaks. CA(t50) is used as the initial wave function on the upper surface. It isrepresented on the Fourier grid and propagated on the ex-cited PES. In practice, with C A(t50) being real, C(t) can be rewritten which allows a shorter time propagation,27 C~t!5^CA~0!uCA~t!& 5KCAS2t 2DCASt 2DL5SCASt 2DD2 . ~7! This scheme is applied to obtain the time autocorrelation function for the absorption spectrum calculation. In a time-dependent picture, the Raman spectrum can be expressed as28 I~vI,vS!;vIvS3E 2‘1‘ eivt^R~vI!uR~vI,t!&dt, ~8! where the Raman wave function R(vI) is deﬁned as the half Fourier transform of the propagated wave packet, R~vI!5E 01‘ eiE8t2GtCA~t!dt ~9! and R~vI,t!5mXAe2iHˆ XtR~vI! ~10!is the Raman wave function propagated on the lower poten- tial energy surface ~usingHˆX).E85E01vI,E0is the en- ergy of the initial state and vIis the laser frequency. In the following calculations the wavelengths 368.9 and 360.3 nmare used. 6v5vI2vS1E0andvSis the frequency of the scattered light. The parameter Gis introduced to damp the wave function to avoid too long time propagation and is setto 15 cm 21. In our calculation a total time of about 1 ps and a time step 0.15 fs are used to obtain the Raman wave func-tion. The convergence concerning these parameters has beenchecked. Similar to the procedure for obtaining the absorption spectrum, the autocorrelation function of the Raman wavefunction is multiplied with a damping function before theFourier transform is performed. Thus the ﬁnal expression forobtaining the Raman spectrum can be written as I ~vI,vS!;vIvS3E 2Tmax1Tmaxeivt2G8t^R~vI!uR~vI,t!&dt, ~11! where G8has been set to 15 cm21andTmaxis the total propagation time, here set to 1.2 ps. The damping functionsapproximately reproduce the Lorentzian width of the peaksand make it possible to obtain the spectra with a ﬁnite timepropagation. For the calculations, 64 364332 grid points are used ~64 grid points along each radial coordinate and 32 long the angular coordinate !. The grid ranges used are @1.8,3.9 #in atomic units for R 1andR2and@1.9,3.1 #in radians for w. Convergence has been checked by comparing with the re-sults obtained by doubling the number of the grid pointsalong each variable but keeping the grid range ﬁxed. Thegrid range has also been checked for convergence. III. RESULTS AND DISCUSSION A. Absorption spectra Barinovs, Markovic ¸, and Nyman19calculated the absorp- tion spectrum of the OClO molecule using a similar numeri-cal method as we use here but employing hypersphericalcoordinates. The ab initio PESs of the gound X 2B1and ex- citedA2A2states were used in their calculation. The initial wave packet was propagated for a rather short time giving aspectrum with vibrational peaks of 53 cm 21full width at half maximum ~FWHM !. It was found that the PESs used well reproduce the intensities of the experimental absorptionpeaks for different progressions but also that the positions ofthe calculated peaks were somewhat shifted to longer wave-lengths. In the present work, the absorption spectrum is ob-tained from a long time autocorrelation function and thepeaks are therefore narrower, allowing the calculated resultsto be compared with an absorption spectrum obtained underjet-cooled conditions. 7 The upper panel of Fig. 1 shows the presently calculated A2A2ÃX2B1absorption spectrum of the OClO molecule using the ab initio PES.13The absorption spectrum using XGPES ~Ref. 14 !is displayed in the lower panel of Fig. 1. The autocorrelation function has been damped by an expo-nential function with G 0520 cm21before Fourier trans-054316-3 OClO molecule J. Chem. Phys. 122, 054316 (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: 128.114.34.22 On: Tue, 02 Dec 2014 22:12:05forming to obtain the absorption spectrum. We also show the experimental absorption spectrum29at medium resolution in Fig. 2. Comparing the results in these two ﬁgures, we ﬁndthat both potential energy surfaces approximately reproducethe overall shape of the absorption spectrum. None of them, however, is capable of exactly reproducing the experimentalresults, assuming that the J50 and the constant transition dipole moment approximations are good. In particular, theenvelope of the spectrum of the ab initio PES is too broad and the XGPES overrates the vibrational progression involv-ing bend excitation ( n,1,0) and underrates that involving an- tisymmetric stretch excitation ( n21,0,2). The notation ( v1,v2,v3) is used to denote the symmetric, bend, and anti- symmetric normal modes. The overall shape of the spectrumis better reproduced by XGPES whose maximum of the vi-brational progression appear at (10,0,0) and whose peaks atshort wavelengths vanish faster. Inclusion of a geometry-dependent transition dipole moment could change these con-clusion. Particularly on the short wavelength side, the calculated absorption peaks are too narrow, reﬂecting that it is impos-sible to reproduce the broadening of all vibrational peakswith a single lifetime, i.e., a single value of the dampingparameter. In the high energy excitation range, fast dissocia-tion from the A 2A2state occurs. ab initio data have only been calculated in the region near the equilibrium. The ab-sorption spectrum in the shorter wavelength region shouldtherefore not be used for judging the quality of the ab initio data. Barinovs, Markovic ¸, and Nyman, 19noted that to repro- duce the positions of the experimental absorption peaks, theab initiovalue for T e, 2.65 eV, had to be increased. We ﬁnd that to correctly reproduce the position of the ﬁrst vibrationalpeak ~0,0,0!,T ehas to be increased by 185.6 cm21for theab initioPES and by 185.0 cm21for XGPES. In 1990, Richard and Vaida reported a high resolution absorption spectrum of the OClO molecule using jet-cooledFourier transform ultraviolet spectroscopy. 7An expanded portion of the jet-cooled spectrum was presented in theirwork, which is reproduced here in the upper panel of Fig. 3.This makes a closer examination of the PESs in the lowenergy region possible. The corresponding portions of thecalculated absorption spectra using the ab initioPES and the XGPES are shown in the middle and bottom panels of Fig. 3,with a resolution G 0515 cm21. From the ﬁgure, we see that both PESs reproduce the symmetric stretch vibrational pro-gression ( n,0,0) well. However, the vibrational progression involving the bend excitation ( n,1,0) is better reproduced by theab initio PES than by XGPES. Especially for the vibra- tional progression involving the antisymmetric stretch exci-tation, the calculated spectrum using the ab initioPES agrees better with the experimental observation. As discussedabove, the XGPES overrates the activity in the bend motionand underrates the activity in the antisymmetric stretch mo-tion. The positions of the vibrational peaks for the XGPES agree much better with the experimental positions than thoseof theab initio PES. This is expected as the XGPES was developed in order to reproduce the experimental vibrationalenergies. 14The vibrational energy levels, read from the vi- brational peaks of the calculated absorption spectra, arelisted in Table I. From the table, it is seen that the vibrationaleigenenergies of our time-dependent absorption spectrumagree well with those of the variational calculation. FIG. 1. Calculated absorption spectra for the OCl35O molecule for the A2A2ÃX2B1transition on the ab initioPES~upper panel !and the revised PES ~bottom panel !, XGPES. In both cases, a damping corresponding to convolution with a Lorentzian function of 20cm21FWHM was performed. Each inset enlarges a small part of the spectrum. FIG. 2. The experimental medium-resolution absorption spectrum of theOClO molecule, reproduced from Ref. 29.054316-4 Sun, Lou, and Nyman J. Chem. Phys. 122, 054316 (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: 128.114.34.22 On: Tue, 02 Dec 2014 22:12:05Comparing the calculated spectra shown in Fig. 3 with those in Fig. 2 of Ref. 14, there are substantial differences inthe intensities of the peaks. The intensities of the transitionsare more sensitive to the convergence of the calculationsthan are the eigenvalues. This is particularly true in thepresent case since the equilibrium geometries of the A 2A2 andX2B1states of the OClO molecule differ substantially and there is strong coupling between the symmetric and an-tisymmetric stretch modes of the A 2A2state. We have paid special attention to this in our convergence checks. We havealso been in contact with Xie and Guo in regards to thedifferences but it is not clear where they come from.B. Raman spectra In 1999 Esposito et al.reported the Raman spectrum of gaseous OClO at ambient temperature on resonance with theX 2B12A2A2transition at excitation wavelengths of 368.9 and 360.3 nm.6In Fig. 4, the calculated Raman spectra using theab initio PES and XGPES are shown and in Fig. 5 the experimental results are shown. Although the calculated results do not include the rota- tional dynamics, insight into the features of the two PESs canbe obtained by comparing with the experimental observation,assisted by the conclusions drawn above in regards to theabsorption spectra. From Fig. 4, it can be seen that both the ab initio PES and XGPES reasonably reproduce the experimental Ramanspectra even though neither PES reproduces them in detail,assuming that the Condon and J50 approximations are valid. Similar to the results for the absorption spectra, XG-PES underrates the activity in the antisymmetric mode whichresults in a too small peak at 2 v3. From the calculated absorption and Raman spectra we have seen that both the ab initio PES and the XGPES de- scribe the A2A2state well. These surfaces have large anhar- monicity in the antisymmetric stretch mode, resulting fromstrong coupling with the symmetric stretch. 11–13This calcu- FIG. 3. Expanded vibrational peaks at low energy of the calculated absorp- tion spectra of the OCl35O molecule with a resolution of 15 cm21are shown in the middle ~ab initioPES!and bottom ~XGPES !panels. The correspond- ing experimental observation is shown in the upper panel, which also in- cludes absorption from the isotope OCl37O.TABLE I. Theoretical and experimental vibrational energy levels (cm21) for the transition A2A2ÃX2B1. (v1v2v3)Expt.aAb initio PESbAb initio PEScXGPESdXGPESe 000 0.0 0.0 0.0 0.0 0.00 010 288.1 280.7 280.5 286.8 286.79100 708.6 698.2 698.4 708.0 708.03 110 991.6 974.5 974.8 991.4 991.39 200 1407.8 1390.5 1390.2 f1408.9 1408.73 102 1581.4 1583.9 1582.9 1578.6 1579.16210 1688.6 1662.5 1662.6 1688.9 1688.66300 2101.1 2075.6 2076.0 2102.6 2102.23202 2264.2 2260.3 2265.0 2264.13310 2378.1 2344.6 2379.4 2378.74400 2788.7 2756.3 2789.0 2788.68302 2942.7 2932.0 2943.7 2943.68410 3061.3 3021.0 3062.2 3061.80500 3470.1 3431.2 3468.6 3468.30402 3617.5 3601.3 3618.2 3617.97510 3738.2 3693.0 3738.7 3737.85600 4146.0 4103.1 4141.7 4141.03502 4288.9 4270.4 4287.3 4287.11610 4409.9 4360.6 4408.7 4408.28700 4816.6 4769.3 4808.6 4808.49602 4954.3 4936.5 4951.2 4951.19710 5077.1 5022.4 5072.5 5071.84800 5481.5 5431.1 5469.6 5469.31702 5614.7 5598.4 5610.4 5610.29810 5734.5 5681.3 5730.2 5729.57 aReference 7. bAcorrection energy of 185.6 cm21has been added to the ab initiovalue for Te~which is 2.65 eV !. cReference 13. dAcorrection energy of 185.0 cm21has been added to the ab initiovalue for Te. eReference 14. fAccording to the work of Peterson and our calculation, Xie and Guo should have misassigned this level.054316-5 OClO molecule J. Chem. Phys. 122, 054316 (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: 128.114.34.22 On: Tue, 02 Dec 2014 22:12:05lations thus suggest that the OClO spectra can be interpreted without invoking the PES model which has non- C2vequilib- rium geometry and a double minimum along the antisym-metric stretch. 7,10Neither the ab initio PES nor the XGPES reproduces the fundamental activity in the anti-symmetricstretch seen in the small but visible peaks at 1105 ( v3) and 1553 cm21(v21v3) of the Raman spectra recorded at am- bient temperature, but it is possible that the Coriolis couplingis responsible for this.6,30TheJ50 calculations presented here do not include Coriolis coupling, which prevents usfrom deﬁnitely addressing this issue. IV. SUMMARY In this paper, we have calculated absorption and Raman spectra for the A2A2ÃX2B1transition of OClO. As upper surfaces, the ab initio A2A2PES of Peterson and the by Xie and Guo revised form of this PES, XGPES, were employed.For the ground state, the ab initio X 2B1PES of Peterson was used. In the discussions this surface is assumed accurate andany inaccuracies are considered to pertain to the upper sur-faces. A time-dependent wave packet model in Radau coor-dinates employing the split-operator propagator togetherwith FFT is used in the calculations. Comparing the calculated results with the experimental observations, we found that although the XGPES correctlygives the positions of the vibrational peaks and a better over-all shape of the absorption spectrum, the ab initio PES re- produces the detailed vibronic structure at long wavelengthbetter. Especially, the ab initio PES well represents the sig- niﬁcant overtone activity in the antisymmetric stretch mode,which has been experimentally observed in absorption andRaman spectra. The Raman spectrum which was used forcomparison with our theoretical results was recorded at am-bient temperature, while in our numerical model the J50 approximation was used. To draw deﬁnite conclusions aboutthe PESs for the A 2A2state of the OClO molecule, further calculations, including a geometry-dependent transition di-pole moment and nonzero total angular momentum, shouldbe performed. Yet, even though the presently investigatedsurfaces overall are quite accurate, there is room for furtherimprovement. FIG. 4. Calculated resonance Raman spectra of gaseous OClO obtained for excitation wavelengths of 368.9 nm~top!and 360.3 nm ~bottom !using the ab initio PES ~leftpanel !andXGPES ~rightpanel !.Thecorrectionsto T ewhich have been used in the calculations are given in Table I. FIG. 5. The experimental Raman spectra, corresponding with the calculatedresults in Fig. 4 ~reproduced from Ref. 6 !.054316-6 Sun, Lou, and Nyman J. Chem. Phys. 122, 054316 (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: 128.114.34.22 On: Tue, 02 Dec 2014 22:12:05ACKNOWLEDGMENTS This work was started at Go ¨teborg University and ﬁn- ished at Dalian Chemical Physics Institute. The authorsgreatly appreciate the helpful calculations of Professor Xieregarding convergence of his CPLwork, whereby they couldcorrect their mistakes in programming the XGPES. Stimulat-ing and useful discussions with Professor Guo are also heart-ily acknowledged, part of which lead to the inclusion of theRaman spectrum subsection in the present paper. Z.S. thanksDr. T. R. Stedl for his helpful discussion about the discrep-ancy between the calculated absorption spectra using differ-ent numerical methods. Financial support from the NationalScience Foundation ~Grant No. 29833080 !, the Knowledge Innovation Program of the Chinese Academy of ScienceGrant, and the Science Research Council of Sweden are ac-knowledged. 1P. J. Reid, J. Phys. Chem. A 106, 1473 ~2002!. 2K. A. Peterson and H.-T. Werner, J. Chem. Phys. 105, 9823 ~1996!. 3H. F. Davis and Y. T. Lee, J. Chem. Phys. 105, 8142 ~1996!. 4A. Furlan, H. A. Scheld, and J. R. Huber, J. Chem. Phys. 106, 6538 ~1997!. 5R. F. Delmdahl, D. H. Park, and A. T. J. B. Eppink, J. Chem. Phys. 114, 8339 ~2001!. 6A. P. Esposito,T. Stedl, H. Jonsson, P. J. Reid, and K.A. Peterson, J. Phys. Chem. A 103,1 7 4 8 ~1999!. 7E. C. Richard and V. Vaida, J. Chem. Phys. 94, 153 ~1991!. 8J. B. Coon, J. Chem. Phys. 14,6 6 5 ~1946!.9J. B. Coon, J. Mol. Spectrosc. 1,8 1~1957!. 10J. B. Coon, F. A. Cesani, and C. M. Loyd, Discuss. Faraday Soc. 35,1 1 8 ~1963!. 11J. C. D. Brand, R. W. Redding, and A. W. Richardson, J. Mol. Spectrosc. 34, 399 ~1970!. 12A. W. 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1.3651231.pdf	Oscillation threshold of a clarinet model: A numerical continuation approach Sami Karkara)and Christophe Vergez Laboratory of Mechanics and Acoustics, CNRS, UPR 7051, 31 chemin J. Aiguier, 13402 Marseille Cedex 20, France Bruno Cochelinb) E´cole Centrale Marseille, Po ˆle de l’E ´toile, Technopo ˆle de Cha ˆteau-Gombert, 38 rue Fre ´de´ric Joliot-Curie, 13451 Marseille Cedex 20, France (Received 2 December 2010; revised 17 March 2011; accepted 17 March 2011) This paper focuses on the oscillation threshold of single reed instruments. Several characteristics such as blowing pressure at threshold, regime selection, and playing frequency are known to change radi- cally when taking into account the reed dynamics and the ﬂow induced by the reed motion. Previous works have shown interesting tendencies, using analytical expressions with simpliﬁed models. In thepresent study, a more elaborated physical model is considered. The inﬂuence of several parameters, depending on the reed properties, the design of the instrument or the control operated by the player, are studied. Previous results on the inﬂuence of the reed resonance frequency are conﬁrmed. Newresults concerning the simultaneous inﬂuence of two model parameters on oscillation threshold, re- gime selection and playing frequency are presented and discussed. The authors use a numerical con- tinuation approach. Numerical continuation consists in following a given solution of a set ofequations when a parameter varies. Considering the instrument as a dynamical system, the oscillation threshold problem is formulated as a path following of Hopf bifurcations, generalizing the usual approach of the characteristic equation, as used in previous works. The proposed numerical approachproves to be useful for the study of musical instruments. It is complementary to analytical analysis and direct time-domain or frequency-domain simulations since it allows to derive information that is hardly reachable through simulation, without the approximations needed for analytical approach. VC2012 Acoustical Society of America . [DOI: 10.1121/1.3651231] PACS number(s): 43.75.Pq [NHF] Pages: 698–707 I. INTRODUCTION Woodwind instruments have been intensively studied since Helmholtz in the late 19th century. In particular, the question of the oscillation threshold has been the focus of many theoretical, analytical, numerical and experimentalstudies. 1–7Following the works of Backus and Benade, Wor- man8founded the basis of the small oscillation theory for clarinet-like systems, and Wilson and Beavers9(W&B) greatly extended his work on the oscillation threshold. Thompson10mentioned the existence of an additional ﬂow, related to the reed motion, and the importance of thereed resonance on intonation as early as 1979, but only recent literature such as Kergomard and Chaigne 11(chap. 9), Silva et al .12and Silva13extended the previous works of Wilson and Beavers to more complex models, taking into account the reed dynamics and the reed-induced ﬂow. Whereas the static regime of the clarinet (i.e., when the player blows the instrument but no note is played) can be derived analytically, the stability of this regime and the blowing-pressure threshold at which a stable periodic solutionoccurs is not always possible to derive analytically, depending on the complexity of the model under consideration. In thelatter two references, the authors analyze the linear stability of the static regime through a characteristic equation. It is a per- turbation method based on the equations written in the fre-quency domain, that suppose an arbitrarily small oscillating perturbation (with unknown angular frequency x) around the static solution. Substituting the linear expressions of the pas-sive components—involving the acoustical impedance Z deﬁned as: P(x)¼Z(x)U(x) and the reed’s mechanical trans- fer function Ddeﬁned as: X(x)¼D(x)P(x), where P(x), U(x) are the acoustical pressure and volume ﬂow at the input end of the resonator and X(x) is the reed tip displacement— into the coupling equation U¼fðX;PÞlinearized around the static solution and balancing the ﬁrst harmonic of the oscillat- ing perturbation, one gets the characteristic equation that the angular frequency xand the blowing pressure cmust satisfy at the instability threshold. As pointed out by Silva, 13the convergence towards an oscillating regime with angular frequency xafter destabili- zation of the static regime is not certain and requires to con- sider the whole non linear system and compute periodic solutions beyond the threshold, in order to determine the na-ture of the Hopf bifurcation (direct or inverse). However, in the case of a direct bifurcation, the oscillation threshold (for a slowly increasing blowing pressure) corresponds to the so-lution of the characteristic equation with the lowest blowinga)Also at: Aix-Marseille University, 3 place Victor Hugo, 13331 Marseille Cedex 03, France. Author to whom correspondence should be addressed. Electronic mail: karkar@lma.cnrs-mrs.fr b)Also at: Laboratory of Mechanics and Acoustics, CNRS, UPR 7051, 31chemin J. Aiguier, 13402 Marseille Cedex 20, France 698 J. Acoust. Soc. Am. 131(1), Pt. 2, January 2012 0001-4966/2012/131(1)/698/10/$30.00 VC2012 Acoustical Society of Americapressure cth. Periodic oscillations emerge at this point with an angular frequency xththat depends only on the other model parameters values. In the present paper, we propose a different method for computing the oscillation threshold that proves to be faster,more general and more robust: numerical continuation. Nu- merical continuation consists in following a given solution of a set of equations when a parameter varies. To theauthor’s knowledge, it is the ﬁrst time this numerical method is applied to the physics of musical instruments. While time- domain or frequency-domain simulations allow to studycomplex physical models with at most one varying parame- ter, the method proposed in this work allows to investigate the simultaneous inﬂuence of several parameters on the os-cillation threshold. The general principle of the method is the following: we ﬁrst follow the static regime while increasing the blowingpressure and keeping all other parameters constant and detect the Hopf bifurcations. Then, we follow each Hopf bifurcation when a second parameter lis allowed to vary ( lbeing, for instance, the reed opening parameter or the reed resonance frequency). Finally, the resulting branch of each bifurcation is p l o t t e di nt h ep l a n e( l,c th) for determination of the oscillation threshold and of the selected regime, and in the plane ( l,xth) to determine the corresponding playing frequency. The paper is structured as follows. In Sec. II, a physical model of a clarinet is reviewed. The method is described in Sec. III, deﬁning the continuation of static solutions and Hopf bifurcations. In Sec. IV, the results are reviewed. First, basic results on the reed-bore interaction are compared with the previous works (W&B9and Silva et al .12). Then, extended results are presented, showing how the control ofthe player and the design of the maker inﬂuence the ease of play and the intonation. II. PHYSICAL MODEL OF SINGLE REED INSTRUMENTS The model used in this study is similar to the one used in Silva et al.12It is a three-equation physical model that embeds the dynamical behavior of the single reed, the linearacoustics of the resonator, and the nonlinear coupling due to the air jet in the reed channel. It assumes that the mouth of the player (together with the vocal tract and lungs) providesan ideal pressure source, thus ignoring acoustic behavior upstream from the reed. A. Dynamics of the reed The reed is indeed a three-dimensional object. However, due to its very shallow shape, it can be reduced to a 2D-plateor even, considering the constant width, a 1D-beam with vary- ing thickness. Detailed studies of such models have been con- ducted (see for instance Avanzini and van Walstijn 14for a distributed 1D-beam model) and lumped models have been proposed. Mainly, two lumped models are widely used: (1) a one degree of freedom mass-spring-damper system, accounting for the ﬁrst bending mode of the reed; (2) a simpler one degree of freedom spring, assuming a quasi-static behavior of the reed.Most papers studying clarinet-like systems use the sec- ond model. This kind of simpliﬁcation is made assumingthat the reed ﬁrst modal frequency is far beyond the playing frequency, thus acting as a low-pass ﬁlter excited in the low frequency range, almost in phase. However, some authors(see for instance W&B 9or Silva13) showed that the reed dy- namics, even rendered with a simple mass-spring-damper os- cillator, cannot be ignored because of its incidence on thephysical behavior of the dynamical system. Thus, in the present study, the reed dynamics is rendered through a lumped, one-mode model. It reﬂects the ﬁrst bend-ing mode of a beam-like model with a reed modal angular frequency x r, a modal damping qr, and an effective stiffness per unit area Ka. The reed is driven by the alternating pressure difference DP¼Pm–P(t) across the reed, where Pmis the pressure inside the mouth of the player and P(t) is the pressure in the mouthpiece. Given the rest position of the tip of the reed h0 and the limit h¼0 when the reed channel is closed, the tip of the reed position h(t) satisﬁes the following equation: 1 x2 rh00ðtÞ¼h0/C0hðtÞ/C0qr xrh0ðtÞ/C0DP Ka: (1) Contact with the mouthpiece could also be modeled, using a regularized contact force function (as presented by the authors in a conference paper15). It will not be discussed or used in this paper, as large amplitude oscillations will not be discussed. However, it is important to note that such phe- nomenon cannot be ignored when the whole dynamic rangeof the system is under study. B. Acoustics of the resonator The acoustic behavior of the resonator of a clarinet is here assumed to be linear. It is modeled through the inputimpedance Z in¼P/U, where Pis the acoustical pressure and Uthe acoustical volume ﬂow at the reed end of the resona- tor. To keep a general formulation, we use a (truncated)complex modal decomposition of the input impedance (as described by Silva 13) of the form: ZinðxÞ¼PðxÞ UðxÞ¼ZcXNm n¼1Cn jx/C0snþC/C3 n jx/C0s/C3 n/C18/C19 : (2) This leads, in the time domain, to a system of Nmdifferential equations, each governing a complex mode of pressure at the input of the resonator: P0 nðtÞ¼CnZcUðtÞþsnPnðtÞ; (3) where Zc¼qc/Sis the characteristic impedance of the cylin- drical resonator, whose cross-sectional area is S. The total acoustic pressure is then given by: PðtÞ¼2XNm n¼1Re½PnðtÞ/C138: (4) In practice, the modal coefﬁcients ( Cn,sn) can be computed in order to ﬁt any analytical or measured impedance J. Acoust. Soc. Am., Vol. 131, No. 1, Pt. 2, January 2012 Karkar et al.: Oscillation threshold of a clarinet model 699spectrum to the desired degree of accuracy (depending on the total number of modes Nm). In the present study, we use an analytical formulation for a cylinder of length L¼57 cm (if not stated otherwise), radius r¼7 mm, taking into account the radiation at the output end and the thermoviscous losses due to wall friction, as given by Silva.13The effect of tone holes is not considered in this work. C. Nonlinear coupling As described by Hirschberg16and further conﬁrmed experimentally by Dalmont et al.17and Almeida et al.,18the ﬂow inside the reed channel forms a jet that is dissipated by turbulence in the larger part of the mouthpiece leading to the following nonlinear equation: UjðtÞ¼signðDPÞWhðtÞﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2DPjj qs ; (5) where Wis the width of the reed channel, assumed to be con- stant, and Ujhas the sign of DP. According to this model, the direction of the ﬂow depends on the sign of the pressure drop across the reed DP. D. Reed motion induced flow Moreover, when an oscillation occurs (a note being played), the reed periodic motion induces an acoustic vol-ume ﬂow in addition to the one of the jet, as early described by Thompson 10and lately studied by Silva et al.12Writing Srthe reed effective area Sr, the resulting ﬂow Urreads: UrðtÞ¼/C0 Srdh dtðtÞ: (6) Notice the negative sign, due to the chosen representation where the reed tip position his positive at rest and null when closing the reed channel. Some authors11,19use another notation, the equivalent length correction DL, related to Sras DL¼qc2 KaSSr: We prefer to stick with the notation Sr, but we will give the corresponding values of DLfor comparison, when necessary. E. Global flow The global ﬂow entering the resonator then reads UðtÞ¼signðDPÞWhðtÞﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2DPjj qs /C0Srdh dt: (7) F. Dimensionless model The above-described multi-physics model involves several variables. Expressed in SI units, their respective values are of very different orders of magnitude. However, as we willuse numerical tools to solve the algebro-differential system composed of Eqs. (1),(3),(4), and (7), it is a useful precau- tion to use a dimensionless model. In the dimensionless model, each variable should be di- vided by a typical value, in order to scale the range of possi-ble values for that variable as close as possible to [0,1]. Choosing the static difference of pressure necessary to close the reed channel P M¼Kah0to scale all pressures, and the reed tip opening at rest h0to scale the reed tip position, the fol- lowing new dimensionless variables and parameter are deﬁned: xðtÞ¼hðtÞ=h0 yðtÞ¼h0ðtÞ=v0 pnðtÞ¼PnðtÞ=PMforn¼1;2; :::Nm pðtÞ¼PðtÞ=PM uðtÞ¼UðtÞZc=PM where v0¼h0xris a characteristic velocity of the reed. The system of algebraic and differential equations [Eqs. (1),(3),(4), and (7)] can be rewritten as follows (time- dependence is omitted, for a more concise writing): 1 xrx0¼y 1 xry0¼1/C0xþp/C0c/C0qry p0 n¼Cnuþsnpnfor n¼1;2; :::Nm p¼2PNm n¼1ReðpnÞ u¼signðc/C0pÞfxﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ c/C0pjjp /C0v0Zc PMSry;8 >>>>>>>>>>>>>>< >>>>>>>>>>>>>>:(8) where c¼P m/PMis the dimensionless blowing pressure and f¼ZcWﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2h0=qKap is the dimensionless reed opening pa- rameter. The reader will notice that the parameter f, related to the maximum ﬂow through the reed channel, mainlydepends on the geometry of the mouthpiece, the reed me- chanical properties, as well as the player’s lip force and posi- tion on the reed that control the opening. In the following sections we will compute the minimal mouth pressure c thnecessary to initiate self-sustained oscil- lations from the static regime and show how this threshold ismodiﬁed when a second parameter of the model varies. When players change the way they put the mouthpiece in their mouth, several parameters of our model are assumed tovary: the control parameter f, the reed modal damping q r, its modal angular frequency xr, as well as the effective area Sr of the reed participating to the additional ﬂow. We will ﬁrst investigate the inﬂuence of the dimensionless parameter krL¼xrL/c, comparing with previous works. Then the inﬂu- ence of f,qr, and Srwill also be investigated. III. METHODS: THEORETICAL PRINCIPLES AND NUMERICAL TOOLS In this section, the method used is brieﬂy reviewed from a theoretical point of view. Practical application of the method will be carried out in Sec. IV. 700 J. Acoust. Soc. Am., Vol. 131, No. 1, Pt. 2, January 2012 Karkar et al.: Oscillation threshold of a clarinet modelA. Branch of static solutions We consider an autonomous nonlinear dynamical sys- tem of the form: u0ðtÞ¼FðuðtÞ;kÞ; (9) where u(t) is the state vector and kis a chosen parameter of the system equations. Letu0[Rnbe a ﬁxed point of this system for k0: F(u0,k0)¼0. Under the hypothesis that u0is a regular solu- tion, there exists a unique function u(k) that is solution of the previous equation for all kclose to k0,w i t h u(k0)¼u0.( S e e Doedel20for formal deﬁnitions, as well as for more details about regular solutions and what is meant by “close.”) In other words, for kclose to k0,there is a continuum of equilibrium points that passes by u0and the branch of solu- tions is represented by the graph ui¼f(k), where uiis a com- ponent of the state vector u. B. Continuation of static solutions Considering the differential equations in system (8),i t can be rewritten in the form u0¼F(u,k) where uis the state vector and kis one of the equation parameters (for instance the blowing pressure c). Thus, a static solution is given by the algebraic system: F(u,k)¼0, where the algebraic equa- tions of system (8)can now be included. The branch of static solutions is computed numerically using a classical path following technique based on Keller’s pseudo-arc length continuation algorithm21as implemented in the softwares AUTO (Doedel and Oldeman22) and MANLAB (Karkar et al.23), both freely available online. For comprehen- sive details about continuation of static solutions using these two numerical tools, please refer to the conference article.15 C. Hopf Bifurcation Let us consider a static solution uof system (9). The sta- bility of this equilibrium point is given by eigenvalues of thejacobian matrix of the system: F uðu;kÞ¼@Fi @uj/C20/C21 : If all the eigenvalues of Fuhave a strictly negative real part, then the equilibrium is stable. If any of its eigenvalues has astrictly positive real part, then the equilibrium is unstable. Several scenarios of loss of stability, along the branch u¼f(k) are possible. One of them is the following: a unique pair of complex conjugate eigenvalues crosses the imaginary axis at ( ix th,/C0ixth) fork¼kth. In that case, the system is said to undergo a Hopf bifur- cation. It means that a family of periodic orbits starts from this point u(kth), with an angular frequency xth. As for the clarinet model, choosing the blowing pressure cas the vary- ing parameter, the ﬁrst Hopf bifurcation encountered when c is increased from 0 is the oscillation threshold: it is the mini- mal blowing pressure cthneeded to initiate self-sustainedoscillations (i.e., to play a note) when starting from zero and increasing quasi-statically the blowing pressure. D. Branch and continuation of Hopf bifurcations This deﬁnition of a Hopf bifurcation can be written as an extended algebraic system G¼0 where Greads: Fðu;k;lÞ Fuðu;k;lÞ /T//C018 < ://C0jx/ (10) kandlare two parameters of interest (for instance the blow- ing pressure cand the reed opening parameter f), and /is the normed eigenvector associated with the purely imaginaryeigenvalue jx. Assuming that a given solution X 0¼(u0,/0,x0,k0,l0) is a regular solution of G¼0, there exists a continuum of solutions X(l)¼[u(l),/(l),x(l),k(l),l] near X0. Thus, the function X(l) is a branch of Hopf bifurcations of our ini- tial dynamical system. Using the same continuation techniques as in Sec. III B , this branch of Hopf bifurcations can be computed. The reader is kindly referred to classical literature on the subject forproper theoretical deﬁnitions and other details about the con- tinuation of Hopf bifurcations (see for instance Doedel 20). IV. RESULTS In the current section, the method described in the previ- ous section is applied to the physical model presented in Sec. IIto investigate the oscillation threshold of a purely cy- lindrical clarinet. The resonator modal decomposition has been computed with the MOREESC24software, using 18 modes. The corresponding input impedance spectrum isshown on Fig. 1. For comparison, the impedance spectrum given by the analytical formulae of Backus 1and used in Wil- son and Beavers9is also plotted. Important differences FIG. 1. (Color online) Impedance spectrum (reduced modulus) of a purely cylindrical bore of length L ¼57 cm taking into account thermoviscous losses and radiation and truncated to 18 modes as used in this study (—), an- alytical impedance spectrum without modal decomposition (– /C1–), imped- ance spectrum used in Wilson and Beavers (Ref. 9) (– –). J. Acoust. Soc. Am., Vol. 131, No. 1, Pt. 2, January 2012 Karkar et al.: Oscillation threshold of a clarinet model 701appear concerning frequency dependent damping (peaks heights) and harmonicity (peaks positions). A. Reed-bore interaction 1. Comparison with previous results Previous works of Wilson and Beavers9and Silva et al.12showed how the resonance of the reed competes with the acoustical modes of the resonator for the existence of self-sustained oscillations: oscillation thresholds and emer-gent frequencies were then measured and numerically com- puted, for different values of the dimensionless product k rL (kr¼xr/cis the corresponding wavenumber of the reed modal frequency, and Lis the length of the bore—which is approximately the quarter of the ﬁrst acoustical mode wave- length) called “tube parameter” in W&B.9Notice that in these studies, the parameter krwas kept constant and Lwas varied. These results showed that several regimes can be selected, depending on the product krL, only if the reed damping is very small. Figure 2shows similar computations using our method, with the same model parameters as was used for the Fig. 1of Silva et al.,12only using a different impedance (as explained above). A secondary x-axis, on top of each plot, shows the corresponding ratio of the reed natural frequency xrto the ﬁrst resonance frequency of the bore x1¼Im(s1). Note that instead of varying Lfor a given xr, the ﬁgure was computed by varying xrfor a given L. The method advantage, here, is that no analytical development of the impedance spectrum is needed, unlike what was done by W&B to derive the charac- teristic equation. Thus, any other impedance could be used,for instance a measured one.Figure 3illustrates how the oscillation threshold is deduced from the previous ﬁgure: for a given abscissa, it cor-responds to the lowest of the ﬁve Hopf bifurcation branches, each corresponding to a register. In contrast with W&B, this ﬁgure clearly shows that even in the case of a stronglydamped reed ( q r¼0.4), a regime selection occurs with vary- ingxr. The term “strongly damped” was used by W&B for that value of qr, but it seems to be a realistic value for a clari- net reed coupled to the player’s lip (see van Walstijn and Avanzini25and Gazengel et al.26for numerical and experi- mental studies of the reed-lip system). A minimum of thethreshold blowing pressure (lower plot) appears a little above x r¼x1, i.e., when the ﬁrst air column resonance frequency is close to the reed one. The frequency of a given mode (upperplot) is close to the corresponding passive resonance fre- quency x n¼Im(sn), for medium and high values of krL,b u t tends to the reed modal frequency for low values of krL. 2. Relevance of the tube parameter k rL While results of the previous ﬁgure are in good agree- ment with Silva’s results (when thermoviscous losses are taken into account, as in Fig. 3of Silva et al.12), several dif- ferences lead to question the relevance of the representation: ifkrLis a characteristic parameter of the model, varying L for a given xrshould be equivalent to varying xrwith a ﬁxed value of L. To answer this question, we recomputed the ﬁrst regime for the same values of krLbut with L¼14.69 cm (one then needs to recompute the modal coefﬁcients of the input im- pedance). The results are plotted in Fig. 4: considering the upper plot, the frequency seems to be independent of L,a s long as the product krLis kept constant (the plain line and dashed line are exactly superimposed); however, considering the lower plot, the blowing pressure does not behave in thesame way, for identical values of k rL, depending on the value of L. Therefore, the tube parameter krLis not a charac- teristic invariant parameter of the model. One good reason for that is that in our impedance spec- trum, the peaks are inharmonic and have a frequency de- pendent quality factor and magnitude. Then varying Ldoes not preserve the impedance peaks height and width, nor their spacing, which leads to a different balance in the competi- tion with the reed resonance. B. Simultaneous influence of reed damping and modal frequency In previous works,9,13ﬁgures similar to Fig. 2were plot- ted for two cases, small and large values of the reed damping qr, which revealed very different behaviors. Because the method proposed in this paper leads to very short computation time, it is quite easy to loop the computa- tion for a series of qrvalues. For sufﬁciently close values of qr, this allows to draw a tridimensional plot in which the sur- face represents the critical value of the dimensionless blow- ing pressure ccorresponding to a given Hopf bifurcation as a function of two parameters, the reed damping qrand the fre- quency ratio xr/x1, as illustrated Fig. 5for the Hopf bifurca- tion of the ﬁrst register. The plotted surface corresponds to a FIG. 2. Critical blowing pressure and frequency of each Hopf bifurcation (labeled from 1 to 5) as a function of the “tube parameter” krL(as used in W&B), or the frequency ratio xr/x1(secondary x-axis on top). (bottom) Critical blowing pressure c. (top) Frequency ratio h¼x/xrof the oscilla- tions; passive resonances of the bore hn¼xn/xrare plotted for comparison (–/C1–). Physical constants and parameters of the model were chosen accord- ing to Fig. 1in Silva et al. (Ref. 12):q¼1.185 g/L, c¼346m/s, L¼57 cm, r¼7 mm, qr¼0.4,f¼0.13, Sr¼0, varying xr. 702 J. Acoust. Soc. Am., Vol. 131, No. 1, Pt. 2, January 2012 Karkar et al.: Oscillation threshold of a clarinet modelHopf bifurcation locus when two parameters are varied (con- versely to Fig. 2where only one parameter is varied). The two plain, thick curves correspond to the limits of the do- main considered: qr¼0.05 and qr¼1. This ﬁgure allows to investigate the transition between large and light damping. It appears that for large values of xr/x1, the critical blowing pressure is nearly independent of qr: a closer look reveals a slight increase with qr.T h i si sn o t surprising since xr/x1/C291 corresponds to the case where the ﬁrst resonance frequency of the resonator is very small com-pared to xr. Hence, only very large values of qr(qr/C291) would contribute signiﬁcantly to an increase of the pressure threshold. However, for smaller values of xr/x1, the value of qrbecome determinant. In the 2D manifold, there is a valley which becomes deeper when qrdecreases. For a given qr,t h e bottom point of the valley (called c0in Silva et al.12)c o r r e - sponds to the minimum threshold. It is plotted as a dot-dashedline on the plane ( x r/x1¼0), whereas the corresponding ab- scissaxr x1j0is plotted as a dot-dashed line in the plane c¼0. It shows that the minimum value c0is an increasing function of qr. The same conclusion holds forxr x1j0. Notice that the chosen range for the values of qrseems quite realistic, according to literature on the subject: in a nu-merical model, van Walstijn and Avanzini 25reported param- eters values equivalent to qr¼0.24 for one playing condition, whereas Gazengel et al.26found experimental val- ues going from qr¼0.05 for the bare reed to qr¼1.54 for a high lip pressure on the reed. Thus, the lower value qr¼0.05 corresponds to the limit case of a very resonant reed with nolip pressing on it, and the higher value q r¼1.00 is high enough to cover a fairly good range of lip pressures. Now we illustrate that the results obtained on this model allow to estimate the range of validity of analytical formula obtained in approximated cases. For instance, in Fig. 6,t h e minimum c0obtained through numeric al continuation (without approximation), is compared w ith two analytical formula from Silva13corresponding to different approximations: no losses in the cylindrical bore (here plo tted with dotted line, corresponds to Eq. (14) in Silva et al.12) and a single-mode resonator with losses (dot-dashed line, Eq. (19) in Silva et al.12). It appears that only one mode with viscothermal losses leads to a more precise result than the undamped formulation FIG. 3. (Color online) Oscillation threshold: blowing pressure, regime selection and frequency with respect tokrL. The oscillation threshold, for a given abscissa, is given by thelowest blowing pressure of the ﬁve Hopf bifurcation branches (thick line). (bottom) Blowing pressure threshold c th. (top) Frequency ratio hth¼xth/xrof the oscillations. Complete branches of the ﬁve bifur- cations are reminded (dashed lines).Same model parameters were used as Fig. 2. A secondary x-axis on top of each plot gives the value of the frequency ratio x r/x1. FIG. 4. Comparison of the ﬁrst regime for L¼57 cm (—) and L¼14.69 cm (– –). (bottom) Dimensionless blowing pressure cof the bifurcation. (top) Corresponding dimensionless frequency ratio h¼x/xr; the ﬁrst passive res- onance of the bore is reminded ( /C1/C1/C1). J. Acoust. Soc. Am., Vol. 131, No. 1, Pt. 2, January 2012 Karkar et al.: Oscillation threshold of a clarinet model 703with all the modes. However, the relative deviation from our results still reaches 10%. However, the oscillation threshold is not always given by the Hopf bifurcation corresponding to the ﬁrst register. Considering not only the ﬁrst register, but the ﬁrst ﬁve regis-ters, leads to Fig. 7(a). Note that the number of registers is not arbitrary: at the right end of the ﬁgure, where x r/x1’10 (krL’16), the sixth resonance ( w6’11w1) is well beyond the reed resonance and thus cannot set up self-sustained oscillations because its Hopf bifurcation is beyond c>1. More registers should be considered if greater xrvalues were to be explored. For sake of clarity. c0andxr x1/C12/C12/C12 0are not plotted. The plain lines in the plane ( qr¼0.05) shows similar behavior as in Fig. 2and has already been discussed. The other black plain lines depict a very different situation where the ﬁrst register always correspond to the lowest c.Figure 7(b)shows the oscillation threshold, as deﬁned by the ﬁrst Hopf bifurcation encountered when increasing the blowing pressure cfrom 0, as a function of the reed damping qrand the frequency ratio xr/x1. It is deduced from the previous one the same way Fig. 3was deduced from Fig. 2. Intersections between the ﬁve surfaces result in an oscillation threshold surface with several local minima. It clearly shows how the reed damping, which can be control by the player with his lower lip, plays a key role in the register selection, as previously reported by W&B9and Silva et al.12It is also clearly visible that the range of qrfor which a given register exists decreases with the index of this register. For instance, considering the ﬁfth register, it canonly be selected for q r<0.3. The frequencies corresponding to the different registers are also calculated and plotted in Fig. 8. The frequency at threshold appears to be an increasing function of qr, but the inﬂuence of qrdoes not look signiﬁcant in this 3D represen- tation. When qrgoes from 0 to 1, the typical relative fre- quency deviations from the bore resonances are less than 1%. However, a frequency shift more than 4% can be observed for the ﬁrst register around xr/x1¼1.1. Such a FIG. 5. (Color online) Surface giving the critical blowing pressure cof the Hopf bifurcation corresponding to the ﬁrst register, with respect to the reed damping parameter qrand the reed modal frequency xr(divided by the ﬁrst bore resonance x1). The plain curves correspond to the limits of the chosen range for qr, i.e., qr¼0.05 and qr¼1. The dot-dashed curve in the plane xr/x1¼0 corresponds to the minimum c0, and the dot-dashed curve in the plane c¼0 corresponds to the value of xr/x1at which it occurs. FIG. 6. Minimum threshold c0as a function of qr. Our numerical results (—), without approximation. Analytical results using two approximations: no loss (/C1/C1/C1), single-mode resonator with losses (– /C1–). FIG. 7. (Color online) ( a) Similar plot as in Fig. 5for the ﬁrst ﬁve registers: the critical blowing pressure cof each Hopf bifurcation is plotted with respect to the reed damping parameter qrand frequency ratio xr/x1. The plain curves correspond to the smallest and highest qrwhere a bifurcation occurs on the chosen range. ( b) The oscillation threshold extracted from Fig. 7(a): blowing pressure at threshold cth, deﬁned by the lowest of the ﬁve surfaces plotted in Fig. 7(a). 704 J. Acoust. Soc. Am., Vol. 131, No. 1, Pt. 2, January 2012 Karkar et al.: Oscillation threshold of a clarinet modelratio is quite unusual for a clarinet, but it is of great interest for reed organ-pipe manufacturer, where the reed natural fre- quency is close to the bore resonance and the damping is very small. C. Influence of the control parameter f Let us remind here the deﬁnition of this parameter: f¼WZ cﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2h0=qKap . Its variations are mainly related to the changes of the reed opening parameter h0during the play. Those are directly driven by changes of the player’s lip pres-sure and position on the reed. Thus it is a very important control parameter of the model. Notice that players modify both q randfwhen changing the embouchure. Figure 9shows the inﬂuence of fon the blowing pres- sure and frequency at the oscillation threshold (expressed as the relative deviation to the corresponding bore resonancefrequency). This control parameter happens to be critical for the regime selection: from very low values up to f¼0.17, the ﬁrst regime (with a frequency close to the ﬁrst acousticalmode of the bore f 0) is selected, while the fourth regime ( f’ 7f0) is selected for higher values. The control parameter falso has noticeable inﬂuence on the frequency of the oscillations at threshold: whereas the ﬁrst regime frequency deviation is less than 0.3% (5 cents) in the range of fwhere it is the selected regime, the fourth regime frequency deviation is as high as 2.7% (46 cents) for f¼0.8. This is a very important feature that has been high- lighted in a paper by Guillemain et al.,27where the lip stress on the mouthpiece of a saxophone were measured while a player was playing, showing an adjustment of the lip stress on the reed in order to correct the tone shortly after the be-ginning of the oscillations. It could also explain the difﬁcult reproducibility of measurements when ﬁtting a clarinet or saxophone mouthpiece in an artiﬁcial mouth. The frequency deviations are monotoneous, decreasing functions of f, almost linear on the range of interest. Also, when ftends towards 0, the frequency at threshold tends tothe corresponding bore passive resonance frequency. This result was to expect, since the boundary condition at the input end tends to a Neumann condition (inﬁnite impedance). However, for very low values of f, the blowing pressure threshold c thquickly increases, and eventually reaches the unit value which is the also the static closing threshold. Inthe case where c th>1, the reed channel is always closed and no sound is possible. D. Concurrent influence of qrand f Figure 10shows the blowing pressure threshold with respect to the reed damping qrand to the control parameter f. For a given pair ( qr,f), the oscillation threshold is given by the lowest surface among the ﬁve 2D manifolds corre-sponding to the different registers. FIG. 8. (Color online) Dimensionless frequency h¼x/xrat the bifurcation for the ﬁrst ﬁve registers, with respect to the reed damping qrand the fre- quency ratio xr/x1. FIG. 9. (Color online) Variations of the oscillation threshold with respect to (bottom) critical, dimensionless blowing pressure c; the lowest curve deﬁnes the oscillation threshold cthat a given abscissa. (top) Relative frequency deviation Dx/xn¼(x–xn)/xnbetween the frequency at threshold xthand the corresponding acoustical mode frequency xn¼Im(sn). Model parame- ters: qr¼0.4,fr¼1500 Hz, Sr¼0c m2. FIG. 10. (Color online) Oscillation threshold with respect to the reed damp- ing parameter qrand the control parameter f. The blowing pressure thresh- oldcthis calculated the same way as in Fig. 7(b). The surface shading indicates the selected regime. J. Acoust. Soc. Am., Vol. 131, No. 1, Pt. 2, January 2012 Karkar et al.: Oscillation threshold of a clarinet model 705Because the player’s lower lip pressure and position on the reed, induce variations of both fandqrat the same time (the lip pressure being positively correlated with qrand nega- tively with f), it is interesting to follow the oscillation thresh- old along the horizontal plane f¼1–qr. On the one hand, for small values of fandqrclose to unity (i.e., a high lip pressure) the ﬁrst regime is clearly selected; on the other hand, for a high value of fand small damping qr(i.e., a relaxed lip on the reed), it is the ﬁfth regime that is clearly selected. In between, all regimes are successively selected except the second regime that is noticeably absent of the ﬁgure. E. Influence of the reed motion induced flow In the numerical results presented so far, the reed induced ﬂow was not taken into account ( Sr¼0 in all previous ﬁg- ures). However, as pointed out by Silva et al.,12it is important to take into account this additional ﬂow, in order to predict accurately the emergent frequency at oscillation threshold. Figure 11shows the inﬂuence of the ﬂow induced by the motion of the reed, through the value of the equivalent area Srof the reed that participates to this ﬂow. Previous works by Dalmont,19Kergomard,11or Silva12,13report val- ues around DL¼10 mm, which corresponds, for a reed stiff- ness Ka¼8.106Pa/m and a bore of radius r¼7m m ,t o Sr¼0.86 cm2. Thus, we scaled the variations of Srbetween 0 and 2 cm2, using parameter value corresponding to a relaxed embouchure, in order to well illustrate its inﬂuence. An important frequency deviation is shown as Sr increases, as well as a regime selection “cascade”: the selected register is successively the ﬁfth, the fourth, thethird, and ﬁnally for high values of S rthe ﬁrst. Finally, com- paring with Fig. 10(in the plane f¼0.75), the reed induced ﬂow acts in a similar manner as the reed damping on theblowing pressure threshold and the regime selection. V. CONCLUSION In this paper, a clarinet physical model is investigated using numerical continuation t ools. To the authors’ knowl-edge, it is the ﬁrst time that such method is used to compute the oscillation threshold variations with respect to severalmodel parameters. The reed dynamics and its induced ﬂow are shown to have critical inﬂuence on the regime selection and the minimal blowing pressure necessary to bifurcate from thestatic regime and establish a stea dy-state periodic oscillation: a note. Previous works already gave useful insights concerning the inﬂuence of some model parameters on ease of play andintonation. The present work conﬁrms and extends these results to the case of a more complex model. Moreover, the method used allows one to invest igate variations of two param- eters at the same time, instead of one, like in previous studies. In this approach, the parameters of the model are assumed to be constant or to undergo quasi-static variations. However,in real situation, the player c an modify some control parame- ters (e.g., blowing pressure, lip stress on the reed) at a time- scale that might sometimes be comparable to the oscillationperiod. Guillemain 27reported measured variations of con a time scale of a few milliseconds only, which is comparable to the period of an oscillation at 150 Hz. However, despite sucha limitation, the results provided through numerical continua- tion are out of reach for direct time-domain simulations. Whereas the main results presented here concern a clari- net model, it should be noted that the method itself is very general. No hypotheses (other than linear behavior) is made on the resonator. Thus, other resonators can be studied by ﬁt-ting the modal decomposition to its input impedance spec- trum. For instance, extending to the case of the saxophone or taking into account the tone holes only requires to computethe corresponding modal decomposition of the input imped- ance. Even the physical model can be modiﬁed. It only has to be written as a set of ﬁrst order ordinary differential equations(and additional algebraic equations, if necessary). The method could also be used to compare models with each other. In comparison with the method of the characteristic equation, the computations are much faster: Silva reported 10 min of computation per branch, whereas it lasts only a few seconds in the present case. Moreover, the continuationalgorithm used is very robust: strong variations (as variations ofcfor low k rLin Fig. 2) do not require special care and are computed in a straightforward way. The same continuation method applied to the continua- tion of periodic solutions, as described by the authors in a con- ference paper,15also allows to compute the entire dynamic range of a given model of wind instrument, without any addi- tional simpliﬁcation, unlike previous works (see Dalmont28 et al .). From that perspective, the numerical continuation approach seems promising for the global investigation of the behavior of a given physical model of musical instrument. Applications of this work to instrument making are pos- sible: for instance, modiﬁcations of the geometry of the reso- nator can be studied in terms of their inﬂuence on ease of play and intonation. Other applications concerning mappingstrategies for sound synthesis are also of interest. Indeed, the estimation of the parameters of a model is a difﬁcult task, especially when the parameter values are allowed to vary, inorder to reproduce typical behaviors of the modeled instru- ment through sound synthesis. On the one hand, direct meas- urements on a real player are most of the time highly FIG. 11. (Color online) Variations of the oscillation threshold with respect toSr: critical blowing pressure cand relative frequency deviation from the bore resonances Dx/xnfor each of the ﬁve Hopf bifurcations. Parameters of the model: qr¼0.3,f¼0.75, fr¼1500 Hz. 706 J. Acoust. Soc. Am., Vol. 131, No. 1, Pt. 2, January 2012 Karkar et al.: Oscillation threshold of a clarinet modelcomplicated (for example, xr,qr,o rh0even when the lips of the player is pressed on the reed). On the other hand, inverseproblems still appear to be limited to the estimation of pa- rameter values from a synthesized sound. Thus, the approach presented in this paper offers the possibility to know in advance the inﬂuence of the parameter values on some key features of the model behavior around the oscillation threshold: ease of play, regime selection andﬁne intonation. Therefore, mapping strategies could be developed for sound synthesis applications. They would con- sist in binding different parameters in order to let a playermodify one of them while maintaining the same playing fre- quency (at threshold) for a given note for example, or while preserving the ease of play on a whole register. 1J. Backus, “Small-vibration theory of the clarinet,” J. Acoust. Soc. Am. 35, 305–313 (1963). 2A. H. Benade and D. J. Gans, “Sound production in wind instruments,” Ann. N.Y. Acad. Sci. 55(1), 247–263 (1968). Available at 10.1111/ j.17496632.1968.tb56770.x 3J. Saneyoshi, H. Teramura, and S. Yoshikawa, “Feedback oscillations inreed woodwind and brasswind instruments,” Acustica 62, 194–210 (1987). 4N. Fletcher and T. Rossing, The Physics of Musical Instruments (Springer, Berlin, 1991), pp. 345–494. 5N. H. Fletcher, “Autonomous vibration of simple pressure-controlledvalves in gas ﬂows,” J. Acoust. Soc. Am. 93, 2172–2180 (1993). 6Y. M. Chang, “Reed stability,” J. Fluids Struct. 8, 771–783 (1994). 7A. Z. Tarnopolsky, N. H. Fletcher, and J. C. S. Lai, “Oscillating reed valves—An experimental study,” J. Acoust. Soc. Am. 108, 400–406 (2000). 8W. E. Worman, “Self-sustained nonlinear oscillations of medium ampli- tude in clarinet-like systems,” Ph.D. thesis, Case Western Reserve Univer- sity, Cleveland, Ohio, 1971. 9T. A. Wilson and G. S. Beavers, “Operating modes of the clarinet,”J. Acoust. Soc. Am. 56, 653–658 (1974). 10S. C. Thompson, “The effect of the reed resonance on woodwind tone production,” J. Acoust. Soc. Am. 66, 1299–1307 (1979). 11A. Chaigne and J. Kergomard, Acoustique des Instruments de Musique (Acoustics of Musical Instruments) (Belin, Paris, 2008), Chap. 5, pp. 202–223. 12F. Silva, J. Kergomard, C. Vergez, and J. Gilbert, “Interaction of reed andacoustic resonator in clarinet-like systems,” J. Acoust. Soc. Am. 124, 3284–3295 (2008). 13F. Silva, “E ´mergence des auto-oscillations dans un instrument de musique a` anche simple (sound production in single reed woodwind instruments),” Ph.D. thesis, Aix-Marseille University, Marseille, France, 2009.14F. Avanzini and M. van Walstijn, “Modelling the mechanical response of the reed-mouthpiece-lip system of a clarinet, part I. a one-dimensional dis- tributed model,” Acta Acust. United Acust. 90, 537–547 (2004). 15S. Karkar, C. Vergez, and B. Cochelin, “Toward the systematic investiga- tion of periodic solutions in single reed woodwind instruments,” in Pro- ceedings of the 20th International Symposium on Music Acoustics , Associated Meeting of the International Congress on Acoustics, Interna- tional Commission for Acoustics, Sydney, Australia (2010). 16A. Hirschberg, Mechanics of Musical Instruments ,i n CISM Courses and Lectures (Springer, New York, 1995), number 355, Chap. 7, pp. 291–369. 17J.-P. Dalmont, J. Gilbert, and S. Ollivier, “Nonlinear characteristics of single-reed instruments: Quasistatic volume ﬂow and reed opening meas- urements,” J. Acoust. Soc. Am. 114, 2253–2262 (2003). 18A. Almeida, C. Vergez, and R. Causse, “Experimental investigation of reed instrument functioning through image analysis of reed opening” Acta Acust. United Acust. 93, 645–658 (2007). 19J.-P. Dalmont, B. Gazengel, J. Gilbert, and J. Kergomard, “Some aspects of tuning and clean intonation in reed instruments,” Appl. Acoust. 46, 19–60 (1995). 20E. J. Doedel, “Lecture notes on numerical analysis of nonlinear equations,” URL http://indy.cs.concordia.ca/auto/notes.pdf (last viewed 3/16/2011). 21H. B. Keller, “Numerical solution of bifurcation and nonlinear eigenvalueproblems,” in Applications of Bifurcation Theory (Academic Press, New York, 1977), pp. 359–384. 22E. J. Doedel and B. E. Oldeman, “AUTO-07P: Continuation and, bifurca- tion software for ordinary differential equations,” Concordia University, Montreal, Canada, URL http://indy.cs.concordia.ca/auto/ (last viewed 3/16/2011). 23S. Karkar, R. Arquier, B. Cochelin, C. Vergez, O. Thomas, and A. Laza-rus, “Manlab 2.0, an interactive continuation software,” available at http:// manlab.lma.cnrs-mrs.fr (last viewed 3/16/2011). 24F. Silva, “Moreesc, modal resonator-reed interaction simulation code,” available at http://moreesc. lma.cnrs-mrs.fr (last viewed 3/16/2011). 25M. van Walstijn and F. Avanzini, “Modelling the mechanical response ofthe reed-mouthpiece-lip system of a clarinet, part II: A lumped model approximation,” Acta Acust. United Acust. 93, 435–446 (2004). 26B. Gazengel, T. Guimezanes, J.-P. Dalmont, J.-B. Doc, S. Fagart, and Y. Leveille, “Experimental investigation of the inﬂuence of the mechanicalcharacteristics of the lip on the vibrations of the single reed,” in Proceed- ings of the International Symposium on Musical Acoustics , Barcelona, Spain, 2007. 27P. Guillemain, C. Vergez, D. Ferrand, and A. Farcy, “An instrumentedsaxophone mouth-piece and its use to understand how an experienced musician plays,” Acta Acust. United Acust. 96, 622–634 (2010). 28J-P. Dalmont, J. Gilbert, and J. Kergomard, “Reed instruments, from small to large amplitude periodic oscillations and the helmholtz motion ana- logy,” Acta Acust. 86, 671–684 (2000). J. Acoust. Soc. 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1.5101003.pdf	J. Appl. Phys. 126, 053901 (2019); https://doi.org/10.1063/1.5101003 126, 053901 © 2019 Author(s).High frequency properties of [Co/Pd] n/ Py multilayer films under different temperatures Cite as: J. Appl. Phys. 126, 053901 (2019); https://doi.org/10.1063/1.5101003 Submitted: 23 April 2019 . Accepted: 13 July 2019 . Published Online: 01 August 2019 Yurui Wei , Chenbo Zhao , Xiangqian Wang , Huiliang Wu , Xiaolei Li , Yueyue Liu , Zhaozhuo Zeng , Jianbo Wang , Jiangwei Cao , and Qingfang Liu ARTICLES YOU MAY BE INTERESTED IN Analysis of carrier lifetimes in N + B-doped n-type 4H-SiC epilayers Journal of Applied Physics 126, 055103 (2019); https://doi.org/10.1063/1.5097718 Comparisons of electrical/magneto-transport properties of degenerate semiconductors BiCuXO (X = S, Se and Te) and their electron-phonon-interaction evolution Journal of Applied Physics 126, 055108 (2019); https://doi.org/10.1063/1.5102141 Thermal boundary conductance of two-dimensional MoS 2 interfaces Journal of Applied Physics 126, 055107 (2019); https://doi.org/10.1063/1.5092287High frequency properties of [Co/Pd] n/Py multilayer ﬁlms under different temperatures Cite as: J. Appl. Phys. 126, 053901 (2019); doi: 10.1063/1.5101003 View Online Export Citation CrossMar k Submitted: 23 April 2019 · Accepted: 13 July 2019 · Published Online: 1 August 2019 Yurui Wei,1Chenbo Zhao,1Xiangqian Wang,1,2Huiliang Wu,1Xiaolei Li,1Yueyue Liu,1 Zhaozhuo Zeng,1 Jianbo Wang,1,3 Jiangwei Cao,1 and Qingfang Liu1,a) AFFILIATIONS 1Key Laboratory for Magnetism and Magnetic Materials of the Ministry of Education, Lanzhou University, Lanzhou 730000, People ’s Republic of China 2Key Laboratory of Sensor and Sensor Technology, Institute of Sensor Technology, Gansu Academy of Sciences, Lanzhou 730000, China 3Key Laboratory for Special Function Materials and Structural Design of the Ministry of the Education, Lanzhou University, Lanzhou 730000, People ’s Republic of China a)Author to whom correspondence should be addressed: liuqf@lzu.edu.cn ABSTRACT High frequency properties of exchange-coupled multilayers are important to develop future fast switching spintronic devices. Here, we report an experimental investigation of temperature-dependent high frequency properties in [Co/Pd] n/Py multilayer thin ﬁlms. The results demonstrate that the linewidth varies with the number of cycles at room temperature. However, the damping slightly decreases with increas-ing repetitions of Co/Pd. By ﬁtting the relationship between the linewidth and the angle (the out-of-plane azimuthal angle of the external magnetic ﬁeld), we found that a similar two-magnetron scattering e ﬀect becomes stronger when the number of Co/Pd cycles increases. For the (Co/Pd) 10/NiFe sample, the linewidth became larger at 9 GHz and 16 GHz with the decrease of temperature. Our ﬁndings help com- prehend the high frequency properties of exchange-coupled multilayer thin ﬁlms and are useful for fast switching magnetic devices. Published under license by AIP Publishing. https://doi.org/10.1063/1.5101003 I. INTRODUCTION High frequency studies of magnetic multilayer materials, includ- ing soft, perpendicular, and exchange-coupled ﬁlms, are of most importance for ultra-low power and high performance spintronicdevices. 1–6It has been demonstrated that the magnetization dynamics of exchange coupled multilayers can be governed by static magnetic properties and microstructure of the ﬁlm.3The exchange-coupled multilayers are consisted of strong perpendicular magnetic anisot-ropy (PMA) ﬁlms and soft ﬁlms. There have been several studies of systems with mixed anisotropies where the exchange coupling can monitor the magnetic properties (Fe 55Pt45/Ni80Fe20,7NiFe/Co,8 [Co/Pd]-CoFeB,9[Co/Pd] 8-NiFe,10[Co/Ni]-NiFe,11[Co/Pd]-NiFe,12,13 [Co/Pd]-Co-Pd-NiFe,14and [Co/Pt]-NiFe15–16). Barman et al. system- atically investigated the high frequency dynamics of [Co/Pt] n, [Co/Pd] 8,F e 55Pt45/Ni80Fe20, and NiFe/Co,4–6,8especially the tunable ultrafast spin dynamics by all-optical study in [Co/Pd]-NiFe multi- layers.13Bollero et al. also investigated that both the number ofCo/Pt repetitions in the multilayer and the NiFe thickness have an inﬂuence on the magnitude of the loop shift and the in-plane and out-of-plane coercivity.16Furthermore, Tryputen et al. studied the magnetic structure and anisotropy of [Co/Pd] 5/NiFe multilayers and found that the anisotropy of the [Co/Pd] 5/NiFe multilayer depends strongly on the thickness of the NiFe layer, and the damping decreases with increasing NiFe thickness.3However, limited work has been reported on how the repetitions of Co/Pd in the multilayers aﬀect the high frequency properties of [Co/Pd] n/NiFe multilayers under di ﬀerent temperatures. In this work, we investigate the repetitions of Co/Pd on the magnetic anisotropy and the high frequency properties of exchange-coupled [Co/Pd] n/Pyﬁlms at di ﬀerent temperatures. We found that the damping at room temperature slightly decreases with increasing repetitions of Co/Pd. Furthermore, the resonancelinewidth increases gradually with a decrease of temperature at 9 GHz and 16 GHz.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 126, 053901 (2019); doi: 10.1063/1.5101003 126, 053901-1 Published under license by AIP Publishing.II. EXPERIMENTAL METHODS The multilayer ﬁlms were consisted of composition: Ta(10 nm)/Pd(5 nm)/[Co(0.3 nm)/Pd(0.8 nm)] n/Py(15 nm)/Ta(3 nm), where Py denotes permalloy (i.e., Ni 81Fe19), n is the number of Co/Pd repetitions [as shown in Fig. 1(a) ]. The ﬁlms were deposited onto a thermally oxidized Si substrate by magnetron sputtering. The basepressure was below 2 × 10 −7Torr, and the Ar gas pressure was kept at 5 m Torr during deposition. The Py and Co layers were deposited byDC magnetron sputtering, while the Ta and Pd layers were deposited by radio-frequency (RF) sputtering. The thin Ta(3 nm) layer was deposited to avoid oxidation of the ﬁlms. The static magnetic proper- ties were characterized by vibrating sample magnetometry (VSM,Lakeshore 7304, USA). The ﬁlm crystallinity was checked by X-ray diﬀraction (XRD). Ferromagnetic resonance measurement (FMR) at room temperature was performed using electron spin resonance (ESR, JEOL ’s JES-FA300), and the low temperature FMR was used by am o d i ﬁed multifunctional insert of physical property measurement system with a coplanar waveguide (PPMS). III. RESULTS AND DISCUSSION Figure 1(b) shows the XRD pro ﬁles for [Co/Pd] n/Py multilayer ﬁlms. Two di ﬀraction peaks correspond to the Pd (111) and Py (111) main peaks. When the number of Co/Pd repetitions nincreases from 2 to 10, the Pd (111) peak enhances gradually andplays a dominant role in the increase of PMA strength. In addition,the main di ﬀraction peak of Co overlaps with the Pd(111) peak, and we did not observe other peaks of Co. Our results are consis- tent with the literature. 17 Out-of-plane hysteresis loops measured by VSM are shown in Figs. 2(a) and2(b)for [Co/Pd] nmultilayers and coupled [Co/Pd] n/Py stacks with a Pd bu ﬀer layer thickness of 5 nm, respectively. As shown in Fig. 2(a) , all the hysteresis loops show a near square hys- teresis with high remanence, which indicates a strong out-of planePMA. The coercivities Hc increase with the repetition n, and thevalues vary from 717 Oe to 1698 Oe. The perpendicular anisotropy originates from the dominating interfacial anisotropy when theferromagnetic layer thickness is very small (e.g., 0.4 nm Co). 18 Figure 2(b) shows the measured hysteresis loop of [Co/Pd] n/Py with n varying from 2 to 10. With the increase of repetition n, theout-of-plane hysteresis loops represent small steps but not veryperfect rectangular ratio. This phenomenon was attributed to theexistence of the soft layer of permalloy. In exchange-coupled mul- tilayers, the switching ﬁeld of the hard magnetic layer was reduced by a soft magnetic layer that was exchange coupled to thehard layer. 19In addition, the coercivities are smaller than that of [Co/Pd] nmultilayers with the same repetition. Figure 2(c) shows in-plane hysteresis loops of the coupled [Co/Pd] n/Py multilayers. In comparison with out-of-plane hysteresis loops, a weak exchange bias as well as the small coercivity is observed. It can beseen that the in-plane exchange bias gradually increases with theenlargement of repeats number n (except that cycles are lowerthan 8 cycles) [inset in Fig. 2(c) ]. This phenomenon was attrib- uted to the interface pinning e ﬀect, which leads to some uncom- pensated spins at the interface of Co/Pd and Py. 20 In order to comprehend the magnetization dynamics mechanism of [Co/Pd] n/Py multilayers, the dynamic properties of [Co/Pd] n/Py multilayers at room temperature are ﬁrst studied using electron spin resonance (ESR). Here, we use JEOL ’s JES-FA300 electronic spin resonance instrument, and the resonant frequency is con-stantly 9 GHz. Figure 3(a) shows the measured con ﬁguration, rela- tive orientation of the magnetization M, the applied dc magnetic ﬁeldH, and experimental coordinate systems. Mis the magnetiza- tion vector. His the external magnetic ﬁeld, which is tilted with respect to the ﬁlm plane. θ HandθMare the out-of-plane azimuthal angle of the external magnetic ﬁeld and the magnetization vector with respect to z axes, respectively. The rf ﬁeld is the source that provides the microwave. Moreover, the rf ﬁeld is perpendicular to the external dc magnetic ﬁeld. For the FMR measurement, based on the previous study,21–23the resonance frequency of the magnetic thinﬁlm with out of plane anisotropy is given by23 ω γ/C18/C192 ¼HsinθH sinθM/C0Hk/C20/C21 [Hcos(θH/C0θM)/C04πMscos 2 θM], (1) FIG. 1. (a) Schematic illustration of an exchange-coupled T a/Pd/[Co/Pd] n/Py/T a multilayers structure. (b) The XRD patterns for Ta/Pd/[Co/Pd] n/Py/T a multilayers.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 126, 053901 (2019); doi: 10.1063/1.5101003 126, 053901-2 Published under license by AIP Publishing.where ω=2πf=2π⋅9 GHz is the angular frequency of rf ﬁeld, γ/2π= 2.94 GHz/kOe is the gyromagnetic ratio, and Hkis the eﬀective anisotropy ﬁeld. All the samples were measured by the angular dependence of the ferromagnetic resonance. A signi ﬁcant natural resonance peak is found in each sample and the peak position moves to the lower ﬁeld with the increase of angle θH.Figure 3(b) is a representative ﬁgure of di ﬀerent angles (corresponding to n = 10), and the result for other repetition numbers is similar to Fig. 3(b) .Figures 3(c) –3(e) show the resonance ﬁeld of [Co/Pd] n/Py multilayers (n = 2, 6, 10) as af u n c t i o no fa n g l e θH.T h ea n g l e θH= 0° represents that the applied magnetic ﬁeld is normal to the ﬁlm plane. According to the above formula, e ﬀective demagnetizing ﬁeld 4 πMeﬀ, anisotropy ﬁeldHk, and g-factor ( γ¼gμB=/C22h) can be obtained [shown in Fig. 3(f) and Table I ]. With the increase of repetition n, the e ﬀective demagnetiz- ingﬁeld decreases and the anisotropy ﬁeld increases because the pinning e ﬀect aﬀects the magnetic moment precession process of [Co/Pd] n/Py multilayers.3The red lines in Figs. 3(c) –3(e) have the values which are obtained by ﬁtting the calculated resonance ﬁeld to the corresponding experimental ones (solid dots). It can be seen thata satisfactory agreement has been obtained between the ﬁtting and the experiment. Also, the value of g-factor was deduced through the ﬁtting data (as shown in Table I ). The g-factor can be expressed as 24 g¼2me eμSþμL hSiþh Li, (2) where μSandμLmean spin and orbital magnetic moments, me/erep- resents the mass-to-charge ratio of the electron, and hSiandhLi mean spin and orbital angular momentum, respectively. The g-factoris obviously constant in all our samples due to the same thicknessand the microstructure of Py. Figure 4 shows the angular dependences of FMR linewidth. For the angular dependence of the FMR linewidth, the nonlinear magnetic ﬁeld pinning e ﬀect should be considered due to the strong magnetic anisotropy of samples. By analysis Figs. 4(a) –4(e), it can be seen that the linewidth strongly depends on θ Hand there is a peak of linewidth maximum at θH≈13° for all ﬁve samples. The measured FMR linewidths in this work are analyzed consider-ing three di ﬀerent contributions, 25–28 ΔH¼ΔHGilbertþΔHinhomþΔHTMS, (3) where ΔHdenotes the total linewidth of the FMR signal, ΔHGilbert denotes the intrinsic Gilbert damping, ΔHinhom denotes the inho- mogeneous linewidth broadening, and ΔHTMSdenotes two-magnon scattering broadening. The two-magnon scattering (TMS) linewidth is related to the angle θHand the magnetic anisotropy. The line- width due to intrinsic damping is derived as ΔHGilbert¼2αωﬃﬃ 3p γ, which is simply proportional to the frequency.26The inhomogeneous line- width broadening can be expressed as follows:28 ΔHinhom¼@Hr @θH/C12/C12/C12/C12/C12/C12/C12/C12Δθ Hþ@Hr @4πMeff/C12/C12/C12/C12/C12/C12/C12/C12Δ4πM eff, (4) where ΔθHrepresents the spread of the crystallograghic axes among various grains and Δ4πMeﬀrepresents the magnitude of the inho- mogeneity of the e ﬀective magnetization, which might be due to the local demagnetizing ﬁeld or anisotropy ﬁeld. In the process of FIG. 2. (a) Out-of plane hysteresis loop of (Co/Pd) nmultilayers measured by VSM (n = 2, 4, 6, 8, 10). (b) and (c) Out-of-plane and in-plane hysteresis loop of[Co/Pd] n/Py multilayers measured by VSM. The insets in (b) indicate the coer- civities of out-of-plane via repetition numbers of the coupled [Co/Pd] n/Py multi- layers. The insets in (c) indicate the relationship between the in-plane exchange bias ﬁeld and repetitions of the coupled [Co/Pd] n/Py multilayers.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 126, 053901 (2019); doi: 10.1063/1.5101003 126, 053901-3 Published under license by AIP Publishing.Fig. 4 ﬁtting, ΔθHandΔ4πMeﬀare free parameters for estimating Hr derivatives. Two-magnon scattering linewidth, based on theoretical description and the free energy density model, can be written as25,28 ΔHTMS¼2ﬃﬃ ﬃ 3pΓ(H,θH)s i n/C01ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ /C0H2 H1þ4πMeffcos(2 θ) cos2θs , (5) H1¼Hrcos(θH/C0θM)/C04πMeffcos 2 θM, (5a) H2¼Hrcos(θH/C0θM)/C04πMeffcos2θM, (5b) 4πMeff¼4πMSþH?,( 5 c ) where H⊥is the perpendicular anisotropy ﬁeld.Γ(H,θH)i st h e ﬁtting value changed with value and orientation of external ﬁeld.Γ0was extracted from Γ(H,θH), which is regarded as an eﬀective parameter to measure the magnitude of TMS. The ﬁtting parameters of the intrinsic Gilbert damping and Γ0areshown in Table I . The intrinsic Gilbert damping decreases from 0.011 to 0.009, which remains basic constant. It is found that the inhomogeneous linewidth is not changed obviously. Moreover, Γ0 rises from 4 mT to 14 mT, which is dominant at 8 or 10 cycles [shown in Fig. 4(f) ]. To investigate the relationship between high frequency prop- erties of [Co/Pd] n/Py multilayer ﬁlms and the temperature, the ferromagnetic resonance spectra of [Co/Pd] 10/Py multilayer thin ﬁlms at di ﬀerent temperature were measured with a physical property measurement system (Quantum Design) possessing abroadband ferromagnetic resonance setup using a coplanar wave-guide. The working frequency range was in the 2 –18 GHz, and the temperature control was used from 400 K to 1.9 K. During the measurement, the static magnetic ﬁeld was perpendicular to the plane of ﬁlm and the microwave magnetic ﬁeld was applied in the plane of ﬁlm. Figures 5(a) and 5(c) show the temperature dependence of ferromagnetic resonance absorption spectra at the frequency 9 GHz and 16 GHz, respectively. As can be seen from the paragraph, theresonance ﬁeld gradually decreases as the temperature decreases from 300 K to 50 K, and the intensity of ferromagnetic resonance absorption also decreases. These experimental results are ﬁtted using the Lorentz equation 29 S1S0(ΔH)2 (ΔH)2þ(H/C0Hres)2, (6) where S0is the constant describing the coe ﬃcient for the transmit- ted microwave power, ΔHis the half linewidth, His the external FIG. 3. (a) The coordinate system used for the measurement of FMR. (b) The resonance absorption as a function of the applied ﬁeld in the [Co/Pd] 10/Py multilayer at dif- ferent θH. (c)–(e) The angular dependence of the resonance ﬁelds of [Co/Pd] n/Py multilayers (n = 2, 6, 10). The angle 0° represents the applied magnetic ﬁeld normal to theﬁlm plane. (f ) The ﬁtting effective resonance ﬁeld and demagnetizing ﬁeld as a function of repetitions of the coupled [Co/Pd] n/Py multilayers. TABLE I. Magnetic parameters obtained from FMR ﬁtting. Sample(Co/ Pd) 2-Py(Co/ Pd) 4-Py(Co/ Pd) 6-Py(Co/ Pd) 8-Py(Co/ Pd) 10-Py 4πMeff(Gs) 7100 6700 6300 6000 6500 g-factor 2.06 2.08 2.0 2.0 2.0Γ 0(mT) 4 3 3 12 14 Α 0.011 0.011 0.01 0.009 0.009Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 126, 053901 (2019); doi: 10.1063/1.5101003 126, 053901-4 Published under license by AIP Publishing.FIG. 4. (a)–(e) The angular dependence of FMR linewidth (dark dots) and the ﬁtting data (colorful lines) of different samples as a function of θH. (f ) Two different contribu- tions of FMR linewidths as the repetition of the coupled [Co/Pd] n/Py multilayers. FIG. 5. (a) and (c) Ferromagnetic res- onance spectra of Co/Pd] 10/Py multi- layer ﬁlms at frequencies 9 GHz and 16 GHz under different temperatures. (b) and (d) FMR linewidths as a func-tion of temperatures at frequencies9 GHz and 16 GHz, respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 126, 053901 (2019); doi: 10.1063/1.5101003 126, 053901-5 Published under license by AIP Publishing.magnetic ﬁeld, and Hresis the resonance ﬁeld. The solid lines repre- sent the ﬁtting results shown in Figs. 5(a) and5(c). It can be seen thatﬁtting results are in good agreement with the experiment. As we know, the linewidth of ferromagnetic resonance is related tothe changes of damping. Therefore, we can better understand thehigh frequency characteristics of our samples by the variation of ferromagnetic resonance linewidth. Figures 5(b) and5(d) show the FMR linewidth ΔHat diﬀerent temperatures. Obviously, the ΔH increases when the temperature decreases. By comparing the varia-tion of resonance linewidth at 9 GHz and 16 GHz, we can clearlysee that the ΔHat 16 GHz is larger than that at 9 GHz. This phe- nomenon can be attributed to the ferromagnetic relaxation process. 30In fact, we have extracted the damping coe ﬃcient but not presented in the paper. The Gilbert damping at 300 K is calcu-lated to be 0.0067 ± 0.0001. As the temperature decreases, the FMRlinewidth ΔHincreases and the corresponding Gilbert damping increases from 300 K to 50 K. The enhanced damping could be related to a thermally induced spin reorientation for the surfacemagnetization of the Py layer. 30 IV. CONCLUSION In summary, we have investigated the high frequency proper- ties of exchange-coupled [Co/Pd] n/Py multilayers. By ferromagnetic resonance at room temperature, we found that the linewidth varies with the number of cycles and the damping is basically unchanged.A similar two-magnetron scattering e ﬀect becomes stronger when the number of Co/Pd cycles reaches n = 8. 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Rep. 6, 22890 (2016).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 126, 053901 (2019); doi: 10.1063/1.5101003 126, 053901-6 Published under license by AIP Publishing.
1.4902443.pdf	Ultrafast magnetization switching by spin-orbit torques Kevin Garello, Can Onur Avci, Ioan Mihai Miron, Manuel Baumgartner, Abhijit Ghosh, Stéphane Auffret, Olivier Boulle, Gilles Gaudin, and Pietro Gambardella Citation: Applied Physics Letters 105, 212402 (2014); doi: 10.1063/1.4902443 View online: http://dx.doi.org/10.1063/1.4902443 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/105/21?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Large voltage-induced modification of spin-orbit torques in Pt/Co/GdOx Appl. Phys. Lett. 105, 222401 (2014); 10.1063/1.4903041 Enhanced spin-orbit torques in Pt/Co/Ta heterostructures Appl. Phys. Lett. 105, 212404 (2014); 10.1063/1.4902529 Chiral magnetization textures stabilized by the Dzyaloshinskii-Moriya interaction during spin-orbit torque switching Appl. Phys. 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Downloaded to IP: 131.111.164.128 On: Wed, 24 Dec 2014 16:09:44Ultrafast magnetization switching by spin-orbit torques Kevin Garello,1,a)Can Onur Avci,1Ioan Mihai Miron,2,3,4Manuel Baumgartner,1 Abhijit Ghosh,1St/C19ephane Auffret,2,3,4Olivier Boulle,2,3,4Gilles Gaudin,2,3,4 and Pietro Gambardella1 1Department of Materials, ETH Z €urich, H €onggerbergring 64, Z €urich CH-8093, Switzerland 2Universit /C19e Grenoble Alpes, SPINTEC, 38000 Grenoble, France 3CEA, INAC-SPINTEC, 38000 Grenoble, France 4CNRS, SPINTEC, 38000 Grenoble, France (Received 2 September 2014; accepted 6 November 2014; published online 24 November 2014) Spin-orbit torques induced by spin Hall and interfacial effects in heavy metal/ferromagnetic bilayers allow for a switching geometry based on in-plane current injection. Using this geometry,we demonstrate deterministic magnetization reversal by current pulses ranging from 180 ps to ms in Pt/Co/AlO xdots with lateral dimensions of 90 nm. We characterize the switching probability and critical current Icas a function of pulse length, amplitude, and external ﬁeld. Our data evidence two distinct regimes: a short-time intrinsic regime, where Icscales linearly with the inverse of the pulse length, and a long-time thermally assisted regime, where Icvaries weakly. Both regimes are consistent with magnetization reversal proceeding by nucleation and fast propagation ofdomains. We ﬁnd that I cis a factor 3–4 smaller compared to a single domain model and that the incubation time is negligibly small, which is a hallmark feature of spin-orbit torques. VC2014 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4902443 ] Magnetization switching is a topic of fundamental inter- est as well as of practical relevance for the development of fast, non-volatile data storage devices. In recent years, current-induced switching of nanosized magnets has emergedas one of the most promising technologies for the realization of a scalable magnetic random access memory (MRAM). 1In the so-called spin transfer torque (STT)-MRAM, a spin-polarized current ﬂowing through a pinned magnetic layer induces a torque on the storage layer that counteracts the magnetic damping. 2,3STT switching can be made faster by increasing the injected current or choosing materials with low damping. However, when the magnetization of the reference and free layer are at rest, parallel or anti-parallel, the STT iszero. The resulting non-negligible incubation delay, governed by thermally activated oscillations, limits ultrafast switching and induces a broad switching time distribution. 4Several sol- utions have been explored to reduce the incubation delay, such as biasing STT devices with a hard axis ﬁeld4or adding an out-of-plane polarizer to an in-plane free layer.5This has led to switching times as low as 50 ps in metallic spin valves6–8and 500 ps in magnetic tunnel junctions (MTJ).9 Despite such progress, the development of STT-MRAM for ultrafast applications such as cache memories remains prob- lematic. Fast switching requires large current through the thin oxide barrier of a MTJ, which leads to reliability issues andaccelerated aging of the barrier. Spin-orbit torque (SOT)-induced switching, generated by the ﬂow of an electrical current in the plane of a ferromag-netic/heavy metal (FM/HM) bilayer, offers an interesting al- ternative to STT. 10Theoretical11,12and experimental10,13–19 studies have evidenced signiﬁcant antidamping Tk/m /C2ðy/C2mÞand ﬁeld-like T?/m/C2ySOT components in such systems, which originate from either the bulk spinHall effect in the HM layer or interfacial Rashba-type spin- orbit coupling, or a combination of these effects. Tkis respon- sible for the switching of the magnetization m. As this torque is directed parallel to yfor a current directed along x,Tk destabilizes both directions of the magnetization and the application of a bias ﬁeld along the current direction is required to stabilize one magnetic conﬁguration over theother. Consequently, switching is bipolar with respect to both current and bias magnetic ﬁeld. 10SOT has proven very effec- tive to switch the magnetization of perpendicular10,20,21and in-plane magnetized layers,14,15as well as to control the motion of domain walls in FM/HM heterostructures.22–24In FIG. 1. (a) Schematic of the experimental setup. (b) Current pulses of differ- ent duration detected in transmission. (c) Magnetization switching of sample s1 induced by positive and negative current pulses with current density Ip¼1.65 mA and sp¼210 ps, averaged over 100 pulses. Bxis swept only once from þ0.65 to /C00.65 T.a)Electronic mail: kevin.garello@mat.ethz.ch 0003-6951/2014/105(21)/212402/5/$30.00 VC2014 AIP Publishing LLC 105, 212402-1APPLIED PHYSICS LETTERS 105, 212402 (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: 131.111.164.128 On: Wed, 24 Dec 2014 16:09:44SOT devices with perpendicular magnetic anisotropy (PMA), asTkis always perpendicular to the magnetization, the incubation delay of the switching process is expected to be minimum. Moreover, SOT allows for the separation of the read and write current paths in an MTJ, avoiding electricalstress of the tunnel barrier during writing. Based on these considerations, novel SOT-MRAM architectures have been proposed 25,26and the switching of in-plane14,15,27and out-of- plane MTJ has been recently demonstrated.28There is, how- ever, no systematic study of SOT switching on a sub-ns timescale. In this letter, we investigate the probability of SOT-induced magnetization reversal of perpendicularly magnetized Pt/Co/AlO xdots as a function of current pulse width, ampli- tude, and external magnetic ﬁeld on timescales ranging from 180 ps to ms. Pt(3 nm)/Co(0.6 nm)/AlO xlayers with PMA were de- posited by magnetron sputtering and patterned into square dots on top of Pt Hall bars, as described in Ref. 10. We pres- ent results for three different samples of lateral sizes1¼90 nm, s2 ¼95 nm, and s3 ¼102 nm, as measured by scanning electron microscopy. These samples have a satura- tion magnetization M s/C258.7/C2105A/m (measured before patterning) and an effective anisotropy ﬁeld Bk/C252K/Ms /C0l0Ms/C251 T. Figure 1(a)shows a schematic of the measure- ment setup. In order to ensure the transmission of fast pulseswithout signiﬁcant reﬂection due to the large resistance of the Pt contacts ( /C242kX), a 100 Xresistor is connected in par- allel with the sample. A 100 k Xseries resistor prevents spreading of the current pulses into the Hall voltage probes. An in-plane bias magnetic ﬁeld ( B x), determining the switch- ing polarity for a given current polarity,10is applied along the current line, with a tilt of 0.5/C14towards zin order to favor a homogeneous magnetization when no current pulses are applied. The perpendicular component of the magnetizationis measured via the anomalous Hall resistance ( R AHE¼0.45 Xat saturation) using a low DC current of 20 lA. A bias tee separates the current pulses and the DC current. All measure-ments are performed at room temperature. To study theswitching probability distribution, we proceed as follows: ﬁrst, a positive 0.7 mA “reset” pulse of 20 ns duration is used to initialize the magnetization direction. Second, a negative “write” pulse of length s pand amplitude Ipis applied. RAHEis measured a few milliseconds afte r each pulse. The switching probability is deﬁned as P¼½Rwrite AHEðIp;sp;BxÞ/C0Rreset AHEðBxÞ/C138= DRAHEðBxÞaveraged over 100 trials. DRAHEðBxÞis the differ- ence between the Hall resistance of the up and down states measured during a sweep of Bxat the same ﬁeld at which the switching is performed. Switchi ng diagrams are constructed by varying two out of the three free parameters sp,Ip,a n d Bxwhile the other one is kept constant. Figure 1(c)shows the magnetic state of sample s1 after applying write pulses with sp¼210 ps and Ip¼1.65 mA (open orange circles) as a function of Bx. The magnetization after the reset operation is shown as solid black squares. Bxis swept in steps from /C00.65 to 0.65 T. At each ﬁeld step, RAHE is determined as described above. Switching can be experi- mentally observed in the hysteretic range delimited by the co- ercive ﬁeld of the Co layer ( Bc/C250.45 T). The orange and black curves indicate that for Bx>0 a current Ip>0 switches the magnetization downwards and Ip<0 switches it upwards, whereas for Bx<0 the effect of the current polarity is reversed. This behavior is typical of SOT and similar to thatreported for single pulses ranging from tens of ns to lsin devices with size varying from 200 to 1000 nm. 10,20,21,28 Since switching occurs on such short timescales and considering the analogy between orthogonal-STT devices and SOT (polarization of the spin current perpendicular to the magnetization), effects related to the magnetization pre-cession are expected to be important when varying s pand Ip.7,29Moreover, macrospin simulations show that T? (equivalent to an effective ﬁeld along y) promotes oscilla- tions of the magnetization with periods up to ns, thus induc- ing precessional switching even for high damping constants such as a¼0.5. We therefore measured the switching proba- bility as a function of spand Ip, as well as of Bx, which, besides being necessary for switching, inﬂuences the SOT- induced dynamics. Figures 2(a)and2(b)show representative measurements of Pas a function of spandBx, respectively, for different values of Ip. By repeating such measurements over a grid of ( Bx,sp) and ( Bx,Ip) pairs, we construct the switching diagrams reported in Figures 2(c) and2(d). The red (blue) color represents high (low) switching probability. In both diagrams, the range of successful switching eventsgrows monotonically as either I p,sporBxincrease. We observe that the white boundary region representing interme- diate Pvalues is relatively narrow. Moreover, we do not observe oscillations of Pbeyond this boundary as a function ofIporsp, as would be expected for precessional switch- ing.7,29In fact, SOT-induced magnetization reversal in our samples is deterministic and bipolar with respect to either ﬁeld or current down to sp¼180 ps. Ipandspdetermine the energy dissipation during the switching process and the speed at which this can be achieved for a given bias ﬁeld. Figure 3shows the critical switching current Ic, deﬁned at P¼90%, as a function of sp measured over eight orders of magnitude in pulse duration forBx¼91 mT. We ﬁnd that there are two very different regimes: at short-time scales ( sp<1 ns), Icincreases stronglyFIG. 2. Switching probability of s1 as a function of (a) sp(Bx¼91 mT) and (b)Bx(sp¼210 ps) at different current amplitudes. Two-dimensional dia- grams of the switching probability showing successful (red) and unsuccess- ful (blue) events measured as a function of (c) spand Bxfor ﬁxed Ip¼1.5 mA and (d) IpandBxfor ﬁxed sp¼210 ps.212402-2 Garello et al. Appl. Phys. Lett. 105, 212402 (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: 131.111.164.128 On: Wed, 24 Dec 2014 16:09:44when reducing sp, whereas on longer time scales ( sp/C211ls), Ichas a weak dependence on sp. This behavior is qualita- tively similar to that observed in STT devices30–32and asso- ciated with an intrinsic regime where the switching speed depends on the efﬁciency of angular momentum transferfrom the current to the magnetic layer and a thermally assisted regime in which stochastic ﬂuctuations help the magnetization to overcome the reversal energy barrier. We focus ﬁrst on the short-time regime. In this limit, I c is inversely proportional to sp, as shown in Figure 4. Similar behavior is observed for samples s1–s3, indicating that thes /C01 pdependence is speciﬁc to the switching process rather than to a particular sample. In analogy with STT,31,32we model Icas Ic¼Ic0þq sp; (1) where Ic0is the intrinsic critical switching current and qis an effective charge parameter that represents the number ofelectrons that needs to be pumped into the system before reversal occurs, describing the efﬁciency of angular momen- tum transfer from the current to the spin system. From theﬁt shown in Fig. 4, we obtain I c0¼0.58 mA ( jc0¼1.76 /C2108Acm/C02) and q¼2.1/C210/C013C. This linear relation- ship holds for different Bx[Fig. 4]. When increasing Bxfrom 91 to 146 mT, qdecreases by about 13%, whereas Ic0increases from 0.58 to 0.61 mA. Further proof that the linear dependence of Icons/C01 pis general to the switching distribu- tion and not dependent on the deﬁnition of the critical cur- rent is reported in the inset of Fig. 4, showing that all the switching probability curves measured for sp<1 ns, plotted as a function of the scaled angular momentum ( Ic/C0Ic0)sp/q, fall onto the same curve. The experimental Ic0can be compared with that expected from monodomain SOT-induced magnetization re- versal,33given by the condition TkðIc0Þ¼ð Bk=2/C0Bx=ﬃﬃﬃ 2p Þ. This torque is often expressed in terms of an effective spin Hall angle heff SHasTk¼½/C22h=ð2eÞheff SH=ðMstFMÞ/C138j, where tFMis the thickness of the FM layer and the current density jis assumed to be uniform throughout the FM/HM bilayer,33,34 which is a reasonable assumption for Co/Pt. heff SHis a useful parameter to compare results from different experiments, but does not correspond to the bulk spin Hall angle of the HM layer, as it takes into account neither the ﬁnite spin diffusionlength in the HM nor FM/HM interface effects. Here, by considering the ratio T k/j¼6.9 mT/107Acm/C02ðheff SH¼0:11Þ obtained from harmonic Hall voltage measurements ofPt(3 nm)/Co(0.6)/AlO xdots in the quasistatic, low current (j/C20107Acm/C02) limit,16we estimate Ic0/C252.05 mA. This value is about 3.5 times larger compared to the experiment.In order to match the critical current of our samples to the macrospin prediction, h eff SHshould be about 0.4, an unreason- ably large value for Pt.35Asspis too fast for thermally assisted switching, this comparison suggests that the magnet- ization reverses by a more current-efﬁcient process than coherent rotation of a single magnetic domain. Further support for this hypothesis comes from macro- spin simulations of SOT switching in the sub-ns regime using the Landau-Lifshitz-Gilbert equation (not shown),which reveals that I c/C24s/C0b pwith b/C252 rather than b¼1a s found in the experiment. This behavior differs from the mac- rospin dynamics of perpendicular magnetic layers inducedby STT, for which our simulations conﬁrm the linear scaling (b¼1) found in Ref. 32. The difference between SOT and STT stems from the competition between T kand the anisot- ropy torque, which tend to align the magnetization, respec- tively, along yand z, whereas in the STT case, they both tend to align it towards z. The inconsistency between macrospin models and our experiment suggests that magnetization reversal occurs by domain nucleation and propagation. In such a scenario, oncea reverse domain nucleates due to the T kandT?, switching is achieved by the propagation of a domain wall through the dot. Since the domain wall velocity is proportional to j, the critical switching current is expected to be proportional to s/C01 p, in agreement with our results in the short-time regime and Eq. (1). In this case, the “effective charge” qis inversely proportional to the domain wall velocity and can be inter- preted as the angular momentum required to switch the entire dot once the reversal barrier of a portion of the sample hasbeen overcome. The ratio between domain wall velocity and current density can be estimated by taking the width wof the sample as the distance that a domain wall has to travel beforeswitching occurs and divide it by the time srequired to cover this distance. This time can be estimated as s¼q=jS, so that v=j¼wS=q¼137 (m/s)/10 8Acm/C02, where S¼w(tFMþtHM)10 FIG. 3. Critical switching current of sample s2 as a function of pulse dura- tion measured with Bx¼91 mT. The green solid line is a ﬁt to the data in the short-time regime ( sp<1 ns) according to Eq. (1). The red dashed line is a ﬁt to the data in the thermally activated regime ( sp/C211ls) according to Eq. (2). The blue dash-dotted line represents the intrinsic critical current Ic0. FIG. 4. Critical switching current of sample s1 as a function of 1/ spfor dif- ferent values of Bx. The thin red line shows a linear ﬁt to the short-time data (1/sp>1 GHz) measured at Bx¼91 mT using Eq. (1). Inset: Pin the short- time regime as a function of sp(Ic/C0Ic0)/q. The red line represents an aver- age ﬁt of all the curves using a sigmoidal function.212402-3 Garello et al. Appl. Phys. Lett. 105, 212402 (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: 131.111.164.128 On: Wed, 24 Dec 2014 16:09:44is the cross section of the FM/HM bilayer. This ratio increases with the increase in Bx, as would domain wall speed, and is in quite good agreement with the large current- induced domain wall velocities (100–400 m/s) reported onsimilar structures. 22,23We further note that micromagnetic simulations studies of FM/HM bilayers with large spin-orbit interaction proposed similar magnetization reversal scenar-ios, 36–39pointing out also the important role played by the chirality of the walls.22–24,39 In the thermally assisted region ( sp/C291 ns), Icis pre- dicted to be34 Ic¼Bk 4j TkSp/C02bx/C0ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 8 nln/C0sp s0ln 1/C0PðÞ/C18/C19 /C08/C04b2 x/C04bxp/C04ðÞ þp2vuuut0 BB@1 CCA; (2) where n¼BkMsw2tFM=2kBTis the thermal stability factor, bx¼Bx=Bk, and s0the thermal attempt time. Although this expression is derived analytically in the framework of a mac- rospin model, we ﬁnd that it ﬁts reasonably well to our data(dashed line in Fig. 3). The ﬁt, performed for s pbetween 1 ls and 10 ms by taking s0¼1 ns (estimated from the inﬂection point of the curve in Fig. 4),bx¼0.091, and P¼0.9, gives n¼110. As for sample s3 n/C25700 at room temperature, the smaller value of nderived from the ﬁt indicates that the Co layer is not reversing as a monodomain, in agreement with theconclusions drawn from the short-time regime and similar to perpendicularly magnetized nanopillars. 31,40An important result from this analysis is that the intercept of the ﬁt in thethermally assisted region (dashed line in Fig. 3) and the intrin- sic current determined in the short-time regime (dash-dotted line) gives the incubation time of the switching process, 31,32 which we ﬁnd to be negligibly small ð/C2410/C02062sÞ. Due to the weak dependence of Iconspin the thermally assisted regime, this result is largely independent of the function used to ﬁt thedata. In conclusion, we have demonstrated non-stochastic bipolar switching of 90 nm magnetic dots induced by SOTusing in-plane injection of current pulses down to 180 ps, and we conﬁrm that the incubation time is negligibly small. This makes SOT-based heterostructures a promising candi-date for ultra-fast recording applications such as MRAMs and cache memories. Similar to STT, we ﬁnd that the de- pendence of the critical switching current on the pulse lengthcan be divided into a short-time (intrinsic) regime and a long-time (thermally assisted) regime. For s p<1 ns, the crit- ical switching current is inversely proportional to sp, con- trary to the precessional behavior expected of a single domain magnet and consistent with a scenario where the switching speed is determined by domain wall propagation.In the single domain limit, the ratio between the SOT and STT critical current scales as 33ISOT c0=ISTT c0¼1 2ag heff SHtFMþtHM w, where a large spin polarization gand low damping afavor STT, whereas a large heff SHand the smaller cross section of the current injection line favor SOT. Our results indicate that ultrafast SOT switching may compare more favorably to STT when domain propagation is involved.This work was supported by the European Commission under the 7 thFramework Program (Grants No. 318144, No. 2012-322369), the Swiss National Science Foundation (Grant No. 200021-153404), the French GovernmentProjects Agence Nationale de le Recherche (ANR-10- BLAN-1011-3, ANR-11-BS10-0008), and the European Research Council (StG 203239). 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1.5128795.pdf	J. Chem. Phys. 151, 214103 (2019); https://doi.org/10.1063/1.5128795 151, 214103 © 2019 Author(s).Excited states via coupled cluster theory without equation-of-motion methods: Seeking higher roots with application to doubly excited states and double core hole states Cite as: J. Chem. Phys. 151, 214103 (2019); https://doi.org/10.1063/1.5128795 Submitted: 22 September 2019 . Accepted: 11 November 2019 . Published Online: 02 December 2019 Joonho Lee , David W. Small , and Martin Head-Gordon ARTICLES YOU MAY BE INTERESTED IN Multireference configuration interaction and perturbation theory without reduced density matrices The Journal of Chemical Physics 151, 211102 (2019); https://doi.org/10.1063/1.5128115 RPA natural orbitals and their application to post-Hartree-Fock electronic structure methods The Journal of Chemical Physics 151, 214106 (2019); https://doi.org/10.1063/1.5128415 Nonadiabatic quantum transition-state theory in the golden-rule limit. II. Overcoming the pitfalls of the saddle-point and semiclassical approximations The Journal of Chemical Physics 151, 214101 (2019); https://doi.org/10.1063/1.5131092The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Excited states via coupled cluster theory without equation-of-motion methods: Seeking higher roots with application to doubly excited states and double core hole states Cite as: J. Chem. Phys. 151, 214103 (2019); doi: 10.1063/1.5128795 Submitted: 22 September 2019 •Accepted: 11 November 2019 • Published Online: 2 December 2019 Joonho Lee,a) David W. Small, and Martin Head-Gordona) AFFILIATIONS Department of Chemistry, University of California, Berkeley, California 94720, USA and Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA a)Electronic addresses: linusjoonho@gmail.com and mhg@cchem.berkeley.edu ABSTRACT In this work, we revisited the idea of using the coupled-cluster (CC) ground state formalism to target excited states. Our main focus was targeting doubly excited states and double core hole states. Typical equation-of-motion (EOM) approaches for obtaining these states struggle without higher-order excitations than doubles. We showed that by using a non-Aufbau determinant optimized via the maximum overlap method, the CC ground state solver can target higher energy states. Furthermore, just with singles and doubles (i.e., CCSD), we demonstrated that the accuracy of ΔCCSD and ΔCCSD(T) (triples) far surpasses that of EOM-CCSD for doubly excited states. The accuracy of ΔCCSD(T) is nearly exact for doubly excited states considered in this work. For double core hole states, we used an improved ansatz for greater numerical stability by freezing core hole orbitals. The improved methods, core valence separation (CVS)- ΔCCSD and CVS- ΔCCSD(T), were applied to the calculation of the double ionization potential of small molecules. Even without relativistic corrections, we observed qualitatively accurate results with CVS- ΔCCSD and CVS- ΔCCSD(T). Remaining challenges in ΔCC include the description of open-shell singlet excited states with the single-reference CC ground state formalism as well as excited states with genuine multireference character. The tools and intuition developed in this work may serve as a stepping stone toward directly targeting arbitrary excited states using ground state CC methods. Published under license by AIP Publishing. https://doi.org/10.1063/1.5128795 .,s I. INTRODUCTION A conceptually simple approach to solving the Schrödinger equation is to diagonalize the Hamiltonian represented by the many- particle basis set spanning the entire Hilbert space. While this full configuration interaction (FCI) approach (or exact diagonal- ization) is formally exact, it becomes quickly unfeasible due to the exponentially growing dimension of the Hilbert space.1 Coupled-cluster (CC) theory, which is usually limited to singles and doubles (i.e., CCSD), has been a popular approximate solver to the Schrödinger equation. Unlike truncated CI methods, truncated CC methods are size-consistent and therefore can be reliably applied to large systems and reach the thermodynamic limit. Most of the CC applications have been focused on approximating the ground state(GS) of systems, and therefore CC methods are usually considered to be ground state methods.2 There is a way to compute excitation energies of CC wave- functions based on the equation-of-motion (EOM-CC)3formalism or the linear response (LR-CC)4formalism. EOM-CCSD is one of the widely used CC excited state (ES) methods, which provides very accurate excitation energies for states dominated by single- excitations. The accuracy of EOM-CCSD for valence single excita- tions is about 0.1–0.2 eV. However, EOM-CCSD commonly fails to predict excitation energies for states with a significant amount of double-excitation character, and the typical error is about 1 eV or even greater than this. These failures could be avoided if the desired excited state is in a different irreducible representation from that of the ground state, since one could just employ a ground J. Chem. Phys. 151, 214103 (2019); doi: 10.1063/1.5128795 151, 214103-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp state CCSD calculation. However, if there is no point group sym- metry in the system or the desired state is in the same irreducible representation, this workaround is no longer an option. The fail- ure of EOM-CCSD for doubly excited states is largely due to the lack of relaxation of doubles amplitudes, which can be usu- ally achieved by having triple excitations (i.e., EOM-CCSDT).5,6 Since EOM-CCSDT has a cost, which scales O(N8), much research has been dedicated to improving the double-excitation gaps of EOM-CCSD by approximating the effect of connected triples either via anO(N8)scaling method with a smaller prefactor or via an O(N7)scaling method. These methods include EOM-CCSDT-n,7,8 EOM-CCSD(T),7EOM-CCSD( ˜T),8EOM-CCSD(T′),8CC3,9,10and CCSDR(3).11 Another challenging class of excited states for EOM-CC are core-ionized states. In most cases, these core ionization energies can be well described by the EOM with ionization potential (EOM- IP) approaches.3,12However, one has to obtain a large number of eigenvectors to cover the energy range for core ionizations, which can be very time-consuming for an O(N5)method. In principle, one may use a modified Davidson procedure that can target high- lying interior roots.13However, due to the strong coupling between core states and continuum states, numerical instability is often encountered.14There are tricks to remedy this problem to an extent via core valence separation (CVS),14–17but it does not solve the inherent drawbacks of EOM-IP-CCSD. In other words, CVS-EOM- IP-CCSD fails when EOM-IP-CCSD fails. In particular, for some core-ionized states, the EOM with excitations up to doubles is not sufficient. Recently, there has been increasing interest in so-called ΔSCF methods18–23as an alternative to the linear-response mean-field approaches such as CI singles (CIS) and time-dependent density functional theory (TDDFT).24In this category, the most popu- lar approach is based on the maximum overlap method (MOM) developed by Gilbert, Besley, and Gill.22The resulting approximate excited states from ΔSCF are not orthogonal to the approximate ground state. This seems to be suboptimal since the exact excited state should be orthogonal to the exact ground state. However, extensive benchmarks have so far suggested that the nonorthog- onality of approximate wavefunction methods is not problematic to get good energetics. Furthermore, it is possible to diagonalize the Hamiltonian with those nonorthogonal determinants to obtain orthogonal states in the end. This approach is called nonorthogonal CI (NOCI).25,26 Similar in spirit to ΔSCF, it is possible to obtain approxi- mate solutions to exact excited states using the CC wavefunction parametrization. We call this approach ΔCC, and this is the focus of our work. In ΔCC, one computes the ground state CCSD (GS- CCSD) and an excited state CCSD (ES-CCSD) energies and takes a difference between them to compute the corresponding excita- tion gap. Performing an ordinary ground state CCSD calculation on an excited reference determinant leads to a desired ES-CCSD energy. Just like ΔSCF targets an excited SCF solution, ΔCC targets an excited CC solution that starts from an excited reference state. We emphasize that ΔCC is not a new approach and has been known in the literature for a while.27–39In particular, there are seminal works by Kowalski and co-workers that attempt to find higher roots in CC methods using the homotopy method.31,35They also established connections between these roots and excited states in FCI for modelsystems such as H 4.ΔCC has been underappreciated because of the obscure nature of CC amplitude solutions. In particular, the higher roots of the CC amplitude equation are difficult to assign to a spe- cific state. It is also often very difficult to converge the CC amplitude equation, and multiple CC roots sometimes correspond to the same FCI state.38 While these drawbacks make ΔCC not so appealing in general, we will show that ΔCC can be an accurate tool for excited states that are dominated by one Hartree-Fock (HF) state. CCSD with per- turbative triples [CCSD(T)] is a de facto standard method for the ground state of systems with one dominant determinant. One may expect CCSD(T) to work well as long as the underlying electronic structure has only one dominant determinant, which does not need to be the ground state. In such cases, we expect the excited state CCSD(T) energies to be quite accurate and even similar in qual- ity to that of the ground state calculation. We found excited states dominated by one double-excitation to be a perfect candidate for this approach. This is largely because the state assignment becomes much easier since it is dominated by one determinant. As men- tioned earlier, EOM-CCSD fails to describe such states with dom- inant double-excitations so ΔCC can be an excellent alternative with the same O(N6)cost. It is also worthwhile to note that ΔCC has been used in the literature to compute core ionization energies.40–43Similarly to the double excitations, this is due to the ease of assigning proper states as well as relatively more stable amplitude iterations. The ampli- tude convergence can often become problematic, but this issue can be completely removed by a CVS-like treatment, which freezes core hole orbitals as proposed in Ref. 43. The resulting CVS- ΔCC is a good computational tool for targeting core-ionized states at the cost of ground state CCSD calculations while retaining the full flexibility of the CC wavefunction. In this work, we will focus on the computation of double ionization potentials (DIPs), which currently not many methods are able to compute. In particular, the CVS implementation of EOM-DIP-CCSD44,45is unavailable at the time of writing this manuscript. Furthermore, we will illus- trate that EOM-DIP-CCSD does not retain the full flexibility of CCSD, and it is not an exact approach for computing electronic energies for 2-electron systems when starting from a 4-electron reference. The goal of this paper is to (1) revive the idea of ΔCC with the emphasis on targeting doubly excited states and double core hole (DCH) states and (2) present numerical data on small molecules to support this idea. II. THEORY A. Coupled-cluster theory as an arbitrary root solver Coupled-cluster (CC) wavefunctions use an exponential parametrization, ∣Ψ⟩=eˆT∣Φ0⟩, (1) where ˆTis the CC cluster operator defined as ˆT=∑ μtμˆτμ, (2) J. Chem. Phys. 151, 214103 (2019); doi: 10.1063/1.5128795 151, 214103-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp with ˆτμbeing the excitation operator, which creates \| Φμ⟩from the reference determinant \| Φ0⟩, and tμis the cluster amplitude. The CC ansatz then follows ˆH∣Ψ⟩=ECC∣Ψ⟩, (3) assuming that \| Ψ⟩is an eigenstate of ˆHandECCis the corresponding eigenvalue (i.e., energy), ECC=⟨Φ0∣ˆH∣Ψ⟩. (4) The amplitudes { tμ} are obtained by solving tμECC=⟨Φμ∣ˆH∣Ψ⟩, (5) where \| Φμ⟩denotes the μth excited state determinant, ∣Φμ⟩=ˆτμ∣Φ0⟩. (6) Up to this point, we have not assumed whether we are trying to approximate the ground state or one of the excited states. In fact, the only assumption that has been made is that the state \| Ψ⟩is an eigenstate of a given Hamiltonian. The bias toward the reference state \| Φ0⟩built in the expo- nential parametrization controls which state we are targeting. The exponential parametrization is expanded to eˆT=ˆ1 +ˆT+ˆT2 2!+⋯. (7) Since nonstrongly correlated systems typically have amplitudes smaller than 1, the largest component in a usual CC wavefunction is the reference state \| Φ0⟩. This is known for model problems due to the work by Kowalski and co-workers.31However, with the advances inΔSCF methods,22it is meaningful to revisit this idea for more complex chemical systems. We will denote such excited CC states as ES-CC states, whereas the ground state CC state will be referred to as the GS-CC state. The energy difference between the GS-CC and ES-CC states defines the ΔCC approach for electronic excitation energies. This viewpoint can also be easily extended to number- changing excitations such as ionization potential (IP) and electron attachment (EA). It is important to note two main limitations of these ΔCC methods for electronic excited (EE) states, IP, and EA. First, tradi- tional CC (TCC) methods are not capable of describing strong elec- tron correlation, so ΔTCC methods are limited to states of single- reference character. Those with multireference character need more sophisticated CC approaches that can handle strong correlation. Examples of such approaches include the CC valence bond with sin- gles and doubles (CCVB-SD),46,47parametrized CCSD (pCCSD),48 and distinguishable cluster SD (DCSD),49. For the purpose of this paper, we will focus on the application of TCC approaches to states described well by a single determinant. Applying more advanced CC approaches to multireference problems will be an interesting topic for future study. We will refer to TCC simply as CC for the rest of this paper. Second, the computation of transition properties such as oscil- lator strengths and transition dipole moments is not straightforward and seems to scale exponentially with system size. Any transition properties between GS-CCSD and ES-CCSD states should techni- cally involve a CC state for both bra and ket (first-order derivativesfor each). Moreover, orbitals of GS-CCSD are not orthogonal to any orbitals of ES-CCSD in general. The evaluation of transition proper- ties therefore formally scales exponentially with system size if done exactly. This contrasts with EOM approaches where the bra state is not a CC state, instead it is only a linear wavefunction with the same set of orbitals as the GS-CCSD state. One may consider lin- earizing both of the CC states to evaluate transition properties to get an approximate answer, but the exact evaluation of such properties is still highly desirable. For the purpose of this work, we will compute only energies and leave the computation of transition properties to the future study. Furthermore, ΔCC methods often require auxiliary calculations to guide the selection of the reference determinant for a desired state. This can often be hinted by other economical approaches such as MP2 or CC2, but there is currently no generally applicable approach to this problem. We also note that ΔCC methods can often lead to artificial spatial symmetry breaking when targeting single core-hole (SCH) states in symmetric molecules. This is another manifesta- tion of their inability to describe excited states with multireference character. B. Equation-of-motion coupled-cluster theory For a given ground state CC wavefunction, one can solve a Hamiltonian eigenvalue problem in the linear response space. We first define the CC Lagrangian, L(λ,t)=⟨˜Ψ(λ)∣ˆH∣Ψ(t)⟩, (8) where the subscript Cimplies that it involves only “connected” diagrams50and the bra is defined as ⟨˜Ψ(λ)∣=⟨Φ0∣(1 +ˆΛ), (9) with the deexcitation operator ˆΛbeing ˆΛ=∑ μλμˆτ† μ. (10) Evidently, we have L=ECCfortsuch that the CC amplitude equation [Eq. (5)] is satisfied. Then, the equation-of-motion (EOM) Hamiltonian (or the CC Jacobian) can be derived from the linear response of this Lagrangian,4 Jμν=∂L(λ,t) ∂λμ∂tμ∣ t=t0, (11) where t0is a set of amplitudes that satisfies the “ground state” CC amplitude equation. Since Lis a linear function of λ,Jis independent fromλ. The EOM is linear response because it is a derivative of an energy expression with respect to the wavefunction parameter for both bra and ket. In EOM-CCSD, Eq. (11) is formed in the space of singles and doubles. Evidently, EOM-CCSD cannot describe any excited states that mainly contain triples and higher excitations. What may not be immediately obvious is that EOM-CCSD, in practice, cannot describe excited states with strong double excitation character. In our view, there are two aspects of Eq. (11) that should be high- lighted: (1) orbitals are determined for the ground-state SCF calcula- tion and are fixed and (2) the CC amplitudes, t, are also determined J. Chem. Phys. 151, 214103 (2019); doi: 10.1063/1.5128795 151, 214103-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp for the ground state and are also fixed. This naturally imposes con- straints on EOM calculations, and reducing the effects of those con- straints requires higher-order excitations (in this case triples). This has, of course, been well-known in the community, and a method such as EOM-CC(2,3) is motivated by this observation.51EOM- CC(2,3) takes the ground state CCSD wavefunction and forms the CC Jacobian in the space of singles, doubles, and triples. This is not really a linear-response method since it goes beyond the ground state parameter space, but it has shown to improve the accuracy of EOM-CCSD greatly, especially for states with strong double exci- tation character. As we will see later, without any nonperturbative connected triples, ΔCCSD and ΔCCSD(T) can perform significantly better than EOM-CCSD. This formalism can be extended to Fock space to treat the num- ber of electrons different from that of the ground state. The method that is relevant to the present work is the EOM ionization poten- tial (EOM-IP) methods. In EOM-IP-CCSD, the “singles” operator (1p) removes an electron and the “doubles” operator (1h2p) removes two electrons from occupied orbitals and adds an electron to one of the unoccupied orbitals. By performing EOM-IP-CCSD on an N- electron system, one can obtain the energies of the corresponding (N−1)-electron system and therefore ionization potentials. Remov- ing an electron from a molecule must be accompanied by sufficient orbital relaxation. This is implicitly done by the 1h2p operator which resembles the singles operator for ( N−1) systems. Interestingly, EOM-IP-CCSD effectively has only “singles”-type excitations from a (N−1) reference state via the 1h2p operator and no higher excita- tions. Therefore, the flexibility of EOM-IP-CCSD is smaller than that of CCSD or EOM-CCSD in terms of describing correlation between electrons. A similar conclusion can be drawn for EOM double IP (EOM- DIP) methods. EOM-DIP-CCSD employs the “singles” operator (2p) which removes two electrons and the “doubles” operator (1h3p) which removes three electrons from occupieds and adds an electron back to virtuals. From an ( N−2) reference state, this EOM-CC state effectively has only “singles” excitations. A majority of those singles would account for orbital relaxation, and only little correlation effect would be gained from using EOM-DIP-CCSD. The limited flexibility of EOM-DIP-CCSD can be most clearly understood by considering a model problem that contains four electrons and four orbitals. If we apply EOM-DIP-CCSD to this system, one would generate some determinants within the two- electron Hilbert space but not all. In Fig. 1(b), we explicitly show four determinants that are unreachable via EOM-DIP-CCSD if one FIG. 1 . (a) Reference determinant for a 4-electron system and (b) four determi- nants in the 2-electron sector that are unreachable via EOM-DIP-CCSD. They are each 2p4h excitations from the reference, but EOM-DIP-CCSD allows at most 1p3h excitations.uses the ground state determinant with four electrons shown in Fig. 1(a). This is somewhat disappointing because CCSD is exact for 2-electron systems. EOM-DIP-CCSD is not exact for 2-electron sys- tems when starting from a 4-electron reference. On the other hand, if one were to compute DIPs of a 4-electron system via ΔCCSD, at least the 2-electron system energy is exactly treated via CCSD. The remaining error is then solely from the CCSD error in the ground state. A spin-flip EOM method, EOM-SF-CCSD, can access a dif- ferent spin-manifold by spin-flipping from a higher spin-manifold to a lower spin-manifold. This is most commonly used to describe diradical systems,3but one may use it to describe doubly excited states. Not every doubly excited state can be accurately described by EOM-SF-CCSD. Generally speaking, doubly excited states which promote at least one electron to lowest unoccupied molecular orbital (LUMO) is within the scope of EOM-SF-CCSD. This is most obvi- ous from an example for a singlet doubly excited state. One starts from a triplet ground state which typically singly occupies highest occupied molecular orbital (HOMO) and LUMO. Therefore, at the reference level, there is already an electron promoted to the LUMO, which may help to describe doubly excited states accurately via spin- flips. The same logic applies to the EOM double EA (EOM-DEA) method. In EOM-DEA-CCSD, doubly excited states that involve the promotion of two electrons to the LUMO can be within the scope of the method. Similarly to EOM-SF-CCSD, this is because one starts from a ( N+ 2)-electron determinant which typically doubly occu- pies an orbital that corresponds to the LUMO of the N-electron determinant. III. H 2: A PROOF-OF-CONCEPT EXAMPLE For the ground state of H 2, CCSD is exact since it includes all possible excitations of two electrons in the system. Similarly, EOM-CCSD is exact for every state of H 2and therefore with con- ventional ground-state CCSD (i.e., GS-CCSD) and EOM-CCSD one can get all of the electronic states of H 2exactly for a given basis set. We will show that it is possible to reproduce those exact energies with the excited-state CCSD (i.e., ES-CCSD) method without severe numerical issues. FIG. 2 . All of the states with MS= 0 of H 2in the STO-3G basis set computed with GS-CCSD, EOM-CCSD, and ES-CCSD. There are two S= 0 excited states (labeled by 1 and 2) and one S= 1 state. Note that ES-CCSD follows EOM-CCSD exactly for all of the excited states. J. Chem. Phys. 151, 214103 (2019); doi: 10.1063/1.5128795 151, 214103-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3 . Reference determinants to obtain the (a) GS-CCSD, (b) ES-CCSD ( S= 0), 1 and ES-CCSD ( S= 1), and (c) ES-CCSD ( S= 0), 2 energies in Fig. 2. First, in Fig. 2, we present the results for H 2with the STO-3G basis set. With this basis set, there are only four states in the MS= 0 sector. All of these states can be obtained by running CCSD calcula- tions with a carefully chosen reference determinant along with initial guess amplitudes. For the ground state and the doubly excited state (1σu)2, the MP2 amplitude guess was used. For the singly excited states (singlet and triplet), we use a guess of Tll hh=±1, respectively, where Tll hhdenotes the doubles amplitude for 1 σg→1σu. These refer- ence determinants are also shown in Fig. 3. This strategy was enough to obtain the numerical data presented in Fig. 2. The same principles can be applied to a larger basis set cal- culation (cc-pVTZ)52as shown in Fig. 4. A distinct feature of the targeted ES-CCSD method is that it follows a root of the same char- acter throughout the potential energy surface (PES). This is most obvious from the ES-CCSD state obtained from the (1σg)1(2σg)1 reference shown in Fig. 4. At R= 0.5 Å, it starts as the 6th excited state of EOM-CCSD, stays on the same state, and even- tually becomes the third excited state of EOM-CCSD as the bond gets elongated. Around R= 1.25 Å, an avoided crossing appears between the third and fourth excited states. The GS-CCSD ener- gies for these two states switch near this avoided crossing. This is natural for a targeted excited state since it follows a state of desired character. FIG. 4 . Electronic energies of H 2with cc-pVTZ: the singlet ground state from GS- CCSD, three singlet excited states from ES-CCSD, 10 EOM-CCSD singlet states, one triplet excited state from ES-CCSD, and the triplet ground state from ES-CCSD and EOM-CCSD for MS= 0. Note that EOM-CCSD states are plotted with lines and they are not labeled for simplicity. ES-CCSD follows a root in EOM-CCSD in all cases.IV. APPLICATIONS TO DOUBLY EXCITED STATES We consider three molecules (CH 2, ethylene, and formalde- hyde) and investigate their singlet dark excited states. Those excited states are characterized by promoting two electrons from HOMO to LUMO. A procedure to perform targeted ES-CCSD calculations is as follows: 1. Perform a ground state HF calculation. 2. Promote two electrons from HOMO to LUMO and converge this non-Aufbau determinant via the MOM algorithm. 3. Perform the CCSD or CCSD(T) calculations on top of the optimized non-Aufbau determinant. In the third step, numerical instability may occur when starting from MP1 amplitudes as a guess. As a simple fix to this problem, we rescale the MP1 guess by a factor of 0.5 or less. In the future, one may use regularized MP1 amplitudes as a guess.53–56Furthermore, we start to update the amplitudes via the direct inversion of the iter- ative subspace (DIIS) algorithm57–59from the beginning as opposed to starting after a few iterations. With these tricks, we were able to converge the ES-CC calculations in this work without numerical difficulties. A. CH 2(11A1→21A1) The ground state of methylene (or carbene) is triplet. The singlet ground state (11A1) for CH 2is therefore an excited state. We optimized the geometry of CH 2on this electronic surface with ωB97X-D and the def2-QZVPPD basis set. Interestingly, the next excited state with the same term symbol (i.e., 21A1) has strong double excitation character. This doubly excited state is dominated by a closed-shell single determinant, and therefore it is a per- fect candidate for the ΔCC methods. Furthermore, it is possible to perform brute-force methods such as the semistochastic heat- bath CI (SHCI) method and a second-order perturbation correc- tion (SHCI+PT2) on this system.60As such, we compare ΔCCSD and EOM-CCSD against near-exact SHCI results. We employed the frozen-core approximation for the results presented in this section. In Fig. 5, the excitation energies for the (11A1→21A1) tran- sition are presented for various methods computed with the aug- cc-pVQZ basis set. For Δmethods, we used a reference that doubly occupies the 11A1LUMO. In other words, we used a reference with a transition of (3a1)2→(1b2)2. As shown in Fig. 5(a), the use of this non-Aufbau state with ground state orbitals yields ill-behaved ΔSCF andΔMP2 energies. This erratic behavior does not appear in the case ofΔCCSD and ΔCCSD(T) due to the singles operator. EOM-CCSD shows an error of 1.89 eV, which is much larger than its typical error for valence single excitations. In contrast, ΔCCSD and ΔCCSD(T) show remarkably accurate excitation energies whose errors are less than 0.1 eV. This is because the 21A1state is mainly dominated by one closed-shell determinant, which can be accurately described by CCSD. In Fig. 5(b), we examine the effect of orbital relaxation for the Δmethods. The non-Aufbau determinant was subsequently opti- mized to lower the HF energy using the MOM algorithm.22ΔSCF andΔMP2 improve significantly when using an orbital-optimized excited state determinant. However, in the case of ΔCCSD and ΔCCSD(T), the results are more or less the same as before. This is J. Chem. Phys. 151, 214103 (2019); doi: 10.1063/1.5128795 151, 214103-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 5 . The (11A1→21A1) excitation energies of CH 2obtained from various meth- ods with the aug-cc-pVQZ basis set. The excited states for the ΔSCF,ΔMP2, and ΔCCSD methods are based on a non-Aufbau state using (a) ground state orbitals and (b) a metastable SCF state optimized via MOM, respectively. The statistical error associated with SHCI+PT2 is 0.00032 eV which is negligible on the scale of these plots. expected because the important effect of orbital optimization can be incorporated through single excitations. In passing, we note that other variants of EOM-CCSD (SF, DEA, and DIP) may work well for this excited state. In particular, from the triplet ground state, one single spin-flip would be sufficient for EOM-SF-CCSD to access this state. States mainly dominated by one spin-flip are usually accurately described by EOM-SF-CCSD. It will be interesting to investigate these variants and assess their accuracy in the future. B. Loos and co-workers’ benchmark set: Ethylene and formaldehyde With advances in brute-force approaches, it is now possible to produce high-quality benchmarks for small molecules. An example of such benchmarks is Loos and co-workers’ recent study where they used a brute-force selected CI (sCI) approach61to produce reference energies for doubly excited states of a total of 14 small molecules: acrolein, benzene, beryllium, carbon dimer, carbon trimer, ethy- lene, formaldehyde, glyoxal, hexatriene, nitrosomethane, nitroxyl, pyrazine, and tetrazine. Interestingly, most of these molecules exhibit multireference character in the doubly excited state, and they are challengingfor single-reference CCSD to describe properly. For instance, Loos and co-workers considered the 2 s2→2p2excitation in Be. This state has three equally important determinants, 2 p2 x, 2p2 y, and 2 p2 z. Therefore, this state is of multireference by nature and not easy to describe with CCSD. More complications arise for butadiene and its isoelectronic species, acrolein and glyoxal. These molecules have a low-lying dark state that has three significant configura- tions, one of which is a doubly excited configuration. This dark state is likewise beyond the scope of conventional single-reference CCSD. On the other hand, the doubly excited states of ethylene and formaldehyde were found to be well described by a single determi- nant. Therefore, these are perfect candidates for the ΔCC approach. Given the CH 2results discussed above, we obtained the excitation gaps for Δmethods with optimized ground-state and excited-state orbitals. In the following benchmarks, we shall compare our results against benchmark numbers reported by Loos and co-workers.61For smaller basis sets, they produced near-exact excitation gaps based on sCI with second-order correction (sCI+PT2) and extrapolated full CI (exFCI) methods.62This should be adequate in assessing the quality of Δmethods for small basis sets. For larger basis sets, Loos and co-workers produced EOM-CC3 excitation energies. As we will see, EOM-CC3 is less accurate than ΔCCSD(T) when com- pared to sCI+PT2 and exFCI in smaller basis sets. Nevertheless, EOM-CC3 is a widely used iterative O(N7)correlated excited state method that can yield qualitatively correct excitation gaps for dou- bly excited states. As such, we will also compare ΔCC methods to EOM-CC3. In the case of ethylene, as shown in Fig. 6, EOM-CCSD signif- icantly overestimates the gap by 2–3 eV. This highlights the failure of EOM-CCSD for doubly excited states. The doubles amplitudes, R2, in EOM-CCSD are not enough to describe this state, and it is necessary to incorporate triple excitations to reach reasonable accu- racy as for instance in EOM-CCSDT with aug-cc-pVDZ. The role of quadruples is relatively unimportant in this case. The use of a reference determinant, (1b1u)2→(1b2g)2, yields remarkably accu- rate excitation energies with ΔCCSD and ΔCCSD(T). The largest T1 andT2amplitudes from the ES-CCSD calculation are 0.0472 and 0.1640, respectively. These small amplitudes mean that the underly- ing excited state is largely dominated by the reference non-Aufbau determinant. ΔCCSD(T) excitation energies are within the error bar of exFCI. Compared to sCI+PT2, the errors are 0.01 eV and 0.03 eV for aug-cc-pVDZ and aug-cc-pVTZ basis sets, respectively. This is better in accuracy than EOM-CC3 whose error is 0.49 eV for both basis sets. FIG. 6 . The (11Ag→21Ag) excitation energies of ethylene for various basis sets. EOM-CC3, EOM-CCSDT, EOM- CCSDTQ, sCI+PT2, and exFCI results were taken from Ref. 61. The error bars on ex-FCI are 0.01 eV and 0.06 eV for aug-cc-pVDZ and aug-cc-pVTZ basis sets, respectively. The numbers on each method indicate the excitation energy (eV) for the largest basis set available for that method. J. Chem. Phys. 151, 214103 (2019); doi: 10.1063/1.5128795 151, 214103-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 7 . The (11A1→31A1) excitation energies of formaldehyde for various basis sets. EOM-CC3, EOM-CCSDT, EOM-CCSDTQ, sCI+PT2, and exFCI results were taken from Ref. 61. The error bars on ex-FCI are 0.01 eV and 0.03 eV for aug-cc-pVDZ and aug-cc- pVTZ basis sets, respectively. The num- bers on each method indicate the excita- tion energy (eV) for the largest basis set available for that method. Since the doubly excited state of ethene is well described by a single determinant, relatively accurate gaps from ΔSCF are not unexpected. What is surprising is the striking underestimation of the gap in ΔMP2. One would think that for problems for which a single determinant is qualitatively correct MP2 should perform well. While this is commonly true, in the case of ΔMP2, the orbital opti- mization of a non-Aufbau determinant often leads to a very small gap between HOMO and LUMO.63Consequently, the MP2 corre- lation energy for such determinants would be heavily overestimated (i.e., more negative than it should be). As an attempt to remedy this problem, we applied a recently developed regularized MP2 method (κ-MP2).53–56Since small energy gaps will be damped away, the resulting correlation energy is stable even for those non-Aufbau determinants. As shown in Fig. 6, Δκ-MP2 excitation energies are similar to those of ΔCCSD, which highlights the utility of κ-MP2 for excited state simulations. In Fig. 7, we present another successful application of ΔCC methods. The doubly excited state of formaldehyde is largely dom- inated by one determinant. Similarly to previous examples, EOM- CCSD significantly overestimates the excitation gap. The error of EOM-CCSD is about 4–5 eV in this case. EOM-CCSDT greatly improves but incorporating quadruples (i.e., EOM-CCSDTQ) is necessary to reach near-exact results. Directly targeting the excited state with CCSD and CCSD(T) using a non-Aufbau reference deter- minant [ (2b1)2→(2b2)2] handles this state nearly exactly. The largest T1and T2amplitudes from the ES-CCSD calculation are −0.1157 and 0.1620, respectively, which implies that the underly- ing excited state is largely dominated by the reference non-Aufbau determinant. With aug-cc-pVTZ, ΔCCSD and ΔCCSD(T) yield an error of 0.07 eV compared to sCI+PT2. ΔCCSD overestimates, whereas ΔCCSD(T) underestimates the gap. This is better than EOM-CC3, which overestimates the gap by 0.81 eV. Given the accu- racy of ΔCCSD(T), we conclude that the role of connected quadru- ples in describing this state can be made negligible with a properly chosen reference deterimnant. We also note that ΔSCF produces a qualitatively correct gap and ΔMP2 does not exhibit the overcorre- lation problem previously shown in the case of ethylene. Therefore, Δκ-MP2 does not offer any improvement. In fact, Δκ-MP2 performs about 0.2 eV worse than ΔMP2. C. Summary In summary, not every doubly excited state requires an explicit treatment for triples unlike what was stated in Loos and co-workers’ work.61It is only those states that are dominated by more than onedeterminant that require a more sophisticated treatment than single- reference CCSD and CCSD(T). For doubly excited states with one dominant determinant, we showed that CCSD and CCSD(T) can directly target such states by simply employing a non-Aufbau deter- minant as a reference state. The errors of ΔCCSD and ΔCCSD(T) were found to be less than 0.1 eV for the systems considered in this work. V. APPLICATIONS TO DOUBLE CORE HOLE STATES Core-ionized states are another class of excited states that can be effectively handled by ΔCC methods. In fact, this was noted in the literature several times40–43and was recently revived by Zheng and Cheng.43In particular, Zheng and Cheng benchmarked sin- gle core hole (SCH) states for various small molecules and found about 0.13 eV standard deviation for ΔCCSD(T) in the ioniza- tion energies with respect to experimental values. Interested read- ers are referred to Ref. 43 for further information about their work. What we will focus in this work is the use of ΔCCSD(T) for double core hole (DCH) states. Following the prescription by Zheng and Cheng for SCH states, we first obtain an ( N−2) electron ref- erence state and freeze two unoccupied core orbitals for numeri- cal stability. The removal of unoccupied core orbitals is similar in spirit to the CVS14–16,64,65treatment in EOM-CC, and it explicitly prevents the CC wavefunction from collapsing to the ground state of the same number of particles. In our case, a double excitation from the HOMO to the unoccupied core orbitals would yield much lower energy than the desired core-ionized state. This is the source of numerical instability. The approach which freezes core hole orbitals will be referred to as CVS- ΔCC. Investigating DCH states to probe chemical environment was first proposed by Cederbaum and co-workers.66–68Compared to SCH states, DCH states are much more sensitive to the chemical environment. A classic example that illustrates this point is the series of hydrocarbons, C 2H2, C2H4, and C 2H6.66Creating a SCH state by removing an electron from a carbon atom in these molecules results in IPs that differ only by tens of electronvolts from each other. On the other hand, DIPs exhibit a difference over 4 eV or so per C–C bond. This highlights the utility of DCH states in probing the chem- ical environment. Since Cederbaum’s proposal, DCH states have also been experimentally realized.69–77In particular, two-site DCH (TSDCH) states are sensitive to the chemical structure, so obtaining TSDCH states in experiments has become a focus.73,75A single-site DCH (SSDCH) state can be readily obtained from a closed-shell J. Chem. Phys. 151, 214103 (2019); doi: 10.1063/1.5128795 151, 214103-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp (N−2) reference determinant, whereas TSDCH states are inher- ently of open-shell singlet character. In this section, we will study both kinds of DCH states and apply the ΔCC approach to obtain their electronic energies. The algorithm to obtain DCH states in CVS- ΔCC is as fol- lows: 1. Perform an SCF calculation on an N-electron system. 2. Localize core orbitals (and optionally valence orbitals sepa- rately) if there is more than one atom for the chemical element of interest. We employed Boys localization78for this step, and other localization schemes are also possible. 3. Identify core orbitals that will be made to be unoccupied. 4. Remove two electrons from hand-selected core orbitals, and perform an ( N−2)-electron SCF calculation using the MOM algorithm or Newton’s method. 5. Perform a CCSD calculation on the converged ( N−2)-electron reference. Note that it is necessary to freeze the two core hole orbitals in this step for numerical stability. We note that the CVS approach naturally requires frozen-core calculations for the N-electron ground state calculation. A. Single-site double core hole states We will investigate the SSDCH states of five small molecules, CO, CH 4, NH 3, N 2, and CO 2, and compare the DIP values computed from CVS- ΔCCSD and CVS- ΔCCSD(T) with those ofexperiments. All geometries were obtained from geometry optimiza- tion withωB97X-D79and aug-cc-pCVTZ.52,80 In Table I, we present DIPs for SSDCH states using ΔCC methods with increasing the size of basis set. In the case of ion- izing two electrons from a carbon atom, we observe roughly 2 eV of correlation effects in CO. On the other hand, the correction effect plays a smaller role for CH 4. In both molecules, increasing the size of the basis set reduces the DIPs. CO is well within the error bar of the experimental value, partly because the experimental error bar is quite large. For CH 4, CVS- ΔCCSD and CVS- ΔCCSD(T) exhibit an error on the order of 1 eV. Due to the lack of relativis- tic effect treatment in our calculations, this error is not so sur- prising81and we will leave more thorough benchmarks for future study. Nonetheless, ΔSCF, CVS- ΔCCSD, and CVS- ΔCCSD(T) all yield the correct trend that CO’s DIP is several eVs higher than CH 4’s DIP. This qualitative conclusion holds even at the SCF level. For SSDCH states involving nitrogen core vacancies, we inves- tigated NH 3and N 2. Their experimental estimates are about 10 eV apart. Similar to the previous cases, we observe smaller DIPs with larger basis sets. With the aug-cc-pCV5Z basis set, we observe about 1.5 eV error for all the methods. The result improves as we go from SCF to CVS- ΔCCSD(T). A major source of error is again the lack of relativistic effects. Nevertheless, all these meth- ods successfully capture qualitative differences between these two chemical species. Specifically, the DIPs of NH 3and N 2differ by about 10 eV. TABLE I . Double ionization potentials (eV) for single-site double core hole states. aCVXZ (X = T, Q, 5) is a short form for aug-cc-pCVXZ. Experimental values are obtained from Refs. 72 and 75. Molecule Ionization Basis set ΔSCF CVS- ΔCCSD CVS- ΔCCSD(T) Expt. CO C 1s−2aCVTZ 667.55 665.88 665.76 668(4) aCVQZ 667.24 665.36 665.20 aCV5Z 667.20 665.29 665.12 CH 4 C 1s−2aCVTZ 650.77 650.59 650.64 651.5(5) aCVQZ 650.49 649.88 649.92 aCV5Z 650.45 649.74 649.77 NH 3 N 1s−2aCVTZ 890.86 891.22 891.33 892.0(5) aCVQZ 890.54 890.53 890.77 aCV5Z 890.49 890.37 890.48 N2 N 1s−2aCVTZ 901.18 901.75 901.83 902.6(5) aCVQZ 900.84 901.18 901.24 aCV5Z 900.79 901.06 901.13 CO O 1s−2aCVTZ 1174.74 1175.92 1176.23 1178.0(8) aCVQZ 1174.32 1175.23 1175.54 aCV5Z 1174.25 1175.08 1175.39 CO 2 O 1s−2aCVTZ 1172.16 1172.48 1172.62 1173(2) aCVQZ 1171.75 1171.81 1171.96 aCV5Z 1171.68 1171.67 1171.81 J. Chem. Phys. 151, 214103 (2019); doi: 10.1063/1.5128795 151, 214103-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp In the case of oxygen, we investigated CO and CO 2. The DIPs of these molecules were experimentally shown to differ by about 5 eV. This qualitative difference is well described by all of the meth- ods examined here. However, the quantitative agreement between CVS-ΔCC methods and experimental values was only within several electronvolts as before in other systems. In passing, we note that we neglected strong correlation present in double core hole states. Specifically, in N 2, there are two possi- ble ways to obtain a DCH state on a nitrogen atom. Likewise, there are two equally important choices for CO 2for generating a DCH state on an oxygen atom. One may think that this would require mixing of two such references. In fact, this effect was studied in the context of NOCI with singles for simulating core-valence exci- tations in our group.82Given the quantitative agreement between CVS-CCSD(T) and experimental values for IPs reported in Ref. 43, we suspect that such a strong correlation effect may not be important in simulating core-ionized states. This can also be found in experi- mental results where single core hole states are usually localized on one atom even when there are multiple atoms of the same chemical element.83 B. Two-site double core hole states Unlike SSDCH, TSDCH states are all open-shell states. Two core holes are created at different atomic sites, and therefore two unpaired open-shell electrons will remain. There are two possi- ble ways to spin-couple these two electrons: singlet and triplet. We will obtain rough energetics of these states by employing a broken-symmetry HF reference state whose ⟨S2⟩is close to 1.0. Such a reference state is well-suited for Yamaguchi’s approxi- mate spin projection (AP).84With AP, one can obtain a spin-pure energy for the singlet state. Specifically, ES=0=EBS−(1−α)ES=1 α, (12)where EBSis the broken-symmetry state energy, ES=0is the singlet energy, ES=1is the triplet energy, and the coefficient αis given by α=⟨S2⟩S=1−⟨S2⟩BS ⟨S2⟩S=1−⟨S2⟩S=0, (13) which uses the ⟨S2⟩values of BS and S= 1 states (assuming ⟨S2⟩S=0=0). Clearly, Eq. (12) requires not only a broken-symmetry calculation but also a MS= 1 calculation to obtain ES=1and⟨S2⟩S=1. Within our CVS approach, an MS= 1 TSDCH calculation would require a different number of frozen core and virtual orbitals for αandβspin sectors. This is uncommon to run for most quantum chemistry packages available at the moment, and therefore we will leave the use of AP for TSDCH states for future study. The TSDCH states of four small molecules, CO, CO 2, N 2, and N 2O, are investigated. We report the DIP values computed from CVS- ΔCCSD and CVS- ΔCCSD(T) and compare them with those of experiments in Table II. All geometries were obtained from geometry optimization with ωB97X-D and aug-cc-pCVTZ. First, we study the TSDCH states where one core hole is local- ized on carbon and the other one is localized on oxygen in CO and CO 2. Experimentally, these two molecules have DIPs that are 6 eV apart from each other. The difference is quite small at the SCF level as two values are only 2 eV apart. With CVS- ΔCCSD, the difference becomes 3.2 eV and CVS- ΔCCSD(T) yields a dif- ference of 3.5 eV. While these are not quantitatively accurate, they all still correctly reproduce the qualitative behavior observed experimentally. Next, we investigate the TSDCH states in N 2and N 2O by creat- ing one core hole on each nitrogen. The DIPs of these two molecules are only 2 eV apart and almost within the experimental error bar from each other. Nonetheless, our goal is to reproduce the fact that the DIP of N 2is slightly larger than the DIP of N 2O. At the SCF level, the trend is reversed. With ΔSCF, N 2has a DIP that is 2 eV lower than that of N 2O. With CVS- ΔCCSD and CVS- ΔCCSD(T), a correct trend is reproduced. At the CCSD level, the DIP of N 2is TABLE II . Double ionization potentials (eV) for two-site double core hole states. For Δmethods, the numbers in parentheses indicate the corresponding ⟨S2⟩value. aCVXZ (X = T, Q) is a short form for aug-cc-pCVXZ. Experimental values were taken from Ref. 75. Molecule Ionization Basis set ΔSCF CVS- ΔCCSD CVS- ΔCCSD(T) Expt. CO C 1s−1, O 1s−1aCVTZ 853.77 (1.31) 854.94 (1.15) 854.96 855 (1) aCVQZ 853.58 (1.32) 854.75 (1.15) 854.76 aCV5Z 853.54 (1.32) 854.72 (1.15) 854.73 CO 2 C 1s−1, O 1s−1aCVTZ 851.71 (1.15) 851.68 (1.19) 851.51 849 (1) aCVQZ 851.53 (1.15) 851.46 (1.19) 851.28 aCV5Z 851.49 (1.15) 851.42 851.24 N2 N 1s−1, N 1s−1aCVTZ 834.06 (1.16) 835.34 (0.38) 835.50 836 (2) aCVQZ 833.90 (1.16) 835.12 (0.39) 835.27 aCV5Z 833.86 (1.16) 835.07 (0.39) 835.22 N2O N 1s−1, N 1s−1aCVTZ 836.27 (1.25) 834.99 (1.23) 834.49 834 (2) aCVQZ 836.11 (1.26) 834.82 (1.23) 834.30 aCV5Z 836.07 (1.26) 834.79 834.27 J. Chem. Phys. 151, 214103 (2019); doi: 10.1063/1.5128795 151, 214103-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp only 0.3 eV higher than that of N 2O, whereas the difference becomes 0.9 eV at the CCSD(T) level. As we can see, even without the relativistic treatment and spin- projection, we observe good qualitative agreement between CVS- ΔCC methods and experiments. It will be valuable to revisit these systems with proper relativistic corrections and Yamaguchi’s AP and try to observe a quantitative agreement between theory and experiments. We note that ⟨S2⟩at both SCF and CCSD levels lies between 0 and 2, which asserts that these states are suitable for Yamaguchi’s AP. C. Summary In this section, we applied the ΔCC method to both single- site and two-site double core hole states. Similarly to the doubly excited states studied in Sec. IV, there was no difficulty encoun- tered as long as the underlying CC state we are targeting is single- reference by nature. Furthermore, the satisfactory numerical sta- bility was ensured by freezing core holes for CC calculations. The resulting CVS- ΔCC methods were tested on a variety of small molec- ular systems. While it was difficult to make a quantitative compar- ison between CVS- ΔCC and experiments due to the lack of rela- tivistic treatment and large error bars in experimental values, both CVS-ΔCCSD and CVS- ΔCCSD(T) captured qualitative trends observed in experiments even when ΔSCF failed to do so. Fur- thermore, given small differences between CVS- ΔCCSD and CVS- ΔCCSD(T), it appears that the role of electron correlation can be fully captured at the CVS-CCSD level. We were not able to make comparisons to other available approaches because EOM- DIP-CCSD cannot obtain those highly excited core hole states at a reasonable cost. Furthermore, a production-level CVS-EOM-DIP- CCSD implementation is currently unavailable. VI. CONCLUSIONS In this work, we revisited the long-standing idea of using the coupled-cluster (CC) wavefunction to directly target excited states that may be beyond the scope of equation-of-motion (EOM) approaches. In particular, we focused on using CC with singles and doubles (CCSD) to describe (1) doubly excited states and (2) double core hole states. For doubly excited states, we show that it is possible to directly target an excited state through the ground state formalism of CCSD without numerical difficulties as long as the targeted state is dom- inated by one determinant. We achieve this simply by employing a non-Aufbau reference determinant that is orbital-optimized at the mean-field level via the maximum overlap method. A directly targeted CCSD and CCSD with a perturbative triples [CCSD(T)] excited state was shown to yield excellent excitation gaps for CH 2, ethylene, and formaldehyde. In particular, ΔCCSD(T) was shown to yield near-exact excitation gaps when compared to brute-force methods in a small basis set. This is quite promising since EOM- CCSD typically exhibits an error greater than 1 eV for these states. Furthermore, ΔCCSD(T) was found to be more accurate than EOM third-order approximate CC (EOM-CC3). Likewise, double core hole states (DCHs) can be directly obtained from the ground state CCSD formalism. This is also done by using a non-Aufbau reference determinant that has double coreholes. To ensure numerical stability, those core holes were frozen in correlation calculations. The resulting ΔCC ansatz is referred to as core-valence separation (CVS)- ΔCC and was benchmarked over double ionization potentials (DIPs) of small molecular systems (CO, CO 2, N 2, N 2O, and NH 3). Without relativistic corrections, CVS-ΔCCSD and CVS- ΔCCSD(T) were not able to reach quantita- tive accuracy when compared to experimental values. Nonetheless, they were able to estimate correct trends even when the mean-field method ( ΔSCF) could not. With the success of ΔCC described here, some interesting new directions become apparent. A more thorough investiga- tion of open-shell singlet states in conjunction with Yamaguchi’s spin-projection will be interesting. Currently, for valence excita- tions, we investigated states dominated by a closed-shell determi- nant. Two-site DCHs were investigated without spin-projection. With spin-projection, a broader class of states will be accessible and spin-pure energies for two-site DCHs can be obtained. Sec- ond, the use of more sophisticated CC methods such as the CC valence bond with singles and doubles (CCVB-SD) for targeting excited states with a multireference character will be interesting. Finally, in addition to core-ionized states, targeting core-valence excited states will be a promising candidate to apply the techniques described in this work. Some of these are currently underway in our group. 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1.4998269.pdf	Topological trajectories of a magnetic skyrmion with an in-plane microwave magnetic field Chendong Jin , Chengkun Song , Jinshuai Wang , Haiyan Xia , Jianbo Wang , and Qingfang Liu Citation: Journal of Applied Physics 122, 223901 (2017); View online: https://doi.org/10.1063/1.4998269 View Table of Contents: http://aip.scitation.org/toc/jap/122/22 Published by the American Institute of PhysicsTopological trajectories of a magnetic skyrmion with an in-plane microwave magnetic field Chendong Jin,1Chengkun Song,1Jinshuai Wang,1Haiyan Xia,1Jianbo Wang,1,2 and 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 30 July 2017; accepted 10 November 2017; published online 8 December 2017) Magnetic skyrmions are stable and topologically protected spin textures which have been observed in several chiral magnetic materials, and the resonant excitations of magnetic skyrmions have become a hot research topic for potential applications in future microwave devices. In this work,we investigate in-plane microwave-induced topological dynamics of a magnetic skyrmion in a nanodisk by using micromagnetic simulations. It is found that the resonant excitations of the sky- rmion are elliptical dynamics which contain counterclockwise and clockwise modes by applyingdifferent frequencies of the microwave ﬁeld. The conversion between these two elliptical modes is achieved by a transition to linear vibration. In addition, we demonstrate that the off-centered process of the skyrmion can be controlled by applying different phases of the microwave ﬁeld.Finally, we discuss the different topological excitations of four types of skyrmions. Our results pre- sent the understanding of topological skyrmion dynamics and may also provide a method to control skyrmions in nanodevices. Published by AIP Publishing. https://doi.org/10.1063/1.4998269 INTRODUCTION Skyrmions are topologically protected particle-like spin conﬁgurations which can exist stably in chiral magneticmaterials. 1,2Magnetic skyrmions are ﬁrst discovered in bulk non-centrosymmetric B20-type ferromagnets such as MnSi,3 FeCoSi (Ref. 4), and FeGe,5and these types of skyrmions are called Bloch skyrmions due to the presence of the bulk Dzyaloshinskii-Moriya interaction (DMI).1,6Recently, N /C19eel skyrmions are observed in ultrathin ﬁlms due to the presenceof interfacial DMI in proximity of heavy metals with strongspin-orbit coupling, 7and these skyrmion lattices have been experimentally observed by spin-polarized scanning tunnel- ing microscopy (STM) in a Fe monolayer grown in Ir (111).8 Meanwhile, it is numerically demonstrated that the antiferro- magnetic skyrmion can be manipulated in antiferromagnetic materials.9–11Besides, many numerical simulations indicate the probability of the existences of skyrmions withoutDMI, 12,13which indicate that DMI is not the necessary con- dition for the existences of skyrmions. Since the existences of skyrmions have been demon- strated experimentally, magnetic skyrmions have drawn a lot of attention for the huge potential applications in spintronicdevices, 14–18microwave generators,15and high-density infor- mation storage19due to their stable small size1of 10–100 nm. Besides, magnetic skyrmions can also be driven by a lowcurrent density 3,20of 10/C06Am/C02. Spin-polarized current is an effective mean to control the motion of magnetic sky- rmions.14,21Moreover, the microwave magnetic ﬁeld is another promising method to manipulate skyrmions6,22–24due to the less Joule heating produced by the microwave magnetic ﬁeld. The clockwise (CW) and counterclockwise(CCW) collective fashions of Bloch skyrmion lattice havebeen demonstrated when applying an in-plane ac magneticﬁeld. 25However, it is unclear how the two modes transform to each other and how the skyrmion responds to differentphases of the microwave magnetic ﬁeld. Therefore, in this work, we investigate the resonant excitation of a single N /C19eel skyrmion in the nanodisk and demonstrate that the N /C19eel sky- rmion has the elliptical CW and elliptical CCW modes. Moreimportantly, we provide a perspicuous description that thetransformation of the two modes is achieved by a transitionto linear vibration with the change in the frequency of the in-plane microwave. In addition, we investigate the off-centeredprocess of the skyrmion with different phases of microwaves.At last, we investigate the different excitations of four typesof skyrmions. METHODS We consider a 2D Heisenberg model with the nearest- neighbor ferromagnetic exchange, interfacial DMI, uniaxial anisotropy along the zaxis, and Zeeman ﬁeld. Therefore, the Hamiltonian of the system can be expressed as H¼/C0 AX hi;ji~mi/C1~mjþDijX hi;ji~mi/C2~mj/C0/C1 /C0KuX i^z/C1~mi ðÞ2/C0~HtðÞ/C1X i~mi; (1) where Ais the ferromagnetic exchange interaction, Dijis the DMI vector, which can be written as Dij¼D^rij/C2^z,Kuis the uniaxial anisotropy interaction, ~HðtÞis a time-dependenta)Author to whom correspondence should be addressed: liuqf@lzu.edu.cn. Tel.:þ86-0931-8914171. 0021-8979/2017/122(22)/223901/6/$30.00 Published by AIP Publishing. 122, 223901-1JOURNAL OF APPLIED PHYSICS 122, 223901 (2017) magnetic ﬁeld which is applied along the þxdirection, and ~mis the normalized magnetization ~m¼M=Ms. The dynam- ics of the magnetic skyrmions are governed by the Landau- Lifshitz-Gilbert (LLG) equation using the Object Oriented MicroMagnetic Framework (OOMMF) public code,26and the LLG equation can be described as d~m dt¼/C0c~m/C2~Heffþa Ms/C1~m/C2d~m dt; (2) where adenotes the Gilbert damping, which is set to 0.5. ~Heffis the total effective ﬁeld which is given by ~Heff¼/C0@H @~m: (3) We consider a nanodisk with a diameter of 100 nm and a thickness of 1 nm. The mesh size is 1 nm /C21n m/C20.5 nm. The skyrmion is initially formed in the center of the nanodisk as the ground state. The material parameters are chosen simi-lar to Ref. 7: saturation magnetization M S¼5.8/C2105A/m, exchange constant A¼1.3/C210/C011J/m, uniaxial anisotropy constant Ku¼0.8/C2106J/m3, and DMI constant Dis ﬁxed at 4m J / m2. RESULTS AND DISCUSSION The spin swirl structure of a topological skyrmion can be conﬁrmed by the skyrmion number which is deﬁned by the following formula:27,28 Q¼1 4pðð qdxdy ;q¼~m/C1@~m @x/C2@~m @y/C18/C19 ; (4) where qis the topological density. We ﬁrst created a N /C19eel skyrmion in the center of the nanodisk as shown in Fig. 1(a), which displays the initial magnetization distribution of a sin-gle skyrmion in the nanodisk. Note that, due to the presence of the boundary effect and sufﬁciently large DMI (4 mJ/m 2 in this work), the skyrmion can exist stably in the center ofthe nanodisk. The discussions of the skyrmion nucleation and its stability have been investigated in our previous work. 15In addition, other works in the literature also demon- strate that a skyrmion can locate stably in a nanodisk.6,7,24 Figure 1(b) shows the topological density distribution of the nanodisk, which corresponds to the magnetization distribu- tion shown in Fig. 1(a), from the center to the boundary; the topological density qﬁrst increases from zero to themaximum value (0.8350 /C21016m/C02) in the region where the z-axis magnetization component is zero, and then, qgradu- ally decreases to /C00.21/C21016m/C02at the border of the nano- disk. The total skyrmion number of the single N /C19eel skyrmion Q¼1. Following, we applied a microwave magnetic ﬁeld ~HðtÞ¼H0sinð2pfÞalong the þxdirection on the nanodisk. The amplitude of the microwave ﬁeld is 10 mT, and the fre- quency of the microwave ﬁeld fvaries from 1 to 50 GHz. Then, we investigate the topological dynamic behaviors of the skyrmion with an in-plane microwave magnetic ﬁeld,and the position of the skyrmion core is deﬁned by the guid- ing center (R x,Ry) instead of the geometric center,29which makes it more accurate to determine the trajectory of theskyrmion. The guiding center is given by R x¼ðð xqdxdy ðð qdxdy;Ry¼ðð yqdxdy ðð qdxdy; (5) where qis the topological density that is shown in Eq. (4). We obtain the topological trajectories of the skyrmion (snap- shot shown in Fig. 1) with different frequencies of the micro- wave magnetic ﬁeld by using the guiding center, as shown inFig.2. Figure 2(a)shows that the guiding center ﬁrst moves out of the nanodisk center and then rotates with a CCWmode with the microwave frequency of 2 GHz, and the majoraxis 2a, the minor axis 2b, and the theta represent the majoraxis of the steady elliptical trajectory of the skyrmion, theminor axis of the steady elliptical trajectory of the skyrmion,and the angular between the major axis and the -xaxis, respectively. Here, we clarify that the skyrmion can return to the center of the nanodisk in nano-seconds (about 1 ns) afterswitching off the microwave ﬁeld due to the presence of theboundary effect. Figure 2(b) shows that the excitation mode is still CCW for the microwave frequency of 5 GHz, andthe minor axis 2b of 5 GHz is obviously less than 2 GHz.When the frequency of the microwave magnetic ﬁeld isabove 7 GHz, the steady rotation transforms to a CW modeas shown in Figs. 2(c)and2(d), which describe the topologi- cal precession of the skyrmion with the microwave ﬁeld of 20 GHz and 40 GHz, respectively. Furthermore, we discuss the elliptical rotation modes of the skyrmion, which isexcited by a microwave magnetic ﬁeld with different fre-quencies as shown in Fig. 2(e). For 1 GHz /C20f/C206 GHz, the resonant excitation of the skyrmion is CCW mode, and boththe major axis 2a and the minor axis 2b decrease with the FIG. 1. (a) Relaxed state of a single sky- rmion in a nanodisk with D¼4m J / m2, and the color map of the out-of-plane component of magnetization is given at the top-right corner. (b) The correspond-ing distribution of topological density of the skyrmion. The in-plane sine-function microwave magnetic ﬁeld is applied along the þxdirection.223901-2 Jin et al. J. Appl. Phys. 122, 223901 (2017)increase in the frequency. For 6 GHz <f<7 GHz, the minor axis 2b approaches to zero, which means the elliptical CCW rotation of skyrmion transforming to linear vibration. For7 GHz /C20f/C2050 GHz, the major axis 2a continually decreases with the increase in the frequency, but the minor axis 2b ﬁrstincreases to 0.56 /C210 /C010m at the frequency of 30 GHz from 0 m, which means the excitation mode converting to CW rotation from linear vibration, and then gradually decreasesto 0.56 /C210 /C010m at the frequency of 50 GHz. Moreover, the theta ﬁrst increases to 59.23 degree at the frequency of 6 GHz and then decreases after 7 GHz. (a)–(d) shown in Fig. 2(e) (marked in red) correspond to those four trajectories shown as Figs. 2(a)–2(d) , respectively. Here, readers may notice that the size of the topological trajectories is on theorder of 0.1 nm, which is smaller than the mesh size (1 nm), and then doubt the reliability of the results. We are reason- ably conﬁdent that the numerical results are reliable becausethe topological trajectories are deﬁned by the guiding centerrather than the geometric center. Therefore, the small size of the topological trajectories does not have an effect on the reliability of the results but just represents the weak reso-nance strength of the skyrmion. In order to further demon-strate the reason we mentioned above, we also investigatethe inﬂuence of microwave magnetic ﬁeld amplitude on the topological trajectories of the skyrmion as shown in Figs. 3(a),3(b), and 3(c). It is found that the shape of the topologi- cal trajectories is kept unchanged with different amplitudesof the microwave magnetic ﬁeld, but the size of the topologi- cal trajectories is increased signiﬁcantly with the increase in the amplitude of the microwave magnetic ﬁeld. Figures 3(d), 3(e),3(f), and 3(g) display the magnetization distribution of the skyrmion marked in Fig. 3(c) for the amplitude of 500 mT. The corresponding topological densities are shown in Figs. 3(h),3(i),3(j), and 3(k), respectively. The previous subsection reveals that the dynamics of the skyrmion can be controlled by different frequencies ofthe microwave magnetic ﬁeld. In addition, the phase of the microwave magnetic ﬁeld ( u) also has a strong inﬂuence onthe topological dynamics of the skyrmion, as shown in Fig. 4. The microwave magnetic ﬁeld frequency is ﬁxed at 2 GHz, and its phase is increased from 0 to 2 pwith a step size of p/4. Figures 4(a),4(b), and 4(c) show that the sky- rmion moves out of the nanodisk center along the bottom-right major axis and then rotates with a CCW mode with u¼0,p/4, and p/2, respectively. With the increase in u, the skyrmion moves out of the nanodisk center along the top-leftmajor axis and keeps the CCW mode as shown in Figs. 4(d), 4(e),4(f),a n d 4(j). With the further increase in uto 7p/4, the skyrmion moves out of the nanodisk center along the bottom- right major axis again as shown in Fig. 4(h). Therefore, the rotation direction of the skyrmion remains unchanged withthe variation in u, but the off-centered process of the sky- rmion is changed with different u. Due to the sign of the interfacial DMI, N /C19eel skyrmions have four stable structures, which can be distinguished by c andp.cindicates the direction of the in-plane component of magnetization with the value of 0 and p, which represent the in-plane component of magnetization pointing to the bound- ary and the skyrmion center, respectively. pindicates the polarization of the skyrmion core with the values of 1 and/C01, which represent the z-axis component of magnetization of the skyrmion core positive and negative, respectively. For D¼4 mJ/m 2, the skyrmion structures of ( p, 1) and (0, /C01) can exist stably in the nanodisk as shown in Figs. 5(a) and 5(b), respectively. For D¼/C04 mJ/m2, the stable structures of the skyrmion are (0, 1) and ( p,/C01) as shown in Figs. 5(c) and5(d), respectively. Figures 5(e),5(f),5(g), and 5(h)show the topological density corresponding to the magnetizationshown in Figs. 5(a),5(b),5(c), and 5(d), respectively. It is found that the topological density is determined by pand unrelated to c.The topological number of the skyrmion is 1 for p¼1 and changes to /C01 when p¼/C01. Following, we investigate the corresponding topological excitations of fourskyrmions by applying a magnetic microwave with a ﬁxedfrequency of 2 GHz, as shown in Figs. 5(i),5(j),5(k), and 5(l). It can be seen that pdetermines the rotation mode of the FIG. 2. (a), (b), (c), and (d) show the trajectories of the guiding center with the frequency of the microwave mag- netic ﬁeld ﬁxed at 2, 5, 20, and40 GHz, respectively. The major axis 2a, the minor axis 2b, and the theta are deﬁned in Fig. 2(b), representing the major axis of the steady elliptical tra- jectory of the skyrmion, the minor axis of the steady elliptical trajectory of the skyrmion, and the angular between themajor axis and the -xaxis, respec- tively. (e) The major axis, the minor axis, and the theta of the elliptical tra- jectory as a function of the frequency of the microwave magnetic ﬁeld. The excitation of the skyrmion shown in Fig.1is CCW for the frequency below 6 GHz and CW for the frequency above 7 GHz.223901-3 Jin et al. J. Appl. Phys. 122, 223901 (2017)skyrmion. For example, the excitation of the skyrmion is the CCW mode for p¼1 as shown in Figs. 5(i)and5(k). For p¼/C01, the rotation mode of the skyrmion is changed to CW as shown in Figs. 5(j)and5(l). Moreover, chas an inﬂuence on the off-centered process of the guiding center. For c¼p [as shown in Figs. 5(i)and5(l)], the skyrmion always movesout of the nanodisk center from the right section of the nano- disk along the major axis. For c¼0 [as shown in Figs. 5(j) and5(k)], the skyrmion always moves out of the nanodisk center from the left section of the nanodisk along the majoraxis. Then, we can get a clear understanding of the excita- tions of the four N /C19eel skyrmions. FIG. 4. The trajectories of the guiding center for different u(the phase of the microwave magnetic ﬁeld) with f(the frequency of the microwave magnetic ﬁeld) ﬁxed at 2 GHz and the amplitude of the microwave ﬁeld ﬁxed at 10 mT. FIG. 3. (a), (b), and (c) show the trajectories of the guiding center with the amplitude of the microwave magnetic ﬁeld ﬁxed at 50, 100, and 500 mT, respec - tively. (d), (e), (f), and (g) display the magnetization distribution of the skyrmion marked in (c). (h), (i), (j), and (k) show the topological densit y corresponding to the magnetization shown in (d), (e), (f), and (g), respectively.223901-4 Jin et al. J. Appl. Phys. 122, 223901 (2017)CONCLUSIONS In summary, we investigate in-plane microwave- induced topological dynamics of a ( p, 1) structure N /C19eel sky- rmion in the nanodisk and ﬁnd that both elliptical CCW andCW modes can be observed, the resonance excitation of thistype of skyrmion is the CCW mode with a low-frequency magnetic microwave ﬁeld and transforms to the CW mode with a high-frequency magnetic microwave ﬁeld, and theconversion between the two types of elliptical modes isachieved by a transition to linear vibration. Moreover, theoff-centered process of the skyrmion can be controlled bydifferent phases of the microwave magnetic ﬁeld, and the skyrmions always move out of the nanodisk center along the major axis of the elliptical trajectory. At last, we contrast thetopological dynamics of four N /C19eel skyrmions by applying a ﬁxed magnetic microwave ﬁeld. The results show that p determines the rotation mode of the skyrmion and chas an inﬂuence on the off-centered process of the guiding center. Our results contribute to the understanding of topologicaldynamics of skyrmions in the nanodisk and also may providea method to manipulate skyrmions in nanodevices. ACKNOWLEDGMENTS This work was supported by the National Science Fund of China (11574121 and 51371092). 1N. Nagaosa and Y. Tokura, Nat. Nanotechnol. 8, 899–911 (2013). 2S. Seki, X. Yu, S. Ishiwata, and Y. Tokura, Science 336, 198–201 (2012).3F. Jonietz et al. ,Science 330, 1648–1651 (2010). 4W. M €unzer, A. Neubauer, T. Adams, S. M €uhlbauer, C. Franz, F. Jonietz, R. Georgii, P. B €oni, B. Pedersen, M. Schmidt, A. Rosch, and C. Pﬂeiderer, Phys. Rev. B 81, 041203 (2010). 5X. Yu, N. Kanazawa, Y. Onose, K. Kimoto, W. Zhang, S. Ishiwata, Y. Matsui, and Y. Tokura, Nat. Mater. 10, 106–109 (2011). 6B. Zhang, W. Wang, M. Beg, H. Fangohr, and W. Kuch, Appl. Phys. Lett. 106, 102401 (2015). 7J. Sampaio, V. Cros, S. Rohart, A. Thiaville, and A. Fert, Nat. Nanotechnol. 8, 839–844 (2013). 8S. Heinze, K. von Bergmann, M. Menzel, J. Brede, A. Kubetzka, R. Wiesendanger, G. Bihlmayer, and S. Bl €ugel, Nat. Phys. 7, 713–718 (2011). 9X. Zhang, Y. Zhou, and M. Ezawa, Sci. Rep. 6, 24795 (2016). 10J. Barker and O. A. Tretiakov, Phys. Rev. Lett. 116, 147203 (2016). 11C. Jin, C. Song, J. Wang, and Q. Liu, Appl. Phys. Lett. 109, 182404 (2016). 12M. V. Sapozhnikov and O. L. Ermolaeva, Phys. Rev. B 91, 024418 (2015). 13L. Sun, R. X. Cao, B. F. Miao, Z. Feng, B. You, D. Wu, W. Zhang, A. Hu,and H. F. Ding, Phys. Rev. Lett. 110, 167201 (2013). 14A .F e r t ,V .C r o s ,a n dJ .S a m p a i o , Nat. Nanotechnol. 8, 152–156 (2013). 15S. Zhang et al. ,New J. Phys. 17, 023061 (2015). 16J. Iwasaki, M. Mochizuki, and N. Nagaosa, Nat. Nanotechnol. 8, 742–747 (2013). 17Y. Zhou and M. Ezawa, Nat. Commun. 5, 4652 (2014). 18X. Zhang, M. Ezawa, and Y. Zhou, Sci. Rep. 5, 9400 (2015). 19N. S. Kiselev, A. N. Bogdanov, R. Sch €afer, and U. K. R €oßler, J. Phys. D: Appl. Phys. 44, 392001 (2011). 20J. Iwasaki, M. Mochizuki, and N. Nagaosa, Nat. Commun. 4, 1463 (2013). 21S.-Z. Lin, C. Reichhardt, and A. Saxena, Appl. Phys. Lett. 102, 222405 (2013). 22Y. Dai, H. Wang, T. Yang, W. Ren, and Z. Zhang, Sci. Rep. 4, 6153 (2014). FIG. 5. Four skyrmions can stably exist in the nanodisk with the different signs of the interfacial DMI. (a), (b), (c), and (d) display the magnetizatio n distribution of the four skyrmions ( p,1 )( 0 , /C01) (0, 1), and ( p,/C01). (e), (f), (g), and (h) show the topological density corresponding to the magnetization shown in (a), (b), (c), and (d), respectively. (i), (j), (k), and (l) give the corresponding trajectories of the four skyrmions by applying a magnetic microwave with a ﬁxed frequ ency of 2 GHz.223901-5 Jin et al. J. Appl. Phys. 122, 223901 (2017)23W. Wang, M. Beg, B. Zhang, W. Kuch, and H. Fangohr, Phys. Rev. B 92, 020403 (2015). 24J.-V. Kim, F. Garcia-Sanchez, J. Sampaio, C. Moreau-Luchaire, V. Cros,and A. Fert, Phys. Rev. B 90, 064410 (2014). 25M. Mochizuki, Phys. Rev. Lett. 108, 017601 (2012).26M. J. Donahue and D. G. Porter, OOMMF User’s Guide Version 1.0 (NIST, Gaithersburg, MD, 1999). 27C. Moutaﬁs, S. Komineas, and J. A. C. Bland, Phys. Rev. B 79, 224429 (2009). 28Y. Yamane and J. Sinova, J. Appl. Phys. 120, 233901 (2016). 29N. Papanicolaou and T. Tomaras, Nucl. Phys. B 360, 425–462 (1991).223901-6 Jin et al. J. Appl. Phys. 122, 223901 (2017)
5.0010926.pdf	Appl. Phys. Lett. 117, 042401 (2020); https://doi.org/10.1063/5.0010926 117, 042401 © 2020 Author(s).Chirality-dependent asymmetric vortex core structures in a harmonic excitation mode Cite as: Appl. Phys. Lett. 117, 042401 (2020); https://doi.org/10.1063/5.0010926 Submitted: 17 April 2020 . Accepted: 14 June 2020 . Published Online: 27 July 2020 Hee-Sung Han , Sooseok Lee , Dae-Han Jung , Myeonghwan Kang , and Ki-Suk Lee COLLECTIONS This paper was selected as Featured ARTICLES YOU MAY BE INTERESTED IN High mobility large area single crystal III–V thin film templates directly grown on amorphous SiO2 on silicon Applied Physics Letters 117, 042103 (2020); https://doi.org/10.1063/5.0006954 Propagation of spin waves through a Néel domain wall Applied Physics Letters 117, 022405 (2020); https://doi.org/10.1063/5.0013692 Nonlinear losses in magnon transport due to four-magnon scattering Applied Physics Letters 117, 042404 (2020); https://doi.org/10.1063/5.0015269Chirality-dependent asymmetric vortex core structures in a harmonic excitation mode Cite as: Appl. Phys. Lett. 117, 042401 (2020); doi: 10.1063/5.0010926 Submitted: 17 April 2020 .Accepted: 14 June 2020 . Published Online: 27 July 2020 Hee-Sung Han, Sooseok Lee, Dae-Han Jung, Myeonghwan Kang, and Ki-Suk Leea) AFFILIATIONS School of Materials Science and Engineering, Ulsan National Institute of Science and Technology (UNIST), Ulsan 44919, South Korea a)Author to whom correspondence should be addressed: kisuk@unist.ac.kr ABSTRACT Chirality of the magnetic vortex plays an essential role in dynamic excitations of the magnetic vortex structure. In a harmonic excitation of the vortex gyrotropic motion, it has been known that the chirality determines its phase to the driving force. From our micromagneticsimulations, we ﬁnd an additional role of chirality in the harmonic excitation of the vortex gyration. The shear deformation of the three-dimensional structure of the vortex core is determined by the chirality of the vortex. We conﬁrm that this is due to the gyrotropic ﬁeld. For the same vortex core motion with the same polarization but with opposite chirality, it turns out that the opposite gyrotropic ﬁeld is formed at the spiral magnetization in the vicinity of the vortex core structure. Published under license by AIP Publishing. https://doi.org/10.1063/5.0010926 Magnetic nanostructure, including magnetic skyrmions and vortices, have attracted much interest owing to their non-trivial topo- logical behaviors. 1–4Generally, it is useful to classify the magnetic nanostructures according to the topological charges such as polarityand chirality since they can provide not only the topological robust-ness to magnetic nanostructures but also different dynamics responses. 4–6For example, depending on the polarity, the deﬂection direction of moving skyrmion and the rotating sense of vortex gyra-tion are determined. 5,6 In the magnetic vortex, chirality cand polarity pare deﬁned by the curling direction of the in-plane magnetization and the direction of out-of-plane magnetization at the center, vortex core (VC), respectively:7,8c¼þ1(/C01) for counterclockwise (clockwise) curling magnetization direction and p¼þ1(/C01) for upward (downward) VC orientation. The magnetic vortex structure, despite its tiny core size, has high thermal stability and can be controlled efﬁciently through theresonant excitation of its inherent dynamic modes, 9–12making it a prominent candidate for multi-bit memory,12–14logic devices,15and magnonic crystals.16,17Recently, the magnetic vortex has attracted much attention since its tunable nonlinear dynamics can be beneﬁcialfor bioinspired neuromorphic computing. 18,19 Most of the studies on the magnetic vortex have focused on its dynamics in a two-dimensional (2D) disk-shaped geometry where the magnetization gradient along the thickness direction is negligibly small. However, the thicker the sample, the higher the increase in magnetiza-tion gradients along the thickness direction. 8,20This characteristicmakes the dynamics of the magnetic vortex more complex compared with the dynamics of the magnetic vortex in a 2D disk-shaped geome- try. The spin-wave modes of the magnetic vortex are expanded into higher-order spin-wave modes along the thickness direction with theformation of the nodes of spin-wave modes in the internal region. 21–25 The magnetic vortex shows nonuniform dynamics in the thickness, allowing the elongation of the VC although there are no nodes.26–29 This indicates that the VC is no longer a rigid structure, but it allows a ﬂexible oscillation in three-dimensional (3D) elements. Changing the dimensionality of the ferromagnetic element makes the dynamic char- acter of the VC much richer. Consequently, the fundamental under-standing of the dynamics of the magnetic vortex in 3D elements is essential for developing multifunctional spintronic devices based on the magnetic vortex structure. Generally, the cof the magnetic vortex in a conﬁned disk inﬂuen- ces only the phase of its motion under a harmonic excitation, whilethepdetermines the rotation sense of the gyrotropic motion of VC. 4,5 In this work, however, we show that the c-dependence occurs in the dynamically elongated 3D VC structure driven by harmonic rotatingmagnetic ﬁelds from micromagnetic simulations. To perform micromagnetic simulations, we used the mumax 3 code to numerically solve the Landau–Lifshitz–Gilbert equation: @m=@t¼/C0c0ðm/C2HeffÞþð a=MsÞðm/C2@m=@tÞ,w i t hl o c a lm a g - netization unit vector m, effective ﬁeld Heff, and phenomenological damping parameter a.30–32We considered a 300 nm-diameter Permalloy (Py, Ni 80Fe20)d i s kw i t ht h i c k n e s s L¼40 and 80 nm. Appl. Phys. Lett. 117, 042401 (2020); doi: 10.1063/5.0010926 117, 042401-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplWe selected the magnetic vortex with p¼þ1 as initial magnetization. The following standard material parameters for Py were used: exchange constant Aex¼13 pJm/C01,Ms¼860 kA /C1m/C01,a¼0.01, c0¼1.76/C21011rad/C1T/C1s/C01, and unit cell size of 2 /C22/C25n m3.A sa n equilibrium magnetic conﬁguration, the magnetic vortex structure was formed in a Py disk. Figure 1(a) shows the 3D magnetization conﬁguration of the magnetic vortex structure with c¼þ1 .T h er e ds u r f a c ei st h ei s o s u r - face for mz¼0.8 and mz>0.8 in the volume surrounded by this iso- surface, which indicates the VC boundary. The VCs on the top and bottom surfaces have divergent and convergent radial magnetizationcomponents, whereas the VC in the middle layer does not have radial magnetization components as shown in Figs. 1(b) and1(c).T h em a g - netization near VC was spiraled with a barrel-shaped VC structureunlike the magnetic vortex conﬁguration in a thin disk, whichdecreases the demagnetization energy by screening the surface charges. 8,20 To explore the deformation of the VC structure during harmonic excitations near the resonance, we obtained the frequency spectra ofthe phase difference ( d) between the magnetic ﬁeld and VC position and the radius ( r) of the VC gyrotropic motion for the top, middle, and bottom layers, as shown in Fig. 1(d) . A counterclockwise (CCW)- rotating magnetic ﬁeld, HCCW¼H0½cosð2pftÞ^xþsinð2pftÞ^y/C138with a frequency fand an amplitude H0¼10 Oe, was used for the harmonic excitation of the gyrotropic motion, including resonance.5As shown inFig. 2 , the overall gyrotropic motions of the VC for c¼þ1 in both of the disks with L¼40 and 80 nm show the typical resonance phe- nomena: the excitation amplitude, which is the radius of the VCmotion in the gyrotropic motion, was maximized at resonance fre-quencies of 940 MHz for L¼40 nm and 1140 MHz for L¼80 nm. The phase changes from /C0p/2 to þp/2 for both thickness cases. 5The responses of each layer, however, are clearly different between the twocases. For L¼40 nm, dandrfor all layers are almost same [ Fig. 2(a) ], which reveals that the VC structure is rigid throughout the thicknessin its gyrotropic motion. By contrast, the VC structure is no longer uniform for L¼80 nm as shown in the different frequency spectra of dandraccording to the layer [ Fig. 2(b) ]. Interestingly, the phase difference of the bottom layer (d Bot) is always larger than the phase difference of the middle layer (dMid), whereas the phase difference of the top layer ( dTop) is smaller than dMid.T h eg a pb e t w e e n dTopanddBotincreases with increasing f. This ﬁnding shows that the nonuniform phase is strongly dependenton the gyrotropic frequency of the VC. As shown in the frequencyspectra of r, the radius of the middle layer ( r Mid) is always smaller than those of the top and bottom layers ( rTopand rBot). At the resonance, the difference between rMidandrTophas their maximum values. These differences of the gyrotropic radius reﬂect the dynamical response ofthe VC by the rotating magnetic ﬁeld. Figure 3(a) shows the 3D structure of the VC in the resonant excitation of the gyrotropic motion. It reveals the dramatic deforma- tion of the VC structure in a steady-state motion: the VC structuresheared straightly along the tangential direction of its gyrotropic orbit[Fig. 3(c) ], thus resulting in the gradual changes of the phase ( d)a n d FIG. 1. (a) Magnetic vortex in the circular disk of diameter D¼300 nm and thick- ness L¼80 nm with the magnetization distribution on the x¼0 plane. The red sur- face at the center of the disk is the VC that represents the isosurface of mz¼0.8. (b) Magnetic vortex structure at the top, middle, and bottom layers. (c)Magnetization distribution on the x¼0 plane. The color represents the out-of-plane magnetization component. (d) Schematic illustration of the rotating-ﬁeld-driven mag- netic vortex. The arrows on the phase of the VC on the top, middle, and bottomlayers represent d Top,dMid, and dBot, respectively. The gyrotropic radius on the top, middle, and bottom layers is represented by rTop,rMid, and rBot, respectively.FIG. 2. Plot of the phase difference d(left) and gyrotropic radius r(right) vs fin the circular disk of diameter D¼300 nm and thickness L¼(a) 40 nm and (b) 80 nm. The magenta, orange, and cyan circular (square) dotted lines represent the phase of VCdTop,dMid,a n d dBot(gyrotropic radius rTop,rMid,a n d rBot) on the top, middle, and bottom layers, respectively. The dotted lines are the eigenfrequency for the gyrotropicmode of the magnetic vortex.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 042401 (2020); doi: 10.1063/5.0010926 117, 042401-2 Published under license by AIP Publishingradius ( r) of the VC gyration along the thickness. This ﬁnding also explains the same amount but opposite offset of the dTopand dBot from the dMidand the lowest value of rMidinFig. 2 . It should be noted that the thickness of the disk signiﬁcantly affects the VC deformation, while the diameter of the disk does not signiﬁcantly affect the VC deformation (see supplementary material A)]. Interestingly, the deformation does not vary monotonically to thethickness. A dramatic deformation of VC starts to occur at 70-nmthickness. The sheared VC structure in the resonant excitation is similar to the asymmetric Bloch wall structure observed in rectangularmagnetic elements. 8,33,34Out-of-plane magnetization components in the cross section of the yellow box in Figs. 3(a) and3(b) clearly show the typical asymmetrical curling magnetization conﬁgurationand the N /C19eel caps between VCs on the top and bottom surfaces, which has been observed in the asymmetric Bloch wall. 8,35 Moreover, this dynamically formed asymmetric Bloch wall shows t h es a m ed e p e n d e n c eo nt h e c. the VC on the top layer is lagging behind it on the bottom layer for c¼þ1[Fig. 3(a) ], whereas the bottom one is lagging behind the top one for c¼/C01[Fig. 3(b) ], which reveals the c-dependence on the dynamically deformed VC s t r u c t u r eu n d e rt h eh a r m o n i ce x c i t a t i o n s .O w i n gt ot h e1 8 0 - d e g r e e rotational symmetry, the magnetic vortex can be classiﬁed to cp¼þ1a n d cp¼/C01,7and the harmonic excitation of the vortex shows the 180/C14rotational symmetry about the axis of the driving force for the same value of cp. Similarly, the same 180/C14rotational symmetry should appear in the dynamic deformation of the VC inthe harmonic excitation. As an example, the lagging of the VC onthe bottom layer behind it on the top layer for p¼/C01a n d c¼/C01corresponds to the 180 /C14rotational symmetry about the driving ﬁeld axis of the lagging of the VC on the top later behind it on the bottom layer (see the simulation results for p¼/C01i n supplementary material B). To understand the c-dependence on the dynamical deformation of the VC structure, we calculated a gyrotropic ﬁeld distribution for the moving VC structure. The gyrotropic ﬁeld is the kinetic part of the effective ﬁeld, which is derived from the time-derivative term of theLLG equation. 9,36,37It is a very useful concept for understanding the dynamical deformation of the magnetic structure compared withthe static structure, wherein the virtual magnetic ﬁeld is considered due to the motion of the magnetic structure. 9,11,37–41As the gyrotropic ﬁeld in steady-state motions with the constant velocity tcan be expressed as ðm/C2ðt/C1r ÞmÞ=c0, one can easily obtain the distribu- tion of the gyrotropic ﬁeld from the magnetization gradient and veloc-ity. To elucidate the effect of the gyrotropic ﬁeld on the VC structure in its motion, we ﬁrst obtained the magnetization conﬁguration, hav- ing shifted VC from the center by applying the in-plane magnetic ﬁeld.Figure 4(a) shows the in-plane magnetic ﬁeld-driven VCs. The stream- line indicates the magnetization near the VC, and the color representsthe out-of-plane magnetization component. As this magnetization conﬁguration was not deformed by the in-plane magnetic ﬁeld-driven motion, the VC did not shear. Then, we assumed that the shifted VCmoves along þx-direction with the velocity t¼t x^x. From the magne- tization conﬁguration and velocity t, the gyrotropic ﬁeld near the VC FIG. 3. Snapshot of the steady-state gyrotropic motion of VC for (a) c¼1 and (b) c¼/C0 1 driven by the rotating magnetic ﬁeld of f¼1140 MHz (eigenfrequency of the gyrotropic mode of magnetic vortex) in the circular disk of diameter D¼300 nm and thickness L¼80 nm. The large arrow crossing the disk is the counterclockwise rotating magnetic ﬁeld. The red, green, and blue lines represent the trajectories of the VCs on the top, middle, and bottom layers, respectively. The magnetization inthe yellow plane is described in the inset of (a) and (b). (c) and (d) Top view of (a)and (b). The magenta, orange, and cyan circles represent the positions of the VCs on the top, middle, and bottom layers, respectively. FIG. 4. (a) Distribution of the gyrotropic ﬁeld near the VC when the VC is shifted along the y-direction toward the lower edge, and the VC is assumed to move at a velocity t¼tx^xforc¼1 (left) and c¼/C0 1 (right). The magnetization distribution near the VC is visualized using the streamline. The magenta and cyan arrowsrepresent the out-of-plane components of the gyrotropic ﬁeld. Large black arrowsrepresent the moving direction of the VC. The inset is the magnetic vortex structure in the middle plane for c¼1 (left) and c¼/C0 1 (right). (b) VC driven by the rotating magnetic ﬁeld of eigenfrequency and H 0¼10 Oe for c¼1 (left) and c¼/C0 1 (right). (c) Schematic illustration of the elongation of the VC driven by the gyrotropic ﬁeldforc¼1 (left) and c¼/C0 1 (right).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 042401 (2020); doi: 10.1063/5.0010926 117, 042401-3 Published under license by AIP Publishingdistribution was obtained, as shown in Fig. 4(a) . The magenta and cyan arrows on the spiral magnetization indicate the gyrotropic ﬁelddue to the motion. Regardless of c, the intensity of the gyrotropic ﬁeld o nt h ei n t e r n a lr e g i o ni sm u c hl a r g e rt h a nt h a to nt h es u r f a c e s .Furthermore, the region near the center on the y-axis was applied by the downward gyrotropic ﬁeld, whereas the region near the edge was applied by the upward gyrotropic ﬁeld. The in-plane compo- nent of the gyrotropic ﬁeld is the opposite direction of the movingdirection for c¼þ1, whereas it is the same direction of the moving direction for c¼/C01.Figure 4(b) shows the snapshot of the elon- gated VC driven by the rotating magnetic ﬁeld of eigenfrequencyand H 0¼10 Oe, which shows the elongation of the VC. Compared with the shifted VC [ Fig. 4(a) ] ,t h es t r e a m l i n er e p r e s e n t i n gt h e magnetization near the VC is tilted when it is dynamically elongated. The tilted direction at c¼þ1 is opposite to the tilted direction at c¼/C01d u et ot h eg y r o t r o p i cﬁ e l dd i s t r i b u t i o n . Figure 4(c) shows the schematic diagram of the gyrotropic ﬁeld distribution. The gyrotropic ﬁeld occurs as soon as the VC moves.The axis surrounded by spiral magnetization near the VC forc¼þ1 was forced to rotate CCW, whereas the axis for c¼/C01w a s forced to rotate CW by the gyrotropic ﬁeld. Thus, the shear of the VC is c-dependent, allowing the formation of the asymmetric Bloch wall. The direction of the gyrotropic ﬁeld corresponds to thedynamically formed asymmetric Bloch wall. In conclusion, we ﬁnd the dynamic deformation of the VC structure in the harmonic excitation and its c-dependence. From calculations of the gyrotropic ﬁeld distribution, the gyrotropic ﬁeldtilts the spiral magnetization near the VC, which gives rise to theshear deformation of the VC. It turns out that the gyrotropic ﬁeld,which relies on the direction of the motion and the direction of the spiral magnetization (the chirality), leads to the c-dependence on the deformation of the VC structure in its dynamic motion. Ourﬁndings not only provide fundamental understanding of the three-dimensional dynamic behavior of the magnetic vortex structure,but also open up a rich variety of vortex dynamics which can beapplicable to design neuromorphic or programmable spintronicdevices based on vortex nano-oscillators. See the supplementary material for additional micromagnetic simulation results for the chirality-dependence on the dynamical deformation of the VC structure. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government(MSIT) (Nos. 2016M3D1A1027831 and 2019R1A2C2002996). Itwas also supported by the 2019 Research Fund (No. 1.190038.01) of UNIST (Ulsan National Institute of Science and Technology). DATA AVAILABILITY The data that support the ﬁndings of this study are available from the corresponding author upon reasonable request. REFERENCES 1N. Nagaosa and Y. Tokura, Nat. Nanotechnol. 8(12), 899–911 (2013). 2S.-G. Je, H.-S. Han, S. K. Kim, S. A. Montoya, W. Chao, I.-S. Hong, E. E. Fullerton, K.-S. Lee, K.-J. Lee, M.-Y. Im, and J.-I. Hong, ACS Nano 14(3), 3251–3258 (2020).3N. Gao, S. G. Je, M. Y. Im, J. W. Choi, M. Yang, Q. Li, T. Y. Wang, S. Lee, H. S. Han, K. S. Lee, W. Chao, C. Hwang, J. Li, and Z. Q. Qiu, Nat. Commun. 10(1), 5603 (2019). 4S.-B. Choe, Y. Acremann, A. Scholl, A. Bauer, A. Doran, J. St €ohr, and H. A. Padmore, Science 304(5669), 420–422 (2004). 5K.-S. Lee and S.-K. 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1.359940.pdf	Numerical simulation of streamer–cathode interaction Igor Odrobina and Mirko Černák Citation: J. Appl. Phys. 78, 3635 (1995); doi: 10.1063/1.359940 View online: http://dx.doi.org/10.1063/1.359940 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v78/i6 Published by the American Institute of Physics. Related Articles Residual turbulence from velocity shear stabilized interchange instabilities Phys. Plasmas 20, 012301 (2013) Enhanced reliability of drift-diffusion approximation for electrons in fluid models for nonthermal plasmas AIP Advances 3, 012108 (2013) Relaxation-time measurement via a time-dependent helicity balance model Phys. Plasmas 20, 012503 (2013) A computational approach to continuum damping of Alfvén waves in two and three-dimensional geometry Phys. Plasmas 19, 122111 (2012) Boundary conditions for plasma fluid models at the magnetic presheath entrance Phys. Plasmas 19, 122307 (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 20 Jan 2013 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsNumerical simulation of streamer-cathode interaction * lgor Odrobina and Mirko &r&ka) Institute oj* Physics, Faculty of Mathematics and Physics, Come&s Umiversity, Mlynskb Do&a F2, 842 15 Bratislava, Slovakia (Received 16 January 1995; accepted for publication 1 June 1995) A self-consistent fluid model has been used to analyze streamer arrival at the cathode and its transformation to the stationary cathode fall in a positive point-to-plane corona discharge in N, at 26.7 kPa. The model is based on a description of the electron and the ion kinetics by one-dimensional continuity equations coupled with Poisson’s equation. The ions and electrons are assumed to be limited to a cylindrical channel with fixed radius and the field is computed using the method of disks. The computed current induced by the streamer-cathode interaction with a small cathode probe is compared with that measured experimentally. The cathode probe signal consists of an initial sharp current spike due to the displacement current followed, some 20 ns later, by a lower current hump due to the ion arrival at the cathode. The current signal is relatively insensitive to changes in the secondary electron emission coefficients. The results obtained indicate that the intense ionization and associated light flash experimentally observed near the cathode at the streamer arrival are not, as generally accepted, due to an intense electron emission but due to a sudden increase in the multiplication factor and a release of electrostatic energy accumulated in the streamer channel-cathode system. 0 1995 American Iditute of Physics. I. INTRODUCTION At gas pressures above roughly 10 kPa, the sequence of events leading to an arc formation consists of the bridging of the gap by primary streamers, and the subsequent heating of the initial channel created by the streamers. The transition between these two stages is determined by the arrival of the primary positive streamer at the cathode resulting in the for- mation of an active cathode region (spot) capable of feeding an increasing current of electrons into the discharge channel. The understanding of the streamer-cathode interaction leading to the cathode spot formation is of considerable prac- tical interest, for example, in view of the recent pulse streamer corona applications for pollution control.“* How- ever, as pointed out by Creyghton et al.,?- the distance be- tween theoretical models and applications is still too large to predict applicable results from theory. There are a few theoretical studies of the transition from primary streamers to an arc3-s and none has dealt with a detailed description of the formation of the cathode region. The recent computer simulations of the initial streamer dis- charge stage have only been continued to the point when the primary streamer approaches the cathode,36-9 or the grid resolution was not fine enough to describe the cathode region evolution in full detail.4V5 As for the later arc-formation dis- charge stage, none of the theoretical models!~‘“~” describing the filamentary glow-to-arc transition takes into account the cathode region and its influence on the discharge develop- ment. Probably the only theoretical model to date that pro- vides a detailed description of the cathode region and its influence on the discharge plasma channel properties is the one-dimensional computer simulation model by Belasri et al. l2 The one-dimensional approximation, however, is not 3Electronic mail: cernak@fmph.uniba.sk adequate for the streamer-to-arc transition, where the dis- charge has a small cross section. In an attempt to bridge the gap between the two groups of theoretical models mentioned above, in this article we present a computer simulation model of the transformation of the primary streamer head to the glow-discharge-type cathode region. In our simulation model the evolution of electrons and positive ions is described by one-dimensional continuity equations, with the space charge electric fields determined by the disk method.r3 The electrons are assumed to be in equi- librium with the local electric field. The model is similar in principle to the model described in Ref. 9, with emphasis given here to examination of the role of various physical processes in the cathode region formation. For the purpose of comparison with experiment the basis for the model is de- rived from experimental measurementsi of the current in- duced in a small cathode probe hit by the streamer in a short positive point-plane gap in N2 at a pressure of 26.7 kPa. The good agreement obtained between the discharge be- havior studied experimentally by other authors’4-‘” and those obtained by the numerical simulations indicates that our model provides an adequate physical picture of streamer arrival at the cathode. The results reveal that the dominant component of a sharp current spike induced by the streamer arrival in a cathode probe is the displacement current and that this current signal is not very sensitive to cathode emis- sion properties. The conductive current due electron emis- sion processes and to positive ion collection by the cathode contributes negligibly to this current spike. The conductive current becomes the dominant part of the cathode probe cur- rent some lo-20 ns after the cathode-probe current spike. This is~ in contrast to the commonly held beliefr6,17 that the streamer arrival at the cathode is associated with a sudden burst of electrons leading to the neutralization of the positive charge in the streamer head. J. Appt. Phys. 78 (6), 1.5 September 1995 0021-8979f95/78(6)/3635/8/$6.00 0 199.5 American Institute of Physics 3635 Downloaded 20 Jan 2013 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsII. THEORY The present model is a self-consistent model of the elec- trical discharge development in positive point-plane gap at a near atmospheric pressure. We will use the term “streamer” to refer to the ionizing wave that sustains its propagation by field enhancement due to finite curvature of its front. In our model we suppose the streamer propagates in the channel with given finite radius. A. The basic equations The electron and ion motion is described by one- dimensional continuity equations 2 %+ f (New,)-De s=Sj (1) @I where t is the time; x is the distance from the anode; N, and Nj are electron and positive ions densities; w, and Wi are the electron and positive ion drift velocities, and D, is the elec- tron diffusion coefficient. The source term is denoted by Si . The diffusion of positive ions is assumed to be negligible. As pointed out by Phelps and Pitchford,L8 if the steep gradients are present the number of high-energy electrons in N, can be reduced by backward diffusion. This is why we have introduced into the term Sj a coefficient A, estimating the fraction of the electrons that may be regarded as high- energy or “hot” electrons r 1 for 50 (a) Sf= ViANe ) A= l- 2 for O-C31 (b), 0 for ?l (c) (31 vi= a] w,I is the ionization frequency, (Y is the ionization coefficient, and wed is the negative diffusive velocity defined as (4) The coefficient A in Eqs. [3(a)] and [3(c)] describes two extreme situations, (a) no backward diffusion is present, (c) the drift flux just compensates the backward diffusion and no hot electrons are present. The terms [3(a)] and [3(c)] are the limit cases of the term [3(b)], which gives a source term identical with that used by Boeuf.” The streamer front formation and propagation due to field enhancement by space charge can be simulated cor- rectly only by solving Poisson’s equation, at least in two dimensions. However, such two-dimensional simulations of high fields and steep density gradients, which appear in the calculation as the streamer approaches the cathode, require excessive computation time. This is why we have used the “one and one-half” approach devised by Davies et a1.13 for streamers, where the finite radius of the discharge channel is taken into account by dividing the discharge into disks and computing the axial field from e x’ -x (5) where a0 is the dielectric constant, x is the distance from the anode, xc is the cathode position, a is the streamer radius, and r is the net charge density. The anode tip is simulated by a charged disk that is kept at a fixed applied potential. The cathode is supposed to be a perfectly conducting infinite plane electrode. This boundary condition is implemented by including “images” of the charges in the gap and that of the anode tip, which are reflected at the cathode surface, into the above equation. The electrodes are supposed to be perfectly absorbing for charged particles so that the ionic flux from both elec- trodes and the electron flux from anode are set to zero. The electron flux from the cathode consists of electrons emitted due to incoming ions and photoelectrons. Neglecting the photon absorption in the gas, the flux density of photoelec- trons is given byt3 .JiKt>=r, -$ Ji exp( - y) I XC x Sj(X’,t’)Sl(X)dX’dt’, x‘4 (6) where y, is the efficiency factor for the release of photoelec- trons per ionizing impact in the gap (i.e., the secondary pho- toemission coefficient) and Cl is a geometric factor. r-=8.3 ns is the radiation delay taken from Ref. 20. The flux of electrons emitted from the cathode due to incoming ions is given by Ja(t)=-YjnTi(XCrt)Wi(XC,t). (7) The photon and ion secondary emission coefficients have been estimated to be y,=5X low3 and yi= 10v2?l The anode tip is assumed to collect the entire current and this can be evaluated from Sato’s equation2* Niwi--N,W,+D, ~ E, dX, (8) where a is the channel radius estimated to be 0.6 mm in the conditions being considered. UA =4 kV is the applied voltage and EL is the Laplacian field intensity. The current collected by the cathode probe, of radius r=2 mm, is calculated using Maxwell’s first equation, i.e., where Q, is the total charge induced in the cathode probe. The lirst and second term in Bq. (9) correspond to the con- ductive and displacement currents, respectively. 3636 J. Appl. Phys., Vol. 78, No. 6, 15 September 1995 I. Odrobina and M. Cernik Downloaded 20 Jan 2013 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsThe electron-impact ionization coefficient and electron scribe the charge density development in the radial direction. mobility in nitrogen are approximated according to Ref. 23. This can result in inaccurate computed values of the cathode For pressure P in Torr and electric field E in V/m probe current Ip if the streamer changes its radius near the cu=57OP exp(-26000PIE) (m-l), surface of the cathode. In our computationsZ26 allowance was made for an increase of the discharge radius with the ,uu,=29/P (m2Nls). distance from the anode. The computed current wave forms, The electron diffusion coefficient De= 72.51 P (m/s) was chosen to correspond to a mean electron energy of 2.5 eV. The ion drift velocity is approximated bya w;=O.2C1-4X 10-5EIP)EIP known only approximately, and other possible-electron &is- however, were not found to be very sensitive to the channel shape, and the discharge behavior was not changed significantly.“7 The secondary emission coefficients yi and y” are for EIPs8 kVlrnA’orr, wi=12.5m-2.4X lO?(EIP) for EIP> 8 kVlmfTorr. sion processes, as for example field emission, are not in- cluded in the model. In the succeeding sections we will discuss the streamer- cathode interaction in terms of the multiplication factor M, the total ionization rate S,, the power dissipated in system P, , the stored energy change PC, and the power incoming from external circuit P, , which are defined as follows: C. Numerical method and stability criteria The system of Eqs. (l)-(5) can be integrated numeri- cally as follows: The electron continuity equation is split into a parabolic part plus a hyperbolic part incorporating the source term. During the one time step At the hyperbolic equation with the source term is solved in a Lagrangian frame, moving with the electron drift velocity. The moved density profile is mapped back to the original fixed frame. This procedure is very similar to that proposed by Davies et al. I3 Subsequently, the standard implicit Crank-Nicolson method is applied to solve the parabolic part of the decoupled continuity equation. The ion continuity equation is treated in a similar way. M= /::a(x)exp( jI/(x’)dx’)dx, St= fir,= s XC Stx)dx, XA (11) (12) (i3) pE= UAIA > (14) where U(x) is the potential on the gap axis and QA is the net charge accumulated on the anode tip. As the position of the ionizing wavefront xs we have used the center of ionization defined as xs= xS(x)dx S,. / B. Validity of the model Several points concerning the applicability of the model must be noted. The model used is an equilibrium model, i.e., the trans- port coefficients are assumed to depend on space and time only through the local value of the electric field. As will be shown later, the discharge development depends on the total multiplication factor rather than on the local values of the ionization coefficient. Also, in each stage of the discharge the cathode sheath length is much larger than an electron mean ionization path. Because of this, in agreement with Belasri et aZ.,l2 we believe that the equilibrium approach used pro- vides realistic results. The more important deficiency of the model is its “one- and-a-half” dimensionality. The disk method does not de- Although this scheme is only of first order of precision in regions with low drift velocity, the estimated value of numeric diffusion is more than one order of magnitude lower than that of physical diffusion in the region near the cathode where the refined mesh was used. The main advantage of the proposed scheme is that the hyperbolic part is solved by the conservative explicit method, which is absolutely stable in time. In the calculations presented here we used about lo4 time steps and a nonuniform mesh consisting of 1300 mesh points with very fine spatial resolution in the vicinity of the cathode. The refinement factor, i.e., the ratio of the largest and smallest space step, was about 60. Because the continuity equations and Poisson’s equation are solved explicitly, the time step At must be limited by the value of the dielectric relaxation time28 At< co q(NiPi+NePu,) * (16) During the formation of a quasi-neutral anode region the density of electrons is changing rapidly and the stability con- dition (16), being based on the explicit estimation, does not guarantee stability of the computation. Test runs indicated that a stronger stability criterion had to be used at higher applied voltages, where the electron density at the anode is replaced by the ion density: J. Appl. Phys., Vol. 78, No. 6, 15 September 1995 I. Odrobina and M. CernGk 3637 Downloaded 20 Jan 2013 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsAt< e” 4(Pi+PCLelNi * XA (17) When the streamer approaches the cathode surface a few times stronger time-step limitation has to be respected, so that the more precise calculation of A in Eq. (3) stabilizes the solution. The space step is limited by the criterion29 (a+l-&~~)Ax<L 08) where the first term in brackets is the limitation due to the spatial exponential electron growth, and the second term ex- presses a limitation arising from the fact that the mesh size must be small enough to limit the change jAw,/w,( to a reasonable value. Ill. RESULTS Results have been calculated for nitrogen at a pressure of 26.7 kPa. In correspondence with the experimental condi- tions in Ref. 13 the electrode spacing S=lO mm, the gap voltage lJ=4 kV, and 4 mm diam of the cathode probe are used. The inter-electrode space is supposed to be initially free of charge carriers. The streamer radius a was chosen to be 0.6 mm. Taking into account that the streamer radius de- pends inversely on the gas pressure,3o this value is consistent with the literature reports ranging from 0.06 to 0.17 mm at atmospheric pressure.30T31 t=ll22.38 ns x (mm) 60- E z iii rg c i- 2 E- 2 iii 6 c zt- 2 g zs 5 p Y i- 2 gj I Lif $ 5 i- 2 t=ll44.33 ns I (9) t=ll68.59 ns I - r FIG. 1. Electron and positive ion density (solid), electric field strength (dashed) and ionization rate (dotted) vs distance from the anode point at various times at cathode vicinity, (a) 1122.38 ns- before the maximum of M, (b) 1125.93 ns- at the point in time when the multiplication factor M is a maximum, (c) 1126.7 ns- at point where ionization rate, velocity, and probe current are maximal. The value of E is also greatly increased., (d) 1128.06 ns- wave front near the cathode, (e) -1132.94 ns- the separation of ions from the wave front, (t) 1144.33 ns- the ion current maximum, (g) 1168.59 ns- the stationary cathode region of an abnormal glow discharge. 3638 J. Appl. Phys., Vol. 78, No. 6, 15 September 1995 I. Odrobina and M. cernik Downloaded 20 Jan 2013 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions‘; UJ z 0 t cfr r; (D 0 % 20 10 0 ill0 1120 1130 1140 1150 1160 time (ns) HG. 2. Time development of the electron multiplication factor M ‘(solid), electron current from the cathode due to photoemission .I,,, electron current at the cathode due to ion impact J, (dashed), &total ionization rate (dotted). A. Streamer ignition and quasi-stationary propagation At the beginning of calculations, a low electron flux of 1.13x10” s-i starts to flow into the inter-electrode space from the cathode. After some 875 ns, these electrons arrive at the high field region near the anode and a plasma region is formed by ionization near the anode. Subsequently, the streamer starts to propagate, reaching a velocity of about 2.5X106 cm/s. During this discharge phase, an additional flux of electrons due to the secondary photoemission is re- leased from the cathode surface. At t = 1090 ns the streamer front makes contact with the first of these electrons and its movement is accelerated to a quasi-stationary propagation with a velocity of -4X lo6 cm/s. The electron and ion den- sity distributions, together with the corresponding ionization rate and electric field intensity values for this quasi- stationary streamer propagation are shown in Fig. l(a). B. Streamer-cathode contact As can be seen from Fig. l(b), when the streamer ap- proaches the cathode, the field in the cathode vicinity comes to rise. In consequence, the multiplication region spreads it- self more ahead of the streamer front, resulting in an increase in the multiplication factor M. The enhancement of the multiplication factor results in a rapid enhancement of the ionization and in increase of the streamer velocity to a maximum of 3.7X lo7 cm/s. The mul- tiplication factor M reaches its maximum approximately 0.8 ns before the streamer velocity and total ionization maxima [see Fig. l(c)]. The delay apparently corresponds to an aver- age electron transit time from the cathode. Comparing the results in Figs. 2-5, it can be seen that the maxima of the total ionization S, , cathode probe current ZP , energy dissipa- tion PD , and the streamer velocity W, coincide within -0.2 ns. From Fig. 4 it can be seen that at this moment the ion- ization is fed mainly from the energy accumulated in the capacitive system between the streamer and the cathode. At the point in time when the probe current reaches its maxi- mum, the photoelectron flux Jph begins to increase rapidly because of the intense ionization, reaching its maximum in a few ns (see Fig. 2). After the streamer passes its velocity z E. A- -% a 80, . , . , . , . ,80 40 0 1110 1120 1130 1140 1150 1160 time (ns) FIG. 3. Time development of the total probe ZP , anode I, and conductive probe 4 currents. E is the intensity of electrical field at the axis of the probe. maximum, the total ionization decreases rapidly as a result of a reduction of electron multiplication path and saturation of the a-coefficient. C. Transformation to stationary cathode fall After the streamer-cathode contact illustrated by Fig. l(a)-l(c), the structure developed near the cathode surface differs from a typical streamer front structure. For this rea- son, we will use the terms “ionizing wave” and “wave front” to refer to the streamer and streamer front structure occurring in the later stages. Beyond the point in time corresponding to the probe current peak, the ionizing wave decelerates its propagation and the ion density in its front increases [Fig. l(d)]. When the wave velocity falls below the ion drift velocity in the wave front, the ions begin to drift to the cathode surface [Fig. l(e)]. The ions separated from the wave front, together with those created by ionization ahead of the wave front, form a group drifting towards the cathode surface. As can be seen from Fig. l(f), the ion flux to the cathode reaches its maxi- mum 18 ns after the cathode probe current peak. The con- duction current of the incoming ions (see Ii in Fig. 3) is partly compensated by changes in the displacement current and, consequently, results only in a wide low hump on the cathode probe current wave form. Under certain conditions, the currents I,, ZP , and Ii, the internally stored energy PC and dissipated energy PD, can exhibit damped oscillations (not properly seen in Figs. 3 and 4). After the oscillations are damped, a stationary cathode fall, such as that shown in Fig. 1 (g), is established. IV. DISCUSSION A. The role of energy balance, rultiplication factor, and photoemission In the streamer-cathode inijkaction In the case of a stationary cathode fall, the total ioniza- tion rate S, is determined by the product of M and the total flux of electrons emitted from the cathode surface Jph + Ji . This differs from the nonstationary part of our simulations, where, during the ionizing wave propagation, the values of J. Appl. Phys., Vol. 78, No. 6, 15 September 1995 I. Odrobina and M. Cernik 3639 Downloaded 20 Jan 2013 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions200 z a” 100 a” 0 2 -100 -200 1110 1120 1130 1140 1150 1160 time (ns) FIG. 4. Time development of total input power P, , (solid) total dissipated energy P, (dotted), and change of energy stored in capacitive system channel-cathode PC (dashed). Negative values of PC mean that the internal energy of the channel has decreased. S, are several times lower than this product (as can be esti- mated from Fig. 2). The primary reason for this is not the difference between the electron flux entering the wave front and J,, f .Zi , but a reduction in the multiplication path for electrons [that is in the exponent of M, see Eq. (l)] due to the opposite movement of the wave. This multiplication path reduction by the wave movement internally stabilizes the streamer propagation veIocity. As the ionizing wave approaches the cathode [see Figs. l(a) and l(b)] the fietd distribution in the wave front-cathode space becomes more uniform, the wave front potential is nearly constant, and the average value of EIN in the ionizing region between the wave front and cathode increases. During these early stages of the streamer approach to the cathode, the average value of E/N increases to the Stoletow-Philips pointP233 where the ionizing efficiency (i.e., the number of ionizing collisions made by an electron passing potential drop of 1 V) ii maximal. As a consequence of this, the ap- proaching of streamer head with constant potential would rest& in several orders of magnitude increase in the multi- plication factor. This, however, is never fully realized since the increase in M leads almost immediately to a roughly proportional increase in the total ionization rate and, conse- quently, to an enhancement of the power dissipated in system PD. Since the power incoming from external circuit P, is limited by the streamer channel resistance of some 75 k!& the increase in P, is fed mainly from the change of the electrostatic energy stored in system PC (see Fig. 4). This results in a fast decrease of the streamer potential seen in Fig. 5 and, consequently, confines effectively the growth of the multiplication factor and the total ionization. Up to the Stoletow-Philips point the decrease in the electrostatic energy, and consequently in the streamer head potential, acts like a negative feedback limiting the growth of electric field, electron multiplication factor, and total ioniza- tion. The ionization rate S,, and thereby also the streamer head velocity W, , is confined to the values at which the dissipation of energy due to the ionization in multiplication region can be balanced with the energy influx from the chan- nel and with the decrease in stored electrostatic energy (see time (ns) FIG. 5. Time dependence of streamer head velocity W, (solid) computed as the time derivative of ionizing wave front position [Eq. (15)] and potential U, (dashed) computed at the same position. Fig. 4). Thus, during the streamer-cathode interaction, the values of S, and W, are determined by the energy available for dissipation and not directly by values of the electron multiplication factor M. This is why the streamer-cathode interaction is only weakly dependent on values of the sec- ondary photoemission coefficient y, (see Fig. 6): Any change, over a large scale of y, , can be effectively compen- sated by an opposite change in the multiplication factor, re- sulting in almost the same value of total ionization and power dissipation. After passing the Stoletow-Philips point [Fig. l(c)], the situation in the multiplication region begins to correspond to that in the cathode fall of an abnormal glow discharge. The wave propagation beyond this point in time will result in a decrease in the multiplication factor even without a further decrease of the wave front potential. Therefore the total ion- ization, propagation velocity, and probe current begin to de- crease, and the energy dissipation is no longer the main mechanism limiting the ionization. The higher y, the closer to the cathode is this turning point. Nevertheless, as during the earlier discharge stages, the changes in y, have little effect on the cathode probe current Zp (see Fig. 6). 80 L time (5nsldiv) FIG. 6. Comparison of current responses lP (solid) and I, (dashed) for two values of (1): y,=O.O05 and (2) r,=O. (The rest+ were computed with a constant weak flux J0=1.13X10u s-’ of electrons from the cathode.) 3640 J. Appl. Phys., Vol. 78, No. 6, 15 September 1995 1. Odrobina and M. cern&k Downloaded 20 Jan 2013 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsB. Comparison with experimental observations Since, for pressures above say 10 kPa, the streamer dis- charge behavior prior to the glow-to-arc transition in a small positive point-to-plane gap in nitrogen is very similar to that in a&s4 our model is believed to qualitatively describe the streamer-cathode contact also in air at near-atmospheric pressures, where the vast majority of the experimental stud- ies mentioned below have been made. As to the question of what happens when the streamer makes contact with the cathode, the view generally accepted by workers in the field of corona and spark discharges”‘5 + seems to be similar to that of Kondo and Ikuta? “The ap- proach of the primary wave to the cathode gives an increase of electron supply and the closest approach allows a sudden decrease of the wave head potential due to the drastic in- crease of the secondary ionization coefficients y, joined with yi action”. This view is, apparently, based on the results of optical observations of the streamer-cathode contact using a photon counting method’” that, similar to the streak camera records,s’ have revealed a bright light flash generated at the cathode by the streamer arrival. The results of optical observations16’35 are compatible with a sudden increase of the ionization rate near the cathode surface at the streamer arrival seen in Figs. l(c) and l(d). However, in contrast to Ref. 16, our results reveal that even if the streamer arrival is associated with an increase in elec- tron emission current (Jph and Ji in Fig. 2) the primary rea- son for the sudden increase of ionization is the increase of electron multiplication due to field enhancement and recon- figuration. The increasing ionization results in acceleration of the streamer head propagation and, as a consequence, in . rapid increase of the streamer head-cathode distributed ca- pacitance. Since the charging of the streamer head is limited by the streamer channel resistance, this rapid increase in the capacitance results in the rapid decrease of the streamer head potential. Similarly, the cathode probe current spike measured at the streamer arrival is due to the dishlacement current caused largely by the temporal development of the streamer- cathode distributed capacitance, and not by “a burst of elec- trons released by a combined effect of photoemission and field emission processes” as suggested by Inoshima et a1.36 Also, the assumption of Achat et al.17 that at the moment of the streamer-cathode contact “the current released from the cathode is associated with an electron current injected within the lilsimentary discharge, whereas the anode current which corresponds to collection of electrons is much. smaller” does not seem to be entirely valid in light of our theoretical model. The possible occurrence of field emission at the streamer-cathode contact has been suggested by Inoshima et aZ.36 and was also indicated by Nasser’s experimental results.s7 Our results, where (see Fig. 3) the computed field intensity at the cathode surface reaches values of the order of 10’ V/cm, indicate that the occurrence of field emission pro- cesses (in the broadest sense of the word) is possible. They, however, would occur some 10 ns after the probe current spike and not during the current spike rise. 0 time (5nsldiv) FIG. 7. Comparison of the experimental (1) and computed (2) lp current wave forms. The experimental IP wave form was measured for the first streamer in a system with 0.2 mm anode diameter point, 10 mm gap, and 4 mm diam central cathode probe. The applied voltage was 4 kV, and the pressure of nitrogen was 26.7 kPa. The experimental setup and method of measurements were the same as described in Ref. 36. As in the theoretical model, the discharge was initiated by a very weak irradiation of the cathode with uv light. Discussing the streamer-cathode interaction, it is inter- esting to refer to the experimental comparisons of current signal, induced by the streamer arrival in the probe coated by a dielectric polymeric layera having very low secondary electron emission, with that induced in the uncoated probe. The comparisons did not reveal any readily discernible-effect of the polymeric coating on the probe current signal. This provides additional support for our conclusion that the cath- ode probe current spike formation is not critically debendent on cathode electron emission. One of the implications of our model is that the flux of positive ions to the cathode reaches its maximum some IO-20 ns after the initial spike of the cathode probe current (see Fig. 3). This delay provides a possible explanation for the spectroscopic observation of Johnson et a1.38 that, at the streamer-cathode contact, the intensity of the cathode metal lines is not proportional to the cathode probe current except after a time of 50 ns. Besides the above discussed qualitative correspondence of the simulation results with experimental observations, we believe that the theoretical model used, despite its simplicity, is sufficient to reproduce some important details of the streamer-cathode contact also quantitatively. This is can be seen by comparing the computed cathode probe current with that measured in similar conditions14 shown in Fig. 7. The agreement in the general shape of the current wave forms in Fig. 7 is satisfactory, especially when it is noted that no allowance is made for the finite bandwidth of the experi- mental measurement system. The most important discrep- ancy between the computed and experimental wave forms is that the width of the first peak of computed wave form is narrower than that of the measured peak. It is probably due to an expansion of the streamer head near the cathode surface observed experimentally for both positive wire-plane and parallel-plane gaps in Refs. 2 and 39, respectively. Such streamer head expansion can be explained as fol- lows: During the streamer propagation far from the cathode, the highest value of the multiplication factor M is at the J. Appl. Phys., Vol. 78, No. 6, 15 September 1995 I. Odrobina and M. eernik 3641 Downloaded 20 Jan 2013 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsstreamer axis. At the streamer-cathode contact, however, af- ter a point in time when M at the streamer axis passes its maximum (t=1125.93 ns in Fig. 2) the ionization on longer multiplication paths far from the axis become important. In realistic conditions this can result in a streamer head expan- sion along the cathode surface. As has been illustrated in Refs. 25 and 26, a better matching to the experimental cur- rent wave forms can be achieved by setting the one- dimensional Eqs. (1) and (2) in a nonconstant radius dis- charge channel. ACKNOWLEDGMENTS This work was supported by a grant from the Slovak Ministry of Education and Science. Thanks are due to Pro- fessor A. J. Davies and Dr. E. M. van Velhuizen for stimu- lating discussions. ‘G. E. Vogtlin and B. M. Penetrante, in Non-Thermal Plasma Techniques for Pollution Control, edited by B. M. Penetrante and S. H. Schultheis (NATO Springer, 1992). ASI Series, p. 187. ‘Y. L. M. Creyghton, E. M. van Veldhuizen, and W. R. Rutgers, IEEE Pmt.-Sci. Meas. Technol. 141, 141 (1994). ‘S Ganesh, A. Rajaboshanam, and S. K. Dhali, J. Appl. Phys. 72, 3957 (i992). ‘J. E Loiseau, Ch. Manassis, N. Spyrou, and B. Held, in Proceedings offhe 4th International Symposium on High Pressure Low Temperature Plasma Chemistry, edited by J. D. Skalni, Bratislava, Slovakia, 1993 (Hakone), p. 183. 5D. Braun, V. Gibalov, and G. Pietsch, Plasma Sources Sci. Technol. 1, 166 (1992). 6M. C. Wang and E. E. Kunhardt, Phys. Rev. A42,2366 (1990). 7J. T. Kennedy, Ph.D. thesis, Emdhoven University of Technology, 1995. “P. Sdgur, D. Djermoune, and E. Marode, Proceedings of the 10th Intema- tional Conference on Gas Discharges and Their Applications, edited by W. T. Williams, Swansea, Wales, 1992 (GD), p. 908. “R. Morrow, IEEE Trans. Plasma Sci. 19, 86 (1991). ‘OF. Bastien and E. Marode, J. Phys. D: Appl. Phys. 12, 249 (1979). “R. S. Sigmond, J. Appl. Phys. 56, 1355 (1984). “+4. Belasri, J. P. Boeuf, and L. C. Pitchford, J. Appl. Phys. 74, 1553 (1993). 13A. J. Davies, C. S. Davies, and C. J. Evans, Proc. IEEE 118, 816 (1971). 14M. cern& and T. Hosokawa (unpublished results). “P. Bertault, J. Dupuy, and A. Gilbert, J. Phys. D: Appl. Phys. 25, 661 (1992). - - “K. Kondo and N. Ikuta, J. Phys. Sot. Jpn. 59, 3203 (1990). 17S. Achat, T. Teisseyre, and M. Marode, J. Phys. D: Appl. Phys. 25, 661 (1992). “A. V. Phelps and L. C. Pitchford, Phys. Rev. A 31, 2932 (1985). 19J. P. Boeuf, Phys. Rev. A 26, 2782 (1987). “W. Legler, 2. Phys. 173, 169 (1963). ‘I H. Tagashira (private communication). 22N. Sato, J. Phys. D: Appl. Phys. 13, L3 (1980). ‘3S. K. Dhali and P. F. Williams, J. Appl. Phys. 62, 4696 (1987). “K. Yoshida and H. Tagashira, J. Phys. D: Appl. Phys..9,491 (1976). *‘I. Odrobina and M. Eern&, in Ref. 4, p. 16.5. 261. Odrobina and M. ~ernitk, Proceedings of 21st International Conference on Phenomena in Ionized Gases, edited by G. Ecker, U. Arendt, and J. BGseler, Bochum Germany, 1993, p. 458. “I. Odrobina and M. eernik, Czech. J. Phys. 42, 303 (1992). “M. S. Barnes, T. J. Colter, and M. E. Elta, .I. Appl. Phys. 61, 81 (1987). ‘91. Odrobina, Ph.D. dissertation, Comenius University, Bratislava, 1994 (in English). 30G. A. Dawson and W. P. Winn, Z. Phys. 183, 159 (1964). 31A. Gilbert and F. Bastien: J. Phys. D: Appl. Phys. 22, 1078 (1989). 32L. B. Loeb, Basic Processes of Gaseous Ekctronics (University of Cali- fornia Berkeley, 1955), pp. 664-671. 33 Sigmond, in Efectn’cal Breakdown of Gases, edited by J. M. Meek and J. D. Graggs (Wiley, London, 1978), p. 330. 34M. eern&, E. Marode, and L Odrobina, in Ref. 26, p. 399. 35A. Gilbert and J. Dupuy, in Ref. 8, p. 452. 36M Inoshima, M. eem&, and T. Hosokawa, Jpn J. Appl. Phys. 29, 1165 (1990). 37E. Nasser, J. Appl. Phys. 37, 4712 (1966). 38P. C. Johnson, G. Berger, and M. Goldman, J. Phys. D: Appl. Phys. 10, 2245 (1977). 39P. Stritzke, I. Sander, and H. Raether, J. Phys. D: Appl. Phys. 10, 2285 (1977). 4 . 3642 J. Appl. Phys., Vol. 78, No. 6, 15 September 1995 I. Odrobina and M. eernik Downloaded 20 Jan 2013 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions
10.0001716.pdf	Low Temp. Phys. 46, 932 (2020); https://doi.org/10.1063/10.0001716 46, 932 © 2020 Author(s).Peculiarities of IV-characteristics and magnetization dynamics in the φ0 Josephson junction Cite as: Low Temp. Phys. 46, 932 (2020); https://doi.org/10.1063/10.0001716 Submitted: 22 July 2020 . Published Online: 30 September 2020 Yu. M. Shukrinov , I. R. Rahmonov , and A. E. Botha ARTICLES YOU MAY BE INTERESTED IN Critical current of dc SQUID on Josephson junctions with unconventional current-phase relation Low Temperature Physics 46, 919 (2020); https://doi.org/10.1063/10.0001714 Thermal stability of nanocrystalline and ultrafine-grained titanium created by cryomechanical fragmentation Low Temperature Physics 46, 951 (2020); https://doi.org/10.1063/10.0001719 Experimental research of condensation processes occurring under laser ablation in superfluid helium and vacuum Low Temperature Physics 46, 896 (2020); https://doi.org/10.1063/10.0001711Peculiarities of IV-characteristics and magnetization dynamics in the w0Josephson junction Cite as: Fiz. Nizk. Temp. 46,1 1 0 2 –1109 (September 2020); doi: 10.1063/10.0001716 View Online Export Citation CrossMar k Submitted: 22 July 2020 Yu. M. Shukrinov,1,2,3 ,a)I. R. Rahmonov,1,4and A. E. Botha3 AFFILIATIONS 1BLTP, Joint Institute for Nuclear Research, Dubna, Moscow Region 141980, Russia 2Dubna State University, Dubna 141980, Russia 3Department of Physics, University of South Africa, Florida, Johannesburg 1710, South Africa 4Umarov Physical Technical Institute, TAS, Dushanbe 734063, Tajikistan a)Author to whom correspondence should be addressed: shukrinv@theor.jinr.ru ABSTRACT Thew0junction demonstrates a rich variety of dynamical states determined by parameters of the Josephson junction and the intermediate ferromagnetic layer. Here we find several peculiarities in the maximal amplitude of magnetic moment ^my, taken at each value of the bias current, which we correlate to the features of the IV-characteristics of the w0junction. We show that a kink behavior in the bias current (voltage) dependence of ^myalong the IV-characteristics is related to the changes in the dynamical behavior of the magnetization precession in the ferromagnetic layer. We also demonstrate a transformation of the magnetization specific trajectories along the IV-curve, magnetization composite structures, and hysteretic behavior in the bias current dependence of ^my. Due to the correlations between features of^myand the IV-characteristics, the presented results open a way for the experimental testing of the peculiar magnetization dynamics which characterize the w0junction. Published under license by AIP Publishing. https://doi.org/10.1063/10.0001716 1. INTRODUCTION Thew0Josephson junction1with the current-phase relation Is=Icsin(w−w0) is becoming an interesting and important topic in condensed matter physics.2,3The superconductor-ferromagnet- superconductor (SFS) w0junctions where the phase shift w0is proportional to the magnetic moment perpendicular to the gradi- ent of the asymmetric spin-orbit potential, demonstrate a number of unique features important for superconducting spintronics andmodern informational technologies. 2,4–8This coupling between phase and magnetic moment of the ferromagnetic layer allows one to manipulate the internal magnetic moment using the Josephson current.1,9The magnetic moment also might pump current through thew0phase shift. It leads to the appearance of the dc component of superconducting current in w0Josephson junction.7,9,10 The application of dc voltage to the w0junction produces current oscillations and consequently magnetic precession. As shown in Ref. 9, this precession may be monitored by the appear- ance of higher harmonics in the current-phase-relation (CPR) aswell as by the presence of a dc component of the superconducting current that increases substantially near the ferromagnetic reso- nance (FMR). The authors stressed that the magnetic dynamics of the SFS w0junction may be quite complicated and strongly anharmonic. In contrast to these results, very simple characteristictrajectories magnetization precession in the m y-m x,m z-m x, and mz-m yplanes were recently discovered in Ref. 10. To distinguish specific shapes, some were named as “apple ”,“sickle ”,“mush- room ”,“fish”,“moon ”, etc. Recent experiments11–13have measures an anomalous Josephson effect, validating the w0junction model and pointing to its potential uses in a variety of technologies that rely on supercon- ducting spintronics.2,14The specific nature of the coupling that occurs in the w0junction allows one to manipulate the internal magnetic moment via the Josephson current, and conversely.1,9,15 Previous simulations of the w0junction demonstrated how the application of an external electromagnetic field could be used to tune the character of the magnetic moment precession over current intervals corresponding to specific Shapiro steps.10We alsoLow Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 46,000000 (2020); doi: 10.1063/10.0001716 46,000000-932 Published under license by AIP Publishing.demonstrated the appearance of a dc component in the superconducting current and clarified how it influences the IV-characteristics within the resonance region, i. e., the region where the precession frequency is close to that of the Josephson fre-quency. The effects of Gilbert damping and spin-orbit coupling onIV-characteristics and magnetization precession were also studied. We also studied factors that could affect the magnetization reversal by the superconducting current in the w 0junction.6,8The physics ofw0junction was found to have many features in common with the famous Kapitza pendulum problem.7Most recently we were able to provide analytical criteria for magnetization reversal from predicting the conditions under which the reversal can occur.16In the Introduction of Ref. 16, we have given a detailed review of recent experimental and theoretical developments relating to the w0 junction, and we also discuss the choice of materials available for its practical realization. This present work is an extension of some preliminary results.17Here we provide a more detailed investigation of the com- plicated dynamics that results from the unique interaction betweenthe superconducting current and magnetic moment in the w 0junc- tion. Interspersed with chaotic dynamics we find several windows of regular dynamics. In certain ranges of bias current, there are stable states of the magnetization precession, which we were able tocharacterize by the very specific shapes they make through projec-tions of the phase trajectories. We also simulate how the maximal amplitude of ^m ychanges as the bias current is swept along the IV-characteristic. We find a kink behavior in the bias current (voltage) depen- dence of the maximal ^mywhich origin is related to the change of the dynamical behavior of the magnetization in ferromagnetic layer. The characteristic trajectories in the my—mx,mz—mx, and mz—myplanes were recently discussed in Ref. 10. However, the kinks that occur in ^my, and specifically their origin, have not been discussed before. As was alluded to in Ref. 10, these characteristic trajectories offer a unique possibility to control the magnetization dynamics via an external bias current. Here we show that similar shaped kinks have a common origin due to the underlying dynam-ics. We also demonstrate how specifically shaped trajectories trans-form from one shape into another, the manifestation of compositestructures, and hysteretic behavior. 2. MODEL AND METHODS In the considered SFS structure the superconducting phase difference wand magnetization Mof the ferromagnetic (F) layer are coupled dynamical variables. The system of equations describ-ing their dynamics is obtained from the Landau-Lifshitz-Gilbert(LLG) equation, the expression for the bias current of the resistively and capacitively shunted junction (RCSJ) model, and the Josephson relation between the phase difference and voltage. Magnetization dynamics is described by the LLG equation 18 dM dt¼/C0γM/C2Heffþα M0M/C2dM dt/C18/C19 , (1) where γis the gyromagnetic ratio, αis Gilbert damping parameter, M0=\|M\|, and Heffis the effective magnetic field. Here we haveused the model developed in Refs. 9and 15, where it is assumed that the gradient of the spin-orbit potential is along the easy axis of magnetization, which is taken to be along z. In this case effective magnetic field is determined by Heff¼K M0Grsinw/C0rMy M0/C18/C19 ^yþMz M0^z/C20/C21 , (2) where Kis the anisotropic constant, G=EJ/(KV),Vis the volume of Fl a y e r , EJ=Φ0Ic/( 2π) is the Josephson energy, Φ0is the flux quantum, Icis the critical current, r=lvso/vFis parameter of spin-orbit coupling, l=4hL//C22hvF,Lis the length of F layer, hi st h ee x c h a n g ef i e l d of the F layer, the parameter vso/vFcharacterizes a relative strength of spin-orbit interaction. We note that the second term inside of the sine function, i.e., rMy/M0is the above mentioned phase shift w0. In order to describe the full dynamics of SFS structure the LLG equations should be supplemented by the equation for phase differ- encew, i.e., equation of RCSJ model for bias current and Josephson relation for voltage. According to the extended RCSJ model,15which takes into account time derivative of phase shift w0, the current flowing through the system in underdamped case is determined by I¼/C22hC 2ed2w dt2þh 2eRdw dt/C0r M0dM y dt/C20/C21 þIcsinw/C0r M0My/C18/C19 , (3) where Iis the bias current, CandRa r et h ec a p a c i t a n c ea n dt h er e s i s t - ance of the Josephson junction, respectively. The Josephson relation for voltage is given by /C22h 2edw dt¼V: (4) Using (1)–(4), we can write the system of equations in nor- malized variables, which describes the dynamics of w0junction as mx¼ωF 1þα2{/C0mymzþGrm zsin(w/C0rmy) /C0α[mxm2 zþGrm xmysin(w/C0rmy)]} : _my¼ωF 1þα2{mxmz/C0α[mym2 z/C0Gr(m2 zþm2 x)sin(w/C0rmy)]}, _mz¼ωF 1þα2{/C0Grm xsin(w/C0rmy) /C0α[Grm ymzsin(w/C0rmy)/C0mz(m2 xþm2 y)]}, (5) _V¼1 βc[I/C0Vþr_my/C0sin(w/C0rmy)], _w¼V; where mx,y,z=Mx,y,z/M0and satisfy the constraintP i¼x,y,zm2 i(t)¼1,βc¼2eIcCR2is the McCumber parameter. In order to use the same time scale in the LLG and RCSJ equations,we have normalized time to the ω /C01 e, where ωc¼2eIcR//C22h, and ωF=ΩF/ωcis the normalized frequency of ferromagnetic resonance (ΩF=γΚ/M0). Bias current is normalized to the critical current Ic and voltage V—to the Vc=IcR. The system of equations (5)isLow Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 46,000000 (2020); doi: 10.1063/10.0001716 46,000000-933 Published under license by AIP Publishing.solved numerically using the fourth-order Runge-Kutta method at fixed value of current I= 0 in time interval [0,1500] with the timestep δt= 0.005. We use the initial conditions mx=0 , my=0 mz=1 , V=0 ,w= 0, and as the results, we obtain mi(t),V(t), and w(t) as the functions of time. Then the value of bias current is increased for the current step the δI= 0.00005 and procedure is repeating. The obtained values of mx,mymz,Vandwat time t= 1500 for the current I, are used as the initial conditions for the value of I+δI. During the calculation we have increased a bias current until Imaxand then decreased to zero. In order to calculate the IV-characteristic, we average the voltage in time interval [200,1500] at each value of I. To investigate the resonance behav- ior of the system we calculate the maximal amplitude of magneticmoment in time domain ^m yat each value of the bias current I a n dp l o ti ta saf u n c t i o n ^my(I). 3. KINKS IN THE IV-CHARACTERISTICS AND THEIR ORIGIN Due to interaction between the superconducting current and the magnetization in the ferromagnetic layer, the w0Josephson junction exhibits a rich, complicated dynamics, which can be strongly anharmonic and even chaotic.9,10,15On the other hand, as has been demonstrated in Ref. 20, the precession of the magnetic moment in some current intervals along IV-characteristics may be relatively simple and harmonic.10Here we concentrate on the inter- action between the Josephson current and ferromagnetic layer mag- netization and on some of the peculiarities of the magnetizationdynamics which may be manifested in the experimentally measuredIV-characteristics of such systems. The magnetization dynamics is characterized by its maximal amplitude ^m ytaken at each value of the bias current along the IV-characteristics.InFig. 1 we present a part of IV-characteristics together with the maximal amplitude ^mywith decrease in bias current at I>Ic. All calculations in this paper were done at the following parametersof the system: G=1 , r=1 ,β c= 25, α= 0.01. Along with chaotic parts (see, particularly, the left side of the figure), reflectingcomplex magnetization precessions, we see a regular variation of ^m ywith the bias current. We note, that the positions of the pecu- liarities in the IV-characteristics coincide with the positions of the specific behavior of maximal amplitude of ^myas a function of bias current. An interesting feature of this ^my(I) dependence are the kinks shown by the arrows. To stress these kink peculiarities, we show in Fig. 2 the V-dependence of the maximal amplitude ^myalong the IV-characteristics of the w0junction at three different values of the ferromagnetic resonance frequency: ωF= 0.4, 0.5, and 0.6. In all cases we can see very clear the kinks on either side of the resonance frequency ωFin the R2and R3regions. Such a shift in the kink position indicates their relation to the ferromagnetic resonance. Akink in the region R 1is also manifested, but it has no symmetric counterpart due to the transition of the Josephson junction to thezero voltage state. One of the the main purposes of the present paper is to explain the origin of the kink. FIG. 1. Part of the IV-characteristic of the w0junction in the ferromagnetic reso- nance region ( ωF= 0.5) together with the maximal amplitude ^mywith decrease in bias current along the IV-curve. Arrows show the kinks in the ^my(I) dependence. FIG. 2. V-dependence of the maximal amplitude ^mywith a decrease of the bias current along the IV-characteristics of the w0junction in the ferromagnetic reso- nance region at different values of the resonance frequency, indicated bydashed lines.Low Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 46,000000 (2020); doi: 10.1063/10.0001716 46,000000-934 Published under license by AIP Publishing.InFig. 3 we demonstrate the magnetization trajectories in the my−mxplane and the corresponding results of FFT analysis of the temporal dependence of myin the regular region R1,a tI= 0.95 (a), (b) and I= 0.75 (c), (d), i.e., to the right and left sides of the kink, respectively. We find that the kink is the bifurcation point betweenthe two types of trajectories, i.e., as the system goes from period one to period two behavior. InFigs. 3(b) and3(d) we present results of FFT analysis of the time dependence of the magnetization component m yat different bias currents corresponding to the dynamics before and after thekink. At I = 0.75 an additional frequency f=f J/2 appears in com- parison to the case at I= 0.95, confirming the period doubling. Different types of magnetization trajectories in the my-m x, mz-m x, and mz-m yplanes, realized along the IV-characteristics were found in Ref. 10, such as “apple ”,“sickle ”,“mushroom ”, “fish”, and “moon ”, called like that for distinctness. But the kinks in^myand their origin were not discussed at that time. It was men- tioned there that the specific trajectories demonstrate a unique pos-sibility of controlling the magnetization dynamics via external biascurrent. Here we show the similarity in the appearance of the dif-ferent kinks and stress that their origin is related to the transforma- tion in the magnetization dynamics. In Fig. 4 we demonstrate the magnetization trajectories around the kink in the R 2region, which present the “apple ”type at I= 0.6 before the kink and the “mush- room ”type after kink at I= 0.555. The results of FFT analysis [see Figs. 4(b) and4(d)] show the doubling of the period of trajectories in case of the “mushroom ”. Actually, such a transformation of the “apple ”-type trajectory to the “mushroom ”type happens over a large bias current interval, and we will discuss this in the next section. In Fig. 5 we first show time dependence of the myvery close to the kink, just at current FIG. 3. Magnetization trajectories in my-m xplane for regular region R1and results of FFT analysis of the temporal dependence of my. The value of current at the corresponded points is indicated in the figures. FIG. 4. Magnetization trajectories in mz-m xplane for regular region R2and results of FFT analysis of the temporal dependence of my: (a), (b) at I= 0.6; (c), (d) at I= 0.555. FIG. 5. (a) Time dependence of the myatI= 0.5704 and I= 0.5702; (b) FFT analysis of time dependence of myatI= 0.5704; (c) The same as (b) at I= 0.5702.Low Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 46,000000 (2020); doi: 10.1063/10.0001716 46,000000-935 Published under license by AIP Publishing.step before (I= 0.5704) and after ( I= 0.5702) the kink. We see that the difference is related to the modulation of the time dependenceafter the kink. FFT analysis [ Figs. 5(b) and 5(c)] again confirms this transformation in the dynamics of the system through theappearance of the corresponding small peaks. InFig. 6 we demonstrate that the kink in the region R 3is related to the transformation of the “fish”-type trajectory, realized at I=0 . 4 4 , to the “double fish ”trajectory, at I= 0.416. Results of FFT analysis presented in Figs. 6(b) and6(d) show the doubling of the magnetiza- tion precession period in the case of “double fish ”-type trajectory. The presented results thus demonstrate that the kinks in ^my, in all three of the considered regular regions R1,R2and R3,a r e related to period doubling bifurcations of the specific precessiontrajectories. 4. TRANSFORMATION OF THE MAGNETIZATION TRAJECTORIES AND COMPOSITE DYNAMICS The application of dc voltage to the w 0junction produces current oscillations and consequently magnetic precession. As shown in Refs. 9and15, this precession may be monitored by the appearance of higher harmonics in the current-phase relation aswell as by the presence of a dc component in the superconductingcurrent. The latter increases substantially near the ferromagnetic resonance (FMR). In contrast to Konschelle and Buzdin, 9,15who stressed that the magnetic dynamics of the SFS w0junction may be quite complicated and strongly anharmonic, in Ref. 10we demon- strated that the precession of the magnetic moment in certain current intervals along IV-characteristics may be very simple and harmonic.In this section we discuss two peculiarities of the magnetiza- tion dynamics. The first peculiarity is related to the transformationsthat occur between the two types of the trajectories. We foundthat such transformations happen continuously. As we have seen above, the region R 2within bias current interval [0.54675,0.6259] demonstrates a kink behavior at I= 0.5703. Going down along IV-characteristic, we observe first the “apple ”-type trajectory, and then after the kink a “mushroom ”type, i.e., in the mz—mxplane. InFig. 7(a) we show the trajectory of the magnetization at the boundaries of the interval [0.5703,0.6259] and see that the ampli-tude of m zandmxis increased with a decrease in I. At the point where the magnetization pass the point with the mx= 0, we can see a scroll structure. On the boundary of the above mentioned two intervals, the scroll structure at I= 0.5703 is widen- ing. It is demonstrated in Fig. 7(b) , where a zoomed part of the tra- jectories for two boundary values of bias current is presented. Inboth cases we see a scroll, which reminded letter “e”. After the kink is passed, the scroll is transformed and reminds one of a reflected letter “e”now. In the current interval [0.54675,0.5703] the “apple ”- type trajectory continuously transforms to the “mushroom ”one. This transformation is demonstrated in Fig. 7(c) , where the trajec- tories for three values of current are presented. In Fig. 7(d) the zoomed part of trajectories closed to the point m x= 0 is shown [this region is marked with the dashed ellipse in Fig. 7(c) ]. As we can see, the scroll structure disappears continuously throughout thetransformation. The second peculiarity is related to the creation of the com- posite type of the trajectory. An example of such magnetization dynamics appears in the current interval [0.6268,0.6288] between FIG. 7. Transformation of the “apple ”-type trajectory to the “mushroom ”one near kink at I= 0.5703. (a) “Apple ”-type trajectory before kink; (b) Enlarged part of (a), demonstrating scroll; (c) Trajectory transformation after kink; (d) Enlargedpart of (c) near scroll. FIG. 6. Magnetization trajectories in my—mxplane for regular region R3and results of FFT analysis of temporal dependence of my. The value of current at the corresponded points is indicated in the figures.Low Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 46,000000 (2020); doi: 10.1063/10.0001716 46,000000-936 Published under license by AIP Publishing.regions R2and R3and is demonstrated in Fig. 8 . We see a realiza- tion of composite-type trajectory, i.e., different type of precessionsare realizing in during one period: “apple ”, large, small “left and right mushrooms ”. Starting from the point S, in during the first 10000 time units the magnetization describes first an “apple ”, then large “right mushroom ”[see Fig. 8(a) ], then it continues a preces- sion along small “left mushroom ”, small “right mushrooms ”and large “left mushrooms ”[see Fig. 8(b) ]. We stop recording of the time dependence close to the end of the period after 20000 timeunits (see point E). We note that the composite structures of differ-ent type trajectories may appear in different parts ofIV-characteristics by changing the system ’s parameters. 5. HYSTERESIS IN ^m y(I) DEPENDENCE Another peculiarity we found in the current dependence of the maximal magnetization of the w0junction is an appearance a sequence of hysteresis at different ferromagnetic resonance frequen-cies. In Fig. 9 we show the bias current dependence of ^m yfor three frequencies: (a) ωF= 0.5, (b) ωF= 1 and (c) ωF= 1.5. They demon- strate that the difference in the bias current dependence of ^myfor increase and decrease the bias current is appeared. We see that atω F= 0.5 the McCumber type of hysteresis known for IV-characteristics of underdamped Josephson junctions is mani- fested only, while with increase in frequency an additional hystere- sises start manifest themselves [see Fig. 9(b) ]. At ωF= 1.5 we observe a rather large and very pronounced hysteresis at I>Ic. There are two more small hysteresises which we do not mark inFigs. 9(b) and 9(c). An important point is that this peculiarity in the bias current dependence of ^m ymanifests itself in the corresponded IV-characteristics also. Below we stress such a manifestation in the theIV-curve, particularly, we show in Fig. 10 the enlarged part of IV-characteristics which demonstrate the manifestation of the start- ing and ending points of hysteresis, indicated in Fig. 9(c) . We show there the stating point [ S1,seeFig. 10(a) ] of the largest hysteresis atωF= 1.5, which is ending with a chaotic behavior at point E1 [Fig. 10(b) ]. We also demonstrate the manifestation of the ending point E2 of another big hysteresis at ωF= 1.5 in Fig. 10(c) . FIG. 8. Realization of composite type trajectory at I= 0.6268: during the first 10000 time units; (b) during the second 20000 time units. Points SandEmean starting and ending points of recording. FIG. 10. Manifestation of the hysteretic behavior in the IV- characteristic near points, marked in Fig. 9(c) . (a) Starting point of the first hysteresis; (b) Ending point of the first hysteresis; (c) Ending point of the second large hysteresis. FIG. 9. Bias current dependence of min the increase (dotted) and the decrease (solid) the bias current for: (a) ωF= 0.5, (b) ωF= 1, (c) ωF= 1.5.Low Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 46,000000 (2020); doi: 10.1063/10.0001716 46,000000-937 Published under license by AIP Publishing.The IV-characteristics are measured experimentally, so the results presented above open a way for experimental testing of the peculiarities pronounced in the magnetization dynamics of w0 Josephson junctions. The question concerning the details of the hysteresis appearance and its dependence on the parameters of thesystem will be addressed elsewhere. 6. CONCLUSIONS We have studied the interaction between the superconducting current and magnetic moment in a w 0junction and investigated the maximal amplitude of the magnetization momement ^my(I), where Iis the bias current. We found a kink behavior in the bias current (voltage) dependence of ^my(I) along the IV-characteristics. Analysis of the magnetization precession dynamics and trajectories revealed that the origin of the kinks can be related to changes in the dynamical behavior of the magnetization precession in ferro-magnetic layer. Found effects concerning the transformation of themagnetization specific trajectories along the IV-curve, magnetiza- tion composite structures, and hys- teretic behavior in the bias current dependence of ^m yopen several interesting directions for future investigations. Due to the correlations between the discov-ered features of ^m yand the IV-characteristics, the presented results open a way for the experimental testing of the peculiar magnetiza-tion dynamics which characterize the w 0junction. We note that in our model the interaction between the Josephson current and the magnetization is determined by theparameter G=E J/(KV), which describes the ratio between the Josephson energy and the magnetic anisotropy energy andspin-orbit interaction. The value of the Rashba- type spin-orbit interaction parameter in a permalloy doped with platinum 21up to 10%, in the ferromagnets without inversion symmetry, like MnSi orFeGe, usually estimated in the range 0.1 –1, the value of the Γin the material with weak magnetic anisotropy K∼4⋅10 −5K⋅A−3,23and a junction with a relatively high critical current density of (3/C1105/C05/C1106)A⋅cm−223is in the range 1 –100. It gives the set of ferromagnetic layer parameters and junction geometry that make itpossible to reach the values used in our numerical calculations, forthe possible experimental observation of the predicted effect. ACKNOWLEDGMENTS The reported study was partially funded by the RFBR research projects nos. 18-02-00318, 18-52 45011-IND. Numerical calcula- tions have been made in the framework of the RSF project no. 18-71-10095. Yu. M. S. and A. E. B. gratefully acknowledge supportfrom the University of South Africa ’s visiting researcher program and the SA-JINR collaboration. REFERENCES 1A. Buzdin, Phys. Rev. Lett. 101, 107005 (2008). 2J. Linder and J. W. A. Robinson, Nat. Phys. 11, 307 (2015). 3O. Durante, R. Citro, C. Sanz-Fernández, C. Guarcello, I. V. Tokatly, A. Braggio, M. Rocci, N. Ligato, V. Zannier, L. Sorba, F. S. Bergeret, F. Giazotto, E. Strambini, and A. Iorio, Nat. Nanotechnol. (2020). 4A. I. Buzdin, Rev. Mod. Phys. 77, 935 (2005). 5A. A. Golubov, M. Y. Kupriyanov, and E. 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Konschelle and A. Buzdin, Phys. Rev. Lett. 102, 017001 (2009); F. Konschelle and A. Buzdin, Phys. Rev. Lett. 123, 169901 (2019). 16A. A. Mazanik, I. R. Rahmonov, and A. E. Botha, and Y. M. Shukrinov, Phys. Rev. Appl. 14, 014003 (2020). 17Y. M. Shukrinov, I. R. Rahmonov, and A. E. Botha, Nanophysics and Nanoelectroncs: Proceedings of the XXVI International Symposium, Nizhniy Novgorod State University, Nizhniy Novgorod, 10 –13 March 2020 ,V o l .1 ,p .2 1 . 18E. M. Lifshitz and L. P. Pitaevskii, Course of Theoretical Physics, Theory of the Condensed State, Butterworth Heinemann, Oxford, 1991 , Vol. 9. 19D. Rabinovich, I. Bobkova, A. Bobkov, and M. Silaev, Phys. Rev. Lett. 123, 207001 (2019). 20Y. M. Shukrinov, M. Nashaat, I. R. Rahmonov, and K. Kulikov, JETP Lett. 110, 160 (2019). 21A. Hrabec, F. J. T. Gonçalves, C. S. Spencer, E. Arenholz, A. T. N ’Diaye, and R. L. Stamps, And C. H. Marrows, Phys. Rev. B 93, 014432 (2016). 22A. Y. Rusanov, M. Hesselberth, J. Aarts, and A. I. 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1.3551729.pdf	Dependence of nonlocal Gilbert damping on the ferromagnetic layer type in ferromagnet/Cu/Pt heterostructures A. Ghosh, J. F. Sierra, S. Auffret, U. Ebels, and W. E. Bailey Citation: Applied Physics Letters 98, 052508 (2011); doi: 10.1063/1.3551729 View online: http://dx.doi.org/10.1063/1.3551729 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 Thickness dependence of spin torque ferromagnetic resonance in Co75Fe25/Pt bilayer films Appl. Phys. Lett. 104, 072405 (2014); 10.1063/1.4865425 Tuning exchange bias in ferromagnetic/ferromagnetic/antiferromagnetic heterostructures [Pt/Co]/NiFe/NiO with in-plane and out-of-plane easy axes J. Appl. Phys. 109, 043906 (2011); 10.1063/1.3553414 A way to measure electron spin-flipping at ferromagnetic/nonmagnetic interfaces and application to Co/Cu Appl. Phys. Lett. 96, 022509 (2010); 10.1063/1.3292218 Spin-transfer effects in nanoscale magnetic tunnel junctions Appl. Phys. Lett. 85, 1205 (2004); 10.1063/1.1781769 Materials dependence of the spin-momentum transfer efficiency and critical current in ferromagnetic metal/Cu multilayers Appl. Phys. Lett. 83, 323 (2003); 10.1063/1.1590432 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: 130.113.86.233 On: Wed, 10 Dec 2014 01:34:30Dependence of nonlocal Gilbert damping on the ferromagnetic layer type in ferromagnet/Cu/Pt heterostructures A. Ghosh,1J. F . Sierra,1S. Auffret,1U. Ebels,1and W. E. Bailey2,a/H20850 1SPINTEC, UMR (8191) CEA/CNRS/UJF/Grenoble INP; INAC, 17 rue des Martyrs, 38054 Grenoble Cedex, France 2Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York 10027, USA /H20849Received 24 November 2010; accepted 8 January 2011; published online 2 February 2011 /H20850 We have measured the size effect in the nonlocal Gilbert relaxation rate in ferromagnet /H20849FM/H20850 /H20849tFM/H20850/Cu/H208493n m /H20850/H20851/Pt/H208492n m /H20850/H20852/Al/H208493n m /H20850heterostructures, FM= /H20853Ni81Fe19,Co 60Fe20B20,pure Co /H20854.A common behavior is observed for three FM layers where the additional relaxation obeys both a strict inverse power law dependence /H9004G=Ktn,n=−1.04 /H110060.06 and a similar magnitude K =224/H1100640 MHz·nm. As the tested FM layers span an order of magnitude in spin diffusion length /H9261SD, the results are in support of spin diffusion rather than nonlocal resistivity as the origin of the effect. © 2011 American Institute of Physics ./H20851doi:10.1063/1.3551729 /H20852 The primary materials parameter that describes the tem- poral response of magnetization M to applied ﬁelds H is theGilbert damping parameter, /H9251, or relaxation rate G=/H20841/H9253/H20841Ms/H9251. An understanding of the Gilbert relaxation, particularly instructures of reduced dimension, is an essential question foroptimizing the high speed/GHz response of nanoscale mag-netic devices. Experiments over the last decade have established that the Gilbert relaxation of ferromagnetic ultrathin ﬁlms exhib-its a size effect, some component of which is nonlocal. Both /H9251/H20849tFM/H20850=/H92510+/H9251/H11032/H20849tFM/H20850andG/H20849tFM/H20850=G0+G/H11032/H20849tFM/H20850increase sev- eral fold with decreasing ferromagnet /H20849FM/H20850ﬁlm thickness, tFM, from near-bulk values /H92510,G0fortFM/H1140720 nm. More- over, the damping size effect can have a nonlocal contribu-tion responsive to layers or scattering centers removedthrough a nonmagnetic /H20849NM/H20850layer from the precessing FM. Contributed Gilbert relaxation has been seen from other FMlayers 1as well as from heavy-element scattering layers such as Pt.2 The nonlocal damping size effect is strongly reminiscent of the electrical resistivity in ferromagnetic ultrathin ﬁlms.Electrical resistivity /H9267is size-dependent by a similar factor over a similar range of tFM; the resistivity /H9267/H20849tFM/H20850is similarly nonlocal, dependent upon layers not in direct contact.3–5It is prima facie plausible that the nonlocal damping and nonlocal electrical resistivity share a common origin in momentumscattering /H20849with relaxation time /H9270M/H20850by overlayers. If the non- local damping arises from nonlocal scattering /H9270M−1, however, there should be a marked dependence upon the FM layertype. Damping in materials with a short spin diffusion length /H9261 SDis thought to be proportional to /H9270M−1/H20849Ref. 6/H20850; the claim for “resistivity-like” damping has been made explicitly forNi 81Fe19by Ingvarsson et al.7For a FM with a long /H9261SD,o n the other hand, relaxation Gis either nearly constant with temperature or “conductivity-like,” scaling as /H9270M. Interpretation of the nonlocal damping size effect has centered instead on a spin current model8advanced by Tserk- ovnyak et al.9An explicit prediction of this model is that the magnitude of the nonlocal Gilbert relaxation rate /H9004Gis onlyweakly dependent upon the FM layer type. The effect has been calculated10as /H9004G=/H20841/H9253/H208412/H6036/4/H9266/H20849geff↑↓/S/H20850tFM−1/H208491/H20850 where the effective spin mixing conductance geff↑↓/Sis given in units of channels per area. Ab initio calculations predict a very weak materials dependence for the interfacial param-eters g ↑↓/S, with /H1100610% difference in systems as different as Fe/Au and Co/Cu and negligible dependence on interfacialmixing.11 Individual measurements exist of the spin mixing con- ductance, through the damping, in FM systems Ni 81Fe19,12 Co,13and CoFeB.14However, these experiments do not share a common methodology, which makes a numerical compari-son of the results problematic, especially given that Gilbertdamping estimates are to some extent model-dependent. 15In our experiments, we have taken care to isolate the nonlocaldamping contribution due to Pt overlayers only, controllingfor growth effects, interfacial intermixing, and inhomoge-neous losses. The only variable in our comparison of nonlo-cal damping /H9004G/H20849t FM/H20850, to the extent possible, has been the identity of the FM layer. Gilbert damping /H9251has been measured through ferromag- netic resonance /H20849FMR /H20850from/H9275/2/H9266=2–24 GHz using a broadband coplanar waveguide with broad center conductorwidth w=400 /H9262m, using ﬁeld modulation and lock-in detec- tion of the transmitted signal to enhance sensitivity. Mag-netic ﬁelds H Bare applied in the ﬁlm plane. The Gilbert damping has been separated from inhomogeneous broaden-ing in the ﬁlms measured using the well-known relation /H9004H pp/H20849/H9275/H20850=/H9004H0+/H208492//H208813/H20850/H9251/H9275//H20841/H9253/H20841. We have ﬁt spectra to Lorenzian derivatives at each frequency, for each ﬁlm, to extract the linewidth /H9004Hppand resonance ﬁeld Hres;/H9251has been extracted using linear ﬁts to /H9004H/H20849/H9275/H20850. For the ﬁlms, six series of heterostructures were deposited of the form Si /SiO 2/X/FM/H20849tFM/H20850/ Cu/H208493n m /H20850/H20851/Pt/H208493n m /H20850/H20852/Al/H208493n m /H20850,F M = /H20853Ni81Fe19/H20849“Py” /H20850, Co60Fe20B20/H20849“CoFeB” /H20850,pure Co /H20854, and tFM=2.5,3.5,6.0, 10.0,17.5,30.0 nm, for 36 heterostructures included in the study. Samples were deposited by dc magnetron sputteringon thermally oxidized Si /H20849100/H20850substrates with typical depo- a/H20850Electronic mail: web54@columbia.edu.APPLIED PHYSICS LETTERS 98, 052508 /H208492011 /H20850 0003-6951/2011/98 /H208495/H20850/052508/3/$30.00 © 2011 American Institute of Physics 98, 052508-1 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: 130.113.86.233 On: Wed, 10 Dec 2014 01:34:30sition rates of 0.5 Å/s and Ar pressures of 2.0 /H1100310−3mbars. For each ferromagnetic layer type, FM, one thickness series tFMwas deposited with the Pt overlayer and one thickness series tFMwas deposited without the Pt overlayer. This makes it possible to record the additional damping /H9004/H9251/H20849tFM/H20850 introduced by the Pt overlayer alone, independent of size effects present in the FM/Cu layers deposited below. In thecase of pure Co, a X=Ta /H208495n m /H20850/Cu/H208493n m /H20850underlayer was necessary to stabilize low-linewidth ﬁlms; otherwise, depo- sitions were carried out directly upon the in situ ion-cleaned substrate. Field-for-resonance data are presented in Fig. 1. The main panel shows /H9275/H20849HB/H20850data for Ni 81Fe19/H20849tFM/H20850. Note that there is a size effect in /H9275/H20849HB/H20850: The thinner ﬁlms have a substantially lower resonance frequency. For tFM=2.5 nm, the resonance frequency is depressed by /H110115 GHz from /H1101120 GHz resonance HB/H112294 kOe. The behavior is ﬁtted to the Kittel relation /H20849lines /H20850/H9275/H20849HB/H20850 =/H20841/H9253/H20841/H20881/H20849HB+HK/H20850/H208494/H9266Mseff+HB+HK/H20850, where HKis the effective ﬁeld from induced anisotropy, found to be /H1102110 Oe in all layers and the inset shows a summary of extracted 4/H9266Mseff/H20849tFM/H20850data for the three different FM layers. Samples with /H20849open symbols /H20850and without /H20849closed symbols /H20850Pt over- layers show negligible differences. Linear ﬁts according to 4/H9266Mseff/H20849tFM/H20850=4/H9266Ms−/H208492Ks/Ms/H20850tFM−1allow the extraction of bulk magnetization 4 /H9266Msand surface anisotropy Ks;w eﬁ n d 4/H9266MsPy=10.7 kG, 4 /H9266MsCoFeB=11.8 kG, and 4 /H9266MsCo =18.3 kG and KsPy=0.69 erg /cm2,KsCoFeB=0.69 erg /cm2, and KsCo=1.04 erg /cm2. The value of gL/2=/H20841/H9253/H20841//H20849e/mc/H20850, /H20841/H9253/H20841=2/H9266·/H208492.799 MHz /Oe/H20850·/H20849gL/2/H20850is found from the Kittel ﬁts subject to this choice, yielding gLPy=2.09, gLCoFeB=2.07, andgLCo=2.15. The 4 /H9266MsandgLvalues are taken to be size- independent and are in good agreement with bulk values:Extracted 4 /H9266Msvalues are slightly larger /H20849by 2%–9% /H20850than those measured by calibrated vibrating sample magnetom-etry in separate depositions of thick ﬁlms, and g Lvalues are typical for the literature.FMR linewidth as a function of frequency /H9004Hpp/H20849/H9275/H20850is plotted in Fig. 2. The data for Py show a near-proportionality with negligible inhomogeneous component /H9004H0/H113494O e even for the thinnest layers, facilitating the extraction of in-trinsic damping parameter /H9251. The size effect in /H9251/H20849tFM/H20850ac- counts for an increase by a factor of /H110113, from /H92510Py =0.0067 /H20849G0Py=105 MHz /H20850for the thickest ﬁlms /H20849tFM =30.0 nm /H20850to/H9251=0.021 for the thinnest ﬁlms /H20849tFM =2.5 nm /H20850. The inset shows the line shapes for ﬁlms with and without Pt, illustrating the broadening without signiﬁcant frequency shift or signiﬁcant change in peak asymmetry. A similar analysis has been carried through for CoFeB and Co /H20849not pictured /H20850. Larger inhomogeneous linewidths are observed for pure Co, but homogeneous linewidth still ex-ceeds inhomogeneous linewidth by a factor of three over thefrequency range studied, and inhomogeneous linewidthsagree within experimental error for the thinnest ﬁlms withand without Pt overlayers. We extract for these ﬁlms /H92510CoFeB=0.0065 /H20849G0CoFeB=111 MHz /H20850and/H92510Co=0.0085 /H20849G0Co =234 MHz /H20850. The latter value is in very good agreement with the average of easy- and hard-axis values for epitaxial fcc Co ﬁlms measured up to 90 GHz, G0Co=225 MHz.16 We isolate the effect of Pt overlayers on the damping size effect in Fig. 3. Values of /H9251have been ﬁtted for each deposited heterostructure: Each FM type at each tFMfor ﬁlms with and without Pt overlayers. We take the difference/H9004 /H9251/H20849tFM/H20850for identical FM /H20849tFM/H20850/Cu/H208493n m /H20850/Al/H208492n m /H20850depo- sitions with and without the insertion of Pt /H208493n m /H20850after the Cu deposition. Data, as shown on the logarithmic plot in themain panel, are found to obey a power law /H9004 /H9251/H20849tFM/H20850=Ktn with n=−1.04 /H110060.06. This is in excellent agreement with an inverse thickness dependence /H9004/H9251/H20849tFM/H20850=KFM /tFM, where the prefactor clearly depends on the FM layer, highest for Py and lowest for Co. Note that efforts to extract /H9004/H9251/H20849tFM/H20850=Ktnwith- out the FM /H20849tFM/H20850/Cu baselines would meet with signiﬁcant errors; numerical ﬁts to /H9251/H20849tFM/H20850=KtFMnfor the FM/H20849tFM/H20850/Cu /Pt structures yield exponents n/H112291.4. Expressing now the additional Gilbert relaxation as /H9004G/H20849tFM/H20850=/H20841/H9253/H20841Ms/H9004/H9251/H20849tFM/H20850=/H20841/H9253FM/H20841MsFMKFM /tFM, we plotω/2π(Ghz) H(Oe)B1 / t (nm)FM-1Py CoCoFeB FIG. 1. /H20849Color online /H20850Fields for resonance /H9275/H20849HB/H20850for in-plane FMR, FM =Ni81Fe19, 2.5 nm /H11349tFM/H1134930.0 nm; solid lines are Kittel ﬁts. Inset: 4/H9266Mseff for all three FM/Cu, with /H20849ﬁlled circles /H20850and without /H20849open squares /H20850Pt overlayers.ω/2π(Ghz)ΔH(Oe)pp H(Oe)BΔHppHres 1400 1500 1600 1700 FIG. 2. /H20849Color online /H20850Frequency-dependent peak-to-peak FMR linewidth /H9004Hpp/H20849/H9275/H20850for FM=Ni81Fe19,tFMas noted, ﬁlms with Pt overlayers. Inset: Lineshapes and ﬁts for ﬁlms with /H20849ﬁlled circles /H20850and without /H20849open squares /H20850 Pt overlayers, FM=Ni81Fe19/H20849right /H20850, CoFeB /H20849left/H20850.052508-2 Ghosh et al. Appl. Phys. Lett. 98, 052508 /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: 130.113.86.233 On: Wed, 10 Dec 2014 01:34:30/H9004G·tFM in Fig. 4. We ﬁnd /H9004G·tPy=192/H1100640 MHz, /H9004G·tCoFeB =265/H1100640 MHz, and /H9004G·tCo=216/H1100640 MHz. The similarity of values for /H9004G·tFMis in good agreement with predictions of the spin pumping model in Eq. /H208491/H20850, given that interfacial spin mixing parameters are nearly equal indifferent systems. The similarity of the /H9004G·t FMvalues for the different FM layers is, however, at odds with expectations from the“resistivity-like” mechanism. In Fig. 4,inset , we show the dependence of /H9004G·t FMupon the tabulated /H9261SDof these lay- ers from Ref. 17. It can be seen that /H9261SDCois roughly an order of magnitude longer than it is for the other two FM layers,Py and CoFeB, but the contribution of Pt overlayers todamping is very close to their average. Since under the re-sistivity mechanism, only Py and CoFeB should be suscep-tible to a resistivity contribution in /H9004 /H9251/H20849tFM/H20850, the results im- ply that the contribution of Pt to the nonlocal damping size effect has a separate origin.Finally, we compare the magnitude of the nonlocal damping size effect with that predicted by the spin pumping model in Ref. 10. According to /H9004G·tFM=/H20841/H9253/H208412/H6036/4/H9266/H20849geff↑↓/S/H20850 =25.69 MHz·nm3/H20849gL/2/H208502/H20849geff↑↓/S/H20850, our experimental /H9004G·tFM and gLdata yield effective spin mixing conductances geff↑↓/S/H20851Py /Cu /Pt/H20852=6.8 nm−2,geff↑↓/S/H20851Co /Cu /Pt/H20852=7.3 nm−2, and geff↑↓/S/H20851CoFeB /Cu /Pt/H20852=9.6 nm−2. Note that these ex- perimental values are roughly half those reported in Ref. 2 for Py/Cu/Pt. The Sharvin-corrected form in the realistic limit of /H9261SDN/H11271tN11is/H20849geff↑↓/S/H20850−1=/H20849gF/N↑↓/S/H20850−1−1 2/H20849gN,S↑↓/S/H20850−1 +2e2h−1/H9267tN+/H20849g˜N1/N2↑↓/S/H20850−1. Using ideal upper-bound interfa- cial conductances and bulk resistivities, 14.1 nm−2/H20849Co/Cu /H20850, 15.0 nm−2/H20849Cu/H20850, 211 nm−2/H20849bulk /H9267Cu,tN=3nm /H20850, and 35 nm−2/H20849Cu/Pt /H20850 would predict a theoretical geff,th.↑↓/S/H20851Co /Cu /Pt/H20852=14.1 nm−2, as reported in Ref. 2. Our results could be reconciled with the theory through the as- sumption of more resistive interfaces, plausibly reﬂective of disorder at the Cu/Pt interface /H20849e.g., g˜Cu /Pt↑↓/S/H1122910 nm−2/H20850. To summarize, a common methodology, controlling for damping size effects and intermixing in single ﬁlms, has al-lowed us to compare the nonlocal damping size effect indifferent FM layers. We observe for Cu/Pt overlayers thesame power law in thickness t −1.04/H110060.06, the same materials independence but roughly half the magnitude that is pre-dicted by the spin pumping theory of Tserkovnyak in thelimit of perfect interfaces. 10The rough independence on FM spin diffusion length, shown here for the ﬁrst time, arguesagainst a resistivity-based interpretation for the effect. We thank Y . Tserkovnyak for discussions. We would like to acknowledge the U.S. NSF-ECCS-0925829, the BourseAccueil Pro no. 2715 of the Rhône-Alpes Region, the FrenchNational Research Agency /H20849ANR /H20850Grant No. ANR-09- NANO-037, and the FP7-People-2009-IEF program no.252067. 1R. Urban, G. Woltersdorf, and B. Heinrich, Phys. Rev. Lett. 87, 217204 /H208492001 /H20850. 2S. Mizukami, Y . Ando, and T. Miyazaki, Phys. Rev. B 66, 104413 /H208492002 /H20850. 3B. Dieny, J. Nozieres, V . Speriosu, B. Gurney, and D. Wilhoit, Appl. Phys. Lett. 61, 2111 /H208491992 /H20850. 4W. H. Butler, X.-G. Zhang, D. M. C. Nicholson, T. C. Schulthess, and J. M. MacLaren, Phys. Rev. Lett. 76,3 2 1 6 /H208491996 /H20850. 5W. E. Bailey, S. X. Wang, and E. Y . Tsymbal, Phys. Rev. B 61,1 3 3 0 /H208492000 /H20850. 6V . Kamberský, Can. J. Phys. 48,2 9 0 6 /H208491970 /H20850. 7S. Ingvarsson, L. Ritchie, X. Y . Liu, G. Xiao, J. Slonczewski, P. L. Trouil- loud, and R. H. Koch, Phys. Rev. B 66, 214416 /H208492002 /H20850. 8R. H. Silsbee, A. Janossy, and P. Monod, Phys. Rev. B 19,4 3 8 2 /H208491979 /H20850. 9Y . Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 /H208492002 /H20850. 10Y . Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin, Rev. Mod. Phys. 77, 1375 /H208492005 /H20850. 11M. Zwierzycki, Y . Tserkovnyak, P. J. Kelly, A. Brataas, and G. E. W. Bauer, Phys. Rev. B 71, 064420 /H208492005 /H20850. 12S. Mizukami, Y . Ando, and T. Miyazaki, J. Magn. Magn. Mater. 239,4 2 /H208492002 /H20850;Jpn. J. Appl. Phys. 40,5 8 0 /H208492001 /H20850, and references therein. 13J.-M. L. Beaujour, W. Chen, A. Kent, and J. Sun, J. Appl. Phys. 99, 08N503 /H208492006 /H20850; J.-M. L. Beaujour, J. H. Lee, A. D. Kent, K. Krycka, and C.-C. Kao, Phys. Rev. B 74, 214405 /H208492006 /H20850, and references therein. 14H. Lee, L. Wen, M. Pathak, P. Janssen, P. LeClair, C. Alexander, C. K. A. Mewes, and T. Mewes, J. Phys. D 41, 215001 /H208492008 /H20850. 15R. D. McMichael and P. Krivosik, IEEE Trans. Magn. 40,2/H208492004 /H20850. 16F. Schreiber, J. Pﬂaum, Z. Frait, Th. Müghe, and J. Pelzl, Solid State Commun. 93, 965 /H208491995 /H20850. 17J. Bass and W. P. Pratt, J. Phys.: Condens. Matter 19, 183201 /H208492007 /H20850;C . Ahn, K.-H. Shin, and W. Pratt, Appl. Phys. Lett. 92, 102509 /H208492008 /H20850,a n d references therein.t(nm)FMt (nm)FMα 0.0010 0.00050.00500.0100 Py CoFeB Co01 02 0 3 00.0100.0150.020 Py(t )/Cu Py(t )/Cu/ PtFM FM Δα FIG. 3. /H20849Color online /H20850Inset:/H9251noPt/H20849tFM/H20850and/H9251Ptfor Py after linear ﬁts to data in Fig. 2. Main panel: /H9004/H9251/H20849tFM/H20850=/H9251Pt/H20849tFM/H20850−/H9251noPt/H20849tFM/H20850for Py, CoFeB, and Co. The slopes express the power law exponent n=−1.04 /H110060.06. x x x x FIG. 4. /H20849Color online /H20850The additional nonlocal relaxation due to Pt overlay- ers, expressed as a Gilbert relaxation rate—thickness product /H9004G·tFMfor Py, CoFeB, and Co. Inset: Dependence of /H9004G·tFMon spin diffusion length /H9261SDas tabulated in Ref. 17.052508-3 Ghosh et al. Appl. Phys. Lett. 98, 052508 /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: 130.113.86.233 On: Wed, 10 Dec 2014 01:34:30
1.4953058.pdf	Nucleation and interactions of 360# domain walls on planar ferromagnetic nanowires using circular magnetic fields F. I. Kaya , A. Sarella , D. Wang , M. Tuominen , and K. E. Aidala, Citation: AIP Advances 6, 055025 (2016); doi: 10.1063/1.4953058 View online: http://dx.doi.org/10.1063/1.4953058 View Table of Contents: http://aip.scitation.org/toc/adv/6/5 Published by the American Institute of PhysicsAIP ADV ANCES 6, 055025 (2016) Nucleation and interactions of 360◦domain walls on planar ferromagnetic nanowires using circular magnetic fields F. I. Kaya,1A. Sarella,1D. Wang,2M. Tuominen,2and K. E. Aidala1,a 1Department of Physics, Mount Holyoke College, South Hadley, MA, 01075, USA 2Department of Physics, University of Massachusetts, Amherst, MA, 01003, USA (Received 29 March 2016; accepted 19 May 2016; published online 25 May 2016) We propose a mechanism for nucleation of 360◦domain walls (DWs) on planar ferromagnetic nanowires, of 100 nm width, by using circular magnetic fields, and find the minimal spacing possible between 360◦DWs. The extent of the stray field from a 360◦DW is limited in comparison to 180◦DWs, allowing 360◦DWs to be spaced more closely without interactions than 180◦DWs, which is potentially useful for data storage devices. We use micromagnetic simulations to demonstrate the positioning of 360◦DWs, using a series of rectangular 16 ×16 nm2notches to act as local pinning sites on the nanowires. For these notches, the minimum spacing between the DWs is 240 nm, corresponding to a 360◦DW packing density of 4 DWs per micron. Understanding the topological properties of the 360◦DWs allows us to understand their formation and annihilation in the proposed geometry. Adjacent 360◦DWs have opposite circulation, and closer spacing results in the adjacent walls breaking into 180◦DWs and annihilating. C2016 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 /1.4953058] I. INTRODUCTION Manipulating magnetic domain walls (DWs) in patterned ferromagnetic nanostructures and understanding their behavior are necessary to achieve proposed logic1and data storage devices.2 Racetrack memory proposes the use of current driven transverse 180◦DWs, which interact over a range of about 2 .5µm.3,4In contrast, 360◦DWs form an almost closed flux magnetic state, substantially reducing the interaction between neighboring DWs. For this reason, 360◦DWs have been proposed to serve as bits for data storage in a magnetic racetrack device.5A 360◦DW can be viewed as consisting of two 180◦DWs, and whether bringing together two transverse 180◦DWs results in annihilation or a 360◦DW depends on the topological edge charges of the 180◦DWs.6,7 Recent studies demonstrated that spatially constrained 180◦DWs can be used as mobile interfaces that manipulate magnetic nanoparticles for biological applications.8,9The ability to split the 360◦ DWs into constituent 180◦DWs or annihilate the 360◦DWs creates novel opportunities to capture and release particles and to combine and separate particles. Current driven motion of 360◦DWs in magnetic stripes is predicted to be di fferent from that of 180◦DWs, exhibiting a qualitatively di fferent breakdown process that does not depend on the applied magnetic field, as well as oscillations in width for specific frequencies of ac current.10,11 While the response of 180◦DWs in wires has been investigated, experimental verification of current driven motion of 360◦DWs has been challenging. Reliable nucleation and manipulation mecha- nisms are needed to study the properties of the 360◦DWs and to develop devices. Most proposals involve an injection pad with a rotating in-plane field,10–14with the exception of Gonzalez Oyarce et al.15Here, we propose a versatile technique to controllably nucleate 360◦DWs at arbitrary loca- tions using a circular field centered in close proximity to a planar nanowire, allowing for the study of 360◦DWs in a wire. aElectronic mail: kaidala@mtholyoke.edu. 2158-3226/2016/6(5)/055025/6 6, 055025-1 ©Author(s) 2016. 055025-2 Kaya et al. AIP Advances 6, 055025 (2016) FIG. 1. (a) Initialization of the nanowire and top-down view of the circular field. (b) Snapshot showing the nucleation of a 360◦DW and two 180◦DWs. (c) Relaxed state of a single 360◦DW. The color scale in (b) and (c) indicate the orientation of the moments along the x-axis: Red points to the right, blue to the left. Green and blue arrows help identify the topological winding of the DWs. II. SIMULATIONS We perform micromagnetic simulations using the OOMMF16package to iteratively solve the Landau-Lifshitz-Gilbert equation. In this simulation, we limit ourselves to soft-magnetic permalloy (Ni80Fe20), which is a commonly used material in the experimental research. Other materials like polycrystalline Ni, Fe or Co are expected to behave similarly, albeit, with more pinning due to the magnetocrystalline anisotropy and higher fields required for nucleation. The typical nanowire dimen- sions used in the simulations are approximately 10 ,000×100 nm2and the material parameters are for permalloy: Ms=8×105A/m,A=1.3×10−11J/m. The cell size is 4 nm along the three axes, there is no crystalline anisotropy, the damping parameter is 0.5, and simulations are run at 0 K. The magnetization state evolves until structures reach an equilibrium state, wheredM dt<0.1 deg/ns. Figure 1(a) shows the mechanism for nucleating a 360◦DW. We initialize the nanowire by magnetizing it along the negative x-axis with a large in-plane magnetic field of 160 mTor higher. We apply a circular field, simulated as if from a current in an infinitely long wire that flows out of the page, which produces a field that decreases as 1 /r, where ris the distance from the center of the field. A current of 21 mA, which is located at a distance of r=48 nm from the nanowire along the positive y-axis, corresponds to a field of 87 .5 mT at the top edge of the wire. Such circular fields can be experimentally implemented via the tip of an Atomic Force Microscope (AFM),17 which can be positioned at arbitrary locations to follow the pattern of fields described in this paper. Alternatively, the procedure could be realized by fabricating wires above each notch and by passing current through those wires perpendicular to the plane of the nanowire. III. RESULTSANDDISCUSSION A. Nucleationof360◦DWs The circular field centered above the wire exerts a torque on the magnetic moments, nucleating a 360◦DW in the nanowire directly below the center of the circular field (Fig. 1(b)). The moments055025-3 Kaya et al. AIP Advances 6, 055025 (2016) directly below the center of the magnetic field experience the smallest torque (theoretically zero in a perfect structure at zero temperature, since they are aligned opposite to the applied field), while the other moments feel stronger torques to align with the field. Two 180◦DWs are created on either side of the 360◦DW. The simultaneous nucleation of two 180◦DWs on either side of the 360◦DW is a topological consequence, and is described in Bickel et al. for rings.18We characterize the 180◦ DWs as “up” or “down,” conveniently revealed by whether the moments at their center are pointing in positive or negative y, indicated by the green or blue arrows in Figure 1(b). We use the same terminology for the 360◦DWs, which can be “up-down” or “down-up” de- pending on the constituent 180◦DWs (read from left to right). Figure 1(b) is a snapshot in time during the nucleation of a 360◦DW, while the circular field is still applied. The half integer winding numbers of the topological edge charges6,7are indicated as well. At the nucleation of the 360◦DW, two topological defects with charge −1/2 are created on the top edge of the nanowire (Fig. 1(b)), and two +1/2 charges are created at the bottom. Two switched (red) domains appear on either side of the 360◦DW, aligning with the applied field. Two 180◦DWs must also be created (at the farther edge of the switched domain), and these carry the opposite topological charges, +1/2 on the top and−1/2 on the bottom. The total winding number of the wire is zero, as required. Given our counterclockwise (CCW) field and the resulting down-up 360◦DW, the 180◦DW that emerges to the right of the 360◦DW is an up 180◦DW, while the one to the left is a down 180◦DW. If a down 180◦DW joins with another down 180◦DW, the topological edge charges sum up to zero on the top and the bottom, hence the DWs annihilate. Similarly, the joining of two up 180◦DWs results in annihilation. When the applied circular field is removed, the wire in Figure 1 relaxes to the state shown in Figure 1(c). The 180◦DWs are pushed to the side until they encounter the end of the wire and annihilate. This is generally not the case for longer wires in which the 360◦DW slides towards one of the 180◦DWs and eventually annihilates into a single 180◦DW as a result of the summation of the topological charges. In order to pin 360◦DWs on the nanowire, a series of rectangular notches of 16 ×16 nm2area are introduced with an inter-notch distance of 280 nm (Fig. 2). Using notches to pin enables a wide range of notch size and geometry to optimize the pinning strength compared to other geometries like corners in a zig-zag shaped stripe. The length ( y-axis) of the DW is reduced at the notches, thereby reducing the energy of the DWs and facilitating pinning at the notches. Figure 2 demon- strates the sequence of steps required to generate a series of 360◦DWs with opposite circulation at adjacent notches. Once the nanowire is saturated along the negative x-axis as shown in Figure 2(a), FIG. 2. The sequence of steps for packing 360◦DWs of opposite circulation at adjacent notches. Red dotted lines indicate the center of the CCW circular field. (a) Initialization. (b) Resulting state after applying 21 mA above notch I. (c) Resulting state after applying 21 mA above notch III, creating a second 360◦DW directly below, and two 180◦DWs, one of which joins the 180◦DW at notch IIto form a 360◦DW. (d) Resulting state after applying 21 mA above notch V.055025-4 Kaya et al. AIP Advances 6, 055025 (2016) the first clockwise (CW) 360◦DW is nucleated at notch Iby passing a current of 21 mA vertically above the notch. As a result, an up 180◦DW pins at notch II, while the down 180◦DW slides to the end of the wire and annihilates due to the field gradient at the edge. The second 360◦DW at notch IIIis injected by following the same procedure. The simultaneously nucleated down 180◦DW to the left pairs with the up 180◦DW that was earlier nucleated and pinned at notch II, forming a CCW 360◦DW as shown in Figure 2(c). Similarly, the circular field is positioned at notch Vto nucleate the CW 360◦DW at Vand form the CCW 360◦DW at IV. The magnetization circulation of the packed DWs at notches ItoValternate between CW and CCW circulation, as shown in Figure 2(d). We have successfully simulated the positioning of 360◦DWs at adjacent notches with 260 nm and 240 nm inter-notch distances. As the notches are spaced more closely, the field strength is higher at notches adjacent to where the 360◦DW is nucleated. The 180◦DWs do not pin at the adja- cent notch but are instead pinned two notches away. The second nucleated 360◦DW must also be formed an additional notch away. It is straightforward to push these nucleated DWs to neighboring notches, by e ffectively unravelling the 360◦DW into two 180◦DWs with the correct strength field, and then pushing the 180◦DWs with an appropriate strength field. B. Interactionsandminimumspacingof360◦DWs The 360◦DWs interact and annihilate if the distance between adjacent notches is ≤220 nm. Figure 3 shows why the previously described procedure fails when the notches are too close together, at 220 nm. We first nucleate the down-up 360◦DW at I, creating an up 180◦DW at II. We nucleate the second 360◦DW at notch IV, as the 180◦DW at notch IIis too close to notch III and prevents the nucleation of a down 180◦DW to the left of notch III. We instead nucleate the down-up 360◦DW at notch IV, and see that the down 180◦DW moves to notch IIIand the 180◦ DWs at IIandIIIare interacting, shown in Figure 3(a). Figure 3(b) shows that by applying a CCW field at notch III, we can temporarily form a tight 360◦DW pinned at notch III. However, once the field is removed (Fig. 3(c)), the 360◦DWs at notch IIIandIVannihilate one another due to their topological charges. E ffectively, the two down constituent 180◦DWs are adjacent, attract each other, and annihilate. The two up 180◦DWs remain at notches IIIandIV. Therefore, a distance of FIG. 3. Failure mechanism when positioning opposite circulation 360◦DWs at the inter-notch distance of 220 nm. (a) 360◦ DWs are formed at notches IandIV. The 180◦DWs interact but do not come together at a single notch. (b) Temporarily applying a CCW field above notch IIpushes the two 180◦DWs into a tight 360◦DW at notch III. (c) When the field is removed, the constituent down DWs are su fficiently close to interact and annihilate, leaving the two up DWs at their respective notches.055025-5 Kaya et al. AIP Advances 6, 055025 (2016) ≤220 nm between notches prevents packing of 360◦DWs at adjacent notches on a nanowire, using this technique and geometry. The minimum spacing between 360◦DWs is determined in part by the notch size and shape. Deeper notches allow closer packing but require stronger fields to de-pin the DWs. We have suc- ceeded in simulating a dense packing of 360◦DWs at adjacent notches with 220 nm spacing by using 16 ×32 nm2rectangular notches. The procedure in this case di ffers slightly due to the stronger pinning of 180◦and 360◦DWs at deeper notches. Additionally, if we control the topology of the adjacent 360◦DWs so that they are all of the same circulation, the failure mechanism changes and we can pack the 360◦DWs more densely. This can be accomplished by annihilating the DW with the circulation that we do not want by using a strong local field above that DW. For example, a strong enough (85 mA) CCW field at notch I in Figure 3(a) annihilates the 360◦DW pinned at I. We can then shift the other 360◦DWs by unravelling them into two 180◦DWs and pushing the 180◦DWs. For 16 ×64 nm2rectangular notches, we can successfully pack 360◦DWs with the opposite circulation at 180 nm spacing between the notches. More work remains to be done to better understand the e ffects of the geometry of the notches and the circulation of adjacent 360◦DWs and their e ffects on packing density.19 The proposed mechanism to nucleate 360◦DWs using the tip of an AFM provides significant flexibility in studying the behavior of 360◦DWs. An alternative method would be to use prefabri- cated wires perpendicular to the plane of the ferromagnetic wire, positioned above each notch where we center the circular field in our simulations. Conceivably, the presence or absence of the 360◦DW could be used as the bit, or possibly the circulation of the 360◦DW. Geometry would be optimized to reduce the current density and power consumption while maintaining a close packing density, and we do not anticipate quantitative agreement between the predicted fields to nucleate the 360◦ DWs as the simulations were performed at 0 K. The readout might be similar to racetrack mem- ory,3,4requiring a spin-polarized current to move the 360◦DWs. Generally, there will be a trade-o ff between strong pinning providing closer packing and weak pinning requiring smaller fields and current to move the DWs. IV. CONCLUSIONS In summary, we propose a mechanism to nucleate 360◦DWs at arbitrary locations determined by notches along an in-plane ferromagnetic nanowire. A circular field that decreases as 1 /rand is centered directly above a notch along the y-axis will nucleate one 360◦DW and two 180◦DWs at that notch. Careful consideration of the series of circular fields allows us to nucleate 360◦DWs with opposite circulation at adjacent notches as close as 240 nm, providing a packing density of about four DWs per micron in the permalloy nanowire simulated with 16 ×16 nm2rectangular notches. ACKNOWLEDGEMENTS The authors acknowledge the support by NSF grants No. DMR 1208042 and 1207924. Simu- lations were performed with the computing facilities provided by the Center for Nanoscale Systems (CNS) at Harvard University (NSF award ECS-0335765), a member of the National Nanotechnol- ogy Infrastructure Network (NNIN). 1G. Hrkac, J. Dean, and D. A. Allwood, Philos. Trans. R. Soc. A 369, 3214–3228 (2011). 2S. S. P. Parkin, M. Haysahi, and L. Thomas, Science 320, 190 (2008). 3M. Haysahi, L. Thomas, R. Moriya, C. Rettner, and S. Parkin, Science 320(2008). 4L. Thomas, M. Hayashi, R. Moriya, C. Rettner, and S. Parkin, Nat. Commun. 3, 810 (2012). 5A. L. G. Oyarce, Y . Nakatani, and C. H. W. Barnes, Phys. Rev. B 87, 214403 (2013). 6A. Pushp, T. Phung, C. Rettner, B. P. Hughes, S.-H. Yang, L. Thomas, and S. S. P. Parkin, Nature Phys. 9, 505 (2013). 7A. Kunz, Appl. Phys. Lett. 94, 132502 (2009). 8M. Donolato, P. Vavassori, M. Gobbi, M. Deryabina, M. F. Hansen, V . Metlushko, B. Ilic, M. Cantoni, D. Petti, S. Brivio, and R. Bertacco, Adv. Mater. 22, 2706 (2010). 9A. Sarella, A. Torti, M. Donolato, M. Pancaldi, and P. Vavassori, Adv. Mater. 26, 2384–2390 (2014). 10M. Diegel, E. Mattheis, and R. Halder, IEEE Trans. Magn. 40, 2655–2657 (2004).055025-6 Kaya et al. AIP Advances 6, 055025 (2016) 11M. D. Mascaro and C. A. Ross, Phys. Rev. B 82, 214411 (2010). 12Y . Jang, S. R. Bowden, M. Mascaro, J. Unguris, and C. A. Ross, Appl. Phys. Lett. 100, 062407 (2012). 13L. D. Geng and Y . M. Jin, J. Appl. Phys. 112, 083903 (2012). 14T.-C. Chen, C.-Y . Kuo, A. K. Mishra, B. Das, and J.-C. Wu, Phys. B Condens. Matter 476, 1 (2015). 15A. L. Gonzalez Oyarce, J. Llandro, and C. H. W. Barnes, Appl.Phys. Lett. 103, 222404 (2013). 16M. J. Donahue and D. G. Porter, OOMMF users Guide, Version 1.0, Interagency Report NISTIR 6376 (National Institute of Standards and Technology, 1999). 17T. Yang, N. R. Pradhan, A. Goldman, A. S. Licht, Y . Li, M. Kemei, M. T. Tuominen, and K. E. Aidala, Appl. Phys. Lett. 98, 242505 (2011). 18J. E. Bickel, S. A. Smith, and K. E. Aidala, J. Appl. Phys. 115, 17D135 (2014). 19F. I. Kaya, A. Sarella, K. E. Aidala, D. Wang, and M. Tuominen, AIP Advances 6, 056408 (2016).
1.2749469.pdf	Magnetic anisotropies in ultrathin iron films grown on the surface-reconstructed GaAs substrate B. Akta, B. Heinrich, G. Woltersdorf, R. Urban, L. R. Tagirov, F. Yldz, K. Özdoan, M. Özdemir, O. Yalçin, and B. Z. Rameev Citation: Journal of Applied Physics 102, 013912 (2007); doi: 10.1063/1.2749469 View online: http://dx.doi.org/10.1063/1.2749469 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/102/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Interface magnetism of iron grown on sulfur and hydrogen passivated GaAs(001) J. Appl. Phys. 111, 07C115 (2012); 10.1063/1.3677822 Determination of magnetic anisotropies in ultrathin iron films on vicinal Si(111) substrate by the ferromagnetic resonance Appl. Phys. Lett. 96, 142511 (2010); 10.1063/1.3396077 A study on ferromagnetic resonance linewidth of single crystalline ultrathin Fe film grown on GaAs substrate J. Appl. Phys. 101, 09D120 (2007); 10.1063/1.2712297 Interface atomic structure and magnetic anisotropy in ultrathin Fe films grown by thermal deposition and pulsed laser deposition on GaAs(001) J. Appl. Phys. 101, 09D110 (2007); 10.1063/1.2711071 Anisotropy of epitaxial Fe films grown on n-type GaAs by electrodeposition J. Appl. Phys. 95, 6546 (2004); 10.1063/1.1667434 [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: 84.117.74.208 On: Sun, 13 Apr 2014 12:58:28Magnetic anisotropies in ultrathin iron ﬁlms grown on the surface-reconstructed GaAs substrate B. Akta ş Gebze Institute of Technology, 41400 Gebze-Kocaeli, Turkey B. Heinrich,a/H20850G. Woltersdorf, and R. Urban Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada L. R. Tagirov Gebze Institute of Technology, 41400 Gebze-Kocaeli, Turkey and Kazan State University, 420008 Kazan, Russian Federation F. Yıldız, K. Özdo ğan, M. Özdemir, O. Yalçin, and B. Z. Rameev Gebze Institute of Technology, 41400 Gebze-Kocaeli, Turkey /H20849Received 5 March 2007; accepted 10 May 2007; published online 11 July 2007 /H20850 Magnetic anisotropies of epitaxial ultrathin iron ﬁlms grown on the surface-reconstructed GaAs substrate were studied. Ferromagnetic resonance technique was exploited to determine magneticparameters of the ﬁlms in the temperature range of 4–300 K. Extraordinary angular dependence ofthe FMR spectra was explained by the presence of fourfold and twofold in-plane anisotropies. A strong in-plane uniaxial anisotropy with magnetic hard axis along the /H2085111 ¯0/H20852crystallographic direction is present at the GaAs/Fe /H20849001 /H20850interface while a weak in-plane uniaxial anisotropy for the Fe grown on Au has its easy axis oriented along /H2085111¯0/H20852. A linear dependence of the magnetic anisotropies as a function of temperature suggests that the strength of the in-plane uniaxial anisotropy is affected by the magnetoelastic anisotropies and differential thermal expansion ofcontacting materials. © 2007 American Institute of Physics ./H20851DOI: 10.1063/1.2749469 /H20852 I. INTRODUCTION The interest in ultrathin magnetic multilayers has been steadily increasing since they are building blocks in spintron- ics applications such as data storage devices and magneticrandom access memories. The magnetic anisotropies of thinﬁlms are of crucial importance in understanding the physicsof magnetic nanostructures. Ferromagnetic resonance /H20849FMR /H20850 is a very accurate and straightforward technique, allowingone to determine magnetic anisotropy ﬁelds of ultrathin mag-netic ﬁlms. 1,2In this paper, we study the magnetic anisotro- pies in single GaAs/15Fe/20Au /H20849001 /H20850, GaAs/15Fe/20Cr /H20849001 /H20850, GaAs/16Fe/9Pd/20Au /H20849001 /H20850, and double GaAs/15Fe/Au/ 40Fe/20Au /H20849001 /H20850iron layer structures grown on the surface- reconstructed GaAs /H20849001 /H20850single-crystalline substrate wafers. The integers represent the number of atomic layers. It will beshown that the interface-induced anisotropies can be used totailor the overall magnetic properties of ultrathin ﬁlm struc-tures. In our FMR experiments, we observed unconventionaltriple-mode FMR spectra allowing one to discriminate be-tween various in-plane magnetic anisotropies. Computer ﬁt-ting of the angular and frequency dependent FMR spectra inthe temperature range of 4–300 K allowed us to determinethe cubic, uniaxial, and perpendicular components of themagnetic anisotropies and establish directions of the easyand hard axes in the individual layer /H20849s/H20850. The magnetic anisotropies are discussed in terms of the interface and bulkanisotropies including magnetoelastic energy arising due tothe lattice mismatch and differential thermal expansion of the metallic materials employed in these structures. II. SAMPLE PREPARATION Single 20Au/15/Fe/GaAs /H20849001 /H20850, 30Au/15Fe/ GaAs /H20849001 /H20850, 20Cr/15Fe/GaAs /H20849001 /H20850, 20Au/9Pd/16Fe/ GaAs /H20849001 /H20850, and double 20Au/40Fe/40Au/15Fe/GaAs /H20849001 /H20850 iron layer ultrathin ﬁlm structures were prepared by molecu- lar beam epitaxy /H20849MBE /H20850on/H208494/H110036/H20850reconstructed GaAs /H20849001 /H20850 substrates. The integers represent the number of atomic lay- ers. A brief description of the sample preparation procedureis as follows. The GaAs /H20849001 /H20850single-crystalline wafers were sputtered under grazing incidence using 600 eV argon-iongun to remove native oxides and carbon contaminations.Substrates were rotated around their normal during sputter-ing. After sputtering the GaAs substrates were annealed atapproximately 580–600 °C and monitored by means of re-ﬂection high energy electron diffraction /H20849RHEED /H20850until a well-ordered /H208494/H110036/H20850reconstruction appeared. 3The /H208494/H110036/H20850 reconstruction consists of /H208491/H110036/H20850and /H208494/H110032/H20850domains: the /H208491/H110036/H20850domain is As-rich, while the /H208494/H110032/H20850domain is Ga rich. The Fe ﬁlms were deposited directly on the GaAs /H20849001 /H20850 substrate at room temperature from a resistively heated pieceof Fe at the base pressure of 1 /H1100310 −10Torr. The ﬁlm thick- ness was monitored by a quartz crystal microbalance and bymeans of RHEED intensity oscillations. The deposition ratewas adjusted at about 1 ML /H20849monolayer /H20850/min. The gold layer was evaporated at room temperature at the deposition rate ofabout 1 ML/min. The RHEED oscillations were visible for a/H20850Electronic mail: bheinric@sfu.caJOURNAL OF APPLIED PHYSICS 102, 013912 /H208492007 /H20850 0021-8979/2007/102 /H208491/H20850/013912/8/$23.00 © 2007 American Institute of Physics 102, 013912-1 [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: 84.117.74.208 On: Sun, 13 Apr 2014 12:58:28up to 30 atomic layers. Films under study were covered by a 20-ML-thick Au /H20849001 /H20850or Cr cap layer for protection in am- bient conditions. More details of the sample preparation aregiven in Ref. 3. III. MODEL AND BASIC FORMULAS FOR FERROMAGNETIC RESONANCE The FMR data are analyzed using the free energy expan- sion similar to that employed in Ref. 4, ET=−M·H+/H208492/H9266M02−Kp/H20850/H925132+K1/H20849/H925112/H925122+/H925122/H925132 +/H925132/H925112/H20850+Kucos2/H20849/H9272−/H9272/H2085111¯0/H20852/H20850. /H208491/H20850 The ﬁrst term is the Zeeman energy in the external dc mag- netic ﬁeld, the second term is the demagnetization energyterm including the effective perpendicular uniaxial aniso-tropy, the third term is the cubic anisotropy energy, and thelast term is the in-plane uniaxial anisotropy energy with the symmetry axis along the /H2085111 ¯0/H20852crystallographic direction.5 /H9251i’s represent the directional cosines6of the magnetization vector Mwith respect to the crystallographic axes /H20849/H20851100 /H20852, /H20851010 /H20852and /H20851001 /H20852/H20850of the Fe /H20849001 /H20850ﬁlm /H20851or GaAs /H20849001 /H20850sub- strate /H20852, and M0is the saturation magnetization. The relative orientation of the reference axes, sample sketch, and variousvectors relevant in the problem are given in Fig. 1. The FMR condition is obtained by using a well known equation, 7 /H20873/H92750 /H9253/H208742 =/H208731 M0/H115092ET /H11509/H92582/H20874/H208731 M0sin2/H9258/H115092ET /H11509/H92722/H20874 −/H208731 M0sin/H9258/H115092ET /H11509/H9272/H11509/H9258/H208742 , /H208492/H20850 where /H92750=2/H9266/H9263is the circular frequency /H20849determined by the operating frequency /H9263of the ESR spectrometer /H20850,/H9253is the gyromagnetic ratio, and /H9258and/H9272are the polar and azimuthal angles of the magnetization vector Mwith respect to thereference axes. The absorbed magnetic energy is caused by the Gilbert damping and is proportional to the out-of-phase rfsusceptibility. 1Standing spin-wave excitations in our ﬁlms were not considered because the ﬁlm thickness is too small/H20849/H1101120–80 Å /H20850. The strength of magnetic anisotropies is ob- tained by computer ﬁtting of the experimental data using Eq. /H208492/H20850. In the in-plane FMR studies, the polar /H9258and/H9258Hangles were ﬁxed at /H9258,/H9258H=/H9266/2. The azimuthal angle of magnetiza- tion/H9272was obtained from the static equilibrium condition for the given angle /H9272Hof the external magnetic ﬁeld. The angle /H9272Hwas varied from zero to /H9266. Then, the set of equations for the in-plane geometry reads Hsin/H20849/H9272−/H9272H/H20850+1 2H1sin 4/H9272−Husin 2 /H20849/H9272−/H9272/H2085111¯0/H20852/H20850=0 , /H20873/H92750 /H9253/H208742 =/H20875Hcos/H20849/H9272−/H9272H/H20850+4/H9266Meff+1 2H1/H208493 + cos 4 /H9272/H20850 −2Hucos2/H20849/H9272−/H9272/H2085111¯0/H20852/H20850/H20876/H20851Hcos/H20849/H9272−/H9272H/H20850 +2H1cos 4/H9272−2Hucos 2 /H20849/H9272−/H9272/H2085111¯0/H20852/H20850/H20852. /H208493/H20850 The effective magnetization Meffincludes contribution from the perpendicular anisotropy: 2 /H9266Meff=2/H9266M0−Kp/M0. The anisotropy ﬁelds are deﬁned as follows: H1=K1/M0,Hu =Ku/M0. The angle /H9272/H2085111¯0/H20852=/H9266/4 is the angle between the easy direction of the cubic and hard axes of the uniaxial anisotropy. For the out-of-plane FMR, the azimuthal angle /H9272His ﬁxed either at /H9272H=3/H9266/4/H20851easy axis, i.e., the dc magnetic ﬁeld was rotated in the /H2084911¯0/H20850plane /H20852or/H9272H=/H9266/4/H20851hard axis, i.e., the dc magnetic ﬁeld was rotated in the /H20849110 /H20850plane /H20852, while the polar angle /H9258Hwas varied from zero to /H9266/2. The polar and azimuthal angles of the magnetization were ob-tained from the static equilibrium condition corresponding tothe minimum free energy of the system. The set of equationsfor the out-of-plane measurements from the easy axis direc-tion reads Hsin/H20849 /H9258−/H9258H/H20850−2/H9266Meffsin 2/H9258 +1 4H1sin 2/H9258/H208493 cos 2 /H9258+1/H20850=0 , /H20873/H92750 /H9253/H208742 =/H20875Hcos/H20849/H9258−/H9258H/H20850−4/H9266Meffcos 2/H9258 +1 2H1/H20849cos 2/H9258+ 3 cos 4 /H9258/H20850/H20876 /H11003/H20875Hcos/H20849/H9258−/H9258H/H20850−4/H9266Meffcos2/H9258 +1 4H1/H208498 cos 2 /H9258− 3 sin22/H9258/H20850+2Hu/H20876. /H208494/H20850 FIG. 1. The sketch of the samples studied in the paper.013912-2 Akta şet al. J. Appl. Phys. 102, 013912 /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: 84.117.74.208 On: Sun, 13 Apr 2014 12:58:28IV. EXPERIMENTAL RESULTS A. General measurement procedure FMR measurements were carried out using a commer- cial Bruker EMX X-band ESR spectrometer equipped by an electromagnet which provides a dc magnetic ﬁeld up to22 kG in the horizontal plane. The FMR measurements werecarried out in the range of 9.5 GHz. A small amplitudemodulation of the dc ﬁeld is employed to increase the signalto noise ratio. The ﬁeld-derivative absorption signal was re-corded in the temperature range of 4–300 K. An OxfordInstruments continuous helium-gas ﬂow cryostat was usedfor cooling. The temperature was controlled by a commercialLakeShore 340 temperature-control system. A goniometerwas used to rotate the sample around the sample holder inthe cryostat tube. The sample holder was perpendicular to thedc magnetic ﬁeld and parallel to the microwave magneticﬁeld. The samples were placed on the sample holder in twodifferent conﬁgurations. For the in-plane angular studies, theﬁlm was attached horizontally at the bottom edge of thesample holder. During rotation, the normal to the ﬁlm planeremained parallel to the microwave ﬁeld, but the external dcmagnetic pointed along different directions with respect tothe sample axes. This geometry is not conventional and givesan admixture of the in- and out-of-phase rf susceptibilitycomponents. Symmetric FMR peaks were obtained in con-ventional geometry in which the dc and microwave magneticﬁelds are always in the ﬁlm plane. Some FMR measurementswere done in conventional geometry which have shown thatthe FMR ﬁelds obtained in unconventional geometry do notdiffer from those obtained in conventional geometry. For theout-of-plane FMR measurements, the samples were attachedto a ﬂat platform which was cut with the normal perpendicu-lar to the sample holder. Upon rotation of the sample holder,the microwave component of the ﬁeld remained always inthe sample plane, whereas the dc ﬁeld was rotated from thesample plane toward the ﬁlm normal. B. FMR in the single ferromagnetic layer samples 1. In-plane FMR measurements For sample 20Au/15Fe/GaAs /H20849001 /H20850, Fig. 2/H20849a/H20850illustrates the temperature dependence of the in-plane FMR spectra for the dc magnetic ﬁeld H/H20648/H20851110 /H20852. A single and relatively nar- row FMR signal was observed at very low magnetic ﬁelds in the entire temperature range. Starting from 300 K, the reso-nance ﬁeld steadily shifted from /H11011320 down to about 150 G at 5 K. The FMR linewidth increased with decreasing tem-perature. Contrarily, the measurements along the /H2085111 ¯0/H20852direction have shown that the in-plane FMR spectrum unexpectedly consists of three signals /H20851labeled by P1,P2, and P3, Fig. 2/H20849b/H20850/H20852. As far as we know, it is a unique observation. Usually, a single resonance or two resonance peaks are expected fromultrathin-ﬁlm /H2084915 ML /H20850ferromagnetic layers /H20849see, for ex- ample, Refs. 8–11and also in Ref. 2, Figs. 2 and 3 /H20850. Spin- wave modes in ultrathin ﬁlms are not observable in this fre-quency range. The temperature evolution of the FMR spectrum along the /H2085111 ¯0/H20852direction is shown in Fig. 2/H20849b/H20850.These three peaks were present in the entire temperature range. The high-ﬁeld signal has largest intensity at all tem-peratures. At room temperature /H20849RT/H20850, the two low-ﬁeld peaks overlapped and merged into the single, somewhat distortedFMR line. With decreasing temperature, the low-ﬁeld signalseparated into two signals, see Fig. 2/H20849b/H20850. The high-ﬁeld sig- nal shifted gradually to higher ﬁelds upon lowering the tem-perature. At T=4–5 K, the splitting of the FMR peaks reached /H110111700 G. Notice that the high-ﬁeld mode for H /H20648/H2085111¯0/H20852shifted in the opposite direction to the spectrum in theH/H20648/H20851110 /H20852direction. This suggests that the easy magnetic axis is along the /H20851110 /H20852crystallographic direction, and /H2085111¯0/H20852 is the hard magnetic axis. The detailed study of magnetic anisotropies was carried out by rotating the dc magnetic ﬁeld in the plane of the ﬁlm.The angular dependence of FMR at RT is shown in Fig. 3. The number of absorption peaks was clearly varied with thein-plane angle of the dc ﬁeld. The intensity of the FMRsignals was also angular dependent. The overall angular pe-riodicity is 180°. This implies that the sample has at least uniaxial in-plane symmetry. The unusual three-componentFMR spectra require an additional anisotropy. It will beshown that the cubic anisotropy of Fe was needed to obtainthe observed three-peak FMR spectra. The magneticanisotropies obtained by FMR are sometimes frequencydependent. 12In order to check this point, the FMR measure- ments were also carried out in the frequency range of9–36 GHz at RT using our high-frequency extension mod-ules. The right-hand-side inset in Fig. 3shows the angular variation of the in-plane resonance ﬁeld measured at24 GHz. The left-hand-side inset shows the FMR ﬁeld as afunction of microwave frequency. A nearly parabolic depen-dence on the microwave frequency clearly indicates that theperpendicular anisotropy ﬁeld /H208494 /H9266Meff/H20850is larger than the in- ternal anisotropy ﬁelds in this frequency region. The aniso- tropy ﬁelds were found independent of the microwave fre-quency. Computer ﬁtting of the FMR data for the 20Au/15Fe/GaAs /H20849001 /H20850allows one to determine the in-plane magnetic anisotropies. The results of this ﬁtting are dis- FIG. 2. Temperature dependence of the in-plane FMR spectra taken for H/H20648/H20851110 /H20852/H20849a/H20850andH/H20648/H2085111¯0/H20852/H20849b/H20850. Sample is 20Au/15Fe/GaAs /H20849001 /H20850.013912-3 Akta şet al. J. Appl. Phys. 102, 013912 /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: 84.117.74.208 On: Sun, 13 Apr 2014 12:58:28played in Fig. 3. The calculated resonance ﬁelds H /H11013Hin planeresare shown in solid circles and the measured FMR ﬁelds are represented by open symbols. The ﬁtting param- eters are given inside the ﬁgure. The in-plane angular depen-dence of the resonance ﬁeld was ﬁtted simultaneously withthe out-of-plane FMR measurements /H20849see next subsection /H20850. The observed three FMR peaks in the vicinity of the magnetic hard axis are the consequence of competition be-tween the cubic and uniaxial anisotropies with different di-rections of the easy axis. In zero dc ﬁeld, the magnetizationis along the axis corresponding to the minimum of energy.For /H20841K u/H20841/H11022K1, the easy axis is along the /H20851110 /H20852direction. By applying an external ﬁeld along the hard axis /H2085111¯0/H20852, the mag- netization started to rotate toward the closest /H20851100 /H20852axis /H20849the easy axis of the fourfold anisotropy /H20850and consequently the uniaxial anisotropy decreased its energy, but the fourfold an-isotropy and Zeeman energies got their contributions in-creased. The presence of three resonant peaks along the hardmagnetic axis indicates that the competition between theuniaxial and fourfold anisotropies ﬁrst results in an increaseof the precessional frequency of FMR with increasing ap-plied ﬁeld, but the precessional frequency eventually reachesa maximum, /H9275res, and decreases when the magnetization is gradually rotated to the hard axis, see Fig. 4. When the mag- netic moment is eventually aligned along the hard axis, thenat that point the internal ﬁeld is zero and consequently theprecessional frequency is zero. The system becomes mag-netically soft. The initial increase in the precessional fre-quency with increasing ﬁeld is obvious from the right insetof Fig. 3. The resonant ﬁeld corresponding to the magnetic moment oriented along /H20851110 /H20852is higher than that required for the magnetic moment oriented along the /H20851100 /H20852axis. This means that the precessional frequency along the easy axis FIG. 3. In-plane angular dependence of FMR spectra for the 20Au/15Fe/GaAs /H20849001 /H20850sample at room tem- perature and /H9263=9.497 GHz. Here and in Figs. 5and7, the zero of the in-plane /H9272angle is shifted to the /H2085101¯0/H20852 axis to bring the /H2085111¯0/H20852hard-axis feature to the middle of the ﬁgure. Notice that the three FMR peaks are re-solved for a narrow range of angles around the hardmagnetic axis. Right-hand-side inset, the same angulardependence for 24 GHz, and left-hand-side inset, thefrequency dependence of the resonance ﬁeld measuredfor the easy /H20849lower /H20850and hard /H20849upper /H20850directions, respectively. FIG. 4. Dependence of the magnetization angle on the applied magneticﬁeld for the 20Au/15Fe/GaAs /H20849001 /H20850sample at room temperature and /H9263 =9.497 GHz. The curve was drawn for the in-plane angle for the magnetic ﬁeld/H9272H=45°. The curve labeled by “1” is the magnetization angle with respect to the easy axis /H20851100 /H20852/H20849corresponds to /H9272=135° /H20850, while the curve labeled by “2” is the angle of magnetization with respect to the magneticﬁeld applied at /H9272H=45°.013912-4 Akta şet al. J. Appl. Phys. 102, 013912 /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: 84.117.74.208 On: Sun, 13 Apr 2014 12:58:28/H20851110 /H20852is lower than that along the /H20851100 /H20852direction. In a narrow angle range, three resonant peaks can be observed, see Fig. 3 along the vertical dash line. The three peaks were observedwhen the angle between the magnetization and the /H20851100 /H20852axis was in the range of 105°–60°. The third peak, correspondingto the critical microwave resonant frequency, disappearedwhen the angle of the magnetization was very close to 90°that corresponds to the cubic axis located between the easy /H20851110 /H20852and hard /H2085111 ¯0/H20852magnetic axes, see Fig. 4. 2. Out-of-plane FMR measurements We have also made complementary out-of-plane FMR measurements when the dc magnetic ﬁeld was rotated from the hard /H2085111¯0/H20852axis in the ﬁlm plane toward the normal di- rection to the ﬁlm plane. The angular dependence of the FMR ﬁeld as a function of the polar angle is shown in Fig. 5 by open squares and triangles. As expected, double-peakFMR spectra are observed for this geometry. The separationbetween the two modes steadily increases with approachingthe ﬁlm normal. The simultaneous ﬁtting of the in- and out-of-plane angular dependencies of the FMR spectra allow oneto determine precisely the strength of crystalline anisotropiesandgfactor. The results of the ﬁtting using Eqs. /H208494/H20850for the out-of- plane FMR measurements of the 20Au/15Fe/GaAs /H20849001 /H20850 sample are plotted in Fig. 5in solid circles. The ﬁtting pa- rameters are given in the ﬁgure. The out-of-plane FMR al-lows one to determine the gfactor. The calculations were done using g=2.09 and led to a fairly good agreement with the measurements at all angles in the in-plane and out-of-plane geometries, and at all temperatures and frequencies. The temperature dependence of the magnetic anisotropies forthe 20Au/15Fe/GaAs /H20849100 /H20850and 30Au/15Fe/GaAs /H20849001 /H20850 samples is shown in Fig. 9/H20849a/H20850. 3. Inﬂuence of the cap layer material For the samples 20Cr/15Fe/GaAs /H20849001 /H20850and 20Au/9Pd/16Fe/GaAs /H20849001 /H20850, the in-plane geometry mea- surements have shown a drastic decrease of the uniaxial component in the in-plane anisotropy, see Fig. 6/H20849a/H20850. A near absence of the in-plane uniaxial anisotropy in these measure-ments indicates that the Cr cap layer results in an almostcomplete canceling of the uniaxial anisotropy induced by theGaAs substrate. At the same time the principal axis of theresidual uniaxial anisotropy is rotated about 23° away from the original /H2085111 ¯0/H20852direction corresponding to the GaAs/15Fe/Au /H20849001 /H20850sample. This indicates that the pres- ence of Cr overlayer resulted in two weak uniaxial in-plane anisotropies oriented along the /H20851100 /H20852and /H20851110 /H20852crystallo- graphic directions. Another sample with a composite caplayer, 20Au/9Pd/16Fe/GaAs /H20849001 /H20850, revealed only a minor inﬂuence of the palladium interlayer on the magnetic aniso- tropy of the iron ﬁlm /H20851see Fig. 6/H20849b/H20850and compare with Fig. 3/H20852. However, the Fe ﬁlm in 30Au/15Fe/GaAs /H20849001 /H20850had a no- ticeably lower in-plane uniaxial anisotropy ﬁeld than that in 20Au/15Fe/GaAs /H20849001 /H20850, showing again that the in-plane uniaxial anisotropy is dependent on the thickness of capping layer and is a complex property of the entire structure. C. FMR in the double-layer samples After measuring FMR in the single iron layers, we stud- ied the spin-valve type, double-layer structures. Figure 7 shows temperature evolution of the in-plane FMR spectra forthe 20Au/40Fe/40Au/15Fe/GaAs /H20849001 /H20850sample. The data from the single-layer sample, 20Au/15Fe/GaAs /H20849001 /H20850, were used as a reference to identify the origin of the individual FMR peaks. The spectra in Fig. 7/H20849a/H20850have been recorded along the easy /H20851110 /H20852axis for the ﬁrst, 15-ML-thick iron layer, and the spectra in Fig. 7/H20849b/H20850have been recorded along FIG. 5. Out-of-plane angular dependence of the resonance ﬁeld for the GaAs/15Fe/20Au /H20849001 /H20850sample. The dc ﬁeld was applied in the /H20849110 /H20850plane. The measurements were carried out at room temperature /H20849RT/H20850. FIG. 6. Inﬂuence of the cap layer. The in-plane angular dependence of the resonance ﬁeld: /H20849a/H20850chromium cap layer /H20849/H9263=9.487 GHz /H20850and /H20849b/H20850composite Pd/Au cap layer /H20849/H9263=9.510 GHz /H20850. The angle /H9254in the inset /H20849a/H20850is measured between the residual uniaxial anisotropy hard axis and the /H20851100 /H20852axis. Mea- surements were carried out at RT.013912-5 Akta şet al. J. Appl. Phys. 102, 013912 /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: 84.117.74.208 On: Sun, 13 Apr 2014 12:58:28the hard axis /H2085111¯0/H20852of the ﬁrst layer. The reference spectra from the 20Au/15Fe/GaAs /H20849001 /H20850sample can be found in Figs. 2/H20849a/H20850and2/H20849b/H20850, respectively. In the measured temperature range from 5 to 291 K, three FMR absorption peaks were present for the spectrarecorded at the /H20851110 /H20852direction /H20851Fig. 7/H20849a/H20850/H20852.A sw ed on o t expect any marked interlayer exchange coupling or magne-tostatic interaction through the 40-ML-thick gold layer, thecontribution of the ﬁrst, 15-ML-thick iron layer to the mul-ticomponent FMR spectra can be easily identiﬁed /H20851see label- ing in Figure 7/H20849a/H20850/H20852by comparison with the measurements on the single-layer sample, Fig. 2/H20849a/H20850. A double peak spectra /H20849at higher magnetic ﬁelds /H20850from the 40-ML-thick layer allow one to conclude that the /H20851110 /H20852direction is a hard magnetic axis for this layer. A partly developed third peak at verysmall ﬁelds is observed only close to room temperature. Onecan see only its tail, and that was in agreement with thecalculated FMR peaks, see Fig. 8.The ﬁtting of the full angular dependence of FMR of the second 40 ML layer revealed that the hard axis of theuniaxial anisotropy term in the second, 40-ML-thick ironlayer, is switched 90° with respect to the hard axis of that inthe ﬁrst, 15-ML-thick iron layer. The ﬁtting parameters givenin Fig. 8have shown that the in-plane uniaxial anisotropy of the 40-ML-thick layer is drastically reduced and has the op-posite sign compared with the ﬁrst, 15-ML-thick iron layer. The FMR spectra recorded along the /H2085111 ¯0/H20852direction, see Fig.7/H20849b/H20850, show a four-peak structure in the main domain of temperatures. Three of them can be identiﬁed as a hard-axisspectra of the ﬁrst, 15-ML-thick iron layer /H20851see labeling in Fig. 7/H20849b/H20850and compare with Fig. 2/H20849b/H20850/H20852. As expected, the single-peak FMR spectrum of the 40-ML-thick layer clearly indicates that /H2085111¯0/H20852is the magnetic easy axis. D. Temperature dependence of the anisotropy ﬁelds and discussion of results The temperature dependence of the magnetic anisotro- pies are shown in Fig. 9. The effective magnetization Meff includes perpendicular anisotropy /H20851see Eq. /H208491/H20850/H20852, and therefore Meffis reduced compared with the bulk magnetization /H20851/H110111.71 kG /H20849Ref. 13/H20850/H20852by/H11011400–500 G at RT for the single iron layer samples. The effective magnetization increasedwith decreasing temperature. The Curie temperature of bulkiron is about 980 °C. Assuming that the 15-ML-thick Fe ﬁlmhas its Curie point close to 980 °C, the saturation magneti-zation would have been increased only by /H1101164 G in the temperature range from 300 to 5 K. 13Therefore, the ob- served decrease in Meffby /H11011300 G would require that the temperature dependence of Meffhad to be caused mostly by the decreasing value of the perpendicular uniaxial ﬁeld withdecreasing temperature. The uniaxial perpendicular aniso-tropy at RT is inversely proportional to the ﬁlm thickness andtherefore it is reasonable to assume that it originates from thebroken symmetry at the Fe/GaAs /H20849001 /H20850and Au/Fe /H20849001 /H20850in- terfaces, see Ref. 14. The magnetic anisotropies in Fe usually increase with decreasing temperature. However, by coolingthe strain in the ﬁlm changed by differential thermal expan- FIG. 9. Temperature dependence of the magnetic parameters: /H20849a/H20850for the single-layer sample and /H20849b/H20850for the double-layer sample. FIG. 7. The in-plane FMR spectra of the double-layer sample, 20Au/40Fe/40Au/15Fe/GaAs /H20849001 /H20850, for two orientations of magnetic ﬁeld with respect to the crystallographic axes: /H20849a/H20850dc ﬁeld is parallel to the easy axis of the ﬁrst, 15-ML-thick iron layer; /H20849b/H20850dc ﬁeld is parallel to the hard axis of the ﬁrst layer. The dash lines are guides for the reader’s eye. Themeasurements were carried out at /H9263=9.51 GHz. FIG. 8. The angular dependence of the in-plane resonance ﬁeld for thedouble-layer sample at room temperature and /H9263=9.51 GHz: open symbols, experimental data, and solid symbols, results of the ﬁtting.013912-6 Akta şet al. J. Appl. Phys. 102, 013912 /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: 84.117.74.208 On: Sun, 13 Apr 2014 12:58:28sion, see Table I, can result in interface magnetoelastic con- tributions which can be responsible for the observed decreaseof the uniaxial perpendicular anisotropy. The values of the in-plane uniaxial and cubic anisotro- pies increase with decreasing temperature. The sign of thecubic anisotropy parameter is positive, making the principalcrystalline directions /H20855100 /H20856easy magnetic axes. For the uniaxial anisotropy K u/H11022K1, the in-plane magnetic hard axis is along the /H2085111¯0/H20852crystallographic direction. The in-plane fourfold uniaxial anisotropy does not show any surprises as a function of the ﬁlm thickness. It decreases with decreasingﬁlm thickness due to the presence of the interface fourfoldanisotropy which has an opposite sign to that of the bulkcubic anisotropy. The most surprising behavior is found for the in-plane uniaxial anisotropy. It is interesting to point out complexitiesrelated to the in-plane uniaxial anisotropy in GaAs /H20849001 /H20850 structures. 15The origin of the large in-plane interface uniaxial anisotropy in GaAs/Fe /H20849001 /H20850has so far not been clearly understood. The hard magnetic axis lies along the /H2085111¯0/H20852crystallographic direction which is parallel to the dan- gling bonds of As terminated /H208492/H110036/H20850and pseudo- /H208494/H110036/H20850-reconstructed GaAs /H20849001 /H20850substrates.3However, Moosbühler et al.16have shown that the strength and sign of the in-plane uniaxial surface anisotropy are not affected by aparticular reconstruction of the GaAs template. A genuineGa-rich /H208494/H110036/H20850reconstruction results in almost the same uniaxial anisotropy as that observed in the /H208492/H110036/H20850As-rich reconstruction. Therefore, it is hard to believe that the source of this anisotropy lies in the chemical bonding between thedangling bonds of As and Fe. This point of view is further supported by our results using Cr /H20849001 /H20850overlayer. Cr /H20849001 /H20850 layer grown over a 15-ML-thick Fe /H20849001 /H20850ﬁlm can almost entirely remove the in-plane uniaxial anisotropy, see Fig.6/H20849a/H20850. The in-plane uniaxial anisotropy was also found to be dependent on the thickness of the capping Au layer in the20,30Au/15Fe/GaAs /H20849001 /H20850structures, see Fig. 9/H20849a/H20850. These results imply again that the interface chemistry alone be- tween the As and Fe interface atoms cannot be the source ofthe in-plane anisotropy. There is about −1.5% misﬁt betweenlattice parameters of Fe, Au, and GaAs substrates. Fe ﬁlmsgrown on GaAs /H20849001 /H20850are under a compressive strain. Calcu- lations by Mirbt et al. 17have suggested that an in-plane in- terface shear /H20849of the order of 2% /H20850can be established at the Fe/GaAs /H20849001 /H20850structures. A signiﬁcant in-plane lattice shear was observed by Xu et al.19in Fe/InAs /H20849100 /H20850structures and Thomas et al. in relatively thick Fe ﬁlms grown onGaAs /H20849001 /H20850.18The in-plane shear can lead to an in-plane uniaxial anisotropy due to the magnetoelastic parameter B2 /H20849Ref. 20/H20850with the uniaxial magnetic axis oriented along one of the /H20855110 /H20856directions. The in-plane uniaxial anisotropy in the 40Fe /H20849001 /H20850ﬁlm surrounded by the Au /H20849001 /H20850 layers /H2085120Au/40Fe/40Au/15Fe/GaAs /H20849001 /H20850/H20852changed the easy in- plane uniaxial axis to /H2085111¯0/H20852. Au has a larger /H20849001 /H20850square mesh than that of Fe /H20849001 /H20850by 0.9%. The GaAs /H20849001 /H20850mesh is smaller by 1.4% than that of Fe /H20849001 /H20850, see Table I. The Fe ﬁlm on GaAs /H20849001 /H20850is under contraction while Fe on Au /H20849001 /H20850 is under tension. That can result in a reversal of the sign ofinterface shear in the 20Au/40Fe/40Au /H20849001 /H20850structure com- pared to that at the GaAs/Fe /H20849001 /H20850interface. This would im- ply that for Fe/Au /H20849001 /H20850, the surface cell length of Fe along /H20851110 /H20852would be larger than the surface cell length along /H2085111¯0/H20852. Magnetoelastic coupling in 20Au/40Fe/40Au /H20849001 /H20850 can then lead to an interface uniaxial anisotropy with the easy axis along the /H2085111¯0/H20852direction. The Cr /H20849001 /H20850square mesh is 1.7% larger than that of Fe /H20849001 /H20850. This lattice mismatch is almost twice of that found in Fe/Au /H20849001 /H20850. Therefore, one can argue that the shear at the Fe/Cr /H20849001 /H20850interface can be larger than that at the Fe/Au /H20849001 /H20850. It can be argued that the shear at the Fe/Cr /H20849001 /H20850interface can result in a large enough in-plane uniaxial anisotropy compensating the in-plane uniaxial anisotropy from the GaAs/Fe /H20849001 /H20850interface in agreement with our measurements on the 20Cr/15Fe/GaAs /H20849001 /H20850sample, see Fig. 6/H20849a/H20850. However, the in-plane uniaxial anisotropy was not changed by a Pd layer, see Fig. 6/H20849b/H20850. The Pd square mesh is 4.6% smaller than that of Fe. Therefore, one can expect a larger in-plane uniaxialanisotropy in 20Au/9Pd/16Fe/GaAs /H20849001 /H20850compared to that measured in Au/Fe/GaAs /H20849001 /H20850. Only a marginal enhance- ment of 20% was found, see Figs. 6/H20849b/H20850and3. This can be caused by a large lattice mismatch between the Fe /H20849001 /H20850and Pd/H20849001 /H20850lattice meshes. Perhaps in this case the Pd square lattice mesh relaxes its strain right from the ﬁrst atomic layerand consequently affects the interface shear only marginally. It is interesting to note that all magnetic anisotropies were found linearly dependent on temperature within the ex-perimental error, see Fig. 9. The almost linear temperature dependence of the perpendicular uniaxial anisotropy hasbeen observed also in Ref. 21. This suggests that there could be common physical grounds behind this uniﬁed universalbehavior. The strain between Fe /H20849001 /H20850and GaAs /H20849001 /H20850de- creases by /H1101140% from RT to 4 K. This is an estimate based on using known thermal expansion coefﬁcients for the bulkTABLE I. Thermal expansion and lattice parameters. Material /H20849structure /H20850Thermal expan. /H2084910−6K−1/H20850 Lattice parameter a=b=c=/H20849Å/H20850 Effect on iron layer 298 K 523 K 1273 K Iron /H20849bcc-Fe /H20850 11.8 15.0 24.0 2.8665 GaAs /H20849ZnS structure /H20850 5.73 5.654/2=2.827 Compressive strain Gold /H20849fcc-Au /H20850 14.2 14.6 16.7 4.078/ /H208812=2.892 Tensile strain Chromium /H20849bcc-Cr /H20850 6.2 2.91 Large tensile strain Palladium /H20849fcc-Pd /H20850 11.8 12.2 13.9 3.891/ /H208812=2.759 Compressive strain013912-7 Akta şet al. J. Appl. Phys. 102, 013912 /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: 84.117.74.208 On: Sun, 13 Apr 2014 12:58:28Fe, Au, and GaAs crystals, see Table I. The observed in- crease in the in-plane uniaxial anisotropy ﬁeld by approxi-mately 60% from RT to liquid He temperature, see Fig. 9, suggests that the difference in the surface cell length along /H20851110 /H20852and /H2085111¯0/H20852closely followed the relaxation of stress with decreasing temperature. One may attribute the temperature dependence of the magnetic anisotropies to the temperaturedependence of the magnetoelastic parameters B 1and B2. This can be applicable to the in-plane and perpendicularuniaxial anisotropies. However, the same linear dependenceon temperature was found also for the in-plane fourfold an-isotropy, see Fig. 9and yet there is no magnetoelastic term known for the fourfold magnetic anisotropy. 15Therefore, it is unlikely that the temperature dependence of the magneto-elastic energy on its own can be a possible explanation forthe observed linear dependence of the fourfold magnetic an-isotropy on temperature, as shown in Fig. 9. V. CONCLUSION We studied the magnetic anisotropies of epitaxial, crys- talline ultrathin iron ﬁlms grown on the surface-reconstructed /H208494/H110036/H20850GaAs /H20849001 /H20850substrate. The ferromag- netic resonance technique has been explored extensively to determine magnetic parameters of the studied ﬁlms in thetemperature range from 4 to 300 K. The triple-peak FMRspectra were observed, allowing an accurate extraction ofmagnetic anisotropies using computer simulations of the ex-perimental data. The measured samples have shown strongperpendicular and in-plane uniaxial anisotropies in theAu/Fe/GaAs /H20849001 /H20850ﬁlms. The fourfold in-plane anisotropy decreases with the ﬁlm thickness due to a presence of the interface fourfold contribution which has an opposite sign tothat of the bulk cubic anisotropy. The most surprising behav-ior is found for the in-plane uniaxial anisotropy induced byreconstruction of the GaAs substrate surface. It is argued thatthe in-plane uniaxial anisotropy in Au,Cr,Pd/Fe/GaAs /H20849001 /H20850 and Au/Fe/Au/Fe/GaAs /H20849001 /H20850structures is more likely af- fected by the interface shear strain. The experiment shows that the Cr /H20849001 /H20850layer grown over a 15-ML-thick Fe /H20849001 /H20850 ﬁlm can almost entirely remove the in-plane uniaxial aniso-tropy. The ﬁtting of the angular dependence of FMR of thedouble magnetic layer sample, 20Au/40Fe/40Au/15Fe/GaAs /H20849001 /H20850, revealed that the easy /H20849hard /H20850axis of the uniaxial anisotropy term in the second, 40-ML-thick iron layer is switched 90° with respect to the easy /H20849hard /H20850axis of the 15-ML-thick iron layer. It has been shown that the surface re- construction of the GaAs substrate and various combinationof materials in the multilayer structure can be used for tai-loring of the magnetic anisotropies in spin-valve-like, doubleferromagnetic layer structure. ACKNOWLEDGMENTS This work was supported in part by the Gebze Institute of Technology, Grant No 03-A12-1, and Russian Ministry ofEducation and Science. One of the authors /H20849B.H. /H20850thanks the Canadian National Science Engineering Research Council/H20849NSERC /H20850and Canadian Institute for Advanced Research /H20849CIAR /H20850for a generous and valuable scientiﬁc research sup- port. 1B. Heinrich and J. F. Cochran, Adv. Phys. 42,5 2 3 /H208491993 /H20850. 2M. Farle, Rep. Prog. Phys. 61, 755 /H208491998 /H20850. 3T. L. Monchesky, B. Heinrich, R. Urban, K. Myrtle, M. Klaua, and J. Kirshner, Phys. Rev. B 60, 10242 /H208491999 /H20850. 4S. McPhail, C. M. Gürtler, F. Montaigne, Y. B. Xu, M. Tselepi, and J. A. C. Bland, Phys. Rev. B 67, 024409 /H208492003 /H20850. 5M. Dumm, F. Bensch, R. Moosbühler, M. Zölﬂ, M. Brockmann, and G. Bayreuther, in Magnetic Storage Systems Beyond 2000 , NATO Science Series II: Mathematics, Physics and Chemistry, edited by G. C. Hadji-panayis /H20849Kluwer Academic, Dordrecht, 2001 /H20850, Vol.41, pp. 555–558. 6A. G. Gurevich and G. A. Melkov, Magnetic Oscillations and Waves /H20849CRC, New York, 1996 /H20850, chap. 2. 7H. Suhl, Phys. Rev. 97, 555 /H208491955 /H20850. 8J. J. Krebs, F. J. Rachford, P. Lubitz, and G. A. Prinz, J. Appl. Phys. 53, 8058 /H208491982 /H20850. 9Yu. V. Goryunov, N. N. Garifyanov, G. G. Khaliullin, I. A. Garifullin, L. R. Tagirov, F. Schreiber, Th. Mühge, and H. Zabel, Phys. Rev. B 52, 13450 /H208491995 /H20850. 10Th. Mühge et al. , J. Appl. Phys. 81, 4755 /H208491997 /H20850. 11T. Toli ński, K. Lenz, J. Lindner, E. Kosubek, K. Baberschke, D. Spoddig, and R. Mechenstock, Solid State Commun. 128, 385 /H208492003 /H20850. 12G. Woltersdorf and B. Heinrich, Phys. Rev. B 69, 184417 /H208492004 /H20850. 13Numerical Data and Functional Relationships in Science and Technology , Landolt-Börnstein, New Series, Vol. III, Pt. 19A /H20849Springer, Heidelberg, 1986 /H20850. 14R. Urban, G. Woltersdorf, and B. Heinrich, Phys. Rev. Lett. 87, 217204 /H208492001 /H20850. 15G. Wastlbauer and J. A. C. Bland, Adv. Phys. 54, 137 /H208492005 /H20850. 16R. Moosbuehler, F. Bensch, M. Dumm, and G. Bayreuther, J. Appl. Phys. 91, 8757 /H208492002 /H20850. 17S. Mirbt, B. Sanyal, C. Ischeden, and B. Johansson, Phys. Rev. B 67, 155421 /H208492003 /H20850. 18O. Thomas, Q. Shen, P. Schieffer, N. Tournerie, and B. Lépine, Phys. Rev. Lett. 90, 017205 /H208492003 /H20850. 19Y. B. Xu, D. J. Freeland, M. Tselepi, and J. A. Bland, Phys. Rev. B 62, 1167 /H208492002 /H20850. 20D. Sander, Rep. Prog. Phys. 62,8 0 9 /H208491999 /H20850. 21Kh. Zakeri, Th. Kebe, J. Lindner, and M. Farle, J. Magn. Magn. Mater. 299,L 1 /H208492005 /H20850; Phys. Rev. B 73, 052405 /H208492006 /H20850.013912-8 Akta şet al. J. Appl. Phys. 102, 013912 /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. 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1.2946039.pdf	AIP Conference Proceedings 34, 138 (1976); https://doi.org/10.1063/1.2946039 34, 138 © 1976 American Institute of Physics.Wall States in Ion-Implanted Garnet Films Cite as: AIP Conference Proceedings 34, 138 (1976); https://doi.org/10.1063/1.2946039 Published Online: 24 March 2009 T. J. Beaulieu , B. R. Brown , B. A. Calhoun , T. Hsu , and A. P. Malozemoff ARTICLES YOU MAY BE INTERESTED IN Theory of Bloch-line and Bloch-wall motion Journal of Applied Physics 45, 2705 (1974); https://doi.org/10.1063/1.1663654 Intrinsic pinning behavior and propagation onset of three-dimensional Bloch-point domain wall in a cylindrical ferromagnetic nanowire Applied Physics Letters 102, 112405 (2013); https://doi.org/10.1063/1.4794823 Direct observation of the spin configurations of vertical Bloch line Applied Physics Letters 98, 052510 (2011); https://doi.org/10.1063/1.3549694138 WALL STATES IN ION-IMPLANTED GARNET FILMS T. J. Beaulleu, B. R. Brown, B. A. Calhoun, T. Hsu IBM Corporation, San Jose, California 95193 and A. P. Malozemoff IBM Corporation, Yorktown Heights, New York 10598 ABSTRACT A model is presented which explains the wall states, and transitions between states, observed for bubbles propagating under the influence of an in-plane field in ion-lmplanted garnet films. We have identified nine transitions in which the deflection angle changes, six of which are accounted for by switching of the capping layer and injection of a Bloch point (BP). It was necessary to invoke three different wall structures for S=+I bubbles depending upon the relative orientation of bubble velocity and the in-plane field. Two different S=I/2 bubbles containing one BP are identified which differ in location of the Bloch point and the associated capping structure. There is one state containing two vertical Bloch lines and two BP's, one on each line; and there is a single S=0 state. I. INTRODUCTION The winding number S of a bubble is related to the angle X between the field gradient and the direction of motion by the equation 1 8SV sinx = ydAH (I) where y is the gyromagnetic ratio, V is the bubble velo- city, and AH the field difference across the bubble dia- meter d. In garnet films which have been ion-implanted to suppress hard bubbles, 2 the number of different wall structures which can exist is quite limited. At zero in-plane field, only bubbles with S=+I are stable; and at sufficiently large values of in-plane field, Hp, only S=0 bubbles are stable. 3 At intermediate values of H,, additional states corresponding to fractional S values have been observed.4, 5 Some observations of the changes in deflection angle X, as an in-plane field perpendicular to the field gradient was increased, have been reported. 6 State changes for an in-plane field parallel to the drive gradient have also been reported in as-grown EuYIG. 7 We have measured the deflection angles for both increasing and decreasing in-plane fields for several different orientations of the in-plane field and have observed nine discrete, irreversible changes in deflec- tion angle. Following a brief description of our experimental techniques, we present these results for the in-plane field parallel and perpendicular to the drive field gradient. Section IV discusses the dif- ferent mechanisms responsible for these state changes, and Section V covers the detailed wall structures of S=+I bubbles moving in an in-plane field. II. EXPERIMENTAL PROCEDURE Two different methods for measuring deflection angles were used. The overlay technique 8 requires no additional device fabrication and is, therefore, con- venient for surveying large numbers of samples. For small displacements, the drive gradient is nearly con- stant, and mobility changes are easyto observe. The deflectometer 9 provides much higher angular resolution at the expense of drive-fleld non-uniformities and the necessity for device fabrication. Most of the data in this work were taken using the deflectometer at ambient temperature. Bubbles were sequentially propagated back and forth by means of pulsing pairs of current con- ductors which were deposited over a 1 ~m thick spacer layer on the garnet surface. Conductors were i ~m thick, 8.5 ~m wide, spaced on 15 ~m centers. Nominal propagation pulse width was 4 ~s with i ~s rise and fall time. State transitions can be recognized by discrete changes in deflection angle. Deflection angle X and in-plane field orientation ~ are measured clock- wise from VH z when looking along the direction of H B. Except as noted, all measurements were made on ion-implanted chips of YSmCaGe garnet with properties as given in Table I. TABLE I Properties of YI.92Sm0.1Ca0.98Fe4.02Geo.98012 Chip Implantation Dosage 2 x 1014 Ne/cm 2 Implantation Energy 80 Kev Thickness h (by SEM) 5.95 pm Strlpe-Width Ws 5.53 ~m Characteristic Length s 0.598 ~m Collapse Field H o 95.30e * 4~Ms 176 G Q 4.54 K u 5650 ergs/em 3 K 1 -2050 ergs/cm3 g 2.042 0.062 * As-grown wafer ** FMR III. OBSERVATIONS OF DEFLECTION ANGLE The observed deflection angles as a function of in-plane field Hp for HpilVHz are given in Fig..l for a drive current of 35 mA (AH~3.40e nominal drive across the 6 pm diameter bubble). As Hp increases, the S=+I deflection angle decreases slightly until, at Hp=ll00e, transition AI occurs in which S=+I+S=0, Reduction of Hp causes no change in deflection angle of S=0 until Hp~40 Oe where transition BI occurs corres- ponding to (0,2)+(1/2,2,1). The three digit notation (S,s signifies a revolution number S, % vertical Bloch lines, and p Bloch points. When no ambiguity can arise, we use abbreviated notation in which s and/or p are not specified. The 1/2 state shows a gradual increase in deflection angle as Hp is reduced. At Hp near zero, 1/2 converts to (i,0); and at large Hp, i/2 converts to (0,2), transition CI, at the same field as for S=+I decay, AI. The feature in Fig. i we wish "~ 30 S=+ 1 a -~ 20 o 10 A1 a 0 HC1 HC2 0 20 40 60 80 100 120 In-Plane Field (Oe) Fig.1. Bubble deflection angles versus in-plane field for Hp]] V H z in an on- mplanted 5.95/Lrn thick chip of Y1.92 Sm0.1 Ca0.98 Fe4.02 Ge098 O12 with a drive current of 35 mA at a bias field of 88 Oe. to emphasize is that for HpIIVHz, transition AI always results in +1+(0,2). The fine structure of S=+I, transitions FI and F2, will be discussed in Section V. Rotating Hp to give HpIVH z gives the results in Fig. 2, where all other parameters are identical to those of Fig. i. The S=+I deflection angle increases monotonically with Hp until A2 converts S=+I into a state we label (1/2,2,1)* or 1/2" for short. Although the deflection angles of 1/2 and 1/2" are identical, the observation that the two states are formed and destroyed by different transitions shows that they are different states. Increasing Hp to 140 Oe converts 1/2" to (0,2). Transitions B1 and CI are similar to Fig. i. If, before transition E, the value of Hp is reduced, 1/2" will convert by transition B2 to (1,2,2). As will be discussed, this state consists of two verti- cal Bloch lines (VBL's) with one Bloch point (BP) on each llne. The (1,2,2) state is stable between transi- tions D2 and C2 and can only be obtained from 1/2" by means of transition B2. Even though 1/2" and (1,2,2) could not be created for HpIIVHz, these states are stable for this orientation of Hp and can be obtained by generating with HpiVH z and then rotating ~. We have found that the behavior of a given state does not depend on the conditions of its creation. We emphasize that for HplVHz, the decay of S=+I always leads to 1/2", not (0,2) as for ~IIVHz. -- 4O E3 v 30 CD < 20 t- O ~10 0 S=+1 1H and o ~-- ~, ~1~2 ii ~L.D2 !D1 (1/2,2,1)&(1/2,2,1)* i .~ (1,2,2) , , HCI 20 40 60 A2 (1/2,2,1)* --r--~-~ E HC2 80 100 120 140 In-Plane Field (Oe) Fig.2. Bubble deflection anglesfor HplV H z forthesamechip asin Fig.l. AIIotherconditionsareidenticalto tho~ in Fig.1. The state S=+I in Figs. i and 2 has a deflection angle of 24 4egrees at Hp=0. Equation (i) gives a X of 22.7 degrees for S=+I if we use the measured wall mobility of 1040 cm/sec-Oe. The wall mobility calcu- lated from the FMR value of e=0.062 is 1912 cm/sec-Oe. Before treating the detailed dependence of deflec- tion angle on Hp for bubbles having fractional S-state, we must account for those effects which are independent of S. Since X for S=+I at Hp=0 decreases only slightly as the drive current is increased from 35 to 50 mA, we will ignore coerclvity in what follows. Putting V=(~/2)AHcosx, where ~ is the wall mobility, we can rewrite Eq. (i) in the more convenient form 4pS tanx = 7d (2) The collapse field was observed to be a function of the magnitude and orientation of the in-plane field. The resulting variation of bubble diameter at constant bias gives rise to changes in deflection angle. We assume the change in diameter is independent of S. In addi- tion, Eq. (2) is only valid for circular bubbles. The in-plane field causes elliptical deformation of the bubble 9 which produces changes in deflection angle so that the observed XT=Xg+XE, where Xg is given by Eq. (2) and XE is proportional to the deformation of the bubble. For ~ either parallel or perpendicular to VHz, it can be shown by an extension of the analysis in reference 9 that 139 tanxT = (i + ke) tanXg (3) where e is the elliptlclty and k is a proportionality constant which is independent of S. Two other results of the analysis can be used to evaluate e. The deflec- tion angle of S=0 bubbles is given by 3 X(0) = ~ e sln2~ (4) where ~ is the orientation of Hp. Measurements of X(0) for the chip of Figs. 1 and 2 as a function of ~ for ~=80 Oe gave s The difference in deflection angle for +i bubbles with Hp perpendicular or parallel to VH z is given by 3 x I - Xll = ~ ~ sin2x 0 (5) where X0 is the deflection angle for Hp=O. Measurements of +i bubbles gave E=0.24 at Hp=80 Oe. In con- trast to Josephs et al,6 these results indicate that elliptical deformation of the bubble accounts for most of the variation of the +i deflection angle with Hp. Using Eq. (3) and the fact that Xg is given by Eq. (2), we can correct the measured X for a bubble of effective S-value, Se, using the equation tanx(S e) = S e 9 tanx(+l) (6) where we measure tanx(+l) for each value of Hp for which we wish to calculate X(Se). Hasegawa 4 has cal- culated S e for bubbles containing one and two BP's. For 1/2 bubbles, Se=i/2-1zI/h where z is the displacement of the BP from the mldplane of the film. For (1,2,2) bubbles containing two BP's, Se=l-21zl/h. Thestable location of a BP occurs where Hp and the radial stray field of the bubble sum to zero. Figure 3 compares the measured deflection angles (points) and the calculated curves using Eq. (6) and the S e values as described above. The agreement un- ambiguously identifies 1/2 and 1/2" as containing one BF, and (1,2,2) as containing two BP's. 24 20 a _~16 t- <12 f- o ~ " 8 o 4 % -+ -. ~~ Hp 1 VH z ,2,2) 9 (1/2,2,1) ~ ~176176 o ~2 o (1/2,2,1)* - - _ I I I 2'0 4b ' 6'0 ' 80 '100 In-Plane Field (Oe) Fig.3. Comparison of measured deflection angles (points) and calcu- lated curves versus in-plane field for the implanted YSmCaGe film of Fig.l. Drive current 35 mA, bias field 88 Oe. IV. TRANSITION MECHANISMS A. Cap-Swltchlng Processes From Fig. 4, we observe that transitions AI, A2, CI, and C2 occur at approximately the same value of Hp, roughly independent of orientation ~. Similar behavior is observed for transitions B1 and B2. A common process is suggested for these two groups of transitions, and because of the lack of ~-dependence, it is clear that the gyrotropic forces play little role in this process. We identify four transitions AI, A2, CI, and C2 as a cap-swltch process in which the ion-lmplanted capping 140 layer of the film becomes saturated. Such a process will occur when Hp dominates the stray field in the capping layer. The low field processes BI and B2 are the reverse cap-switch process in which the capping layer switches from a saturated configuration to a formation having a closure domain which is compatible with the stray field of the bubble. I0 We introduce the nomenclature "capped Bloch line" to indicate a Bloch line which terminates under the portion of the capping layer which switches; i.e., under the closure domain. For cap-switch processes, we postulate injec- tion of a BP onto a capped BL whenever the cap switches. The required energy presumably arises from the change in magnetostatic energy accompanying the cap-switch itself. Considering the exchange interaction between the capping layer and the underlying VBL's, it is clear on topological grounds that a BP singularity must exist any time the cap magnetization opposes the magnetization in the VBL. 3 Schematic representations of two pairs of states which undergo cap-switches are shown in Fig. 5. The effect of the in-plane field is to stabilize two verti- cal Bloch lines on the extremities of the bubble as determined by the in-plane field direction. Transitions BI and B2 involve injection of a BP onto the capped VBL as the cap switches from the saturated to the umbrella configuration which has the closure domain. The high field processes CI and C2 cause the injection of one additional BP onto a VBL which already contains a BP. The two BP's on the same VBL annihilate each other, thus completing processes CI and C2. Also in Fig. 5, we see that 1/2 and 1/2" have structures which differ in cap configuration and the location of the BP's on different lines. 140 I 9 ^^ -"- "'~..-~. 9 r o A2:1 -'> 1/2" A tuu -"9="~-(~--~s / = C2: (1,2,2) --~ 1/2" (~ =LOAI: 1 -+0 v 8o "5 _~ 6o > 40j ^ ~ _~ <~. ~[.B2 1/2" (122) 1~ ~ s o~[<>al 0 1/2 20t hD1 1/2~(1 0) (3~'~~ ,----I D2 : (1,2,2) -- (1,0) 90 = 120 ~ 150 ~ 180 ~ Orientation of Hp(~) Fig.4. Dependence of the in-plane field at which state switching occurs on the azimuthal orientation ~ of the in-plane field. Ion-implanted YSrnCaGe chip of Fig.l, drive current 35mA, bias field 88 Oe. Angles are measured clockwise from VHz, looking along the direction of H B. We expect cap-switch processes to be affected by the level of ion-implantation. Chips of a 3.88 ~m film of YI. 52Eu0.3-Tm0.3Ca0.88Ge0.88Fe4.12012 (4~Ms=240 G, %=0.62 ~m, Ho=103 Oe) were implanted with 2xi014 Ne+ ions/cm2 using energies of 25, 50, and 80 Kev. The upper cap-switch field HC2 decreased slightly from 115 Oe for 25 Kev to 109 Oe for 80 Kev. The lower cap-switch field HCI increased from 31 Oe to 56 Oe for 25 and 80 Kev, respectively. Transition DI, which is not a cap-switch, showed no variation with implantation level. II We have ignored dynamic effects in our discussion of the cap-switching process. In general, the difference between the upper and lower cap-switch fields decreases Hp (a) . " /H B C1 ~ (1/2,2,1) (c) 41---- 9 4---- ~ 11,2,2) ,BP ~- q - BP' C2 ~.B2j (b) (0,2,0) (d) (1/2,2,1)* Fig.5. Schematic representations of four states derived by cap-switch- ing processes. Transitions C1 and C2 occur at Hp= HC2; and transit- ions B1 and B2 occur at Hp= HCI. as bubble velocity increases, but there are significant differences in this behavior for different garnet com- positions. For example, in our YSmCaGe garnet, the high field transition decreases with velocity much more rapidly than the low field transition increases. In YEuTmCaGe garnets, the converse is true. B. Bloch Point Annihilation With the structure of the 1/2" bubble shown in Fig. 5d, it is natural to identif~ the transition 1/2"->0 (E in Fig. 2) as the annihilation of the BP as it approaches the lower surface (no capping layer) of the garnet film. In a stationary bubble for the YSmCaGe chip studied, E occurs at Hp=143 Oe and, as shown in Fig. 4, is only slightly dependent on $ when the bubble is in motion. Thus, the gyrotropic forces play little role in this transition. From the field at which transition E occurs stati- cally and the condition that the BP is located where Hp equals the radial stray field from the bubble, we calculate that the BP is 0.23 ~m from the surface when E occurs. Slonczewskil2 has pointed out that a Bloch line is perturbed by the presence of a BP for a distance of about one linewidth. Thus, the perturbed wall magne- tization reaches the surface when the BP is half a line- width from the surface. For our sample, ~=0.598 ~m and Q=4.54 so that ~A/2=~/QI/2=0.22 ~m. This agreement is probably fortuitous, but it does support the proposed mechanism for transition E. C. Bloch Line Annihilation From Figs. i, 2, and 4, we notice that transition DI displays very strong ~-dependence which is shown in more detail in Fig. 6. In all cases, 1/2 was observed to decay to (i,0) when two conditions were met: (I) Hp was reduced below a critical value, and (2) t~he~bubble velocity had a positive component along the HpXH B direc- tion. From the inset of Fig. 6, we notice that such a velocity places the mobile VBL near the unstable g-force node. The gyrotropic force acting on a VBL has been calculated by Slonczewski 13 --I F G = s x V (7) where ~ is along the bias field direction and the sign is (+) if the sense of the VBL is the same as that of the adjacent wall magnetization. Equating this gyro- tropic force on the VBL to the In-plane field restoring force 2M~AHp, we find to lowest order the critical wall velocity for the 1/2§ transition v w = A 9 y 9 H (8) P where A is the wall width parameter and y is the gyromagnetic ratio. Using the fact that the linear wall mobility ~=yA/u, we can rewrite Eq. (8) in terms of the drive field AH acting across the bubble diameter AH = 2~H (9) P _%._% _.% where ~ is the Gilbert damping parameter. For V=HpXHB, once Hp falls below the value given by Eq. (9), the 1/2 bubble will decay by line annihilation to (i,0). The detailed dependence on in-plane fleld orientation is given in Fig. 6. The asyB~etry about r ~ is attri- O (1/2,2,1)-'>(1,0~ BP .e \| B 1 7 o ~l o (-#" ~/ o 1270~ V~.px.B j '~ o>O 301- 18 ~ .,~ ,l& 20 .: ,%.A: 10 l ,,oSO ''~ 9 9 S=+1 Trajectory 01 i , ~ I , i , 10 ~ 30 ~ 50 ~ L~ 190 ~ 210 ~ 230 ~ 250 ~ In-Plane Field Orientation i , [ , 70 ~ 90 ~ 11; ~ 270 ~ 290 ~ Fig.& Variation of the critical in-plane field for the 1/2 ~ (1,0) transit- ion with azimuthal orientation of Hp. Bubble velocity was always as shown in the inset when decay was obse~ed. buted to the radial velocity ~>0 present in our deflec- tometer. The resulting gyrotropic force F G ~ either assists (r ~ or opposes (~>90 ~ the g-force FG, V due to the translational velocity. Hence, for @<90 ~ we expect less stability for 1/2 than for ~>90 ~ For r ~ 1/2 becomes stable down to Hp~0 for ~ between ii0 ~ and 1600 . Figure 7 plots the critical Hp versus drive current and field for Hp aligned at @=i00 ~ perpendicular to the 1/2 trajectory. The nominal drive 40 I "G o 30 9 c 20 10 / AH = 2eHp / ~ lO 2o ,dI AI 0 0.96 1.92 2.88 3.84 4.8 AH(Oe) Fig.7. Dependence of the critical field of Fig.6 on drive current ! for Hp 1 Y (r 100~ 141 AH on the bubble is calculated midway between the powered conductors and ignores ~. The curve is a plot of Eq. (9) with slope corresponding to ~=0.06 (from FMR) which is in reasonable agreement with ~=0.ii from the mobility measurement. Transition D2, in which (1,2,2)+S=I, displays only slight @-dependence, again suggesting a quasi-static process. As Hp->0, the BP's on the two VBL's of the (1,2,2) bubble approach the mid-plane and the lines become gyrotropically inactive. These lines are, how- ever, magnetostatically attractive and would be expected to unwind as they collide. V. DEPENDENCE OF S=+I WALL STRUCTURE ON H AND V P In this section, we discuss the influence of an in- plane field on the wall structure of the S=+I bubble. 14 Depending on the direction of propagation, and the actual drive conditions, two structures other than the simple unichiral (i,0,0) structure are identified. Both structures are necessary in order to explain the experimental observations. Referring to transitions AI and A2 of Figs. i and 2, we observe that the nominal S=+I bubble decays to either 1/2" or (0,2) as Hp is increased through the upper cap-switch field HC2. This dynamic behavior provides a clue to the detailed Bloch linestructure of the +I bubble in the presence of an in-plane field. It is known that an in-plane field directed anti- parallel to the magnetization in a planar domain wall can quasi-statically twist the spins near both surfaces to form Bloch loops at a threshold field given by 14 H = 4 2r h -I (i0) P which is 80e in the YSmCaGe sample. Once formed, these loops give rise of a 2~ horizontal Bloch line (HBL) which is subject to the usual gyrotropic forces.15 For a cylindrical domain, this 2~ HBL formation leads quite naturally to the wall structure shown in Fig. 8a, which we label IH. For the velocity shown, the gyrotropic force on the 2~ HBL is upward. The opposite initial chirallty would result in the 2~ HBL located on the back of the bubble with the gyrotropic force still directed upward. The capping layer, due to a large exchange repulsion, resists punch-thru at the top sur- face for drive currents used in this study. As the cap (a) Hp Vo IHB (b) '0 p (c) ,qu -- 4--- "4--- B~- 4-- t Fig.8. (a) The 1H bubble showing a 2# horizontal Bloch line. (b) tran- sient state which exists just after 1H undergoes a cap-switch at P. The dotted circle denotes the region where unraveling of the upper and low- er loops is expected as the BP advances. (c) The final state 1/2". 142 of the 1H bubble switches at H~=HC2 , a BP is introduced at point P, as shown in Fig. 8~. This BP travels down the upper Bloch loop as shown, reversing the sign of the llne as it progresses. The two Bloch loops unravel behind the moving BP and the resulting state, 1/2", shown in Fig. 8c, has the BP on the front VBL (with res- pect to the direction of Hp) even though the cap-switch occurred at the back of the bubble. This BP position is stable with respect to the bubble stray field; whereas, no position on the back or right-hand VBL is stable. The IH structure and its high-field cap-switch provide a natural explanation of transition A2 of Fig. 2 in which +1+1/2" for V=~pXH B. The question arises as to how to explain the cap-switch AI of Fig. 1 in which +i§ for Hpli?H z. Dynamic results cannot dls- tinguish between g-forces on IH during the cap-switch process and the possibility of a different structure for the parent S=+I bubble. In order to sort out the detailed wall structure as a function of bubble velocity and in-plane field orientation, we make use of fact that even quasi-static switching of the ion-implanted capping layer introduces a Bloch point onto the under- lying Bloch line. From Fig. 4, it is clear that the center of the operating margin for the YSmCaGe chip is Hp ~70 Oe. Thus, if we stop an S--+I bubble operating under a certain Hp, and raise Hp to 135 Oe, the static cap-switch field for this chip, we can then reset Hp=70 Oe and propagate the bubble gently in order to determine its final state by means of its deflection angle. This technique greatly simplifies the identi- fication of states because in the absence of g-forces, the 2~ HBL relaxes to the film mid-plane; i.e., it does not disappear as in the case of the loops present in the dynamic conversion process.16 Thus, any time we observe a final state 1/2" after a static cap- switch, we deduce that the parent S=+1 bubble was IH. When the static cap-switch process was applied to bubbles initially propagating with HpllVHz, we observed the following: For 1=35 mA and Hp~ 55 Oe, transition F2 of Fig. i, final states i/2" were observed for S=+I initially propagating either parallel or anti-parallel to Hp. Larger values of Hp lead to (0,2) for either direction of velocity. Increasing the drive to 50 mA caused F2 to be reduced to ~35 Oe. For an explanatlon of these results, we refer to Fig. 9 which shows the (a) Hp,V Z H B (b) (c) 4 -- (-~ +) Fig.9. (a) Sheared Bloch loop structure which occurs for 1H propa- gating with V IIH D. (b) Punch-thru if drive or Hp is sufficiently large. (c) Final stateo L- which exists after punch-thru and collapse of the lower loop in (b). IH bubble propagating with V parallel to Hp. In the vicinity of HC2 , Hp is large, so to first order, the loops of the IH bubble are held on the slde of the bubble as shown. Because of the nodes in the g-force which exist on the sides of the bubble where Vn=0, we see that the loops tend to be sheared for V parallel to Hp. For V anti-parallel to Hp, the g-forces reverse direction. Thus, half of the loop is being pushed down, favoring punch-thru, for either direction of velocity. Once punch-thru at the lower surface has occurred, as In Fig. 9b, and the velocity is again reduced to zero, the lower Bloch loop collapses leaving a bubble as shown in Fig. 9c, which is labeled (1,2,0)- or o-. The complementary state, labeled o+, in which the two VBL's point inward, is not compatible with the cap magnetization for the direction of bias field shown. 17 Figure 10 shows the result of a static cap-switch of the o- bubble. The BP is introduced (a) (b) 4 -p BP Hp HB ~ ~' 9 ~I-- -- .4--- 9 ql.- -- .,l--- ,II-I -- il (-P-- Fig.10. (a) The o- bubblejustafterundergoing a cap-switch at P. (b) Theresultingfinal ~ate(0,2). below point P as the cap switches, and it travels down the VBL and is lost at the bottom surface since there is no statically stable location for the BP on that line. The resulting state is seen to be (0,2). Thus, whenever we observe a final state (0,2), we deduce that the parent S=+I bubble must have been a-. Returning to the earlier results for HplIVHz, we conclude that for 1=35 mA and Hp~ 55 Oe, the parent S=+I bubble is IH for bubble velocity either parallel or anti-parallel to Hp. For larger values of Hp, the parent S=+I bubble must have been o-, based on the results of the static cap-switch. Raising I to 50mA resulted in punch-thru of IH-~- whenever Hp exceeded 35 Oe. Thus, not only does increased drive favor punch- thru, but so too does increased Hp. The physical rationale for increased Hp favoring punch-thru is unclear but may be associated with the fact that Hp compresses the 2~ HBL thereby enabling it to more closely approach the film surface before the repulsive potential is felt. Returning to the case of HpiVHz, there is the possibility of punch-thru of the 2~ HBL at the lower surface of the~H bubble_~ _~ if the velocity in Fig. 8a is reversed to V=-(HnxHR). Observations of the static cap-switch process ~--for HpiVH z yielded the followlng: For I=35 mA and Hp~35 Oe, transition F2 of Fig. 2, the final state observed after a static cap-switch was 1/2" for ~= in~i~at%ng that the parent state was ~H._~ For Hp>35 Oe, V~pX~ B yielded 1/2"; whereas, V=-(HpXHB) yielded (Q,2).,..dafter the static cap-swltch. This indicates that V=-(HpXH B) causes punch-thru at the lower film surface in which IH-~-. If the drive current is increased to 1=50 mA, the c_c_~rre~po_~nding critical Hp for IH-~- conversion for V=-(HpXHB) is reduced to about 20 Oe. These results are remarkable in that they indicate that the S=+I bubble has two different structures depending upon the dlrection~of_motion for HpIVH z (and sufficiently large Hp). For_~V=~ S=+l is actually the IH bubble; whereas, for V=-(EpX~B) , it is the o- bubble. Even one step is sufficient to cause conver- sion between IH and ~-, depending upon the direction of motion. The process of IH-~O- is punch-thru, as already discussed. A proposed mechanism for the reverse pro- cess O-§ is shown schematically in Fig. ii. The VBL labeled (+) in Fig. lla is gyrotropically unstable for ~=~pX~ B. In addition, we note that if this (+) line lies on the right-hand side of the bubble, its magneti- zation is unfavorably oriented with respect to Hp. We expect this magnetostatic misalignment energy to be reduced as the (+) VBL rotates toward the front of the bubble, since ~>0. We also expect nucleation of a horizontal Bloch loop to occur behind the rotating (+) line. At a later instant, t2, shown in Fig. lib, the lower portions of the two VBL's collide; and since they are unwinding, we expect formation of the IH structure shown in Fig. llc. (a) o'_ V r~._. ( ~ i'>0 t = t~ 1- (b) Hp 4t Vo IHB (c) 1 H _ ~ /"~T~ t=t 2 Fig.11. Proposed conversion of o_~ 1H. The detailsare di~ussedin thetext. For large values of Hp, we do ~o~ f~nd the q- bubble stable even for one step if V=HpXH B. This appears to be in conflict with the related transition 1/2+(1,0) discuss ed~in~Figs. 6 and 7, for which 1/2 is stablized for V=HpXH B if Hp is sufficiently large. At the present time, we do not have a satisfactory explanation for this difference. For conditions under which we expect to have IH or a- depending upon the direction of propagation of the S=+I bubble, we have observed two important differences confirming the existence of two distinct states. The mobility of IH is about 20 percent less that that of ~- if the measurement is made using propagation pulse durations comparable to the expected transit time of the 2~ HBL (0.2-0.5 ps). Stopping the respective bubbles, reducing Hp to zero, and subjecting them to a square collapsing bias field pulse, we ob- served that IH did not jump, but q- did jump in the direction of previous motion. The theory of bias Jumps will be discussed elsewhere. 18 Further confirmation of the existence of the IH bubble was obtained by subjecting a unichiral (I,0,0) bubble to the static cap-switch process. Quasi- statically increasing Hp to HC2 always resulted in a final state 1/2" as confirmed by propagation of the final state, and by subsequent observation of transi- tion B2 as Hp was reduced to HCI. 143 VI. SUMMARY We have found nine different transitions in ion- implanted garnet films, six of which are accounted for by cap-switching and injection of a BP. We found it necessary to invoke three different wall structures for S=+I bubbles depending upon the direction of motion in an in-plane field. We have determined the static and dynamic conditions under which these different wall structures are stable. There are two 1/2 states con- taining one BP which differ in the location of the BP and in the associated capping structure. There is one state containing two BP's, one on each line; and there is a single S=0 state. These results are applicable at sufficiently low velocities such that, for wall magnetization approxi- mately parallel to the in-plane field, there is no generation of Bloch loops. At higher velocities, there will be additional complications. ACKNOWLEDGEMENTS The authors are grateful for many helpful discus- sions with P. Dekker, J. C. Slonczewski, B. E. Argyle, and S. Maekawa, and with R. L. White of Stanford University. We are also grateful to J. Engemann for permission to quote his unpublished results, and to D. Y. Saiki and D. Johnson to device fabrication. REFERENCES i. J.C. Slonczewski, A. P. Malozemoff, and 0. Voegeli, AlP Conf. Proc. iO, 458 (1972). 2. R. Wolfe, J. C. North, and Y. P. Lal, Appl. Phys. Lett. 22, 683 (1973). 3. T. Hsu, AlP Conf. Proc. 24, 624 (1974). 4. R. Hasegawa, AlP Conf. Proc. 24, 615 (1974). 5. D.C. Bullock, AlP Conf. Proc. 18, 232 (1973). 6. R.M. Joseph, B. F. Stein, and W. R. Bekebrede, AlP Conf. Proc. 29, 65 (1975). 7. 0. Voegeli, C. A. Jones, and J. A. Broom, Paper 5A-6 (21st Annual Conf. on Magnetism and Magnetic Materials, 1975), to be published. 8. B.R. Brown, AlP Conf. Proc. 29, 69 (1975). 9. T.J. Beaulieu and B. A. Calhoun, Appl. Phys. Lett. 28, 290 (1976). i0. R. Wolfe and J. C. North, Appl. Phys. Lett. 25, 122 (1974). ii. J. Engemann (unpublished). 12. J. C. Slonezewskl, AlP Conf. Proc. 24, 613 (1974), and private communication. 13. J. C. Slonczewski, J. Appl. Phys. 45, 2705 (1974). 14. P. Dekker and J. C. Slonczewski, to be published. 15. A. A. Thiele, J. Appl. Phys. 45, 377 (1974). 16. F. B. Hagedorn, J. Appl. Phys. 45, 3129 (1974). 17. G. R. Henry and J. Gitschier, private communication. 18. A. P. Maloz~moff and S. Maekawa, to be published in J. Appl. Phys.
1.3460132.pdf	Structural stability versus conformational sampling in biomolecular systems: Why is the charge transfer efficiency in G4-DNA better than in double-stranded DNA? P. Benjamin Woiczikowski, Tomáš Kubař, Rafael Gutiérrez, Gianaurelio Cuniberti, and Marcus Elstner Citation: J. Chem. Phys. 133, 035103 (2010); doi: 10.1063/1.3460132 View online: http://dx.doi.org/10.1063/1.3460132 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v133/i3 Published by the AIP Publishing LLC. 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 19 Jun 2013 to 128.135.12.127. 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_permissionsStructural stability versus conformational sampling in biomolecular systems: Why is the charge transfer efﬁciency in G4-DNA betterthan in double-stranded DNA? P. Benjamin Woiczikowski,1Tomáš Kuba ř,1Rafael Gutiérrez,2Gianaurelio Cuniberti,2and Marcus Elstner1,a/H20850 1Department for Theoretical Chemical Biology, Karlsruhe Institute of Technology, D-76131 Karlsruhe, Germany 2Institute for Materials Science and Max Bergmann Center of Biomaterials, Dresden University of Technology, D-01062 Dresden, Germany /H20849Received 26 March 2010; accepted 15 June 2010; published online 19 July 2010 /H20850 The electrical conduction properties of G4-DNA are investigated using a hybrid approach, which combines electronic structure calculations, molecular dynamics /H20849MD /H20850simulations, and the formulation of an effective tight-binding model Hamiltonian. Charge transport is studied bycomputing transmission functions along the MD trajectories. Though G4-DNA is structurally morestable than double-stranded DNA /H20849dsDNA /H20850, our results strongly suggest that the potential improvement of the electrical transport properties in the former is not necessarily related to anincreased stability, but rather to the fact that G4 is able to explore in its conformational space alarger number of charge-transfer active conformations. This in turn is a result of the non-negligibleinterstrand matrix elements, which allow for additional charge transport pathways. The higherstructural stability of G4 can however play an important role once the molecules are contacted byelectrodes. In this case, G4 may experience weaker structural distortions than dsDNA and thuspreserve to a higher degree its conduction properties. © 2010 American Institute of Physics . /H20851doi:10.1063/1.3460132 /H20852 I. INTRODUCTION For the past two decades there has been a revival of interest in issues related to charge migration in DNA basedoligomers. On one side, charge transfer /H20849CT /H20850in DNA is sup- posed to play a key role in self-repair processes of DNAdamage in natural conditions, i.e., via oxidative stress. 1On the other side, DNA may have a huge potential applicationﬁeld in the development of complex nanoscale electronicdevices with self-assembling properties, since self-assembly,recognition, and fully automatic synthesis ofoligonucleotides 2–4can allow for building almost any imaginable two and three dimensional DNA motifs/H20849e.g., DNA-Origami /H20850. 5–7 Concerning charge transport, several experiments in the past years have obtained quite controversial results rangingfrom insulating, 8over semiconducting9to even metallic-like behavior,10–12which was apparently related to the difﬁculty of establishing well-controlled experimental setups for mea-suring the electrical characteristics of such complex biomol-ecules. Nevertheless, several experiments, which appearedrecently, 12–14have shown electrical currents in the nanoam- pere range despite differences in studied base sequences andexperimental setups. One central issue which has emerged bythe theoretical treatment of charge transport is the necessityto consider the conformational ﬂuctuations of the molecularframe not as a weak perturbation but rather as a crucial factorpromoting or hindering charge motion. 15–20Concerning thismatter considerable work has been done on the effects of structural ﬂuctuations and energetics on CT, not only in pep-tide nucleic acid /H20849PNA /H20850and DNA, but in biomolecules in general by the groups of Waldeck and Beratan. 21–23 It has been meanwhile shown that it is possible to syn- thesize DNA derivatives, which do not necessarily have thedouble-strand structure of natural DNA. Thus, C and G-richDNA strands are able to form four-stranded quadruplexstructures, like the i-motif structure composed of two parallelhemiprotonated duplexes intercalated into each other into ahead-to-tail orientation, and the G4-quadruplex /H20849also known as G4-DNA /H20850which is formed by either one, two, or four G-rich DNA strands in a parallel or antiparallel orientation. 24 The latter is supposed to play an important role in somebiological processes such as in telomeric DNA regions,where it inhibits telomerase and human immunodeﬁciencyvirus integrase. 25Additionally, G4-DNA is known to interact with various cell proteins that cause diseases such asBloom’s and Werner’s syndromes. 26Moreover, G4-DNA is cytotoxic toward tumor cells and, therefore, might be a keyfor the design of anticancer drugs. 27Eventually, G-quadruplexes are found to be thermally more stable thandouble-stranded DNA /H20849dsDNA /H20850. 28Gilbert and Sen29–31re- solved the x-ray structure of the ﬁrst G-quadruplex and pro-posed the formation of these unique structures in the pres-ence of monovalent alkali ions, see also Fig. 1/H20849a/H20850. Consequently, G4-DNA can be regarded as stacks composedof individual planar G-tetrads /H20849or G-quartets /H20850as shown in Fig.1/H20849b/H20850, each formed by four guanines bound together by a/H20850Electronic mail: marcus.elstner@kit.edu.THE JOURNAL OF CHEMICAL PHYSICS 133, 035103 /H208492010 /H20850 0021-9606/2010/133 /H208493/H20850/035103/12/$30.00 © 2010 American Institute of Physics 133, 035103-1 Downloaded 19 Jun 2013 to 128.135.12.127. 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_permissionseight hydrogen bonds via Hoogsteen pairing.32For detailed reviews on the structural diversity and special mechanical properties of G4-DNA based molecules, see Refs. 33and34. Recently, Porath et al.35showed the larger polarizability of G4-DNA compared to dsDNA, when attached to gold sur- faces. This might be an indicator of improved CT propertiesand the potential advantage of using G4-DNA rather thandsDNA in molecular scale devices. Eventually, the higherconductance in G4-DNA could be attributed to the increasedstructural stability and higher number of overlapping /H9266-orbitals. Furthermore, experimental work by Kotlyar et al.36,37revealed that up to 300 nm long G4-wires could be synthesized revealing a considerable stability even in the ab- sence of metal ions. Also, theoretical studies have been performed on the sta- bility and rigidity of G4-DNA. Molecular dynamics /H20849MD /H20850 simulations by Špa čková et al.38,39could conﬁrm the experi - mental results. For one thing, G4-DNA is more rigid thandsDNA and for another, monovalent ions within the quadru-plex cavity are necessary for the stability of short G4-DNA.On the other hand, MD simulations 40pointed out the stabilityof longer G4 molecules /H20849with 24 G-tetrads /H20850in the absence of metal ions which is consistent with the synthetic procedureof Refs. 36and37. Electronic structure calculations by Di Felice and co-workers 41–46showed a higher degree of delo - calization of the electronic states in G4 than in dsDNA.Moreover, the presence of metal ions may contribute addi-tional states supporting CT, though this is a still unresolvedissue. Guo et al. 47,48recently showed using the Landauer theory /H20849coherent transport /H20850that G4-DNA exhibits much larger delocalization lengths at the band center compared todouble-stranded poly /H20849G/H20850DNA. Surprisingly, it was also found out that the delocalization length can be even en-hanced via environment-induced disorder through the back-bones. Though these studies assume a static atomic structure,they nevertheless suggest that disorder plays an importantnontrivial role in mediating CT in DNA. Similarly, it is known from other biomolecules such as proteins that electron transport is dominated by nonequilib-rium ﬂuctuations which have lately been analyzed by Bal-abin et al. 49As demonstrated in previous studies, idealized static structures are not representative when considering CT properties, since dynamical as well as environmental effectswere shown to be too important. 50–54It was indicated that dynamic disorder has dramatic effects, since it suppresses CT in homogeneous sequences on the one hand, but canenhance CT in heterogeneous sequences, on the other. Forinstance, in our previous work, 20the conductance of the Dickerson dodecamer /H20849sequence: 5 /H11032-CGCGAATTCGCG-3 /H11032/H20850 was found to be almost one order of magnitude larger insolution /H20851quantum mechanics/molecular mechanics /H20849QM/ MM /H20850/H20852than in vacuo . Interestingly, similar ﬁndings have re- cently been shown by Scheer and co-workers, for they ob-tained for a heterogeneous sequence /H20849i.e., A and G bases are present /H20850with 31 base pairs 74two orders of magnitude larger conductance in solution compared to the in vacuo measurements.13Furthermore, it could be shown that only a minor part of the conformations is CT-active.20Therefore, neglecting these signiﬁcant factors or assuming a purely ran- dom disorder distribution can lead to a considerable loss of avital part of CT-relevant structural and electronic informa-tion. Relying on our previously developed methodologies to deal with CT in different DNA oligomers, 19,20,55we present here a detailed investigation of charge transport in G4-DNA. The paper is organized as follows. In Sec. II details of theMD simulations and of the electronic structure approach aredescribed. Subsequently, the MD and the electronic structuredata are analyzed in Secs. III A and III B, respectively. Fi-nally in Sec. III C, CT results for G4 molecules are com-pared with those of dsDNA in solution and in vacuo . More importantly, we point out signiﬁcant factors which are re-sponsible for the enhanced conductance in G4 molecularwires. II. METHODOLOGY A. Starting structures and simulation setup The molecules used in this work are based on the x-ray crystal structure of a tetrameric parallel-stranded quadruplex FIG. 1. /H20849a/H20850Tetrameric G4 x-ray crystal structure /H20851244D /H20849Ref. 56/H20850/H20852with four units /H20849TG4T/H208504: backbones are indicated as light blue ribbons, central coor- dinated sodium ions as dark blue spheres, terminal thymine and guanineresidues are shown in gray and orange, respectively. The highlighted qua-druplex is used as starting structure and also to generate longer G4 mol-ecules /H20849G 12/H208504and /H20849G30/H208504./H20849b/H20850Lewis structure of a single G4-tetrad with C4h symmetry containing a monovalent metal ion in its center. Guanines are bound together by eight hydrogen bonds via Hoogsteen pairing /H20849Ref. 32/H20850. The metal ion is coordinated either coplanar by four or cubic by eight O6oxygen atoms of the respective guanines which depends on the ionic radiusof M +.035103-2 Woiczikowski et al. J. Chem. Phys. 133, 035103 /H208492010 /H20850 Downloaded 19 Jun 2013 to 128.135.12.127. 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/H20849244D /H20850formed by the hexanucleotide sequence d /H20849TG4T/H20850in the presence of sodium ions. The structure has been resolved at 1.2 Å resolution by Laughlan et al.56As shown in Fig. 1/H20849a/H20850, it contains two pairs of quadruplexes /H20849TG4T/H208504, for which each pair is stacked coaxially with opposite polarity at the 5 /H11032ends. Moreover, nine sodium ions are located inside each stack of quadruplexes illustrating well orderedG4-DNA constructs. However, the terminal thymine resi-dues, shown in gray, were not completely resolved becauseof high thermal disorder. 56As indicated in Fig. 1only one of these four quadruplexes is taken as our basis structure, which corresponds to the parallel strands A, B, C, and D from thePDB ﬁle 244D. This quadruplex, from now on denoted as/H20849TG 4T/H208504, will be used for simulations and calculations. Fur- thermore, two longer G4 quadruplexes with 12 and 30 tet- rads, denoted as /H20849G12/H208504and /H20849G30/H208504, are generated by omitting the terminal thymine residues and adding subsequently G4 tetrads with a distance of 3.4 Å and twisted by 30°. Forcomparison, corresponding double-stranded B-DNAs withbase sequence 5 /H11032-TGGGGT-3 /H11032, poly /H20849G/H20850denoted as G 12and G30, respectively, and a sequence containing 31 base pairs74 /H20849Scheer /H20850are built with the make-na server.57 Depending on the helix length the G4 and dsDNA mol- ecules are solvated in a rectangular box with 4000–11 000water molecules using the TIP3P model 58and periodic boundary conditions are applied. The DNA molecules are centered in the box with a distance of at least 1.5 nm to thebox edges. In order to neutralize the total charge of the sys-tem, due to the negatively charged backbones, an appropriatenumber of sodium counterions /H20849Na +/H20850are added. Concerning the quadruplexes, we carried out four simulations for /H20849TG4T/H208504and /H20849G12/H208504, respectively, ﬁrst without any central monovalent alkali ions within the quadruplex, and then in presence of either lithium /H20849Li+/H20850, sodium /H20849Na+/H20850, or potassium /H20849K+/H20850ions within the respective G4 molecules. Therefore, ei- ther three /H20849/H20849TG4T/H208504/H20850or 11 /H20849/H20849G12/H208504/H20850of these ions are subse- quently placed in the center between two G4 tetrads coordi- nated by O6 oxygen atoms /H20851see also Fig. 1/H20849b/H20850/H20852. Nevertheless, the longer /H20849G30/H208504quadruplex is simulated only twice in the absence and presence of central sodium ions. All simulations are carried out with the GROMACS software package59using the AMBER parm99 forceﬁeld60including the parmBSC0 extension.61After a standard heating-minimization protocol followed b ya1n s equilibration phase, which is discarded afterward, we performed 30 ns /H2085150 ns for /H20849G30/H208504/H20852MD simu- lations with a time step of 2 fs. Snapshots of the molecular structures were saved every 1 ps, for which the CT param-eters were calculated with the self-consistent-charge density-functional tight-binding /H20849SCC-DFTB /H20850fragment orbital /H20849FO /H20850 approach as described in Sec. II B. B. Electronic structure In this section we will shortly describe how the elec- tronic structure of G4-DNA molecules is mapped to a coarse-grained transfer Hamiltonian using the FO approach. Themethod has already successfully been applied to dsDNAmolecules. 19,20,55,62A more detailed description of the meth -odology can be found in Ref. 63. To begin with, the elec- tronic structure of the DNA system is written in an effectivetight-binding basis /H20849fragment basis /H20850as H=/H20858 i/H9255iai†ai+/H20858 ijTij/H20849ai†aj+ H.c. /H20850. /H208491/H20850 The onsite energies /H9255iand the nearest-neighbor hopping in- tegrals Tijcharacterize, respectively, effective ionization en- ergies and electronic couplings of the molecular fragments.The evaluation of these parameters can be done very efﬁ-ciently using the SCC-DFTB method 64combined with a FO approach17 /H9255i=− /H20855/H9278i/H20841HˆKS/H20841/H9278i/H20856/H20849 2/H20850 and Tij0=/H20855/H9278i/H20841HˆKS/H20841/H9278j/H20856. /H208493/H20850 The indices iandjcorrespond to the fragments of the qua- druplexes which, in this model, are constituted by single gua-nine bases. Accordingly, the molecular orbitals /H9278iand/H9278jare the respective highest occupied molecular orbitals /H20849HOMOs /H20850 which are obtained by performing SCC-DFTB calculationsfor these isolated fragments, i.e., the individual guaninebases. Using a linear combinations of atomic orbitals ansatz /H9278i=/H20858/H9262c/H9262i/H9257/H9262, the coupling and overlap integrals can be efﬁ- ciently evaluated as Tij0=/H20858 /H9262/H9263c/H9262ic/H9263j/H20855/H9257/H9262/H20841HˆKS/H20841/H9257/H9263/H20856=/H20858 /H9262/H9263c/H9262ic/H9263jH/H9262/H9263 /H208494/H20850 and Sij=/H20858 /H9262/H9263c/H9262ic/H9263j/H20855/H9257/H9262/H20841/H9257/H9263/H20856=/H20858 /H9262/H9263c/H9262ic/H9263jS/H9262/H9263. /H208495/H20850 H/H9262/H9263andS/H9262/H9263are the Hamilton and overlap matrices in the atomic basis set as evaluated with the SCC-DFTB method,which is derived from DFT as a second order approximationof the DFT energy with respect to the charge density. For theFO approach, only the electronic part is used. Moreover,SCC-DFTB makes use of precalculated atomic orbital matrixelements H /H9262/H9263, which are computed by solving the atomic Kohn–Sham equations. Note that in the standard DFTB ap-proach these atomic orbitals are slightly compressed, 64while for the calculation of electronic couplings, we used uncom- pressed orbitals, since a proper description of the wave-function tails is essential for accurate couplings. 63 Note that since Tij0is built from nonorthogonal orbitals /H9278iand/H9278j, which is for various problems not suitable, we apply the Löwdin transformation65 T=S−1/2T0S−1/2. /H208496/H20850 The effect of the environment, i.e., the electrostatic ﬁeld of the DNA backbone, the water molecules, and the counterions/H20849including the central alkali ions within the quadruplex /H20850,i s taken into account through the following QM/MM Hamil-tonian, which enters Eq. /H208494/H20850,035103-3 Charge transfer in G4-DNA J. Chem. Phys. 133, 035103 /H208492010 /H20850 Downloaded 19 Jun 2013 to 128.135.12.127. 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_permissionsH/H9262/H9263=H/H9262/H92630+1 2S/H9262/H9263/H9251/H9252/H20873/H20858 /H9254/H9004q/H9254/H20849/H9253/H9251/H9254+/H9253/H9252/H9254/H20850 +/H20858 AQA/H208731 rA/H9251+1 rA/H9252/H20874/H20874, /H208497/H20850 where /H9004q/H9254are the Mulliken charges of atoms /H9251and/H9252in the QM region which is polarized by the surrounding MMcharges Q A, i.e., the DNA backbone, counterions, and water molecules. S/H9262/H9263/H9251/H9252are the overlap matrix elements of atomic orbitals /H9262and/H9263on atoms /H9251and/H9252,/H9253/H9251/H9254represents the effec- tive damped Coulomb interaction between the atomiccharges /H9004q /H9254on atoms /H9251and/H9252/H20849/H9254is the index running over QM atoms /H9251and/H9252/H20850, and ris the distance between QM atoms /H20849/H9251and/H9252/H20850and the MM charges QA. The coupling to the environment is therefore explicitly described via the interactions with the QAcharges. In the following, we will denote the calculation setup based on thecomplete expression in Eq. /H208497/H20850as QM/MM; neglecting the last term will be denoted as “vacuo.” Note that this approachdeviates from standard QM/MM methods, since the dynam-ics is completely driven by the MM subsystem, and theQM/MM part is only invoked on top of the purely classicalgeometries /H20849MD snapshots /H20850, while in standard QM/MM ap- proaches, the forces in the QM region are computed from theQM method as well. Nevertheless, the important point hereis that the QM energies are computed within the QM/MMapproximation, i.e., taking fully the electrostatic interactionwith the MM part into account, which is crucial to describethe interaction with solvent and counterions. 20,62,63 The electronic parameters /H9255iandTijare evaluated for every snapshot of the simulations. Subsequently, the transferHamiltonian is used to calculate the CT in the quadruplexesas described in Sec. II C. C. Charge transport through a four-stranded quadruplex Once the transfer Hamiltonian has been constructed, the transport properties will be calculated using Landauer theoryfor each snapshot along the MD trajectories. For this, weconsider a two-terminal setup where the G4 oligomer is con-tacted to left /H20849L/H20850and right /H20849R/H20850metallic electrodes through the four terminal bases. A central quantity to be computed is themolecular /H20849G4/H20850Green function, which can be obtained through a Dyson equation G −1/H20849E/H20850=E1−H−/H9018L−/H9018R. /H208498/H20850 Here/H9018Land/H9018Rare self-energy matrices characterizing the coupling to the electrodes. To simplify the calculations we donot take the full energy dependence of the self-energies intoaccount, but rather use the so-called wide band limit,where /H9018 Land/H9018Rare replaced by energy-independent parameters66,67/H20849/H9018L/H20850lj=−i/H9253L/H9254lk/H9254jk /H20849k= 1,2,3,4 /H20850, /H20849/H9018R/H20850lj=−i/H9253R/H9254lk/H9254jk /H20849k=N−3 ,N−2 ,N−1 ,N/H20850. We set /H9253Land/H9253Rto 1 meV. Within the Landauer approach, the transmission function T/H20849E/H20850for a given set of electronic parameters is then obtained as T/H20849E/H20850=T r /H20851/H9003LG/H9003RG+/H20852, /H208499/H20850 where /H9003Land/H9003Rare the broadening matrices calculated as the anti-Hermitian part of /H9018L/R, /H9003L/R=i/H20851/H9018L/R−/H9018L/R+/H20852. /H2084910/H20850 Using the former expressions, the conformational /H20849time /H20850de- pendent electrical current will be simply calculated by I/H20849U,t/H20850=2e h/H20885dE/H20873f/H20873E−EF−eU 2/H20874 −f/H20873E−EF+eU 2/H20874/H20874T/H20849E,t/H20850. /H2084911/H20850 TheI-Ucharacteristics presented in this work should be in- terpreted only qualitatively, for the Fermi energy /H20849EF/H20850is ar- tiﬁcially placed as average of the onsite energies for all gua- nine sites and for each snapshot, respectively. The readershould note that the current could exhibit quite differentshapes depending on where E Fis located. However, in our model only the current-voltage gap is affected by EF, whereas the maximum current obtained at high voltages isnot altered. The transmission function and the current are evaluated for every snapshot of the MD simulation in order to obtainstatistical average quantities. However, we are aware of thelimits of the coherent transport model which is valid only inthe adiabatic regime. Hence, we assume that the time scalesof the CT process are shorter than the fastest dominant struc-tural ﬂuctuations. This issue has also been addressed inRef. 20. III. RESULTS A. MD simulations In this section we sum up the most important results obtained from MD simulation for G-quadruplexes /H20849G12/H208504in the absence and presence of central alkali ions. More detailed information and further structural analysis, also for /H20849TG4T/H208504 and /H20849G30/H208504, can be found in Ref. 78. To begin with, the root mean square deviation /H20849RMSD /H20850 calculations, given in Fig. 2, exhibit a much higher rigidity, i.e., less structural disorder for G-quadruplexes /H20849G12/H208504in the presence of central ions compared to dsDNA G 12. However, without central ions the quadruplex appears not to be in equi-librium, for the RMSD increases for the whole simulationtime. As can be seen from molecular snapshots after 30 ns inFig.3, the structure is not entirely destroyed. The inner tet- rads roughly maintain a G-quadruplex form, whereas theouter ones are considerably disordered. On the other hand,the quadruplexes in the presence of alkali ions reveal highlyregular four-stranded structures. Generally, the ions prefer035103-4 Woiczikowski et al. J. Chem. Phys. 133, 035103 /H208492010 /H20850 Downloaded 19 Jun 2013 to 128.135.12.127. 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_permissionsdifferent coordinations within the quadruplex cavity, that is, lithium favors the planar position within the tetrads whilesodium and potassium are most likely to be found in thecenter between adjacent tetrads. Nevertheless, lithium andsodium are more ﬂexible than potassium ions, therefore al-lowing for longitudinal mobility along the quadruplex. Thisissue is analyzed in more detail in Tables SI and SII in Ref.78. The ﬁndings from the MD simulation about the higher rigidity of G4-DNA as well as the preferred locations ofcentral ions within the quadruplex are in perfect agreementwith those obtained by Špa čková et al. 38,68as well as by Cavallari et al.40 B. Electronic parameters 1. Molecular states involved in hole transfer Similar to our previous work on dsDNA,63the HOMOs are used to describe the hole transfer process in G4-DNA. To validate the SCC-DFTB electronic structure of idealized G4tetrads as well as snapshots from classical MD trajectories,benchmark calculations with Hartree-Fock /H20849HF /H20850and densityfunctional theory /H20849DFT /H20850methods were carried out. As a re- sult, the four highest occupied molecular states for the ide-alized tetrad have /H9266-symmetry and are rather delocalized over the four G bases, for the energy range between them isvery close as indicated in Table I. Moreover, these states appear to be linear combinations of HOMOs for the isolatedG bases. By contrast, once disorder is introduced, i.e., via struc- tural distortion and/or via the electrostatic surrounding com-posed by the solvent, backbone, and counterion charges, theenergy range between HOMO to HOMO-3 becomes signiﬁ-cantly larger; thus these states are rather localized onto thesingle guanine bases. Furthermore, the MO energies areshifted due to the electrostatic potential. Especially the cen-tral alkali ions seem to have a large impact on the MO ener-gies due to their close distance to the G bases. However, aQM treatment of the central ions revealed no contribution tothe hole transfer states. As a consequence, the electronic structure is mapped onto the single guanine bases rather than onto whole tetrads,which also reduce the computational costs immensely. Theeffect of central ions is captured by a classical treatment. Acomplete analysis of molecular states including MO energiesand visualizations as well as the treatment of central ionsboth quantum and molecular mechanically is shown in detailin Tables SIII–SV in Ref. 78. 2. Onsite energies and electronic couplings In this section we analyze CT parameters onsite energies /H9255iand electronic couplings Tijobtained for the simulations of /H20849TG4T/H208504and compare them to those for the crystal structure 244D as well as for an idealized G4 stack. The parameters /H9255i and Tijwere calculated as described in Sec. II B. Before showing the results, ﬁrst the applied fragment methodologyis introduced. The scheme in Fig. 4shows different types of electronic couplings T ijpresent in G4-DNA, from now on denoted as T1, T2, and T3. Here, T3 represents electroniccouplings within the respective G4-tetrads in-plane, whereasT1 and T2 denote intra- and interstrand couplings length-ways to the quadruplexes which occur on either one strand orbetween two strands, respectively. In contrast to dsDNA, ex-periments suggest that also competing horizontal CT can oc-cur in G4-DNA. 69Clearly, a charge can follow several path - ways along the quadruplex strongly dependent on those threecouplings. This should be an advantage compared to dsDNA.Even in small G4 stacks there is a large number of electroniccouplings. We will see that only a minor part of them will bevital for CT in G4-DNA. For instance, T3 couplings are usu- FIG. 2. RMSD of G4 and poly /H20849G/H20850DNA. /H20849a/H20850Parallel stranded quadruplex /H20849G12/H208504in the absence and presence of centrals ions Li+,N a+, and K+./H20849b/H20850 Comparison with double-stranded G12. FIG. 3. Molecular snapshot of parallel stranded quadruplex /H20849G12/H208504after 30 ns MD simulation in absence and presence of centrals ions Li+/H20849green spheres /H20850,N a+/H20849blue spheres /H20850, and K+/H20849purple spheres /H20850.TABLE I. Energies of HOMOs for a idealized G4 tetrad with C4h-symmetry, comparison among DFTB, DFT, and HF, for the latter the 6–31G /H20849d,p /H20850basis set is used, all values in eV. Method HOMO HOMO-1 HOMO-2 HOMO-3 HOMO-4 DFTB /H110024.588 /H110024.593 /H110024.593 /H110024.597 /H110024.870 PBE /H110024.356 /H110024.383 /H110024.383 /H110024.408 /H110025.267 B3LYP /H110025.170 /H110025.196 /H110025.196 /H110025.222 /H110026.637 HF /H110027.777 /H110027.803 /H110027.803 /H110027.828 /H1100210.827035103-5 Charge transfer in G4-DNA J. Chem. Phys. 133, 035103 /H208492010 /H20850 Downloaded 19 Jun 2013 to 128.135.12.127. 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_permissionsally quite small, especially that the diagonal in-plane cou- plings in Fig. 4/H20849G1-G3, G2-G4, …/H20850are generally negligible. Also, most of T2 couplings are small, except those betweenadjacent strands /H20849G1-G6, G2-G7, …/H20850, since these are rather close to each other. The largest couplings are certainly foundfor T1 /H20849intrastrand /H20850and they are well comparable to those for double-stranded poly /H20849G/H20850DNA. 63A summary of the average electronic couplings and onsite energies for the simulations of/H20849TG4T/H208504compared to the crystal structure /H20849244D /H20850as well as to the corresponding dsDNA, i.e., the two central guanines in 5 /H11032-TGGGGT-3 /H11032, is given in Table II. As has been discussed above, the central ions stabilize the G4 structure signiﬁcantly; the structural ﬂuctuations ofG4 with ions are much lower compared to G4 without ionsor dsDNA. The effect of ions on the electronic parameters istwofold: ﬁrst of all, the onsite energies are shifted down byabout 0.5 eV but, surprisingly, the enhanced structural stabil-ity does not lead to smaller ﬂuctuations of the onsite param-eters. They are in the order of 0.4 eV, similar to the situationsin dsDNA, as discussed recently. 62These large onsite energy ﬂuctuations are introduced by the solvent /H20849and not by ﬂuc- tuations of the DNA/G4 structure itself /H20850; therefore, they are not affected by the higher structural rigidity. The onsite en-ergy ﬂuctuations in vacuo , i.e., neglecting the last term in Eq. /H208497/H20850, reduce to 0.1–0.15 eV which corresponds to structuralﬂuctuations of the DNA bases and is in agreement with ﬁnd- ings reported by Hatcher et al. 21and Řehaet al.71In contrast, the ﬂuctuations of electronic couplings have the same mag- nitude as the averages themselves. More importantly, theydepend sensitively on DNA conformation, i.e., base stackingand structural ﬂuctuations, and only marginally on the sol-vent. These ﬁndings have been reported by severalgroups. 21,50,62,70 The T1 values, on the other hand, are even lower in the G4 structures with ions. This would indicate that these struc-tures conduct even less when compared to four strands ofdsDNA. However, the T2 and T3 values in G4 are still large,indicating that interstrand transfer can occur quite frequently.This opens a multitude of pathways for charge transport inG4 /H20849compared to dsDNA /H20850, which will be the key to under- stand G4 conductivity, as discussed in more detail below. As a ﬁrst result, we do not see any indication that the higher structural stability of G4 with central ions will lead toa higher conductivity due to a reduced dynamical onsite dis-order or due to increased electronic couplings, i.e., becauseof somehow better stacking interactions owing to the moreregular structure. Therefore, the higher conductivity musthave different reasons. The more stable structure leads tosmaller couplings, in contrast to prior expectations. 35A more detailed analysis of onsite energies and electronic couplings is given in Tables SVI–SXI in Ref. 78. C. Coherent transport in G4-DNA The parameters /H9255andTijanalyzed in the previous sec- tion are now used to evaluate the charge transport propertiesusing the approach described in Sec. II C. Concerning thecalculation of I-Ucharacteristics, it is important to point out that the Fermi energies E Fare not explicitly calculated, rather they are artiﬁcially placed as average of the onsiteenergies for each snapshot. For details, see Sec. II C. Thereader should be aware that the I-Ushape could be quite different depending on E F. To begin with, reference calculations of transmission function T/H20849E/H20850and current I/H20849U/H20850are performed on static struc- tures for both idealized G4 and dsDNA models and /H20849TG4T/H208504 based on the x-ray structure 244D. The data can be found in Figs. S3 and S4 in Ref. 78. Predominantly, we are interested in ensemble averages, for single snapshots or static structurecannot elucidate the CT process in DNA. Subsequently, wewill compare the CT properties of G4-DNA with those of dsDNA and distinguish the differences in CT efﬁciency by FIG. 4. Scheme for electronic couplings Tijin G4-DNA shown for the two innermost tetrads in /H20849TG4T/H208504: representatively, T1 indicates intrastrand, T2 interstrand, and T3 in-plane couplings. TABLE II. Average onsite energies /H20855/H9255/H20856and electronic couplings T1, T2, and T3 with standard deviations for the two central guanine tetrads of /H20849TG4T/H208504. Energies obtained from MD simulations in absence and presence of ions are compared to the crystal structure /H20849244D /H20850, an idealized G4 dimer stack, and also to the corresponding dsDNA structure, i.e., the two central guanines in 5 /H11032-TGGGGT-3 /H11032. Note that averaging is carried out not only along the MD time series but also over 8 an d 2 G bases for G4 and dsDNA, respectively. All values in eV. Type Ideal 244D MD no ions MD Li+MD Na+MD K+dsDNA /H9255/H11002 4.895 −4.905 /H110060.062 −4.812 /H110060.368 −5.339 /H110060.370 −5.400 /H110060.350 −5.201 /H110060.354 −4.790 /H110060.371 T1 0.028 0.051 /H110060.011 0.039 /H110060.028 0.031 /H110060.021 0.031 /H110060.021 0.029 /H110060.020 0.052 /H110060.034 T2 0.001 0.012 /H110060.002 0.010 /H110060.013 0.022 /H110060.014 0.015 /H110060.012 0.013 /H110060.010 0.004 /H110060.005a T3 0.009 0.007 /H110060.003 0.009 /H110060.009 0.007 /H110060.005 0.006 /H110060.004 0.007 /H110060.004 0.012 /H110060.008b aInterstrand coupling for 5 /H11032-3/H11032orientation /H20849G/C /H20850, for 3 /H11032-5/H11032/H20849G/C/H20850the coupling is 0.013 /H110060.014 eV. bCoupling within the WCP between G and C, note there is an energy gap of 0.4 eV.035103-6 Woiczikowski et al. J. Chem. Phys. 133, 035103 /H208492010 /H20850 Downloaded 19 Jun 2013 to 128.135.12.127. 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_permissionsmeans of conformational analysis. Finally, we determine the effects of the MM environment and in particular of the cen-tral ions on CT in G4 quadruplexes. 1. G-quadruplex versus dsDNA Previous studies revealed that idealized static structures cannot exhibit reasonable CT properties in DNA, since dy-namical as well as environmental effects were shown to becrucial. 50–54Dramatic effects are induced by dynamic disor - der, for it suppresses CT in homogeneous sequences on theone hand, but can also enhance CT in heterogeneous /H20849ran- dom /H20850sequences on the other. Moreover, only a minority of conformations appears to be CT-active as has been indicatedin Ref. 20. Therefore, neglecting these signiﬁcant factors or assuming purely random distributions for dynamical disorderleads to a considerable loss of a vital part of the CT in DNA.On this account the CT properties are evaluated for everysnapshot along classical MD trajectories which then leads toensemble averaged quantities, that is, the average transmis-sion function /H20855T/H20849E/H20850/H20856and the current /H20855I/H20849U/H20850/H20856. In Fig. 5/H20855T/H20849E/H20850/H20856and /H20855I/H20849U/H20850/H20856are shown for both quadru- plex molecules /H20849with central sodium ions /H20850, the two central tetrads of /H20849a/H20850/H20849TG 4T/H208504and /H20849b/H20850/H20849G8/H208504as well as their corre- sponding double-stranded poly /H20849G/H20850analogs G 2and G 8.75Be- cause of the substantial onsite energy ﬂuctuations of about0.4 eV due to the DNA environment and dynamics, the trans-mission function reveals large broadening for both DNA spe-cies. The transmission maxima for the quadruplexes areshifted to lower energies by about 0.3 eV due to the presenceof the central sodium ions. Basically, the average transmis-sion is strongly reduced compared to the idealized staticstructures in Fig. S4 in Ref. 78. Nevertheless, the maximum of/H20855T/H20849E/H20850/H20856for the two central tetrads in /H20849TG 4T/H208504is almost ﬁve times larger compared to G 2. Accordingly, the maximum cur- rent is about 4.4 times larger in the quadruplex.Considering the octamers in Fig. 5/H20849b/H20850the conductance difference between G4 and poly /H20849G/H20850even increases. Here, the average transmission in the relevant energy range for thequadruplex is to a great extent two orders of magnitudelarger than those for the poly /H20849G/H20850sequence. The poly /H20849G/H20850 spectra show much larger spikes at certain energies. Thesespikes indicate the strong impact of dynamics and may beexplained by the existence of few CT active conformations,which dominate the average transmission, i.e., few confor-mations, which feature a high transmission. This may indi-cate that even longer sampling /H20849more than 30 ns /H20850would be required to converge the spectra. Notwithstanding, it alsoreveals that in dsDNA the average current is dominated to amuch larger degree on few nonequilibrium structures, asconcluded from our earlier work. 20As a result, the average current /H20855I/H20849U/H20850/H20856for /H20849G8/H208504is almost two orders of magnitude larger than for G 8suggesting that the enhancement of CT in G4 with respect to poly /H20849G/H20850might grow with increasing DNA length. a. Length dependence. Thus, additional sets of CT cal- culations for longer DNA species with 14 and 20 tetrads /H20849base pairs /H20850, respectively, are carried out. The corresponding data were obtained from MD simulations of a quadruplex/H20849G 30/H208504/H20849including central Na+/H20850, and ds DNAs G 30and a het- erogeneous sequence containing 31 sites74recently used by Scheer et al.13in a CT-measurement /H20851Note, for CT calcula - tions only the 14 /H2084920/H20850central sites are used /H20852. As expected, the transmission strongly decreases for both DNA species. How-ever, in G4 this effect is not as strong as in dsDNA. Forinstance the current for /H20849G 14/H208504is only about one order of magnitude lower than for /H20849G8/H208504, although expanding to /H20849G20/H208504the current drops signiﬁcantly additional three orders of magnitude. In contrast, /H20855I/H20849U/H20850/H20856for poly /H20849G/H20850and the “Scheer” sequence decreases by more than ten orders of magnitude by increasing the number of base pairs from 14 to20. This indicate there is a much stronger distance depen-dence of CT in dsDNA; hence the notion of coherent CT forlonger molecular wires might be considerably more likely inG4 than in dsDNA. Admittedly, the Landauer formalismused in this work performs well for short DNA species /H20849less than ten sites /H20850, where the transport is assumed to be at least partially coherent. On the other hand, it clearly fails for longDNA sequences; hence the CT results for the longer mol-ecules should be interpreted only qualitatively and with cau-tion. For instance, the currents obtained for the 14mer and20mer of both dsDNA molecules poly /H20849G/H20850and the Scheer sequence are orders of magnitude smaller than picoamperewhich is far beyond any measurable range. The completedata are given in Fig. S5 in Ref. 78. 2. Analysis of CT differences in G4 and dsDNA The signiﬁcant conductance difference of G4 and ds- DNA may not be attributed to the fact that G4 is composedof four poly /H20849G/H20850like wires. To analyze this further the CT in /H20849G 8/H208504with central sodium ions /H20849full MD /H20850is compared to various models, in which /H20849i/H20850only intra- and interstrand76 couplings are nonzero /H20849intra+inter /H20850and /H20849ii/H20850the quadruplex is separated into its four single strands, for which the intras- FIG. 5. Average transmission /H20855T/H20849E/H20850/H20856and current /H20855I/H20856obtained from MD simulation: comparison between G-quadruplex and double-stranded poly /H20849G/H20850 DNA: /H20849a/H20850/H20849G2/H208504/H20849Na+/H20850and G2, i.e., the two central tetrads /H20849base pairs /H20850of /H20849TG4T/H208504/H208495/H11032-TGGGGT-3 /H11032/H20850, respectively. /H20849b/H20850/H20849G8/H208504/H20849Na+/H20850and G8. Generally, the two last tetrads /H20849base pairs /H20850at the 5 /H11032and 3 /H11032end are not considered for CT calculations to avoid end effects, although the simulations were donewith 12 tetrads and base pairs for G4 and dsDNA, respectively.035103-7 Charge transfer in G4-DNA J. Chem. Phys. 133, 035103 /H208492010 /H20850 Downloaded 19 Jun 2013 to 128.135.12.127. 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_permissionstrand transport is calculated independently and added up af- terward /H20849/H90184s/H20850. Furthermore the results are analyzed with ref- erence to G 8/H208494xpG /H20850. Note that, for comparison, the CT quantities of poly /H20849G/H20850are multiplied by 4. As it appears from Fig. 6, the average transmissions for the two models, full MD and intra+inter in the spectral sup-port region, clearly reveal the largest plateaus, which are alsoshowing similar peak structures. However, both quadruplexmodels exhibit similar moderate ﬂuctuations. By contrast,the sum of the four single G4 strands /H20849/H90184s/H20850shows much larger ﬂuctuations, comparable with those for G 8/H208494xpG /H20850. Moreover, the average transmission for /H90184s is signiﬁcantly reduced and is even slightly lower than 4xpG. Interestingly,in/H90184s are barely CT-active conformations, i.e., single domi- nating peaks like in 4xpG which sometimes even outreachthe maximum transmission of the quadruplex /H20849full MD /H20850. This might reﬂect the structural differences between quadruplexesand dsDNA, since the four strands in the quadruplex are notas ﬂexible as those in poly /H20849G/H20850/H20849see also RMSD ﬂuctuations in Fig. 2/H20850; hence in poly /H20849G/H20850the structural phase space is much larger and therefore, several high-transmissive struc-tures can arise. On the other hand, the more rigid quadruplexcovers only a smaller conformational phase space which in-deed ensures a large number of structures showing moderateCT properties for each single strand, respectively. Thisclearly indicates that the most important factor for the en-hanced conductance in G4 is the interstrand couplings be-tween the four strands in the quadruplex. Thus, if there areconformations in which the four isolated channels are nottransmissive, there is a considerable probability that CTmight occur via coupling between the individual stands. Ascan be extracted from Table IIthose interstrand couplings are sufﬁciently large with about 0.01–0.02 eV. As a conse-quence, at each snapshot there is a substantial amount ofpathways over which the CT might occur through the qua-druplex. These ﬁndings are supported by the I-V character-istics in Fig. 6. The current for the intra+inter model almost matches that of the full MD with 2.0 and 2.3 nA, respec-tively. If we switch off the interstrand couplings in the qua-druplex /H20849/H90184s/H20850the current will drop down to 0.08 nA. A slightly larger current of 0.11 nA is obtained for 4xpG. For further insights into the different CT properties be- tween G4 and dsDNA, we make use of conformation analy-sis. For that purpose, we investigate the amount of confor-mations which dominate the average CT. Additionally, thedistribution of transmissions for the multitude of conforma-tions is explored for both /H20849G 8/H208504and G 8. The results are pre- sented in Fig. 7. Panel /H20849a/H20850evidently indicates that there are substantially more conformations contributing to the averageCT in G4 than in poly /H20849G/H20850. Consequently, virtually every tenth G4 conformation is CT-active /H20849about 3000 out of 30 000 /H20850, whereas only 128 /H20849again out of 30 000 /H20850single non- equilibrium structures characterize the average CT inpoly /H20849G/H20850. Second, as demonstrated in the transmission probability distribution functions /H20851/H20849PDFs /H20850in Fig. 7/H20849b/H20850/H20852, the distribution width for the G4 quadruplex is considerably narrower com-pared to 4xpG and /H90184s. Note the x-axis is scaled logarith- mic. In addition, the G4 PDF is signiﬁcantly shifted to highertransmission. This underscores that the majority of poly /H20849G/H20850 structures is not transmissive; yet a few single conformationsare responsible for the average CT. On the other hand, for/H90184s the major part of conformations reveal higher transmis- sion than in 4xpG, but the single dominating conformationsare missing. Clearly, this explains that these single high-transmissive poly /H20849G/H20850conformations are the reason for a bet- FIG. 6. Effect of electronic couplings on CT in /H20849G8/H208504/H20849Na+/H20850: average transmission and current calculated /H20849i/H20850for the full time series of couplings obtained from MD /H20849full /H20850,/H20849ii/H20850only intra- and interstrand couplings are nonzero /H20849intra+inter /H20850,/H20849iii/H20850the quadruplex is separated into its four single strands /H20849/H90184s/H20850, for which the intrastrand transport is calculated independently and added up afterward, and /H20849iv/H20850for G8multiplied by 4 /H208494xpG /H20850. FIG. 7. Conformation analysis for /H20849G8/H208504/H20849Na+/H20850and G8,/H20849a/H20850number of con- formations that make up 90% of the average transmission maximum /H20855Tmax/H20856, /H20849b/H20850PDFs of Tmaxfor /H20849G8/H208504/H20849Na+/H20850and the two models 4xpG and /H90184s as used in Fig. 6.035103-8 Woiczikowski et al. J. Chem. Phys. 133, 035103 /H208492010 /H20850 Downloaded 19 Jun 2013 to 128.135.12.127. 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_permissionster conductivity in 4xpG compared to /H90184s. The higher aver- age T1 values in dsDNA are due to few highly conductingconformations. The smaller average of the G4 T1 couplingstherefore resembles a more stable structure, however, notleading to a higher conductance, as could be argued before-hand. The advantage of G4 over dsDNA is due to the exis-tence of non-negligible interstrand couplings in G4. Theamount of high-transmissive structures is remarkably in-creased compared to double-stranded poly /H20849G/H20850DNA. 3. Electrode connection effects In most of the conductance experiments for DNA the molecules are only connected with one strand to the respec-tive left and right contacts. For instance, in a very recentexperiment by Scheer et al. 73a single stranded nucleotide with sequence 5 /H11032-/H20849T/H11569G3/H20851TTAGGG /H208523T/H11569/H20850-3/H11032/H20849T/H11569denotes modiﬁed thymine residues /H20850, which is known to form stable quadruplexes, has been attached between two contacts.Therefore, the question arises whether the CT in an all-parallel stranded quadruplex differs if only two or even onestrand of the quadruplex are coupled to the left and rightcontacts, respectively. For that purpose, the CT in /H20849G 8/H208504 /H20849Na+/H20850is calculated for various contact models as illustrated in the scheme in Fig. 8:/H20849i/H20850all four strands are connected to the left and right electrodes, respectively /H208491–4 /H20850,/H20849ii/H20850only one strand is connected to both contacts /H208491-intra /H20850, and /H20849iii/H20850–/H20849v/H20850 one strand is attached to the left electrode while one of theremaining is contacted to the right one /H2084912-inter, etc. /H20850. The reader should note that there is no atomistic description ofthe electrodes in our model, rather the wide band limit isapplied as described in Sec. II C. As can be seen in Fig. 9, reducing the number of con- nected strands from 4 to 1 leads to a decrease in /H20855T/H20849E/H20850/H20856for the relevant energy range of about one order of magnitude. However, the transmission for the 1-intra and inter models isvery similar, indicating a minor signiﬁcance for which strandor strands are connected to the contacts. This ﬁnding is alsosupported by the I-V characteristics, for the current rangesaround 0.2 nA for these models which is roughly one orderof magnitude smaller as though all four strands are attached/H208491–4 model /H20850. Interestingly, there seems to be an increase in the transmission ﬂuctuations if the quadruplex is contactedthrough only one strand at each end. Notwithstanding, thetransmission plateau for the one-stranded /H20849intra and inter /H20850 contacted models is still about 1.5 orders of magnitude largercompared to those for poly /H20849G/H20850resulting in an average current which is again one order of magnitude smaller with 0.025nA. Thus our results suggest that independent on the variouscontact linking schemes of G4 and dsDNA, one might expecta higher conductivity for the quadruplex. Certainly, the opti-mal conductance for all-parallel stranded G4-DNA is ensuredif all four strands are coupled to the contacts. 4. Effect of DNA environment on transport The major part of the dynamical disorder is induced by the QM/MM environment, i.e., the last term in Eq. /H208497/H20850which is built of the MM charges of DNA backbone, solvent, andcounterions. Previous results have indicated that the disorderdue to the DNA environment might not only suppress CT inhomogeneous sequences such as poly /H20849G/H20850, rather it is able to enhance CT in random sequences like the heterogeneousDickerson dodecamer. 20Very recent experiments by Scheer et al. conﬁrmed this notion, since they found the current for a 31 base pair sequence74to be two orders of magnitude smaller in vacuo than in aqueous solution.13Therefore, it might be interesting to investigate what happens if we switch of the environment for CT in G4-DNA, also with respect tothe effect in dsDNA. As can be seen in Fig. 10/H20849a/H20850, the trans- mission maximum for /H20849G 8/H208504in vacuo is about two orders of magnitude larger than with the QM/MM environment. Be- sides, its broadening and also the ﬂuctuations are signiﬁ-cantly reduced and the plateau is shifted to higher energiesdue to the neglect of the electrostatic interaction with thesodium ions within the quadruplex. The transmission forpoly /H20849G/H20850in vacuo shows basically the same features, for the FIG. 8. Modeling different types of connections to the left and right elec- trodes: /H20849a/H20850all four G strands are connected /H208491–4 /H20850,/H20849b/H20850only one strand is connected with both termini /H208491-intra /H20850,a n d /H20849c/H20850one strand is connected to the left electrode while one of the remaining is contacted to the right one /H2084912- inter, etc. /H20850. FIG. 9. Effect of electrode connections on CT in /H20849G8/H208504/H20849Na+/H20850: average transmission and current calculated for the various contact models from Fig.8. Comparison with poly /H20849G/H20850.035103-9 Charge transfer in G4-DNA J. Chem. Phys. 133, 035103 /H208492010 /H20850 Downloaded 19 Jun 2013 to 128.135.12.127. 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_permissionsbroadening is likewise strongly reduced and the maximum is located in the same energy range as for G4 in vacuo , al- though it is two orders of magnitude smaller. In general, thetransmission for G4 including the QM/MM environmentclearly reveals the largest broadening which might indicatethat the central sodium ions have an additional strong inﬂu-ence on the dynamic disorder due to longitudinal mobilitywithin the quadruplex, which is not the case in dsDNA. De-spite all that, the current at high voltages is larger for G4with QM/MM than for poly /H20849G/H20850in vacuo . Basically, for heterogeneous sequences /H20849not shown here /H20850 a reduced transmission in vacuo is found, which is caused by energy gaps between A and G states. Notwithstanding, forDNA molecules with no static energy gaps such as double-stranded poly /H20849G/H20850and G4-DNA /H20849both with uniform DNA bases /H20850, the QM/MM environment is most likely to increase the dynamical disorder, thus will suppress CT compared tothe vacuo model. As a result, there is no signiﬁcant differ-ence in the effect of the DNA environment on CT for G4 andpoly /H20849G/H20850DNA. This is also underscored by conformation analysis given in panels /H20849b/H20850and /H20849c/H20850in Fig. 10, which indicate that there are considerably more CT-active conformations in vacuo than with QM/MM environment for both DNA species G4 and poly /H20849G/H20850. Moreover, for G4 in vacuo nearly every conformation appears to be high-transmissive, since the av-erage maximum transmission /H20855T max/H20856increases almost lin- early with the number of conformations. Thus, the CT in vacuo is only marginally affected by single nonequilibrium conformations, rather the whole ensemble of G4 conforma-tions seems to be CT active which is demonstrated in thePDF of transmission. 5. Effect of channel ions on transport One last issue remains when considering CT in G4-DNA that is the effect of ions within the quadruplex channel. Thestructural inﬂuence of these ions has already been addressedin detail in Sec. III A. As could be seen in Table SIV in Ref.78we did not ﬁnd the central ions to contribute states in therelevant energy range for hole transfer in G4; therefore, the effect will only be investigated electrostatically. In Fig. 11 the average transmission and current is shown for the qua-druplex simulations of /H20849G 8/H208504in the absence and presence of either lithium, sodium, and potassium ions. Obviously, the transmission maximum is only marginally affected by thepresence of different types of central ions. Furthermore, asexpected the transmission function is shifted to lower ener-gies if central ions are present. However, for potassium thetransmission is to be found slightly reduced, also the broad-ening is not as large as for the other species. This is alsoreﬂected in the PDF of transmission maxima which can befound in Fig. S6 in Ref. 78. As a consequence, the average current for the simulation with central potassium ions is halfas large as for those with lithium and sodium ions whichmight be attributed to the different mobilities of Li +and Na+ compared to K+. Interestingly, there is no signiﬁcant differ- ence for CT in the absence and presence of ions, although itis known from Sec. III A that the G4 molecules without cen-tral ions exhibit signiﬁcant destabilization. Once more thissupports the notion that the enhanced conductance in G4may not exclusively be explained in terms of higher struc-tural rigidity, i.e., less dynamical disorder, rather it is themultitude of CT pathways via interstrand couplings thatbrings on an increased number of high-transmissive confor-mations. Apparently, those interstrand and in-plane couplingsT2 and T3 are not altered by the presence of central ions, i.e.,by a more rigid quadruplex structure. IV. DISCUSSION AND CONCLUSION In this work, we have investigated the conductivity of G4 with respect to dsDNA using classical MD simulations,combined QM/MM methods to compute CT parameters and FIG. 10. Inﬂuence of MM environment on CT: comparison between /H20849G8/H208504 /H20849Na+/H20850and G8./H20849a/H20850Average transmission and current. /H20849b/H20850Number of confor- mations that make up 90% of the average transmission maximum /H20855Tmax/H20856and c/H20850PDFs of Tmax. FIG. 11. Inﬂuence of central ions on CT: calculation of the average trans- mission and current for /H20849G8/H208504in the absence and presence of monovalent central ions Li+,N a+, and K+.035103-10 Woiczikowski et al. J. Chem. Phys. 133, 035103 /H208492010 /H20850 Downloaded 19 Jun 2013 to 128.135.12.127. 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_permissionsLandauer theory to compute transmission and I-Ucharacter- istics. These properties have been evaluated for phase-spaceensembles of G4 and dsDNA structures. The approach adopted here to calculate the transport characteristics at a given time /H20849Landauer theory /H20850has clear limitations related to the underlying assumption of tunnelingtransport. Though this may be an efﬁcient pathway over veryshort segments of the molecules under study, its validity be-comes questionable with increasing length. This becomesquite evident when investigating sequences as used in recentexperiments, e.g., by Scheer et al. , 13where our methodology would predict currents order of magnitude smaller than those found in the experiments. Notwithstanding, the Landauer re-sults on dsDNA, concerning promoting modes and CT pa-rameter ﬂuctuations, are in agreement with ﬁndings of pre-vious work by Kuba řet al. 72on the simulation of incoherent hopping in DNA, in which the hole wave-function has been propagated using the time dependent Schrödinger equation.We are currently developing a methodology in order to de-scribe transport in complex materials in a more generalway. 19,55 Nevertheless, the Landauer calculations reported here give interesting insight into particular properties of G4, es-pecially when compared to dsDNA. First of all, G4 withcentral ions is structurally much more stable than dsDNAand has more /H9266-contacts along the chain.35At ﬁrst sight, one may argue that this leads to improved electrical conduction; however, it turns out that the large ﬂuctuations found fordsDNA lead to highly conducting structures, which dominatethe transport. Therefore, a dsDNA molecule conducts betterthan one strand of G4, or equivalently, the higher conductionof G4 is not due to an increased structural stability of itssingle strands. The phase space of the single strands stillcontains a vast majority of conformations, which are not CTactive. It is the ability of G4 to allow for a large number ofconformations due to the interstrand couplings T2 and T3which make this species better conducting. If the pathwayalong one strand is blocked, e.g., when one T ijalong the chain vanishes, many other conduction channels may be vi-able due to interstrand hopping. At the end, G4 has a muchlarger number of CT active conformations than four dsDNA /H20851poly /H20849G/H20850/H20852offers. This is the basis of the higher conductivity of G4. This advantage is even maintained when contactedonly at one G-site, instead of four sites. The structural differences between “CT-active” and “CT-silent” conformations are quite subtle; therefore it isquite difﬁcult to characterize high-conducting conformationsin terms of structural properties. Generally, CT-active confor-mations are characterized by good stacking /H20849i.e.,T ijfor vari- ous pathways along the quadruplex are large /H20850and low onsite energy disorder /H20849i.e., all /H9255iequal /H20850as discussed in detail in Ref. 20. As a second point, dsDNA or G4 may be exerted to strain due to the contacting procedure. Here, clearly thehigher stability of G4 may help to maintain a conductingconformation, while the conduction in dsDNA may be muchmore easily disrupted. 73 In a recent work, we have predicted dsDNA with non- uniform sequences /H20851i.e., not poly /H20849G/H20850or poly /H20849A/H20850/H20852to conductbetter in solution than in gas phase. 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Ed. 49, 3313 /H208492010 /H20850. 745-thiol-dG-GGC GGC GAC CTT CCC GCA GCT GGT ACG GAC. 75Note, generally, the two last tetrads /H20849base pairs /H20850at the 5 /H11032and 3 /H11032end are not considered for CT calculations to avoid end effects, although the simulations were done with 12 tetrads and base pairs for G4 and dsDNA,respectively. 76Only T2 interstrand couplings between adjacent strands are considered. See also the scheme in Fig. 4. 77Note, that this only applies when the whole structure is homogeneous. If other bases such as A or T enter the sequence, the behavior may be the opposite. 78See supplementary material at http://dx.doi.org/10.1063/1.3460132 for further data analysis.035103-12 Woiczikowski et al. J. Chem. Phys. 133, 035103 /H208492010 /H20850 Downloaded 19 Jun 2013 to 128.135.12.127. 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
1.1708053.pdf	311 and 311 Damped Spherical Ferrimagnetic Resonance Modes Oscar J. Van Sant Citation: Journal of Applied Physics 37, 4422 (1966); doi: 10.1063/1.1708053 View online: http://dx.doi.org/10.1063/1.1708053 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/37/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in InSb(111)3×1: New surface reconstruction J. Vac. Sci. Technol. B 14, 957 (1996); 10.1116/1.589183 Dynamical conversion of optical modes in garnet films induced by ferrimagnetic resonance J. Appl. Phys. 62, 648 (1987); 10.1063/1.339794 Modes in SphericalMirror Resonators J. Appl. Phys. 36, 1306 (1965); 10.1063/1.1714301 Ferrimagnetic Resonance Modes in Spheres J. Appl. Phys. 30, 687 (1959); 10.1063/1.1735216 Multiplicities of the Uniform Precessional Mode in Ferrimagnetic Resonance J. Appl. Phys. 29, 324 (1958); 10.1063/1.1723118 [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: Wed, 03 Dec 2014 00:02:214422 W. E. FLANNERY AND S. R. POLLACK rence of the abnonnal samples may be associated with traps in the oxide. The deviations from linearity in Figs. 8 and 9 were noted previously by Standley and Maisse1.4 Since the low-temperature tunnel current was subtracted from the total current in Fig. 9, all of the current should have been Schottky current. Therefore, there is no explana-tion for the observed deviation based on electrode limited Schottky emission. This problem, therefore, has not been resolved by the present investigations. The low-voltage behavior shown typically in Fig. 8 is also not explicable on the basis of Simmon's Schottky emission analysis and again indicates the possible presence of trap excitations. JOURNAL OF APPLIED PHYSICS VOLUME 37, NUMBER 12 NOVEMBER 1966 311 and 311 Damped Spherical Ferrimagnetic Resonance Modes OSCAR J. VAN SANT U. S. Naval Ordnance Laboratory, Silver Spring, Maryland (Received 1 April 1966) The technique employed by Fletcher and Bell to obtain the potential functions of the various modes in undamped ferrimagnetic spheres has been extended to the damped case. Solutions for the potential func tions, induced magnetic-moment vectors, and the power absorbed by the sphere in the 311 and 311 modes are given when the Gilbert equation of motion is used to obtain the effects of damping. Sketches of these modes are shown along with graphs which describe the motion of the rf magnetic-moment vectors induced in the sphere. INTRODUCTION IN solid-state microwav_e amplifiers employing YIG spheres the 311 and 311 modes have been used for parametric amplifier operation because they require a minimum of pump field. The adjustment of the YIG sphere and the probe in the cavity is very critical for optimum operation of such an amplifier. For these reasons the behavior of the 311 and 311 modes have been the primary target of an investigation summarized by this paper. CALCULATIONS OF THE DAMPED SOLUTIONS The following solutions involve essentially a notation similar to that employed by Fletcher and BelU How ever, some exceptions are as follows. The equation of motion for the magnetic-moment vector per unit volume M in a magnetic field intensity H is as- sumed to be, M=-'Y(M xH), (1) which has a negative sign instead of a positive one preceding 'Y = / e/ mc /. In this case the complex time factor of H becomes exp( -iwt) rather than exp(iwt). This still gives/,2 (2a) (2b) where hx and hll are the components of the rf field and the factors K and v are now assumed to be complex rather than real [see Eqs. (23a)-(23d)J. Other excep tions are that in the expression for the internal potential, y"nintm= P "m(~)p "m(COST]) X[Gnm cosm¢+H"m sinm¢], (3) and the external potential, y"noutm=rnP"m(cos9)[Anm cosm¢+B"m sinm¢] + P" m( cos9) [Dnm cosm¢+ F nm sinm¢ ]r-n-l, (4) the coefficients A"m, Bnm, Dnm, Fnm, Gnm, and Hnm are assumed to be complex rather than as indicated by Fletcher and Bell.1 The general solution given by (3) is a solution of the partial differential equation/,2 (a¥ a2if;) a2if; (l+K) -+-+-=0. ax2 ay2 az2 (5) From the boundary conditions one can obtain the real and imaginary parts of Gnm and H "m in terms of the real and imaginary parts of Anm and Bnm, that is, Gnrm= [anmA nrm+bnmA nim+CnmBnrm+dnmBnim] (Znm)2, Gnim= [ -b"mA nrm+anmAnim-d"mBnrm+cnmBnr](Znm)2, H "rm= [-cnmAnrm-dnmA n;m+anmBnrm+b"mBnr] (znm)2, H nim= [dnmA nrm-cnmA nim-bnmBnrm+anmBn;mJ(znm)2, (6a) (6b) (6c) (6d) 1 P. C. Fletcher and R. O. Bell, J. App!. Phys. 30, 687 (1959). 2 L. R. Walker, J. App!. Phys. 29, 318 (1958). [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: Wed, 03 Dec 2014 00:02:21DAM P E D S P HER I CAL FER RIM A G NET I eRE SON A NeE MOD E S 4423 where where and where anm= enm[(enm)2+ (fnm)2+ (gnm)2-(hnm)2J-2Jnmgnmhnm, bnm= Jnm[(fnm)2- (gnm)2+ (hnm)2+ (enm)2J-2gnmhnmenm, cnm= -gnm[(gnm)2+ (hnm)2+ (enm)2-(fnm)2J+2h nmenmjnm, dnm= hnm[(hnm)2- (enm)2+ (fnm)2+ (gnm)2]- 2enmjnmgnm, Jnm= [(n+ 1)P nim(eo)+eoiP nrm(eo)'+eorP nr(eo)']/ (2n+ 1)an, gnm= m[viP nrm( eo)+vrP nim( eo) J/ (2n+ 1)an, hnm=m[vrP nrm(eo)- ViP nr(eo)]/(2n+ 1)an, (Znm)2= [(enm)4+ (fnm)4+ (gnm)4+ (hnm)'-8e nmJnmgnmhnm+ 2(enm)2 (fnm)2 (7a) (7b) (7c) (7d) (8a) (8b) (8c) (8d) + 2 (enm)2(gnm)2- 2 (enm)2 (hnm)2-2 (fnm)2 (gnm)2+ 2 (fnm)2 (hnm)2+ 2 (gnm)2(hnm)2]-t, (9) where a is the radius of the sphere. In the above expressions the rand i subscripts indicate the real and imaginary parts, respectively, and eo2= 1+K-1• Let us assume that With Knm and Lnm so defined we can now obtain for the time varying internal potential, where (anm-dnm)Lnm- (bnm+cnm)Knm Xnm=tan 1 • (anm-dnm)K nm+ (bnm+cnm)Lnm For the 3, 1 modes it follows that, (10) (lla) (llb) (13) (14) (15) (16a) (16b) (17a) (17b) [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: Wed, 03 Dec 2014 00:02:214424 where and where OSCAR J. VAN SANT Cal= -[9 (zi)2/16?r(-K)ia ][ (aa1_Jal)2+ (bal+ca1)2]i[(Kal)2+ (Li)2]i, Ca-I= [9 (Za-I)2/230411"( -K)!aJ[ (aa-1-da-I)2+ (ba-1+ca-1)21![ (Ka-l)2+ (La-I)2]!, CALCULATION OF THE POWER ABSORPTION (18a) (18b) (19) To observe the ferromagnetic resonance effect in spheres, one often observes the power spectrum. The average power absorbed per unit volume by the sphere U nffl in emu may be obtained from the expression, (20) where V is the volume of the sphere. For the 3, 1 modes this turns out to be, 81w(za1)4[ (aal-dal)2+ (bal+cal)2][ (K al)2+ (Ll)2] --------------{ (Ki-Jli)[16(K.2+K;)+40K,+25J+ 10K;), 2811"1 Klaa2 (2la) 81w(za-1)4[ (aa-Lda-1)2+ (ba-1+ca-1)2][ (K a-1)2+ (La-1)2] Ua-J= {(Ki-Jli)[16(Kr2+K;)+40Kr+25]+lOKi}. (2lb) 28(12)411" 1 K laa2 In order to give an example of the results of the solutions of the damped case, the Gilbert equation of motion3 shall be assumed to obtain the real and imaginary parts of K and JI. It can be shown that the Gilbert equation of motion can be expressed in the form of . M= -'Yo(M xH)+'Y(A/M)[M x (H xM)], (22) where 02+A2=O. From (22) one can obtain, when A~<l, Kr= (QH2-Q2)QH/[(QJ?_Q2)2+4A 2Q2QJ?J, Ki= A (QJ?+Q2)Q/[ (QJ?_Q2)2+4A2Q2QH 2J, l'r= (QH2-Q2)Q/[ (Q~-Q2)2+4A2Q2Q~J, JI;= 2AQ2QH/[ (QH2-Q2)2+4A2Q2!2H2J, (23a) (23b) (23c) (23d) where Q=w(47r'YM)-t, (24a) !2H= (Ho-411"M/3)/47rM, (24b) where Ho is the large static magnetic field intensity applied in the z direction. Table I shows some example calculations when A=O.OOOl, QH=0.796, and 47rM=1780 Oe. Here it is seen that Qnm,....,I/A and that for a given value of K and L, the power absorbed by the uncoupled 311 mode is about 4X 102 times that of the uncoupled 311 mode. The table also shows that the average absorbed power density varies as a.4 It turns out that for the assumed value of A, the resonant frequency iT of the damped 310 mode equals that of the undamped mode to about six figures and the resonant frequencies of the damped 311 TABLE 1. Calculations of the average absorbed power density U nm, the amplitude factor Cn'", and Qnm of the 3, 1 modes in terms of the sphere radius and potential constants (Knm)2+ (Lnm)2= [An'" [2= [Bnm [2 when X=0.0001, flB = 0.796, and 4n-M = 1780 Oe. Cnm Unm fl ""f. a2[ (Knm)2+ (Lnm)2]' ')'a4[ (Knm)2+ (Lnm)2] n m r flB (Me/sec) (Oe) Qnm 3 1 0 1.161748 4607.9 -4.29X1Q2 3.28 X 105 4.95X1Q3 3 I 1 1.439407 5709.2 4.19XlOl 5.71X108 4.71X1Q3 3 1 1 1.457125 5779.5 -8.20X1Q2 2.28X10· 4.29X1OZ 3 T. L. Gilbert and J. M. Kelly, Proceedings of the American Institute Electrical Engineers Conference on Magnetism, Pittsburgh, June 1955, pp. 253--263. 4 P. C. Fletcher, 1. H. Solt, and R. O. Bell, Phys. Rev. 114, 739 (1959), [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: Wed, 03 Dec 2014 00:02:21DAM P E D S P HER I CAL FER RIM A G NET I eRE SON A NeE MOD E S 4425 and 3 i1 modes equal those of the undamped modes to at least seven figures. The table also shows calculations of c"mja2[(K"m)2+ (L"m)2Jl. This factor is helpful in comparing values of M n m in terms of the values of the external potential constants An'" and Bnm and the radius of the sphere since (K"m)2+CLnm)2= IA,,'nI2= IB"mI2. 311 AND 311 RESONANCE MODES When I Ki/Krl, I vJvrl «1, an approximate picture of the orientations of the physical rf components of the induced magnetic dipole moments per unit volume mn",m and mnym may be obtained by first writing expressions (16a)-(16b) in the form: M3Hc31= [a31-~31+iIN] exp[i(XaLwt)], e2Sa) MayljCal= [ -i(aal+~al)-tin exp[i(Xa1-wt)], (2Sb) M ax-l/Ca-l= [aa-l-~a-I-itia-l] X exp[i(Xa-L wt)], (2Sc) M ay-IjCa-l= [i(aa-l+~a-l)-tia-l] X exp[i(Xa-I-wt)], (2Sd) where aal= (1-njnIl){20(1+K) (ka)2 -10[(k 1)2+ (k2)2]-4K), (26a) aa-l= (1+njnH){20(1+K)(k a)2 -10[(k 1)2+ (k2)2]-4K}, (26b) ~al=S(1+njnH)[(kl)2- (k2)2], (27a) fa-l = 5 (l-njnIl)[(kl)L (k2)2], (27b) tia1= 10(1 +njnIl)k 1k2, e28a) tia-l= 10(1-njn Il)k1k2, (28b) where kl=xja, k2=y/a, k3=z/a, (29) and where Ki and vdn the right sides of Eqs. (26a)-(28b) have been neglected. Expressions for mnxm and mnym may be obtained by taking the real parts of M nxm and M nym, respectively, which are, m3ijC}= (a,I-fal) cos (wt-Xal) +til sin (wt-xa1), (30a) mayljCal= -tial cos (wt-xa1)-(aal+fal) Xsin(wt-xh (30b) ma,,-l/Ca-l= (a3-1-fa-I) cos(wt-xa-l) -tia-l sin (wt-xa-I), (30e) may-IjC 3-1= -tia-l cos (wt-Xa-l) + (agl+E3-1) Xsin(wt-Xa-1). (30d) For the remaining discussion K is considered to be real wherever it directly appears. As long as -K~ 1.25, it is helpful to consider that m" m consists of two components in the following manner, (31) where maa)/C31=t(aal-Eal) cos(wt-Xa l) -J(al+€al) sin(wt-xa l), (32a) mal/Cal = ti31[i sin(wt-xal)- J cos(wt-xa l)], (32b) maa,.-ljC3-1= i(aa-I-fa-I) cos (wt-Xa-1) + J(aa-I+€3-1) sin(wt-Xa-1), (33a) mati-IjCa-'= -tia-Iet sin(wt-xa-l)+ J cos(wt-xa-l)]. (33b) Figure 1 is a conventional sketch (not to scale) of the loci of msa,NCal in various parts of the plane z=O. The horizontal axis serves as both the geometrical x axis and the axis for maa,.x1jCsl in unspecified arbitrary units. Similarly, the vertical axis serves as both the geometricaly axis and the axis for m3a.,./jCal in un specified arbitrary units. The largest circle is the perimeter of the cross section of the sphere of radius a. The loci of maa,.' are either circles or ellipses having their semi-axes lying along and perpendicular to the radius. For a given radius, En m varies as I cos2¢ /. The solid straight vectors indicate. the position of ma", NC31 when wt = Xal• The cross section of the sphere is divided into nine areas by its intersection with four hyper boloids given by, S(l-~~) (kl)2+5(3-~) (k2)2 nIl nIl +(1-n:)[4K-20(1+K) (ka)2J=O, (34a) 5(3 -~) (k1)2+ 5(1-3n) (k2)2 nIl nlI All of the vectors in the shaded areas travel in a counter clockwise direction, and conversely, all of the vectors in the unshaded areas travel in a clockwise direction as indicated by the arrows on the circles and ellipses. At the center of the sphere the locus of maa./ is circular in the clockwise direction. As one travels along the x axis from the center, the loci become more and more elliptical until point A is reached where the motion is linear in the x direction. Point A is the point at which the motion is reversed. Past point A, the motion is now counterclockwise with the loci becoming more circular. At point B the locus is circular and past point B the loci become more and more elliptical but this time with [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: Wed, 03 Dec 2014 00:02:214426 OSCAR ]. VAN SANT y FIG. 1. Conventional diagram of the loci of the motion of maaj/Ca1 in the 311 mode in the plane z=O. x the semi-minor axis lying along the x axis. On the circle having radius OB and along the 45° axes, the motion is circular. Points C, where the hyperbolas intersect, are points of reversal of motion and also the only points in the cross section where the motion of maa,/ is zero. More generally, the hyperboloids given by (34a) and (34b) are surfaces of reversal of motion of ma",.l and the intersections of these hyperboloids are lines where the motion of maa) is zero. Using the convention described for Fig. 1, Fig. 2 is a sketch (also not to scale) showing the loci of the y FIG. 2. Conventional diagram of the loci of the motion of mapl/Cal in the 311 mode in the plane z=O. motion of ma/NCal in the plane of z=O. All of the loci are circular with the motion in the clockwise direction. In the x= 0 and y= 0 planes the motion is zero, and along the 45° axes the magnitude of ma/l! actually varies as the square of the distance from the origin. The points of constant amplitude of mai form hyperbolas having the x and y axes asymptotes. The straight vectors show that the position of ma/ll/Cl is in the negative y direc tion in the first and third quadrants and in the positive y direction in the second and fourth quadrants when wt=xa1• The places where the motion of ma/l) is zero do not coincide anywhere with the places where the motion of maa) is zero and so at no point in the sphere in the 311 mode is the motion of mal zero. Figure 3 is a conventional sketch showing the loci of motion of ma",.-l/Ca-l in the plane of z=O. The cross y -M-+--X FIG. 3. Conventional diagram of the loci of the motion of maa ...... I/Ca-I in the 311 mode in the plane z=O. section of the sphere is divided into six areas by its inter section with two ellipsoids given by 5( 3+ n:) (kl)2+ 5( 1 + ~:) (k2)2 +( 1+ n:)[4K-20(1+K) (ka)2J= O. (35b) [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: Wed, 03 Dec 2014 00:02:21DAM P E D S P HER I CAL FER R I :vi A G NET I eRE SON A NeE MOD E S 4427 All of the motion in the four shaded areas is in the clockwise direction and all of the motion in the two unshaded areas is in the counterclockwise direction. As in Fig. 1 all of the loci of ma<>,,-l are either circles or ellipses with the semi-axes lying along and perpen dicular to the radius. The straight solid vectors indicate the position of maa,,-l/Ca-1 when wt= Xa-1• The locus of the motion at the origin is circular. As one travels along the x axis from the origin along the path OAB the motion becomes elliptical as in the case of Fig. 1 except that the motion is in the counterclockwise direc tion from point 0 to A and clockwise from point A to B. At point B the motion is circular and past point B the motion becomes more and more elliptical until point D is reached where the motion is linear and parallel to y FIG. 4. Conventional diagram of the loci of the motion of m3~-I/C3-1 in the 3i1 mode in the plane z=O. the y direction. Point D is a point of reversal of motion whereby from point D to the perimeter the motion is in the counterclockwise direction. As in the case of Fig. 1, the motion is circular on the circle having a radius OB and along the 45° axes. Points C, where the ellipses intersect, are points of reversal of motion and the only points in the cross section where the motion of ma<>,,-l is zero. More generally, the ellipsoids given by (35a) and (35b) are surfaces of reversal of the motion of maa,,-l. Figure 4 (not to scale) shows the loci of the motion of ma/l-I/Ca-l. This figure is similar to Fig. 2 except that the direction of motion of ma/l-1jCa-1 is reverse that of maljCi, and the direction of mar1jCa-l when wt=Xa-l is opposite that of ma/l1jCl when wt= Xal. As in the case of Fig. 2 the motion of ma/l-l is zero in the x= 0 and y= 0 planes. The places where the motion of ma/l-1 is zero do not coincide anywhere with the places where the 18 0'" 16 "-'" .,'14 -'" IE "-12 0 U 10 0 -" u-8 0 <J) 6 w x « 4 ~ w <J) 2 o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 kl FIG. 5. Plot of the semi-axes of the loci of the motion of rna, ;;C31 sketched in Fig. 1 along the x axis as a function of kl when OH=0.796 and O/OH= 1.46. motion of maa, ,-1 is zero and so at no point in the sphere in the 3il mode is the motion of ma-1 zero. Figure 5 shows a plot of the semi-axes of the loci of the motion of maer, NCal along the x axis of Fig. 1 when the sphere is in the 311 resonant mode with values of A= 0.0001 and rlH= 0.796. Point A in Fig. 5 corresponds to point A in Fig. 1 where the eccentricity of the ellipse is zero. The two curves cross at point B which corresponds to point B in Fig. 1 where the circle is a special case of an ellipse with equal semi-axes. 26 24 22 20 -'" 18 .!2 -'" IE 16 t5 <J) f-14 ~ 12 z o ~ 10 o lJ "-8 o ~ 6 ::> f- Z 4 '-" « ::; 2/---_L c ~ o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 DISTANCE IN UNITS OF a FIG. 6 .. Plo~ of the magnitudes of ffi3a,;;Ci and mallei sketched III FIgs. 1 and 2, respectively, along a 45° axis as a function of the distance from the center of the sphere when OH=0.796 and o/oH= 1.46. [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: Wed, 03 Dec 2014 00:02:214428 OSCAR ]. VAN SANT I'" 16 <.> "-"',14 _l:< '''' tE 12 "-0 10 0 0 8 -...J "-0 (II w 6 - x 4 <t ~ 2 w <Il o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 kl. FIG. 7. Plot of the semi-axes of the loci of the motIOn of maa.,-I/C 3-1 sketched in Fig. 3 along the x axis as a function of kJ when OH=0.796 and O/OH=1.44. Figure 6 shows plots of the magnitudes of the com ponents of mNC31 along the 45° axes of Figs. 1 and 2 when the sphere is in the 311 resonant mode with values of >-=0.0001 and QH=0.796. Point C where the curve of m3a,,I/C31 is zero corresponds to point C in Fig. 1. The other curve in Fig. 6 shows the magnitude of m3~I/C31 sketched in Fig. 2 varying as the square of the distance from the center along the 45° axis. Figure 7 shows a plot of the semi-axes of the loci of the motion of m3a,,-1/C 3-1 along the x axis of Fig. 3 when the sphere is in the 3i1 mode with values of ,r<> u ::::.. '''' IE "-0 <Il I-z w z 0 a. ~ 0 u "-0 w 0 ::) I-Z <!> <t ~ 16 14 12 10 8 6 4 2 I m;~.t/c;11 o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 DISTANCE IN UNITS OF a FIG. 8. Plot of the magnitudes of maa, ,-l/Ca-l and mar1/Ca-1 sketched in Figs. 3 and 4, respectively, along a 450 axis as a function of the rustance from the center of the sphere when OH= 0.796 and O/OH= 1.44. >-=0.0001 and QH=0.796. Points A and D in Fig. 7 correspond to points A and D in Fig. 3 where the eccentricities of the ellipses are zero. The two curves cross at point B which corresponds to point B in Fig. 3 where the circle is a special case of an ellipse with equal semI-axes. Figure 8 shows plots of the magnitudes of the com ponents of m3-I/C3-1 along the 45° axes of Figs. 3 and 4 when the sphere is in the 3 i 1 resonant mode wi th values of ;\=0.0001 and f211=0.796. Point C where the curve of m3a.,-I/C 3-1 is zero corresponds to point C in Fig. 3. The other curve in Fig. 8 shows the magnitude of ma!lI/Ca-l sketched in Fig. 4 varying as the square of the distance from the center along the 45° axis. CONCLUSIONS The damped solutions of the resonant modes enable the amounts of power absorbed by the 3, 1 modes to be calculated and compared as in Table I. Since p n-m(X)/ P nm(x) = r(n-m+ 1)/r(n+m+ 1), it can be shown from (12)-(21b) and (7a)-(9) that the ratio of the power absorbed by an nmr mode to that of the corresponding nmr mode, in cases where neither mode is intentionally favored over the other as far as positioning of the sphere is concerned, will probably be approximately, Un-m/unm,.,{f(n-m+1)/f(n+m+1)J2. (36) Expression (36) explains why in data similar to that taken by Fletcher, Solt, and Bell,4 the absorption peak of an nmr mode will usually be very much smaller than that of the corresponding nmr mode. Since the phase of the rf component of the induced magnetic dipole moment is a function of position in the sphere and since it is not zero at any point in the sphere, conventional methods of showing the configuration of the fields of a resonant mode cannot be employed. How ever, when the rf component of the induced magnetic dipole moment is assumed to be composed of the two components given by the right side of (31), the resonant state can be described by giving the loci of the motion of these individual components in physical space as was done in Figs. 1 through 8. ACKNOWLEDGMENT The author thanks E. T. Hooper and A. D. Krall for their many helpful discussions on the subject. [This article is copyrighted as indicated in the article. 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1.94308.pdf	Operation of an inductively ballasted helical TECO2 laser D. J. Biswas, P. K Bhadani, P. R. K. Rao, U. K. Chatterjee, A. K. Nath, and U. Nundy Citation: Applied Physics Letters 43, 224 (1983); doi: 10.1063/1.94308 View online: http://dx.doi.org/10.1063/1.94308 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/43/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Pulsed operation of a segmented longitudinal discharge CO2 laser without ballast impedance Rev. Sci. Instrum. 56, 2021 (1985); 10.1063/1.1138411 Operation of a cw CO2 laser without a ballast resistance Rev. Sci. Instrum. 54, 775 (1983); 10.1063/1.1137445 Improved performance of a ballast resistance helical transversely excited CO2 laser with water vapor and low ionization potential additives instead of helium J. Appl. Phys. 53, 5469 (1982); 10.1063/1.331479 Erratum: Modes of operation of waveguide TE CO2 lasers J. Appl. Phys. 50, 8265 (1979); 10.1063/1.327281 Modes of operation of waveguide TE CO2 lasers J. Appl. Phys. 50, 3102 (1979); 10.1063/1.326389 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: 129.120.242.61 On: Sun, 23 Nov 2014 04:19:42Operation of an inductively ballasted helical TE-C0 2 laser D. J. Biswas, P. K. Bhadani, and P. R. K. Rao Multidisciplinary Research Scheme, Physics Group, Bhabha Atomic Research Centre, Bombay-400 085, India U. K. Chatterjee, A. K. Nath,a) and U. Nundy Laser Section, Bhabha Atomic Research Centre, Bombay-400 085, India (Received 22 February 1983; accepted for publication 23 May 1983) We report here the successful operation of a Beaulieu-type TE-C02 laser using equalizing inductances. An improvement of 50% or more in the efficiency was observed over resistance ballast system. PACS numbers: 42.S5.Dk, 42.60.By, 52.80.Hc Transversely excited carbon dioxide lasers of the Beau lieu type! are simple in construction and with a helical elec trode geometry, have a radial gain profile which naturally favors a TEMoo mode.2 They have been used as oscillators in a chain consisting of double discharge amplifiers.3,4 The use of these lasers at high repetition rates when moderate energy per pulse is required is attractive because they are less prone to arcing!,5 compared to UV preionized lasers. (During our experiments it was observed that Beaulieu type systems could be operated reliably without arcing even when more than 2% of oxygen was deliberately introduced into the gas mixture.) This is probably because the discharge is segment ed and the individual segments are current limited by ballast resistances.s However, the efficiency of a Beaulieu type sys tem is low (2_5%)2 primarily due to the power loss in the ballast resistances. 5 In this letter we report the operation of a helical TE CO2 laser where each pin pair was ballasted with an appro priate inductance, partially eliminating the loss in the ballast resistances. The laser chamber is formed by a 64-cm-long perspex tube of 5-cm i.d. with 85 pairs of diametrically oppo site pin electrodes having a gap of 2.5 cm. In the first set of experiments each pin which formed the anode was ballasted through a series resistance of approximately 400 n. In the next set of experiments each ballast resistance was replaced by an inductance of approximately 40 !tH. A zinc selenide plane mirror of 90% reflectivity and a 4-m radius of curva ture gold plated concave mirror of nearly 100% reflectivity separated by nearly 100 cms, formed the optical cavity. The laser was energized by charging a O.OI-flF condenser to suit able voltages and then switching it with a thyratron (EG & G Model HY -5). A resistive voltage divider probe was used to measure the voltage across the discharge, and the discharge current was estimated by measuring the voltage across a small resistance connected in series with the discharge 100p.6 Energy of the laser pulse was measured by a pyroelectric joule meter (Lumonics Model 20 0167). With the resistively ballasted laser operated at 2-Hz re petition rate we first measured the output energy per pulse as a function of the operating pressure. The results are shown in Fig. 1. Current and voltage of the discharge were measured at the pressure for which output energy was maximum. They are plotted in Fig. 2. Next we ballasted the laser inductively .) Now on leave at Department of Electrical Engineering, University of Al berta, Edmonton, Canada T6G2El. and repeated the similar studies, results of which are also shown in Figs. 1 and 2. For all these measurements, CO2:N2:He: 1: 1:4 gas mixture was used and the discharge condenser was charged to 24 k V. The value of each induc tance was chosen to offer an impedance of about 300 n. The upper limit of this value was determined by the conditions for critical damping ofthe whole circuit. However, the cir cuit loop inductances when added to this made the discharge a little under damped (Fig. 2, broken line). From Fig. 1 it can be seen that in a resistively ballasted system output energy has a stronger dependence on gas pres sure compared to inductively ballasted system. This can be understood as follows: It has been experimentally observed that any change in the gas pressure also changed the dis charge impedance. In a resistively ballasted system this obvi ously means that the energy that goes into the discharge load also changes with gas pressure. However, in an inductively ballasted system all the energy stored into the condenser goes into the discharge load since there is no dissipation in the inductances. Therefore, ballast-resistance systems will have stronger dependence on gas pressure compared to bal last-inductance system. From the figure it can also be ob served that the output energy in ballast-inductance system is at least 50% more than that in the ballast-resistance system. The reason for this improvement becomes obvious when Fig. 2 is analyzed. From this figure the energy going into the discharge can be calculated.5 In case of ballast inductance, 198 t 180 1162 144 ~ 126 108 >- ~ 90 w z 72 w ~4 36 18 o o INOUCTANCE A RESISTANCE 50 100 150 200 250 300 350 400 450 PRESSURE (TORR) FIG. I. Dependence of laser pulse energy on the gas pressure. In case of ballast inductance energy falls abruptly at 450 Torr because of the onset of arcing at this pressure. (Energy stored in the discharge condenser::::2.9 J.) 224 AppL Phys. Lett. 43 (3), 1 August 1983 0003-6951/83/150224-02$01.00 © 1983 American Institute of Physics 224 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: 129.120.242.61 On: Sun, 23 Nov 2014 04:19:4225KV {\ , , , • . , , r , ::> lKA t o ·2 ·4 FIG. 2. Discharge voltage and current pulse shapes. Full lines, with resis tance; broken lines, with inductance. nearly all the energy stored in the condenser goes into the discharge while approximately 3/5 of the stored energy goes into the discharge in case of ballast resistance. We also wanted to compare laser outputs correspond ing to operations with ballast resistance and ballast induc tance when the same energy was coupled with the discharge. 225 Appl. Phys. Lett., Vol. 43, No.3, 1 August 1983 For this we operated the inductively ballasted system by charging the condenser to a correspondingly lower voltage (19 kV). There was no significant difference in output ener gies even though the voltage and current shapes were differ ent in the two cases. Beam pattern in both cases fluctuated between lower order modes. A computer simulation (modi fied version of the model by Andrews et al.7) that uses vol tage dependent excitation ratesH also confirmed these experi mental observations. In conclusion, we have operated a Beaulieu type TE CO2 laser both with ballast resistance and ballast induc tance. An improvement of 50% or more in the efficiency has been observed in the later case. In application, where a well defined beam, preferably in the TEM()() mode, is required, the effective efficiency of such a system may become com parable to that ofUV preionized system. This is because in a UV preionized system the volume utilized in a low order mode is much smaller than the excited volume. If we add to this the ease of reliable arc-free sealed-off operation, this la ser has a decided advantage. The authors acknowledge the help given by N. S. Shi karkhane in the experiment. They also acknowledge the technical assistance of S. L. Songire and R. A. Nakhwa. I A. 1. Beaulieu, App!. Phys. Lett. 16, 504 (1970). 2R. Fortin, M. Gravel, and R. Tromblay, Can. 1. Phys. 49,1783 (1971). 'F. Rheault, J. Lachambre, 1. Gilbert, R. Fortin, and M. Blauchard, Vll International Quantum Electronics Conference, Montreal, Quebec, Paper Q 2,1972. 4A. Girard and H. Pepin, Opt. Commun. 8, 68 (1973). 5R. Fortin, Can. 1. Phys. 49, 257 (1971). bU. K. Chatterjee, N. S. Shikarkhane, and U. Nundy, Rev. Sci. Instrum. 52, 618 (1981). 7K. J. Andrews, P. E. Dyer, and D. J. James, J. Phys. E 8, 493 (1975). "0. P. Judd, 1. Appl. Phys. 45, 4572 (1974). Biswasetal. 225 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: 129.120.242.61 On: Sun, 23 Nov 2014 04:19:42
1.3672076.pdf	Domain wall propagation in micrometric wires: Limits of single domain wall regime V. Zhukova, J. M. Blanco, V. Rodionova, M. Ipatov, and A. Zhukov Citation: J. Appl. Phys. 111, 07E311 (2012); doi: 10.1063/1.3672076 View online: http://dx.doi.org/10.1063/1.3672076 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v111/i7 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 24 Aug 2013 to 131.170.6.51. 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_permissionsDomain wall propagation in micrometric wires: Limits of single domain wall regime V . Zhukova,1,a)J. M. Blanco,2V . Rodionova,1,3M. Ipatov,1and A. Zhukov1,4 1Departamento de Fı ´sica de Materiales, UPV/EHU, 20018 San Sebastian, Spain 2Departamento de Fı ´sica Aplicada, EUPDS, 20018 San Sebastian, Spain 3Faculty of Physics, Moscow State University, Leninskie Gory, Moscow, Russia 4IKERBASQUE, Basque Foundation for Science, 48011 Bilbao, Spain (Presented 2 November 2011; received 19 September 2011; accepted 23 October 2011; published online 22 February 2012) We measured magnetic domain propagation and local domain wall (DW) nucleation in Fe-Co-rich amorphous microwires with metallic nucleus diameters from 2.8 to 18 lm. We found that manipulation of magnetoelastic energy through application of applied stresses, changing of magnetostriction constant, and variation of internal stresses through changing the microwiresgeometry affects DW velocity. We observed uniform or uniformly accelerated DW propagation along the microwire. The abrupt increasing of DW velocity on v(H) dependencies correlates with the location of the nucleation place of the new domain wall. VC2012 American Institute of Physics . [doi: 10.1063/1.3672076 ] Last few years controllable and fast domain wall (DW) p r o p a g a t i o ni nt h i nm a g n e t i cw i r e s (planar and cylindrical) become a topic of intensive research.1–3Certain efforts have been paid for achieving high DW velocity, v, within a wire, considering its great importance for proposed applications.1 On the other hand, faster DW propagation at a relatively low magnetic ﬁeld, H, has been reported for cylindrical glass coated amorphous wires with typical diameters about 10–20 lm,3,4although there are recently reported fabrication of thinner (with diameters of order 1–5 lm) microwires.5 Glass-coated micrometric wires present a few quite pecu- liar magnetic properties, such as the magnetic bistability (MB)and the giant magnetoimpedance (GMI) effects. 6,7Amorphous microwires with positive magnetostriction are quite suitable for studies of the single DW dynamics because of ideally cy-lindrical cross-section and MB, related with large and single Barkhausen jump, attributed to the fast magnetization switch- ing inside the inner single domain. 6There are results, indicat- ing that this DW is relatively thick and has a complex structure.8,9 Elevated internal stresses originated from the simultane- ous solidiﬁcation of ferromagnetic nucleus surrounded by the glass coating have been commonly accepted.7,10,11 Therefore, although these microwires exhibit large DW velocities, the inﬂuence of such stresses on DW dynamics in these samples must be relevant. Additionally, recently the effect of real structure consist- ing on DW nucleation on local defects has been described.12 Consequently, in this paper we are trying to reveal the effect of magnetoelastic anisotropy and DW nucleation on defects on DW dynamics in amorphous magnetically bistable microwires. DW propagation is measured by using Sixtus Tonks-like experiments, as described elsewhere.6,9 We studied microwires of Co 56Fe8Ni10Si10B16, Co41.7Fe36.4Si10.1B11.8,F e 55Co23B11.8Si10.2, Fe70B15Si10C5,Fe72.75Co2.25B15Si10, and Fe 16Co60Si11B13compositions with positive magnetostriction constant and diameters of me- tallic nucleus from 2,8 to 22 lm. It is worth mentioning, that the magnetostriction constant, ks, in system (Co xFe1/C0x)75 Si15B10changes with xfrom/C05/C210/C06atx¼1, to ks/C25 40/C210/C06atx/C250.2.13 We produced microwires with different ratio of metallic nucleus diameter and total diameter, D,q¼d/D. This allowed us to control residual stresses, since the strength ofthe internal stresses is determined by ratio q. 10,11,14In this way we studied the effect of magnetoelastic contribution on DW dynamics controlling the magnetostriction constant,applied and/or residual stresses. Moreover the distributions of the local nucleation ﬁelds, H N, were measured for the same microwires using the method described in Ref. 13. Considerable increasing of the switching ﬁeld is observed when the ferromagnetic metallic nucleus diameter decreasesfrom 15 to 1,2 lm (one order), as observed early [ 12]. The increase of coercivity for low microwire diameters should be attributed to the magnetoelastic energy arisingfrom enhanced internal stresses when ratio qis small. 14 v(H) dependencies for Co 41.7Fe36.4Si10.1B11.8 micro- wires with different q-ratios are shown on Fig. 1. From Fig. 1we can observe that at the same Hvalues the DW velocity is higher for microwires with higher q- ratio, i.e., when the strength of internal stresses is lower.14 The magnetoelastic energy, Kme, is given by Kme/C253=2ksr; (1) where r¼riþrais the total stress, riare the internal stresses, raare the applied stresses, and ksis the magneto- striction constant.11 We measured v(H) dependencies applying stress. In this case, stress applied to metallic nucleus has been eval- uated as previously described in Ref. 7. As can be observed from Fig. 2, under tensile stress application reduction of DWa)Electronic mail: valentina.zhukova@ehu.es. 0021-8979/2012/111(7)/07E311/3/$30.00 VC2012 American Institute of Physics 111, 07E311-1JOURNAL OF APPLIED PHYSICS 111, 07E311 (2012) Downloaded 24 Aug 2013 to 131.170.6.51. 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_permissionsvelocity, v,i nC o 41.7Fe36.4Si10.1B11.8microwires has been observed. Usually it is assumed that DW propagates along the wire with a velocity: v¼SðH/C0H0Þ; (2) where Sis the DW mobility and H0is the critical propaga- tion ﬁeld. It is worth mentioning that observed by us DW velocity values exceeds estimated maximum velocity (Walker limit) for DW propagation, which can be estimated from: v¼cD aH; (3) where D- is the DW width, c¼2.2/C2105m/As, a-damping parameter. The DW dynamics in viscous regime is determined by a mobility relation (2), where Sis the DW mobility given by S¼2l0Ms=b; (4) where bis the viscous damping coefﬁcient, l0is magnetic permeability of vacuum. Usually two contributions to viscous damping bhave been considered and generally accepted.4 The ﬁrst is determined by the micro-eddy currents circulating nearby moving DW. However, the eddy current parameter beis considered to be negligible in high-resistive materials, like amorphous microwires, which additionallyhave quite thin diameters. The second generally accepted contribution of energy dissipation is magnetic relaxation damping, b r,related to a delayed rotation of electron spins. This damping is related to the Gilbert damping parameter, afrom (3)and is inversely proportional to the DW width dw, br/C25aMs=cD/C25MsðKme=AÞ1=2; (5) where cis the gyromagnetic ratio, Ais the exchange stiffness constant, Kmeis given by Eq. (1). Consequently, magnetoelastic energy, Kme, can affect DW mobility, S, what we experimentally observed in few Co-Fe-rich magnetically bistable microwires (Figs. 1and2).Considering aforementioned, we can suggest, that DW velocity, v, should decrease with stress and magnetostriction constant increasing. As recently observed,4,9,15increasing the magnetic ﬁeld abrupt increase of DW velocity or even oscillations accompa- nied by a slow drift of the wall are usually observed in micro- wires and strips with submicron dimensions. This increase hasbeen interpreted in different ways, considering Walker-like behavior, 4collective-coordinate approach15although recently such deviations from linear v(H) dependence have been attrib- uted to the nucleation of additional DW on defects at elevated magnetic ﬁelds.9We realized comparison of results on DW dynamics with measurements of local nucleation ﬁelds in thesame branch of Fe-rich microwires of the same composition. We observed different kinds of v(H) dependencies. In the microwires from the ﬁrst group the DW velocity valuesmeasured by the ﬁrst pair of pick-up coils, V 1/C02and by the second pair of pick-up coils, V2/C03, almost identical values. This means that in these microwires the DW propagates uni-formly within the microwires Fig. 3(a). The samples from the second group exhibit increasing of the DW velocity FIG. 1. v(H) dependencies for Co 41.7Fe36.4Si10.1B11.8microwires with dif- ferent ratios q. FIG. 2. v(H) dependencies for Co 41.7Fe36.4Si10.1B11.8 microwires ( d/C25 13,6mm,D/C2524,6mm,q/C250,55) measured under application of applied stresses, ra. FIG. 3. (Color online) Typical v(H) dependencies of the domain measured in different samples of magnetically bistable amorphous Fe 74B13Si11C2 microwires exhibiting (a) uniform and (b) accelerated DW propagation.07E311-2 Zhukova et al. J. Appl. Phys. 111, 07E311 (2012) Downloaded 24 Aug 2013 to 131.170.6.51. 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_permissionswhile it moves along the microwire, as evidenced by detecta- ble difference between the V1/C02and the V2/C03and by differ- ent slope on V(H) dependence, as shown in Fig. 3(b). This means that these microwires do not exhibit uniform DW propagation. The difference observed on dependencies of local nuclea- tion ﬁelds versus position of nucleation coil, l,m e a s u r e di n these microwires is in the amplitude and the width of the mini-mums (Fig. 4). The local nucleation ﬁelds distribution shown in Fig. 4(a) is typical for the microwires exhibiting the uniform DW prop-agation [i.e., exhibiting v(H) dependence shown in Fig. 3(a)], while the local nucleation ﬁelds distribution shown in Fig. 4(b) with larger ﬂuctuations amplitude corresponds to the case of Fig. 3(b) typical for microwires exhibiting increasing of the DW velocity while it moves along the microwire. Taking into account observed correlation (Figs. 3and4) we can deduce, that larger defects (higher defects density or efﬁciency) typical for the second group of the microwires are related with exhibiting accelerated DW propagation. To determine the limits of single DW propagation re- gime, we compared the local nucleation ﬁelds distribution with v(H) dependencies measured by coils 1-2 and 2-3. Likein the case considered in the work, 9we observed, that when the applied magnetic ﬁeld has reached minimum nucleation ﬁeld HN(for coils pair 1-2 Hn/C25168 A/m at l¼52lm), the abrupt increase of DW velocity, V1/C02, is observed. At the same time, V2/C03did not show any jump on V2/C03(H) depend- ence (Fig. 5). Consequently we assume that the jump observed at H/C25168 A/m is related with new domain nuclea- tion and propagation of two more DW in the opposite direc- tions toward the wire’s ends. Similar correlations weobserved for each studied samples. Consequently, observed above dependencies allow us to manipulate DW dynamics in magnetically bistable micro-wires, considering magnetoelastic anisotropy and real struc- ture of microwires. We observed uniform and uniformly accelerated DW propagation along the microwire. Magnetic ﬁeld value, cor- responding to the jump on v(H) dependence correlates with the minimum nucleation ﬁeld, which determines thresholdbetween single and multiple DW propagation regimes. We experimentally observe d that the magnetoelastic energy signiﬁcantly affected DW dynamics in magnetically bistable microwires. We assume that in order to achieve higher DW propagation velocity and enhanced DW mobility special attention should be paid to decreasing of magnetoelasticenergy. Applied and internal stresses result in decreasing of DW velocity. Within single DW regime v(H) dependence can be manipulated through magnetoelastic anisotropy, i.e., throughmetallic nucleus composition and strength of internal stresses. This work was supported by EU ERA-NET program under project “SoMaMicSens” (MANUNET-2010-Basque-3),by Spanish Ministry of Science and Innovation, MICINN, under project MAT2010-18914 and by the Basque Govern- ment under Saiotek 11 MIMAGURA project (S-PE11UN087). 1M. Hayashi et al.,Phys. Rev. Lett. 97, 207205 (2006). 2D. A. Allwood et al.,Science 309, 1688 (2005). 3A. Zhukov, Appl. Phys. Lett. 78, 3106 (2001). 4R. Varga et al.,Phys. Rev. B 76, 132406 (2007). 5V. Zhukova et al.,Sens. Actuators B 126, 232 (2007). 6V. Zhukova et al.,J. Appl. Phys. 106, 113914 (2009). 7A. Zhukov, Adv. Funct. Mat. 16, 675 (2006). 8P. A. Ekstrom and A. Zhukov, J. Phys. D: Appl. Phys. 43, 205001 (2010). 9M. Ipatov et al.,J. Appl. Phys. 106, 103902 (2009). 10H. Chiriac et al.,Phys. Rev. B 42, 10105 (1995). 11V. Zhukova, M. Ipatov, and A. Zhukov, Sensors 9, 9216 (2009). 12M. Ipatov et al.,Physica B 403, 379 (2008). 13H. Fujimori et al.,Jpn. J. Appl. Phys. 15, 705 (1976). 14H. Chiriac et al.,J. Magn. Magn. Mater. 254-255 , 469 (2003). 15D. J. Clarke et al.,Phys. Rev. B 78, 134412 (2008). FIG. 4. Typical distributions of the local nucleation ﬁelds measured in dif- ferent samples of magnetically bistable amorphous Fe 74B13Si11C2micro- wires for (a) uniform and (b) accelerated DW propagation. 1, 2, and 3 are the position of the pick-up coils. FIG. 5. Correlation of local nucleation ﬁelds distribution (a) and V(H) depend- encies in magnetically bistable amorphous Fe 74B13Si11C2microwire exhibiting accelerated DW propagation, 1, 2, 3 are the positions of the pick-up coils.07E311-3 Zhukova et al. J. Appl. Phys. 111, 07E311 (2012) Downloaded 24 Aug 2013 to 131.170.6.51. 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
1.3224883.pdf	Microwave resonance in nanocomposite Ji Ma , Jiangong Li , Xia Ni , Xudong Zhang , and Juanjuan Huang Citation: Appl. Phys. Lett. 95, 102505 (2009); doi: 10.1063/1.3224883 View online: http://dx.doi.org/10.1063/1.3224883 View Table of Contents: http://aip.scitation.org/toc/apl/95/10 Published by the American Institute of Physics Microwave resonance in Fe/SiO 2nanocomposite Ji Ma, Jiangong Li,a/H20850Xia Ni, Xudong Zhang, and Juanjuan Huang MOE Key Laboratory for Magnetism and Magnetic Materials and Institute of Materials Science and Engineering, Lanzhou University, Lanzhou 730000, People’s Republic of China /H20849Received 4 April 2009; accepted 17 August 2009; published online 10 September 2009 /H20850 A broad resonance band in the 1–16 GHz range observed in Fe /SiO 2nanocomposite results from the coexsistence of natural resonance and exchange resonance. The natural resonance appears at5.91 GHz and can be related to the core spins in the Fe nanoparticles, whereas the exchangeresonance appears at 11.01 GHz and can be associated with the surface spins of the Fe nanoparticlesin the Fe /SiO 2nanocomposite. Both resonance frequencies depend on the surface anisotropy of the Fe nanoparticles, which can be affected by the Fe particle size, and can be tuned by adjusting theFe particle size. © 2009 American Institute of Physics ./H20851doi:10.1063/1.3224883 /H20852 Recently, much attention has been paid to nanocompos- ites of ferromagnetic nanoparticles dispersed in dielectricmatrixes due to their potential applications as microwaveabsorbers. 1–5In general, such materials exhibit a single natu- ral resonance peak in gigahertz range.2–4However, the theo- retical study predicted that exchange resonance will occur as the particle size is smaller than 100 nm;6and such a predic- tion has been veriﬁed qualitatively in monodisperse magneticparticles such as FeCoNi particles with particle size of 60 nmand Co particles with particle size of 45 nm. 7Based on these results, the nanocomposite materials may exhibit the coexist-ence of natural resonance and exchange resonance in giga-hertz range, which is beneﬁcial for broadband microwaveabsorption. So it is necessary to study whether the nanocom-posite materials show the coexistence of these two reso-nances in gigahertz range and which factors inﬂuence thenatural and exchange resonance frequencies. In this letter, wereport the coexistence of natural resonance and exchangeresonance in the Fe /SiO 2nanocomposite. To characterize such a double resonance behavior, the natural resonance andexchange resonance were resolved in the permeability spec-tra by ﬁtting Landau–Lifshitz–Gilbert equation. 8The surface effects of the Fe particles on both resonance frequencies arediscussed. The Fe /SiO 2nanocomposite was prepared by milling the mixture of Fe 2O3and Si powders for 40 h in the stainless steel vial with stainless steel balls under an argon atmosphereby a Fritsch P4 planetary ball mill. Ball-to-powder weightratio is 20:1. The main disk revolution speed and relativerotation speed ratio of the vial to the main disk are 300 rpmand/H110022, respectively. The structure and morphology of the nanocomposite were characterized by x-ray diffraction/H20849XRD /H20850on Rigaku D/Max-2400 with Cu K /H9251radiation and transmission electron microscopy /H20849TEM /H20850on JEOL JEM 3010. The static magnetic properties were measured by LakeShore 7304 vibrating sample magnetometer with a maximumﬁeld of 15 kOe. The toroidal samples used for microwavemeasurements were prepared by mixing the Fe /SiO 2nano- composite powder and parafﬁn. The volume fraction of theFe /SiO 2nanocomposite in the mixture is 40%. The scatter- ing parameters /H20849S11andS21/H20850were measured on the toroidalsamples by an Agilent Technologies E8363B network ana- lyzer in 0.1–18 GHz range. The complex permeability /H20849/H9262 =/H9262/H11032−j/H9262/H11033/H20850were determined from the scattering parameters. The XRD pattern of the Fe /SiO 2nanocomposite /H20851Fig. 1/H20849a/H20850/H20852shows a diffuse peak of the amorphous SiO 2and sev- eral diffraction peaks corresponding to the Fe grains with abcc structure. The average Fe grain size is 11.2 nm, esti-mated from the integral width of the Fe diffraction peaks with strain and instrumental broadening eliminated by Wil-son’s method. The bright ﬁeld and dark-ﬁeld TEM images inFig. 1reveal that the nearly equiaxial Fe nanoparticles mainly with sizes between 8 and 16 nm are randomly dis-persed in the SiO 2matrix. The average Fe particle size, de- termined by ﬁtting the particle size distribution /H20851Fig. 1/H20849d/H20850/H20852 with a log-normal distribution, is 11.7 nm which consistswith the average Fe grain size, indicating the Fe nanopar-ticles are single crystals. a/H20850Author to whom correspondence should be addressed. Electronic mail: lijg@lzu.edu.cn. FIG. 1. /H20849a/H20850The XRD pattern of the Fe /SiO2nanocomposite milled for 40 h, /H20849b/H20850the bright ﬁeld, and /H20849c/H20850dark ﬁeld TEM micrographs of the Fe /SiO2 nanocomposite, and /H20849d/H20850the particle size distribution of the Fe particles in the Fe /SiO2nanocomposite.APPLIED PHYSICS LETTERS 95, 102505 /H208492009 /H20850 0003-6951/2009/95 /H2084910/H20850/102505/3/$25.00 © 2009 American Institute of Physics 95, 102505-1 The relative complex permeability spectra of the Fe /SiO 2nanocomposite-parafﬁn mixture sample /H20851Fig. 2/H20849a/H20850/H20852 shows that the real part /H9262/H11032decreases with increasing fre- quency in the 0.1–18 GHz range and the imaginary part /H9262/H11033 exhibits a broad resonance band in the 1–16 GHz range witha shoulder at around 6 GHz. Generally, the microwave mag- netic loss of magnetic particles originates from hysteresis,domain wall resonance, eddy current effect, natural reso-nance, and exchange resonance for particles smaller than 100nm. In our case, the contributions of magnetic hysteresis anddomain-wall resonance can be excluded due to the weak ap-plied ﬁeld and the Fe particle size smaller than the Fe singledomain size of about 20 nm. 9The eddy current loss is related to thickness /H20849d/H20850and electric conductivity /H20849/H9268/H20850of the compos- ite and can be described by10/H9262/H11033/H20849/H9262/H11032/H20850−2f−1=2/H9266/H92620d2/H9268/3/H20849/H92620 is the vacuum permeability /H20850. If magnetic loss only results from eddy current loss, the /H9262/H11033/H20849/H9262/H11032/H20850−2f−1value should be con- stant as the frequency changes. Since the /H9262/H11033/H20849/H9262/H11032/H20850−2f−1value for our sample varies with frequency /H20851Fig. 2/H20849b/H20850/H20852, the eddy current loss can be precluded. Thus, the magnetic loss in theFe /SiO 2nanocomposite-parafﬁn mixture should arise from the natural resonance and exchange resonance. According tothe Kittel equation,11the nature resonance frequency /H20849fR/H20850for the spherical Fe particles is estimated to be 1.6 GHz. Com- pared to this estimated fRvalue and the reported fRvalues such as 6 GHz for Fe/C nanocomposites3and 7.2 GHz for ZnO-coated Fe nanoparticles,1the frequency /H20849fmax /H20850at which the/H9262/H11033maximum appears /H2084910.6 GHz /H20850is probably too high to correspond to fRfor our sample. So the natural resonance in our sample should appear at about 6 GHz where the shoulderappears in the /H9262/H11033curve. The fmaxis very likely to correspond to the exchange resonance frequency /H20849fex/H20850since the Fe par- ticle size is small enough to result in the occurrence of ex- change resonance. In particular, the Fe nanoparticles with anaverage size of 11.7 nm could possess a high f exdue to the dependence of fexonR−2/H20849Ris particle radius /H20850.6Conse- quently, the /H9262/H11033spectrum should be an overlap of natural resonance and exchange resonance. To understand such over-lap behavior, the /H9262/H11033curve will be ﬁtted with the linear over- lap of two resonance bands /H20849C1and C2/H20850by ﬁtting the Landau–Lifshitz–Gilbert equations8 /H9262/H11032=B+/H20858 i=12 Ii/H208511− /H20849f/fi/H208502/H208491−/H9251i2/H20850/H20852 /H208511− /H20849f/fi/H208502/H208491+/H9251i2/H20850/H208522+4/H9251i2/H20849f/fi/H208502/H208491/H20850 and /H9262/H11033=/H20858 i=12 Ii/H20849f/fi/H20850/H9251i/H208511+ /H20849f/fi/H208502/H208491+/H9251i2/H20850/H20852 /H208511− /H20849f/fi/H208502/H208491+/H9251i2/H20850/H208522+4/H9251i2/H20849f/fi/H208502, /H208492/H20850 where fis frequency, fiis the spin resonance frequency, /H9251iis the damping coefﬁcient, and Iiis the intensity of the band. The solid lines in Figs. 2/H20849c/H20850and2/H20849d/H20850represent the best ﬁt of the experimental data; and the ﬁtted results are listed in TableI. The C 1andC2resonance bands are assigned to the natural resonance and the exchange resonance, respectively, whichare discussed as follows. For natural resonance, f Rfor spherical particles is de- ﬁned as fR=/H92530Heffwith Heff=2Keff/Ms, where /H92530=/H9253/2/H9266is the gyromagnetic ratio, Heffis the effective anisotropy ﬁeld, Msis the saturation magnetization, and Keffis the effective anisotropy constant.11Keffcan be calculated from the coer- civity /H20849Hc/H20850of Fe /SiO 2nanocomposite by the equation12 Hc=0.96Keff Ms/H208751−/H2087325kBT KeffV/H208740.77/H20876 /H208493/H20850 provided the effective anisotropy is uniaxial. Here, kBis the Boltzmann constant, Vis particle volume, and Tis tempera- ture. The Hcvalue is 495 Oe obtained from the hysteresis loop /H20851Fig. 3/H20849a/H20850/H20852. Since the /H9004mplot13in Fig. 3/H20849b/H20850indicates dipolar interactions between the Fe nanoparticles, the term ofK effVin Eq. /H208493/H20850can be rewritten as /H20849Keff+HdMs/H20850V, where Hd is the mean dipolar ﬁeld, under a linear approximation based on the results14that the energy barrier of magnetization re- versal can be increased by dipolar interactions for the ﬁne-particle system. Using the parameters of T=298 K, M s =1.7/H11003103emu /cm3for Fe, and Hd=256 Oe estimated by the remanence coercivity,15the evaluated Keffvalue is 2.07 /H11003106erg /cm3, which is close to the values reported for Fe granular solids.9,16Then fRis calculated to be 6.82 GHz, which consists with the ﬁtted C1resonance frequency /H208497.47 GHz /H20850. Taking the damping of spin motion into account, fmax FIG. 2. /H20849Color online /H20850/H20849a/H20850The relative permeability /H20849/H9262/H20850of the Fe /SiO2 nanocomposite/parafﬁn mixture sample as function of frequency, /H20849b/H20850the /H9262/H11033/H20849/H9262/H11032/H20850−2f−1values for mixture sample as function of frequency, /H20849c/H20850the ﬁtting curves of the imaginary part of permeability /H20849/H9262/H11033/H20850, and /H20849d/H20850the calcu- lated curves of real part /H20849/H9262/H11032/H20850of permeability.TABLE I. Fitting and calculated parameters for permeability dispersion spectra /H20851f/H20849fit/H20850and f/H20849cal/H20850are the ﬁtted and calculated resonance frequency, respectively. fmaxis the frequency at which the /H9262/H11033maximum value appears in the ﬁtted curves. /H9251is the damping coefﬁcient. /H20852 f/H20849fit/H20850 /H20849GHz /H20850fmax/H20849fit/H20850 /H20849GHz /H20850f/H20849cal/H20850 /H20849GHz /H20850 /H9251 Natural resonance 7.47 5.91 6.82 0.70 Exchange resonance 11.78 11.01 11.33 0.38102505-2 Ma et al. Appl. Phys. Lett. 95, 102505 /H208492009 /H20850 in the C1curve is calculated to be 5.59 GHz through the fmax=fR//H208491+/H92512/H208501/2relation17with the calculated fRand the ﬁtted damping coefﬁcient /H20849/H9251/H20850, which is close to the ﬁtted fmaxvalue of 5.91 GHz. For exchange resonance, fexfor spherical particles is ex- pressed by6 /H11006fex=/H92530/H20873C/H92622 R2Ms+H0−4/H9266 3Ms+2K1 Ms/H20874, /H208494/H20850 where C=2Ais the exchange constant, H0is the applied ﬁeld, K1is the magnetocrystalline anisotropy constant, and /H9262 is the eigenvalue of the derivative of spherical Bessel func- tion jn/H20849/H9262/H20850. However, owing to the existence of surface an- isotropy /H20849Ks/H20850of nanoparticles, the dependence of fexonRis usually weaker than the predicted R−2.7,18In our case, the power of Ris evaluated to be about /H110021.928 from the Ks value estimated by the Keff=K1+3Ks/Rrelation19using the method provided in Ref. 6. Thus the calculated fexvalue is 11.33 GHz, which consists with the ﬁtted C2resonance fre- quency /H2084911.78 GHz /H20850and close to the ﬁtted fmaxvalue /H2084911.01 GHz /H20850. The agreement of the ﬁtted and calculated results reveals the coexistence of natural and exchange resonances in theFe /SiO 2nanocomposite. Such double resonance behavior is related closely to the surface effects of the Fe nanoparticles.First, the surface effects make the arrangement of surfacespins different from that of core spins. The core spins remainparallel to each other due to the strong exchange couplingamong them, whereas the surface spins deviate from paral-lelism due to the competition of surface anisotropy and ex-change interactions. Such difference in spin arrangementleads to the different spin precessions in the Fe nanoparticles,which may result in the coexistence of natural resonance/H20849uniform precession modes /H20850related to the core spins and exchange resonance /H20849nonuniform precession modes /H20850associ- ated with the surface spins. Second, the surface effects intro-duce the surface anisotropy that affects the resonance fre-quencies. For natural resonance, due to the contribution of K s to the Keff, the fmaxvalue /H208495.91 GHz /H20850is larger than the the-oretical value /H208491.6 GHz /H20850. According to fmax=fR//H208491+/H92512/H208501/2, fR=2/H92530Keff/Ms, and Keff=K1+3Ks/R, when Rdecreases and/or Ksincreases, fmaxwill shift to high frequency. There- fore, one can tune natural resonance frequency by changingK sthrough adjusting Ror the particle/matrix interface. For exchange resonance, although the surface anisotropy weak-ens the R −2dependence of fex, it has been found that fexstill increases as Rdecreases.7In our case, the natural and ex- change resonance frequencies can be regulated by adjustingthe Fe particle size in the Fe /SiO 2nanocomposite through controlling the milling time. In conclusion, the Fe /SiO 2nanocomposite exhibits a broad resonance band in the 1–16 GHz range due to thecoexistence of natural resonance and exchange resonance.The double resonance behavior may result from the differentspin precessions in the Fe nanoparticles due to the differentarrangements of surface and core spins in the Fe nanopar-ticles. Both resonance frequencies depend strongly on thesurface anisotropy of the Fe nanoparticles. For natural reso-nance, the increased surface anisotropy resulting from thereduced Fe particle size shifts the resonance frequency tohigh frequencies. For exchange resonance, the decrease inparticle size lead not only to an increase in resonance fre-quency but also to an increase in surface anisotropy whichweakens the R −2dependence of exchange resonance fre- quency on particle radius. This work was supported by the International S&T Co- operation Program /H20849ISCP /H20850of the Chinese Ministry of Sci- ence and Technology under Grant No. 2008DFA50340 andthe National Natural Science Foundation of China underGrant No. 50872046. 1X. G. Liu, D. Y . Geng, H. Meng, P. J. Shang, and Z. D. Zhang, Appl. Phys. Lett. 92, 173117 /H208492008 /H20850. 2J. R. Liu, M. Itoh, and K. Machida, Appl. Phys. Lett. 83,4 0 1 7 /H208492003 /H20850. 3J. R. Liu, M. Itoh, T. Horikawa, and K. Machida, J. Appl. Phys. 98, 054305 /H208492005 /H20850. 4X. F. Zhang, X. L. Dong, H. Huang, Y . Y . Liu, W. N. Wang, X. G. Zhu, B. Lv, J. P. Lei, and C. G. Lee, Appl. Phys. Lett. 89, 053115 /H208492006 /H20850. 5X. F. Zhang, X. L. Dong, H. Huang, B. Lv, J. P. Lei, and C. J. Choi, J. Phys. D 40, 5383 /H208492007 /H20850. 6A. Aharoni, J. Appl. Phys. 81,8 3 0 /H208491997 /H20850. 7G. Viau, F. Fiévet-Vincent, F. Fiévet, P. Toneguzzo, F. Ravel, and O. Acher, J. Appl. Phys. 81, 2749 /H208491997 /H20850. 8A. Aharoni, Introduction to the Theory of Ferromagnetism /H20849Clarendon, Oxford, 1996 /H20850, p. 181. 9S. Gangopadhyay, G. C. Hadjipanayis, B. Dale, C. M. Sorensen, K. J. Klabunde, V . Papaefthymiou, and A. Kostikas, Phys. Rev. B 45,9 7 7 8 /H208491992 /H20850. 10S. B. Liao, Ferromagnetic Physics (3) /H20849Science, Beijing, 2000 /H20850,p .1 7 . 11C. Kittel, Phys. Rev. 73,1 5 5 /H208491948 /H20850. 12H. Pfeiffer, Phys. Status Solidi A 118, 295 /H208491990 /H20850. 13D. E. Speliotis and W. Lynch, J. Appl. Phys. 69, 4496 /H208491991 /H20850. 14J. L. Dormann, D. Fiorani, and E. Tronc, Adv. Chem. Phys. 98,2 8 3 /H208491997 /H20850. 15M. El-Hilo, K. O’Grady, T. A. Nguyen, P. Baumgart, and I. L. Sanders, IEEE Trans. Magn. 29, 3724 /H208491993 /H20850. 16S. H. Liou and C. L. Chien, J. Appl. Phys. 63, 4240 /H208491988 /H20850. 17A. Chevalier, J. L. Mattei, and M. Le Floh’h, J. Magn. Magn. Mater. 215, 66 /H208492000 /H20850. 18A. F. Bakuzis, P. C. Morais, and F. Pelegrini, J. Appl. Phys. 85,7 4 8 0 /H208491999 /H20850. 19F. Bødker, S. Mørup, and S. Linderoth, Phys. Rev. Lett. 72,2 8 2 /H208491994 /H20850. FIG. 3. /H20849Color online /H20850/H20849a/H20850The hysteresis loop of the Fe /SiO2nanocompos- ite milled for 40 h /H20849inset is the magniﬁed plot near the origin /H20850and /H20849b/H20850the /H9004mplot for the Fe /SiO2nanocomposite.102505-3 Ma et al. Appl. Phys. Lett. 95, 102505 /H208492009 /H20850
1.2176890.pdf	Simulated domain wall dynamics in magnetic nanowires Andrew Kunz Citation: Journal of Applied Physics 99, 08G107 (2006); doi: 10.1063/1.2176890 View online: http://dx.doi.org/10.1063/1.2176890 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 Field- and current-induced domain-wall motion in permalloy nanowires with magnetic soft spots Appl. Phys. Lett. 98, 202501 (2011); 10.1063/1.3590267 Magnetic imaging of the pinning mechanism of asymmetric transverse domain walls in ferromagnetic nanowires Appl. Phys. Lett. 97, 233102 (2010); 10.1063/1.3523351 Vortex domain wall chirality rectification due to the interaction with end domain spin structures in permalloy nanowires Appl. Phys. Lett. 95, 252501 (2009); 10.1063/1.3275576 Combined electrical and magneto-optical measurements of the magnetization reversal process at a domain wall trap. Appl. Phys. Lett. 94, 103113 (2009); 10.1063/1.3098359 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: 141.212.109.170 On: Tue, 09 Dec 2014 21:23:07Simulated domain wall dynamics in magnetic nanowires Andrew Kunza/H20850 Department of Physics, Marquette University, Milwaukee, Wisconsin 53233 /H20849Presented on 3 November 2005; published online 28 April 2006 /H20850 The simulated domain wall dynamics in rectangular 10 nm thick, 2000 nm long Permalloy wires of varying width is presented. In the absence of an applied ﬁeld the static domain wall length is foundto be linearly dependent to the width of the nanowire. As magnetic ﬁelds of increasing strength areapplied along the wire’s long axis, the domain wall motion changes from a uniform reversal to asteplike reversal. The onset of the stepping motion leads to a decrease in the domain wall speed. Bycontinuing to increase the ﬁeld it is possible to decrease the time between steps increasing thedomain wall speed. The critical ﬁeld associated with the crossover from uniform to nonuniformreversal decreases as the wire width increases. © 2006 American Institute of Physics . /H20851DOI: 10.1063/1.2176890 /H20852 I. INTRODUCTION Scientiﬁcally magnetic nanowires offer the opportunity to study magnetic phenomena between atomic and bulklimits. 1Technologically the nanowires show promise for ap- plications in high-density recording2and spintronic sensing devices.3To be useful in devices it is necessary to both un- derstand the domain wall motion and control the switchingbehavior of the magnetic moments inside the wire. 4Electron spin currents have been shown to excite domain wall motionin nanowires, leading to some nonlinear effects 5and interest- ing observed domain wall velocity characteristics.6In this paper micromagnetic simulations of the domain wall dynam-ics of a simpliﬁed model in which the walls are moved withexternal magnetic ﬁelds are presented. It is found that thepresence of an external ﬁeld can lead to irregular domainwall motion and an increase in the overall switching time. II. SIMULATION DETAILS The nanowires are simulated by numerically integrating the three-dimensional Landau-Lifshitz equation with Gilbertdamping /H20849LLG /H20850. 7The LLG equation describes the preces- sional motion of individual magnetic moments due to allinternal and external ﬁelds. In this paper the rectangularnanowires are 2000 nm long and 10 nm thick, with widthsthat vary from 50 to 200 nm. The simulated material wasPermalloy with a saturation magnetization of 800 emu/cm 3, an exchange constant of 1.3 /H1100310−6erg/cm, no crystalline anisotropy, and a Gilbert damping parameter of 0.08.8The nanowire is discretized into identical cubic blocks of uniformmagnetization, with dimensions of 2.5 and 5.0 nm on edge.The discretization size had no noticeable effect on the re-sults. The large number of elements allows for a detailedview of the domain wall structure. Subpicosecond time stepswere used to simulate the dynamics of the domain wall underthe inﬂuence of externally applied magnetic ﬁeld.III. RESULTS The static domain structure was determined for a domain wall located at the center of the wire. A sharp head to head,or transverse, wall was placed into the center of the wire.The magnetic moments were allowed to relax in zero exter-nal ﬁeld to a stable conﬁguration. The inset of Fig. 1 showsthe top view of the magnetic structure for a transverse wall,where every other magnetic moment in both in-plane direc-tions is represented by an arrow and by the gray scale. Thedomain wall is characterized by a vortex with an axis per-pendicular to the length of the nanowire. The energy is mini-mized when the magnetic moments on either side of the wallpoint out of the material. Domain walls tend to stabilizewhen the moments at the ends of the walls point perpendicu-larly out of the material. 9As the width of the nanowire is increased from 50 to 200 nm the stable length of the domainwall grows proportionally as shown in Fig. 1. The domainwall length is typically found to be about 60% of the widthof the wire. An external ﬁeld is applied to the equilibrium structure, a/H20850Electronic mail: andrew.kunz@marquette.edu FIG. 1. Plot of the simulated domain wall width as a function of wire width in zero applied ﬁeld showing a linear relationship. The inset ﬁgure is a topview of a 300 nm long segment containing the static magnetic structure ofthe transverse wall in a 100 nm wide wire where every second moment inboth in-plane directions is shown. The circular inset indicates the relationbetween the gray scale and the direction of the in-plane component ofmagnetization.JOURNAL OF APPLIED PHYSICS 99, 08G107 /H208492006 /H20850 0021-8979/2006/99 /H208498/H20850/08G107/3/$23.00 © 2006 American Institute of Physics 99, 08G107-1 [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: 141.212.109.170 On: Tue, 09 Dec 2014 21:23:07directly along the long /H20849x/H20850axis of the wire, moving the wall down the wire’s length. Independent of the wire’s width, applied ﬁelds under 100 Oe set the domain wall into steady-state propagation down the wire in the direction of the ap-plied ﬁeld. Figure 2 shows a time lapse sequence of themotion of the domain wall in a 100 nm wide wire with anapplied ﬁeld of 100 Oe. Figure 3 is a plot of M x/Msas a function of time corresponding to the same wire and ﬁeld.The linear relationship further conﬁrms the steady-state wallmotion down the length of the wire. Figure 3 also shows the change in M x/Msas a function of time in the same 100 nm wide wire for an applied ﬁeld of140 Oe. The motion is quite different and can be character-ized by the onset of plateaus where M x/Msis constant in time. Figure 4 is a time lapse representation of the domainwall motion from about 2000 to 4000 ps, corresponding tothe motion between subsequent plateaus in Fig. 3. In thisrepresentation the plateaus correspond to metastable stateswhere the domain wall structure is similar to the zero appliedﬁeld equilibrium structure shown in Fig. 1 /H20849t=1984 ps /H20850. En- ergetically speaking this period of time corresponds to peri-ods of increased exchange energy which needs to be over- come by the external ﬁeld. The exchange energy is relativelyconstant as the central wall vortex moves toward the wire’sedge aligning the moments within the wall /H20849t=2446 ps /H20850.A s the vortex reaches the edge there is a large drop in the ex- change energy, and the wall begins to travel down the wire/H20849t=2907 ps /H20850. The wall then travels uniformly down the wire but with one end of the wall leading the other /H20849t=3369 ps /H20850. This expanding of the domain wall increases the wall energy. When the leading edge of the domain wall gets too far ahead,a vortex is nucleated at the trailing edge and another plateauis reached /H20849t=3851 ps /H20850. The process continues in this man- ner, with the domain wall traveling in a ratcheting manner down the wire. It is noted that the simulated nanowires haveperfectly smooth edges so there are no inherent pinning siteson the wire, unlike what might be expected to happen experi-mentally. To quantify the results it is helpful to look at the speed at FIG. 5. /H20849Color online /H20850Plot of the domain wall speed as a function of applied ﬁeld for three different wire widths. The initial linear increase in speedcorresponds to uniform motion along the wire. The peak and subsequent dipappear when the walls begin ratcheting down the wire. The speed increasesagain when the applied ﬁeld is strong enough to quickly unpin the walls. FIG. 2. Domain wall evolution in a 100 nm wide wire in the presence of a100 Oe ﬁeld to the right. The domain wall travels uniformly down the wirein the direction of the ﬁeld. A third of the wire’s total length is shown. FIG. 3. Mx/Msas a function of time corresponding to the motion of domain walls in a 100 nm wide wire for 100 and 140 Oe ﬁelds. The xdirection is along the long axis of the wire. The plateaus are a result of induced dynam-ics within the domain wall region by the increasing magnetic ﬁeld. FIG. 4. Domain wall evolution in a 100 nm wide wire in the presence of a140 Oe ﬁeld to the right. The domain wall ratchets down the wire in anonuniform manner with intervals of uniform motion and intervals of dy-namics only within the wall. A third of the wires total length is shown.08G107-2 Andrew Kunz J. Appl. Phys. 99, 08G107 /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: 141.212.109.170 On: Tue, 09 Dec 2014 21:23:07which the domain wall moves down the wire. The results are summarized in Fig. 5, the domain wall speed as a function ofapplied ﬁeld for three different wire widths. Figure 5 showsthat the domain wall speed increases as the magnetic ﬁeldincreases as long as the applied ﬁeld is less than 100 Oe forall wire widths. As the external ﬁeld is further increased, acritical ﬁeld is reached at which point the motion ceases tobe uniform. The nonuniformity in the motion causes the do-main wall speed to decrease. Increasing the ﬁeld furthercauses the speed to decrease even more before it begins tospeed up again for still larger ﬁelds. This increase in speedstill involves ratcheting behavior, but the length of time spentin the metastable plateaus decreases due to the increasedZeeman energy. IV. CONCLUSIONS A detailed series of three-dimensional LLG simulations has been carried out to study the evolution of domain wallswithin a magnetic nanowire under the inﬂuence of an appliedﬁeld. As the ﬁeld strength is increased the domain wall mo-tion undergoes a transition from uniform to nonuniform char-acterized by a ratcheting of the domain wall down the lengthof the wire. Calculations of the domain wall speed over the range of ﬁelds studied show similar behavior to that ob-served in spin-torque experiments, with an initial increase indomain wall speed followed by a decrease. This decrease inspeed is also characterized by the onset of nonuniform wallmotion. ACKNOWLEDGMENTS The author would like to thank the Helen Klingler Col- lege of Arts and Sciences and the Marquette University Sum-mer Faculty Fellowship Program for supporting this work. 1J. Shi, S. Gider, K. Babcock, and D. D. Awschalom, Science 271,9 3 7 /H208491996 /H20850. 2C. A. Ross et al. , Phys. Rev. B 62,1 42 5 2 /H208492000 /H20850. 3G. Prinz and K. Hathaway, Phys. Today 48/H208494/H20850,2 4 /H208491995 /H20850. 4R. Wieser, U. Nowak, and K. D. Usadel, Phys. Rev. B 69, 064401 /H208492004 /H20850. 5P.-B. He and W. M. Liu, Phys. Rev. B 72, 064410 /H208492005 /H20850. 6A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu, and T. Shinjo, Phys. Rev. Lett. 92, 077205 /H208492004 /H20850. 7LLG Inc., Tempe, AZ 85282, USA. 8Standard Problem No. 4, http://www.ctcms.nist.gov/rdm/mumag.org.html 9G. D. Skidmore, A. Kunz, C. E. Campbell, and E. D. Dahlberg, Phys. Rev. B70, 012410 /H208492004 /H20850.08G107-3 Andrew Kunz J. Appl. Phys. 99, 08G107 /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: 141.212.109.170 On: Tue, 09 Dec 2014 21:23:07
1.2163845.pdf	Precessional and thermal relaxation dynamics of magnetic nanoparticles: A time- quantified Monte Carlo approach X. Z. Cheng, M. B. A. Jalil, H. K. Lee, and Y. Okabe Citation: Journal of Applied Physics 99, 08B901 (2006); doi: 10.1063/1.2163845 View online: http://dx.doi.org/10.1063/1.2163845 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 Thermal fluctuations of magnetic nanoparticles: Fifty years after Brown J. Appl. Phys. 112, 121301 (2012); 10.1063/1.4754272 Reversal of single-domain magnetic nanoparticles induced by pulsed magnetic fields J. Appl. Phys. 103, 07F502 (2008); 10.1063/1.2829594 Numerical study for ballistic switching of magnetization in single domain particle triggered by a ferromagnetic resonance within a relaxation time limit J. Appl. Phys. 100, 053911 (2006); 10.1063/1.2338128 Influence of surface anisotropy on the magnetization precessional switching in nanoparticles J. Appl. Phys. 97, 10J302 (2005); 10.1063/1.1845891 Magnetization reversal of iron nanoparticles studied by submicron Hall magnetometry J. Appl. Phys. 93, 7912 (2003); 10.1063/1.1557827 [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: 129.22.67.107 On: Mon, 24 Nov 2014 06:23:18Precessional and thermal relaxation dynamics of magnetic nanoparticles: A time-quantiﬁed Monte Carlo approach X. Z. Chenga/H20850and M. B. A. Jalil Department of Electrical and Computer Engineering, National University of Singapore, 4 Engineering Drive 3, Singapore 117576, Singapore H. K. Lee and Y . Okabe Department of Physics, Tokyo Metropolitan University, 1-1 Minami-Osawa, Hachioji-shi,Tokyo 192-0397, Japan /H20849Presented on 31 October 2005; published online 20 April 2006 /H20850 A hybrid Monte Carlo /H20849MC /H20850method is proposed to study the full magnetization dynamics of a magnetic nanoparticle, comprising damping, thermal ﬂuctuations, and precessional motion involvedin a magnetization reversal process. The precessional motion is an athermal process, and has beenneglected in previous MC schemes used to model magnetization dynamics. We introduce it into ourhybrid method by adding a precessional step of the appropriate size to the random walk of theheat-bath Metropolis MC method. This hybrid MC method is applied for the study of the role ofprecession in the magnetization switching process of a magnetic nanoparticle under the inﬂuence ofan oblique ﬁeld. We numerically predict two distinct behaviors in the switching processcorresponding to the high and low damping limits. © 2006 American Institute of Physics . /H20851DOI: 10.1063/1.2163845 /H20852 I. INTRODUCTION Theoretical understanding and computational simulation of magnetization reversal process are essential in modelingvarious applications of magnetism at the submicron scale.For instance, thermally activated magnetization reversal atremanence of magnetic grains in hard-disk media determinesthe storage lifetime of the media. In general, the computa-tional modeling of a magnetization reversal process is per-formed by integrating the well-known Langevin dynamicalequation, known as the Landau-Liftshitz-Gilbert /H20849LLG /H20850 equation, i.e., dm dt=−/H92530Hk 1+/H92512·m/H11003heff−/H9251/H92530Hk 1+/H92512/H20851m/H11003/H20849m/H11003heff/H20850/H20852, /H208491/H20850 where m=M/Msis the unit magnetic moment normalized by the saturated magnetization constant Ms, the effective ﬁeld heffis normalized by the anisotropy ﬁeld Hk, and /H9251and/H92530 are the damping constant and gyromagnetic constant, respec- tively. In the conventional LLG equation the effectiveﬁeld is obtained from the energy gradient, i.e., h eff =−/H208491/2KuV/H20850/H11612mE, where Kuis the anisotropy constant and V is the particle volume. At ﬁnite temperatures, heffcontains an additional stochastic term h/H20849t/H20850, which is a white-noise term representing the thermal ﬂuctuations,1and has the following properties as /H20855h/H20849t/H20850/H20856=0 ,/H20855hi/H208490/H20850·hj/H20849t/H20850/H20856=/H9251kBT /H208491+/H92512/H20850KuV/H9254ij/H9254/H20849t/H20850, /H208492/H20850 where iandjdenote Cartesian components x,y, and z,kBis the Boltzmann constant, and Tis the temperature. Previously, based on the stochastic LLG equation, Brown2and Aharoni3 have derived analytical expressions for the switching time inthe uniaxial case /H20849i.e., applied ﬁeld and easy axis are parallel to one another /H20850. Subsequently, Coffey et al. 4derived the ana- lytical switching time for the more general oblique case,where the applied ﬁeld deviates from the easy axis orienta-tion. The latter result is based on the assumption that thedamping constant is small, so that precessional motion con-tributes signiﬁcantly to the magnetization reversal process.The study of precessional motion is also essential in model-ing the switching process of tilted perpendicular recordingmedia, which has recently been proposed as an alternative toperpendicular media, due to its higher thermal stability. Insuch a media, precessional dynamics play a major role be-cause the applied external ﬁeld is in the oblique orientationcompared to the easy axis of the magnetic grains. Thus, it is essential to have a means of investigating the precessional contribution to stochastic magnetization rever-sal. This precessional motion is closely intertwined with thedamping motion and thermal ﬂuctuations. For instance, achange of the damping parameter /H9251not only affects the rela- tive ratio between the precessional and damping contribu-tions but also modiﬁes the amplitude of the thermal ﬂuctua-tions in the magnetization /H20851see Eqs. /H208491/H20850and /H208492/H20850/H20852. Although the stochastic LLG equation offers an avenue for studying allthree contributions to the magnetization dynamics, it is com-putationally intensive. An alternative stochastic modelingmethod is via the Monte Carlo /H20849MC /H20850scheme, which is more efﬁcient especially for long-time simulation. 5This is espe-a/H20850Electronic mail: g0300882@nus.edu.sgJOURNAL OF APPLIED PHYSICS 99, 08B901 /H208492006 /H20850 0021-8979/2006/99 /H208498/H20850/08B901/3/$23.00 © 2006 American Institute of Physics 99, 08B901-1 [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: 129.22.67.107 On: Mon, 24 Nov 2014 06:23:18cially useful for predicting the lifetime of storage media, which is dependent on long time-scale thermal relaxation ofmagnetization. The MC methods also offer the possibility ofscalability 6which is not possible with the LLG equation. However, the MC methods, in general, suffer from the dis-advantage of having the time variable calibrated in terms ofMC steps /H20849MCSs /H20850instead of real physical time units. 7–9 Recently, however, Nowak et al.8proposed a time- quantiﬁed Monte Carlo /H20849TQMC /H20850method for magnetization dynamics, where the MCS time steps are converted to realtime units. They apply the heat-bath Metropolis random-walk algorithm, and obtain the time-quantiﬁcation factor byconsidering the mean-square deviation of the magnetizationabout the energy minima. However, one major omission oftheir MC method is the energy-conserving /H20849or so-called “athermal” /H20850precessional motion. Due to this omission, the time quantiﬁcation factor is strictly valid only in the limit ofhigh damping constant, in which the precessional motion iseffectively suppressed. This is conﬁrmed by a subsequentanalysis by Chubykalo et al. , 9which shows that the Nowak TQMC method breaks down at low values of /H9251. II. MODEL In this paper, we proposed a modiﬁed /H20849hybrid /H20850TQMC method which does not exclude the precessional motion. Inthis hybrid MC method, a MC step consists of a random-walk trial step with of size R/H20849similar to Nowak’s method /H20850, and a precessional step of appropriate size of /H9021/H20849 /H9251/H20850. A key factor for the success of this hybrid method lies in the deter- mination of the correct step size /H9021/H20849/H9251/H20850. A reasonable but somewhat nonrigorous derivation can be obtained based on the time quantiﬁcation factor of Ref. 8, where 1 MCS corre-sponds to a real time unit of /H9004t 0=/H20851/H208491+/H92512/H20850/H9262s/20/H9251/H92530kBT/H20852R2. Thus, the precessional walk step can directly be determined to have a magnitude and direction of /H9004m=−/H92530Hk 1+/H92512/H9004t0·m/H11003h=−/H9252KuV 10/H9251R2·m/H11003h /H11013−/H9021/H20849/H9251/H20850·m/H11003h. /H208493/H20850 A detailed proof of the validity of this hybrid MC method to represent the precessional motion will be given elsewhere. Inthis hybrid MC method, the precessional motion is separatedindependently from the damping motion and thermal ﬂuctua-tion. Thus we can investigate the inﬂuence of precessionalmotion on the magnetization reversal process by simplychanging the damping constant /H9251. By contrast in the LLG method, all the three motions /H20849damping, precessional, and thermal ﬂuctuations /H20850are closely interlinked. III. RESULTS AND DISCUSSION In the following, we apply the hybrid TQMC method on a speciﬁc case of a noninteracting single domain particlewith its easy axis’ direction lying along the zaxis. An ob- lique applied ﬁeld h, normalized by H kis added in the x-z plane at an angle of /H9272with respect to the zaxis. Thus, we can write down the total energy of the system asE/H20849/H9258/H20850 2KuV=−1 2cos2/H9258−hcos/H20849/H9272−/H9258/H20850, /H208494/H20850 where /H9258is the angle between the magnetic moment and the z axis. To understand the role of precession in inducing a mag-netization reversal, we need to consider the energy proﬁle E vs /H9258based on Eq. /H208494/H20850, as shown in Fig. 1. Initially, the mo- ment of the particle is ﬂuctuating stochastically about theminima A. The random walk due to thermal ﬂuctuations has a ﬁnite probability of increasing the particle’s energy. Bycontrast, the precessional motion is an energy-conservingmotion, which does not lead to any change in the energy ofthe particle. Thus, the precessional motion will have no con-tribution to the switching process when the energy level islower than the peak point B. After some time, the random walk of the magnetic moment will cause the energy of thesystem to reach E B, the energy level of B. The average time interval /H20849which we term as /H9270R, the random-walk delay time /H20850 for this to occur is a function of temperature, which deter-mines the size of thermal ﬂuctuations, and the energy barrierheight. It is independent of the precessional motion. Note that the system does not necessarily undergo mag- netization reversal, once it has attained the energy E B. This is because in general the solution E/H20849/H9258,/H9278/H20850=EBtraces out a closed curve in the /H20849/H9258,/H9278/H20850space /H20851say/H9258=f/H20849/H9278/H20850, which is also the trajectory of the precessional motion /H20852, where /H9258and/H9278are the axial and azimuthal orientations of the moment. Switch-ing only occurs when E=E Band/H9258=/H9258B. We investigate the two extreme cases of /H20849a/H20850very low damping and /H20849b/H20850very high damping conditions. For low damping condition, the mag-netic moment precesses around rapidly. Thus, all pointsalong the path f/H20849 /H9278/H20850are quickly accessed, including the point B. Once point Bis reached, the system is in an unstable equilibrium, and rapidly transits or switches to the otherminima C. Thus, in the low damping limit, the switching time /H9270Lis predominantly due to the random-walk delay time /H9270Rrequired to raise the energy from the local minima EAto EB. In the high damping limit, however, the precessional motion along f/H20849/H9278/H20850is so slow that before the system has managed to reach point B, the random-walk ﬂuctuations have caused the system to change to another /H20849usually a relaxation to lower /H20850energy level. Thus, in this case, we require the FIG. 1. Energy vs magnetization orientation /H9258. The parameters used are easy axis orientation /H9272=/H9266/4 and applied ﬁeld =−0.32.08B901-2 Cheng et al. J. Appl. Phys. 99, 08B901 /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: 129.22.67.107 On: Mon, 24 Nov 2014 06:23:18random walk not just to bring the system to energy EBbut to reach the speciﬁc point Bas well. This requires a longer time interval, which we term as /H9270H. In general, for an intermediate damping constant /H9251between the two limits, the switching time/H9270will such that /H9270L/H11021/H9270/H11021/H9270H. We now numerically conﬁrm the role of precession in switching, which have been qualitatively described above.First we deﬁne the switching time as the time for the mag-netic moment to reach zero along the easy axis. Figure 2shows switching time in units of /H20849 /H92530Hk/H20850−1versus damping constant. For the simulation parameters, KuV/kBT=15 and R=0.03. There is an optimum switching time at /H9251=0.3 and switching time increases at both small and large dampingconstants. From our qualitative discussion, the switchingmechanism at small damping constant is predominately dueto precession, and at large damping constant switching ismainly due to thermal ﬂuctuations. To see the relative effectsof precession and thermal ﬂuctuations more clearly, we plotthe switching time in units of Monte Carlo steps /H20849Fig. 3 /H20850. Note that Monte Carlo steps are not in units of real time andtheir conversion into real time depends on /H9251. Figure 3 shows saturation in the switching times for small and large dampingconstant. This observation is consistent with our qualitativeargument. At small damping constant, switching is due to therandom-walk delay time /H9270R/H20849measured in MC steps /H20850, which is independent of damping constant. At large damping val-ues, there is little precession and switching is due to therandom-walk delay /H9270Hwhich is also independent of damp- ing. We also ﬁnd that the damping constant only affects the switching time threshold, but not the reversal behavior, sinceall switching curves show almost the same gradient in Fig. 3during reversal. This may be understood by the fact that thereversal process is an energy relaxation to minima Cand is independent of any precessional motion. Another feature ofFig. 3 is the presence of distinct magnetization oscillations,especially for the curves corresponding to low damping fac-tors. These oscillations occur prior to the actual magnetiza-tion reversal. This may be explained by the fact that themoment precesses about the minima during the random-walkdelay required to excite it to the required energy level E B.The magnetization oscillation thus can be observed if the simulation time is of the order of the precession period. Aquick check of the time /H9270pcorresponding to the ﬁrst peak shows that it varies linearly with the damping constant. Thisis in accordance to the fact that /H9270pis inversely proportional to the precessional frequency, while the latter itself is inverselyproportional to the damping constant. IV. CONCLUSION We propose a hybrid Monte Carlo /H20849MC /H20850scheme which combines the previous heat-bath Metropolis random-walkMC with an additional precessional step of an appropriatesize. Using this modiﬁed MC method, we investigate the roleof precessional motion in magnetization reversal process.Our calculations reveal an upper and lower limit to the re-versal time corresponding to the high and low damping con-stants. We also observe distinct magnetization oscillationsprior to the actual switching event, for the case of low damp-ing constants. These numerical ﬁndings are explained quali-tatively, based on the energy proﬁle of the system. ACKNOWLEDGMENT This work was supported by the National University of Singapore Grant No. /H20849R-263-000-329-112 /H20850. Two of the au- thors /H20849H.K.L. and Y.O. /H20850are supported by a Grant-in Aid for Scientiﬁc Research from the Japan Society for the promotionof science. 1W. F. Brown, Phys. Rev. 130, 1677 /H208491963 /H20850. 2W. F. Brown, IEEE Trans. Magn. 15,1 1 9 6 /H208491979 /H20850. 3A. Aharoni, Phys. Rev. 177, 793 /H208491969 /H20850. 4W. T. Coffey, D. S. F. Crothers, J. L. Dormann, Yu. P. Kalmykov, E. C. Kennedy, and W. Wernsdorfer, Phys. Rev. Lett. 80, 5655 /H208491998 /H20850. 5H. K. Lee, Y. Okabe, X. Cheng, and M. B. A. Jalil, Comput. Phys. Commun. 168, 159 /H208492005 /H20850. 6H. K. Lee and Y. Okabe, Phys. Rev. E 71, 015102 /H208492005 /H20850. 7X. Z. Cheng, M. B. A. Jalil, H. K. Lee, and Y. Okabe, Phys. Rev. B 72, 094420 /H208492005 /H20850. 8U. Nowak, R. W. Chantrell, and E. C. Kennedy, Phys. Rev. Lett. 84, 163 /H208492000 /H20850. 9O. Chubykalo, U. Nowak, R. Smirnov-Rueda, M. A. Wongsam, R. W. Chantrell, and J. M. Gonzalez, Phys. Rev. B 67, 064422 /H208492003 /H20850. FIG. 3. Magnetization component along zaxis as a function of time /H20849in units of MCS /H20850. The damping constant /H9251is varied from 1/64 to 4 /H20849top to bottom /H20850, with a multiplication factor of 2 between the adjacent curves. Inset ﬁgure:Switching time /H20849in units of MCS /H20850as a function of damping constant /H9251. FIG. 2. Switching time /H20849in real time units /H20850as a function of damping constant.08B901-3 Cheng et al. J. Appl. Phys. 99, 08B901 /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: 129.22.67.107 On: Mon, 24 Nov 2014 06:23:18
1.575273.pdf	Summary Abstract: Magnetic properties of thin metal cluster films B. Andrien, J. Haugdahl, and D. R. Miller Citation: Journal of Vacuum Science & Technology A 6, 1865 (1988); doi: 10.1116/1.575273 View online: http://dx.doi.org/10.1116/1.575273 View Table of Contents: http://scitation.aip.org/content/avs/journal/jvsta/6/3?ver=pdfcov Published by the AVS: Science & Technology of Materials, Interfaces, and Processing Articles you may be interested in Summary Abstract: Ceramic thin films from cluster beams J. Vac. Sci. Technol. A 6, 1770 (1988); 10.1116/1.575291 Summary Abstract: Properties of hydrocarbon polymer films containing metal clusters J. Vac. Sci. Technol. A 5, 1913 (1987); 10.1116/1.574489 Summary Abstract: Thin films for magnetic recording technology: A review J. Vac. Sci. Technol. A 3, 657 (1985); 10.1116/1.572973 Summary Abstract: Cluster formation and the percolation threshold in thin Au films J. Vac. Sci. Technol. A 1, 438 (1983); 10.1116/1.571940 Summary Abstract: Photoelectron spectra of metal molecular clusters J. Vac. Sci. Technol. 17, 221 (1980); 10.1116/1.570441 Redistribution subject to AVS license or copyright; see http://scitation.aip.org/termsconditions. Download to IP: 140.254.87.149 On: Sat, 20 Dec 2014 02:10:08M. Mansurlpur: Summary Abstract: Magneto-optical recording in thin films puter simulations based on models that postulate variations as low as ± 10% of the nominal values over length scales of a few hundred angstroms have successfully reproduced the experimental observations. Random spatial variations of magnetic parameters give rise to "soft spots," i.e., regions that are reverse magnetized before the rest of the film and therefore act as nucleation centers. They also create local potential wells and energy barriers that trap the walls and stabilize the domains. The same random variations, how ever, may be responsible for jagged domain boundaries. The most dramatic confirmation of theoretical predictions con cerning the magnetic structure within thermomagnetically recorded domains has come from Lorentz transmission elec tron microscopy with resolutions approaching 100 A.7 The write and erase processes are similar in many respects but the erasure has its own peculiarities. It is believed that thermomagnetic erasure of a written domain consists of nu cleation from within and wall destabilization and collapse from without. In general, the required external fields for writing and erasure are different. Consequently, recording a block of data must be preceded by an erase cycle; this is a significant drawback for the erasable optical disk technolo gy. Shieh and Kryder, however, have recently demonstrated that in certain materials it is possible to write and erase with-out external fields, simply by using a long laser pulse for writing and a short pulse for erasing. x The dynamics of this process, however, are not well understood at the present time. Detailed analyses of domain wall dynamics in amorphous media are currently under way9 and large-scale computer simulations based on the Landau-Lifshitz-Gilbert equation of magnetization dynamics 10 are expected to provide further insights into the nature of these complex and fascinating phenomena. 'M. Mansuripur, J. Opt. Soc. Am. 3. 2086 (1986). 'M. Mansuripur, App\. Opt. 26, 3981 (1987). 'M. Mansuripur and G. A. N. Connell, J. App!. Phys. 54,4794 (1983). 4M. Mansuripur, G. A. N. Connell, and J. W. Goodman, J. AppJ. Phys. 53, 4485 (1982). 'M. Mansuripur, 1. Appl. Phys. 61, 1580 (1987). "T. W. McDaniel and M. Mansuripur, IEEE Trans. Magn. 23, 2943 ( 1987). '1. c. Suits, R. H. Geiss, C.l. Lin, D. Rugar, and A. E. Bell, 1. Appl. Phys. 61,3509 (1987). "H-P. D. Shieh and M. H. Kryder, App!. Phys. Lett. 49, 473 (1986). '1M. Mansuripur and T. W. McDaniel, 1. App!. Phys. (in press). iliA. P. Malozemoffand J. C. Sioncl.ewski, Magnetic Domain Walls in Bub ble Materials (Academic, New York, 1979). Summary Abstract: MagnetiC properties of thin metal cluster films B. Andrien, J. Haugdahl, and D. R. Miller Department a/ Applied Mechanics and Engineering Sciences, Unil'ersity a/California-San Diego, La Jolla, California 92093 (Received 18 September 1987; accepted 7 October 1987) Weare using a continuous thermal source to form metal cluster beams of Fe and Ni to grow thin films in ultrahigh vacuum (UHV). The magnetic properties of thin metal films are generally understood I and are of great interest to the magnetic recording industry. In order to monitor the growth of these films, and to investigate their surface effects, it is desirable to make magnetization measurements in situ in the UHV environment where the films are grown. We have recently adapted a dynamic induction coil magnetometer2-t to our UHV facility and improved the sensitivity to levels below 10-6 emu, enabling in situ hysteresis loop measure ments on films 100 A thick. Figure I is a schematic of our thin cluster film facility that is presently being completed. The data reported below are from a prototype system of similar design. The free-jet noz zle source is made of high-density graphite or tungsten with a nozzle orifice 0.02-0.05 em in diameter. The source is heat ed by electron bombardment. The metal evaporates into an inert helium carrier gas and subsequently expands into high vacuum. The beam is collimated and passes into an UHV chamber where it can be deposited onto a quartz rod sub strate, which can be heated or cooled by conduction. The beam can be monitored by a thin-film deposition monitor, and energy analyzed by time-of-flight. The mass spectrom eter can examine the beam or thermal desorption spectra of adsorbed gases from the film. The rod can be rotated so that the film can be analyzed with the Auger electron spectrom eter. Similar substrate samples can be removed from the vacuum and analyzed by transmission electron microscopy and electron diffraction. Magnetic measurements are made in situ on the films by translation of the film into one of the two balanced pick-up coils of the pulsed induction magne tometer. A schematic of the magnetometer is included in Fig. 1; the associated electronic circuitry will be discussed elsewhere.4 The current waveform through the 7S-turn field solenoid appears as a damped oscillatory waveform with a frequency of ~ 5.3 kHz, and provides a maximum field of 3000 Oe. The induction pick-Up system is of the conventional form: two identical counterwound pick-up coils placed inside the drive coil. Since the frequency of the driving field is several kHz the pick-up coils require only 40 turns. This small number of turns allows the system bandwidth to be nearly 1 Me. The coils are first balanced to null out the H field. The sample is 1865 J. Vac. Sci. Technol. A 6 (3), May/Jun 1988 0734-2101/88/031865-02$01.00 © 1988 American Vacuum SOCiety 1865 Redistribution subject to AVS license or copyright; see http://scitation.aip.org/termsconditions. Download to IP: 140.254.87.149 On: Sat, 20 Dec 2014 02:10:081866 Andrlen, Haugdahl, and Miller: Summary Abstract: Magnetic properties of thin films 1866 + .. ---.DUARH ROO _VACUIft1 SEAL GLASS TUBE ]]:. I.' --:..:--- H FElJl SOLENOIO COL : ' I . , T\<IO P1O<-UP COL$ ____ ._ --:c-::~ WOUND N SERIES N.G£TOMETEIi . --. '. OPPQSlTION : j, __ ~_-_T'oI'STED PAlR OUT : -II • ~ ____ -r--_~--'-"c:.':.J' ) ,"''':.;..'-'-'-----, OEPOSInON III THIN R.IM ~CE CHAM68' I OW18ER I OEPOSlnON I ~ONlTOR ~Al:s~~r.l /-''! /' : 10-7 TO 10-' // ! 10-10 TO 10-3 rom / : TORR J> ..... 55 SPECTROMETER FIG. 1, Schematic of vacuum facility and dynamic induction magnetometer. then translated into one coil and its output is amplified and integrated, so that it is proportional to M. When the magnet ization versus the drive field are displayed on the oscillo scope, the result is an M-H curve of nested hysteresis loops, as shown in Fig. 2. Since the films are very thin no correc tions are necessary for demagnetization, eddy currents, or heating effects. Figure 2 shows the magnetization hysteresis curve for an Fe cluster film of unmeasured thickness grown in the proto type facility. Using the known saturation induction for Fe, together with our measured flux, we calculate a thickness of 610 A. This thickness agrees to within 10% of that obtained J. Vac. Sci. Technol. A, Vol. 6, No.3, May/Jun 1988 FIG. 2, 41TM vs H for Fe film from cluster source; B, = 21 600 G, B, = 16900 G, and H, = 160 Oe. by the film thickness monitor. Electron micrographs of clus ters deposited on a slide placed in front of this source indicat ed cluster sizes of order 700 A. We intend to use the magne tometer and surface analysis equipment to study the growth of Fe, Ni, and Co cluster films, and the effect of adsorbed gases, Because of the magnetometer's bandwidth, dynamic magnetic effects such as Barkhausen noise, eddy current, and wall resonance effects can also be examined in situ. Acknowledgments: This work was supported by NSF Grant No. CBT-8319762 and ONR NOOO14-87-K-0675. 'B. D. Cullity, introductioJl to Magnetic Materials (Addison-Wesley, Reading, MA, 1972), 'E, C. Crittenden, Jr., A. A. Hudimac, and R. I. Strough, Rev, Sci. Instrum. 22,872 (1951). 'H. Oguey, Rev, Sci. lnstrum, 31, 70] (1960). 4J. B. Haugdahl and D, R, Miller, Rev. Sci, lnstrum, (submitted), Redistribution subject to AVS license or copyright; see http://scitation.aip.org/termsconditions. Download to IP: 140.254.87.149 On: Sat, 20 Dec 2014 02:10:08
1.3371695.pdf	Broad-band ferromagnetic resonance characterization of lossy ferromagnetic metallic elements V. V. Zagorodnii, A. J. Hutchison, S. Hansen, Jue Chen, H. H. Gatzen, and Z. Celinski Citation: Journal of Applied Physics 107, 113906 (2010); doi: 10.1063/1.3371695 View online: http://dx.doi.org/10.1063/1.3371695 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/107/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Phase-resolved x-ray ferromagnetic resonance measurements in fluorescence yield J. Appl. Phys. 109, 07D353 (2011); 10.1063/1.3567143 Interface-related damping in polycrystalline Ni 81 Fe 19 / Cu / Co 93 Zr 7 trilayers J. Appl. Phys. 105, 07D309 (2009); 10.1063/1.3072030 Ferromagnetic resonance study of polycrystalline Fe 1 − x V x alloy thin films J. Appl. Phys. 103, 07B519 (2008); 10.1063/1.2830648 Tunable remanent state resonance frequency in arrays of magnetic nanowires Appl. Phys. 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Celinski1,a/H20850 1Center for Magnetism and Magnetic Nanostructures, UCCS, Colorado Springs, Colorado 80918, USA 2Taras Shevchenko National University of Kyiv, 64 Volodymyrs’ka Street, 01033 Kyiv, Ukraine 3Center for Production Technology, Institute for Microtechnology, Leibniz Universitaet Hannover, An der Universitaet 2, 30823 Garbsen, Germany /H20849Received 2 November 2009; accepted 25 February 2010; published online 2 June 2010 /H20850 We developed a method to analyze broad-band ferromagnetic resonance /H20849FMR /H20850data for rectangular ferromagnetic bars of micron and submicron thicknesses. This method allows one to determine thegyromagnetic ratio, the saturation magnetization, and the damping constant of the measuredstructures. The proposed technique can be used for nondestructive testing of the ferromagneticelements of micro-electro-mechanical system sensors, actuators, and related devices without anyspecial sample preparation. In the developed approach, an analysis of the FMR linewidth is notneeded to determine the damping constant. This method rather utilizes the frequency dependence ofthe demagnetizing factors in the range of 1–40 GHz for the extraction of magnetic parameters. Itsapplication is demonstrated using Ni 81Fe19,N i 45Fe55, and Co 35Fe65specimens as examples. © 2010 American Institute of Physics ./H20851doi:10.1063/1.3371695 /H20852 The ferromagnetic resonance /H20849FMR /H20850method is a univer- sally recognized and well established tool for the analysisand characterization of magnetic materials. It can be used todetermine damping parameters /H20849via the FMR linewidth /H20850, 1–3 the saturation magnetization value4,5the anisotropy constants,6etc. FMR measurements can be carried out on magnetic oxides,1,4,7including half-metallic materials,8and ferromagnetic conductive ﬁlms.2,3,5,9–11Recently, the use of a VNA in FMR measurements /H20849VNA-FMR /H20850has attracted a lot of attention due to many advantages.2,3,5,7,9,10For example, such a setup allows the performance not only of measure-ments at a series of ﬁxed frequencies with a swept magneticﬁeld /H20849as it commonly done in standard FMR measurements with a resonant cavity /H20850, but also measurements in the fre- quency domain /H20849with the applied magnetic ﬁeld kept at con- stant value /H20850. With VNA-FMR, however, ﬁeld-swept mea- surements at multiple frequencies are very convenient whencompared to cavity-based measurements. For example, such broad-band linewidth measurements can separate and deter-mine the intrinsic and extrinsic damping contributions. 12,13 In the past, the majority of FMR measurements on fer- romagnetic materials were carried out on thin continuous orpatterned ﬁlms with thicknesses from /H1101110 Å to /H11011100 nm. Such thicknesses are signiﬁcantly smaller than the skin depthfor measured materials. In such cases, ﬁlms are driven by thenearly uniform high-frequency magnetic ﬁelds. Moreover,the lateral sizes of the samples are usually several orders ofmagnitude greater than the thickness. This makes it possibleto use Kittel’s formula 14as a good approximation for the FMR resonance condition. Difﬁculties in FMR measurements arise when samples are made of relatively thick conductive materials due toshielding and nonuniform magnetic ﬁeld distributions. The depth dependence of the demagnetizing and driving ﬁeldsleads, for example, to a non-uniform broadening of the FMRresonant line and to a shift of the absorption maximum. 15,16 Eddy currents also make a signiﬁcant contribution to FMR damping in conductive ferromagnetic ﬁlms thicker than ap-proximately 100 nm. 5,17As a result, the FMR linewidth broadens drastically, distorts, and diminishes, complicatingthe measurements and analysis. Nevertheless, there is a need to determine the magnetic parameters of different metallic elements of micron and sub-micron thicknesses, such as electroplated ferromagnetic ﬁlmsfor micro-electro-mechanical system /H20849MEMS /H20850 actuators/sensors 18and on-chip inductive components for radio-frequency integrated circuit applications.19These types of samples are difﬁcult to characterize by vibrating-samplemagnetometer or optical methods. For sensors and actuators,an FMR analysis can verify properties such as saturationmagnetization quickly and reliably. These measurementscould be carried out on processed substrates immediately fol-lowing the electroplating process. Sensors most often com- prise Ni 81Fe19due to the low magnetostriction of this alloy, while actuators employ Ni 45Fe55due to its high saturation magnetization. The FMR technique discussed here can indi-cate the actual composition of these alloys in these and simi-lar devices. In this work, we present a model developed to analyze FMR measurements of rectangular ferromagnetic bars. Thebars have large length/thickness and width/thickness aspectratios, and thicknesses on the order of, or larger than, themicrowave skin depth. Employing only the FMR resonantcondition over a wide frequency range /H208491–40 GHz /H20850and waiving analysis of the linewidth, we demonstrate the possi-bility of determining the gyromagnetic ratio, the saturationmagnetization, and the damping constant for such relatively a/H20850Electronic mail: zcelinsk@uccs.edu.JOURNAL OF APPLIED PHYSICS 107, 113906 /H208492010 /H20850 0021-8979/2010/107 /H2084911/H20850/113906/7/$30.00 © 2010 American Institute of Physics 107 , 113906-1 [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: 130.113.111.210 On: Mon, 22 Dec 2014 13:15:07thick and lossy rectangular ferromagnetic elements. In the general case, the classical motion of magnetiza- tion in a magnetic material subject to an applied magneticﬁeld is described by the Landau–Lifshitz /H20849LL/H20850equation 17 /H11509M/H6023 /H11509t=−/H9253M/H6023/H11003H/H6023−/H9253/H9261 M2M/H6023/H11003/H20849M/H6023/H11003H/H6023/H20850, /H208491/H20850 where M/H6023is the magnetization, H/H6023is the effective magnetic ﬁeld,/H9253is the gyromagnetic ratio, and /H9261is the LL dissipation parameter. Considering the magnetization motion in a con-stant dc biasing ﬁeld and a high-frequency driving magnetic ﬁeld, M /H6023andH/H6023can be decomposed according to the form M/H6023=M/H60230+m/H6023V,H/H6023=H/H60230+h/H6023V, /H208492/H20850 where m/H6023Vandh/H6023Vare the dynamic components of the mag- netization and the driving magnetic ﬁeld, respectively, and M/H60230andH/H60230are the constant equilibrium /H20849static /H20850magnetiza- tion and internal constant /H20849dc/H20850effective magnetic ﬁeld. For experiments with microwave excitations in magnetic materi- als,h/H6023Vis ordinarily supplied directly by a microwave reso- nator or a waveguide coupled to the magnetic sample undertest. An accurate description of the motion of the magnetiza- tion must involve damping. Using the phenomenologicaldamping constant /H9251, with the substitutions /H9261→/H9251M//H208491+/H92512/H20850 and/H9253→/H9253H//H208491+/H92512/H20850, the LL equation transforms to the Gil- bert equation /H11509M/H6023 /H11509t=−/H9253H/H20849M/H6023/H11003H/H6023/H20850+/H9251 MM/H6023/H11003/H11509M/H6023 /H11509t, /H208493/H20850 where the value of the gyromagnetic ratio /H9253His different from its initial value /H9253. This can change resonance condi- tions in the case of lossy magnetic materials with /H9251/H113500.05. It is commonly assumed that a relatively small amplitude ofh/H6023V/H20849hV/H11270H0/H20850excites a relatively small oscillations of m/H6023V /H20849mV/H11270M0/H20850and that the equilibrium orientation of the satura- tion magnetization in an isotropic ferromagnet coincides with the biasing magnetic ﬁeld direction. Writing the har-monic driving magnetic ﬁeld in terms of complex amplitudes /H20849h /H6023˙=h/H6023·exp /H20849i/H9275t/H20850,R e /H20849h/H6023˙/H20850=h/H6023V, where /H9275is the frequency of the variable magnetic ﬁeld and neglecting the small products of m/H6023andh/H6023in the Gilbert equation /H20850, one can obtain the linear- ized equation of motion with complex amplitudesi/H9275m/H6023+/H9253Hm/H6023/H11003H/H60230+i/H9251/H9275 M0m/H6023/H11003M/H60230=−/H9253/H20849M/H60230/H11003h/H6023/H20850, /H208494/H20850 where the m/H6023Vtime dependence is identical with that of h/H6023V due to the linearity of the equation, i.e., the complex ampli- tude of oscillatory magnetization is given by m/H6023˙=m/H6023 /H11003exp /H20849i/H9275t/H20850,R e /H20849m/H6023˙/H20850=m/H6023V. The investigated ferromagnetic metallic samples were prepared by electrochemical deposition onto Si substrates,and have the shape of rectangular bars of micron and submi-cron thicknesses. The materials and dimensions of the exam-ined samples are listed in Table I. The fabrication process started with the sputter deposition of a Permalloy seed layeronto a Si substrates. A positive photoresist was spin-coatedonto Permalloy templates. Applying UV lithography, thephotoresist layer was patterned. Before the electrodepositionwas carried out, the resist at the rim of the substrate wasstripped. For the plating process, a pulse current depositionwas used, applying forward and reverse pulses for composi-tions Ni 45Fe55and Ni 81Fe19/H20849see Table II/H20850. For Co 35Fe65,a sequence of forward pulses combined with a dwell periodswithout current was used. For each composition, sampleswith a thickness of 0.4 /H9262m, 1/H9262m, and 2.5 /H9262m were cre- ated. After the electrodeposition, the samples were pla-narized using chemical-mechanical polishing. The thin-ﬁlmprocess ended with an ion beam etching of the seed layer.Afterwards, the wafers were diced into chips. The exampleof the atomic force microscopy /H20849AFM /H20850image is shown in Fig. 1for the Permalloy sample no. 3. The AFM analysis indicated that the average surface roughness was on the orderof 6 nm in all the samples. The maximum measured differ-ence in height for many measurements was about 20 nm.This indicates fair uniformity for samples with thickness onthe order of 1 /H9262m. Due to shape anisotropy, the internal oscillatory mag- netic ﬁeld at any location inside the sample depends on thevariable magnetization distribution. Similarly, the internalstatic magnetic ﬁeld distribution depends on the static mag-netization distribution. Thus, in terms of complex ampli- tudes, H /H60230is written as H/H60230=H/H6023E0−NJSM/H60230and h/H6023ash/H6023=h/H6023ETABLE I. Data on rectangular ferromagnetic bars used in the broad-band FMR experiment. Three different metallic compositions were used. Each bar has 5 mm length, 0.8 mm width, and deposited over the 0.2 /H9262m Permalloy seed layer. Specimen No. 1234 5 67 Composition Ni81Fe19 Ni45Fe55 Co35Fe65 Thickness, /H9262m 0.42 2.25 0.99 2.6 0.03 0.45 2.7 Actual Ni/Fe /H20849Co/Fe /H20850ratio 83/16 85/15 31/69 43/56 N.A. 35/65 32/68 TABLE II. Process parameters for electroplating process. MaterialForward current /H20849mA /H20850Reverse current /H20849mA /H20850Forward time /H20849ms/H20850Reverse time /H20849ms/H20850 Ni45Fe55 800 80 9 1 Ni81Fe19 1000 100 9 1 Co35Fe65 800 0 25 75113906-2 Zagorodnii et al. J. Appl. Phys. 107 , 113906 /H208492010 /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: 130.113.111.210 On: Mon, 22 Dec 2014 13:15:07−NJDm/H6023, where H/H6023E0is the applied static biasing magnetic ﬁeld, NJSis the static demagnetizing tensor, h/H6023Eis the applied driving magnetic ﬁeld, and NJDis the dynamic demagnetizing tensor. We are assuming that it is possible to express the static demagnetizing ﬁeld as − NJSM/H60230and the dynamic de- magnetizing ﬁeld as − NJDm/H6023in terms of these demagnetizing tensors. Such an approach follows Kittel,14except that the components of the demagnetizing tensor of the rectangularbars are averaged values 20/H20849i.e., the internal ﬁeld inside the bar is nonuniform /H20850, and NJSis not identical to NJDdue to the skin effect in conducting metallic ferromagnets. The typicalskin depth in high conductivity ferromagnets at microwavefrequencies is on the order of 0.1 /H9262m, and the dynamic de- magnetizing ﬁeld quickly decays and can be considered neg-ligible far from the surface /H20849a few skin depths /H20850. The interac- tion between the high-frequency magnetic ﬁeld and theferromagnetic metallic material thus occurs mainly close to the surface. In the case of in-plane H /H6023E0the internal biasing ﬁeld H/H60230increases close to the surface20as well as the mag- nitude of h/H6023. The ferromagnetic bars therefore have a higher dynamic demagnetizing ﬁeld induced by the higher drivingﬁeld in that surface layer, and a smaller average static de-magnetizing ﬁeld in the interaction region compared to non-conductive samples with the same dimensions. The orientation of the ferromagnetic bars with respect to the biasing magnetic ﬁeld in all experiments is shown in Fig.2. The z-axis of each bar was along the signal line of the coplanar waveguide in the experimental setup describedlater. That way, the biasing static magnetic ﬁeld coincided with the direction of propagation of the microwave, and amicrowave magnetic ﬁeld was generated in xy-plane due to the high-frequency current. For the microwave frequencies/H208490–40 GHz /H20850used here, the dimensions of the samples and the width of the signal line are substantially smaller than theelectromagnetic wavelength in air or in the coplanar wave-guide. One can therefore neglect the nonuniform distributionof high-frequency current in the signal line under the sampleas well as the corresponding nonuniform distribution of thehigh-frequency magnetic ﬁeld interacting with the sample. In the coordinate system depicted in Fig. 2, the tensors and vectors from Eq. /H208494/H20850are written as NJ S=/H20900NX0 0 0NY0 00 NZ/H20901,NJD=/H20900N11 00 0N22 0 00 N33/H20901, H/H6023E0=/H209000 0 HZ/H20901,M/H60230=/H209000 0 M0/H20901,m/H6023=/H20900mX mY 0/H20901. Therefore Eq. /H208494/H20850becomes i/H9275m/H6023+/H9253Hm/H6023/H11003/H20849H/H6023E0−NJSM/H60230/H20850−/H9253HM/H60230/H11003/H20849NJDm/H6023/H20850 +i/H9251/H9275 M0m/H6023/H11003M/H60230=0 . /H208495/H20850 Equation /H208495/H20850includes different dynamic and static de- magnetizing factors as well as the damping constant, anddiffers from the linearized equation obtained by Kittel. 14 Equation /H208495/H20850can be written as a system of 2 scalar equations with only xand ycomponents of dynamic magnetization, and the corresponding characteristic equation for the studiedsystem is given by /H92752/H208491+/H92512/H20850−/H9275·i/H9251/H9253H/H208512HZ+/H20849N11+N22−2NZ/H20850M0/H20852 −/H9253H2/H20851HZ+/H20849N11−NZ/H20850M0/H20852·/H20851HZ+/H20849N22−NZ/H20850M0/H20852=0 . /H208496/H20850 Because damping is included in Eq. /H208496/H20850, the frequency /H20849or the magnetic ﬁeld /H20850should be treated as a complex value. Using /H9275=/H9275/H11032+i/H9275/H11033, the real part of the complex eigenfre- quency derived from Eq. /H208496/H20850is /H20849/H9275+/H11032/H208502=/H92532HZ2+/H92532/H20849N11+N22−2NZ/H20850M0HZ +/H92532/H20875/H20849N11−NZ/H20850/H20849N22−NZ/H20850−/H92512 4/H20849N11−N22/H208502/H20876M02. /H208497/H20850 The maximal absorption of power in the FMR experiments is observed at the frequency equal to the real part of the com-plex eigenfrequency as it takes place in driven oscillations.The subscript+on the /H9275/H11032indicates that the positive root is taken in ﬁnding the FMR frequency. The damping parameter /H9251is treated here as a constant, independent of frequency and biasing ﬁeld. This is consid-ered to be an acceptable approximation for ferromagneticmetals. 17N11,N22, and NZare treated as frequency dependent parameters due to the frequency dependence of the skin FIG. 1. /H20849Color online /H20850AFM image of the Permalloy sample no. 3 surface /H208493/H110033/H9262m2/H20850. FIG. 2. Geometry and orientation of the ferromagnetic bar on the Si substrate.113906-3 Zagorodnii et al. J. Appl. Phys. 107 , 113906 /H208492010 /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: 130.113.111.210 On: Mon, 22 Dec 2014 13:15:07depth. Attempts to ﬁt the whole frequency range of /H20849/H9275+/H11032/H208502as a function of HZleads to inaccurate results for the gyromag- netic ratio and saturation magnetization. The skin depth de-creases signiﬁcantly at very high frequencies and the demag-netizing factors for a thin ﬁlm approach their limiting values,N 11→4/H9266,N22→0, and NZ→0, becoming virtually indepen- dent of frequency. The FMR frequency-ﬁeld dependence cantherefore be successfully ﬁtted by the Kittel formula in thehigh frequency regime. This is the common way to deter-mine /H9253and M0from FMR measurements when the ﬁlm thickness is much smaller than the skin depth for the entirefrequency range. If FMR measurements are carried out over a wide fre- quency range, starting from the resonant frequency at mini-mal saturation biasing ﬁeld, it is possible to extract magneticparameters in the case of relatively thick metallic ﬁlms. It isalso possible to determine the damping constant since thedemagnetizing factors change with frequency /H20851see the last term of Eq. /H208497/H20850/H20852. This peculiarity is commonly neglected in the traditional approximation where small or zero values of /H9251, and constant demagnetizing factors, are assumed. How- ever, for ferromagnetic metallic ﬁlms with a thickness ex-ceeding several hundreds nanometers, the damping of mag-netic oscillations drastically increases due to eddy currents.Also, the FMR absorption linewidth broadens inhomoge-neously due to the nonuniform electromagnetic ﬁeld and bi-asing ﬁeld distributions inside the probe region. Furthermore,that distribution leads to a nonuniform exchange interactionand subsequent broadening of the FMR linewidthbroadening. 17Due to eddy currents and inhomogeneous broadening, an analysis of the FMR absorption linewidthdoes not provide a reliable value for the damping constant. Inour case, this is further complicated due to the small intensityof the measured FMR absorption. This is due to a weakcoupling between the broad-band coplanar waveguide andthe typically small samples under test. With the additional complications associated with noise, the frequency response of measuring waveguide, etc., onecannot practically separate different contributions to the line-width and easily extract the damping constant from low- intensity FMR absorption. The measured FMR absorptionline is, in fact, the averaged response of the ferromagneticsample to the biasing ﬁeld or frequency change, with all theabove-mentioned factors playing a role. In this respect, themeasured maximum of the resonant absorption is the resultof the sum of all the partial absorptions from different re-gions of the sample, inﬂuenced by different factors which areconstant at ﬁxed frequency. Assuming /H9251has a constant value, and the interrelation of all the above mentioned fac-tors inﬂuencing the measured linewidth are the same at dif-ferent frequencies or biasing ﬁelds, the absorption maximumis related to a certain average biasing ﬁeld and average mi-crowave ﬁeld inside the probe region. These average ﬁeldsdepend on the particular ﬁeld distributions in the probe re-gion, which changes with frequency. For the studied ferromagnetic samples, the skin effect is supposed to be normal. While the skin depth is unknown fora variety of ferromagnetic metals and fabrication techniques,the behavior of the magnetic ﬁeld distribution inside themagnetic elements can be predicted for certain simple geom- etries. For a rectangular bar, the internal magnetic ﬁeld dis-tribution can be calculated 20and the average “effective” in- ternal ﬁeld /H20849i.e., average demagnetizing factors /H20850can be determined.21In the case of large aspect ratios of l/dand w/d/H20849where l,d, and wdenote length, thickness, and width of the bar, respectively /H20850one can determine, based on20and21the average “effective” internal ﬁeld for the parallel biased barand the averaged internal ﬁeld in the surface layer. Compar-ing the results of 20and21in the case of the above mentioned large aspect ratios of l/dand w/d, one can conclude the almost perfect matching of average “effective” internal ﬁeldfor the parallel biased bar of thickness dand the averaged internal ﬁeld in the surface layer of the same thickness din the bar of greater total thickness and the same dimensionsl,w. This allows one to employ the analytical expressions for “effective” demagnetizing factors and calculate those factors,used in Eq. /H208497/H20850. The dynamic demagnetizing factors are dif- ferent from the static ones for the same sample dimensions/H20849i.e.,N 11+N22+NZ/HS110054/H9266/H20850due to different spatial distribution of the surface divergence of the microscopic static and dy-namic magnetization. However, the change of dynamic fac-tors versus frequency follows the change of the correspond-ing static ﬁeld versus frequency because both factors are therepresentation of the averaged internal magnetic ﬁelds in thethin surface layer. The exact depth used for averaging is notcrucial for calculations. One can, for example, consider asurface layer with a thickness of three skin depths as a regionwhere the interaction between the metallic ferromagnet andthe high-frequency magnetic ﬁeld mainly takes place. FIG. 3. Frequency dependence of term a=N11+N22−2NZin the Eq. /H208497/H20850for the FMR frequency. The skin depth is calculated for typical iron parameters. FIG. 4. /H20849Color online /H20850Frequency dependence of term b=/H20849N11−NZ/H20850/H20849N22 −NZ/H20850−/H92512/4/H20849N11−N22/H208502in the Eq. /H208497/H20850for the FMR frequency. The skin depth is calculated for typical iron parameters.113906-4 Zagorodnii et al. J. Appl. Phys. 107 , 113906 /H208492010 /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: 130.113.111.210 On: Mon, 22 Dec 2014 13:15:07The frequency dependencies of the terms in Eq. /H208497/H20850are illustrated in Figs. 3and4, where a=N11+N22−2NZand b=/H20849N11−NZ/H20850/H20849N22−NZ/H20850−/H92512/H20849N11−N22/H208502 4. The expressions for the effective demagnetizing factors from21have been used, and taking values of /H9267=1 /H1100310−7/H9024m for resistivity, and /H9262r=120 for relative perme- ability /H20849purporting iron high-frequency properties as an ex- ample /H20850, to estimate the skin depth. The avalue saturates rapidly when the frequency increases, reaching the limitingvalue of 4 /H9266when the frequency exceeds /H1101510–20 GHz /H20849see Fig. 3/H20850. For ferromagnetic metallic bars with such aspect ratios as used herein, this frequency-dependent behavior of a is expected. Figure 4shows the frequency dependence of b for different values of /H9251. For the same values of /H9267and/H9262r mentioned above, any difference between curves for similar /H9251values starts to be perceptible only when /H9251exceeds ap- proximately 0.01. The frequency ﬁt for bthus reveals /H9251for relatively lossy ferromagnetic ﬁlms. Also, the frequency de-pendence of bbecomes weak above 20 GHz due to the domi- nance of the − /H92512/H20849N11−N22/H208502/4 term. The value of bis thor- oughly unaltered at higher frequencies due to the fact that N11reaches 4 /H9266while N22approaches zero. For high enough values of /H9251,b→4/H92662/H92512as the frequency increases. There- fore, at higher frequencies, a frequency ﬁt of amakes it possible to extract 4 /H9266M0and/H9253using Eq. /H208497/H20850. The frequency ﬁt of b=/H20849/H20849/H9275+/H11032/H208502//H92532−HZ2−aM 0HZ/H20850/M02enables one to extract the value of /H9251. Figure 4shows curves obtained using the exponential sum ﬁtting formula. The value of /H9251can be ex-tracted from ﬁtting curves by that model with a standard error of approximately 0.0016. The coplanar waveguide used in the FMR experiment had a 400 /H9262m wide signal line. The waveguide was centered in the electromagnet gap and connected to an Agilent8722ES S-parameter VNA by coaxial cables via coaxialwaveguide adapters The biasing ﬁeld strength was measuredby a Lakeshore 421 gaussmeter. The microwave scatteringparameters of ensemble were collected and the absorption ofthe ferromagnetic material were extracted in accordance withthe analysis model just described. 2,10Figure 5shows ex- amples of measured FMR spectra at 15, 25, and 36 GHz forthe Permalloy sample no. 3. Note that the distortion of theLorentzian shape of the absorption curve /H20849predicted by Kit- tel’s model /H20850would make analysis of the damping based on the linewidth difﬁcult. Field-swept /H208490–1 T /H20850FMR measurements were per- formed at different frequencies /H208491–40 GHz /H20850, held constant for each run. Note that this mode of measurements is differ-ent from typical FMR experiments performed with a VNAwhere the frequency is swept while the external magneticﬁeld is kept constant. 2The approach used here allows one to avoid the inﬂuence of the frequency response of the wave-guide on the measured signal. The intensity of the FMR sig-nal was on the order of 0.05–0.3 dB, while the total attenu-ation of the microwave carrier signal reaches 25 dB. Thiswas caused by losses in the previously-described ﬁxturesused to connect the VNA ports with the coplanar waveguidecell inside the electromagnet gap. To increase the signal-to-noise ratio, sampling and averaging of the measuredS-parameters were used. The low intensity of the FMR ab- FIG. 5. /H20849Color online /H20850FMR signal as a function of applied ﬁeld at 15, 25, and 36 GHz for the Permalloy sample no. 3. Note a small signal from theseed layer observed at 15 GHz. FIG. 6. Field vs frequency dependence for FMR in the Permalloy structureno. 1. FIG. 7. Field vs frequency dependence for FMR in the Permalloy structureno. 2. FIG. 8. /H20849Color online /H20850Field vs frequency dependence for FMR in the Ni45Fe55structure no. 3.113906-5 Zagorodnii et al. J. Appl. Phys. 107 , 113906 /H208492010 /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: 130.113.111.210 On: Mon, 22 Dec 2014 13:15:07sorption indicates low coupling between the waveguide and the ferrromagnetic bar under the test, so the resonance fre-quency is assumed to be unaffected by any coupling-inducedshift /H20849as described in Ref. 22/H20850. The chosen frequencies were arranged as a series of clusters of frequency triplets. Thefrequency differences inside each triplet were on the order ofa few hundreds of megahertz. This makes it possible to ex-ploit ﬁnite differences by using Eq. /H208497/H20850to calculate aassum- ing small changes in the demagnetizing factors within eachfrequency triplet: a=1 /H92532/H20849/H9275j2−/H9275i2/H20850−/H20849HZj2−HZi2/H20850 M0/H20849HZj−HZi/H20850, where /H9275i,HZi,/H9275j, and HZjare used values of frequencies and resonant ﬁelds. The results of our analysis, based on the considerations described above, are plotted in Figs. 6–12. The extracted data for /H9253,4/H9266M0, and/H9251are shown in Table III. The FMR response from the Permalloy seed layer was observed in sev-eral samples. This appears at lower frequencies for the0.99 /H9262m thick Ni 45Fe55structure no. 3 /H20849see Figs. 5and8/H20850 and was perfectly discernible for the 0.03 /H9262m thick Ni 45Fe55 structure no. 5 /H20849see Fig. 10/H20850. The thinnest Ni 45Fe55structure revealed the FMR resonant absorption only at frequenciesabove 17 GHz due to a relatively small thickness of thestructure in comparison to the actual skin depth. Also, theseed layer response was consistently perceptible in the allFMR measurements of the Co 35Fe65structures, even for the 2.7/H9262m thick sample no. 7. This can signify the uncom- monly low conductivity for the studied Co 35Fe65structures.The data for the 0.03 /H9262m thick Ni 45Fe55structure was not ﬁtted for /H9251extraction due to the practical absence of fre- quency dependence of the demagnetizing factors. Similarlyfor seed layers, /H9251was not extracted due to the different be- havior of demagnetizing factors for buried layers. The satu-ration magnetization was, however, determined for those lay-ers, and the Permalloy seed layer shows reasonable M 0, while the 0.03 /H9262m thick Ni 45Fe55layer, on top of the seed layer, demonstrates abnormally low M0in comparison to thicker ﬁlms of the same composition. Evidently, the actualcomposition of that structure contains more Ni than desiredand intended by the deposition process. Though Permalloystructures nos. 1 and 2 demonstrate reasonable M 0values, they differ depending on the ﬁlm thickness. This is probablydue to a difference in the composition of the ﬁlms introducedduring the electrochemical deposition process. This may alsobe associated with the fact that the thickness of the thin-ﬁlmstructures was determined by proﬁlometer measurements.Any overetching due to the seed layer removal may haveresulted in thickness measurement errors since the ﬁlmheight is measured above the wafer surface level. Even more signiﬁcant differences in M 0and/H9251were ob- served in structures nos. 3 and 4 /H20849see Table III/H20850. On the other hand, both Co 35Fe65structures nos. 6 and 7, have high satu- ration magnetization and exhibit similar values of deter-mined magnetic parameters. The damping constants for all specimens were found to be in the range of 0.025–0.055. These high values are mostlikely caused by the dominance of eddy currents and mag-netic inhomogenities over the intrinsic magnetic loss forfairly thick metallic structures. The accuracy of determining FIG. 9. Field vs frequency dependence for FMR in the Ni45Fe55structure no. 4. FIG. 11. /H20849Color online /H20850Field vs frequency dependence for FMR in the Co35Fe65structure no. 6. FIG. 12. /H20849Color online /H20850Field vs frequency dependence for FMR in the Co35Fe65structure no. 7. FIG. 10. /H20849Color online /H20850Field vs frequency dependence for FMR in the Ni45Fe55structure no. 5.113906-6 Zagorodnii et al. J. Appl. Phys. 107 , 113906 /H208492010 /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: 130.113.111.210 On: Mon, 22 Dec 2014 13:15:07the magnetic parameters depends on the amount of recorded data /H20849the more, the better /H20850for each sample, and is limited by the time available for measurements. For our system, oneﬁeld scan for each frequency can last approximately 10–40min depending on the FMR signal strength. In conclusion, a method was developed which allows the use of braod-band FMR measurements for determining /H9253, 4/H9266M0as well as /H9251for the fairly lossy metallic rectangular bars with thicknesses on the order of micrometers or submi-crometers. Using this technique, an analysis of the FMR line-width is not needed. This method was developed for rectan-gular shaped bars with the proper dimensions/aspect rationeeded to utilize the frequency dependence of the demagne-tizing factors for the extraction of the magnetic parameters.This allows one to study the magnetic properties of the thickferromagnetic elements of sensors, actuators, etc. withoutany special sample preparation. The estimated precision forthe extracted parameters does not completely satisfy the highdemands of material science or high-end material propertiesmonitoring. However, it can be successfully used for nonde-structive test purposes. A similar approach can be applied forother, simply shaped, ferromagnetic elements for which theaverage demagnetizing factors can be evaluated. This work was supported at the UCCS in part by the ARO /H20849Grant No. W911NF-04-1-0247 /H20850and at the Leibniz Universitaet Hannover by the DFG /H20849German Reseach Foun- dation /H20850within the Collaborative Research Center /H20849SFB /H20850516. The authors would like to thank Mr. Ian Harward for care-fully proof reading of the manuscript.1Y . Yu and J. W. Harrell, J. Magn. Magn. Mater. 155,1 2 6 /H208491996 /H20850. 2S. S. Kalarickal, P. Krivosik, M. Wu, C. E. Patton, M. L. Schneider, P. Kabos, T. J. Silva, and J. P. Nibarger, J. Appl. Phys. 99, 093909 /H208492006 /H20850. 3H. Zhang, C. Song, and P. Wang, J. Appl. Phys. 105, 07E716 /H208492009 /H20850. 4Y . Chen, I. C. Smith, A. L. Geiler, C. Vittoria, V . V . Zagorodnii, Z. J. Celinski, and V . G. Harris, IEEE Trans. Magn. 44, 4571 /H208492008 /H20850. 5Y .-C. Chen, D.-S. Hung, Y .-D. Yao, S.-F. Lee, H.-P. Ji, and C. Yu, J. Appl. Phys. 101, 09C104 /H208492007 /H20850. 6V . Golub, K. M. Reddy, V . Chernenko, P. Müllner, A. Punnoose, and M. Ohtsuka, J. Appl. Phys. 105, 07A942 /H208492009 /H20850. 7M. Pasquale, S. Perero, and D. Lisjak, IEEE Trans. Magn. 43, 2636 /H208492007 /H20850. 8H. Sakuma, S. Sato, R. Gomimoto, S. Hiyama, and K. Ishii, IEEJ Trans. Electrical. and Electro. Eng. 2,4 3 1 /H208492007 /H20850. 9I. Neudecker, G. Woltersdorf, B. Heinrich, T. Okuno, G. Gubbiotti, and C. H. Back, J. Magn. Magn. Mater. 307, 148 /H208492006 /H20850. 10C. Bilzer, T. Devolder, P. Crozat, C. Chappert, S. Cardoso, and P. P. Freitas, J. Appl. Phys. 101, 074505 /H208492007 /H20850. 11M. Dıaz de Sihues, C. A. Durante-Rincón, and J. R. Fermín, J. Magn. Magn. Mater. 316, e462 /H208492007 /H20850. 12R. Arias and D. L. Mills, Phys. Rev. B 60, 7395 /H208491999 /H20850. 13K. Lenz, H. Wende, W. Kuch, K. Baberschke, K. Nagy, and A. Janossy, Phys. Rev. B 73, 144424 /H208492006 /H20850. 14C. Kittel, Phys. Rev. 73, 155 /H208491948 /H20850. 15E. Schlömann, Phys. Rev. 182, 632 /H208491969 /H20850. 16R. D. McMichael, J. Appl. Phys. 103, 07B114 /H208492008 /H20850. 17A. G. Gurevich and G. A. Melkov, Magnetization Oscillations and Waves /H20849CRC, New York, 1996 /H20850. 18M. Bedenbecker, R. Bandorf, G. Brauer, H. Luthje, and H. Gatzen, Mi- crosyst. Technol. 14, 1949 /H208492008 /H20850. 19Y . Zhuang, M. Vroubel, B. Rejaei, J. N. Burghartz, and K. Attenborough, J. Appl. Phys. 97, 10N305 /H208492005 /H20850. 20R. I. Joseph and E. Schlömann, J. Appl. Phys. 36, 1579 /H208491965 /H20850. 21A. Aharoni, J. Appl. Phys. 83, 3432 /H208491998 /H20850. 22G. Feher, Bell Syst. Tech. J. 36, 449 /H208491957 /H20850.TABLE III. /H9253,4/H9266M0, and/H9251values for ferromagnetic metallic bars determined using the broadband FMR measurements. Specimen No./H9253/2/H9266 4/H9266M0,G s /H9251 Value Standard error Value Standard error Value Standard error 1 2.95 0.03 9650 60 0.04 0.006 2 2.93 0.01 10260 40 0.035 0.013 2.89 0.03 19700 1800 0.025 0.0054 2.9 0.03 15100 550 0.055 0.0205N i 45Fe55, 0.03 /H9262m 2.94 0.05 9450 450 ¯¯ Py, 0.2 /H9262m/H20849seed layer /H208502.95 0.05 9000 1000 ¯¯ 6 2.92 0.02 20900 1100 0.04 0.0157 2.92 0.02 21600 700 0.05 0.015113906-7 Zagorodnii et al. J. Appl. Phys. 107 , 113906 /H208492010 /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: 130.113.111.210 On: Mon, 22 Dec 2014 13:15:07
1.4972959.pdf	Current induced magnetization dynamics and magnetization switching in superconducting ferromagnetic hybrid ( ) structures Saumen Acharjee and Umananda Dev Goswami Citation: J. Appl. Phys. 120, 243902 (2016); doi: 10.1063/1.4972959 View online: http://dx.doi.org/10.1063/1.4972959 View Table of Contents: http://aip.scitation.org/toc/jap/120/24 Published by the American Institute of Physics Articles you may be interested in Enhanced saturation magnetization in perpendicular L10–MnAl films upon low substitution of Mn by 3d transition metals J. Appl. Phys. 120, 243903243903 (2016); 10.1063/1.4972972 Ponderomotive convection in water induced by a CW laser J. Appl. Phys. 120, 244902244902 (2016); 10.1063/1.4972969 Improving three-dimensional target reconstruction in the multiple scattering regime using the decomposition of the time-reversal operator J. Appl. Phys. 120, 243101243101 (2016); 10.1063/1.4972470 Positive charge trapping phenomenon in n-channel thin-film transistors with amorphous alumina gate insulators J. Appl. Phys. 120, 244501244501 (2016); 10.1063/1.4972475 Current induced magnetization dynamics and magnetization switching in superconducting ferromagnetic hybrid (F jSjF) structures Saumen Acharjeea)and Umananda Dev Goswamib) Department of Physics, Dibrugarh University, Dibrugarh 786 004, Assam, India (Received 25 August 2016; accepted 10 December 2016; published online 28 December 2016) We investigate the current induced magnetization dynamics and magnetization switching in an unconventional p-wave superconductor sandwiched between two misaligned ferromagnetic layersby numerically solving the Landau-Lifshitz-Gilbert equation modiﬁed with current induced Slonczewski’s spin torque term. A modiﬁed form of the Ginzburg-Landau free energy functional has been used for this purpose. We demonstrated the possibility of current induced magnetizationswitching in the spin-triplet ferromagnetic superconducting hybrid structures with a strong easy axis anisotropy and the condition for magnetization reversal. The switching time for such arrange- ment is calculated and is found to be highly dependent on the magnetic conﬁguration along withthe biasing current. This study would be useful in designing the practical superconducting- spintronic devices. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4972959 ] I. INTRODUCTION During over last 15 years, a number of very interesting compounds have been discovered which reveal the coexis-tence of ferromagnetism and superconductivity in the same domain in bulk. 1–6The interplay between the ferromagnetic order and superconductivity thus gains lots of attention froma variety of research communities. 7Among those, some peo- ple were hunting for superconductivity in a ferromagnetic spin valve made up of two ferromagnetic substances sepa-rated by a superconducting element (F jSjF system). In this context, it is to be noted that the spin triplet superconductiv- ity in superconductor jferromagnet (F jS) hybrid structures including F jSjF spin valves is a topic of intense research 8–16 in the theoretical as well as the experimental points of view for almost last two decades. The major interest of the F jS hybrid structures is due to the dissipation less ﬂow of charge carriers offered by the superconducting environment. To completely understand this hybrid structure, it is importantto study the spin polarized transport. Moreover, the transport of spin is closely related to the phenomenon of current induced magnetization dynamics 17 and spin transfer torque (STT).18,19Spin transfer torque, which is the building block of spintronics is based on the prin- ciple that, when a spin polarized current is applied into the fer-romagnetic layers, spin angular momentum is transferred into the magnetic order. It is observed that for a sufﬁciently large current, magnetization switching can occur 20,21in a magnetic layer. Thus, the ﬂow of electrons can be served to manipulate the conﬁguration of the spin valves. Traditionally, a lot of works have been done earlier on current induced magnetiza-tion dynamics and STT on ferromagnetic layers. Soon after, a lot attention has been given to anti-ferromagnetic layers 21–24 also. Making a hybrid structure of a superconductor with a fer- romagnet and the concept of current induced magnetizationdynamics suggest a very interesting venue for combining two different ﬁelds, namely, superconductivity and spintronics.25 A few works have been done earlier on F jS hybrid struc- tures.26–30In Ref. 20, supercurrent-induced magnetization dynamics in Josephson junction with two misaligned ferro-magnetic layers have been studied and demonstrated for the favourable condition of magnetization switching and reversal. Motivated by the earlier works, in this paper, we studied the current induced magnetization dynamics of a supercon-ducting ferromagnet in a hybrid structure of F jS based on the Landau-Lifshitz-Gilbert (LLG) equation with Slonczewski’s torque (LLGS) using the Ginzburg-Landau-Gibb’s free energy functional. The proposed experimental setup is shown in Fig. 1, in which the two ferromagnets are separated by a thin superconducting ferromagnet. The coercive ﬁelds of the ferromagnets are such that, the magnetization is hard in one layer while soft in the other and the orientation of magnetization of the soft ferromagnetic layer is supposed to be misaligned with the hard ferromagnetic layers by an angleh. When the junction is current-biased, it gets spin polarized in the hard layer and thus transfers the angular momentum to the magnetic order. This generates an induced magnetization contributing to the magnetic order. The dynamics of this induced magnetization has been studied by numerically solv-ing the LLGS equation. The paper is organized as follows. In the Section II,a theoretical framework of the proposed setup is developed. The results of our work are discussed in Section IIIby solv- ing the LLGS equation numerically. Finally, we conclude our work in Section IV. II. THEORY To study the current induced magnetization dynamics of a ferromagnetic superconductor with easy axis anisotropy in FjSjF spin valve, we utilized the Landau-Lifshitz-Gilbert (LLG) equation with the Slonczewski’s spin transfer torque (LLGS). The resulting LLGS equation takes the forma)saumenacharjee@gmail.com b)umananda2@gmail.com 0021-8979/2016/120(24)/243902/8/$30.00 Published by AIP Publishing. 120, 243902-1JOURNAL OF APPLIED PHYSICS 120, 243902 (2016) @M @t¼/C0cM/C2Hef f ðÞ þaM/C2@M @t/C18/C19 þT; (1) where cis the gyromagnetic ratio, ais the Gilbert’s damping constant, and Heffis the effective magnetic ﬁeld of supercon- ducting ferromagnet. Tis the current induced spin transfer torque and can be read as20 T¼IfðM/C2½M/C2ðMT/C0/C15MB/C138Þ; (2) where MTandMB, respectively, represents the normalized magnetization vector in the top and bottom magneticlayers of the spin valve and is taken as M T¼(0, 1, 0) and MB¼ð0;cosh;sinhÞsuch that for h¼0, the conﬁguration is parallel and is anti-parallel for h¼p./C15provides the fac- tor of asymmetry in polariza tion in the top and bottom fer- romagnetic layers. The term fis given by f¼/C23/C22hl0 2em0V: (3) Here, eis the electronic charge, /C23is the polarization efﬁ- ciency, /C22his the Planck’s constant, l0is the magnetic perme- ability, m0is the amplitude of magnetization, and Vis the volume of the system. Iis the applied current bias. The effec- tive magnetic ﬁeld of the system can be obtained from the functional derivative of the free energy with respect to the magnetization: Hef f¼/C0dF dM: (4) The free energy functional F(w,M) can be given by31 Fðw;MÞ¼ð d3rfðw;MÞ; (5) where f(w,M) gives the free energy density of a spin-triplet superconductor and can be read as31,32 fw;MðÞ ¼ fSwðÞ þ fFMðÞþfintw;MðÞ þB2 8p/C0B:M;(6) where w(/C17wj;j¼1, 2, 3) is the superconducting order parameter and is a three dimensional complex vector, Misthe magnetization vector, which characterizes the ferromag- netism, fS(w) that gives the superconductivity, while the fer- romagnetic order is described by fF(M). The interaction of the two orders, Mandwis described by the term fint(w,M). The last two terms in Equation (6)account for the contribu- tion of magnetic energy on free energy with magnetic induc-tionB¼Hþ4pM¼r/C2 A. The superconductivity of the system is described by the term f S(w) under the condition H¼0 and M¼0 and can be written as31–33 fSwðÞ ¼ fgradwðÞ þ asjwj2þbs 2jwj4þus 2jw2j2þvs 2X3 i¼1jwj4; (7) where fgradcan be written as32 fgrad¼K1ðDiwjÞ/C3ðDiwjÞþK2½ðDiwiÞ/C3ðDjwjÞ þðDiwjÞ/C3ðDjwiÞ/C138 þ K3ðDiwiÞ/C3ðDiwiÞ (8) with Di¼/C0i/C22h@ @xi/C16/C17 þ2jej cAibeing the covariant derivative, usdescribes the anisotropy of the spin triplet Cooper pair, and the crystal anisotropy is described by vs.asandbsare positive material parameters. The term fF(M)i n(6)describes the ferro- magnetic ordering of the material and is given by31,32 fFMðÞ¼cfX3 j¼1jrjMjj2þafM2þbf 2M4: (9) While the term fint(w,M)i n(6)corresponds to the inter- action of ferromagnetic order with the complex supercon-ducting order and can be written as f intðw;MÞ¼ic0M:ðw/C2w/C3ÞþdM2jwj2; (10) where c0term provides the superconductivity due to ferro- magnetic order, while dterm makes the model more realistic as it represents the strong coupling and can be both positiveand negative values. Rewriting the free energy f(w,M)i na dimensionless form by redeﬁning the order parameters w j ¼b/C01 4s/jeihjandm¼b/C01 4 fM, the free energy (6)takes the form f¼fgradþr/2þ1 2/4/C02t1 /C2½/2 1/22sin2h2/C0h1 ðÞ þ/2 1/23sin2h1/C0h3 ðÞ þ/2 2/23sin2h2/C0h3 ðÞ /C138/C0v/2 1/22þ/2 2/23þ/2 3/21hi þwm2þ1 2m4þ2c1/1/3msinh3/C0h1 ðÞ þc2/2m2/C0v1B:m; (11) where the parameters, r¼as b1 2;w¼af b1 2 f;t1¼us b;v¼vs b,c1¼ c0 b1 2b14 f;c2¼d ðbbfÞ1 2and v1¼b1 4 fwith b¼(bsþusþvs). The coexistence of superconductivity and ferromagne- tism was ﬁrst observed in UGe 2(Refs. 1and34) within a FIG. 1. The proposed experimental setup. An unconventional p-wave type superconductor is sandwiched in between two ferromagnetic layers. The magnetization orientation of the ferromagnetic layers are supposed to be misaligned by an angle h. When a current I is injected, it gets polarized and the transfer of spin torque to the magnetic order causes magnetization dynamics. Different colours of the ferromagnetic layers indicate the level of magnetization. Here, the bottom layer is hard in magnetization.243902-2 S. Acharjee and U. D. Goswami J. Appl. Phys. 120, 243902 (2016) limited pressure range (1.0–1.6 GPa). In the following years, same coexistence was found in URhGe2,34and UCoGe4,34,35 at an ambient pressure, and in UIr36similar to the case of UGe 2, i.e., within a limited pressure range (2.6–2.7 GPa). These Uranium-based (U-based) compounds, with the coex- istence of ferromagnetism and superconductivity, exhibitunconventional properties of ground state in a strongly corre-lated ferromagnetic system. One of the interesting featuresof these U-based ferromagnetic superconductors is that, thistype of superconductivity was found to occur within thevicinity of a quantum critical point (QCP). The critical pres-sure, or the critical chemical composition is referred to as theQCP, where the ordering temperature is tuned to T C¼0K .I t should be noted that in general, the U-based ferromagneticsuperconductors have a very strong easy-axis magneto crys-talline anisotropy. 31,34However, the free energy in Equation (11)is isotropic. To account the contribution of anisotropy in free energy, we introduce a term Kan20resulting in an effec- tive ﬁeld of the form Han¼ðKanmy=M0Þ^y. Here we direct the anisotropy axis in parallel to the y-direction, and contri- bution along the anisotropy axis is being considered. In viewof this, the LLGS Equation (1)takes the form @m @t¼/C0 c" m/C2/C18 2wmþ2m3þ2c1/1/3sinh3/C0h1 ðÞ ^y þ2c2/2m/C0v1BþKanmy M0^y/C19# þam/C2@m @t/C18/C19 þT; (12) where myis the component of malong the anisotropy axis which we direct parallel to y-axis with B¼/C0B0^z. The Equation (12) is a non-linear coupled differential equation in mand can be transformed into the following form: dmx ds¼a/C15Isinhm3 ysðÞþaI1/C0/C15cosh ðÞ m2 ysðÞmzsðÞ þaI1/C0/C15cosh ðÞ m3 zþaIm2 xsðÞ /C2/C15sinhmysðÞþ1/C0/C15cosh ðÞ mzsðÞ/C2/C3 þmysðÞ/C0B0v1þmzsðÞKanþ2wþa/C15IsinhmzsðÞ ð ½ þ2/2c2Þ/C138þ2sinbmzsðÞc1/1/3/C0mxsðÞ /C2mzsðÞ /C15IsinhþaB0v1 ðÞ þam2 ysðÞh /C2Kanþ2wþ2/2c2/C0/C1 þmysðÞ/C0Iþ/C15Icosh ð þ2ac1/1/3sinbÞ/C138=1þa2m2 xsðÞþm2 ysðÞþm2 zsðÞ/C16/C17 hi ; (13) dmy ds¼/C0a/C15Isinhm3 xsðÞþmxsðÞ/C0a/C15Isinhm2 xsðÞþm2 zsðÞ/C0/C1 /C2 þB0v1/C138þm2 xsðÞ/C0Iþ/C15IcoshþamysðÞ/C2 /C2Kanþ2wþ2/2c2/C0/C1 þ2asinbc1/1/3/C138 þmzsðÞmysðÞð/C0/C15Isinh/C0aB0v1þamzsðÞ/C2 /C2Kanþ2wþ2/2c2/C0/C1 ÞþmzsðÞI/C01þ/C15cosh ðÞð þ2asinbc1/1/3Þ/C138=½1þa2ðm2 xsðÞ þm2 ysðÞþm2 zsðÞÞ/C138; (14)dmz ds¼aI1þ/C15cosh ðÞ m3 xsðÞþm2 xsðÞI/C15sinhþaB0v1 ðÞ /C0mxsðÞ½aI1/C0/C15cosh ðÞ m2 ysðÞþaI1/C0/C15cosh ðÞ m2 zsðÞ þmysðÞKanþ2wþ2/2c2/C0/C1 þ2 sinbc1/1/3/C138 þmysðÞ½mysðÞð/C15IsinhþaB0v1/C0amzsðÞ /C2Kanþ2wþ2/2c2/C0/C1 ÞþmzsðÞI/C0/C15Icosh ð /C02asinbc1/1/3Þ/C138=½1þa2ðm2 xsðÞþm2 ysðÞþm2 zsðÞÞ/C138; (15) where b¼(h3–h1) represents the phase mismatch of surviv- ing components of the superconducting order parameter. For arealistic situation, this phase mismatch should not be verylarge and hence we have taken the bto be equal to 0.1 p arbitrarily to have a similarity with the practical situation. AsU-based ferromagnetic superconductors have a very strongmagneto crystalline anisotropy, 31,34to model a realistic super- conducting ferromagnet, the anisotropy ﬁeld can be taken as34 Kan/C24103, the asymmetry factor is taken as /C15¼0.1 with a magnetic induction B0¼0.1 and f¼1. Furthermore, we have set31v1¼w¼0:1;/1¼/3¼/ﬃﬃ 2pand initially c1¼2c2 ¼0.2, which makes the F jSjF spin valve system more realizable. III. RESULTS AND DISCUSSIONS To investigate the magnetization dynamics and switch- ing behaviour quantitatively, we have solved the full LLGSEquations (13)–(15) of the F jSjF system using numerical simulation. The magnetization dynamics and the switchingbehaviour of our system are investigated based on the abovementioned parameters, initially for very weak damping(a/C281) and then for strong damping (up to a¼0.5) with a very small angle of misalignment h¼0.1pand c 1¼2c2 ¼0.2. Furthermore, to solve the Equations (13)–(15)numeri- cally, the time coordinate has been normalized to s¼ct/M0, where M0is the magnitude of the magnetization. Few of the corresponding numerical solutions are shown in Fig. 2for two different current biasing in the ﬁrst four plots. The restof the plots in the ﬁgure show the corresponding parametricgraphs of time evolution of the magnetization components.The plots in left panels show the weak damping regime withGilbert’s damping parameter a¼0.05 for two different choices of current biasing 0.1 mA and 0.252 mA, respec-tively, from top to bottom. While the damping is consideredto be strong with a¼0.5 in the plots of the right panels for the respective current biasings. It is seen that the magnetiza-tion components show quite different behaviours. The com-ponents m xand mzdisplay an oscillating decay until they vanish completely, while on the other hand, the componentm ysaturates with the increasing value of s. It is to be noted that the qualitative behaviour of the components of magneti-zation is similar for different damping parameters. But thequantitative difference is that, in strong Gilbert’s dampingregime, the oscillation of the magnetization components m x andmydie out faster in time scale, while the component mz saturates too rapidly as seen from the right panels of Fig. 2. It is also seen that in strong damping, the reversal of243902-3 S. Acharjee and U. D. Goswami J. Appl. Phys. 120, 243902 (2016) magnetization components myandmzdoes not occur for a current biasing I¼0.252 mA, contrary to the case for the small damping. This result indicates that, it is possible to generate a current induced magnetization reversal of a triplet superconducting ferromagnet in a F jSjF spin valve setup shown in Fig. 1by means of current biasing under weak damping conditions. It is also our interest to see what happens when the parameter c2is increased. To check the inﬂuence of c2on switching mechanism, we investigated the behaviour of the magnetization components for both positive and negative val-ues of c 2keeping the damping parameter and c1ﬁxed for cur- rents, respectively, of I¼0.25 mA and 0.252 mA. It is found that the switching time sgets delayed for c2¼/C045, while on the other hand, we observed a more rapid switching for c2¼45. Moreover, the xandycomponents show a more rapidoscillation for c2¼45 than for c2¼/C045. The components of magnetization under this condition are shown in Fig. 3with the parametric graphs. This result suggests that magnetization reversal is dependent on the strong coupling parameter and the switching of a system is more rapid for positive couplingthan that for the negative coupling parameter as seen. Our one more interest here is to check the inﬂuence of B 0on switching. To investigate this, we have plotted the magnetization components for a higher value of B0¼1.0 in Fig.4. It is seen that under this situation, the switching does not even occur for a current biasing of 1.5 mA as seen fromthe middle panel. In this conﬁguration, the reversal of the magnetization components occurs at a current biasing of 1.65 mA as seen from the right panel of Fig. 4. This suggests that the magnetization switching condition can also be con- trolled by magnetic induction.FIG. 2. (First four plots) The time evo- lution of the normalized components of magnetization with an initial angle of misalignment h¼0.1pand with c1¼2c2¼0.2. The plots in the left panels depicted for the weak damping a¼0.05, while the plots in the right panels are for strong damping a¼0.5 with a current biasing of 0.1 mA and 0.252 mA, respectively, from top to bottom panels. The corresponding parametric graphs representing the behaviour of the magnetization are dis- played in the last four plots.243902-4 S. Acharjee and U. D. Goswami J. Appl. Phys. 120, 243902 (2016) It is to be noted from Figs. 2and3that for the weak damping but with higher current biasing, oscillations and switching for the reversal of respective components of mag- netization are delayed by some factors. In view of this result,it is also important to see explicitly what happens to the spin valve if conﬁguration is changed and what inﬂuence, the spin valve conﬁguration has on the switching time s switch? To answer these two questions, we have studied the switching time sswitch as a function of hrepresenting the angles of misalignments for four different current biasingskeeping the damping factor a¼0.05 as shown in Fig. 5.I t should be noted that the switching time of a magnetization component is deﬁned as the time required by the componentto attain numerically the 0.975 times of its saturated value. 21 One of the important results of this study is that, for theincreasing angle of misalignment, a more rapid switching of corresponding components of magnetization occurs with the increasing value of the current bias. From Fig. 5, it is also seen that, for all current biasing, the switching time shows amonotonic increase with a sharp peak starting from the zero angle of misalignment, providing the most delayed magnetic spin valve conﬁguration at a particular current. The angle ofmisalignment of this most delayed conﬁguration decreases with an increasing value of the biasing current. It is interest- ing to note that the maximum switching time for a particularangle of misalignment increases with an increasing value of current biasing except for the case of 0.2 mA current, at which it is lowest. This particular behaviour at 0.2 mA cur-rent indicates that in the range of smaller angle of misalign- ment ( h/C200.05p), 0.2 mA is the optimum value of biasingFIG. 3. (First four plots) The time evo- lution of the normalized components of magnetization with an initial angle of misalignment h¼0.1p. Here c1¼0.2 with c2¼45 and /C045, respec- tively, in left and right for current bias-ingI¼0.25 mA in the top panels and 0.252 mA for the bottom panels keep- ingB 0¼0.1 constant. The correspond- ing parametric graphs representing the behaviour of the magnetization are shown in the last four plots.243902-5 S. Acharjee and U. D. Goswami J. Appl. Phys. 120, 243902 (2016) current among all for the magnetic spin valve. The data of these results are summarized in Table I. These results as a whole clearly signify that, switching is highly dependent onmagnetic conﬁguration in association with the biasing cur- rent: switching occurs swiftly at a higher angle of misalign- ment and with a higher value of current bias. This suggeststhat the conﬁguration near the anti-parallel ( h¼p) offers rapid switching then the parallel ( h¼0) for all the current biasing, and for higher current, the peak position shifted towards the parallel conﬁguration lowering the switchingtime just after the peak. This is quite obvious as the STTbecomes stronger in this case. Our ﬁnal interest is to study the inﬂuence of a higher magnetic ﬁeld on the component of magnetization. To ana-lyze this, we have plotted the magnetization components with sfor higher values of the magnetic ﬁeld in Fig. 6, keep- ing the biased currents ﬁxed at 0.1 mA and c 1¼2c2¼0.2. It is seen that for B0¼102, the components of magnetizationshow a quite similar behaviour as seen earlier in Fig. 2. However, on the other hand as the magnetic ﬁeld increases, the components of magnetization show quite an irregularbehaviour. For example, for B 0¼103as seen from the top right panel of the Fig. 6, the mycomponent suddenly reverses with a small initial ﬂuctuation and then starts saturating afteran oscillating decay period. This is due to the fact that, as B 0 becomes of the order of the anisotropy ﬁeld, the components of magnetization behave quite differently. In this condition,the component m yreverses and saturates, while mxandmz show an oscillating decay. With a further rise in B0makes the system more unstable in such a way that, with an increas-ing value of B 0, both myandmzcomponents gradually tend to behave almost similarly by retaining the original direction of the mycomponent as seen from the bottom panels of the ﬁrst four plots in the Fig. 6. Because, with a further rise in B0, the magnetic ﬁeld dominates over the anisotropy ﬁeld. It can be easily visualized from the parametric graph shown in the bottom left of panel of Fig. 6, where the motion takes place about the direction of the magnetic ﬁeld. The motionstabilizes itself for more higher values of B 0. We have found that the inﬂuence of magnetic ﬁeld as mentioned above is almost similar for the biasing current and hence a higherFIG. 4. (Top three plots) The time evolution of the normalized components of magnetization with an initial angle of misalignment h¼0.1pfor a magnetic induction B0¼1.0 with c1¼2c2¼0.2 and for weak damping a¼0.05. The plots in left and right depicted the magnetization dynamics for a current biasing of 0.1 mA and 1.65 mA, respectively, while the plot in the middle is for a current biasing of 1.5 mA. The corresponding parametric graphs representing the b ehav- iour of the magnetization are shown in the bottom three plots. FIG. 5. Switching time and its dependence on the spin valve conﬁgurationfor different current biasings.TABLE I. Maximum switching time ( sswitch) and the corresponding mis- alignment angle ( h) for different biasing currents in low damping with a¼0.05. I (mA) hs switch (s) 0.1 0.168 p 0.2995 0.2 0.114 p 0.2734 0.3 0.091 p 0.3022 1.0 0.044 p 0.3280243902-6 S. Acharjee and U. D. Goswami J. Appl. Phys. 120, 243902 (2016) value of magnetic ﬁeld ( B0/C21100) eliminates the effect of biasing current. IV. SUMMARY In this work, we have investigated the current induced magnetization dynamics and m agnetization switching in a superconducting ferromagnet sandwiched between two mis- aligned ferromagnetic layers with easy-axis anisotropy by numerically solving the Landau-L ifshitz-Gilbert-Slonczewski’s equation. For this purpose, we have used the modiﬁed form of the Ginzburg-Landau free energy functional for a triplet p- wave superconductor. We hav e demonstrated about thepossibility of current i nduced magnetization switching for an experimentally realistic parame ter set. It is observed that, for the realization of magnetization switching, a sufﬁcient biased current and moderate ﬁeld are suitable for the case of low Gilbert damping. Although, switching can be delayed for largedamping, however, such a system cannot be used because the system becomes highly unstabl e in such situation, which is unrealistic. It is also to be noted that switching is highly depen-dent on the strong coupling par ameter, and it is seen that the positive value of that offers more rapid switching than that of negative. It is also seen that switching has a high magnetic con-ﬁguration dependence. It shows a monotonic increase for both low and high current in very near to parallel conﬁguration. TheFIG. 6. (First four plots) The time evo- lution of normalized components of magnetization with an initial angle of misalignment h¼0.1p with c1¼2c2¼0.2 and for current biasing I¼0.1 mA for higher values of exter- nal magnetic ﬁeld, viz., B0¼102(top left), B0¼103(top right), B0¼2/C2103 (bottom left), and B0¼5/C2103(bottom right). The corresponding parametric graphs representing the behaviour of the magnetization are shown in the lastfour plots.243902-7 S. Acharjee and U. D. Goswami J. Appl. Phys. 120, 243902 (2016) conﬁguration near anti paralle l offers a more rapid switching than the parallel. Again, it can also be concluded that thedynamics is highly dependent and controlled by the magnetic ﬁeld as it becomes of the order of the anisotropy ﬁeld. 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1.1945253.pdf	NuclearData forAstrophysicalNucleosynthesis: A Japanese+ LANL Activity SatoshiChiba∗,ToshihikoKawano†,HiroyukiKoura∗,Tsuneo Nakagawa∗, TakahiroTachibana∗∗, ToshitakaKajino‡, Shinsho Oryu§,TakehitoHayakawa¶, AkiyukiSeki∗, ToshikiMaruyama∗, TomonoriTanigawa/bardbl,∗,YukinobuWatanabe††, ToshiroOhsaki‡‡, ToruMurata§§andKohsukeSumiyoshi¶¶ ∗JapanAtomic EnergyResearchInstitute,Tokai,Naka, Ibaraki319-1195, Japan †Los Alamos National Laboratory,T16, MS B283, LANL, Los Alamos, NM 87545, U.S.A. ∗∗WasedaUniversity,Nerima, Tokyo177-0044, Japan ‡National Observatory of Japan,Mitaka, Tokyo181-8588, Japan §TokyoUniversityof Science,Noda, Chiba 278-8510, Japan ¶JapanAtomic EnergyResearchInstitute,Kizu, Sagara,Kyoto619-0215, Japan /bardblJapanSociety for the Promotionof Science,Tokyo102-8471, Japan ††KyushuUniversity,Kasuga-Koen,Kasuga, Fukuoka 316-8580, Japan ‡‡TokyoInstitute of Technology,O-okayama, Meguro, Tokyo152-8550, Japan §§AitelCo., Isogo,Yokohama235-0016, Japan ¶¶Numazu Collegeof Technology,Ooka, Numazu 410-8501, Japan Abstract. There are many common features in nuclear data for energy applications and nuclear cosmology/astrophysics, especially the neutron-capture nucleosynthesis. Therefore it is a natural consequence to think that many of the tools that we have developed for the conventional nuclear-data applications can be applied for a development of a database for nuclear cosmology/astrophysics. However, there are also many features that are uncommon to these ﬁelds, so new development is necessarywhenwethinkaboutadatabasefornuclearcosmologyandastrophysics.Suchnewdevelopmentwillthengiveusa new horizon for the conventional nuclear data activities as well. In this paper we will show the present status of our activities inthis direction, putting emphasis on data relevantto neutron capture nucleosynthesis, namely s- and r-processes. INTRODUCTION Nuclear reactions, especially the neutron-induced reac- tions, play essentially important roles both in nuclear- energy applications and astrophysical nucleosynthesis. In fact, about 99% of elements heavier than the iron- nickelgroupinthesolarsystemweresynthesizedbytwo successive neutron-capture reaction chains, namely the s- and r-processes. In the past, the nuclear-data and astrophysical com- munities have engaged in constructing databases on nu- clear structure, reaction, and decay rates independently. There are many common features in the activities of these two communities, but both approaches have spe- ciﬁc strengths and drawbacks that can be compensated by each other. Here, we would like to present the current status of a loosely-bound Japanese + LANL activity to generate a new database for astrophysical nucleosynthesis. As a newcomer,wearecurrentlyputtingemphasisonnuclear- reaction rates for neutron-induced reactions (mostly by phenomenological methods), and nuclear mass and de-cay rates covering some thousands of nuclei needed for r-process calculations. However, special attention is also paidto,e.g.,theoreticaltreatmentofcharged-particlere- actions in light nuclei, properties of nuclei far off the β- stability,andhigh-densitynuclear/hadronmatterthatde- termines the physical conditions of BBN and r-process nucleosynthesis sites. The main contents of our activities are categorized as follows: •Nuclear mass and related data (Waseda/JAERI) •α- andβ-decayrates (Waseda/JAERI) •Fission barrier and ﬁssion half-lives (Waseda/JAERI) •Calculationsoflight-ioninducedreactioncrosssec- tions (LANL/JAERI) •Reaction rates for light nuclei (JAERI/TUS) •Resonance analysis of low-energy charged-particle data (AITEL/JAERI) •Measurements of important reaction rates (JAERI/Titech) •Compilationofastrophysicalreactionrates(JAERI) 1339•Properties of supernova and neutron-star matter (JAERI) •Astrophysical applications, interpretations, and feedbackto nuclear data (NAO/Numazu/JAERI) In this paper, the status of some of our activities is outlined. MODELSFOR NUCLEAR MASS AND DECAYPROPERTIES NuclearMass Model Mostnuclidesrelated tothe r-processnucleosynthesis are located at a very neutron-rich region and are hence unknown. Moreover, even properties of known neutron- rich nuclei are not well understood. In order to construct a database of reaction cross sections, decay branching ratios, and decay modes for such kinds of nuclei, we do relyonmodelsthatreproduceknownexperimentalmass values well. Although there are some mass models that can reproduce experimental masses well, we need, for ourpurpose,notonlymassvaluesbuttheﬁrstderivatives of them, namely the neutron and proton separation ener- gies, accurately. Understanding the properties of nuclear structure for neutron-rich nuclei are also important. In the neutron-rich nuclei, exotic properties such as change of magicities are pointed out. These properties greatly affect their reaction crosssections and decay modes. Among some mass models, there is a class of mass formula that separately takes account of the global fea- ture of nuclear mass and deviationsfrom it as the micro- scopic effect. Our models, which we refer to as KUTY [2] or KTUY03 [3] models, fall into such a category. In our models, the global part is expressed as a function of the proton number Zand neutron number Nin order to consider systematics of atomic masses. The remaining part consists of the macroscopic deformed liquid-drop part and the microscopic part. To obtain the microscopic corrections,wetakeasetofmodiﬁedWoods-Saxon-type potentialsasanuclearmean-ﬁeldpotential.Itisprepared sothatitcanbeappliedtoawideregionofnucleiandto reproduce well the single-particle levels of spherical nu- clei, such as208Pb and132Sn [1]. As for the treatment of deformation, we adopt a phenomenological method so that experimental masses are reproduced well [2]. In our model, as shown in Table 1, not only experimental masses but neutron and proton separation energies are reproduced well. Figure 1 shows the two-neutron separation energies S2n,theexperimentaldataintheupperpanel,andourre- sults in the lower panel. In the upper panel, we see large gapsbetween N=8and10(abbreviatedas“at N=8”),andTABLE 1. RMS deviations of separation energies from experimentaldatafortheirmassmodelsinkeV.Thevalues in the parentheses are the number of nuclei. Mass neutron proton formula Mass Sn S2n Sp S2p Z,N≥2 (1835) (1648) (1572) (1592) (1483) KTUY03 657.7 361.7 466.0 403.1 542.0 Z,N≥8 (1768) (1585) (1515) (1527) (1424) KTUY03 640.8 319.1 391.9 344.4 465.8 FRDM 678.0 416.7 551.6 409.0 514.2 HFBCS 718.0 464.6 506.1 483.3 529.0 40 30 20 10 0S2n (MeV)N=8N=20 N=28 N=50 N=82Exp.(AW95) (even- N) 50 40 30 20 10 0S2n (MeV) 60 50 40 30 20 10 Proton number ZKTUY03 (even- N) N=28N=50 N=82N=20 N=8 FIGURE 1. The two-neutron separation energies S2nof the experimental data (upper panel) and our results (lower panel). We connect nuclei with the same Nby solid lines. In such a ﬁgure, magicities are seen as largegapsbetween twolines. atN=20, 28, 50, and 82, except for the region with very small values of S2n. Similar gaps are seen in the lower panel.Intheveryneuron-richregion,whichcorresponds to the region near the S2n=0line, the gaps of our S2nat N=20, 28, and 50 show substantial decreases, while the gaps atN=16, 32 (or 34), and 58 become larger com- paredwiththeneighboringones.Thechangeofmagicity atN=20 to 16 is already established experimentally[4]. With the use of these mass models, we calculate neutron-capture cross sections, β-decay half-lives, and β-delayed neutron emission in the neutron-rich nuclidic region. 1340NuclearDecay Modes In the heavy and super-heavy nuclidic region, there are nuclei of which α-decay,β-decay, and spontaneous ﬁssion compete among each other. We calculate these partial half-lives over a wide nuclidic region using the KUTY mass formula. As for the β-decay half-lives we take the gross theory in which not only the allowed tran- sitions such as the Fermi and Gamow-Teller transitions but the forbidden transitions are considered [5]. To cal- culate the half-lives, β-decayQ-values are required and we take them from our mass model. As for the spon- taneous ﬁssion, we calculate potential energy surfaces against the nuclear deformations using the method used for the KUTY mass model. The ﬁssion-barrier height is deﬁnedasthehighestsaddlepointfromtheground-state shell energy towards the prolate shapes. To calculate the half-lives,wetaketheone-dimensionalWKBmethodfor avirtual particle on the potential energysurface.That is log10(TSF) =log10µ 1+exp·2 ¯hK¸¶ +log10(NColl)−0.159 +hδoddZ+hδoddN−ΔooδoddZδoddN, (1) with K=Zq 2κ(V(ξ)−Egs)dξ. (2) Here,κisaneffectivemassofavirtualparticlepenetrat- ing the barrier and we now take κ=kµwith a reduced massµofthesymmetricﬁssionfragments.Thepath ξis described by the differential of the deformation parame- tersαas dξ=r0A1/3dα. (3) and taken along the minimum energy trajectory towards theprolateshapes.Amongtheaboveparameters,wetake r0=1.2 fm and NColl=1020.38. The value of kin Eq. (2) is adjusted to reproduce the experimental TSFfor even- even nuclei. Values of a hindrance factor hand an odd- odd correction Δooin Eq. (1) are adjusted for odd- Aand odd-odd nuclei after ﬁxing k. The results are k=6.90, h=3.54, and Δoo=3.0. Figure 2 shows experimental and calculated log10(TSF/(s)) for even-even nuclei. The root- mean-square deviation from experimental values is 3.33 [6]. With the use of the potential energy surfaces, other quantitiesrelatedtotheﬁssionarealsoobtained.Figure3 shows,byblackandwhitesquares,probablenucleifrom whichtheneutron-inducedﬁssionisexpectedtooccurin ourmassmodel.Itshowsthatformostofthenucleirele- vanttother-process,theﬁssionbarrierishigherthanthe neutronseparationenergysothatneutron-inducedﬁssion does not play important roles in this particular example. Therefore, we are going to calculate the β-delayed ﬁs- sion probabilities in this region. These are expected to 25 20 15 10 5 0 -5 -10log10(TSFee/(s)) 172 168 164 160 156 152 148 144 140 Neutron number NU Pu Cm Cf FmNo Rf SgDs (110) Exp(e-e) Th(e-e) ( k =6.90) RMS dev.=3.33 (2000-1/2000) FIGURE 2. Experimental and calculated log10(TSF) for even-evennuclei. giveanend-pointofther-processreactions,anditispos- sibletoaffectabundancesofthemedium-heavynuclei.It must be noticed, however, that the neutron-induced ﬁs- sion may play important roles if other mass models are adopted, or in a certain astrophysicalenvironment. 120 110 100 90 80 70 60Proton number Z 200 180 160 140 Neutron number N (Sn−Bfiss+1MeV)>0 (Sn−Bfiss)>0 Sn and S2n>0 r-process path FIGURE 3. Nuclei having ﬁssion barrier heights Bﬁss smaller than one-neutron separation energies Sn(black). An r-process path estimated from the canonical model with the KUTY mass model and the gross theory of the β-decay is also shown. 1341COMPUTATIONALSCHEME FOR REACTIONRATES Statistical Code and Direct/Semidirect CaptureCode The code system to the calculate nuclear reaction rate consists of the Hauser-Feshbach-Moldauer code CoH, direct/semidirect capture calculation code DSD, and some utility codes. These utility codes generate an input ﬁle for CoH and DSD, and calculate a Maxwell- averaged cross section (MACS). The system also in- cludes databases of nuclear masses, nuclear structure, gamma-raystrength functions, and leveldensities. The CoH code is the optical and statistical Hauser- Feshbach model combined program that calculates particle-induced particle-emission cross sections. A multiple-particle emission is not taken into account since nuclear reactions at low energies (typically below 5 MeV) are important forastrophysicalapplications. TheDSDcodecalculatesthedirect/semidirectneutron capture cross section, which is indispensable when the incident neutron energiesare higher than 1 MeV. BothofthecodessolveaSchrödingerequationforthe opticalpotential,andtheysharethesamebuilt-inoptical potentialmodule.Sinceaglobalparameterizationisnec- essary for a cross-section calculation for a large number of (unstable) nuclides, the code has built-in global opti- cal potentials — Koning-Delaroche Global Potential [7] fornandp, and Lemos’ Potential [8] for the α-particle. Database forInput Parameters Excited-state data, namely the excitation energies, spin,andparitiesofdiscretelevels,areretrievedfromthe Reference Input Parameter Library (RIPL) [9] compiled at IAEA, which is a library containing nuclear-model parameters mainly for the statistical Hauser-Feshbach model calculation. For many unstable nuclides that are involved in the r- processcalculation,theground-statespinandparity( Jπ) are often unknown. We predict them with the Nilsson- Strutinsky-BCS method. These ground-state Jπvalues are prepared as a separate database. For theγ-ray emission, E1,M1, andE2transitions are takenintoaccount.WeemployageneralizedLorentzian- form for the γ-raystrength function givenby fE1(Eγ) =Cσ0Γ0 ×( EγΓ(Eγ,T) (E2γ−E2 0)2+E2γΓ2(Eγ,T)+0.7Γ(Eγ=0,T) E3 0) (4) wheretheconstant Ccanbeobtainedbyanexperimental 2π/angbracketleftΓγ/angbracketright/D0value taken from RIPL-2 if available; other-wiseC=8.68×10−8mb−1MeV−2, the GDR param- etersσ0,Γ0, andEγare calculated with the systematics given in RIPL-2. The strength functions for E2andM1 are also takenfrom the RIPL-2 systematics. Nuclear masses and reaction Q-values are calculated withtheKUTYmassformula[2]ifthedataarenotfound in the Audi-Wapstramass table [10]. The KUTY mass formula is also used to calculate level-density parameters. We adopt the Ignatyuk-type level-densityparameters that include shell effects, a=a∗½ 1+δW U¡1−e−γU¢¾ , (5) wherea∗is the asymptotic level density parameter, U is the excitation energy, δWis the shell correction en- ergy, and γis the damping factor. With the shell correc- tionandpairingenergiestakenfromtheKUTYmassfor- mula,theasymptoticleveldensityparameters a∗become a smooth function of the mass number A, which was es- timatedbasedontheGilbert-Cameron-typelevel-density parametersinRIPL-2.Asourﬁrstattempt,weapplythis systematics to estimate the level density parameters of unstable nuclei. CalculatedCaptureCrossSection The capture cross section of90Zr was calculated with this system as a test case. The calculated capture cross sections are compared with experimental data of Bolde- manet al.[11] and Kapchigashev [12]. The model pa- rameters used were “default” to show a quality of eval- uation with our input parameters, which is shown by the dashed curve in Fig. 4. The evaluated cross section in JENDL-3.3isalsoshowninthisﬁgurebythedot-dashed line. Since resonance parameters are given below about 200 keV in JENDL-3.3, 70-group structure cross sec- tions are shown. About 50% underestimation is seen in our ”default" calculation. Therefore the gamma-ray strength function should be renormalized appropriately to reproduce the experimental data, which is shown by the solid line in Fig. 4. The comparison shown here is, of course, a case for stable nuclides, and those agreements with the experi- mental data do not necessarily ensure that the system gives us reasonable cross sections for unstable nuclides. However, if we anchor the cross-section calculations to the experimental data available, extrapolation of the pa- rametersystematicstotheunstableregionbecomesmore reliable. Figure5showsthesamecalculationbutinthehigher- energyrange.TheDSDcomponentshownbydottedline is very small at low energies in comparison with the 1342 1 10 100 0.01 0.1 190Zr Capture cross section [mb] En [MeV]Boldeman (1982) Kapchigashev (1965) CoH+DSD CoH+DSD (without normalization) JENDL-3.3 FIGURE 4. Comparison of calculated neutron capture cross section for90Zrwith the experimentaldata. 0.01 0.1 1 10 100 0 5 10 15 2090Zr Capture cross section [mb] En [MeV]CoH+DSD DSD JENDL-3.3 FIGURE5. Calculatedneutroncapturecrosssectionfor90Zr in the higher-energy region. The dotted line is calculated with the DSD model, and the solid line is the sum of DSD and Hauser-Feshbachmodel calculations. Hauser-Feshbach component. However, this process be- comes prominent above 10 MeV. Note that JENDL-3.3 also includes the DSD cross section, which is evaluated with a simple systematics. Because the DSD cross sec- tion is only important at higher energies, its contribution to the MACS is expected to be small, as the temperature of interest is typically less than 1 MeV. We have looked into the impact of the DSD process on the MACS by in- cluding/excluding the DSD contribution to the total cap- turecrosssection,andtheresultisshowninFig.6bythe ratio of calculated MACS with DSD to that without the DSDcomponent.WefoundthatthemaximumDSDcon- tribution is less than 3% in this mass range, which will beofnoimportanceforthes-andr-processes.However, the DSD component is much more signiﬁcant for lighter nuclei. ReactionRates forLight Nuclei In the region of light nuclei, clustering aspects play important roles in understanding the reactions among 0.96 0.97 0.98 0.99 1 1.01 1.02 1.03 1.04 0.001 0.01 0.1 1MACS (with DSD) / MACS (without DSD) Temperature [MeV] FIGURE 6. Ratio of MACS for90Zr capture with the DSD component to that without DSD. them. In order to describe the reactions between light nuclei, we are on the way to constructing a systematic computationalschemebasedontheFaddeevtheory[13]. To this aim we construct a database of the effective cluster-clusterinteractionforanarbitrarypairof n,p,d,t, h,andαbasedontheResonatingGroupMethod[14]and the Orthogonality Condition Model [15]. In the case of theN-Npair,weadoptedarealisticN-Ninteractionsuch as the Paris [16] or CD-Bonn [17] potential. Then, we converted them to the momentum representation since our theory is formulated in the momentum space. Such a potential database in the p-space is then utilized in the successive Faddeev calculation for reactions involving three clusters. NUCLEAR-DATADEPENDENCE OF THE R-PROCESSABUNDANCE PATTERNIN A DYNAMICAL R-PROCESSCALCULATION Elementsproducedbyther-processareimportantﬁnger- prints of one of the most dramatic events in the universe [18], namely the type II supernovae (SNeII) or neutron star (NS) - NS mergers. Both kinds of events can pro- vide environments with high neutron densities and high temperature,soheavyelementssuchasactinidesandbe- yond are produced through highly unstable nuclei near the neutron drip line via a chain of neutron-capture pro- cesses. Therefore, the r-process abundances witness the interplaybetweennuclearpropertiesfarfrom β-stability and the appropriate astrophysical environment. Due to these reasons, studies on the r-process give us unique and indispensable information on the astrophysical and nuclear/hadron physics. For example, it is possible to determine the age of the progenitor events that produce the r-process elements since the nucleosynthesis in both SNeII and NS-NS mergers occurs on an instantaneous 13430 50 100 150 200 25010−910−810−710−610−510−410−310−210−1 Mass NumberRelative AbundanceSolar r-process AbundanceKUTY00FRDMHFB2 FIGURE 7. R-process abundance patterns calculated with variousmass tables compared to solar r-processabundance. time scale of around ms to 100 ms, which enables us to estimate the star-formation history, chemical evolu- tion of the galaxy, and the age of the universe. Further- more, the r-process abundance pattern is sensitive to the physical condition of the site where it is cradled. This gives critical information on, e.g., the equation-of-state of hadron/nuclear matter at extremely dense and highly isospin-asymmetric conditions. The masses, structure, and reaction mechanisms involving nuclei near the drip line can be also inferred from the study of the r-process. Actualr-processcalculationsareusuallycarriedoutin two different approaches, namely a model-independent static approach and a dynamical one that follows the ex- pansion of matter in the, e.g., type-II supernovae. Here we report our results on the r-process abundance pat- tern in the latter approach. We follow exactly the same methodasdevelopedbyTerasawa etal.[19],namelythat the same nuclear- and weak-reaction rates, the same β- decayrates,andthesameSNeIIexpansiontrajectoryun- der the neutrino-driven wind were employed except for the use of several different mass tables to see nuclear- datadependence(inthiscasenuclearmassmodel)ofthe r-processabundance. Figure 7 compares the solar r-process abundance pat- tern with those calculated by using three mass models; the KUTY model [2], the Finite-Range Droplet Model [20] (FRDM), and the Hartree-Fock-Bogoliubov model [21]. The calculated abundance data are normalized at the third peak region (A ∼200). We can see interesting similarities and differences. Firstly, all the mass tables can produce nuclei up to the actinide region, and three prominent peaks at A ∼80, 130, and 200 are present. However, the abundance patterns are different for these three mass models. Furthermore, the abundance of ac- tinide nuclei is drastically different among them. This shows clear evidence that the nuclear data, in this exam- ple the mass data, are an important ingredient in under- standing the astrophysicalnucleosynthesis.CONCLUSION In this paper, the present status of a Japanese + LANL activityonnucleardataforastrophysicalnucleosynthesis is described. The results will be combined as a compre- hensive database. It can also act as a database for energy applications. ACKNOWLEDGMENTS The authors are grateful to Drs. P. Möller and S. Goriely who gave critical and informative comments on our ac- tivities. A part of this work was supported by Japanese Nuclear Data Committee. REFERENCES 1. H. Koura,M. Yamada,Nucl. Phys. A671,96 (2000). 2. H.Koura,M.Uno,T.Tachibana,M.Yamada,Nucl.Phys. A674, 44 (2000). 3. H. Koura, T. Tachibana, M. Uno, M. Yamada, submitted to Prog. Theor.Phys. 4. A. Ozawa, T. Kobayashi, T. Suzuki, K. Yoshida, I. Tanihata,Phys.Rev.Lett. 84,5493 (2000). 5. T. Tachibana and M. Yamada, Proc. Int. Conf. on exotic nuclei and atomic masses, Arles, 1995, eds. M. de Saint Simon and O. Sorlin (Editions Frontueres, Gif-sur- Yvette,1995), p.763 , and references therein. 6. H. Koura, TOURS Symposium on Nuclear Physics V (TOURS2003), ed. M. Arnould, et al., (The American Institute of Physics Conference Proceedings 704, 2004) pp60-69. 7. A.J. Koning, J.-P. Delaroche, Nucl. Phys. A713, 231 (2003). 8. O.F.Lemos, Orsay Report, Series A, No.136 (1972). 9. “Handbook for Calculations of Nuclear Reaction Data — Reference Input Parameter Library —,” IAEA-TECDOC- 1034, International Atomic Energy Agency (1998); RIPL-2 Handbook, International Atomic Energy Agency [to be published]. 10. G. Audi, A. H. Wapstra, Nucl. Phys. A595, 409 (1995); A.H. Wapstra, G. Audi, C. Thibault, Nucl. Phys. A729, 129 (2003). 11. J.W. Boldeman, B.J. Allen, A.R. De L. Musgrove, R.L. Macklin, Nucl. Sci. Eng., 82, 230 (1982). 12. S.P.Kapchigashev,At. Ener. 19, 294 (1965). 13. L.D. Faddeev, Zh. Eksp. Theor. Fiz. 39, 1459 (1960) , [Sov.Phys.JETP 12, 1014 (1961)]. 14. K. Wildermuth and Y.C. Tang, A Uniﬁed Theory of the Nucleus, Viewing,Braunscheig (1977). 15. S. Saito, Prog. Theor. Phys. 40, 893 (1968); 41, 705 (1969) 16. J.Haidenbauer,W.Plessas,Phys.Rev.C 30,1822(1984). 17. R. Machleidt, Phys.Rev.C 63, 024001 (2001). 18. F.-K.Thielemann et al.,astro-ph/9802077(1998). 19. M. Terasawa et al., Ap. J.562(2001)470. 20. P.Möller et al., At. Data Nucl. Data Tables5¯9(1995)185. 21. M. Samyn etal., Nucl. Phys. A700, 142(2001). 1344
1.5037958.pdf	Electric field controlled spin waveguide phase shifter in YIG Xi-guang Wang , L. Chotorlishvili , Guang-hua Guo , and J. Berakdar Citation: Journal of Applied Physics 124, 073903 (2018); doi: 10.1063/1.5037958 View online: https://doi.org/10.1063/1.5037958 View Table of Contents: http://aip.scitation.org/toc/jap/124/7 Published by the American Institute of Physics Articles you may be interested in Electric fields shift synthetic garnet spin waves Scilight 2018 , 330005 (2018); 10.1063/1.5051831 Multivariate analysis to evaluate indium behavior at the copper phthalocyanine thin film Journal of Applied Physics 124, 075302 (2018); 10.1063/1.5027912 Characterization of trap states in perovskite films by simultaneous fitting of steady-state and transient photoluminescence measurements Journal of Applied Physics 124, 073102 (2018); 10.1063/1.5029278 Secondary shock wave: Implication for laser ablation inductively coupled plasma mass spectrometry Journal of Applied Physics 124, 073101 (2018); 10.1063/1.5030164 Enhanced ability of defect detection using high voltage time-domain resonance analysis and impedance spectrum Journal of Applied Physics 124, 074501 (2018); 10.1063/1.5022079 Structures and magnetic properties of iron silicide from adaptive genetic algorithm and first-principles calculations Journal of Applied Physics 124, 073901 (2018); 10.1063/1.5036992Electric field controlled spin waveguide phase shifter in YIG Xi-guang Wang,1L.Chotorlishvili,2Guang-hua Guo,1and J. Berakdar2 1School of Physics and Electronics, Central South University, Changsha 410083, China 2Institut f €ur Physik, Martin-Luther Universit €at Halle-Wittenberg, D-06120 Halle/Saale, Germany (Received 30 April 2018; accepted 24 July 2018; published online 16 August 2018) We propose a new type of a spin waveguide in yttrium iron garnet solely controlled by external electric ﬁelds. Spin waves are generated by microwave electric ﬁelds while the shift of the phasebetween spin waves is achieved by means of static electric ﬁelds. The phase shifter operation is based on the magneto-electric coupling and effective Dzyaloshinskii Moriya interaction. The spe- cial geometry of the waveguide imposes certain asymmetry in the dispersion relationships of thespin waves. Depending on the propagation direction, the phases of the spin waves are shifted differ- ently by the external electric ﬁeld. The phase difference is entirely controlled by the driving electric ﬁelds. The proposed phase shifter can be easily incorporated into electronic circuits and in spinwave logical operations. Published by AIP Publishing. https://doi.org/10.1063/1.5037958 I. INTRODUCTION Spin waves (SW) have promising applications in infor- mation processing and communications with low-energy dis-sipations. 1–5In this respect, several elements are important such as SW generation, SW guides, and SW phase shifter.2–4,6A wide range of setups has been already consid- ered and/or realized, such as nano-structure antenna, spin torque oscillator, and domain wall based SW phase shifters.4,6–13The SW device should be ﬂexible for integra- tion into the nanoscale electronic circuits. Insulating materi- als such as Yttrium Iron Garnet (YIG) are currently of great interest.8,14–16In such materials and in the frequency regime of interest, here, we can regard the charge carriers as frozenand focus on the magnon transport. Experimentally, SWs in such materials were intensely studied, for instance, SWs were excited and manipulated by the spin Hall torquein the YIG/heavy metal layers or nonlocal spin value structures. 15–20 Recent studies highlighted the importance of the coupling of YIG to an external electric ﬁeld.21–23The spin orbital inter- action and virtual hopping of oxygen ions result in a ﬁnite netmacroscopic effective electric polarization P¼/C0J ea ESOen;nþ1 /C2ðSn/C2Snþ1Þwith an effective Dzyaloshinskii Moriya (DM) interaction between the magnetic moments of the ionsD¼/C0J ea ESOEen;nþ1.21,22Here, ESO¼/C22h2=2mek2,w h e r e meis the mass of the electron, kis the spin-orbital coupling constant, Jis the exchange coefﬁcient, ais the distance between the magnetic ions, eis the electron charge, en;nþ1is the unit vector connecting the ions, and Snis the spin of the electron. Due to the spin orbital interactions on the dorbitals, kis quite substan- tial. In particular, for the values of the parameters jEj¼100 MV/m, ESO¼3.2 (eV), and a¼12/C210/C010(m), the effective DM constant scales with the exchange constant D/C250:03 J. The concept of the electric control of the spin-wave dis- persion was introduced in Ref. 22. However, until now, only two methods were utilized for the excitation of SW in YIG:nano-antenna and spin transfer torque. Using nano-antenna, it is difﬁcult to excite short-wavelength SWs, 4,7,8while exci- tation by spin transfer torque entails a high current density,and due to the ohmic losses, it is energetically costly.9,15 Here, we propose a design of a SW guide phase shifter based on the magneto-electric coupling in YIG and manipulated solely by an electric ﬁeld. For the activation of the SW, weutilize microwave electric ﬁelds resulting in a time-varying DM interaction and, thus, magnetization rotation at the edges of the YIG stripe. By steering the direction, frequency, andamplitude of the microwave electric ﬁeld, one can control the spatial distribution and the intensity of SWs. Compared to standard tools such as nano-antenna and spin transfer tor-que, 4,7–9,15our method can be exploited to generate propa- gating SW of very short wavelength at low ohmic losses. The alternative phase shifter is designed for an electric ﬁeld with a uniform direction and provides an alternative for manufacturing realistic SW phase shifters and logic opera-tions driven by an electric ﬁeld. This paper is organized as follows: in Sec. II, we intro- duce the model. In Sec. III, with the material parameters of YIG, microwave electric ﬁeld induced SW excitation is numerically calculated and analyzed. The excitation mecha-nism also applies for other magnetoelectric materials, such asðBiRÞ 3ðFeGa Þ5O12in Sec. IV. Besides, a design for the phase shifter driven by static electric ﬁelds is presented in Sec. V, and related results are discussed in Sec. VI. This paper ends with our main conclusions. II. THEORETICAL MODELLING OF SPIN-WAVE EXCITATIONS We consider a YIG nanostripe extending 50 nm in the y- direction and 5 nm in the z-direction. To excite propagating SW, we apply a microwave ﬁeld on the whole sample. Alength of L¼6000 nm in the x-direction is needed to avoid interference between the SWs excited at both edges ( x¼0 and x¼L). The effective polarization is expressed in the continuous limit as follows: P¼c E½ðm/C1r Þm/C0mðr /C1mÞ/C138 and is coupled to the electric ﬁeld via the energy termE elec¼/C0E/C1P. Here, mis the unit magnetization vector of YIG, E¼ðEx;Ey;EzÞ/C2sinð2pftÞis the microwave electric ﬁeld, and cE¼Jea2=ESOis an effective magneto-electric 0021-8979/2018/124(7)/073903/8/$30.00 Published by AIP Publishing. 124, 073903-1JOURNAL OF APPLIED PHYSICS 124, 073903 (2018) coupling constant. The SW dynamics in YIG are governed by the Landau-Lifshitz-Gilbert (LLG) equation supple- mented by the electric ﬁeld term @M @t¼/C0cM/C2Heff/C01 l0MsdEelec dm/C18/C19 þa MsM/C2@M @t:(1) Here, M¼Msm,Msis the saturation magnetization, cis the gyromagnetic ratio, l0¼4p/C210/C07N/A2is the permeabil- ity, and ais the phenomenological Gilbert damping constant. The effective ﬁeld Heff¼2Aex l0MsDmþHdemag þHxxconsists of the exchange ﬁeld, the demagnetization ﬁeld and theexternal magnetic ﬁeld applied along the xaxis, A exis the exchange constant. The applied strong external magnetic ﬁeld allows us to neglect magnetocrystalline anisotropy. Thedemagnetization ﬁeld is given by H demagðrÞ¼/C0Ms 4pð Vrr01 jr/C0r0jmðr0Þdr0: (2) We note that the magneto-electric coupling Eelec ¼/C0E/C1Pmimics an effective Dzyaloshinskii Moriya inter- action (DMI) term. Following the method described in Ref. 24, we adopt boundary conditions relevant for the two- dimensional model @mx @x/C12/C12/C12/C12 @VþcEEy 2AexmyþcEEz 2Aexmz¼0; @my @x/C12/C12/C12/C12 @V/C0cEEy 2Aexmx¼0; @mz @x/C12/C12/C12/C12 @V/C0cEEz 2Aexmx¼0; @mx @y/C12/C12/C12/C12 @V/C0cEEx 2Aexmy¼0; @my @y/C12/C12/C12/C12 @VþcEEx 2AexmxþcEEz 2Aexmz¼0; @mz @y/C12/C12/C12/C12 @V/C0cEEz 2Aexmy¼0:(3) According to Ref. 24, the effective DMI and boundary con- ditions speciﬁed above may grant the magnetization rotationat the edges of the nanostripe. The static electric ﬁeld E ¼ðE x0;Ey0;Ez0Þapplied to the nanostripe couples to the magnetization via the DMI term. The equilibrium magneti-zation distributions near the right edge are shown in Fig. 1 for different electric ﬁelds. Apparently, the magnetization rotation induced by yandzcomponents of the electric ﬁeld is mainly located near the boundary x¼0. The ﬁeld compo- nent E y0uniformly tilts the boundary magnetization towards theyaxis. The component of the electric ﬁeld Ez0tilts the bound- ary magnetization toward the zaxis. Besides, the boundary magnetizations in the upper ( y¼50 nm) and lower ( y¼0) parts are slightly tilted along the yaxis. The rotations with respect to the yaxis in the upper and lower parts are oppo- site. Importantly, the component of the electric ﬁeld Ex0 affects not only the edges but also the magnetization in the whole sample. Namely, the upper and the lower boundary magnetizations are tilted respect to the yaxis in oppositedirections. The amplitude of the rotation of the magnetiza- tion increases with the electric ﬁeld (see Fig. 2). The observed rotation of the boundary magnetization is the com- bined effect of the exchange interaction, the effective DMI,and the demagnetization ﬁeld. Inverting the electric ﬁeld direction reverses the direction of the rotation (not shown). The ground state magnetic order m z/C250;mx/C251 uncovers diverse effects of the different ﬁeld components. Namely, from the equation@my @yj@VþcEEx 2AexmxþcEEz 2Aexmz¼0, we observe that if the electric ﬁeld is applied along the xaxis then @my @yj@V6¼0, while if the ﬁeld is applied along the zaxis @my @yj@V¼0. In the case of the boundary conditions Eq. (3), the effec- tive polarization Pis oriented along the applied electric ﬁeld (see Fig. 3). With an increase in the electric ﬁeld amplitude, the polarization amplitude enhances linearly, as shown in Fig.4.FIG. 1. Magnetization rotation induced by the effective DMI. A static elec- tric ﬁeld E¼ðEx0;Ey0;Ez0Þis applied on the nanostripe. FIG. 2. (a) The averaged Myfor 0 /C20x/C201000 nm at y¼0 (black squares) and 50 nm (red circles) as a function of Ex0. (b) The averaged Myfor 0 /C20y/C2050 nm at x¼0 as a function of Ey0. (c) The averaged Mzfor 0 /C20y /C2050 nm at x¼0 as a function of Ez0. (d) Myatðx¼0;y¼0Þ(black squares) and ðx¼0;y¼50Þnm (red circles) as a function of Ez0.073903-2 Wang et al. J. Appl. Phys. 124, 073903 (2018)III. RESULTS FOR SPIN-WAVE EXCITATION IN YIG In the numerical simulations, we implemented a ﬁnite dif- ference scheme and the coarse-grained nanostripe with the cell size of 5 /C25/C25n m3. The following material parameters of YIG are used: Ms¼1:4/C2105A/m, Aex¼5/C210/C012J/m, damping constant a¼0:01, and cE¼0:9P C / m .21–23The mag- netic ﬁeld Hx¼1/C2105A/m is applied along the xaxis. In what follows, we analyze the SW excited by microwave-electric ﬁeld. We explore the propagation of the SW spreading out from the edges. Proﬁles of SWs excited in the vicinity of the edges bythe microwave electric ﬁeld E¼ðE x;Ey;EzÞ/C2sinð2pftÞwith the frequency f¼7G H za r es h o w ni nF i g . 5. In particular, the Zcomponent of magnetization mzis used for illustration. The amplitude of the SW is maximal in the region of the applied microwave ﬁeld and decays gradually along the xaxis. The SW e x c i t e db yr fﬁ e l d s Ey¼100 MV/m and Ez¼100 MV/m has a uniform n¼0 mode in the ydirection. The SW excited by Ey has a larger amplitude compared to the SW excited by Ez.T h e S We x c i t e db ym i c r o w a v eﬁ e l d Ex¼10 MV/m with frequency f¼20 GHz spreads over the whole sample, while the micro- wave ﬁeld is applied locally near the right edge of the sample. We clearly observe an alternation between maximum and mini-mum values of the SW proﬁle corresponding to the n¼1S W mode in the both xandydirections. The SW exponentially decays in the xdirection due to the Gilbert damping.The observed SW proﬁles can be explained qualitatively in terms of the non-collinear magnetization rotations gener- ated by the magneto-electric coupling. The rotations inducedby the E yandEzﬁeld components are almost uniform along theyaxis and depend on the amplitude and direction of the electric ﬁeld. The time-dependent electric ﬁeld induces oscil-lation in the uniform magnetization near the region x¼0 and generates the n¼0 mode SW. The ﬁeld component E xindu- ces a rotation of the lower (near y¼0) and the upper (near y¼50 nm) boundary magnetizations in opposite directions. The induced oscillations of the magnetization in the upperand lower parts are out-of-phase, while the spin-wave ampli- tude is zero in the center along the yaxis, i.e., the n¼1 mode SW. Depending on the proﬁle of the non-collinearrotation shown in Fig. 1, the induced SW excitation is mainly located at the boundary for E yandEzbut is distributed in the whole region if SW is excited by the ﬁeld component Ex,a s we observe in Fig. 1. We explore the frequency dependence of the SW ampli- tude (see Fig. 6), where Ex¼Ey¼Ez¼10 MV/m. For n¼0 SW mode ( EyandEz), SWs with frequencies lower than 6 GHz are prohibited to propagate in the nanostripesince their frequencies match the frequency bandgap of the system. For n¼1 SW mode ( E x), the threshold cutoffFIG. 3. x,y,andzcomponents ( Px,PyandPz) of the electric polarization P for static electric ﬁelds Ex0¼/C0130 MV/m, Ey0¼/C0130 MV/m, and Ez0 ¼/C0130 MV/m, respectively. FIG. 4. x(a),y(b), and z(c) components of the electric polarization Pas a function of the corresponding electric ﬁeld Ex0(a),Ey0(b), and Ez0(c).FIG. 5. The spatial distribution of the excited spin wave expressed in terms of the zcomponent of the magnetization. The unit of the spin-wave proﬁle is A/m. FIG. 6. Dependence of the amplitude of SWs on the frequency. The appliedmicrowave electric ﬁeld reads E¼ðE x;Ey;EzÞ/C2sinð2pftÞ.073903-3 Wang et al. J. Appl. Phys. 124, 073903 (2018)frequency becomes larger 14 GHz. The threshold cutoff fre- quency can be estimated from the SW dispersion relationship of the ﬁnite size bulk system25,26 x2 SW¼xHþaexxmk2 nþxmPðkntÞk2 y;n=k2 nhi /C2fxHþaexxmk2 nþxm1/C0PðkntÞ ½/C138 g : (4) Here, PðkntÞ¼1/C0½1/C0expð/C0kntÞ/C138=ðkntÞ, and we intro- duced the following notations: xH¼cHx;xm¼cMs;aex ¼2Aex=ðl0M2 sÞ. The SW wave-vectors k2 n¼k2 xþk2 y;nare quantized along xandyaxes ky;n¼np=w, where wis the width of the nanostripe. From Eq. (4), the cutoff frequencies forn¼0 and n¼1 modes are of the order of 5.5 GHz and 13.8 GHz, respectively. These values are in good agreement with the simulation results. Furthermore, by increasing the amplitude of the electric ﬁeld pumping larger amount of theenergy into the system, the amplitudes of the excited SWs are increased, as shown in Fig. 7. The frequency and the amplitude of the SW excited by the microwave ﬁeld compo- nent E xare higher than those excited by other ﬁeld compo- nents EyandEz. The reason is that Exalters the magnetic order in the whole sample, while rotations induced by the components Eyand Ezare local, mainly located near the boundary x¼0. The amplitude of the n¼0 SW mode increases linearly with the electric ﬁeld amplitude Ey. However, the dependence of the SW amplitude on the Ez component of the electric ﬁeld shows a nonlinear character. At the same frequency, the SW excited by Eyhas a larger amplitude compared to the SW excited by Ez. A possible rea- son for the non-linear dependence may lie in the asymmetric rotation of magnetization along the yaxis that increases rap- idly with Ez[Fig. 2(d)]. Therefore, a part of the pumped energy is absorbed by n¼1 mode SW excitation. At 7 GHz, n¼1 mode SW cannot propagate (below the cut-off fre- quency), and the propagating n¼0 mode SW shows a non- linear dependence on the ﬁeld component Ez. When the saturation magnetization Msis very small, the asymmetric rotation of magnetization disappears, and the results obtained, for instance, for iron garnets ðBiRÞ3ðFeGa Þ5O12(R¼Lu and Tm), show a linear depen- dence of the SW amplitude on the electric ﬁeld Ez.In the support of our analysis, we performed further cal- culations and found (results now shown): the amplitude ofthe SW excited by E yincreases linearly with Ey. Microwave ﬁeld Eywith a frequency of 7 GHz excites only the n¼0 mode of the SW. The amplitude of the SW excited by Ez depends nonlinearly on Ez, while two modes n¼0 and n¼1 are excited. The n¼1 mode is not propagating because of a matching of the frequency with the frequency bandgap of thesystem. However, for larger E z, when the n¼1 mode of the SW is enhanced, nonlinearity leads to the emergence of the new n¼1 SW mode with a double frequency f¼14 GHz. Thus, more energy is transferred to the non-propagatingn¼1 SW channel (Fig. 8), and the lack of the energy for propagating n¼0 mode SW violates the linear dependence of the SW amplitude on the E z. IV. RESULTS FOR SPIN-WAVE EXCITATIONS IN (BiR) 3(FeGa) 5O12 The discovered effect is not speciﬁc to YIG but can be explored in a variety of magnetoelectric materials. As anexample, we consider material parameters for iron garnetsðBiRÞ 3ðFeGa Þ5O12(R¼Lu and Tm):23,27MS¼1/C2104A/ m,Aex¼5/C210/C012J/m, cE¼6 pC/m, and the Gilbert damping constant a¼0:03. Using these parameters and Eq. FIG. 7. The dependence of the averaged amplitude of the SW, on the ampli- tude of the electric ﬁeld jEj. The averaging procedure is done for the region 1500 nm <x<2500 nm.FIG. 8. The dependence of the averaged amplitude of the n¼1 SW with a frequency of 14 GHz, on the amplitude of the electric ﬁeld jEzj. Here, the SW is excited by the microwave electric ﬁeld with a frequency of 7 GHz. The averaging procedure is done for the region 1500 nm <x<2500 nm. FIG. 9. The spatial distribution of the excited spin wave expressed in terms of thezcomponent of magnetization. The unit of the spin-wave proﬁle is A/m. Material parameters of ðBiRÞ3ðFeGa Þ5O12(R¼Lu and Tm) are used.073903-4 Wang et al. J. Appl. Phys. 124, 073903 (2018)(4), we obtained estimations of the cut-off frequencies: for then¼0 mode SW, the cut-off frequency is equal to 3.7 GHz and for the n¼1 mode SW, the cut-off frequency is equal to 110 GHz. The results for n¼0 mode SWs at f¼7 GHz, excited by EyandEz(see Fig. 9), are quite simi- lar to the results obtained for YIG. However, due to the smaller saturation magnetization and hence the weakerdemagnetization ﬁeld, the magnetization rotation along the z direction induced by E zbecomes more uniform. The differ- ence between SWs excited by EyandEzreduces, as con- ﬁrmed by the SW frequency spectrum (see Fig. 10). The dependence of the amplitude of SW on the electric ﬁeld is shown in Fig. 11. The cut-off frequency of the n¼1S W mode induced by Exis too large (110 GHz). Therefore, the excited magnetization oscillation cannot propagate in the sample (not shown). V. THEORETICAL MODEL FOR THE SPIN-WAVE PHASE SHIFTER The structure of the SW phase shifter is sketched in Fig. 12. In the numerical simulation, we used material parameters of YIG. The equilibrium magnetization is aligned along theþxaxis parallel to the applied external magnetic ﬁeld Hx ¼2:5/C2105A/m. In the left bifurcation point, the propagat- ing SW symmetrically splits apart into two branches. Thewaves merge together after reaching the right conﬂuencepoint. The only n¼0 SW mode is considered here, and SWs may propagate in the both xorydirections. The analytical study based on the method of the Refs. 25and26elucidates the role of the propagation direction in the dispersion relations. In particular, for the waves propa-gating along the xaxis, the dispersion relationship [Eq. (4)] is still valid, and the effect of the electric ﬁeld is absent.However, for the n¼0 SW propagating along the yaxis, the component E z0of the electric ﬁeld has a major effect xSW¼ðxHþaexxmk2 yÞ1=2/C2fxHþaexxmk2 y þxm1/C0PðkytÞ/C2/C3gg1=262ccEEz0 l0Msky: (5) Here,6corresponds to the SW propagating along the 6y direction. Apparently, a positive (negative) Ez0increases (decreases) the SW wave-vector kyfor SW propagating along /C0yand decreases (increases) the wave vector for SW propagating along þy. Thus by the applying electric ﬁeld ð0;0;Ez0Þin the left half of the waveguide (see Fig. 12), the SW wave vector can be shifted differently for SWs propagat-ing along 6ydirection in the upper and lower branches but unaffected when they propagate along þx. The two waves with different phases merge in the right half of the wave-guide and generate the SW interference. VI. RESULTS FOR THE SPIN-WAVE PHASE SHIFTER The phase shift of the SW induced by the electric ﬁeld is shown in Fig. 13. In the absence of the applied electric ﬁeld, the SWs in the upper and lower branches are symmetric, andthe difference between their phases is zero d/ SW¼0. The applied negative ﬁeld component Ez0generates the phaseFIG. 10. The dependence of the amplitude of SWs on the frequency. The applied microwave electric ﬁeld is E¼ð0;Ey;EzÞ/C2sinð2pftÞ. Material parameters of ðBiRÞ3ðFeGa Þ5O12(R¼Lu and Tm) are used. FIG. 11. The dependence of the averaged amplitude of the SW on the ampli- tude of the electric ﬁeld jEjat the frequency f¼7 GHz. Material parameters ofðBiRÞ3ðFeGa Þ5O12(R¼Lu and Tm) are used.FIG. 12. Schematics of the spin-wave phase shifter and the interference of the spin waves. The region with color represents the YIG sample where the spin wave propagates. The white region represents the vacuum. The propa- gating spin wave is excited locally at the left edge and is detected near the right edge. The static electric ﬁeld is applied in the left half of the structure. The equilibrium magnetization is uniform and is aligned along the þx direction.073903-5 Wang et al. J. Appl. Phys. 124, 073903 (2018)difference between two branches. For the SW with the fre- quency fsw¼18:3 GHz, the phase difference is equal to d/SW¼p=2a t Ez0¼/C0100 MV/m and is twice large (d/SW¼p) for Ez0¼/C0200 MV/m. The detailed analysis indicates a linear increase in the phase difference with Ez0, as shown in Fig. 14(a) . Reversing of the electric ﬁeld inverts the sign of the phase difference d/SW. Furthermore, we ﬁnd that the phase shift does not depend on the frequency of the microwave ﬁeld, as shown in Fig. 15. We do believe this is anew feature of our phase shifter added to the functionalities of other phase shifters.28,29 To understand the obtained results, we utilize Eq. (5). For SWs propagating along the yaxis in the opposite direc- tions, the electric ﬁeld shifts the SW dispersion differently [see Fig. 16(a) ]. For the system sketched in Fig. 12, the induced phase difference can be evaluated approximately as follows: ½kyðþyÞ/C0kyð/C0yÞ/C138ly. Here, kyð6yÞis the wave vec- tor of the SW propagating along the 6ydirection. We further calculate the frequency dependence arising from the term ½kyðþyÞ/C0kyð/C0yÞ/C138[see Fig. 16(b) ]. Apparently, this term only slightly changes with the SW frequency. This slightchange explains the frequency independence of the phase shift in our simulations. Besides, the d/ SWvsEz0curve (Fig. 14) testiﬁes the calculated linear dependence of ½kyðþyÞ /C0kyð/C0yÞ/C138on the electric ﬁeld Ez(see Fig. 17). The phase shift induced by the electric ﬁeld impacts the interference of the SWs in the right half of the wave-guide. The phase difference enhances with the amplitude of the electric ﬁeld E z0, and the interference between SWs becomes less distinct. In particular, for Ez0¼/C0200 MV/m, the phase difference is equal to d/SW¼p. Waves merge together, and the detected SW amplitude diminishes (see Fig. 18). Steering of the electric ﬁeld Ez0and the SW fre- quency modify the phase difference d/SWand the detected SW amplitude, as shown in Figs. 14(b) and 15(b) . The obtained results could serve to guide logical device devel-opment based on the SW phase shift and interference. We note that the term ½k yðþyÞ/C0kyð/C0yÞ/C138increases linearly with the electric ﬁeld. The shift of the spin-wave phase isgiven by ½k yðþyÞ/C0kyð/C0yÞ/C138lyand increases linearly with ly. Therefore, while in our case for the system of the length ly ¼100 nm, the shift of the spin wave by pis achieved by means of the 200 MV/m microwave electric ﬁeld, in the case of a larger, experimentally relevant sample ly¼104 nm, the same shift can be achieved with a smaller ﬁeld 2 MV/m. Instead of applying an electric ﬁeld in the left half of the system, one can also generate the SW phase shift andthe interference by applying an the electric ﬁeld only in the lower (or upper) branch of Fig. 12. By setting a large enough external magnetic ﬁeld Hand the equilibrium magnetization along the þzaxis, the n¼0 SW dispersion relationship can be shifted by means of the ycomponent of the electric ﬁeld E y0.FIG. 14. The phase difference d/SW(a) between the upper and lower branches and the detected spin-wave amplitude (b) (normalized by its maxi- mum value) near the right end of the wave guide (Fig. 12) as a function of the electric ﬁeld Ez0. The frequency of the spin wave is 18.3 GHz.FIG. 13. In the left half of the wave guide, we use the spatial distribution of sinð/SWÞ(represented by color in this ﬁgure) to describe the spin-wave propagation, where /SWis the spin-wave phase angle. Width of the wave guide is w0¼20 nm. The frequency of the spin wave is 18.3 GHz. FIG. 15. The phase difference d/SW (a) between the upper and lower branches and detected spin-wave amplitude (b) (normalized by its maxi- mum value) near the right end of the wave guide (Fig. 12) as a function of the spin-wave frequency fsw.073903-6 Wang et al. J. Appl. Phys. 124, 073903 (2018)xSW¼ðxHþaexxmk2 xÞ1=2/C2fxHþaexxmk2 x þxm1/C0PðkxtÞ ½ /C138gg1=262ccEEy0 l0Mskx: (6) In this case, the phase shift is determined by the term ½kxð0Þ/C0kxðEy0Þ/C138lx, where lxis the length of the upper and lower branches as shown in Fig. 12, while kxðEy0Þandkxð0Þ are the wave vectors with and without the electric ﬁeld Ey0. kxð0Þ/C0kxðEy0Þis calculated by means of Eq. (6). Obviously, the phase shift only slightly changes with the SW frequency (see Fig. 19). This is further testiﬁed by the numerical simu- lation results in Fig. 20. VII. SUMMARY We proposed a spin-wave guide phase shifter manipu- lated solely by external electric ﬁelds and demonstrated itsoperation in the micro structures of YIG. The magneto- electric coupling and the effective DM interaction allows exciting and guiding SWs by electric ﬁelds. With microwaveelectric ﬁelds, the time-dependent DM interaction induces arotation of the magnetization at the edges of the sample anddrives the propagating SW with variable spatial distributions.Depending on the propagation direction of the SWs, theapplied static electric ﬁeld shifts selectively the SW disper- sion relationships. The induced phase difference depends on the geometry of the SW guide. The phase difference can fur-ther be tuned on the direction and the intensity of the staticFIG. 16. (a) The spin-wave dispersion relationship calculated from Eq. (5). 6yrepresent the spin wave propagating along the 6ydirection with the wave vector kyð6yÞ. (b) For Ez0¼/C0100 MV/m, kyðþyÞ/C0kyð/C0yÞas a func- tion of the spin-wave frequency fsw¼xsw. FIG. 17. kyðþyÞ/C0kyð/C0yÞas a function of the spin-wave frequency Ez0for fsw¼18:3 GHz.FIG. 18. The spatial distribution of the spin-wave amplitude (represented by color in this ﬁgure) in the right half of the wave guide. The amplitude in the ﬁgure is normalized to its maximum value. The frequency of the spin wave is 18.3 GHz. FIG. 19. For Ey0¼20 MV/m, kxð0Þ/C0kxðEy0Þas a function of the spin- wave frequency fsw¼xsw=ð2pÞ. FIG. 20. The phase difference d/SWbetween the upper and lower branches.073903-7 Wang et al. J. Appl. Phys. 124, 073903 (2018)electric ﬁeld and practically does not depend on the fre- quency of the microwave ﬁeld. ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (Grant Nos. 11704415, 11674400, and 11374373), the Natural Science Foundation of Hunan Province of China (Grant No. 2018JJ3629), andGerman Science Foundation DFG (Grant Nos. SFB 762 and SFB-TRR 227). 1A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands, Nat. Phys. 11, 453 (2015). 2V. V. Kruglyak, S. O. Demokritov, and D. Grundler, J. Phys. D: Appl. Phys. 43, 264001 (2010). 3K. Alexander, B. Mingqiang, and L. W. Kang, J. Phys. D: Appl. Phys. 43, 264005 (2010). 4B. Lenk, H. Ulrichs, F. Garbs, and M. M €unzenberg, Phys. Rep. 507, 107 (2011). 5S. Neusser and D. Grundler, Adv. Mater. 21, 2927 (2009). 6R. Hertel, W. Wulfhekel, and J. Kirschner, Phys. Rev. Lett. 93, 257202 (2004). 7V. E. Demidov, M. P. Kostylev, K. Rott, J. M €unchenberger, G. Reiss, and S. O. Demokritov, Appl. Phys. Lett. 99, 082507 (2011). 8A. A. Serga, A. V. Chumak, and B. Hillebrands, J. Phys. D: Appl. 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1.2946047.pdf	AIP Conference Proceedings 34, 154 (1976); https://doi.org/10.1063/1.2946047 34, 154 © 1976 American Institute of Physics.Effect of Acoustic Wave on Domain Wall Velocity Cite as: AIP Conference Proceedings 34, 154 (1976); https://doi.org/10.1063/1.2946047 Published Online: 24 March 2009 S. Uchiyama , S. Shiomi , and T. Fujii 154 EFFECT OF ACOUSTIC WAVE ON DOMAIN WALL VELOCITY S. Uchiyama, S. Shiomi, and T. Fujii Nagoya University, Nagoya, Japan ABSTRACT Fundamental equations of motion for the spin and the lattice displacement in the magnetic domain wall are derived by taking the effect of magnetoelastic coupling into consideration. The equations are solved for the case of steady wall moving with a constant velocity under an assumption that the tilting angle of the spin in the wall is small. The solution obtained may interprete the anomaly in the velocity-field relation appearing near the trans- Verse acoustic velocity. The effect of lattice damping is introduced intuitively, and the correct treatment of the effect is left to the further investigation. For small value of the quality factor q, the velocity has two possible values for a certain region of the applied field. ~his might infer the instability of the domain wall. INTRODUCTION The dynamic behavior of the magnetic domain wall has been a subject of many investigations because of the technical importance in bubble device. The velocity and the structure of the wall at high driving field may be one of the very interesting problems. Konishi et al. 1 measured the wall velocity in YFeO 3 up to 25000 m/sec by bubble collapse method and found out three irregular points in velocity-field relation. Konishi, Kawamoto and Wada 2 suggested that the first weak knee like velocity saturation around 4400 m/sec may be the indication of the Walker's critical velocity. 3 We thought, however, Eph=(1/2)[Cll(~Ri~y) 2+ c44{(~Rx/~y)2+(~Rz/~y)2}] , (7) where M is the saturation magnetization, H the applied field, SK u the uniaxial anisotropy constant, A the exchange stiffness, B 1 and B 2 the magnetoelastic coupling constants, Cll and c44 the elastic moduli and Rx, ~, and R z the components of lattice displacement. In deriving these equations, the wall is assumed to have a form of infinite plane parallel to the x-z plane as shown in Fig.l, and the polar coordinates 0, ~ of the magnetization direction as well as R , ~., R z are 9 assumed to depend only on y coordlna~e. Further assumptions are as follows: the drive field is applied in z-direction, the ferromagnetic material has a uni- axial anisotropy with easy axis parallel to z-axis and it has a cubic elastic andmagnetoelastic s!nmnetries for simplicity. Fundamental equations of motion are given by the following equations. 6 + a$sine= -(171/NsSinO)(~E/~) , (B) ~sine - ~0 = C E/ e) , r d~x~ ~Rx = - ~EI~Rx ' 110) ~y+ s~ : - ~E/~ , cn) PRz § ~z = - ~EI~R z , 112) where dot indicates the time derivative, ~ is the Gilbert's damping constant, 8 is the lattice damping that this knee might be from the effect of acoustic wave.constant, y is the gyromagnetic ratio, p is the density, This seemed to be confirmed by the finding of Tsang and and White 4; namely they reported that the velocity of 6E/6~ = ~E/~ - (~/~y)[~E/~(~/~y)] , (13) head-to~he~d walls in YFeO 3 saturated at a value of 4100 m/sec which agreed with the transverse acoustic velocity, where ~ stands for 0, ~, Rx, Ry, or R z. In this paper, the effect of acoustic wave on the wall velocity is investigated theoretically. SOLUTION FOR STEADY MOTION FUNDAMENTAL EQUATIONS OF MOTION Since the coupled equations of motion of spin and lattice are derived from the wall energy variation, the wall energy is considered firstly. After the work of Kittel 5, the energy density E in the wall consists of the Zeeman term Ez, the anisotropy term Ea, the exchange term Eex , the demagnetizing term E d, the magnetoelastic term Emel, and the phonon term Eph, namely E = Ez+ Ea+ Eex + Ed+ Emel + Eph . (i) Each of these terms may be written as follows. E = -M Hcos0 , (2) z s E = K sin20 , (3) a u E = A[(~0/By)2+ sin20(~/~y) 2] , (4) ex Ed= 2~M~sin20sin2~ , (5) Emel = Bl(~Ry/~y)(sin2@sin2~ - 1/3) + 2 B2[(~Rx/~y)sin Osin~cos~ +(~Rz/~y)sinOcosOsin ~] , (6) Combining eqs.(1)-(7),(10) and (13), the following equation is obtained for R x. PRx = c44(~2Rx/~y2)+ B2~(sin20sin~cos~)/~y , (14) where the phonon damping term is dropped for simplicity. In case of steady wall moving in y-direction with a constant velocity v, the solution of R should be a x function of (y-vt), and thus H Fig. l Y 4J Y i Coordinate system of domain wall and spins within wall 155 Rx = v 2 (~2Rx/~y2) 9 (15) Replacing the left hand side of eq. (14) by eq. (15), ~2Rx/Sy22 - (4~S2s/B 2) 6t3 (sin2esin~cos~)/~y , (16) or ~Rx/~y = -(4~M2s/B2 ) ~tsin2esin~cos~ , (17) since ~/~y should vanish for y=-+m or @=0, ~. In eqs. (16) and (17), -v 2 . 2 6t mt/(vt - v2) ' Vmt_- B2/2Ms (3p) 1/2, and vt_ (c44/P)1/2. Here v t is the transverse acoustic velocity. Similarly, following equations are derived from eqs. (ii) and (12). ~R~y= - (4~M2/B1) ~isin2@sin2~ , (18) ~Rz/~y= - (4~M2s/B2) ~tsin@cos@sin~ , (19) where 2 2 6 l- Vml/(V I- v 2) , Vm I_ Bl/2Ss (~Q) 1/2, and Vl_ (Cll/Q) 1/2 Here v I is the logitudinal acoustic velocity. S~bstituting eqs. (17)- (19) into eqs. (8) and (9), the equations of motion for spin are derived as follows. 8+e~sine= - ~M [-2A{sine (~2) +2cosS (~y) '~') s 8y +2~M2s{ (i-~ t) +2 (~t-~l) sin2@sin2~}sinesin2~] (20) ~sin@-@e= ~M [{Ku+A(~)2}sin2@+MsHsine-2A(~2~@2 ) s Y ~y +2wM2{ (1-~ t) +2 (~t-~l) sin2@sin2~}sin2esin2~] (21) As is well known, the structure of the static 180 ~ wall can be expressed by @ = 2tan -I [exp{ (y-s)/A0}] , (22) and ~ = 0 , (23) where s is the y-coordinate of the wall center and ~0~(A/Ku )I/2 is the wall width parameter. For steady wall moving with constant velocity v, it is also known 3,6 that the form of eq.(22) holds though the wall width parameter A differs from A 0 if the magnetoelastic coupling can be neglected, namely @ = 2tan -l[exp{ (y-s)/d}] . (24) 6 As for ~, it is shown to be almost constant independent of the y-coordinate. Therefore ~r = 0 . (25) Let us assume that eqs.(24) and (25) hold also in the present case, then eqs.(20) and (21) become @= -2~I~[Ms[(l-~t)+2(~t-61)sin2@sin2~]sinSsin2~ (26) 2 -~e=2~I~IMs[q{I-(A0/A ) }sin2@+~hsin@ +{(l-6t)+2(6t-~l)sin2esin2~}sin2esin2~], (27) where q~Ku/2~M ~ and h~H/2w~M s Eliminating e from eqs.(26) and (27), we get ~[(l-~t)sin2~-h]-2[(l-~t)sin2~+q{l-(~O/A)2}]cos@ +4~(~t-61)sin3~cos~sin2@-4(~t-~l)sin4~sin2@cos@=O (28) So far as v is constant, it is not possible to satisfy this equation for all values of e. However, if e and are very small compared to one, the third and the 0.8 8 0.6 '~ 0.4 1.0-- 0.2 -- V t Yl ,I 0.01 I 0.2 0.4 0.6 Normalized Field h sVe] 0.01 Walker' bcity 9 .~~176 o~. ! 1 0.8 1 Fig.2 Dependence of wall velocity upon applied'field fourth terms in eq. (28) may be negligible. With this approximation, the next relation is obtained by equating the first term of eq. (28) to be zero. sin2# = h/(l-~ t) (29a) or sin2~ =(1/2) [l-{l-h2/(l-~t)2}i/2] . (29b) In the same way, the wall width parameter A is determined from the second term of eq. (28) as follows. l-~t 2 A = A0[I+ 2--~q~ l-[l-h /(I-6t)2]i/2}]-I/2 (30) Thus the wall velocity v is given by ~A I'rl H/c~ 2 2 %l~rlH/l~(l_ Vmt v t 211/2}i-1/2 . = ~) { i- [l-h2/(I--~---~) - vt-v (31) Since the velocity v is involved in both sides of this equation, self consistent value of v has to be looked for numerically. RESULTS OF NUMERICAL CALCULATION Fiuure 2 shows the dependence of the normarized velocity v ( v = V/2~IYIMsA 0 ) on the normarized field h taking the quality factor q as a parameter. The values of other parameters used here are as follows; vt=vt/==IYIMs%= 060, vt=vt/2~IYIMsA0 = 020 So far as q is much larger than one, the velocity is approximately proportional to the field as typically seen in this figure by q=10. Because of the magneto- 156 elastic coupling, however, there is a region just below where Re means the real part, and e is a parameter v t where no real number solution for v exists. In this expressing the effect of lattice damping. region, the wall may accompany the oscillatory motion. Examples~of the ~umerical calculation are shown in With decreasing q, the irregularity in the v-h relation Fig.3, where vt=0.6, Vmt=0.2, q=l, and s is taken as a becomes more appreciable. When q becomes as small as 0. i, parameter. for example, there appears a region where v has two solutions for a certain value of h. In this connection, we must make mention of the fact that the results shown in the figure are valid only for sin~ i<<I. The arrows shown in the graph indicate the points of sin2r = 0.i, and only the left portion of the arrows in the lower velocity branches satisfy the condition of sin2r ~ 0.i. In order to make the comparison easy, the Walker's velocity (Vmt=0) for q=O.l is shown in the figure by dotted curve. In the case of q=0.01, the lower branch velocity shows noticeable saturation. Beyond the two-values region, the velocity jump to the higher velocity branch. The velocity of this branch is nearly equal to the transverse acoustic velocity independent of the applied field. 0.7 // I 0.5 0.6 Normarized Field h Fig. 3 Effect of lattice damping on v-h relation and comparison with experiment 60 65 70 75 80 I I i # ~.5 Applied Field for Experimental Point (Oe) i.0 i.5 s.o --~.5 0.7 EFFECT OF LATTICE DAMPING The effect of lattice damping is thought to be very important in the region where the contribution from the megnetoelastic energy becomes noticeable. Leaving the correct treatment of the effect in further investigation, the effect is introduced here intuitively, namely the parameter ~t in eq.(30) is replaced by %2 ~2 Vmt Vmt = ~t = Re( v~2- ~2+ ie~ ) ~2_ ~2 2~2..~2 ~2.' (32) v t v +E v /lvt-v ; Closed circles shown in the same figure are the experimental results measured in a YFeO 3 thin plate by means of bubble collapse method in our laboratory. DISCUSSION AND CONCLUSION In relation to the work of Tsang and White 4 , the appearance of the two velocities region in the v-h relation seems to be interesting. According to the present theory, however, the saturation velocity of the lower velocity branch is very smell compared with the transverse acoustic velocity. Furthermore, the velocity of the higher velocity branch coincides with the transverse acoustic velocity. These facts contradict to the results on the head-to-head velocity 4. In addition, the two velocities region appears only in the case of small q value, while for YFeO 3, q is surely larger than one. The anomaly seen in the v-h relation near v = 4.4 (km/sec) 2 seems to be explained by the present theory, although the knee velocity is a little larger than the transverse acoustic velocity different from the theoretical prediction. It should be noted that the longitudinal acoustic wave hardly influences the domeiD wall motion. ACKNOWLEDGEMENT The authors express our sincere thanks to Mr.A.Ikai for his experimental assistance and to Dr.S.Tsunashima and Dr.M.Takayasu for their discussion. REFERENCES i. S.Konishi, T.Miyama, and K.Ikeda,"Domain wall veloci- ty in orthoferrites", Appl.Phys.Letters 27 , 258 (1975) 2. S.Konishi, T.Kawamoto, and M.Wada, "Domain wall ve- locity in YFeO 3 exceeding the Walker critical veloci- ty", IEEE Trans.Magn. MAG-10, 642 (1974) 3. L.R.Walker (unpublished). Described by J.F.Dillon,Jr. in Treatise on Magnetism, edited by G.T.Rado and H. Suhl (Academic, New York, 1963) Vol. III, p.450 4. C.H.Tsang and R.L.White, "Observations of domain wall velocities and mobilities in YFeO3", AIP Conf.Proc. 24, 749 (1974) 5. C.---~ttel, "Interaction of spin waves and ultrasonic waves in ferromagnetic crystals", Phys.Rev. ii0, 836 (1958) 6. N.L.Schryer and L.R.Walker, "The motion of 180~ walls in uniform dc magnetic fields", J.Appl.Phys. 45 5406 (1974)
1.5113536.pdf	Appl. Phys. Rev. 7, 021308 (2020); https://doi.org/10.1063/1.5113536 7, 021308 © 2020 Author(s).Pathways to efficient neuromorphic computing with non-volatile memory technologies Cite as: Appl. Phys. Rev. 7, 021308 (2020); https://doi.org/10.1063/1.5113536 Submitted: 05 June 2019 . Accepted: 01 May 2020 . Published Online: 03 June 2020 I. Chakraborty , A. Jaiswal , A. K. Saha , S. K. Gupta , and K. Roy COLLECTIONS Paper published as part of the special topic on Brain Inspired Electronics Note: This paper is part of the special collection on Brain Inspired Electronics. This paper was selected as Featured ARTICLES YOU MAY BE INTERESTED IN Machine-learning-assisted metasurface design for high-efficiency thermal emitter optimization Applied Physics Reviews 7, 021407 (2020); https://doi.org/10.1063/1.5134792 A comprehensive review on emerging artificial neuromorphic devices Applied Physics Reviews 7, 011312 (2020); https://doi.org/10.1063/1.5118217 Nonlinear topological photonics Applied Physics Reviews 7, 021306 (2020); https://doi.org/10.1063/1.5142397Pathways to efficient neuromorphic computing with non-volatile memory technologies Cite as: Appl. Phys. Rev. 7, 021308 (2020); doi: 10.1063/1.5113536 Submitted: 5 June 2019 .Accepted: 1 May 2020 . Published Online: 3 June 2020 I.Chakraborty,a) A.Jaiswal, A. K. Saha, S. K. Gupta, and K. Roy AFFILIATIONS School of Electrical and Computer Engineering, Purdue University, 465 Northwestern Ave., West Lafayette, Indiana 47906, USA Note: This paper is part of the special collection on Brain Inspired Electronics. a)Author to whom correspondence should be addressed: ichakra@purdue.edu ABSTRACT Historically, memory technologies have been evaluated based on their storage density, cost, and latencies. Beyond these metrics, the need to enable smarter and intelligent computing platforms at a low area and energy cost has brought forth interesting avenues for exploiting non- volatile memory (NVM) technologies. In this paper, we focus on non-volatile memory technologies and their applications to bio-inspired neuromorphic computing, enabling spike-based machine intelligence. Spiking neural networks (SNNs) based on discrete neuronal “actionpotentials” are not only bio-ﬁdel but also an attractive candidate to achieve energy-efﬁciency, as compared to state-of-the-art continuous-valued neural networks. NVMs offer promise for implementing both area- and energy-efﬁcient SNN compute fabrics at almost all levels ofhierarchy including devices, circuits, architecture, and algorithms. The intrinsic device physics of NVMs can be leveraged to emulate dynam- ics of individual neurons and synapses. These devices can be connected in a dense crossbar-like circuit, enabling in-memory, highly parallel dot-product computations required for neural networks. Architecturally, such crossbars can be connected in a distributed manner, bringingin additional system-level parallelism, a radical departure from the conventional von-Neumann architecture. Finally, cross-layer optimizationacross underlying NVM based hardware and learning algorithms can be exploited for resilience in learning and mitigating hardware inaccu-racies. The manuscript starts by introducing both neuromorphic computing requirements and non-volatile memory technologies. Subsequently, we not only provide a review of key works but also carefully scrutinize the challenges and opportunities with respect to various NVM technologies at different levels of abstraction from devices-to-circuit-to-architecture and co-design of hardware and algorithm. Published under license by AIP Publishing. https://doi.org/10.1063/1.5113536 TABLE OF CONTENTS I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 II. GENERIC NEURO-SYNAPTIC BEHAVIORAL AND LEARNING REQUIREMENTS. . . . . . . . . . . . . . . . . . . . . . 4 A. Neurons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 B. Synapses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1. Unsupervised learning . . . . . . . . . . . . . . . . . . . . . 52. Supervised learning . . . . . . . . . . . . . . . . . . . . . . . 6 III. NON-VOLATILE TECHNOLOGIES FOR NEUROMORPHIC HARDWARE . . . . . . . . . . . . . . . . . . 6 A. Phase change devices . . . . . . . . . . . . . . . . . . . . . . . . . 7 1. PCM as neurons . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2. PCM as synapses . . . . . . . . . . . . . . . . . . . . . . . . . 83. PCM crossbars . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 B. Metal-oxide RRAMs and CBRAMs . . . . . . . . . . . . . 10 1. Metal-oxide RRAMs and CBRAMs as neurons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102. Metal-oxide RRAMs and CBRAMs as synapses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3. Metal-oxide RRAM and CBRAM crossbars. . . 13 C. Spintronic devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1. Spin devices as neurons. . . . . . . . . . . . . . . . . . . . 132. Spin devices as synapses . . . . . . . . . . . . . . . . . . . 15 3. Spintronic crossbars . . . . . . . . . . . . . . . . . . . . . . . 16 D. Ferroelectric FETs. . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1. FEFETs as neurons. . . . . . . . . . . . . . . . . . . . . . . . 182. FEFETs as synapses . . . . . . . . . . . . . . . . . . . . . . . 18 3. FEFET crossbars . . . . . . . . . . . . . . . . . . . . . . . . . . 19 E. Floating gate devices . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1. Floating gate devices as neurons . . . . . . . . . . . . 192. Floating gate devices as synapses. . . . . . . . . . . . 203. Floating gate crossbars. . . . . . . . . . . . . . . . . . . . . 20 F. NVM architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 IV. PROSPECTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Appl. Phys. Rev. 7, 021308 (2020); doi: 10.1063/1.5113536 7, 021308-1 Published under license by AIP PublishingApplied Physics Reviews REVIEW scitation.org/journal/areA. Stochasticity—Opportunities and challenges . . . . . 22 B. Challenges of NVM crossbars. . . . . . . . . . . . . . . . . . 22C. Mitigating crossbar non-idealities . . . . . . . . . . . . . . 24D. Multi-memristive synapses . . . . . . . . . . . . . . . . . . . . 24 E. Beyond neuro-synaptic devices and STDP . . . . . . . 25 F. NVM for digital in-memory computing . . . . . . . . . 25G. Physical integrability of NVM technology with CMOS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 V. CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 AUTHORS’ CONTRIBUTION . . . . . . . . . . . . . . . . . . . . . . . . . 26 I. INTRODUCTION The human brain remains a vast mystery and continues to bafﬂe researchers from various ﬁelds alike. It has intrigued neuroscientists by its underlying neural circuits and topology of brain networks that result in vastly diverse cognitive and decision-making functionalities as a whole. Equivalently, computer engineers have been fascinated by the energy-efﬁciency of the biological brain in comparison to the state- of-the-art silicon computing solutions. For example, the Bluegene supercomputer 1consumed mega-watts of power2for simulating the activity of cat’s brain.3This is in contrast to /C2420 W of power account- ing for much more complex tasks including cognition, control, move- ment, and decision making, being rendered simultaneously by the brain. The massive connectivity of the brain fueling its cognitive abili- ties and the unprecedented energy-efﬁciency makes it by far the most remarkable known intelligent system. It is, therefore, not surprising that in the quest to achieve “brain-like cognitive abilities with brain- like energy-efﬁciency,” researchers have tried building Neuromorphic Systems closely inspired by the biological brain (refer Fig. 1 ). Worth noting is the fact that neuromorphic computing not only aims at attaining the energy-efﬁciency of the brain but also encompasses attempts to mimic its rich functional principles such as cognition,efﬁcient spike-based information passing, robustness, and adaptability. Interestingly, both the brain’s cognitive ability and its energy-efﬁciency stem from basic computation and storage primitives called neurons and synapses, respectively. Networks comprising artiﬁcial neurons and synapses have, there- fore, been historically explored for solving various intelligent problems. Over the years, neural networks have evolved signiﬁcantly and areusually categorized based on the characteristic neural transfer function as ﬁrst, second, and third generation networks. 4As shown in Fig. 2 , the ﬁrst generation neurons, called as perceptrons ,4had a step function response to the neuronal inputs. The step perceptrons, however, were not scalable to deeper layers and were extended to Multi-Layer Perceptrons (MLPs) using non-linear functional units.5This is alluded to as the second generation neurons based on a continuous neuronal output with non-linear characteristic functions such as sigmoid5and ReLU (Rectiﬁed Linear Unit) .6Deep Learning Networks (DLNs) as we know it today are based on such second generation neural networks. The present revolution in artiﬁcial intelligence is being currently fueled by such DLNs using global learning algorithms based on the gradientdescent rule. 7Deep learning has been used for myriad of applications including classiﬁcation, recognition, prediction, cognition, and deci- sion making with unprecedented success.8However, a major require- ment to achieve the vision of intelligence everywhere is to enable energy-efﬁcient computing much beyond the existing Deep learning solutions. Toward that end, it is expected that networks of spiking neu- rons hold promise for building an energy-efﬁcient alternative to tradi-tional DLNs. Spiking neural networks (SNNs)—the third generation of neural networks—are based on the bio-plausible neural behavior and communicate through discrete spikes as opposed to the continu-ous valued signal of DLNs. Note that for this paper, we refer the sec- ond generation networks as DLNs and the third generation spiking networks as SNNs. FIG. 1. Neuromorphic computing as a brain-inspired paradigm to achieve cognitive ability and energy-efﬁciency of the biological brain. “Hardware” and “Al gorithms” form the two key aspects for neuromorphic systems. As shown in the right hand side, a generic neuromorphic chip consists of several “Neuro-Cores” interconnec ted through the address event representation (AER) based network-on-chip (NOC). Neuro-Cores consist of arrays of synapses and neurons at the periphery. Non-volat ile technologies including PCM, RRAM, MRAM, and FG devices have been used to mimic neurons and synapses at various levels of bio-ﬁdelity.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 021308 (2020); doi: 10.1063/1.5113536 7, 021308-2 Published under license by AIP PublishingFIG. 2. Three generations of neural networks. First generation (Gen-I) of networks used step transfer functions and were not scalable, and second generatio n (Gen-II) uses transfer functions such as Rectiﬁed Linear Unit (ReLU) that has fueled today’s deep learning networks. The third generation (Gen- III) uses spiking neurons resembling the neural activity of their biological coun- terparts. The three components of an SNN are (1) neurons, (2) synapses, and (3) learning. (1) Neurons: three broad classes of spiking neurons that rese archers attempt to mimic using NVMs are Leaky-Integrate-Fire (LIF), Integrate-Fire (IF), and Stochastic Neurons. (2) Synapses: the key attributes needed for a particular device to fu nction as a synapse are its ability to map synaptic efﬁcacy (wherein a synaptic weight modulates the strength of the neuronal signal) and that they can perform multiplication and dot-product operatio ns. (3) Learning: as shown in the ﬁgure, learning can be achieved either through supervised or unsupervised algorithms. From an NVM perspective, various NVM technologies are being used to m imic neuronal and synaptic function- alities with appropriate learning capabilities. At an architectural level, arrays of such NVMs are connected through the network-on-chip to enable seamless integration of a large neural network.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 021308 (2020); doi: 10.1063/1.5113536 7, 021308-3 Published under license by AIP PublishingFrom the energy-efﬁciency perspective, SNNs have two key advantages. First, the fact that neurons exchange information through discrete spikes is explicitly utilized in hardware systems to enable energy-efﬁcient event-driven computations. By event-driveness, it is implied that only those units in the hardware system are active, which have received a spike, and all other units remain idle reducing theenergy expenditure. Second, such an event-driven scheme also enables Address Event Representation (AER). 9AER is an asynchronous com- munication scheme, wherein the sender transmits its address on the system bus and the receiver regenerates the spikes based on the addresses it receives through the system bus. Thereby, instead of trans- mitting and receiving the actual data, event addresses are exchanged between the sender and the receiver, leading to energy-efﬁcient trans- fer of information. In addition to emulation of neuro-synaptic dynamics and use of event-driven hardware, two notable developments, namely, (1) theemergence of various non-volatile technologies and (2) the focus on learning algorithms for networks of spiking neurons, have accelerated the efforts in driving neural network hardware closer toward achievingboth energy-efﬁciency and improved cognitive abilities. Non-volatile technologies have facilitated area- and energy-efﬁcient implementa- tions of neuromorphic systems. As we will see in Sec. IIIof the manu- script, these devices are of particular interest since they are governed by intrinsic physics that can be mapped directly to certain aspects of biological neurons and synapses. This implies that instead of usingmultiple transistors to imitate neuronal and synaptic behavior, in many cases, a single non-volatile device can be used as a neuron or a synapse with various degrees of bio-ﬁdelity. In addition, a major bene-factor for non-volatile memory (NVM) technologies is that they can be arranged in dense crossbars of synaptic arrays with neurons at the periphery. This is of immense importance since the co-locations ofcompute (neuronal primitives) and storage (synaptic primitives) are inherent characteristics that make the biological brain so effective. Note that this closely intertwined fabric of compute and storage is con-spicuously different from state-of-the-art computing systems that rely on the well-known von-Neumann model with segregated compute and storage units. Additionally, learning algorithms for networks ofspiking neurons has recently attracted considerable research focus. For this paper, we would deﬁne neuromorphic computing as SNN based neural networks, associated learning algorithms, and their hardwareimplementations . In this paper, we focus on non-volatile technologies and their applications to neuromorphic computing. With reference to Fig. 2 ,w e start in Sec. IIby ﬁrst describing the generic neural and synaptic behavioral characteristics that are in general emulated through non- volatile devices. Subsequently, in Sec. III, we describe learning strate- gies for SNNs and associated topologies. With the knowledge of basic neuro-synaptic behavior and learning methodologies, Sec. IVpresents non-volatile memories as the building block for neuromorphic sys-tems. Finally, before concluding, we highlight on future prospects and key areas of research that can further the cause of neuromorphic hard- ware by exploiting non-volatile technologies. II. GENERIC NEURO-SYNAPTIC BEHAVIORAL AND LEARNING REQUIREMENTS One of the key advantages of non-volatile technologies is that their intrinsic device characteristics can be leveraged to map certainaspects of biological neurons and synapses. Let us highlight few repre- sentative behaviors for both neurons and synapses that form the basic set of neuro-synaptic dynamics usually replicated through non-volatile devices. A. Neurons Neural interactions are time varying electro-chemical dynamics that gives rise to brain’s diverse functionalities. These dynamical behaviors in turn are governed by voltage dependent opening and closing of various charge pumps that are selective to speciﬁc ions such as Na þand Kþ.10,11In general, a neuron maintains a resting potential , across its cell membrane by maintaining a constant charge gradient. Incoming spikes to a neuron lead to an increase in its membrane potential in a leaky-integrate manner until the potential crosses a cer- tain threshold after which the neuron emits a spike and remains non- responsive for a certain period of time called as the refractory period .A typical spike (or action potential) is shown in Fig. 3 highlighting the speciﬁc movements of charged ions through the cell membrane. Additionally, it has been known that the ﬁring activity of neurons is stochastic in nature.12,13 Having known the generic qualitative nature of neural function- ality, it is obvious that a resulting model, describing the intricacies of a biological neuron, would consist of complex dynamical equations. In fact, detailed mathematical models such as Hodgkin–Huxley model14 and spike response model have been developed, which closely matchthe behavior of biological neurons. However, implementing such mod- els in hardware turns out to be a complex task. As such, hardware implementations mostly focus on simpliﬁed neuronal models, such as Leaky-Integrate-Fire (LIF) model 15–17shown in Fig. 3 .C o n s e q u e n t l y , the diverse works on mimicking neurons using non-volatile technolo- gies can be categorized into three genres—(1) the Leaky-Integrate-Fire (LIF) neurons, (2) the Integrate-Fire (IF) neurons, and (3) Stochastic- Firing (s-F) neurons. Figure 2 graphically represents the typical neural behavior for each type of neuron, while Fig. 3(c) presents a Venn- diagram highlighting various works based on non-volatile technologies and the corresponding neural behavior that they are based on. •Leaky-Integrate-Fire (LIF) neurons: The membrane potential of an LIF neuron is incremented at every instance when the neuron receives an input spike. In the interval between two spikes, the neuron potential slowly leaks, resulting in the typical leaky- integrate behavior shown in Fig. 2 . If the neuron receives sufﬁ- cient input spikes, its membrane potential crosses a certain threshold, eventually allowing the neuron to emit an output spike. •Integrate-Fire (IF) neurons: The IF neuron is a simpliﬁed versionof the LIF neuron without the leaky behavior. Essentially, an IF neuron increments its membrane potential at every spike main- taining its potential at a constant value between two spikes, as shown in Fig. 2 . IF neurons ﬁre when the accumulated mem- brane potential crosses a pre-deﬁned threshold. •Stochastic-Firing neurons: In contrast to deterministic neurons that ﬁre whenever the neuron crosses its threshold, a stochastic ﬁring neuron ﬁres with a probability, which is proportional to its membrane potential. In other words, for a stochastic neuron, an output spike is emitted with a certain probability, which is a function of the instantaneous membrane potential. In its simplestApplied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 021308 (2020); doi: 10.1063/1.5113536 7, 021308-4 Published under license by AIP Publishingform, a stochastic ﬁring behavior can be modeled by a ﬁring probability, which increases with the input stimulus. However, stochasticity can also be combined with LIF and IF neurons, such that once the neuron crosses the threshold, it only emits a spike based on a probabilistic function. LIF neurons are most widely used in the domain of SNNs. The leaky nature of LIF neurons renders a regularizing effect on their ﬁring rates. This can help particularly for frequency based adaptation mech- anisms that we will discuss in the next section.18IF neurons are typi- cally used in supervised learning algorithms. In these algorithms, the learning mechanism does not have temporal signiﬁcance, and hence, temporal regularization is not required. Stochastic neurons, on the other hand, have a different computing principle. Due to the probabil- istic nature of ﬁring, it can also act as a regularizer and also lead to bet- ter generalization behavior in neural networks. All the aforementioned neurons can leverage the inherent device physics in NVM devices for efﬁcient hardware implementation. B. Synapses Information in biological systems is governed by transmission of electrical pulses between adjacent neurons through connecting bridges, commonly known as synapses. Synaptic efﬁcacy ,r e p r e s e n t i n gt h e strength of connection through an internal variable, is the basic crite- rion for any device to work as an artiﬁcial synapse. Neuro-chemical changes can induce plasticity in synapses by permanentlymanipulating the release of neurotransmitters and controlling the responsiveness of the cells to them. Such plasticity is believed to be the fundamental basis of learning and memory in the biological brain. From the neuromorphic perspective, synaptic learning strategies can be broadly classiﬁed into two major classes: (1) unsupervised learning and (2) supervised learning. 1. Unsupervised learning Unsupervised learning is a class of learning algorithms associated with self-organization of weights without the access to labeled data. In the context of hardware implementations, unsupervised learning relates to biologically inspired localized learning rules where the weight updates in the synapses depend solely on the activities of the neurons on its either ends. Unsupervised learning in spike-based systems can be broadly classiﬁed into (i) Spike Timing Dependent Plasticity (STDP) and (ii) frequency dependent plasticity. Spike timing dependent plasticity (STDP), shown in Fig. 4 ,i sa learning rule, which strengthens or weakens the synaptic weight based on the relative timing between the activities of the connected neurons. This kind of learning was ﬁrst experimentally observed in rat’s hippo- campal glutamatergic synapses.19It involves both long-term potentia- tion (LTP),20which signiﬁes the increase in the synaptic weight2þ,a n d long-term depression (LTD), which signiﬁes a reduction in the synap- tic weight. LTP is realized through STDP when the post-synaptic neu- ron ﬁres after the pre-synaptic activity, whereas LTD results from an FIG. 3. (a) The biological neuron and a typical spiking event. Various ions and the role they play in producing the spiking event are shown. (b) A simpliﬁed neur al computing model highlighting the ﬂow of information from the input of neurons to the output. Spikes from various pre-neurons are multiplied by the correspondin g weights and added together before being applied as an input to the neuron. The neuron shows a typical leaky-integrate behavior unless its membrane potential crosses a c ertain threshold, leading to emission of a spike. (c) The LIF differential equation.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 021308 (2020); doi: 10.1063/1.5113536 7, 021308-5 Published under license by AIP Publishingacausal spiking between the pre-synaptic and post-synaptic neurons, wherein the post-synaptic neuron ﬁres before the pre-synaptic neuron. Mathematically, the relative change in synaptic strength is depen- dent on the timing difference of the post-synaptic and pre-synaptic spikes as dwðDtÞ¼Aþexpð/C0Dt=sþÞifDt>0; (1) ¼A/C0expðDt=s/C0ÞifDt<0: (2) Here, Aþ,A/C0,sþ;s/C0are the ampliﬁcation coefﬁcients and time- constants, respectively, and Dtis deﬁned as the difference between the pre-synaptic and post-synaptic ﬁring instants. STDP has been widely adopted in not only computational neuroscience but also neuromor- phic systems as the de facto unsupervised learning rule for pattern detection and recognition. In conjunction to long-term modiﬁcation of synaptic weights, the physiology of synapses induces yet another type of learning, i.e., frequency dependent plasticity, dependent on the activity of the pre- synaptic potential.21,22Activity-dependent learning can induce two types of changes in the synaptic strength. The change occurring over ashort timescale (hundreds of milliseconds in biological systems) is known as Short-Term Plasticity (STP), while the long-term effects are a form of LTP that can last between hours to years. In general, at a given instance, a pre-synaptic activity induces STP; however, when the pre-synaptic activity reduces, the synaptic efﬁcacy is reverted back to its original state. Repeated stimuli eventually result in LTP in the syn- apses. As STP corresponds to the recent history of activity and LTP relates to long-term synaptic changes resulting from activity over a period of time, they are often correlated with short-term memory (STM) and long-term memory (LTM), respectively, in mammals. 23 2. Supervised learning Although unsupervised learning is believed to form the dominant part of learning in biological synapses, the scope of its applicability is still in its nascent stages in comparison to conventional deep learning. An alternative ex situ learning methodology to enable spike-based processing in deep SNNs is restricting the training to the analog domain, i.e., using the greedy gradient descent algorithm as in conven- tional DLNs and converting such an analog valued neural network to the spiking domain for inferencing. Various conversion algo- rithms24–26have been proposed to perform nearly lossless transforma- tion from the DLN to the SNN. These algorithms address several concerns pertaining to the conversion process, primarily emerging due to differences in neuron functionalities in the two domains. Such con- version approaches have been demonstrated to scale to state-of-art neural network architectures such as ResNet and VGG performing classiﬁcation tasks on complex image datasets as in ImageNet.27More recently, there has been a considerable effort in realizing gradient- based learning in the spiking domain itself28to eliminate conversion losses. III. NON-VOLATILE TECHNOLOGIES FOR NEUROMORPHIC HARDWARE As elaborated in Sec. II, SNNs not only are biologically inspired neural networks but also potentially offer energy-efﬁcient hardware solutions due to their inherent sparsity and asynchronous signalprocessing. Advantageously, non-volatile technologies provide two FIG. 4. Different kinds of learning strategies can be broadly classiﬁed into (i) spiking timing dependent plasticity (STDP), (ii) frequency dependent plasticity, and (iii) gradient- based learning. STDP induces both potentiation and depression of synaptic weights in anon-volatile fashion based on the difference in spike timing of pre-neurons and post-neurons, Dt. Classical STDP assumes an exponential relationship with Dt, as demon- strated by Bi and Poo. 19Other variants of STDP have also been observed in mamma- lian brains. Frequency dependent plasticity manifests itself in the form of short-term plasticity (STP) and long-term potentiation (LTP). The change in the synaptic weight, inthis case, depends on how frequently the synapse receives stimulus. STP and LTP formthe basis of short-term and long-term memory in biological systems. Finally, gradient- based learning is a supervised learning scheme where the change in the synaptic weight depends on gradients calculated from error between the predicted and the ideal output.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 021308 (2020); doi: 10.1063/1.5113536 7, 021308-6 Published under license by AIP Publishingadditional beneﬁts with respect to neuromorphic computing. First, the inherent physics of such devices can be exploited to capture the func- tionalities of biological neurons and synapses. Second, these devices can be connected in a crossbar fashion allowing analog-mixed signal in-memory computations, resulting in highly energy-efﬁcient hardware implementations. In this section, we ﬁrst delve into the possibilities and challenges of such non-volatile devices, based on various technologies, used to emulate the characteristics of synapses and neurons. Subsequently, we describe how crossbar structures of such non-volatile devices can be used for in-memory computing and the associated challenges. A. Phase change devices Phase change materials (PCMs) such as chalcogenides are the front-runners among emerging non-volatile devices—with speculationabout possible commercial offerings—for high density, large-scale storage solutions.31T h e s em a t e r i a l sc a ne n c o d em u l t i p l ei n t e r m e d i a t e states, rendering them the capability of storing multiple bits in a single cell. More recently, PCM devices have also emerged as a promising candidate for neuromorphic computing due to their multi-level stor- age capabilities. In this section, we discuss various neuromorphic applications of PCM devices. 1. PCM as neurons PCM devices show reversible switching between amorphous and crystalline states, which have highly contrasting electrical and optical properties. In fact, this switching dynamics can directly lead to inte- grate and ﬁring behaviors in PCM-based neurons. The device struc- ture of such a neuron comprises a phase change material sandwiched between two electrodes, as shown in Fig. 5(a) . The mushroom FIG. 5. (a) Device structure of a PCM-based IF neuron.29The thickness of the amorphous region (shown in red) represents the membrane potential of the neuron. The inte- grating and ﬁring behaviors for different incident pulse amplitudes and frequencies are shown (bottom). (b) Device structure of a photonic IF neuron based on PCM (GST).30 The input pulses coming through the INPUT port get coupled to the ring waveguide and eventually to the GST element, changing the amorphous thickness. T he output at the “THROUGH” port represents the membrane potential, which depends on the state of the GST element.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 021308 (2020); doi: 10.1063/1.5113536 7, 021308-7 Published under license by AIP Publishingstructure shows the shape of the switching volume just above the region known as the heater. The heater is usually made of resistive ele- ments such as W, and high current densities at the contact interface between the phase change material and the heater cause locally con-ﬁned Joule heating. When the PCM in the neuron is in its initial amor-phous state, a voltage pulse that has an amplitude low enough so as ton o tm e l tt h ed e v i c eb u th i g he n o u g ht oi n d u c ec r y s t a lg r o w t hc a nb eapplied. The resulting amorphous thickness, u a, on application of such a pulse is given as29 dua dt¼/C0vgðRthðuaÞPpþTambÞ;uað0Þ¼u0 (3) where vgis the crystal growth velocity dependent on the temperature determined by its argument RthðuaÞPpþTamb.H e r e , Rthis the thermal resistance and Tambis the interface temperature between amorphous and crystalline regions. The variable, ua,i nE q . (1)can be interpreted as the neuron’s membrane potential where Ppis the input variable controlling the dynamics. On successive application of crystallization pulses, the amorphous thickness, ua, decreases, leading to lower con- ductance and temporal integration of the membrane potential.Beyond a certain threshold conductance level, the neuron ﬁres, or inother words, the PCM changes to a crystalline state. A reset mecha-nism puts the neuron back in its original amorphous state. The afore-mentioned integrate-and-ﬁre characteristics in PCM neurons are accompanied by inherent stochasticity. The stochasticity arises from different amorphous states created by repeated resets of the neuron.Different initial states lead to different growth velocities, which resultin an approximate Gaussian distribution of inter-spike intervals, theinterval between adjacent ﬁring events. Populations of such stochasticIF neurons have also been used in detection of temporal correlation inparallel data streams. 32 Thus far, we have talked about electronic devices mimicking neu- ronal behavior using PCM. Such behavior can also be achieved with Si-based photonic devices with PCM embedded on top of them. 30Such a device is shown in Fig. 5(b) , which consists of a Si microring resonator on the SiO 2substrate with a phase change material, Ge 2Sb2Te5(GST), deposited on top of the ring waveguide. The membrane potential ofsuch a neuron or, in other words, the amorphous thickness of thePCM can be modulated by guiding laser pulses through Si waveguides. Light gets evanescently coupled to the PCM element and changes the thickness of the amorphous region, thereby allowing an optical IF neu-ron based on PCM elements, as shown in Fig. 5(b) (bottom). 2. PCM as synapses We have discussed the ability of PCM to store multiple bits in a single cell. This multi-level behavior of PCM-based devices makesthem a promising candidate to emulate synaptic characteristics. In addition, the large contrast in electrical properties allows for a signiﬁ- cantly high ON/OFF resistance ratio in PCM devices. The same two-terminal structure described in Fig. 5(a) can be used as a synaptic device. The programming of such a synapse is performed through thephase transition mechanism between amorphous and crystallinestates. Amorphization (or “RESET”) is performed by an abrupt melt-quench process, where high and short voltage pulses are applied toheat the device followed by rapid cooling such that the material solidi- ﬁes in the amorphous state. On the other hand, crystallization isperformed when an exponential current above the threshold voltage leads to heating of the material above its crystallization temperature and switches it to the crystalline state, as depicted by the I–Vcharac- teristics in Fig. 6(a) . The crystallization (or “SET”) pulses are much longer as opposed to amorphization (or RESET) pulses, as shown in Fig. 6(b) . Multiple states are achieved by progressively crystallizing the material, thus reducing the amorphous thickness. These multi-level PCM synapses can be used to perform unsu- pervised on-chip learning using the STDP rule. 33LTP and LTD using STDP involves a gradual increase and decrease in conductance of PCM devices, respectively. However, such a gradual increase or decrease in conductance needs to ensure precise control, which is difﬁ- cult to achieve using identical current pulses. As a result, by conﬁgur-ing a series of programming pulses of increasing or decreasing amplitude [ Fig. 7(a) ], both LTP and LTD have been demonstrated using PCM devices. 34–36In this particular scheme, the pre-spikes con- sist of a number of pulses of gradually decreasing or increasing pulses, whereas the post-spike consists of a single negative pulse. The differ-ence between the magnitude of the pre-spike and post-spike due to overlap of the pulses varies with the time difference, resulting in the change in conductance of the synapse following the STDP learning rule. The scheme for potentiation is explained in Fig. 7(a) . A simpliﬁed STDP learning rule with constant weight update can also be imple-mented using a single programming pulse by shaping the pulses appropriately 33as shown in Fig. 7(b) . However, such pulse shaping requires additional circuitry. These schemes rely on single PCM devi- ces representing a synapse. Alternatively, using a “2-PCM” synapse, one can potentially implement LTP and LTD characteristics that canbe independently programmed. Such a multi-device implementation becomes important for PCM technology as the amorphization is an abrupt process, and it is difﬁcult to control the progression of different amorphization states, which poses a fundamental limitation toward realizing both LTP and LTD in a single device. Visual pattern recogni-tion has been demonstrated using such 2-device synapses, which are able to learn directly from event-based sensors. 37While these works focus on asymmetric STDP, which forms the basis of learning spatio- temporal features, PCM synapses can also exhibit symmetric STDP based learning enabling associative learning.38As we had discussed about IF neurons, the difference in optical responsivity of PCMs can also lead to emulation of synaptic behavior on Si-photonic devices. The change in optical transmission in photonic synaptic devices arises from the difference in the imaginary part of the refractive index of PCMs in their amorphous and crystalline states. The gradual increase FIG. 6. (a)I–Vcharacteristics of PCM devices showing SET and RESET points for two states. (b) Pulsing schemes for SET and RESET processes to occur, showingthe temperatures reached due to the pulses.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 021308 (2020); doi: 10.1063/1.5113536 7, 021308-8 Published under license by AIP Publishingin the optical response of PCM elements by modulating the refractive index can be achieved through varying the number of programmingpulses. This has been exploited to experimentally demonstrate unsu- pervised STDP learning in photonic synapses. 39To scale beyond single devices, the rectangular waveguides used in this work can be replacedwith microring resonators to perform unsupervised learning in anatemporal fashion. 40 3. PCM crossbars We have thus far talked about isolated PCM devices mimicking the neuronal and synaptic behaviors. Interestingly, these devices canbe connected in an integrated arrangement to perform in-memory computations involving a series of multiply and-accumulate (MAC) operations. Such operations can be broadly represented as a multipli-cation operation between an input vector and the synaptic weightmatrix, which is key to many neural computations. Vector–matrixmultiplication (VMM) operations require multiple cycles in a standard von-Neumann computer. Interestingly, arranging PCM devices in a crossbar fashion (or in more general terms, arranging resistive memo-ries in a crossbar fashion) can engender a new, massively parallel para-digm of computing. VMM operation, which is otherwise a fairly cumbersome operation, can be performed organically through the application of Kirchoff’s laws as follows. This can be understoodthrough Fig. 8 , where each PCM device encodes the synaptic strength in the form of its conductance. The current through each device is pro-portional to the voltage applied and the conductance of the device.Currents from all the devices in a column get added in accordance with Kirchoff’s current law to produce a column-current, which is a result of the dot-product of the voltages and conductance. Such a dot- product operation can be mathematically represented as I j¼X iViGij; (4) where Virepresents the voltage on the i-th row and Gijrepresents the conductance of the element at the intersection of the i-th row and j-th columns. This ability of parallel computing within the memory array using single-element memory elements capable of packing multiple bits paves the way for faster, energy-efﬁcient, and high-storage neuro-morphic systems. In addition to synaptic computations, PCM crossbars can also be used for on-chip learning that involves dynamic writing into individ- ual devices. However, parallel writing to two-terminal devices in acrossbar is not feasible as the programming current might sneak toundesired cells, resulting in inaccurate conductance updates. To allevi- ate the concern of sneak current paths , two-terminal PCM devices are usually used in conjunction with a transistor or a selector. Such mem-ory cell structures are termed as “1T-1R” or “1S-1R” (shown in Fig. 8 ) and are extensively used in NVM crossbar arrays. Such 1T-1R crossbar arrays can be seamlessly used for on-line learning schemes such asSTDP. To that effect, PCM crossbars were used as one of the ﬁrst of itskind to experimentally demonstrate on-chip STDP based learning, 41,42 and simple pattern recognition tasks were conducted using the arrays. FIG. 7. (a) STDP learning in PCM synapses34by a series of pulses of increasing (decreasing) amplitude demonstrating LTP behavior (left) similar to neuroscientiﬁc experi- ments19(right). Reprinted with permission from Kuzum et al. , Nano Lett. 12(5), 2179–2186 (2012). Copyright 2012 American Chemical Society. (b) STDP learning effected due to overlap of appropriately shaped pulses.33Reprinted with permission from Ambrogio et al. , Front. Neurosci. 10, 56 (2016). Copyright 2016 Author(s), licensed under a Creative Commons Attribution (CC BY) license.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 021308 (2020); doi: 10.1063/1.5113536 7, 021308-9 Published under license by AIP PublishingAlthough these works focused on smaller scale crossbar arrays of size 10/C210, slightly modiﬁed 2T-1R memory arrays have also been explored for in situ learning on a much bigger scale.43Using two tran- sistors enables simultaneous LIF neurons and STDP learning charac- teristics in an integrated fashion. We have discussed how unsupervised STDP learning can be implemented using PCM crossbars. However, on-line learning usingSTDP requires complex programming schemes and is difﬁcult to scale to larger crossbars. On the other hand, networks trained with super- vised learning can be mapped on to much larger PCM crossbar arraysfor inferencing. These neural networks have been experimentally dem-onstrated to perform complex image recognition tasks 44,45with reason- able accuracy. Note that for these works, the supervised learning schemes were implemented with software and the PCM crossbars were used for forward propagation both during training and inferencing. We have discussed how PCM crossbars leverage Kirchoff’s laws to perform neuro-synaptic computations in the electrical domain. In theoptical domain, however, the dot-product operation can be implemented using wavelength-division-multiplexing (WDM). 40,46The input is encoded in terms of different wavelengths, and each synaptic devicemodulates the input of a particular wavelength. The resulting sum is fedto an array of photo-detectors to realize the dot-product operation. PCM technology shows remarkable scalability and high-storage density, making them amenable to efﬁcient neuromorphic systems.However, further material and device research is necessary to truly realize the full potential of PCM-based neuromorphic accelerators. First, the most common PCM devices are based on the chalcogenide material group comprising elements Ge, Sb, and Te due to their high optical contrast, repeatability, and low reﬂectivity. In the GeSbTe sys- tem ranging from GeTe to Sb 2Te3,G e 2Sb2Te5has been identiﬁed as the optimum material composition47,48based on the trade-offs between stability and switching speed. Despite this development, PCMs suffer from signiﬁcantly high write power due to their inherent heat dependent switching and high latency. Second, PCM devices suf- fer from the phenomenon of resistance drift, which is more pro- nounced for high resistance states (HRSs). The resistance drift is the c h a n g ei nt h ep r o g r a m m e dv a l u eo ft h er e s i s t a n c eo v e rt i m ea f t e rp r o - gramming is completed. This has been attributed to structural relaxa- tions occurring shortly after programming.49–51The effect of drift on neural computing has been studied, and possible mitigation strategies have been proposed.52However, the inability to reliably operate PCM devices at high resistance states has an impact on large-scale crossbar operations. In light of these challenges, it is necessary to investigate newer materials that offer more stability and lower switching speeds for efﬁcient and scalable neuromorphic systems based on PCM devices. B. Metal-oxide RRAMs and CBRAMs An alternative class of materials to PCMs for memristive systems are perovskite oxides such as SrTiO 3,53SrZrO 3,54Pr0:7Ca0:3MnO 3 (PCMO),55and binary metal oxides such as HfO x,56TiO x,57and TaO x,58which exhibit resistive switching with lower programming voltages and durations. Such resistive switching is also observed when the oxide is replaced by a conductive element. Two-terminal devices based on these materials form the base of Resistive Random Access Memories (RRAMs). The devices with oxides in the middle are known as metal-oxide RRAMs, whereas the ones with conductive elements are known as the Conductive Bridge RAM (CBRAM). Although the internal physics of these two classes of resistive RAMs is slightly differ- ent, both kinds of devices have a similar behavior and hence applica- bility. In the initial years of research, RRAM was envisaged to be a non-volatile high-density memory system along with CMOS- compatible integration. With signiﬁcant development over the years, various other applications leverage the non-volatility of RRAMs for power and area-efﬁcient implementations. Among these, neuromor- phic computing is a dominant candidate, which exploits the multi- level capability and the analog memory behavior of RRAMs to emulate neuro-synaptic functionalities. In this section, we will discuss how RRAMs can directly mimic neuronal and learning synaptic behaviors using single devices. 1. Metal-oxide RRAMs and CBRAMs as neurons The dynamics of a voltage driven metal-oxide RRAM device was ﬁrst investigated by HP labs in their iconic work on TiO 2,w h i c hi d e n - tiﬁed the ﬁrst device61showing the characteristics of a memristor, pre- dicted by Chua in 1971.62The oxide material can be conceptually split into two regions, a conductive region and an insulating region. The conductance of such a device can be given by its state variable, w, which varies as FIG. 8. Synaptic devices arranged in a crossbar fashion along with selector devices to perform dot-product operations. The input voltages are applied to the differentrows of the crossbars, and the current from each column represents the dot-product, I j¼PViWij, between the input voltages and the conductance, W, of the devices.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 021308 (2020); doi: 10.1063/1.5113536 7, 021308-10 Published under license by AIP PublishingI¼gMðw=LÞVðtÞ;dw dt¼fðwðtÞ;VðtÞÞ: (5) Interestingly, the RRAM device can be used in an integrator circuit as a resistor in parallel to an external capacitance, as shown in Fig. 9 (top), to emulate the LIF characteristics where the conductance of the device can be used as an internal variable.59When the memristor is in its OFF state, the current through the circuit is low, and hence, it doesnot output a spike. Once the memristor reaches its ON state, thecurrent suddenly jumps, which can be converted to analog spike. Thevoltage across the memristor, in that case, obeys the dynamics of a LIFneuron, given by Eq. (1)in Sec. II A. A similar neuron circuit has also been explored for CBRAM devices based on Cu =Ti=Al 2O360[Fig. 9 (bottom)]. Unlike PCMs, to emulate the differential equations of theLIF neuron, an R-Ccircuit conﬁguration is used. If the leaky behavior is not required, the internal state of the neuron or the membranepotential can be directly encoded in the oxygen concentration in the device. By manipulating the migration of oxygen vacancies using post-synaptic pulses, IF neurons can be realized by oxide-baseddevices. 63To that effect, oxide-based devices have been used to design common neuronal models involving leaky behavior, such asthe Hodgkin–Huxley model and leaky IF model. 64 2. Metal-oxide RRAMs and CBRAMs as synapses Much like PCM devices, RRAM devices can also be programmed to multiple intermediate states between the two extreme resistancestates, which are known as the high resistance state (HRS) and the lowresistance state (LRS). This capability of behaving as an analog mem-ory makes RRAMs suitable for mimicking synaptic operations in neu-ral networks. The physics behind emulating such synaptic behaviorrests on soft di-electric breakdown in metal-oxide RRAM devices anddissolution of metal ions in CBRAM devices. The device structure fora metal-oxide RRAM is shown in Fig. 10(a) . In the case of the metal-oxide RRAM, the switching mechanisms can be categorized as (a) ﬁla- mentary and (b) non-ﬁlamentary. The ﬁlamentary switching results due to the formation and rupture of ﬁlamentary conductive paths dueto thermal redox reactions between metal electrodes and the oxide material. The “forming” or SET process occurs at a high electric ﬁeld due to the displacement and drift of oxygen atoms from the lattice.These oxygen vacancies form localized conductive ﬁlaments, which form the basis of ﬁlamentary conduction in RRAM devices. The form- ing voltage can be reduced by thinning down the oxide layer 65and controlling annealing temperatures during deposition.66The RESET mechanism, on the other hand, is well debated, and ionic migration has been cited as the most probable phenomenon.67,68Au n i ﬁ e dm o d e l of RESET proposes that the oxygen ions that drifted to the negative electrode causes the insulator/anode interface to act as a “oxygen reser- voir.”69Oxygen ions diffuse back into the bulk due to a concentration gradient and possibly recombine with the vacancies that form the ﬁla- ment such that material moves back to the HRS. The I–Vcharacteris- tics are shown in Fig. 10(b) where varying SET and RESET pulses lead to different resistance states. In order to emulate synaptic behavior through analog memory states in ﬁlamentary RRAMs, various pro-gramming techniques have been explored. For example, the SET cur- rent compliance can be used to modulate the device resistance by determining the number of conductive ﬁlaments. On the other hand,varying the external stimulus can control the degree of oxidation at the electrode and oxide interface, resulting in a gradual change in resis- tance. 70These analog states in RRAM devices can be exploited to per- form learning on devices using various pulsing techniques. To that effect, the time dependence of synaptic conductance change in STDP learning can be induced by manipulating the shapes of pre-synapticand post-synaptic voltage waveforms, 71,72shown in Fig. 11(a) .S i m i l a r to programming PCM devices, a gradual increase or decrease in con- ductance can be achieved using a succession of identical pulses as well,as shown in the ﬁgure. Such a pulsing scheme, despite requiring a more number of pulses, provides a more granular control over the synaptic conductance, 73,74shown in Fig. 11(b) . Furthermore, adding more peripheral transistors to programming circuits can further enable precise control over STDP. For example, a 2T/1R synapse uses the overlapping window of two different pulses to generate program-ming current to induce time-dependent LTP and LTD. 75I nt h ec a s eo f ﬁlamentary RRAMs, variability in the forming process induces sto- chasticity in resistive switching, which can be leveraged to design sto-c h a s t i c a l l yl e a r n i n gs y n a p s e s .T h es w i t c h i n gp r o b a b i l i t yc a nb e FIG. 9. (a) RRAM59and (b) CBRAM60neuron circuits showing the memristive device RN(below) or RON=OFF (top) in parallel to a capacitor to emulate LIF characteristics. FIG. 10. (a) Basic device structure for RRAM devices consisting of a metal-oxide layer sandwiched between two electrodes. (b) I–Vcharacteristics showing varying SET and RESET points, leading to different resistance states.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 021308 (2020); doi: 10.1063/1.5113536 7, 021308-11 Published under license by AIP Publishingcontrolled by using a higher pulse amplitude. Stochastic synapses have the ability to encode information in the form of probability, thus achieving signiﬁcant compression over deterministic counterparts. Learning stochastically using binary synapses has been demonstratedto achieve pattern learning. 76Unsupervised learning using multi-state memristors can also be performed probabilistically to yield robust learning against corrupted input data.77 Oxides of some transition metals, such as Pr 0:7Ca0:3MnO 3 (PCMO), exhibit non-ﬁlamentary switching as well. This type of switching, on the other hand, results from several possible phenomenasuch as charge-trapping or defect migration at the interface of metal and oxide, which end up modulating the electrostatic or Schottky bar- rier. Although the switching physics in non-ﬁlamentary RRAM devi-ces is different from that in ﬁlamentary RRAMs, the fundamentalbehavior of using these RRAM devices as synapses is quite similar.Non-ﬁlamentary RRAMs can also be programmed using different voltage pulses to exhibit multi-level synaptic behavior. Moreover, vary- ing pulse widths can instantiate partial SET/RESET characteristics,which have been used to implement STDP characteristics in RRAMsynapses. 78,79By encoding the conductance change using the number of pulses coupled with appropriate waveform engineering can enable various kinds of STDP behaviors, explained in Sec. II B, of isolatedRRAM devices showing non-ﬁlamentary switching.80In addition to long-term learning methods, RRAM devices with controllable volatil- ity can also be used to mimic frequency dependent learning, thus enabling a transition from short-term to long-term memory.81By con- trolling the frequency and amplitude of the incoming pulses, STP-LTPcharacteristics have been achieved in WO 3based RRAM synapses.82 In general, higher amplitude pulses in quick succession are required totransition the device from decaying weights to a more stable persistentstate. Such metastable switching dynamics of RRAM devices havebeen used to perform spatiotemporal computation on correlatedpatterns. 83 Thus far, we have discussed how metal-oxide RRAM devices can emulate synaptic behavior. Next, we will discuss CBRAM devices,which also exhibit similar switching behavior by just replacing theoxide material with an electrolyte. The switching mechanism is analo-gous to ﬁlamentary RRAM except that the ﬁlament results in a metal- lic conductive path due to electro-chemical reactions. This technology has garnered interest due to its fast and low-power switching. MostCBRAM devices are based on Ag electrodes where resistive switchingbehavior is exhibited due to the contrast in conductivity in Ag-richand Ag-poor regions. The effective conductance of such a device can be written as 88 FIG. 11. (a) Appropriately shaped pulses representing the post-synaptic and pre-synaptic potential.72The overlap between the two pulses in time leads to STDP learning char- acteristics in the form of the writing current ﬂowing through the device. Reprinted with permission from Rajendran et al. , IEEE Trans. Electron Devices 60(1), 246–253 (2012). Copyright 2013 IEEE. (b) STDP characteristics can also be emulated by passing multiple pulses, repetitively.74Reprinted with permission from Wang et al. ,i n2014 IEEE International Electron Devices Meeting (IEEE, 2014), p. 28. Copyright 2014 IEEE.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 021308 (2020); doi: 10.1063/1.5113536 7, 021308-12 Published under license by AIP PublishingGeff¼1 RONwþROFFð1/C0wÞ; (6) where wdeﬁnes the normalized position of the end of the conducting region at the interface of Ag-rich and Ag-poor regions. The conduc- tance of such a device can also be gradually manipulated to implement STDP using a succession of pulses.88Here, the exponential depen- dence on spike timing is implemented using time-division multiplex- ing where the timing information is encoded in the pulse width. CBRAM based STDP learning has been implemented on-chip using CMOS integrate-and-ﬁre neurons.89As with ﬁlamentary RRAM devi- ces, stochastic behavior in CBRAM devices can also enable low-power probabilistic learning. One such implementation uses the recency of spiking as a measure of manipulating the probability of the device for visual and auditory processing.90Some CBRAM devices also exhibit decay in conductance, which can be leveraged to implement short- term plasticity. Ag2S based synapses also show the properties of sen- sory memory, wherein conductance does not change for some time, before exhibiting STP.91 3. Metal-oxide RRAM and CBRAM crossbars RRAMs are two-terminal devices, similar to PCMs. Hence, like PCMs, RRAM devices can also be arranged into large-scale resistive crossbars, shown in Fig. 8 , for building neuromorphic systems. RRAM crossbar arrays can be integrated seamlessly with CMOS circuits for hybrid storage and neuromorphic systems. To that effect, a 40 /C240 array with CMOS peripheral circuits has been demonstrated to reliably store complex bitmap images.92Such an experimental demonstration is a testimony to the scalability of RRAM crossbars. Leveraging this scalability, studies have proposed RRAM crossbar arrays to perform in situlearning in single layer neural networks.93,94This scalability has been corroborated by the recent development in the process technol- ogy, which have led to the realization of large crossbars of sizes up to 128/C2128 to perform image processing tasks95andin situ learning for multi-layer networks.45The aforementioned works focus on using RRAM as an analog memory. To achieve more stability, RRAM cross- bar arrays have also been used as binary weights in a scalable and par- allel architecture85to emulate a large-scale XNOR network.96Both PCM and RRAM crossbars have been extensively explored at an array-level, and Table I provides a comparative study of different experimental demonstrations. It should be understood that large-scale RRAM crossbars have been primarily explored for non-spiking type networks; however, the compute primitives can be easily ported to realize spike-based computing. We will later discuss NVM architec- tures based on these RRAM crossbars, which show immense potentialto achieve energy-efﬁciency and high density compared to standard CMOS-based computing. Thus far, we have discussed metal-oxide RRAM crossbar arrays. From a scalability point of view, CBRAM crossbars exhibit similartrends. To that effect, high-density 32 /C232 crossbar arrays based on Ag–Si systems have been experimentally demonstrated, which can be potentially used to build neuromorphic circuits. Simulation studiesbased on such Ag–Si systems show signiﬁcant potential of using large-scale crossbars for image classiﬁcation tasks. 97 Of the two classes of materials belonging to the RRAM family, metal-oxide RRAM devices have been more dominantly explored inthe context of developing large-scale neuromorphic circuits. However, despite signiﬁcant progress, RRAM-based devices suffer from signiﬁ- cant variability, particularly in the ﬁlament formation process. On theother hand, non-ﬁlamentary RRAM devices, being barrier-dependent,may lead to trade-offs between stability and programming speed. Overall, further material research is crucial toward making RRAMs viable for large-scale neuromorphic systems. C. Spintronic devices Akin to other non-volatile technologies, spin based devices were conventionally investigated as a non-volatile replacement for the exist- ing silicon memories. What makes spin devices particularly unique as compared to other non-volatile technologies is their almost unlimitedendurance and fast switching speeds. It is therefore not surprising thatamong various non-volatile technologies, spin devices are the only ones that have been investigated and have shown promise as on-chip cache replacement. 98With respect to neuromorphic computing, it is the rich device physics and spin dynamics that allow efﬁcient mappingof various aspects of neurons and synapses into a single device. As we will discuss in this section, spintronics brings in an alternate paradigm in computing by using electron spin as the memory storage variable.The fact that spin dynamics can be controlled by multiple physicsincluding current induced torques, 99domain wall motion,100voltage based spin manipulation,101and elastic coupling adds to the rich device possibilities with spintronics and their applications to neuro- morphic computing. In this section, we would describe key representa-tive works with spin devices showing their applicability as IF-, LIF-,and stochastic neurons, and synaptic primitives. 1. Spin devices as neurons As mentioned earlier, it is the rich spin dynamics that allows mapping of different aspects of biological neurons using a single device. In fact, the simplest and the most well-known spin device—the two-terminal Magnetic Tunnel Junction (MTJ)—can be seen as a TABLE I. NVM Technologies. Technology PCM45RRAM84RRAM85RRAM86RRAM87 Crossbar size 512 /C2512 108 /C254 128 /C2128 128 /C216 512 /C2512 ON/OFF ratio 10 5 N/A 10 N/AArea per operation ( lm 2) 22.12 24 0.05 31.15 N/A Latency (ns) 80 10 13.7 0.6 9.8Energy-efﬁciency (TOPS/W) 28 1.37 141 11 121.38Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 021308 (2020); doi: 10.1063/1.5113536 7, 021308-13 Published under license by AIP Publishingstochastic-LIF neuron. MTJs are composed of two ferromagnetic (FM) nanomagnets sandwiching a spacer layer105as shown in Fig. 12(a) . Nanomagnets encode information in the form of the direction of magnetization and can be engineered to stabilize in two oppositedirections. The relative direction of the two FMs—parallel (P) vs anti-parallel (AP)—results in two distinct resistive states—LOW vs HIGH resistance. Switching the MTJ from the P to the AP state or vice versa can be achieved by passing a current through the MTJ, resulting intransfer of torque from the incoming spins to the FMs. Interestingly,the dynamics of the spin under excitation from a current induced tor-que can be looked upon as a stochastic-LIF dynamics. Mathematically, t h es p i nd y n a m i c so fa nF M ,s h o w ni n Fig. 12(b) , can be expressedeffectively using the stochastic-Landau–Lifshitz–Gilbert–Slonczewski (s-LLGS) equation, @^m @t¼/C0 j cjð^m/C2HEFFÞþa^m/C2@^m @t/C18/C19 þ1 qNsð^m/C2Is/C2^mÞ 1þa2 c@^m @t/C18/C19 ¼/C0 ð ^m/C2HEFFÞþað^m/C2^m/C2HEFFÞ þ1 qNsð^m/C2Is/C2^mÞ (7) where ^mis the unit vector of free layer magnetization, cis the gyro- magnetic ratio for the electron, ais Gilbert’s damping ratio, and HEFF FIG. 12. (a) MTJ-based neuron102showing the device structure (top) and leaky-integrate characteristics (bottom). Sengupta et al. , Sci. Rep. 6, 30039 (2016). Copyright 2016 Author(s), licensed under a Creative Commons Attribution (CC BY) license. The magnetization of the free layer of the MTJ integrates under the inﬂuenc e of incoming current pulses. (b) ME oxide-based LIF neuron103showing the device structure (top) and LIF characteristics (bottom). Reproduced with permission from Jaiswal et al. , IEEE Trans. Electron Devices 64(4), 1818–1824 (2017). Copyright 2017 IEEE. (c) SHE-MTJ-based stochastic neuron102showing the device structure (top) and the stochastic switching characteristics (bottom). Reprinted with permission from Sengupta et al. , Sci. Rep., 6, 30039 (2016); Copyright 2016 Author(s), licensed under a Creative Commons Attribution (CC BY) license. (d) DWM-based IF spiking neuron104showing the device structure (top) and integration and ﬁring behavior (bottom) over time. For incident input spikes, the domain wall moves toward the MTJ at the end, thus decreasing the resistance of the device. When the domain wall is at the end, the resistance reaches its l owest, enough for the neuron ﬁres. Reproduced with permission from Sengupta and Roy, Appl. Phys. Rev. 4(4), 041105 (2017). Copyright 2017 AIP Publishing.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 021308 (2020); doi: 10.1063/1.5113536 7, 021308-14 Published under license by AIP Publishingis the effective magnetic ﬁeld including the shape anisotropy ﬁeld, external ﬁeld, and thermal ﬁeld. This equation bears similarities withthe leaky-integrate-and-ﬁre behavior of a neuron. The last term repre- sents the spin transfer torque (STT) phenomenon, which causes the magnetization to rotate by transferring the torque generated throughthe change in angular momentum of incoming electrons. Interestingly, the ﬁrst two terms can be related to the “leak” dynamics in an LIF neuron, while the last term relates to the integrating behaviorof the neuron as follows. When an input current pulse or “spike” is applied, the magnetization starts integrating or precessing toward the opposite stable magnetization state owing to the STT effect (last term).In the absence of such a spike, the magnetization leaks back toward the original magnetization state (Gilbert damping, second term). Furthermore, due to nano-scale size of the magnet, the switchingdynamics is a strong function of a stochastic thermal ﬁeld, leading to the stochastic behavior. This thermal ﬁeld can be modeled using Brown’s model. 106In terms of Eq. (7), the thermal ﬁeld can be incor- porated into HEFFas a magnetic ﬁeld, Hthermal ¼fﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2akT jcjMsVs ; (8) where fis a zero mean, unit variance Gaussian random variable, V is the volume of the free layer, T is the temperature, and k is theBoltzmann constant. A typical, stochastic-LIF behavior using MTJ is shown in Fig. 12(a) . 102While the two-terminal MTJ does repre- sent the stochastic-LIF dynamics, the very fact that the leaky andintegrate behaviors are controlled by intricate device physics and intrinsic material parameters makes it difﬁcult to control as needed for a large-scale circuit/system implementation. As a result, alter-nate physics such as the magneto-electric switching (ME) has been proposed as stochastic-LIF neurons, wherein the leaky and inte- grating behaviors can be easily controlled through device dimen-sions and associated circuit elements. In ME devices, the voltage induced electric ﬁeld polarization induces a magnetic ﬁeld at the interface of the FM and ME oxide, which induces switching of theFM layer. The ME oxide layer acts a capacitor, and a series resis- tance can enable LIF neuronal dynamics in such a device. The ME switching process is susceptible to noise like the conventional MTJswitching and hence inherently mimics the stochastic dynamics with the LIF behavior. 103By controlling the ME oxide dimension inFig. 12(b) and/or the leaky resistive path, the LIF dynamics can be easily tweaked as per requirement. In essence, we have seen that both current induced MTJ and voltage driven ME switching can act as stochastic-LIF neurons. However, on one hand, currentbased MTJ is difﬁcult to control, while on the other hand, ME switching is still in its nascent stage of investigation and needs extensive material research for bringing the device to mainstreamapplications. Alternatively, at the cost of reduced dynamics, three terminal Spin-Orbit-Torque MTJ (SOT-MTJ) has been used as a reliable sto- chastic spiking neuron while neglecting the leaky-integrate dynam-ics. 102SOT-MTJ is reasonably mature, and also its three terminal nature brings in attractive circuit implications. First, SOT-MTJ is switched by passing a bi-directional current through a heavy-metal(HM) layer, as shown in Fig. 12(c) . When a charge current enters the HM, electrons of opposite spins get scattered to the opposite sides ofthe layer, and a spin-polarized current is generated, which rotates the magnetization in the adjacent MTJ such that the switching probability increases as the magnitude of the input current is increased. This in turn implies that the incoming current passes through a much lower metal resistance and sees a constant metal resistance throughout the switching process as opposed to current based switching in conven- tional two-terminal MTJs. As we will see later, the existence of a low input resistance for the neuron allows easy interfacing with synaptic crossbar arrays. Second, the decoupled read-write path in SOT-MTJs allows for independent optimization of the read (inferencing) and write (learning) paths. A typical SOT-MTJ and its sigmoid-like stochastic switching behavior are shown in Fig. 12(c) . While the aforementioned behaviors depicted in Fig. 12(c) correspond to an SOT-MTJ with a high energy-barrier (10–60 kT), telegraphic SOT-MTJ with an energy-barrier as low as 1 kT has also been explored as stochastic neurons. 107 In addition to smaller magnets, wherein the entire magnet switches like a giant spin, longer magnets known as domain wall mag- nets (DWMs)108have been used as IF neurons. DWMs consist of two oppositely directed magnetic domains separated by a domain wall [see Fig. 12(d) ]. Electrons ﬂowing through the DWM continuously exchange angular momentum with the local magnetic moment. Current induced toque affects the misaligned neighboring moments around the domain wall region, thus displacing the domain wall along the direction of current ﬂow. The instantaneous membrane potential is encoded in the position of the domain wall, which moves under the inﬂuence of post-synaptic input current. The direction of movement is determined by polarity of the incident current. The resulting magnetic polarity can be sensed by stacking a MTJ at an extremity of the DWM, and subsequent thresholding is performed when the domain wall reaches that extremity. The leak functionality in such a neuron can be implemented by passing a controlled current in the opposite direction. A constant current driven leak would result in increased energy con- sumption; as such, voltage driven DWMs based on elastic coupling can be used to reduce the energy consumption.109However, a concern with DWM-based neuromorphic devices is that the motion of domain walls might be pinned by the presence of defects.110To that effect, magnetic skyrmions promise enhanced stability and has been explored in the context of emulating neuromorphic behavior.111In summary, we have described multiple devices and their physics and extent of bio-ﬁdelity, wherein spin is used as the basic state variable. Let us now consider the applicability of spin devices as synaptic elements (Fig. 13 ). 2. Spin devices as synapses Recall that, for PCM and RRAM devices, the existence of multi- ple non-volatile resistance states between the two extreme HIGH andLOW resistances makes them suitable as synaptic elements. On similar lines, spin devices can be engineered to enable a continuous analog resistive stable state between its AP (HIGH) and P (LOW) resistances. This is achieved by stacking an MTJ over DWMs. The position of the domain wall determines the resistance state of the device. In extreme cases, the magnetization direction of the entire DWM aligns with that of the pinned layer, resulting in a LOW resistance state of the device, shown in Fig. 13 . Conversely, the magnetization direction of the DWM in the opposite direction to that of the pinned layer leads to anApplied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 021308 (2020); doi: 10.1063/1.5113536 7, 021308-15 Published under license by AIP PublishingAnti-Parallel (AP) conﬁguration, which deﬁnes the HIGH resistance state of the device. With respect to the position of the domain wall, x, the resistance of the device changes as Geq¼GPx LþGAPL/C0x LþGDW: (9) Here, GP(GAP) is the conductance of the MTJ when the domain wall is at the extreme right (left) of the DWM. GDWis the conductance of the domain wall region, and Lis the length of the DWM. Owing to low write currents, synaptic elements based on DWM devices113can achieve orders of magnitude lower energy consumption over corre- sponding realizations in other non-volatile technologies. Similar to spin neurons, inducing switching using the Spin-Hall effect (SHE)through a heavy-metal below the MTJ, the programming current can be further reduced. DWM-based devices have been explored to mimic the behavior of multi-level synapses in works such as Ref. 114.W i t ha few extra transistors, STDP learning can be enabled by a relatively sim- ple programming scheme as shown in Fig. 13 . 112This scheme lever- ages the exponential characteristics of transistors in the sub-threshold regime. A linearly increasing voltage is applied to the gate of the tran- sistor, M STDP, which is activated when the pre-neuron spikes. When the post-neuron ﬁres, an appropriate programming current passes through the HM layer, which now depends exponentially to the timing difference due to the sub-threshold operation. It is worth noting that although the DWM provides a way to encode multiple stable states in spin devices, the key drawback of such devices is the extremely limitedHIGH–LOW resistance range. The resistance range for spin devices is much lower than their PCM and RRAM counterparts. Encoding mul- tiple states within the constrained resistance range raised functionality concerns considering variability. Alternatively, non-domain wall devices such as two-terminal MTJs of three terminal SHE based MTJs can be used as synapses. In the absence of DWMs, MTJs can only encode binary information, i.e., two resistance states. In such a scenario, stochasticity can play an inter- esting role in realizing multi-level behavior by probabilistic switching.In spin devices, such thermally induced stochasticity can be effectivelycontrolled by varying the amplitude or duration of the programming pulse as shown in Fig. 12(c) . This beneﬁt of controlled stochasticity leads to energy-efﬁcient learning in binary synapses implemented using MTJs. 115,116An advantage of on-chip stochastic learning is that the operating currents are lower than the critical current for switching,thus ensuring low-power operations. Such multiple stochastic MTJscan be represented as a single synapse to achieve an analog weight spectrum. 117These proposals of stochastic synapses based on MTJs have shown applications of pattern recognition tasks on a handwrittendigit dataset. Finally, the precessional switching in the free FM layer in the MTJ inherently represents a dependence of switching on the frequency on programming inputs. On the incidence of a pulse, the magnetiza- tion of the free FM layer moves toward the opposite stable state.However, if the pulse is removed before the switching is completed, it reverts back to its original stable state. These characteristics can be used to represent volatile synaptic learning in the form of STP-LTPdynamics. 118 3. Spintronic crossbars Synapses based on 2-terminal MTJs can be arranged in a crossbar fashion, similar to other memristive technologies. The currents ﬂowing through the MTJs of each column get added in the crossbar and repre- sent the weighted sum of the inputs. Unlike the two-terminal devices,SHE based MTJs, being 3-terminal devices, have decoupled read and write paths. As a result, they require a modiﬁed crossbar arrangement. One major advantage of spin neurons is that current through the syn-aptic crossbars can be directly fed to the current controlled spin neu-rons. As discussed earlier, spin devices suffer from very low ON/OFF resistance ratios compared to other technologies. Hence, despite exper- imental demonstration of isolated synaptic spin devices, 119large-scale crossbar-level neuromorphic implementations have been mostly lim- ited to simulation studies. Such simulation studies have been based on reasonable ON/OFF ratios considering a predictive roadmap.120To that effect, multi-level DWM-based synapses have been arranged in a FIG. 13. STDP learning scheme in the DWM-based spin synapse112using peripheral transistors. The exponential characteristics of STDP are realized by operating MSTDP in the sub-threshold region and applying a linearly increasing voltage at its gate. MSTDP is activated when a pre-neuron spikes, and the programming current (shown in blue) through the transistor is injected into the HM layer (grey) when a post-neuron spikes. Reproduced with permission from Sengupta et al. , Phys. Rev. Appl. 6(6), 064003 (2016). Copyright 2017 American Physical Society.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 021308 (2020); doi: 10.1063/1.5113536 7, 021308-16 Published under license by AIP Publishingcrossbar fashion to emulate large-scale neural networks, both in a fully connected form114and as convolutional networks.121In addition to inferencing frameworks based on spin synapses, STDP based learn- ing112has also been explored at an array-level, as shown in Fig. 14 ,t o perform feature recognition and image classiﬁcation tasks. As dis- cussed earlier, MTJ-based binary synapses require stochasticity for effective learning. They can leverage the inherent stochasticity in the network, and a population of such synapses can perform on-line learn- ing, which not only achieves energy-efﬁciency but also enables extremely compressed networks.116 These simulation-based designs and results show signiﬁcant promise for spin based neuromorphic systems. However, several tech- nological challenges need to be overcome to realize large-scale systems with spin. As alluded to earlier, the ON/OFF ratio between the two extreme resistance states is governed by the TMR of the MTJ, which has been experimentally demonstrated to reach 600% (Ref. 122), lead- ing to an ON/OFF ratio of 7. This is signiﬁcantly lower than other competitive technologies and poses a limitation on the range of synap- tic weight representation at an array level. Second, MTJs can only rep- resent binary information. For multi-bit representation, it is necessary to use domain wall devices or multiple binary MTJs at the cost of area density. However, since synapses in the neural networks usually encode information in an analog fashion, the lack of multi-state representation in MTJs can potentially limit the area-efﬁciency of non-volatile spin devices for neuromorphic applications. The lack of multi-bit precision can be alleviated with architectural design facets such as “bit-slicing.” This involves multiple crossbars with binary devi- ces to represent multiple bits of storage. Despite such provisions, improved sensing circuits along with material exploration to achieve higher TMR is necessary to truly realize the potential of spin devicesas a viable option to emulate synaptic behavior for large-scale neuro- morphic systems. D. Ferroelectric FETs Similar to the phase change and ferromagnetic materials, another member of functional material family is ferroelectric (FE) materials. In addition to being electrically insulating, ferroelectric materials exhibit non-zero spontaneous polarization (P), even in the absence of an applied electric ﬁeld (E). By applying an external electric ﬁeld (more than a threshold value, called the coercive ﬁeld), the polarization direc- tion can be reversed. Such an electric ﬁeld driven polarization switch- ing behavior of FE is highly non-linear (compared to di-electric materials) and exhibits non-volatile hysteretic characteristics. Due to the inherent non-volatile nature, FE based capacitors have been histor- ically investigated for non-volatile memory elements. However, in fer- roelectric ﬁeld effect transistors (FEFETs), an FE layer is integrated at the gate stack of a standard transistor and thus offers all the beneﬁts of CMOS technology in addition to several unique features offered by FE. The FE layer electrostatically couples the underlying transistor. Due to such coupling, FEFETs offer non-volatile memory states by vir- tue of polarization retention of FE. Beside CMOS process compatibil- ity, one of the most appealing features of FEFET based memory is the ability of voltage based READ/WRITE operation, which is unlike the current based READ/WRITE schemes in other non-volatile memory devices (spin based memory and phase change memory). Due to the non-volatility and the intricate polarization switching dynamics of FE, FEFETs have garnered immense interest in recent times as a potential candidate for neuron-mimicking and multi-bit synaptic devices. In FIG. 14. A crossbar arrangement of spintronic synapses connected between pre-neurons A and B and post-neurons C and D, showing peripheral circuits for enabli ng STDP learning.112Reproduced with permission from Sengupta et al. , Phys. Rev. Appl. 6(6), 064003 (2016). Copyright 2017 American Physical Society.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 021308 (2020); doi: 10.1063/1.5113536 7, 021308-17 Published under license by AIP Publishingthis section, we will brieﬂy discuss the recent progress in FEFET based neuro-mimetic devices. 1. FEFETs as neurons The dynamics in a ferroelectric FET device can be used to mimic the functionality of a biological neuron. In a scaled FEFET, if identical sub-threshold pulses (“sub-coercive” in the context of FE) are applied at the gate terminal [shown in Fig. 15(a) (leftmost)], the device remains in the OFF state (since the sub-threshold pulses are insufﬁ- cient for polarization switching). However, after a certain number of pulses are received, the FEFET abruptly switches to the highly conduc-tive state [ Fig. 15(a) (rightmost)]. Such phenomena can be understood as the initial nucleation of nano-domains followed by an abrupt polari- z a t i o nr e v e r s a lo ft h ee n t i r eg r a i nc o n n e c t i n gt h es o u r c ea n dd r a i no f FEFETs. Before the critical threshold is reached, the nucleated nano- domains are not capable of inducing a signiﬁcant charge inversion in the channel, leading to the absence of the conduction path (OFF state). The accumulative P-switching presented in Ref. 125appears to be invariant with respect to the time difference between the consecutive excitation pulses, and therefore, the device acts as an integrator. Moreover, the ﬁring dynamics of such FEFET based neurons can be tuned by modulating the amplitude and duration of the voltage pulse. 123,125However, to implement the leaky behavior, a proposed option is to modulate the depolarization ﬁeld or insertion of a negative inhibit voltage in the intervals between consecutive excitation pulses. Apart from this externally emulated leaky process, an intrinsically leaky (or spontaneous polarization relaxation) process has beentheoretically predicted in Ref. 126. Such spontaneous polarization relaxation has been attributed as the cause of domain wall instabil- ity,126and such a process has recently been experimentally demon- strated in an Hf xZr1-xO2(HZO) thin-ﬁlm.127By harnessing such a quasi-leaky behavior along with the accumulative and abrupt polariza- tion switching in FE, a quasi-leaky-integration-ﬁre (QLIF) type FEFET based neuron can offer an intrinsic homeostatic plasticity. Network level simulations utilizing the QLIF neuron showed a 2.3 /C2reduction in the ﬁring rate compared to the traditional LIF neuron while main- taining the accuracy of 84%–85% across varying network sizes.127 Such an energy-efﬁcient spiking neuron can potentially enable ultra-low-power data processing in energy constrained environments. 2. FEFETs as synapses We have seen how the switching behavior of a FEFET can mimic the behavior of a biological neuron. The switching behavior also pro- duces bi-stability in FEFETs, which makes them particularly suitable for synaptic operations. The bi-stable nature of spontaneous polariza- tion of ferroelectric materials causes voltage induced polarization switching characteristics to be intrinsically hysteretic. The device struc- ture of a FEFET based synapse is similar to a neuronal device as shown inFig. 15(b) (leftmost). The FE electrostatically couples with the underlying transistor. Due to such coupling, FEFETs offer non-volatile memory states by virtue of polarization retention of the ferroelectric (FE) material. In a mono-domain FE (where the FE area is comparable to the domain size), two stable polarization states ( /C0Pa n d þP) can be achieved in the FE layer, which, in turn, yield two different channel FIG. 15. (a) FEFET device structure showing an integrated ferroelectric layer in the gate stack of the transistor (leftmost). A series of pulses can be applied to emulate the inte- grating behavior of neurons and the eventual ﬁring through abrupt switching of the device.123(b) A FEFET synaptic device (leftmost) showing programming pulsing schemes generating the STDP learning curve based on the change in charge stored in the device.124Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 021308 (2020); doi: 10.1063/1.5113536 7, 021308-18 Published under license by AIP Publishingconductances for the underlying transistor. Such states can also be referred to as “low VT” (corresponds to þP) and “high VT”( c o r r e - sponds to /C0P) states.128Even though the polarization at the lattice level (microscopic polarization) can have two values ( þPo r/C0P), in a macroscopic scenario, multi-domain nature of FE ﬁlms (with the areasigniﬁcantly higher than the domain size), multiple levels of polariza- tion can be achieved. Furthermore, the polycrystalline nature of the FE ﬁlm offers a distribution in the polarization switching voltages (coer- cive voltage) and time (nucleation time) in different grains. As a result, a voltage pulse dependent polarization tuning can be obtained suchthat the overall polarization of the FE ﬁlm can be gradually switched. This corresponds to a gradual tuning of channel conductivity (or V T) in FEFETs and can be readily exploited to mimic multi-level synap- ses,124,129in a manner similar to what has already been reported for PCM and RRAMs. As noted above, FEFETs are highly CMOS com- patible, which makes their applications as neuro-mimetic devices quite appealing. Recently, several FEFET based analog synaptic devices have been experimentally demonstrated,124,130,131where the conductance poten- tiation and depression via a gradual VTtuning were obtained by apply- ing a voltage pulse at the gate terminal. However, in the case of identical voltage pulses, the observed potentiation and depressioncharacteristics are highly non-linear and asymmetric with respect to the number of pulses. To overcome such non-ideal effects, different non-identical pulsing schemes were proposed in Ref. 130,w h i c hu t i l i z e a gradual modulation of pulse magnitude or pulse time. Such non- identical pulsing schemes demonstrate a signiﬁcant improvement in potentiation/depression linearity and symmetry. However, if pulses are not identical throughout the programming process, an additionalstep of accessing the weight value is needed every time, and an update takes place so that an appropriate pulse can be applied. This leads to design overheads and may reduce the training efﬁciency. To overcome such detrimental effects, an optimum weight update scheme using identical pulses for improved linearity and asymmetry was experimen-tally demonstrated in a FE-Germanium-NanoWire-FET (FE- GNWFET). 131Based on the experimentally extracted parameters of the FE-GNWFET, simulation of the multi-layer perceptron neural net- work over 1 /C2106MNIST images predicts an on-line learning accu- racy of /C2488%. It should be noted that the underlying physics in potentiation/depression linearity and symmetry enhancement in FE- GNWFETs over the conventional FEFET is still unclear. Hence, thereis a timely demand for further theoretical understanding that can enable aggressive device level engineering for achieving higher linearity and symmetry in FEFET based synaptic devices. FEFET synapses can also be used to enable learning with the STDP based update scheme, which can also be achieved. 124In order to utilize the single FEFET as a two-terminal synapse connected to the pre- and the post-neuron, a resistor is connected between the gate and drain [ Fig. 15(b) (leftmost)] terminals. Thus, the pre-spike is applied to the gate and resistor, while the source and bulk are controlled by the post-neuron. With this synaptic scheme and the spiking waveformdepicted in Fig. 15(b) (middle), the relative spike timing between the pre- and the post-neurons can be converted into voltage-drop across the FEFET. The closer the spiking in the time domain, the larger the voltage-drop, which induces a larger conductivity change in the FEFET. The corresponding STDP pattern showing the potentiation and depression is depicted in Fig. 15(b) (rightmost).3. FEFET crossbars FEFETs utilize the electric ﬁeld driven writing scheme, and such a feature is unique when compared with the Spin-, PCM-, and RRAM-based synaptic devices. Therefore, FEFET based synaptic devi- ces are potential candidates for low-power realization of neuro- mimetic hardware. These transistor-like devices can also be arranged in a crossbar fashion to perform dot-product operations. Simulation studies using the population of neuronal and synaptic devices have been studied for image classiﬁcation tasks.130–132We discussed earlier that the multi-state conductance of FEFETs originates from the multi- domain behavior of the FE layer at the gate stack. However, such multi-domain features of FE (domain size and patterns) are highly dependent on the physical properties of FE (i.e., thickness, grain size, etc.).126As a consequence, in a FEFET synaptic array, the multi-state behavior of FEFETs may suffer from the variability of the FE layer along with the variation induced by underline transistors. Therefore, large-scale implementation of the synaptic array with identical FEFET characteristics will be challenging, which can potentially be overcome with high quality fabrication of FE ﬁlms and variation aware designs. Despite the beneﬁts offered by FEFETs, the technology is still at its nascent stage in the context of neuro-mimetic devices, and crossbar- level implementations will be potentially explored in the future. E. Floating gate devices Most of the aforementioned non-volatile technologies are based on non-Si platforms requiring effective integration and CMOS com- patibility. Si-based non-volatile memories, such as Flash memory, use ﬂoating gate devices134to store data. These devices have seen consider- able commercial use in universal serial bus (USB) ﬂash drives and solid state drives. Owing to their non-volatility, ﬂoating gate devices were one of the ﬁrst devices explored for emulating synaptic behavior in neuromorphic systems. Furthermore, these devices are even more promising because of their standard process technology. In this sub- section, we will discuss how neuro-synaptic functionalities can be effectively mimicked using ﬂoating gate devices. 1. Floating gate devices as neurons A ﬂoating gate (FG) transistor has the same structure as a con- ventional MOSFET, except for an additional electrode between the gate and the substrate, called the ﬂoating gate, shown in Fig. 16(a) . The non-volatility is induced by the charge stored on the ﬂoating gate of the transistor. As the charge stored in the ﬂoating gate increases, the threshold voltage of the transistor decreases, as shown in Fig. 16(b) . This charge storage dynamics can also be leveraged to emulate inte- grating behavior in a leaky IF neuron.133Such a LIF neuron circuit is shown in Fig. 17 . Block A shows the integrating circuit where a charge is injected into the ﬂoating gate by the pre-synaptic current. This mod- ulates the voltage at the ﬂoating gate, VFG, which accounts for the inte- gration. Over time, the charge decays, introducing a leaky behavior. The leak factor is dependent on the tunneling barrier thickness. The balance between charge injection and charge ejection determines the neuron operation. The rest of the circuit performs the thresholding and resetting operation as required by a LIF neuron.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 021308 (2020); doi: 10.1063/1.5113536 7, 021308-19 Published under license by AIP Publishing2. Floating gate devices as synapses Unlike the neuronal behavior, which depends on the charge injection/ejection dynamics of the ﬂoating gate, the synaptic behavior depends primarily on charge storage and its ability to modulate the conductance of the device. The charge storage mechanism is governedby two phenomena known as the Fowler–Nordheim (FN) tunnel- ing 135,136and hot-electron injection (HEI). HEI requires a high posi- tive voltage across the gate and the source such that electrons haveenough kinetic energy to cross the insulating barrier between the ﬂoat- ing gate and the channel. Charge gets trapped in the ﬂoating gate and remains intact even after removal of voltage due to the excellent insu-lating abilities of SiO 2. The other mechanism involves FN tunneling, which stores and removes charge from the ﬂoating gate in a reversible manner. A sufﬁciently high positive voltage between the source andcontrol gate causes the electrons to tunnel into the ﬂoating gate, whereas an equivalent voltage of opposite polarity removes the charge. Charge in the ﬂoating gate increases the threshold voltage of the tran- sistor, thus enabling two stable states in the FG transistor, based on the presence and absence of charge. This can be used to emulate binarysynapses. In addition, due to the analog nature of charge, by manipu- lating the amount of charge stored in the ﬂoating gate, multi-level cells(MLC) are possible. Such multi-level storage capability of FG transis- tors have been heavily used in ﬂash memory technologies. 137,138This analog memory characteristics along with excellent stability and reli- ability, especially for multi-level states, make FG devices promising for emulating analog synaptic behavior. In fact, the earliest proposals ofon-chip synapses with computing and learning abilities were based on FG transistors. 139–141 3. Floating gate crossbars Owing to the integrability with CMOS processes, ﬂoating gate transistors have been used to implement large-scale arrays of program- mable synapses to perform synaptic computations between popula- tions of neurons. The exponential dependence of injection and tunneling currents on the gate and tunneling voltages can be furtherused to perform STDP based weight update in such “single transistor” synapses. 142,143 FG transistors overcome most of the major challenges encoun- tered by the previously discussed non-volatile technologies including reliability and stability. Moreover, the retention time can also be mod-ulated by varying the tunneling barrier of the gate oxide. However, this comes with a trade-off that FG transistors require high voltage for writing and reading. Moreover, unlike the high-density storage offeredby PCM and RRAM technologies, FG transistors consume a larger area. The power-hungriness and area inefﬁciency have thus propelled research toward more energy and area-efﬁcient solutions offered bybeyond-CMOS technologies. F. NVM architecture So far, we have discussed how NVMs, owing to their intrinsic physics, can be exploited as neural and synaptic primitives. A compari-son table of the aforementioned NVM technologies is shown in Fig. 18 . Additionally, we have seen that, at a circuit level, the dense crossbar arrangement and associated analog computations present apromising way forward with respect to in-memory computing. Advantageously, beyond devices and circuits, even at an architectural FIG. 17. Floating gate leaky-integrate-and-ﬁre neuron133showing (a) the integrating circuit, (b) and (c) feedback ampliﬁer circuits for thresholding operation, and (d) reset circuit.133 FIG. 16. (a) Basic ﬂoating gate transistor structure showing the control gate and the ﬂoating gate separated by a blocking oxide layer. (b) Increasing charge in the blocking oxide layer lowers the threshold voltage, VT, of the transistor causing higher current at a particular voltage.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 021308 (2020); doi: 10.1063/1.5113536 7, 021308-20 Published under license by AIP Publishing(or system) level, NVMs and crossbars provide interesting opportuni- ties for energy- and area-efﬁciency. NVMs provide a radical departure from the state-of-the-art von-Neumann machines due to the following two factors: (1) NVM based crossbars are being looked upon by theresearch community as the holy grail for enabling in-memory mas- sively parallel dot-product operations, and (2) the high storage densityoffered by NVMs allows construction of spatial neuromorphic archi- tectures, leading to higher levels of energy, area, and latency improve- ments. 144–147Spatial architectures differ from conventional processors in the sense that the latter rely heavily on various levels of memoryhierarchy, and data have to be shufﬂed back and forth between variousmemory sub-systems over long distances (between on-chip and off- chip memory). As such, the energy and time spent in getting the datain the right level of memory hierarchy, before it can be processed, lead to the memory-wall bottleneck. Since the storage density of NVMs is much larger [a single static random access memory (SRAM) cell stor- ing one bit of data consumes 150F 2area compared to an NVM that can take 4F2space storing multiple bits], they lend themselves easily for distributed spatial architectures. This implies that an NVM basedneuromorphic chip can have a crossbar array that stores a subset of the network weights, and such multiple crossbars can be arranged in a tiled manner, wherein weights are almost readily available within eachtile for processing. Keeping in view the aforementioned discussion, a generic NVM based distributed spatial architecture is shown in Fig. 19 , enable map- ping of neural network applications entirely using on-chip NVM. The FIG. 18. Table showing a comparison of different beyond-CMOS NVM technologies and some representative works on demonstrations and design of neuronal and syn aptic elements in a spiking neural network. Note that neurons and synapses can also be designed using non-volatile ﬂoating gate transistors (discussed in S ec.III E). However, in this table, we focus on beyond-CMOS materials due to their non-standard material stack.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 021308 (2020); doi: 10.1063/1.5113536 7, 021308-21 Published under license by AIP Publishingvarious computing cores with their crossbar arrays are interconnected through network-on-chip (NOC). A distinct characteristic of SNN architecture is event-drivenness. SNNs communicate through spikes, i.e., binary information transfer between neurons. As such, for on-chip NOCs, spike-addresses are communicated between various computecores rather than energy expensive transfer of actual data. 144 Furthermore, only those units are active, which have received a spike,and others remain idle, resulting in added energy-efﬁciency. Note that both spike-based on-chip communication and event-drivenness are direct consequences of SNN based data processing. Distributed archi- tectures based on NVM technologies have been explored heavily to build special-purpose accelerators for both machine learning work-loads such as convolutional neural networks (CNNs), multi-layer per- ceptrons (MLPs), and long short term memories (LSTMs), 145–148as well as SNNs.144,149These works have demonstrated signiﬁcant improvements over CMOS-based general purpose systems such as central processing units (CPU), graphics processing units (GPU), orapplication speciﬁc integrated circuits (ASICs), 150which highlight the potential of neuromorphic computing based on NVM devices. Until now, we have talked about inference-only accelerators that require ﬁxed-point arithmetic, which NVM crossbars are well suited for. In addition, on-chip training based on unsupervised learning has been explored at a primitive level using low-precision devices;151,152 however, training accelerators for large-scale tasks, which use such primitives, have not been demonstrated yet. Moreover, supervised learning, on the other hand, requires ﬂoating-point arithmetic due to small magnitude of weight updates, which is difﬁcult to be captured by ﬁxed-point representation. Architectures, which support training, thus face a signiﬁcant challenge of incorporating such small updates to NVM crossbars. This problem is accentuated especially with limitedendurance and high write latency of some NVM technologies, such as PCMs and RRAMs. Writing into crossbars in parallel using pulse width encoding schemes has been proposed although the scalability of such a technique still needs to be investigated. 153Based on the discus- sion in this section, two important developments that are yet to beseen from the neuromorphic community with respect to architectures based on NVMs are (1) experimental demonstration of large-scale inference-only NVM crossbar systems that can rival their CMOScounterparts, for example, the CMOS based large-scale neuromorphic chip presented in Refs. 154and155, and (2) investigation and estab- lishment of the limits of crossbar based neuromorphic systems for on- chip training keeping in mind the constrained writability of NVMtechnologies. IV. PROSPECTS A. Stochasticity—Opportunities and challenges We have discussed about the promises of NVM technology for emulating neuro-synaptic behavior using single devices. These devices can have inherent variability embedded into their intrinsic physics, which can lead to stochastic characteristics. This is a major advantage from CMOS-based implementations where extra circuitry is requiredto generate stochastic behavior. Stochastic devices derive inspiration from the inherent stochasticity in biological synapses. Such synaptic uncertainty can be used in both learning and inferencing 157in spiking neural networks. This is especially crucial for binary or ternary synap- ses where arbitrary weight update may result in overwriting previouslylearned features. Using stochasticity in binary synapses can vastly improve its feature recognition capabilities. This can be done in both a spatial manner 158where a number of synapses are randomly chosen for weight update or a temporal manner116where learning in a proba- bilistic manner can follow the footsteps of the STDP based synapticweight update algorithm. Stochastic STDP thus enables feature recog- nition with extremely low-precision synaptic efﬁcacy, resulting in compressed networks, 152which has the potential to achieve signiﬁcant energy efﬁciency when implemented on hardware.151Stochastic learn- ing is particularly helpful for low-precision synapses because its adds an analog probabilistic dimension, thus ensuring less degradation in accuracy in low-precision networks. For higher-precision networkswhere the classiﬁcation accuracy does not degrade, stochasticity does not make a signiﬁcant difference. In addition to stochastic learning, we have also discussed how stochastic devices can be used to mimic the functionality of corticalneurons. In PCM devices, stochasticity has been explored in integrate- and-ﬁre neurons 29where multiple reset operations lead to different initial glass states. Although such stochastic IF characteristics can be exploited for robust computing, the overhead for achieving control over such stochasticity remains to be seen. On the other hand, in spindevices, stochastic neurons with sigmoidal characteristics are heavily tunable. These kinds of neurons have been explored both using high energy-barrier (10–60 kT) magnets 102and low barrier magnets (1 kT).107While the resultant sigmoidal behavior looks similar, a 1 kT magnet loses its non-volatility and is more susceptible to variations,leading to more complex peripheral circuit design. 156This results in the peripheral energy dominating the total energy consumption of such devices, which, interestingly, often makes them less energy- efﬁcient than high barrier counterparts ( Fig. 20 ). B. Challenges of NVM crossbars We have also discussed about the promises of NVM technology for emulating neuro-synaptic behavior using single devices. We have shown how these devices can be connected in an integrated crossbar network to perform large-scale neural computing. Although the prom- ise of enabling parallel in-memory computations using crossbar arrays is attractive from the energy- and area-efﬁciency perspective, manynon-ideal devices and circuit behaviors limit their wide scale FIG. 19. A representative neuromorphic architecture based on NVM crossbars as basic compute engines.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 021308 (2020); doi: 10.1063/1.5113536 7, 021308-22 Published under license by AIP Publishingapplicability. These include the variability in RRAM states, which can detrimentally affect the verity of analog computations in synaptic ele- ments. This is primarily due to the uncontrolled nature of the variabil-ity in ﬁlamentary RRAM or CBRAM devices. 159PCM devices on the other hand, in spite of being less prone to variability, suffer from resis-tance drifting due to structural relaxations after the melt-quenchamorphization of the material. 160Resistance drifting primarily affects high resistance states in PCMs and hence adversely impacts the perfor-mance of neural networks especially for ex situ trained networks. 52 Carefully manipulating the highest resistance state of operation using partial resetting pulses can potentially reduce the impact of resistancedrift. 52Spintronic devices are more robust with respect to variability and endurance challenges as compared to RRAM and PCM technolo- gies owing to their stable and controlled switching. However, practicaldevices suffer from low contrast in conductivity between the stableextremities. The low ON–OFF ratio severely affects the mapping ofsynaptic weights when implemented in neural networks and is themajor technical roadblock for synaptic implementations using spindevices. Additionally, all non-volatile devices have energy and latencyexpensive write operations in comparison to conventional CMOSmemories. This in turn limits the energy-efﬁciency of performing on-chip synaptic plasticity that requires frequent write operations. Apart from device variations and limitations, building large-scale crossbars using non-volatile synaptic devices is a major hurdle towardrealizing the goal of neuromorphic accelerators. Crossbar sizes areseverely limited by various factors such as peripheral resistances, para- sitic drops, and sneak paths. Figure 21 shows a schematic of a realistic crossbar with source, sink, and line resistances and peripherals. Whentraining is performed on-chip taking into account the non-ideal cross-bar behavior, such inaccuracies in crossbar computations can be miti-gated to a large extent. However, for neuromorphic systems designedas inference-only engines, it is necessary to perform effective modelingof the crossbar array, which can potentially account for the non-idealities during off-line training and take corrective measures foraccurate crossbar computations. Such modeling can either involve rig-orous graph-based techniques for linear circuits, 161simple equationsinvolving Kirchoff’s laws under certain assumptions,162or even data- dependent ﬁtting.163Considering the minimal effect of IR-drops along the metal lines, equations of a crossbar under the effect of peripheralresistances can be simpliﬁed as I j¼PVi;niGij 1þRsinkPGij; (10) Vi;ni¼Vi1=Rs 1=RsþP 1 RjiþRsink: (11) Here, Ijis the current of the j-th column, Viis the input voltage to the i-th row of the crossbar, ( Rij¼1=GijÞis the resistance/conductance of the synaptic element connecting the i-th row with the j-th column, Vi;niis the degraded input voltage due to the effect of peripheral resis- tances, Rsis the effective source resistance, and Rsinkis the effective sink resistance. These resistances in relation to a crossbar are shown in Fig. 21 . This modeling gives us an intuition about the behavior of crossbars, which can help preserve the computation accuracy. Forexample, lower synaptic resistances result in higher currents, whichresults in larger parasitic drops across the metal line. On the other hand, higher operating resistances might lead to low sensing margins, necessitating the need for expensive peripheral circuitry. The presenceof sneak paths in synaptic crossbars can also adversely affect the pro-gramming process, thus harming the performance of on-chip learningsystems. In addition to non-ideal elements in NVM crossbars, the design of peripheral components such as Digital-to-Analog Converters FIG. 21. A realistic crossbar system showing the peripheral circuits including digi- tal-to-analog converters (DACs) at the input to the crossbar and analog-to-digital converters (ADCs) at the output. Crossbars can possess non-ideal resistance ele-ments such as the source resistance ðR source ), line resistance ( Rline), and sink resis- tance ( Rsink). FIG. 20. A comparison in energy consumption for stochastic spin neurons for various energy-barrier heights.156Reproduced with permission from Liyanagedera et al. , Phys. Rev. Appl. 8(6), 064017 (2017). Copyright 2017 American Physical Society.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 021308 (2020); doi: 10.1063/1.5113536 7, 021308-23 Published under license by AIP Publishing(DACs) and Analog-to-Digital Converters (ADCs) is essential toward building large-scale neuromorphic systems. As shown in Fig. 21 , DACs are used to convert bit-streamed data to voltages, whereas the ADCs convert back the analog voltage outputs from a sample-and- hold array into digital bits. These converters are especially necessary asthe sizes of neural network models are much higher than the size of asingle crossbar. As a result, multiple crossbars are required to representthe entire neural network, which necessitates digital communicationbetween the outputs of individual crossbars. As the crossbar sizeincreases, the precision requirements for ADCs become higher, lead-ing to enormous power consumption, which can potentially reducethe beneﬁts in terms of energy consumption that NVM crossbarsinherently offer. However, the inherent robustness of neural networkstoward computation errors may allow us to design approximateperipheral circuitry based on ADCs with lower precision require- ments. Moreover, efﬁcient mapping of crossbars and introducing pro- grammability in peripheral precision requirements can potentiallypreserve the beneﬁts offered by NVM technology. In light of thesechallenges such as device variations, non-ideal resistances, sneak paths,and peripheral design, careful design space exploration is required toidentify optimum resistances for operation and crossbar sizes of syn-aptic elements along with efﬁcient device-circuit-algorithm co-designfor exploring effective mitigation techniques. C. Mitigating crossbar non-idealities NVM provides a massively parallel mode of computations using crossbars. However, as we have discussed previously, analog comput-ing is error-prone due to the presence of circuit-level non-idealitiesand device variations. Various mitigation techniques have beenexplored to address these computing inaccuracies. Although some ofthese techniques have been demonstrated for artiﬁcial neural net-works, the methodologies still hold true for spike-based neuromorphic computing. The most commonly used methodology to recover the performance of neural networks due to crossbar-level computingerrors is to re-train the network using software models of resistivecrossbars. The re-training approach involves updating the weights ofthe network based on information of non-idealities in crossbars. Thishas been explored for both stuck-at-faults 164and device variations165 where it has been observed that re-training the network with aware-ness about the defect or variation distribution can minimize the effectsof these non-idealities on classiﬁcation performance. Re-training,however, does not recover the performance of an ideal neural network without any non-idealities. The presence of non-idealities in the for- ward path of a neural network may require a modiﬁed backpropaga- tion algorithm to closely resemble the ideal neural network. 162For unsupervised learning algorithms such as STDP, the impact of non-idealities may be signiﬁcantly lower due to the ease of enabling on-linelearning, which can automatically account for the errors. In addition to static non-idealities in the crossbars, the effect of non-linearity and asymmetry of programming characteristics of NVM devices can alsobe detrimental to the performance of the network. Reliable mitigationdue to such programming errors can be performed by novel pulsingschemes. 166,167These pulsing schemes involve modulation of pulse- widths based on the current conductance state, which help restore linearity. Beyond re-training, other static compensation techniques can also be used to recover some system level inaccuracies. For example,the limited ON/OFF ratio and precision of NVM synaptic devices canresult in computational errors, which can be taken care of by effective mapping of weight matrices to synaptic conductance. 168Static trans- formations of weight matrices have been explored to alleviate circuit-level non-idealities. 169This methodology performs gradient search to identify weight matrices with non-idealities that resemble ideal weightmatrices. Most of the compensation techniques adopted to account for computation inaccuracies in NVM crossbars address very speciﬁc problems. A more complete and holistic analysis, modeling, and miti-gation of crossbar non-idealities are necessary to completely under-stand the impact and explore appropriate solutions. D. Multi-memristive synapses Multi-memristive synapses are examples, wherein device limita- tions have been countered by the use of circuit techniques, albeit atadditional area overhead. Figure 22 depicts two illustrations, which use multiple NVM devices to represent one synaptic weight. In Fig. 22(a) , two separate PCM devices were used to implement LTD and LTP sep-arately. Incrementing the PCM device corresponding to LTP increasedthe neuronal input, whereas incrementing the device corresponding toLTP decreased the neuronal input. By this scheme, the authors in Ref. 35were able to simply the peripheral write circuits since only incre- ments in device resistances were required for representing both LTPand LTD plasticity. Note that conventionally using one single devicewould have required write circuits for both incrementing and FIG. 22. (a) Two separate NVM devices used for LTP and LTD, and the resulting output of the synapse is fed to the neuron. (b) Multiple NVM devices connected in para llel to increase the current range of the synapse. (c) Through the use of an arbitrator, any one of the devices is selected for learning.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 021308 (2020); doi: 10.1063/1.5113536 7, 021308-24 Published under license by AIP Publishingdecrementing the PCM device resistance, and given the complex nature of waveforms required to write into PCM devices, this wouldhave led to additional area overhead. In yet another work, more thanone memristors were connected in parallel [ Fig. 22(b) ] 170to allow the increased current range of the overall synaptic cell. For learning, anarbitration scheme was used to select one memristor and program inaccordance with the learning scheme as shown in Fig. 22(c) .W i t hr e f - erence to these examples, we believe that such schemes, wherein device level constraints can be mitigated through the use of clever circuittechniques, can be a key enabler for NVMs in neuromorphic comput-ing without solely relying on better material stack and manufacturingprocesses for improved device characteristics. E. Beyond neuro-synaptic devices and STDP As would be apparent by now, the state-of-the-art in neuromor- phic hardware using non-volatile devices can be characterized in twobroad categories of works—(1) those that tend to mimic the LIF dynamics of a neuron using device characteristics and (2) others that are geared toward synaptic functionalities and associated learningthrough STDP in shallow SNNs. On the other hand, the state-of-the-art on the algorithmic side of neuromorphic computing has taken astep forward beyond LIF dynamics and STDP learning. We have dis-cussed brieﬂy about how supervised learning such as gradient descentcan also be used for spike-based systems. Previously, supervised learn-ing has been performed in the artiﬁcial neural networks (ANN)domain, and trained networks have been converted to SNNs. 27 Although this method has been scaled to complex image recognitiondatasets such as ImageNet, one particular drawback of this scheme ishigh inference latency. To circumvent that, researchers have exploredlearning schemes, which incorporate such gradient descent algorithmsin the spiking domain itself. 28,171,172Moreover, combining unsuper- vised and supervised learning techniques have also been widelyexplored. 173This kind of hybrid learning technique has shown better scalability (to deeper networks) and improved accuracy. We believe that it is important for the hardware community to move beyond mimicking neurons and synapses on shallow SNNs andﬁnd ways and means of executing more dynamic learning schemes on hardware for deeper spiking networks. Such improved learning schemes would inevitably require complex compute operations, whichcould be beyond the intrinsic device characteristics of non-volatiledevices. As such, there is a need to explore systems, wherein computa-tions can be segregated between non-volatile sub-arrays and CMOSbased compute engines, allowing the overall system to beneﬁt bothfrom parallelism offered by NVMs and the compute complexityoffered by CMOS engines. This would also be a key enabler in buildingend-to-end deployable neuromorphic systems (wherein a spike-based sensor is directly interfaced to a neuromorphic processor) that can cater to real life task as in ultra-low energy IoT systems. Such IoT sys-tems not only are important from a research perspective but can alsoprovide a possible commercial niche-application for neuromorphicprocessors based on non-volatile technologies. F. NVM for digital in-memory computing Most of the current works involving neuromorphic computing and emerging devices have concentrated on analog-mixed-signal com-puting. However, the inherent approximations associated with analogcomputing still remain a major technical roadblock. In contrast, one could use digital in-memory computing for implementing on-chip robust SNN networks. These implementations can use various digitaltechniques, as in use of read only memory (ROM) embedded RAM inNVM arrays 174or peripheral circuits based on in-memory digital computations.175Interestingly, these works do not require heavy re- engineering of the devices themselves. As such, they can easily beneﬁtfrom the recent technological and manufacturing advancements driven by industry for commercialization of various non-volatile tech- nologies as memory solutions. Furthermore, in a large neural network, NVM can be used as sig- niﬁcance driven in-memory compute accelerators. For example, layers of the neural network, which are less susceptible to noise, can be accel- erated using analog in-memory computing, while those layers thatneed more accurate computations can be mapped on NVM arrays rendering digital in-memory computing. Thus, ﬁne-grained heteroge- neous in-memory computing (both digital and analog) can be used inunison to achieve both lower energy consumption and higher applica- tion accuracy. It is also well known that NVMs that store data digitally are easier to program as opposed to analog storage, which requiresmultiple “read-verify” cycles. Thus, on-chip learning, which requiresfrequent weight updates, is more amenable to digital or heterogeneous (digital þanalog) computing arrays as opposed to analog storage of data. Additionally, bit errors induced due to digital computing can beeasily rectiﬁed using error correction codes. Thereby, resorting to digi- tal processing for critical or error susceptible computation could help widen the design space for use of NVMs as SNN accelerators. G. Physical integrability of NVM technology with CMOS There are several works on experimental demonstration of in- memory computing primitives based on non-volatile memories, espe-cially RRAM and PCM technologies. 45,84,95NVM devices in most state-of-art RRAM and PCM crossbars are accompanied by a CMOS selector device (like a transistor). Such a 1T-1R crossbar conﬁgurationresolves sneak paths during read and write operations. 176Crossbars based on NVM technologies such as RRAM,177PCM,178and Spintronics179are fully compatible with the CMOS back end of the line (BEOL) integration process. There are some issues that need to be considered. For example, PCM is fabricated in crystalline form, as BEOL integration involves high temperature processes. Although therehave been large-scale demonstrations on RRAM and PCM crossbarswith CMOS peripherals, work on CMOS integration of spintronic devices has been limited to small scale Boolean logic circuits. 179It is to be noted that the limited use of spin devices for the crossbar structureis a result of the low ON–OFF ratio for spintronic devices and not because of compatibility issues pertaining to integration of spin devices with CMOS technology. In fact, the current advancement in processintegration for spin based devices with CMOS technology has led to recent widespread interest for commercial use of spin based read-write memories. 180FEFETs, on the other hand, follow the standard Front End of Line (FEOL) CMOS process. Thus, all the NVM technologiesbeing explored can be physically integrated with CMOS. V. CONCLUSION The growing complexity of deep learning models and the humongous power consumption of standard von-NeumannApplied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 021308 (2020); doi: 10.1063/1.5113536 7, 021308-25 Published under license by AIP Publishingcomputers while implementing such models have led to a three decade long search for bio-plausible computing paradigms. They draw inspi-ration from the elusive energy-efﬁciency of the brain. To that effect,non-volatile technologies offer a promising solution toward realizingsuch computing systems. In this review article, we discuss how therich intrinsic physics of non-volatile devices, based on various technol-ogies, can be exploited to emulate bio-plausible neuro-synaptic func-tionalities in spiking neural networks. We delve into the generic requirements of the basic functional units of SNNs and how they can be realized using various non-volatile devices. These devices can beconnected in an intricate arrangement to realize a massively parallelin-memory computing crossbar structure representing a radical depar-ture from the existing von-Neumann computing model. A huge num-ber of such computing units can be arranged in a tiled architecture torealize extremely area and energy-efﬁcient large-scale neuromorphicsystems. Finally, we discuss the challenges and possible solution ofrealizing neuromorphic systems using non-volatile devices. We believe that non-volatile technologies show signiﬁcant promise and immense potential as the building blocks in neuromorphic systems of the future.In order to truly realize that potential, a joint research effort is neces-sary, right from the materials that would achieve better trade-offsbetween higher stability and programming speeds and exhibit morelinear and symmetric characteristics. This material investigationshould be complemented with effective device-circuit co-design to alle-viate problems of variations and other non-idealities that introduceerrors into neuromorphic computations. Finally, there must be efﬁ-cient hardware-algorithm amalgamation to design more hardware- friendly algorithms and vice versa. With these challenges in mind and possible avenues of research, the dream of achieving truly integratednon-volatile technology based neuromorphic systems should not befar into the future. AUTHORS’ CONTRIBUTION I.C. and A.J. contributed equally to this work. 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1.1359462.pdf	Magnetization dynamics in NiFe thin films induced by short in-plane magnetic field pulses Th. Gerrits, J. Hohlfeld, O. Gielkens, K. J. Veenstra, K. Bal, Th. Rasing, and H. A. M. van den Berg Citation: Journal of Applied Physics 89, 7648 (2001); doi: 10.1063/1.1359462 View online: http://dx.doi.org/10.1063/1.1359462 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/89/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Complex pulsed field magnetization behavior and Walker breakdown in a NiFe thin-film J. Appl. Phys. 108, 073926 (2010); 10.1063/1.3490233 Dopants for independent control of precessional frequency and damping in Ni 81 Fe 19 (50 nm) thin films Appl. Phys. Lett. 82, 1254 (2003); 10.1063/1.1544642 Detection of coherent and incoherent spin dynamics during the magnetic switching process using vector-resolved nonlinear magneto-optics Appl. Phys. Lett. 81, 2205 (2002); 10.1063/1.1508163 Investigation of ultrafast spin dynamics in a Ni thin film J. Appl. Phys. 91, 8670 (2002); 10.1063/1.1450833 Coherent control of precessional dynamics in thin film permalloy Appl. Phys. Lett. 76, 2113 (2000); 10.1063/1.126280 [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: 155.97.178.73 On: Mon, 01 Dec 2014 12:18:50Magnetization dynamics in NiFe thin ﬁlms induced by short in-plane magnetic ﬁeld pulses Th. Gerrits, J. Hohlfeld, O. Gielkens, K. J. Veenstra, K. Bal, and Th. Rasinga) Research Institute for Materials, Toernooiveld 1, 6525 EDNijmegen, The Netherlands H. A. M. van den Berg Siemens AG, ZT MF 1, Paul-Gossen-Strasse 100, 91034 Erlangen, Germany The magnetization dynamics in a thin NiFe ﬁlm was investigated by applying short in-plane magnetic ﬁeld pulses while probing the response using a time-resolved magneto-optical Kerr effectsetup. In-plane magnetic ﬁeld pulses, with duration shorter than the relaxation of the system, weregenerated using a photoconductive switch and by subsequent propagation of current pulses along awaveguide. The ﬁeld pulses with typical rise and decay times of 10–60 and 500–700 ps,respectively, have a maximum ﬁeld strength of 9 Oe, by which Permalloy elements of 16 nmthickness and lateral dimensions of 10 320 mm were excited. The observed coherent precession of a ferromagnetic NiFe system had precession frequencies of several GHz and relaxation times on ananosecond time scale. The dynamic properties observed agree well the Gilberts’s precessionequation and the static magnetic properties of the elements © 2001 American Institute of Physics. @DOI: 10.1063/1.1359462 # I. INTRODUCTION The study of spin dynamics in magnetic media has gained more and more interest in the last few years, as thewriting speed of data on magnetic media is increasing rap-idly and because of the rapid rise of the magnetic randomaccess memory ~MRAM !technology on the basis of the magnetic tunnel effect. As soon as writing times of less than1 ns are reached, spin precession effects will play a dominantrole here. There have been various experimental studies ofultrashort spin dynamics in ferromagnetic media, 1–3showing magnetization collapse and recovery on ultrasfast ~ps!time scales. However, all these studies employed ultrashort in-tense laser pulses that heated the electrons far above equilib-rium temperature. Though of great fundamental interest,such studies do not address the issues that are relevant for thewriting process of magnetic information in recording media.The write ﬁeld pulse pulls the magnetic spin system out of itsequilibrium state. The relaxation process to its new equilib-rium state is determined by the rate of the energy dissipationof the media, i.e., by the Gilbert’s damping constant. Thestudy of this magnetization dynamics can best be done byusing magnetic ﬁeld pulses, that are shorter in time than thetypical relaxation-time constants of the system. Here, we report on investigations on the spin dynamics of a ferromagnetic system following excitation by 10–60 psrise time, 500–700 ps decay time in-plane magnetic ﬁeldpulses, being much shorter than the magnetic relaxationtime. Thus the magnetic response of the system at largepump-probe delays was solely governed by the magneticproperties of the sample. The response was probed by a time-resolved magneto-optical Kerr effect ~MOKE !experiment detected with balanced photodiodes. 4We will show that our experimental approach is optimally suited to study the spindynamics of a weak ferromagnetic system. Similar experi- ments were used to probe the dynamics by a polar ﬁeldpulse 5in a single coil. The great interest of using in-plane ﬁeld pulses to study the dynamics of a ferromagnetic system lies in its impact onthe writing process of MRAM, for which the timing andshape of the in-plane ﬁeld pulses are of decisive importancefor the speed, the reproducibility, and the energy consump-tion of the writing procedure. The design of our device isdone in a way that in principle we can generate pulses ofarbitrary shape. Experiments using in-plane ﬁeld pulses have already been reported previously. However, the ﬁeld pulses in theseexperiments were generated by pulse generators 6and only the rise times were shorter than the relaxation time of thesystem. II. EXPERIMENT The short magnetic ﬁeld pulses were generated by using a GaAs photodiode in combination with two copper elec-trodes structured into a coplanar waveguide. Figure 1 showsa photograph of the device. The inset of Fig. 1 shows aclose-up photograph of the photoconductive switch ~Auston switch 7!. A 100 fs laser pulse is used to pump the switch, which is designed in a ﬁnger structure that enlarges the areafor the excitation of carriers and thus the total current. Thegap between the electrodes is 15 mm. As the pump laser beam hits the device under a certain angle, the electrodeswould cause some dark area within the photoswitch. Thiswould result in a larger resistance and would decrease thegenerated current. Therefore we introduced 10-nm-thin elec-trodes as ﬁrst conducting layer on the GaAs substrate, whichwere separated 5 mm from each other. The thickness of these electrodes is chosen to be smaller than the skin depth of theincoming laser beam. Light can travel through into the GaAs a!Electronic mail: theoras@sci.kun.nlJOURNAL OF APPLIED PHYSICS VOLUME 89, NUMBER 11 1 JUNE 2001 7648 0021-8979/2001/89(11)/7648/3/$18.00 © 2001 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: 155.97.178.73 On: Mon, 01 Dec 2014 12:18:50and excite carriers there. Consequently, the resistance due to dark areas within the switch is reduced. This techniquemakes it possible to change the angle of incidence of thepump beam without changing the resistance of the photo-switch. Figure 1 shows a photograph of the complete waveguide structure. There are two photoswitches, which can be used asone switch only, or can be used in a pump-pump-probe ex-periment, where the voltage on the electrodes can be arbi-trary. By applying opposite voltages and pumping theswitches at different times, one can in principle shorten apulse already generated 8,9and produce arbitrary pulse shapes. Here we restrict ourselves to the excitation of asingle switch only. We designed the device according to amodel 10,11which describes the propagation of the current pulse on the signal line. It describes the attenuation and dis-persion due to the surface impedance of a coplanar strip line,including the dielectrics surrounding it. The generation of large magnetic ﬁeld pulses clearly de- pends on the generated current. The magnetic ﬁeld close tothe surface is proportional to the current density inside theconductor: H5I/2w, wherewis the width of the conductor. In our case, we chose wto be 10 mm. The large photo- switches provide a large current, as the total current dependson the carrier density times the area of the photoswitch. Thecombination of large photoswitches and small signal linesrequires the introduction of a tapering. The latter was de-signed in such a way that the impedance of the waveguidewould not change, by keeping the ratio between the middleline and the spacing constant. 10A change in impedance would cause reﬂections on the signal line, which willbroaden the current pulse and lower the maximum obtainableﬁeld. Figure 2 shows a scheme of our experimental setup. The magnetic response of the system due to the ﬁeld pulse wasprobed by a standard time-resolved pump-probe setup. Withthe probe beam we measured the linear MOKE signal bymeans of the balancing diodes and lock-in technique. Focus-ing was done by a long working distance microscope objec-tive~numerical aperture 0.3 !to a spot size of 5 mmo nt h e NiFe ﬁlm. The use of a long working distance objective wasnecessary to avoid screening of the pump beam. III. RESULTS AND DISCUSSION Figure 3 shows the time-resolved magneto-optical re- sponse of a NiFe ﬁlm element that is subjected to an in-planebias ﬁeld of 94 Oe and, perpendicular to that, an in-planepulse ﬁeld of 9 Oe at the peak. The ﬁgure shows a dampedoscillation with a period of about 400 ps and a damping ofthe order of 1 ns. This dynamics can well be described interms of the Landau–Lifshitz equation with the Gilbertdamping term dM/dt5 g~M3H!2a~M3dM/dt!. ~1! The value of gis given by 2gmB/h, where mBandg are the Bohr magneton and the spectroscopic splitting factor,respectively. The Gilbert damping of the system is repre-sented by a. In our ﬁts to the measured precessions, we took g517.63106(Gs)21and estimated a50.008. In Eq. ~1!H denotes the total ﬁeld within the system. FIG. 1. A photograph of the waveguide, used in our experiment. From the 65365mm photoswitches ~white arrows !, the tapering concentrates the cur- rent onto the 10 mm signal line ~black arrows !. The thin 10 320mm ﬁlm is placed at the end of the tapering. Around the signal line, two big groundﬂats are placed as a waveguide. The inset shows a close-up photograph ofthe photoswitches. Between the ﬁnger structure, another light electrode canbe seen. This is a thin copper electrode that should prevent shadow effectsfrom the larger electrodes. FIG. 2. Scheme of the waveguide with the signal line and the magnetic thinﬁlm lying on it. The 100 fs pump pulse generates the current pulse, which isconcentrated in the tapering. The response of the system is measured by the100 fs probe beam. FIG. 3. Precession of the ferromagnetic NiFe system as measured by atime-resolved pump-probe MOKE experiment. For this measurement thebias ﬁeld was 94 Oe. The solid line shows the LLG simulation for the givensystem. The dashed line shows the estimated magnetic ﬁeld pulse.7649 J. Appl. Phys., Vol. 89, No. 11, 1 June 2001 Gerrits 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: 155.97.178.73 On: Mon, 01 Dec 2014 12:18:50H5Hext1Hs, ~2! whereHextis given by the applied bias ﬁeld ( H0) and the ﬁeld pulse h(t).Hext5@H0,h(t),0#.Hsrepresents the shape anisotropy and includes both the magnetostatic shape andﬁeld-induced anisotropy contribution of the thin ﬁlm ele-ment. The solid line in Fig. 3 is a simulation of the Landau–Lifshitz–Gilbert ~LLG!equation showing excellent agree- ment with the experimental data. To simulate the precessionof the NiFe system, we derived the static magnetic param-eters of the elements from the magnetization curves mea-sured by the longitudinal Kerr effect. Determination of theanisotropy constants of the thin ﬁlm elements with lateraldimensions is of primary interest, as these contribute to thetorque experienced by the dipoles. @cf. Eq. (1) 12#. Our thin ﬁlm elements are oriented on the wafer such that the easyaxis of the uniaxial anisotropy, induced during the ﬁlm depo-sition, coincides with the long axis of the lateral geometries.The hard axis hysteresis loop of the thin ﬁlm element canwell be described by two uniaxial anisotropy constants, withvalues:K155200erg/cm 3;K2523000erg/cm3. These measurements give an effective anisotropy of 2 Oe along thelong axis of the thin ﬁlm. The shape of the magnetic ﬁeld pulse was estimated by ﬁtting it to the dynamics of the system at different bias ﬁelds.It could well be represented by the simple formula h ~t!5h0@12exp~2t/tr!#3exp~2t/tf!, ~3! where tr,tf, andh0are the rise time, decay time, and the peak ﬁeld value, respectively. By ﬁtting this pulse shape toprecessions, obtained for different bias ﬁelds, we determined tr5(35625) ps, tf5(600 6100) ps, and h05(961) Oe. The dashed line in Fig. 3 shows an estimation of the mag-netic ﬁeld pulse. The simulation was done using a pulse of30 ps rise time and 600 ps decay time. In Fig. 3 it can beseen that the precession frequency decreases, during the de-cay of the magnetic ﬁeld pulse. This is due to the fact theprecession frequency is proportional to the total effectiveﬁeld, which can only be enhanced by the magnetic ﬁeldpulses due to their orthogonal orientation to the bias andeffective anisotropy ﬁeld. In addition, the center of preces-sion is shifted towards the direction of the ﬁeld pulse duringthe ﬁeld pulse. As the pulse decays the precession frequencydecreases until it reaches its equilibrium state at which thefrequency is determined by the bias ﬁeld and the anisotropyof the ﬁlm element. The stronger this effective bias ﬁeld, thehigher the precession frequency is. 12In these conditions, the precession axis coincides with the long axis of the element. IV. CONCLUSION We have shown that our technique, involving the com- bination of a photoswitch and a coplanar waveguide, is per-fectly suited to study the spin dynamics in a soft ferromag-netic system. The setup produces short in-plane ﬁeld pulsesof large amplitude, which are short enough to study the dy-namics of a magnetic system. We estimated the ﬁeld pulse tohave a rise time of 10–60 ps, a decay time of 500–700 ps,and a ﬁeld strength of 8–10 Oe at the peak. We could alsoshow that the device in principle is suited to produce veryshort pulses in a pulse time regime, which will be of muchinterest to future MRAM devices. Therefore, we plan to in-vestigate the dynamics of ultrashort magnetization reversalsof the ferromagnetic thin ﬁlm elements by further improvingthe shape and strength of the magnetic ﬁeld pulses. ACKNOWLEDGMENTS The authors are grateful for the good collaboration with all members of ZTMF1 at Siemens and for the helpful guid-ance preparing the devices. This work was part of the re-search program of the Stichting voor Fundamenteel Onder-zoek der Materie ~FOM !and ﬁnancially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek~NWO !and partly supported by the TMR network NOMOKE and the Brite Euram project Tunnel Sense. 1B. Koopmans, M. van Kampen, J. T. Kohlhepp, and W. J. M. de Jonge, Phys. Rev. Lett. 85, 844 ~2000!. 2E. Beaurepaire et al., Phys. Rev. B 58, 12134 ~1998!. 3J. Hohlfeld, E. Matthias, R. Knorren, and K.-H. Bennemann, Phys. Rev. Lett.78, 4861 ~1997!. 4M. R. Freeman, IEEE Trans. Magn. 27, 4840 ~1991!. 5W. K. Hiebert, A. Stankiewicz, and M. R. Freeman, Phys. Rev. Lett. 79, 1134 ~1997!. 6T. J. Silva and T. M. Crawford, IEEE Trans. Magn. 35,6 7 1 ~1999!. 7D. H. Auston, IEEE J. Quantum Electron. 19, 639 ~1983!. 8H. J. Gerritsen, Ph.D. thesis, Technical University of Eindhoven, 1998. 9U. D. Keil, H. J. Gerritsen, J. E. M. Haverkort, and H. J. Wolter, Appl. Phys. Lett. 66, 1629 ~1994!. 10U. D. Keil, D. R. Dykaar, A. F. Levi, R. F. Kopf, L. N. Pfeiffer, S. B. Darak, and K. W. West, IEEE J. Quantum Electron. 28, 2333 ~1992!. 11M. Y. Frankel, S. Gupta, J. A. Valdmanis, and G. A. Mourou, IEEE Trans. Microwave Theory Tech. 39, 910 ~1991!. 12M. Bauer, R. Lopusnik, J. Fassbender, and B. Hillebrands, Appl. Phys. Lett.76, 2758 ~2000!.7650 J. Appl. Phys., Vol. 89, No. 11, 1 June 2001 Gerritset 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: 155.97.178.73 On: Mon, 01 Dec 2014 12:18:50
1.3562299.pdf	Tunable magnetic domain wall oscillator at an anisotropy boundary J. H. Franken,a/H20850R. Lavrijsen, J. T . Kohlhepp, H. J. M. Swagten, and B. Koopmans Department of Applied Physics, center for NanoMaterials (cNM), Eindhoven University of Technology, P .O. Box 513, 5600 MB Eindhoven, The Netherlands /H20849Received 2 December 2010; accepted 8 February 2011; published online 11 March 2011 /H20850 We propose a magnetic domain wall /H20849DW /H20850oscillator scheme, in which a low dc current excites gigahertz angular precession of a DW at a ﬁxed position. The scheme consists of a DW pinned ata magnetic anisotropy step in a perpendicularly magnetized nanostrip. The frequency is tuned by thecurrent ﬂowing through the strip. A perpendicular external ﬁeld tunes the critical current densityneeded for precession, providing great experimental ﬂexibility. We investigate this system using asimple one-dimensional model and full micromagnetic calculations. This oscillating nanomagnet isrelatively easy to fabricate and could ﬁnd application in future nanoscale microwave sources.©2011 American Institute of Physics ./H20851doi:10.1063/1.3562299 /H20852 As predicted theoretically, 1the magnetization of a free magnetic layer in a multilayer nanopillar can oscillate atGHz frequencies caused by the spin transfer torque exertedby a dc spin-polarized current. 2–5These magnetic oscillations at the nanoscale could ﬁnd application in the area of radio-frequency /H20849rf/H20850devices, such as wide-band tunable rf oscilla- tors. However, the fabrication of such nanopillar devices is particularly hard and the frequency and the output powercannot be tuned independently. An alternative oscillating na-nomagnet is a precessing magnetic domain wall /H20849DW /H20850.I ti s already widely known that DWs precess during motion atcurrents /H20849and ﬁelds /H20850above the so-called Walker limit. 6Ob- viously, for a continuously operating oscillator it is vital thatthe DW remains at a ﬁxed position, but for commonly usedin-plane magnetized materials /H20849i.e., Ni 80Fe20/H20850a high current density is needed for Walker precession, leading to undesiredDW displacement motion. Experiments have been reported on rf-driven DW reso- nance phenomena, 7–10but for use as an rf source, a DW device needs to convert a dc current to an rf signal. Recently, several such devices have been proposed in theory,11–13but signiﬁcant obstacles must be overcome before an experimen-tally feasible device can be produced. Perhaps the most vi-able scheme to date was proposed in Ref. 11, using a DW pinned at a constriction in a nanostrip with large perpendicu-lar magnetic anisotropy /H20849PMA /H20850. The key for achieving DW precession at low dc currents is to minimize the energy bar-rier for DW transformation between the Bloch and Néeltypes /H20849Fig. 1/H20850. In wide strips, Bloch walls have the lowest magnetostatic energy, whereas the Néel wall is preferred invery narrow strips. 14By locally reducing the wire width at the constriction, this energy barrier is minimized, leading toa low critical current. However, at the constriction the wirewidth needs to be trimmed to a challenging 15 nm, and alsothe DW needs to be initialized at the correct position, leadingto cumbersome experimental schemes. In this letter, we propose a different scheme, inspired by our recent experimental observation that a DW in a nanowirecan be controllably pinned at a magnetic anisotropy step cre-ated by ion irradiation. 15–17Interestingly, the anisotropy alsocontrols the width of a DW and, therefore, it controls whether the Bloch or Néel wall is stable. One can thus tunethe anisotropy values at both side of the boundary in such away, that a Bloch/Néel wall is stable in the two respectiveregions /H20849Fig.1/H20850. A DW can be pinned exactly at the transi- tion point between the Bloch/Néel stability regions by a dcexternal ﬁeld. At this position, the energy barrier betweenboth walls is minimal and, therefore, oscillations are easilyexcited by dc currents. We study the feasibility of this ap-proach by a one-dimensional /H208491D/H20850model and micromagnetic simulations and discuss its advantages in terms of ease offabrication, experimental ﬂexibility and scalability. To characterize the behavior of this DW oscillator as a function of current and ﬁeld, we ﬁrst investigate its dynamicsusing a 1D model. Starting from the Landau–Lifshitz–Gilbert equation with spin-torque terms and parameterizingthe DW using the collective coordinates q/H20849DW position /H20850, /H9274 /H20849in-plane DW angle /H20850, and/H9004/H20849DW width /H20850,11,14we get /H9004/H20849q/H20850/H9274˙−/H9251q˙=/H9252u+/H9253/H9004/H20849q/H20850 2Ms/H11509/H9280 /H11509q, /H208491/H20850 a/H20850Electronic mail: j.h.franken@tue.nl.ion irradiation u DW energy position qBloch wall stability region transition point H=0 HK0 K1<K0 xy z/c121 /c68Néel wall stability region /c68 DW FIG. 1. /H20849Color online /H20850Sketch of the perpendicularly magnetized strip with a step in the magnetic anisotropy /H20849from K0toK1/H20850and associated DW poten- tials in the absence and presence of an external magnetic ﬁeld. At a properlytuned ﬁeld, the DW energy minimum might shift to the Bloch/Néel transi- tion point, where it is easy to excite DW precession /H9274˙by a spin-polarized current /H20849u/H20850.APPLIED PHYSICS LETTERS 98, 102512 /H208492011 /H20850 0003-6951/2011/98 /H2084910/H20850/102512/3/$30.00 © 2011 American Institute of Physics 98, 102512-1q˙+/H9251/H9004/H20849q/H20850/H9274˙=−u−/H9253/H9004/H20849q/H20850 MsKd/H20849q/H20850sin 2/H9274, /H208492/H20850 where u=/H20849g/H9262BPJ /2eMs/H20850is the spin drift velocity, represent- ing the electric current, with gthe Landé factor, /H9262Bthe Bohr magneton, Pthe spin polarization of the current, Jthe cur- rent density, and ethe/H20849positive /H20850electron charge. Msis the saturation magnetization, /H9253is the gyromagnetic ratio, /H9251is the Gilbert damping constant, /H9252is the nonadiabaticity con- stant, and Kdis the transverse anisotropy. The term /H11509/H9280//H11509qis the derivative of the DW potential energy, which was ob-tained by assuming that the DW retains a Bloch proﬁle sym-metric around its center /H20851m z=tanh /H20849x//H9004/H20850/H20852. Using our geom- etry sketched in Fig. 1, this yields d /H9280/dq=2/H92620MsH−/H20849K0 −K1/H20850sech2/H20851q//H9004/H20849q/H20850/H20852. Here, we have made the additional assumption that the effective perpendicular anisotropy /H20849K =Ku−/H208491/2/H20850/H92620NzMs2/H20850changes instantly from the high value K0to the lower value K1at the position q=0. This is appro- priate if the anisotropy gradient length is smaller than theDW width, which can be achieved using a He +focused ion beam /H20849FIB/H20850.17The transverse anisotropy constant Kdrepre- sents the energy difference between a Bloch /H20849/H9274=0 or /H9266/H20850and Néel /H20849/H9274=/H11006/H9266/2/H20850wall and results from demagnetization ef- fects. Therefore, it depends on the dimensions of the mag- netic volume of the DW, given by the DW width /H9004, the width of the magnetic strip w, and its thickness t. We esti- mate the demagnetization factors Nx,Ny, and Nzof the DW by treating it as a box with dimensions 5.5 /H9004/H11003 w/H11003t.18The effective DW width 5.5 /H9004was determined from micromag- netic simulations: if w/H110155.5/H9004the Bloch and Néel walls have the same energy and the transverse anisotropy Kd =/H208491/2/H20850/H92620/H20849Nx−Ny/H20850Ms2vanishes because Nx/H11015Ny. In the absence of transverse anisotropy /H20849Kd=0/H20850, an ana- lytical solution exists to the system of Eqs. /H208491/H20850and /H208492/H20850. The DW will precess at a constant frequency fproportional to the current,11 2/H9266f=/H9274˙=−u /H9251/H9004,/H20849Kd=0/H20850, /H208493/H20850 while the DW remains at a ﬁxed position /H20849q˙=0/H20850. For the case Kd/HS110050, however, the system is solved numerically. We use parameters typical for a Co/Pt multilayer system, with Ms =1400 kA /m,A=16 pJ /m, and /H9251=0.2. For the moment, we assume only adiabatic spin-torque /H20849/H9252=0/H20850. For the effec- tive anisotropy at the left side of the boundary, we choose K0=1.3 MJ /m3/H20849corresponding to Ku,0=2.5 MJ /m3/H20850.B y ion irradiation, this can be reduced to arbitrarily low valuessuch as K 1=0.0093 MJ /m3/H20849Ku,1=1.2 MJ /m3/H20850at the right of the boundary. For the calculation of the transverse aniso- tropy, we use the geometry w=60 nm and t=1 nm. The very low K1leads to a DW that is wide /H20849/H90041=/H20881A/K1 /H1101541 nm /H20850relative to the wire width, which ensures stability of the Néel wall in the right region, whereas a Bloch wall is stable in the left region /H20849/H90040/H110153.5 nm /H20850. At the boundary, the anisotropy is not constant within the DW volume leading to a nontrivial dependence of /H9004on position q. Under the given assumptions, the derivative of internal DW energy equalsd /H9268DW /dq=/H20849K0−K1/H20850sech2/H20851q//H9004/H20849q/H20850/H20852. By using the fact that /H9268DW=4A//H9004, numerical integration yields /H9004/H20849q/H20850as presented in the inset of Fig. 2/H20849a/H20850. The fact that the DW width depends on the position implicitly leads to a time-dependent DWwidth /H9004, which we take into account by updating /H9004/H20849q/H20850at every integration step. Time variations in Kdare taken into account as well, because it depends on /H9004. Solutions of the precession frequency at various ﬁelds and currents are plotted in Fig. 2/H20849a/H20850. The results differ from the purely linear behavior predicted by Eq. /H208493/H20850in two ways. First of all, because of the energy barrier Kdbetween the Bloch and Néel walls, a critical current density needs to beovercome before precession occurs. Of the curves shown, aﬁeld of 70 mT yields the lowest critical current, so appar-ently this ﬁeld brings the DW close to the Bloch/Néel tran-sition point. The second deviation from linearity is seen athigh current densities, where an asymmetry between nega- FIG. 2. /H20849Color online /H20850/H20849a/H208501D-model solution of DW precession frequency as function of current density at various ﬁelds. Positive /H20849negative /H20850findi- cates clockwise /H20849counterclockwise /H20850precession. Sketches show the potential landscape of the DW and the displacement due to the electron ﬂow. Theinset graph shows the equilibrium DW width as function of position. /H20849b/H20850 Similar to /H20849a/H20850but obtained from micromagnetic simulations. The inset shows snapshots of the spin structure during simulation /H20849 /H92620H=70 mT and u=4 m /s/H20850./H20849c/H20850Critical effective velocity /H20849current /H20850as a function of applied ﬁeld, obtained using the two methods.102512-2 Franken et al. Appl. Phys. Lett. 98, 102512 /H208492011 /H20850tive and positive current densities exists. This arises solely from the change in the DW width: with increasing positive/H20849negative /H20850current density, the equilibrium DW position is pushed to the left /H20849right /H20850, where the DW becomes narrower /H20849wider /H20850. This behavior is sketched in the insets of Fig. 2/H20849a/H20850. To conﬁrm the validity of our 1D approximation, we simulate the same system using micromagneticcalculations. 19The strip is 400 nm long, 60 nm wide, and 1 nm thick and divided into cells of 4 /H110034/H110031n m3. Snap- shots of the spin structure during precession are shown in theinsets of Fig. 2/H20849b/H20850. The results in Fig. 2/H20849b/H20850qualitatively match our simpliﬁed 1D model, with slightly lower frequen-cies. However, the critical current needed for precession issomewhat larger in the simulations as compared to the 1Dmodel, which is shown in Fig. 2/H20849c/H20850, where the ﬁeld depen- dence of the critical current is plotted for both methods. Weattribute this to an observable deviation from the 1D proﬁlein the simulations, which leads to inhomogeneous demagne-tization ﬁelds posing additional energy barriers between theBloch and Néel states. At /H92620H/H1101565 mT, ucrit/H110152m /si s minimized, which corresponds to an experimentally feasiblecurrent density J/H110159/H1100310 10Am−2assuming a spin polariza- tion P=0.56 in Co/Pt.20 Although the nonadiabatic /H9252-term in Eq. /H208491/H20850greatly af- fects the dynamics of moving DWs,6we found only minor consequences for a pinned oscillating DW. Simulations atvarying /H9252could be reduced to a single f/H20849u,H/H20850curve by a simple correction to the external ﬁeld H/L50195=H+/H20849/H9252u//H92620/H9253/H9004/H20850. We argue that this DW oscillator scheme has several advantages over prior schemes. First of all, one does notneed complicated nanostructuring of geometric pinning sites,as FIB irradiation readily creates pinning sites withoutchanging the geometry and with a spatial resolution in thenanometer range when a focused He beam is used. 17Second, initialization of a DW at an anisotropy boundary is inher-ently simple; the area with reduced anisotropy has lower co-ercivity and is, therefore, easily switched by an externalﬁeld. Third, many DW oscillators can be introduced in asingle wire by an alternating pattern of irradiated and nonir-radiated regions, and all DWs can be initialized at the sametime. Fourthly, the external magnetic ﬁeld provides theunique ﬂexibility to tune the critical current needed for pre-cession. The ﬁeld might be cumbersome in device applica-tions, but by correctly tuning the anisotropy K 1a low critical current density at zero ﬁeld is also possible. The main ad-vantage of DW oscillators over the conventional nanopillargeometry is the ability to tune the frequency independent ofthe microwave output power. This can be achieved by lettingthe DW act as the free layer of a magnetic tunnel junction/H20849MTJ /H20850grown on top of the DW and with the approximate dimensions of the DW /H2084920/H1100360 nm 2/H20850, in a three-terminal geometry.13Interestingly, the output power of such a device might exceed that of a conventional spin torque oscillator/H20849STO /H20850, since the DW exhibits full angular precession in con- trast to the small-angle precession of most STOs, at a similarfeature size. An estimate of the output power can be madeusing the parameters of an STO MTJ, 21namely, a low resistance-area product /H208491.5/H9024/H9262m2/H20850, a TMR ratio of 100% and a maximum bias voltage of 0.2 V. Under these assump-tions, we estimate a maximum rf output power Prms =23/H9262W. The output power can be further increased by pro- ducing arrays of DW oscillators which are coupled throughdipolar ﬁelds, spin waves and/or the generated rf current.Simulations show that slightly different DW oscillators inparallel wires indeed oscillate at a common frequency due tostray ﬁeld interaction. 22 In conclusion, we have introduced a DW oscillator scheme, in which a low dc current excites gigahertz preces-sion of a DW pinned at a boundary of changing anisotropy ina PMA nanostrip. The frequency of the precession is tunedby the dc current amplitude. A perpendicular external ﬁeldtunes the critical current needed for precession. The systemis well-described by a 1D model, which gives results almostidentical to micromagnetic calculations. This work is part of the research program of the Foun- dation for Fundamental Research on Matter /H20849FOM /H20850, which is part of the Netherlands Organisation for Scientiﬁc Research/H20849NWO /H20850. We thank NanoNed, a Dutch nanotechnology pro- gram of the Ministry of Economic Affairs. 1J. Slonczewski, J. Magn. Magn. Mater. 159,L 1 /H208491996 /H20850. 2D. Houssameddine, U. Ebels, B. Delaët, 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. 3M. Tsoi, A. G. M. Jansen, J. Bass, W. C. Chiang, V. Tsoi, and P. Wyder, Nature /H20849London /H20850406,4 6 /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 /H20849London /H20850425,3 8 0 /H208492003 /H20850. 5S. Kaka, M. R. Pufall, W. H. Rippard, T. J. Silva, S. E. Russek, and J. A. Katine, Nature /H20849London /H20850437,3 8 9 /H208492005 /H20850. 6A. Thiaville, Y. Nakatani, J. Miltat, and Y. Suzuki, Europhys. Lett. 69, 990 /H208492005 /H20850. 7E. Saitoh, H. Miyajima, T. Yamaoka, and G. Tatara, Nature /H20849London /H20850432, 203 /H208492004 /H20850. 8L. Thomas, M. Hayashi, X. Jiang, R. Moriya, C. Rettner, and S. Parkin, Science 315, 1553 /H208492007 /H20850. 9D. Bedau, M. Kläui, S. Krzyk, U. Rüdiger, G. Faini, and L. Vila, Phys. Rev. Lett. 99, 146601 /H208492007 /H20850. 10S. Lepadatu, O. Wessely, A. Vanhaverbeke, R. Allenspach, A. Potenza, H. Marchetto, T. R. Charlton, S. Langridge, S. S. Dhesi, and C. H. Marrows,Phys. Rev. B 81, 060402 /H208492010 /H20850. 11A. Bisig, L. Heyne, O. Boulle, and M. Kläui, Appl. Phys. Lett. 95, 162504 /H208492009 /H20850. 12J. He and S. Zhang, Appl. Phys. Lett. 90, 142508 /H208492007 /H20850. 13T. Ono and Y. Nakatani, Appl. Phys. Express 1, 061301 /H208492008 /H20850. 14S.-W. Jung, W. Kim, T.-D. Lee, K.-J. Lee, and H.-W. Lee, Appl. Phys. Lett. 92, 202508 /H208492008 /H20850. 15A. Aziz, S. J. Bending, H. Roberts, S. Crampin, P. J. Heard, and C. H. Marrows, J. Appl. Phys. 98, 124102 /H208492005 /H20850. 16R. Lavrijsen, J. H. Franken, J. T. Kohlhepp, H. J. M. Swagten, and B. Koopmans, Appl. Phys. Lett. 96, 222502 /H208492010 /H20850. 17J. H. Franken, M. Hoeijmakers, R. Lavrijsen, J. T. Kohlhepp, H. J. M. Swagten, B. Koopmans, E. van Veldhoven, and D. J. Maas, “Precise con-trol of domain wall injection and pinning using helium and gallium fo-cused ion beams,” J. Appl. Phys. /H20849to be published /H20850. 18A. Aharoni, J. Appl. Phys. 83,3 4 3 2 /H208491998 /H20850. 19M. Scheinfein, LLG micromagnetics simulator. 20A. Rajanikanth, S. Kasai, N. Ohshima, and K. Hono, Appl. Phys. Lett. 97, 022505 /H208492010 /H20850. 21D. Houssameddine, S. H. Florez, J. A. Katine, J.-P. Michel, U. Ebels, D. Mauri, O. Ozatay, B. Delaet, B. Viala, L. Folks, B. D. Terris, and M.-C.Cyrille, Appl. Phys. Lett. 93, 022505 /H208492008 /H20850. 22See supplementary material at http://dx.doi.org/10.1063/1.3562299 for a micromagnetic movie of two stray ﬁeld coupled oscillators.102512-3 Franken et al. Appl. Phys. Lett. 98, 102512 /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
1.2949740.pdf	Spin polarization of amorphous CoFeB determined by point-contact Andreev reflection S. X. Huang, T. Y. Chen, and C. L. Chien Citation: Appl. Phys. Lett. 92, 242509 (2008); doi: 10.1063/1.2949740 View online: http://dx.doi.org/10.1063/1.2949740 View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v92/i24 Published by the American Institute of Physics. Related Articles Lithium implantation at low temperature in silicon for sharp buried amorphous layer formation and defect engineering J. Appl. Phys. 113, 083515 (2013) Differential scanning calorimetric determination of the thermal properties of amorphous Co60Fe20B20 and Co40Fe40B20 thin films Appl. Phys. Lett. 102, 051905 (2013) Analysis of optical properties of porous silicon nanostructure single and gradient-porosity layers for optical applications J. Appl. Phys. 112, 053506 (2012) Post-growth surface smoothing of thin films of diindenoperylene Appl. Phys. Lett. 101, 033307 (2012) Post-growth surface smoothing of thin films of diindenoperylene APL: Org. Electron. Photonics 5, 156 (2012) 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 17 Mar 2013 to 129.89.24.43. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsSpin polarization of amorphous CoFeB determined by point-contact Andreev reﬂection S. X. Huang, T . Y . Chen, and C. L. Chiena/H20850 Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, Maryland 21218, USA /H20849Received 15 May 2008; accepted 3 June 2008; published online 19 June 2008 /H20850 Point-contact Andreev reﬂection measurements reveal that amorphous Co xFe80−xB20/H20849x=20, 40, and 60/H20850alloys possess spin polarization of as much as 65%, much higher than the values of 43%–45% for Co and Fe. This accounts for the high magnetoresistance values in magnetic tunnel junctionsincorporating amorphous CoFeB as the ferromagnetic electrodes. The crystallization of theamorphous alloys substantially reduces the spin polarization. © 2008 American Institute of Physics . /H20851DOI: 10.1063/1.2949740 /H20852 The amorphous ferromagnet CoFeB has been exten- sively exploited in spintronic devices recently. TheAlO x-based magnetic tunnel junctions /H20849MTJs /H20850with CoFeB electrodes show larger tunneling magnetoresistance /H20849TMR /H20850 values than those with crystalline CoFe.1Even larger TMR values have been observed in MgO-based MTJs using bothamorphous 2and crystalline3CoxFe80−xB20; the TMR values display a substantial compositional dependence.3,4Further- more, the critical current densities for spin transfer torqueswitching 5and current-driven domain wall motion experiments6are also greatly reduced when CoFeB, instead of NiFe, has been utilized as the free layer. All of thesephenomena suggest a high spin polarization in amorphousCoFeB, whose value and compositional dependence remainto be determined. The spin polarization /H20849P/H20850of a metal is deﬁned as the normalized imbalance of the density of states /H20849DOS /H20850of the two spin orientations at the Fermi level, P=/H20851N/H20849E F↑/H20850 −N/H20849EF↓/H20850/H20852//H20851N/H20849EF↑/H20850+N/H20849EF↓/H20850/H20852, with N/H20849EF↑/H20850and N/H20849EF↓/H20850, re- spectively, as the spin-up and spin-down densities of states. Experimentally, the measured Pvalue of a material depends on the weighted DOS by certain factors speciﬁc to the ex-perimental techniques, 7which include Fermi velocity for An- dreev reﬂection and tunneling matrix for superconductingtunnel junctions. The DOS in a crystalline solid is the resultof the energy band based on the crystal structure. However,the DOS is not well deﬁned in an amorphous ferromagnet, inwhich only short-range ordering exists. Nevertheless, the conduction electrons in an amorphous ferromagnetic metalremain polarized with a value that can be experimentallymeasured. It is of fundamental interest to measure the spinpolarization of amorphous ferromagnets, especially thosethat exhibit superior magnetoelectronic properties. Amor-phous ferromagnets also offer the prospect of revealing theeffect of crystallization on the spin polarization. In this work, we have determined the spin polarization of amorphous and crystallized CoFeB of various compositionsusing the point-contact Andreev reﬂection /H20849PCAR /H20850. The spin polarization of amorphous Co xFe80−xB20, weakly dependent on composition, has been found to be as high as 65%, muchlarger than those of all the common magnetic metals. Fur-thermore, we have found that the crystallization process sub-stantially reduces the spin polarization of the materials.Thin ﬁlms of Co xFe80−xB20/H20849x=20,40,60 /H20850, 200–500 nm in thickness, have been fabricated on thermally oxidized sili- con substrates by dc magnetron sputtering using compositetargets in an Ar atmosphere of 5 mTorr in a vacuum systemwith a base pressure of 2 /H1100310 −7Torr. Extensive studies have shown that vapor-quenching technique can fabricate amor-phous alloys of very wide composition ranges, much widerthan those by liquid quenching and other techniques. 8,9Rep- resentative physical properties of Co 20Fe60B20, one of the as-deposited samples, are shown in Fig. 1. X-ray diffraction /H20849XRD /H20850of the as-prepared Co 20Fe60B20exhibit a broad pat- tern near 2 /H9258/H1101545°, typical of amorphous materials, as shown by the lower curve in Fig. 1/H20849a/H20850.10The electrical resistivity of the sample, measured by a standard four-probe method, isabout 167 /H9262/H9024cm at 5 K and a weak temperature depen- dence, as shown by the solid symbols in Fig. 1/H20849b/H20850. The as- prepared CoFeB sample is magnetically very soft. Its mag-netization at room temperature, obtained by a vibratingsample magnetometer, displays a narrow hysteresis loop witha coercivity of only 0.7 Oe, as shown by the solid symbols inFig. 1/H20849c/H20850. These properties, observed in the as-deposited samples, are characteristics of amorphous ferromagnets. 10 However, the properties of the CoFeB samples are dras- tically different after annealing for 12 h at 450 °C in a mag-netic ﬁeld of 2 kOe and a vacuum of 3 /H1100310 −6Torr. The diffraction peaks corresponding to a bcc structure now ap-pear in the XRD, as shown by the upper curve in Fig. 1/H20849a/H20850, indicating that the sample has been partially crystallized to abcc structure. The resistivity of the sample greatly reducesfrom 167 to 19 /H9262/H9024cm, whereas the coercivity increases from 0.7 to over 70 Oe. These results clearly indicate that theoriginally amorphous sample has been partially crystallizedmostly into crystalline bcc FeCo alloys, with dramaticallydifferent structural, electrical, and magnetic properties. Next, we describe the spin polarization measurements of these samples using the PCAR. 11,12This technique requires a point contact made between the material in question and asuperconductor such that electrons can be ballistically trans-ported through the contact. The actual contacts are in therange of a few nanometers to a few tens of nanometers, ascan be estimated from the contact resistance. Contacts ofsuch sizes are not purely ballistic, especially for amorphousmaterials, in which the mean free path, as indicated by theresistivity of order 100 /H9262/H9024cm, is only 1–2 nm. The actual contact is thus closer to the diffusive regime than the ballistic regime. Fortunately, it has been previously shown that con-a/H20850Electronic mail: clc@pha.jhu.edu.APPLIED PHYSICS LETTERS 92, 242509 /H208492008 /H20850 0003-6951/2008/92 /H2084924/H20850/242509/3/$23.00 © 2008 American Institute of Physics 92, 242509-1 Downloaded 17 Mar 2013 to 129.89.24.43. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsductance results obtained in the diffusive regime can still be analyzed using the formalism for the ballistic regime exceptwith a rather different interfacial scattering potential, 13which has been approximated as a /H9254function with a scattering strength factor Z.14More importantly, the spin polarization value can still be reliably determined by ﬁtting the conduc-tance results to the theoretical model. 13 Another issue is the extra resistance included in the PCAR method. Since the two voltage leads are at ﬁnite dis-tances from the point contact, some additional resistance R E, in addition to the contact resistance of interest, is also inad-vertently measured. The Andreev reﬂection process, whichoccurs only in the vicinity of the point contact, affects thecontact resistance, but not R E. The extra resistance REcan have a signiﬁcant effect on the overall PCAR spectra espe-cially when the point contact is made on a material with alarge resistivity such as amorphous CoFeB. We have care-fully taken the extra resistance into account in our analysis ina self-consistent manner, the details of which will be submit-ted elsewhere. 15Here we describe the determined spin polar- ization of the amorphous and crystalline CoFeB alloys.Some representative PCAR spectra with different Zfac- tors of contacts with Nb tips on amorphous Co 40Fe40B20and polycrystalline Co 20Fe60B20samples at 4.2 K are shown in Fig. 2. The open circles are the experimental data and the solid lines are the best ﬁt to the ballistic theoreticalmodel.15–17From the superconducting transition temperature, and the well-known BCS relation 2 /H9004/kBTC=3.53, the gap value /H9004of the superconductor Nb wire is 1.42 meV, which can be readily seen in the PCAR spectra using crystallineferromagnets such as Co and Fe with low resistivity and longmean free path. 16In the present case of amorphous CoFeB thin ﬁlms, because of the larger values of RE, the apparent peak separation is substantially larger than 1.42 mV. Thesizable value of R E, which contributes to the total resistance R=RAR+RE, can be addressed by our improved theoretical model incorporating the parameter /H9251=RE/RAR, a factor that measures the relative contribution of the extra resistance.15 All of the data can be well described by the model with thevalues of /H9004/eﬁxed at 1.42 mV and Tﬁxed at the actual temperature, but allowing the spin polarization P, the param- eter /H9251, and the interfacial scattering factor Zto vary, as shown in Fig. 2. Furthermore, by allowing the gap value /H9004/e as a free parameter, we have also obtained /H9004/eclose to 1.42 mV and a similar Pvalue within 10%. Our improved ﬁtting procedure incorporating the effect of a large REcan therefore describe the data very well.15 We have measured and analyzed the PCAR spectra of over 100 contacts on amorphous Co xFe80−xB20/H20849x =20,40,60 /H20850and polycrystalline Co xFe80−xB20/H20849x=20,60 /H20850 FIG. 1. /H20849Color online /H20850Properties of as-deposited and annealed Co20Fe60B20 ﬁlms /H20849annealing condition: 450 °C for 12 h at 2 kOe ﬁeld in vacuum /H20850:/H20849a/H20850 XRD pattern of the as-deposited /H20849bottom /H20850and annealed ﬁlms /H20849top, shifted for clarity /H20850,/H20849b/H20850normalized resistivity as a function of temperature with solid circles for the as-deposited ﬁlm and open circles for the annealed ﬁlm. /H20849The resistivities at 5 K of the as-deposited and annealed ﬁlms are 167 and19 /H9262/H9024cm, respectively. /H20850./H20849c/H20850Normalized magnetization hysteresis loops at room temperature of the as-deposited /H20849solid circles /H20850and annealed ﬁlms /H20849open circles /H20850. FIG. 2. /H20849Color online /H20850Representative PCAR spectra for a Nb tip in contact with amorphous /H20851/H20849a/H20850–/H20849c/H20850/H20852Co40Fe40B20and annealed /H20851/H20849d/H20850–/H20849f/H20850/H20852Co20Fe60B20 ﬁlms with different Zfactors. Open circles are the experimental data and solid lines are the best ﬁt to the data using the purely ballistic model.242509-2 Huang, Chen, and Chien Appl. Phys. Lett. 92, 242509 /H208492008 /H20850 Downloaded 17 Mar 2013 to 129.89.24.43. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionssamples. The Pvalue extracted from the analysis varies for each contact, depending systematically on Zfor each sample, as shown in Fig. 3. In the limit of Z=0, one ﬁnds the intrinsic Pvalues, which are 65%, 63%, and 57% for amorphous CoxFe80−xB20with x=20, 40, and 60, respectively. One notes that these values are substantially larger than those of com-mon magnetic metals such as Fe /H2084943% /H20850,C o /H2084945% /H20850, and Ni /H2084937% /H20850.16The larger values of Pnaturally accounts for the larger TMR value observed in AlO x-based MTJs using CoFeB as the electrodes.1Although the signiﬁcantly lower critical current density for spin transfer torque switching intrilayers5and domain wall motion6using CoFeB as the free layer has been ascribed to the lower Gilbert damping factor /H9251in CoFeB, our results indicate that the higher intrinsic P value in the CoFeB is just as signiﬁcant. The tunneling spinpolarization /H20849TSP /H20850values of some amorphous CoFeB ferro- magnets including Co 40Fe40B20have been measured by the superconducting tunneling junction method and the TSPvalue of 49% is higher than that of CoFe /H2084937% /H20850.18While this fact has been attributed to the better interface formed at thejunction by the amorphous CoFeB, our work indicates thatthe higher intrinsic Pvalue of amorphous CoFeB is as im- portant. Very recent theoretical studies using density func-tional theory indeed indicate an enhanced spin polarizationin amorphous CoFeB, in good agreement with our measure-ments as well as those of superconducting tunnel junctions. 19Amorphous alloys also offer the unique prospect of compar- ing spin polarization in the amorphous state as well as thecrystalline state. Very interestingly, the Pvalue is reduced after crystallizing the sample. As shown in Figs. 3/H20849a/H20850and 3/H20849b/H20850, the Pvalues for all the contacts are lower than the P values of the amorphous samples. The determined intrinsic P value for partially crystallized Co xFe80−xB20is 53% and 52% forx=20 and 60. The TMR value of as-prepared MgO-based MTJs using amorphous CoFeB as electrodes20is only a few percent, but over 500% /H20849Ref. 3/H20850is observed after annealing during which both CoFeB and MgO are crystallized. Accord-ing to the Julliere formula, 21which is often invoked in AlO x tunnel junctions, the TMR value scales with spin polarizationof the electrodes. The lower intrinsic Pvalues observed in crystalline CoFeB unambiguously corroborates that the very high TMR in MgO-MTJs is due to coherent tunneling 22but not enhanced spin polarization. In summary, we have fabricated amorphous and crystal- line Co xFe80−xB20/H20849x=20,40,60 /H20850and determined their spin polarization using PCAR. 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FIG. 3. /H20849Color online /H20850Spin polarization values Pas a function of the barrier strength Zfor/H20849a/H20850amorphous /H20849open squares /H20850and annealed /H20849open circles /H20850 Co20Fe60B20,/H20849b/H20850amorphous /H20849open squares /H20850and annealed /H20849open circles /H20850 Co60Fe20B20, and /H20849c/H20850amorphous Co40Fe40B20. The determined intrinsic spin polarization values are listed in the respective ﬁgures.242509-3 Huang, Chen, and Chien Appl. Phys. Lett. 92, 242509 /H208492008 /H20850 Downloaded 17 Mar 2013 to 129.89.24.43. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissions
1.5007435.pdf	Design of a 40-nm CMOS integrated on-chip oscilloscope for 5-50 GHz spin wave characterization Eugen Egel , György Csaba , Andreas Dietz , Stephan Breitkreutz-von Gamm , Johannes Russer , Peter Russer , Franz Kreupl , and Markus Becherer Citation: AIP Advances 8, 056001 (2018); doi: 10.1063/1.5007435 View online: https://doi.org/10.1063/1.5007435 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 Spin-wave propagation in ultra-thin YIG based waveguides Applied Physics Letters 110, 092408 (2017); 10.1063/1.4976708 Experimental prototype of a spin-wave majority gate Applied Physics Letters 110, 152401 (2017); 10.1063/1.4979840 High-quality single-crystal thulium iron garnet films with perpendicular magnetic anisotropy by off-axis sputtering AIP Advances 8, 055904 (2018); 10.1063/1.5006673 Excitation of the three principal spin waves in yttrium iron garnet using a wavelength-specific multi-element antenna AIP Advances 8, 056015 (2018); 10.1063/1.5007101 A switchable spin-wave signal splitter for magnonic networks Applied Physics Letters 111, 122401 (2017); 10.1063/1.4987007 Spin wave modes of width modulated Ni 80Fe20/Pt nanostrip detected by spin-orbit torque induced ferromagnetic resonance Applied Physics Letters 111, 172407 (2017); 10.1063/1.4999818AIP ADV ANCES 8, 056001 (2018) Design of a 40-nm CMOS integrated on-chip oscilloscope for 5-50 GHz spin wave characterization Eugen Egel,1,aGy¨orgy Csaba,2Andreas Dietz,1 Stephan Breitkreutz-von Gamm,1Johannes Russer,3 Peter Russer,3Franz Kreupl,1and Markus Becherer3 1Chair of Technical Electronics, Technical University of Munich, Munich, Germany 2Faculty of Information Technology and Bionics, P ´azm´any P ´eter Catholic University, Budapest, Hungary 3Chair of Nanoelectronics, Technical University of Munich, Munich, Germany (Presented 10 November 2017; received 2 October 2017; accepted 6 October 2017; published online 1 December 2017) Spin wave (SW) devices are receiving growing attention in research as a strong can- didate for low power applications in the beyond-CMOS era. All SW applications would require an efﬁcient, low power, on-chip read-out circuitry. Thus, we provide a concept for an on-chip oscilloscope (OCO) allowing parallel detection of the SWs at different frequencies. The readout system is designed in 40-nm CMOS technology and is capable of SW device characterization. First, the SWs are picked up by near ﬁeld loop antennas, placed below yttrium iron garnet (YIG) ﬁlm, and ampliﬁed by a low noise ampliﬁer (LNA). Second, a mixer down-converts the radio frequency (RF) signal of 5 50 GHz to lower intermediate frequencies (IF) around 10 50 MHz. Finally, the IF signal can be digitized and analyzed regarding the frequency, amplitude and phase variation of the SWs. The power consumption and chip area of the whole OCO are estimated to 166.4 mW and 1.31 mm2, respectively. © 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/10.1063/1.5007435 I. INTRODUCTION CMOS has been dominating over decades as a technology for low power, low cost and high- volume applications. In the meantime, more and more emerging devices are getting attention in the research as candidates for beyond-CMOS era.1SW based or magnonic devices are considered as a low power alternative to CMOS computing. It can perform both Boolean and non-Boolean operations. A current example of a majority gate, performing logic operations, is demonstrated by Klinger et al.2 Similar to optical computing,3SWs can also perform additional operations using wave phenomena in a more direct way than it is done with Boolean logic.4As shown by Csaba5and Papp,6a non-Boolean computing concept for a Fourier transform calculation can be realized using phase shifting plates. SW devices operate in a wide GHz frequency range,4that makes detection and signal analysis challenging. There are several ways to detect and analyze SW signals, e.g. detection via spin-pumping effect or Brillouin light scattering spectroscopy.4But a low power and low cost SW characteriza- tion equipment, covering a reasonable frequency range of several GHz, is still missing. Hence, we consider a concept for SW on-chip characterization with respect to their frequency, amplitude and phase variations as published previously in Refs. 7 and 8. In this paper we present a modiﬁed SW characterization concept, compared to Ref. 7, with simulation results achieved with a state of the art low power radio frequency (LP-RF) 40-nm CMOS technology. In order to detect smaller SW signal power we split 5 50 GHz range in 9 frequency bands. Simulation results for frequency detection as well as for amplitude and phase transfer characteristics are discussed in Sec. III. aElectronic mail: eugen.egel@tum.de 2158-3226/2018/8(5)/056001/6 8, 056001-1 ©Author(s) 2017 056001-2 Egel et al. AIP Advances 8, 056001 (2018) II. CONCEPT FOR ON-CHIP SPIN WAVE DETECTION In order to convert SWs into electron current, we assume a 50 near ﬁeld loop antenna placed below the dielectric material yttrium iron garnet (YIG) with low SW propagation damping4 (see Fig. 1). Based on micro magnetic simulations, an electrical signal power of 80 to 90 dBm is expected in the loop antenna, as previously published in Ref. 8. Due to a limited bandwidth of a single on-chip antenna, an array of loop antennas can be used for covering the targeted 5 50 GHz frequency range of the SWs, i.e. different frequencies can be picked up by different antennas.9Besides, smaller bandwidth of the circuit components provide better noise ﬁltering. Therefore, we assume a slightly reduced signal amplitude of 5 V in the antenna instead of 15 V , as previously published in Ref. 7. As known, there is a trade-off between noise ﬁgure (NF), chip area and power consumption, which are balanced in the presented design. The modiﬁed OCO is divided in 9 frequency bands, listed in Tab. I. While the mixer is covering the whole frequency range 5 50 GHz with the single design, the LNAs and the VCOs are optimized for smaller frequency ranges in order to achieve better performance against noise. For the sake of simplicity, we are currently using near ﬁeld antennas, but the OCO design works with other SW sensing elements that generate an RF voltage. Conversion of SWs into electrical signal is perhaps the most challenging aspect of SW devices. One needs a compact, integrable way of measuring SW amplitudes/phases/frequencies and do it without an extensive circuitry that would diminish the advantages of SW devices. The challenges associated with magneto-electric interfaces to SWs are described in Refs. 9 and 10. As shown in Fig. 1, a periodic radio frequency (RF) signal, picked up by the antenna, is ampliﬁed by a differential low noise ampliﬁer (LNA). The ultra-wideband mixer down-converts the signal to lower IF of 10 50 MHz. For that purpose, a local oscillator (LO) signal, generated in a voltage controlled oscillator (VCO), is required. Subsequently, an operational ampliﬁer (OpAmp) ampliﬁes the periodic IF signal to higher voltage values. Finally, the ampliﬁed signal can be digitized in an analog to digital converter (ADC) that gives information about the SWs frequency, amplitude change and phase variation. FIG. 1. Block diagram of the on-chip readout circuitry for SW characterization covering 5 50 GHz. SWs picked up by inductive loop antennas, placed below the SW low damping medium, e.g. YIG, are ampliﬁed and mixed to lower frequencies by 40-nm CMOS circuitry for analyzing the SW frequency, amplitude and phase difference at the output pads. TABLE I. Frequency bands of the OCO. The check mark symbolizes a single design of the LNA, mixer and VCO covering the corresponding frequency range. Bands Freq. Range [GHz] LNA Mixer VCO 1 5 9 X XX2 9 13 X 3 13 19 X X 4 19 25.5 X X 5 25.5 33 X X 6 33 39 X X 7 39 44 XX 8 44 48 X 9 48 50 X056001-3 Egel et al. AIP Advances 8, 056001 (2018) FIG. 2. Equivalent circuit model of the near ﬁeld loop antenna with the OCO containing differential LNA, mixer, VCO and OpAmp. The LNA has two amplifying stages with an output driver. The LO frequency of the VCO is adjustable with the voltage V CTRL for a ﬁne tuning and with the switchable capacitors for a coarse tuning. The mixer down-converts the signal to lower IFs. Finally, the signal is additionally ampliﬁed by the OpAmp with 2 stages. The topology of the OCO components depicted in Fig. 2 show the design for the frequency band 6 (33 39 GHz). The OCO designs of the other bands are very similar to the presented one and skipped for simplicity. In order to amplify 90 dBm signal of the loop antenna, we use a fully differential LNA with 2 stages. The output driver provides additional isolation between LNA input and output. Besides, the driver is crucial for impedance matching between the LNA output and the mixer input. Starting with the mixer design in Ref. 11, we extended the circuitry with inductors to achieve better NF and higher conversion gain over the whole frequency range of 5 50 GHz. To create the IF signal at lower frequencies, the frequency difference of the RF and LO signals should be in the MHz range. Hence, a tunable VCO is necessary for the readout circuitry. Fine tuning of the VCO output frequency is controlled by the voltage V CTRL . For a coarse frequency tuning we use appropriately sized switchable capacitors. The ﬁnal ampliﬁcation step is realized with the 2 stage OpAmp bringing IF signal to 100 mV range. III. SIMULATION RESULTS The presented results were obtained by simulations using Cadence Virtuoso with device models of the Global Foundries 40-nm LP-RF technology.12The simulation results are valid for room tem- perature of 300 K and already include the noise of each single device. The interconnect parasitics, which will be impacted by the physical layout, have not yet been included in these simulations. The parasitics, extracted from the layout, will of course affect operating frequencies of the OCO. However, this reﬁnement will be tackled in the next project step. We use transistor models with a low threshold voltage. Implemented resistors operate with sili- cided or unsilicided p+ poly resistor models, depending on required resistance values. For capacitors we use alternative polarity metal oxide metal capacitors (APMOM Cap) as well as metal isola- tor metal capacitors (MIM Cap). Symmetric inductor and center tapped inductor models, with nitride as passivation layer, are deployed from optimal inductor ﬁnder kit provided by Global Foundries. The most critical part of the OCO regarding the NF is the ﬁrst stage of the LNA. Depending on the frequency band, we achieved a NF of the LNAs between 2.4 4 dB. For the 50 matching to the antenna, we have a return loss of each LNA better than 10 dB over the whole frequency range. The achieved gain of the LNAs is around 30 dB (see Fig. 3). We use an active mixer, i.e. the RF signal is additionally ampliﬁed during the conversion to lower frequencies. The conversion gain of the mixer is higher than 12 dB as shown in Fig. 3. Finally, the OpAmp ampliﬁes the IF signal with a gain of more than 30 dB in 10 50 MHz.056001-4 Egel et al. AIP Advances 8, 056001 (2018) FIG. 3. Simulated gain of the LNAs, mixer and OpAmp. SW signals of 5 50 GHz are covered by 7 switchable LNAs with band pass characteristics and ampliﬁcation of approx. 30 dB. The conversion gain of the mixer is more than 12 dB over the whole frequency range. Down-converted RF signals are ampliﬁed by OpAmp with a gain above 30 dB in the IF range from 10 to 50 MHz. The main task of the OCO is the characterization of the SWs regarding frequency, amplitude and phase variations. The frequency detection is demonstrated in Fig. 4. We assume sinusoidal signals with an amplitude of 10 V and frequencies at 5, 10, 15, 20, 30, 35, 40, 45, 50 GHz at the input of the OCO in the 9 bands, respectively. Subsequently, the frequency of the LO signal is swept from 5 50 GHz. Finally, the simulated signal at the output of the OpAmp is ﬁtted to a sinusoidal curve and divided by the root mean square error (RMSE). As a result, we get peaks at the assumed frequencies (see Fig. 4). Due to a jitter of the LO signal, the resolution of the SW frequency detection is limited. We achieve a precision in frequency of 20 MHz. The transfer characteristic of the amplitude at the OCO output signal versus input signal is depicted in Fig. 5. Here we use band 6 for demonstration. The RF signal is set to 35 GHz. The LO FIG. 4. Frequency detection of the SWs by sweeping the LO frequency in 9 frequency bands, listed in Tab. I. Peaks demonstrate set frequencies at 5, 10, 15, 20, 30, 35, 40, 45, 50 GHz in the loop antennas of 1-9 bands, respectively. The Y-axis represents ﬁtted amplitude of the sinusoidal signal measured at the output of the OpAmp and divided by the RMSE. FIG. 5. Amplitude at the OpAmp output by sweeping the amplitude of the voltage in the antenna between 1 50V with 1V steps. Amplitude depedence remain almost linear until 30 V of the antenna signal.056001-5 Egel et al. AIP Advances 8, 056001 (2018) FIG. 6. Phase at the OpAmp output signal dependent on the phase shift of the sinusoidal signal picked up by the antenna. The simulated phase shift curve clearly follows the ideal one. frequency is set to 35.03 GHz. Due to a limited output voltage swing of the OpAmp, the amplitude curve has a linear characteristic until 30 V , that corresponds to the assumed maximum signal power of 80 dBm in the loop antenna.8The simulation results show that the signal power of less than 96 dBm in the 50 loop antenna is detectable with proposed OCO concept, i.e. a signiﬁcant improvement of factor 3 compared to our ﬁrst approach in Ref. 7. Figure 6 shows the phase transfer characteristic between the input of the OCO and the output of the OpAmp. The phase shift due to run-time of the signal through the circuity is compensated here in order to compare the simulated phase shift with the ideal one. The maximum deviation of the simulated phase shift from ideal one is equal to 23. The main reason for phase error is the jitter noise of the VCO. An introduction of a phase locked loop (PLL) circuitry in the OCO design could essentially reduce the phase deviation and will be considered in our future work. IV. CONCLUSION SW based devices are emerging for high-speed and low power signal processing tasks, but the challenge of an effective SW detection remains. The OCO could be an integrated alternative to current spin wave detecting systems with the near ﬁeld loop antenna as a sensing element placed below an insulating magnetic medium such as YIG. Besides, the OCO could be adapted for other magneto-resistive or spin hall effect sensing elements. Simulation results show that the signal power of less than 96 dBm can be detected with the proposed design. Sensing time for SW amplitude and phase is below 1 s and for frequency detection less than 40 s with accuracy of 20 MHz. The OCO shows a further step of a possible realization of the SW on-chip detection with a power consumption of 166.4 mW and chip area of 1.31 mm2. ACKNOWLEDGMENTS Fruitful discussions with S. Kiesel and U. Nurmetov from Technical University of Munich are gratefully acknowledged. 1D. E. Nikonov and I. A. Young, “Benchmarking of beyond-CMOS exploratory devices for logic integrated circuits,” IEEE Journal on Exploratory Solid-State Computational Devices and Circuits 1, 3–11 (2015). 2S. Klingler, P. Pirro, T. Br ¨acher, B. Leven, B. Hillebrands, and A. V . Chumak, “Design of a spin-wave majority gate employing mode selection,” Applied Physics Letters 105(15), 152410 (2014). 3D. C. Feitelson, Optical Computing: A Survey for Computer Scientists (MIT Press, 1992). 4A. V . Chumak, V . I. Vasyuchka, A. A. Serga, and B. Hillebrands, “Magnon spintronics,” Nature Physics 11(6), 453–461 (2015). 5G. Csaba, A. Papp, and W. Porod, “Spin-wave based realization of optical computing primitives,” Journal of Applied Physics 115(17), 17C741 (2014). 6A. Papp, W. Porod, E. Song, and G. Csaba, “Wave-based signal processing devices using spin waves,” 15th International Workshop on Cellular Nanoscale Networks and their Applications (CNNA), pp. 1–2, 2016. 7E. Egel, C. Meier, G. Csaba, and S. Breitkreutz-von Gamm, “Design of a CMOS integrated on-chip oscilloscope for spin wave characterization,” AIP Advances 7(5), 056016 (2017). 8S. Breitkreutz-von Gamm, A. Papp, E. Egel, C. Meier, C. Yilmaz, L. Heiß, W. Porod, and G. Csaba, “Design of on-chip readout circuitry for spin-wave devices,” IEEE Magnetics Letters 8, 1–4 (2017).056001-6 Egel et al. AIP Advances 8, 056001 (2018) 9A. Papp, W. Porod, A. Csurgay, and G. Csaba, “Nanoscale spectrum analyzer based on spin-wave interference,” Scientiﬁc Reports 7(9245) (2017). 10G. Csaba, A. Papp, and W. Porod, “Perspectives of using spin waves for computing and signal processing,” Physics Letters A 381(17), 1471–1476 (2017). 11C.-S. Lin, P.-S. Wu, and H.-Y . Chang, “A 9-50-GHz Gilbert cell down-conversion mixer in 0.13- m CMOS technology,” IEEE Microwave and Wireless Components Letters 16(5), 293–295 (2006). 12Global Foundries, “Product Brief 40nm LP RF Technology,” Global Foundries, 2015.
1.4914033.pdf	Controlling the microwave characteristic of FeCoB-ZnO granular thin films deposited by obliquely sputtering Xiaohong Liu,1Y alu Zuo,2Xueyun Zhou,3Wenchun Li,1Liefeng Feng,1 and Dongsheng Y ao1,a) 1Tianjin Key Laboratory of Low Dimensional Materials Physics and Preparing Technology, Institute of Advanced Materials Physics, Faculty of Science, Tianjin University, No. 92 Weijin Road, Nankai District, Tianjin 300072, China 2The Key Laboratory for Magnetism and Magnetic Materials, Lanzhou University, Lanzhou 730000, People’s Republic of China 3Faculty of Science, Jiujiang University, Jiujiang City 332005, Jiangxi Province, China (Received 29 October 2014; accepted 17 February 2015; published online 9 March 2015) A series of FeCoB-ZnO soft magnetic granular ﬁlms deposited at different oblique angles were pre- pared by magnetron sputtering system. A variable in-plane uniaxial magnetic anisotropy ﬁeld from27.6 Oe to 212 Oe and an adjustable ferromagnetic resonance frequency from 1.89 GHz to 5.3 GHz were obtained in the as-deposited ﬁlms just by increasing the oblique angle from 15 /C14to 56/C14. Frequency line-width and effective Gilbert damping factor were both insensitive to the differentoblique angles ( a effdecreased from 0.036 to 0.03 and Dfdecreased from 1.49 to 1.27), which almost satisﬁed the requirement that fFMR could be tuned independently in a certain frequency range. Besides, the change of dynamic magnetic anisotropy ﬁeld versus oblique angle was illus-trated and analyzed quantitatively compared with the static magnetic anisotropy. VC2015 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4914033 ] I. INTRODUCTION Due to the miniaturization and operation in high fre- quency range (GHz) of electro-magnetic devices, such asmagnetic recording write heads, highly sensitive inductors, and micro-wave transformers, the soft magnetic thin ﬁlms with properly magnetic anisotropy ( H k), high or adjustable ferromagnetic resonance frequency ( fFMR), and high electric resistivity ( q) have been strongly recommended.1,2Thus, many researchers have devoted to improving soft thin ﬁlmsfor wider applications. 3–12For thin ﬁlm with in-plane uniax- ial magnetic anisotropy (IPUMA), it has a resonance fre- quency fFMRbeing proportional to [ Hk(4pMsþHk)]1/2(4pMs is the saturation magnetization),13which indicates that fFMR can be well improved by properly increasing Hk. For single layer ﬁlms, ex situ annealing in magnetic ﬁeld,14in situ dep- osition on prestressed substrate,15and obliquely sputtering16 are frequently used to control the IPUMA ﬁeld. It is wellknown that any post fabrication treatment may affect theother components of entire magnetic ﬁlms and it is difﬁcult to employ the stress induced method in ﬁlms deposited on hard substrates. While it has been certiﬁed that oblique depo-sition can well induce the IPUMA for magnetic transition metal ﬁlms, and oblique angles also affect the intensity of IPUMA. 17Despite the wide application of traditional metal- lic thin ﬁlms in magnetic storage owing to their good high- density response, the relatively lower qcannot effectively suppress the eddy current loss in high frequency range.Besides, f FMRand effective damping factor aeffare needed to be tuned independently from the view of application in someway. Thus, it is essential to obtain large fFMRand maintain aeffstable (without bringing the obvious variation for the aeff) in a large frequency range meanwhile. So, how to real- ize the adjusting of Hk,fFMR, and aeffby a simple method is a valuable work. In our work, FeCoB-ZnO granular ﬁlms were prepared by oblique sputtering. The IPUMA ﬁeld was expanded to 212 Oe effectively and the fFMRreached up to about 5.3 GHz just by changing oblique angles, which is easier to be real- ized. The aeffwas insensitive to the oblique angles, while the fFMRwas well tailored, which was desired from the view of application. Besides, the microwave characteristic was quan- titatively analyzed and discussed by Landau-Lifshitz-Gilbert (LLG) equation combining with the static magnetization. II. EXPERIMENT FeCoB-ZnO granular thin ﬁlms were prepared onto water-cooled single-crystal Si substrate with (100) surface orientation by radio frequency magnetron sputtering system at room temperature. A 3 in. ZnO ceramic target on which 23 FeCoB chips were put regularly was used as the sputter-ing source. The sputtering chamber was evacuated to a base pressure below 8.0 /C210 /C05Pa. During the deposition, Ar gas used as the ambient gas was imported at the ﬂow rate of 30 SCCM (SCCM denotes standard-state cubic centimeter per minute) and the total sputtering gas pressure was maintained at 3.0 mTorr. The sputtering power was kept at 150 W and the average deposition rate was 0.13 nm/s. This series of samples were fabricated by oblique sputtering to induce the IPUMA without any external ﬁeld and the easy axis was per- pendicular to the projection of the oblique deposition direc- tion,19,20which was realized by making the composite targeta)Author to whom correspondence should be addressed. Electronic mail: yaodsh@tju.edu.cn. Tel.: 86 þ022þ27408599; Fax: 86 þ86þ022þ27408599. 0021-8979/2015/117(10)/103902/5/$30.00 VC2015 AIP Publishing LLC 117, 103902-1JOURNAL OF APPLIED PHYSICS 117, 103902 (2015) at various tilted angles (shown in Fig. 1). These oblique angles were controlled as h¼0/C14,1 5/C14,2 6/C14,3 5/C14,4 5/C14, and 56/C14. The total thickness was controlled by ﬁxing the average deposition rate and deposition time, which was veriﬁed by aDektak 8 surface proﬁle-meter for every sample. The composi-tion of these samples was determined by energy dispersivex-ray spectroscopy (EDS). The static magnetic properties weremeasured by a vibrating sample m agnetometer (VSM). The re- sistivity was characterized by traditional four-probe method. The complex permeability measurements of the ﬁlms were carried out with a PNA E8363B vector network analyzer usingthe micro-strip method from 100 MHz to 9 GHz. 21During the measurement, the sample was positioned in the middle ofmicro-strip with inner height of 0.8 mm between upper lineand ground plate, upper line width of 3.94 mm, length of9 mm, and microwave electromagnetic ﬁeld was conducted along the hard axis in-plane of all the ﬁlms. All the measure- ments mentioned were performed for as-deposited sampleswithout any thermal treatment and carried out at roomtemperature. III. RESULTS AND DISCUSSIONS Figure 2presents the static magnetic hysteresis loops measured along easy axis and hard axis of all as-depositedsamples with various oblique angles h. Here, easy axis is per- pendicular to the direction of oblique incidence plane for oursamples fabricated at 15 /C14/C20h/C2056/C14, which is consistent with the result in other papers.20,22The turn of easy axis to paral- lel to the incidence plane, found in obliquely evaporated Ni ﬁlms,23is not observed in all angle range here. There is noobvious in-plane uniaxial anisotropy in Fig. 2(a) and it is veriﬁed that the IPUMA can be effectively induced byoblique sputtering from Figs. 2(a) and2(b), which can be explained by the preferential orientation of crystallite caused by oblique sputtering (Ref. 24). The phenomenon that the loops along hard axis become more slanted and all the loops along the easy axis trend to a rectangle gradually with theincreasing oblique sputtering angle is observed from Figs. 2(b)–2(f) , which marks the gradually increased in-plane uni- axial anisotropy ﬁeld H k-stat. But for the sample deposited at h¼0/C14, a weak perpendicular anisotropy may exist in the ﬁlm according to the shape of the hysteresis loops, which is also found in our previous report about FeNi-ZnO granularﬁlm. 25As the oblique angle increases, in-plane uniaxial ani- sotropy increases and begins to predominate. When h¼15/C14, an obvious in-plane uniaxial anisotropy is obtained and per-pendicular anisotropy exhibits a weaker effect on the mag- netization process, which is reﬂected by the hysteresis loop shape. The weak perpendicular anisotropy gradually disap-pears for the samples fabricated at larger h.Especially, an appropriately large IPUMA H k-stat about 212 Oe is obtained ath¼56/C14in Fig. 2(f). The gradual increase in Hk-statmay be explained by the change of the growth orientation of the co- lumnar microstructure aroused by oblique deposition, which gradually deviates from the normal direction of the substrateas the gradually increasing incidence angle. 20,26,27 Moreover, all samples prepared by oblique sputtering are softer with the relatively smaller coercivities than the sampledeposited at the angle of zero. The further quantitative analy- sis of soft magnetic properties is given later. The oblique angle dependences of easy and hard axis coercivities ( H ceandHch), as well as the resistivity q, for all FeCoB-ZnO granular thin ﬁlms are shown in Fig. 3.I ti s observed obviously that both Hceand Hchdecrease ﬁrstly and then increase when hincreases from 0/C14to 56/C14and they both reach the minimum ( Hch¼2.6 Oe and Hce¼7.9 Oe) at FIG. 1. The schematic sketch of oblique deposition. FIG. 2. In-plane hysteresis loops of FeCoB-ZnO granular thin ﬁlms with various oblique angles h: (a)h¼0/C14, (b)h¼15/C14, (c)h¼26/C14, (d)h¼35/C14, (e) h¼45/C14, and (f) h¼56/C14.103902-2 Liu et al. J. Appl. Phys. 117, 103902 (2015)h¼35/C14. The growing large of coercivity at larger oblique angle may be caused by the inhomogeneous growth of sam-ples. The factors like stress and defects may play an impor- tant role to weak the homogeneity of as-samples deposited at larger angles and ﬁnally lead to gradually increasing coerciv-ity. However, the relatively small coercivity in all angleranges of 15 /C14–56/C14can still make samples keep appropriate soft magnetic property from the view of application. In our case, the resistivity of our samples slightly increases (shownin Fig. 3) from 955 lX/C1cm to 1026 lX/C1cm and is much larger than traditional FeCoB ﬁlms (Ref. 18), indicating that the eddy current loss can be suppressed in high frequency range. Figure 4shows the frequency dependence of effective complex magnetic permeability l¼l 0/C0il00(l0,l00are real and imaginary parts, respectively) for FeCoB-ZnO granularthin ﬁlms. There is no obvious resonance peak for the sampledeposited at h¼0 /C14, which corresponds to the isotropic hysteresis loop shown in Fig. 2(a). It is obviously seen thatthe ferromagnetic resonance frequency fFMR increases with increasing oblique angle h, which is corresponding to the shift of the peak of imaginary permeability toward a higher frequency range. The fFMR is expanded to about 5.3 GHz from 1.9 GHz as the hincreases from 15/C14to 56/C14, which reveals that the fFMRcan be effectively adjusted in a large range just by changing the oblique angle for high and even ultrahigh frequency application. The above behavior of fFMR is mainly caused by the improvement of the magnetic anisot- ropy ﬁeld Hk-stat. Moreover, LLG equation is used for quanti- tative analysis of this result28 dM dt¼/C0c~M/C2~H ðÞ þaef f M~M/C2d~M dt/C18/C19 : (1) Here, ~H,~M,aeffrepresent effective ﬁeld, magnetization, and the dimensionless effective damping coefﬁcient, respec-tively. The cexpresses the gyromagnetic ratio and is 1.9/C210 7Hz/Oe. By solving Eq. (1)with the assumption of macrospin approximation and the presence of only in-planeuniaxial anisotropy in the ﬁlms, the expressions of l 0andl" are obtained as follows: l0¼1þ4pMs4pMsþHk/C0dyn ðÞ 1þa2 ef f/C16/C17 x021þa2 ef f/C16/C17hi þ4pMsþ2Hk/C0dyn ðÞ aef fxðÞ2 x021þa2 ef f/C16/C17 /C0x2hi2 þaef fxc4pMsþ2Hk/C0dyn ðÞ ½/C1382; (2) l00¼4pcMsxaef fc24pMsþHk/C0dyn ðÞ21þa2 ef f/C16/C17 þx2hi x021þa2 ef f/C16/C17 /C0x2hi2 þxaef fc4pMsþ2Hk/C0dyn ðÞ ½/C1382; (3) where x0¼cﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ Hk/C0dynð4pMsþHkÞp ,Hk-dyn,4pMssign the dynamic magnetic anisotropy ﬁeld and saturation magnetiza-tion, respectively. Taking the H k-dynandaeffas the adjustable ﬁtting parameters and taking them into Eqs. (2)and(3), theLLG ﬁtting curves are in good agreement with the experi- mental effective permeability spectra. The oblique angle dependences of Hk-dyn and fFMR are plotted in Fig. 5together with the change of static magnetic FIG. 3. The dependences of coercivities along easy and hard axis and resis- tivity for FeCoB-ZnO granular thin ﬁlms on oblique angles. FIG. 4. The dependence of real ( l0) and imaginary ( l00) parts of the perme- ability spectra on the frequency for the samples deposited by oblique sputter- ing with different incidence angles: (a) h¼0/C14(b)h¼15/C14, (c) h¼26/C14, (d) h¼35/C14, (e)h¼45/C14, (f)h¼56/C14, and the solid lines mean the ﬁtting curves with LLG model.103902-3 Liu et al. J. Appl. Phys. 117, 103902 (2015)anisotropy Hk-stat. It is obviously found that the values of Hk- stat,Hk-dynandfFMRincrease monotonously in all angle range with the gradually increasing ob lique angles. The result shown in Fig. 5(b) can be explained by the equation: fFMR ¼c 2pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Hk/C0dynðHk/C0dynþ4pMsÞp ,w h e r et h e 4pMs/C2413.6 kG. However, there is a discrepancy between Hk-sta testimated from the slope of hard loops and Hk-dynobtained by ﬁtting the per- meability spectra. The discrepancy may be ascribed to the ex-istence of rotational magnetic anisotropy induced by dynamically internal ﬁeld. 29The behaviors of these parameters indicate that soft magnetic an d high frequency properties can be well tuned just by changing oblique angles. Figure 6shows the relations of initial permeability lini0 (the real component of the effective complex magnetic per- meability at low frequency) and maximum value of imagi- nary components lmax"versus the oblique angle for obliquely deposited samples. The phenomenon that both lini0 andlmax"decrease with the increasing oblique angle his observed obviously. However, the minimum value of lini0is beyond 60, indicating that good microwave characteristiccan still be kept. The relation of l ini0and effective magnetic anisotropy Heffcan be expressed by the equation as following:lini0¼1þ4pMs=Hef f: (4) Here, the oblique angle dependence of Heffcan be obtained, which has been shown in Fig. 5(a). It is found that the effec- tive magnetic anisotropy is larger than the static magneticanisotropy, which may be ascribed to rotatable magnetic ani-sotropy (so-called dynamically induced internal ﬁeld) 29at low macro-wave frequency during the permeability measure-ment. It is well known that the l max"can be determined by the equation as following:27 lmax00¼1 2lini0/C01/C0/C1ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ 1þ1 aef f2s ; (5) where the aeffis almost constant, which indicates that the behavior of lmax"shown in Fig. 6is ascribed to the decrease inlini0. The changes of frequency line-width Dfand effective damping factor aeffversus the oblique angle hare shown in Fig.7. It is found that Dfandaeffare both insensitive to the varying angle and decrease slightly with the increasingoblique angle h(a effdecreases from 0.036 to 0.03 and Df decreases from 1.49 to 1.27), which almost satisfy therequirement that f FMRneeds to be adjusted independently by various oblique angles in a large frequency range. The fre-quency line-width Dfcan be calculated by the following equation: 30,31 Df¼caef fð4pMsþ2Hk–dynÞ 2p: (6) Here, 4pMsis around 13.6 kG and much larger than the Hk-dyn. Therefore, the behavior of Dfis mostly ascribed to aeff. It is well known that effective damping parameter aeff consists of the intrinsic Gilbert damping aintand the extrinsic damping aext.aintis determined by the nature of material and origins of aextare complex. In our case, the material is the same for all the samples, so that the intrinsic contributionsare almost the same and the oblique angle dependence of a eff is mostly due to the extrinsic contributions. For the sample deposited at lower angle, both weak perpendicular anisotropyand in-plane anisotropy exist, resulting in magnetic inhomo-geneity and the dispersion of effective anisotropy ﬁeld. So,a extis relatively large for the samples fabricated at low h.A s hincreases, in-plane uniaxial anisotropy begins to FIG. 5. The dependence of (a) static magnetic anisotropy ﬁeld Hk-stat, effec- tive magnetic ﬁeld Heff, and the dynamic magnetic anisotropy ﬁeld Hk-dyn and (b) ferromagnetic resonance frequency fFMR of all the as-deposition samples by oblique sputtering on different oblique angles. FIG. 6. The changes of the initial permeability lini0and the maximum peak value lmax"of imaginary components of the permeability for all FeCoB- ZnO thin ﬁlms with various oblique angles. FIG. 7. The changes of Dfandaeffversus the incidence angle.103902-4 Liu et al. J. Appl. Phys. 117, 103902 (2015)predominate and the magnetic inhomogeneity becomes weaker. Thus, aextslightly decreases at some extent, which ﬁnally results in the slight decrease in aeff. Therefore, the sim- ilar behavior of Dfandaeffis mostly contributed by the pres- ence of extrinsic contributions, including magnetic inhomogeneity. IV. CONCLUSION In summary, the oblique angle dependences of soft mag- netic and high-frequency microwave properties for FeCoB- ZnO granular thin ﬁlms are systematically investigated. Avariable IPUMA ﬁeld from 27.6 Oe to 212 Oe and an adjust-able ferromagnetic resonance frequency from 1.89 GHz to 5.3 GHz were obtained in the as-deposited ﬁlms just by increasing the oblique angle from 15 /C14to 56/C14. The coerciv- ities along easy and hard axis decrease ﬁrstly and thenslightly increase with gradual increasing angle. The resistiv-ityqof our samples can reach about 1016 lX/C1cm, which can properly suppress eddy current loss coming with high fre- quency. The Dfanda effhave not obvious changes ( aeff decreases from 0.036 to 0.03 and Dfdecreases from 1.49 to 1.27), which satisfy the requirement that fFMRandaeffneed to be tuned independently in a large frequency range. Besides, LLG equation combining with the static magnetiza-tion is used to quantitatively analyze the micro-wave charac-teristics of samples sputtered by oblique sputtering. ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (NSFC) under Grant Nos.50901050 and 11204209. 1S. Jin, W. Zhu, R. B. van Dover, T. H. Tiefel, V. Korenivski, and L. H. Chen, Appl. Phys. Lett. 70, 3161 (1997). 2P. J. H. Bloemen and B. Rulkens, J. Appl. Phys. 84, 6778 (1998). 3K. Seemann, H. Leiste, and Ch. Klever, J. Magn. Magn. Mater. 322, 2979 (2010).4I. T. Iakubov, A. N. Lagarkov, S. A. Maklakov, A. V. Osipov, K. N. Rozanov, I. A. Ryzhikov, V. V. Samsonova, and A. O. Sboychakov, J. Magn. Magn. Mater. 321, 726 (2009). 5P. S. Wang, H. Q. Zhang, R. Divan, and A. Hoffmann, IEEE Trans. Magn. 45, 71 (2009). 6M. Gloanec, S. Dubourg, O. Acher, F. Duverger, D. Plessis, and A. Bonneau-Brault, Phys. Rev. B 85, 94433 (2012). 7T. Dastagir, W. Xu, S. Sinha, H. Wu, Y. Cao, and H. B. Yu, Appl. Phys. Lett. 97, 162506 (2010). 8Y. Yang, B. L. Liu, D. M. Tang, B. S. Zhang, M. Lu, and H. X. Lu, J. Appl. Phys. 108, 073902 (2010). 9C. Brosseau, J. B. Youssef, P. Talbot, and A. M. Konn, J. Appl. Phys. 93, 9243 (2003). 10J. B. Youssef and C. Brosseau, Phys. Rev. B 74, 214413 (2006). 11C. Brosseau and P. Talbot, J. Appl. Phys. 97, 104325 (2005). 12C. Brosseau, S. Mall /C19ego, P. Qu /C19effelec, and J. B. Youssef, Phys. Rev. B 70, 092401 (2004). 13C. Kittel, Phys. Rev. 73, 155 (1948). 14O. Acher, J. L. Vermeulen, P. M. Jacquart, J. M. Fontaine, and P. Baclet, J. Magn. Magn. Mater. 136, 269 (1994). 15Y. Fu, Z. Yang, T. Miyao, M. Matsumoto, X. X. Liu, and A. Morisako, Water. Sci. Eng. B 133, 61 (2006). 16X. 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). 17H. Ono, M. Ishida, M. Fujinaga, H. Shishido, and H. Inaba, J. Appl. Phys. 74, 5124 (1993). 18I. Kim, J. Kim, K. H. Kim, and M. Yamaguchi, IEEE Trans. Magn. 40, 2706 (2004). 19Y. Park, E. E. Fullerton, and S. D. Bader, Appl. Phys. Lett. 66, 2140 (1995). 20A. Lisﬁ, J. C. Lodder, H. Wormeester, and B. Poelsema, Phys. Rev. B 66, 174420 (2002). 21Y. Liu, L. Chen, C. Y. Tan, H. J. Liu, and C. K. Ong, Rev. Sci. Instrum. 76, 063911 (2005). 22Y. Hoshi, E. Suzuki, and M. Naoe, J. Appl. Phys. 79, 4945 (1996). 23T. Otiti, J. Mater. Sci. 39, 477 (2004). 24H. Geng, J. Q. Wei, Z. W. Wang, S. J. Nie, H. Z. Guo, L. S. Wang, Y. Chen, G. H. Yue, and D. L. Peng, J. Alloys Compd. 576, 13 (2013). 25X. Y. Zhou, D. S. Yao, W. J. Xie, J. N. Wei, and G. P. Xua, J. Alloys Compd. 533, 58 (2012). 26N. N. Phuoc and C. K. Ong, J. Appl. Phys. 111, 093919 (2012). 27Z. W. Li, G. Q. Lin, L. F. Chen, Y. P. Wu, and C. K. Ong, J. Appl. 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1.1740817.pdf	A Recording Vacuum Grating Spectrometer for the InfraRed E. E. Bell, R. H. Noble, and H. H. Nielsen Citation: Review of Scientific Instruments 18, 48 (1947); doi: 10.1063/1.1740817 View online: http://dx.doi.org/10.1063/1.1740817 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/18/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A New Source of Radiation for InfraRed Spectrometers Rev. Sci. Instrum. 20, 833 (1949); 10.1063/1.1741404 InfraRed Spectrometer Recording System Rev. Sci. Instrum. 19, 915 (1948); 10.1063/1.1741196 A Vacuum Spectrograph for the InfraRed Rev. Sci. Instrum. 19, 861 (1948); 10.1063/1.1741186 A New Source for InfraRed Spectrometers Rev. Sci. Instrum. 13, 63 (1942); 10.1063/1.1769973 A Recording Echelette Grating Spectrometer for the Near InfraRed Rev. Sci. Instrum. 13, 54 (1942); 10.1063/1.1769972 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 141.209.100.60 On: Sun, 21 Dec 2014 02:51:08THE REVIEW OF SCIENTIFIC INSTRUMENTS VOLUME 18, NUMBER 1 JANUARY, 1<)41 A Recording Vacuum Grating Spectrometer for the Infra-Red E. E. BELL, R. H. NOBLE, AND H. H: NIELSEN Mendenhall Laboratory of Physits, The Ohio State University, Columbus, Ohio (Received October 16, 1946) A description is given of a recording vacuum grating spectrograph for the infra-red region of the spectr~rr: from 1 micron to 30 microns. The description contains the details of the recording and amplifymg system and the method for stabilizing the power from the radiant source. Typical records are shown of well-known infra-red spectra. The lines from the two isotopes of HCl are completely separated in the fundamental band obtained in first order. The rotational lines of the 4.3-micron band of CO2 are completely separated in the first-order spectrum. In most parts of the spectrum it is possible to resolve lines separated by intervals less than 1 em-I. 1. INTRODUCTION IMPROVEMENTS in the technique of infra- red spectroscopy have made it possible to operate with slits narrow enough to subtend frequency intervals as small as 0.25 cm~l or less. To obtain a resolution equivalent to this it is necessary to make observations at settings on the spectrometer circle so close together that the equivalent frequency separations are at least as small, and preferably considerably smaller, than the frequency interval subtended by the slit. In this laboratory it has been found advantageous to make observations at intervals on the circle of five seconds of arc. The width of the average infra·red band measured with a grating suitable for the spectral region in which the band lies extends over a space on the circle of from one degree to a degree and a half. At five·second intervals this would require a total of nearly a thousand observations for a single "run." The measurement of infra·red bands at high disper. sion is therefore extremely time consuming and laborious and makes the adoption of some method of self recording very desirable. Self-recording has become an every day event for prism infra-red spectrographs but only a few attempts have been made automatically to record the highly dispersed spectra produced by a grating instrument. In only three previous instances known to the authors has self recording of such spectra been realized. The first of these is the vaCuum grating spectrograph constructed by Randall and Firestone! for the far infra-red. The second is the high dispersion spectrograph I H. M. Randall and F. A. Firestone, Rev. Sci. lnst. 9, 404 (1938). 48 constructed by Smith2 for the region lp to 25,u. A third self-recording spectrometer is'reported3 to have been built by Marcel V. Migeotte at the University of Liege for this same region of the spectrum. Concerning this last instrument no further information is available. In the spectral region l,u to 25,u there are many atmospheric bands, due principally to water vapor and carbon dioxide, which seriouslv inter fere with the study of the band spe~tra of molecules which have bands overlapping these atmospheric absorption regions. Smith has dealt with this problem by surrounding his spec trometer with an airtight box in which the air may be circulated and dried over P205• Though effective, the process is quite slow. In this article we shall describe a self-recording spectrometer of high dispersion where we have preferred to FIG. 1. Photograph of spectrograph showing circle drive, circle viewing microscope, slit controls, and prism setting control. 2 L. G. Smith, Rev. Sci. Inst. 13, 54 (1942). 3 P. Swings, Astrophys, J. 99, 118 (l944), This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 141.209.100.60 On: Sun, 21 Dec 2014 02:51:08R E COR DIN G V A C U U M G RAT I N G S PEe T ROM E T E R FOR I N F R A - RED 49 follow the example of Randall and Firestone by enclosing the spectrograph in a tank from which the water vapor and carbon dioxide may be removed by evacuation. II. THE SPECTROMETER The vacuum chamber in which the spec trometer is housed is made of steel plate of i" thickness and has a kettle-like shape as shown in Fig. 1. The chamber stands on a concrete pier and is supported by three legs which continue through the bottom of the tank and serve as supports for the spectrometer itself to stand on. The tank is evacuated through an opening in the bottom by means of a Cenco H ypervac oil pump. The vacuum chamber may be sealed by means of a cast steel top with a flange on the under side which fits into a groove in the top of the tank. The groove contains a rubber gasket. A vacuum of 0.5 mm Hg is sufficient to remove the atmospheric lines and this evacuation may be obtained in 30 minutes. The spectrometer itself rests on three leveling screws which turn on the three feet which support the vacuum chamber and which extend inside the tank. This insures that any flexing or deformation which the tank may suffer upon evacuation will not disturb the adjustment of the spectrometer. The radiation source is a Nernst filament fitted with platinum terminals in the manner described by Ebers and Nielsen.4 The Nernst filament so constructed can be operated at currents of as high as 1.3 amperes for rather long periods of time. When the spectrometer is not in use the current is turned down, but the filament is never shut off. The life of such a filament is of the order of months. A Nernst lamp, although not as good a blackbody emitter as certain other radiation sources, has the advantage, because of its size, that it may be operated at high tem peratures without needing a cooling jacket. It has never been found necessary to water cool the source. A Nernst filament has a negative temperature coefficient of resistance and has a resistance many times smaller when hot than when cold. If a glower is connected directly across a vol tage 4 Earle S. Ebers and Harald H. Nielsen, Rev. Sci. Inst. 11,429 (1940). C:ONSTANT VOLTAGE TRAN$FOFiM[R FIG. 2. Diagram of glower circuit. source it may, once started, continue to draw more and more current until it burns out. It is customary therefore when using a N ernst glower to place a resistor, which has a positive tem perature coefficient of resistance, in series with it to act as a ballast. In general the temperature coefficient of the ballast resistor is small com pared to the temperature coefficient of the Nernst filament and may to a good approxima tion be neglected. When this is so and the ballast resistance is made equal to the operating resistance of the glower, it may be shown that small fluctuations in the glower resistance produce practically no change in the glower power. The glower is thereby protected against power fluctuations due to changes in glower resistance arising from drafts across it or from glower evaporation, etc. This ballasting method produces, however, a percentage power fluctu ation equal to twice the percentage fluctuation of the voltage at the source. To insure a constant voltage source a constant voltage transformer should be used. The glower power then becomes practically independent of both line and glower fluctuations. In this spectrograph a result equivalent to the foregoing was achieved with half the total power consumption and about two thirds the source voltage by using a capacitive ballast. The circuit is illustrated in Fig. 2 and contains a constant voltage transformer which feeds a variable auto transformer which, in turn, supplies current to the glower and a capacitor connected in series. The capacitance bank and the variable trans former are adjusted so that the voltage across the condenser is equal (in magnitude) to the voltage across the glower at the desired glower operating power. The optical system is shown in Fig. 3. The radiation from the source N is collected by the mirror Ml in the spectrometer from which the light converges to A where the absorption cell is placed. After crossing, the beam is col- This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 141.209.100.60 On: Sun, 21 Dec 2014 02:51:0850 BELL, NOBLE, AND NIELSEN ...................................................... Tl[\ __ .. -... -_ .. -.-.... :. __ :.: .. : ~.~--\J~ lected by the mirror M2 and then focused on the slit Sl by reflection from the plane mirror Ma. Sl is the entrance slit of the monochromator section of the spectrometer and is at the focus of the collimating mirror M4• The light is rendered parallel by the mirror, and it is then dispersed by the 30° prism, P, through which it passes twice. The dispersed radiation energy is collected by M4 which produces a spectrum at S2, the exit slit of the monochromator. The slit S2 is also the entrance slit of the main grating spec trometer. To obtain the maximum purity in the spectrum produced at S2 by the monochromator the entrance slit Sl must be curved, the curva ture being given by the relation:6 p n2f ----coti, 2(n2-1) (1) 6 H. Kayser, Handbuch der Spectroscopie (S. Hirzel, Leipzig, 1900), VoL 1, p. 321. FIG. 3. Schematic diagram of the optical system of the spectro graph. where n is the refractive index of the prism material, f is the focal length of the collimating mirror, and i is the angle of incidence of the light to the prism. The mirror M6 is the main collimating mirror. It is an off-axis paraboloidal section, 10" in diameter and has a focal length of 1 meter. The optical axis passes through the edge of the mirror at O. The monochromatic radiation en tering at S2 is gathered by M6 and reflected on to the grating G as a parallel beam. These are echellete gratings 6" X 8" in size and mounted on a calibrated circle 24" in diameter. The spec trometer is of a Littrow design in which after diffraction the radiation is again collected by the mirror M6 which focuses the spectrum at the exit slit Sa. As the circle turns the spectrum moves across the slit Sa. The exit slit Sa is at the far focus of an elliptic mirror M6 and the infra-red detector T, which in this instrument is a com- This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 141.209.100.60 On: Sun, 21 Dec 2014 02:51:08RECORDING VACUUM GRATING SPECTROMETER FOR INFRA-RED 51 pen sated vVeyrich thermocouple, is at the near focus. The two foci are at distances of 35 and 7 cm, respectively, from the mirror. The radiation energy of the wave-length for which the grating is set passes through the slit S3 and an image reduced about five times is concentrated on one of the receivers of the thermocouple. The receivers are joined in series opposition to minimize drifts due to small temperature fluc tuations. The thermoelectric voltages produced as the spectrum passes across the thermocouple are led out of the spectrometer to the relay and amplifier through a shielded coaxial cable. The widths of the slits of the spectrometer may all be controlled from outside the vacuum chamber. Each control is a vacuum-sealed micrometer which is connected to the slit opening and closing mechanism by means of a flexible cable. The spectrometer is driven by an electric motor through a gear system. The shaft of the gear box connects through a copper sylphon joint with a worm gear meshing into a gear cut around the edge of the 24" circle. The circle may be turned in either direction at four different speeds in addition to one which is very rapid and is used to turn the circle from one region in the spectrum to another. The 24" graduated circle is a very fine one graduated by the Gaertner Scientific Company in Chicago, Illinois. It may be read through a window in the side of the tank by means of a microscope which contains a vernier eyepiece scale. The circle can be read accurately to about ± 1 second of arc. The fre quency position of a line in the spectrum may be obtained in one of two manners. It may be cal culated in the usual manner by means of the relation v=K. csdi, (2) where K.= (n/2d coso), n being the order of the spectrum, d the grating space, a half the angle subtended at the collimating mirror M6 by the centers of the slits S2 and S3 and 8 is the angle between the circle setting for the line being measured and the circle setting for the central or zero-order image. While K. may be calculated it is preferable in practice to evaluate it with the aid of a line (say, a line in the mercury spectrum) for which the frequency is known to great accuracy. A second method which is quite useful is the following. The shaft of a synchro-type generator is attached to one of the shafts of the gear box. This generator drives a synchro-type motor of which the shaft is attached to a Veeder counter. The arrangement is such that one count on the counter corresponds to the rotation of the circle through an angle of 1 minute. A cell containing a gas having absorption lines which lie in the same spectral region as the one that occupied the lines of the molecule under investigation and of which the frequency posi tions are known to a high degree of accuracy is inserted into the light beam. A record of this standard spectrum is then run and a careful account is kept of the counter numbers corre sponding to these standard lines. A calibration curve may then be made by plotting frequency positions against counter numbers. It is found that when the spectrograph is driven in the same direction the calibration curve repeats itself with great faithfulness. This curve may then be used to determine the frequencies of the lines in the spectrum under investigation if a careful record of the Veeder numbers is kept which correspond to the spectral lines. This record may be made manually or with the aid of a fiducial pen at tached to the recorder. ill. THE AMPLIFIER AND RECORDING SYSTEM The recording system6 was designed as a modification of a basic photoelectric bridge circuit1 used by McAllister, Matheson, and SweeneyS for infra-red recording. The function of the recording system is to produce an inked record of the variations in the amount of infra red energy passing through the spectrometer. Fundamentally this is accomplished by use of a thermocouple and a galvanometer. The deflec tions of the galvanometer resulting from varia tions in the energy received by the thermocouple, are watched by a photo-cell circuit. The photo cell circuit produces an output current propor- 6 E. E. Bell, Phys. Rev. 63, 461 (A) (1943). This ab stract states that the Moll galvanometer is bifilar. This was meant to mean that the galvanometer has a taut upper and taut lower suspension. 7 R. W. Gilbert, Proc. Inst. Radio Eng. 24, 1239 (1936). 8 E. D. McAllister, G. L. Matheson, and W. J. Sweeney, Rev. Sci. Inst. 12,314 (1941). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 141.209.100.60 On: Sun, 21 Dec 2014 02:51:0852 BELL, NOBLE, AND NIELSEN p.'~\ \~) ( \~Ol~--~ THERMOCOUPLE 0.10 : ~-'1.5V' I /' O-IOMA. I :.: I -1.5 V. ---I I I I LaN I rr,.-;;;+./Y"',-+-+'t-:--~ R EC. I 6V,32 C.P. I L' f\9"'T"rv--+ ........ t-J--UU-j-"E-A I c:J ~~. I I'", I I oJ <l :.: I <.> Q o @ I I --'~------ tional to the galvanometer deflection, and this current is recorded either by an Esterline-Angus 5 rna recording milliameter or a Leeds and Northrup Speedomax recorder. The recording system consists of four units: a relay (galvanometer and photo-cells), an am plifier and power supply, a control panel, and the Speedomax recorder. These units are connected together with multiconductor cables. Normally the amplifier and the relay are close together in the spectrograph room and the control panel and recorder are. in an adjacent room. The control panel and recorder are in easily movable mount ings and may be placed next to the spectrograph when desirable. ,/ I --' I PWR.I GAL I LIGHT I FIG. 4. Circuit diagram of ther mal relay, ampli fier, and control panel. Circuit val ues are given in ohms and micro farads. The thermocouple is connected to the relay through a copper tubing containing an insulated copper inner conductor which forms a coaxial line. This tubing runs from the thermocouple on the inside of the tank to the relay on the outside and is sealed with picein wax to prevent air leakage into the vacuum chamber. The relay consists of a light source and necessary optical system, a Moll galvanometer,9 a prism and a pair of photo-cells. The relay is shown in Fig. 4. The relay elements are fastened to a heavy metal base and the whole assembly is supported on a i-inch sponge rubber pad on the floor in one corner of the spectrograph room. iW. J. H. Moll, Proc. Phys. Soc. 35, 253 (1923). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 141.209.100.60 On: Sun, 21 Dec 2014 02:51:08R E COR DIN G V A C U U M G RAT I N G S PEe T ROM E T E R FOR I N F R A -RED 53 The lamp is a 32-candlepower headlight bulb the light of which passes through an aperture and lens system so that after reflection from the galvanometer mirror a rectangle of light about 1 cm X 2 cm in size falls on the beam splitting prism. The prism is a cheap 60° glass prism set in the position shown in Fig. 4. Internal reflection at the two back faces divides the light into two beams, each component leaving the prism normal to one of the prism faces. These two emergent light beams fall on a pair of gas-filled photo cells. A deflection of the galvanometer mirror alters the proportion of the light which falls on each of the photo-cells. The galvanometer to prism distance is about 30 cm. At this distance a thermocouple potential of 1 microvolt will pro duce a shift of the image of about 1.6 millimeters. Figure 4 also shows the circuit of the amplifier and control panel. In operation each of the 874 tubes maintains a constant 90-volt potential across itself and serves as a constant potential supply and as a fixed arm in the bridge circuit. The galvanometer is adjusted so that the amounts of light on the two photo-cells are practically equal. If a thermocouple voltage then deflects the galvanometer so that the amount of light on photo-cell (B) is increased, then an increased negative potential is applied to the grid of tube (2). Tube (2) operates as a voltage amplifier with a load consisting of the plate resistance of tube (1) and R in parallel. Consequently an amplified increase in voltage is applied to the control grid of tube (3). The additional current passed by tube (3) as a result of this change in grid poten tial, and not passed by tube (4), is passed through the recording meters. The control panel and rack are shown in Fig. 5. On this rack are mounted the control panel, and Esterline-Angus recording milliammeter, a self synchronous motor and counter for indicating the grating position, a reversing motor switch to operate the grating drive motor, and a switch to operate a shutter in the infra-red beam of the spectrometer. The control panel contains various power and recorder switches, a galvanometer zero adjustment circuit, a calibration circuit, and a feedback circuit to control the sensitivity of the system. The zero adjustment circuit introduces a small voltage into the galvanometer circuit to adjust the light on the photo-cells so that the recording meters may be set at a convenient position on the scale. This circuit is a high impedance source feeding current through a small resistance in series with the galvanometer and thermocouple. The small resistance in the galvanometer circuit is a few feet of stranded copper wire. The poten tial source is a single 1.5-volt dry cell. For ease of adjustment and for a large range both a coarse and a fine control are used. A panel mounted galvanometer shunted to make a 5-0-5 ma milliammeter is permanently installed in the recording meter circuit. This meter serves for the preliminary balancing of the system before the recording meters are switched into the circuit. This meter is not injured when the circuit is completely unbalanced because of the limited current output of the tubes. The calibration circuit operates similarly to the zero adjustment circuit except that decade ranges and a meter is included for measuring the FIG. 5. Photograph of control panel. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 141.209.100.60 On: Sun, 21 Dec 2014 02:51:0854 BELL, NOBLE, AND NIELSEN voltage applied into the galvanometer circuit. This circuit is useful in checking the sensitivity of the system and of the thermocouple. The feedback sensitivity circuit is also shown in Fig. 4. This circuit introduces a small voltage into the galvanometer circuit which is propor tional to the recorded current, (i.e., to the gal vanometer deflection) and in such a direction as to reduce the galvanometer deflection. This nega tive feedback serves to reduce the sensitivity of the system. The amount of the feedback intro duced is controlled by a potentiometer. A switch is included to remove the 5-ohm shunt from this circuit and thereby increase the amount of feedback available so that the central image may be recorded. The effect of this sensitivity negative feedback in reducing the galvanometer deflections is equivalent to a stiffening of the galvanometer fiber. Since the thermocouple has a resistance near the critical damping resistance of the gal vanometer, the addition of negative feedback makes the system under-damped. A capacitance feedback was therefore added to introduce a negative feedback voltage into the galvanometer circuit proportional to the time rate of deflection. The effect of this capacitance feedback is equiva lent to an increase in the electromagnetic damp ing of the galvanometer. After the sensitivity feedback has been adjusted to keep the deflection within the recording meter scC!.le limits the damping feedback is adjusted to damp the system critically. The galvanometer can be deflected by the calibration circuit or by the shutter in the infra-red beam to facilitate this damping adjustment. Care was taken in the construction of the thermocouple galvanometer circuit to insure good copper to copper contacts. and to protect the junction from thermal changes which would cause drifting of the system. A constant voltage transformer was used to feed the recording system in order to minimize line fluctuations. The photo-cells are protected from stray light fluctuations by a composition board shield surrounding the entire relay and amplifier. The drift of the recording system arising entirely in the thermocouple circuit has not been serious. A record of a spectral band is usable so long as the drift does not shift the line peaks or interfere with a judgment of the relative inten sities. The spectral records shown in Figs. 7, 8, and 9 were made after allowing only a few hours for the instrument to come to equilibrium. The records shown in Figs. 7 and 8 were each obtained with about one and one-half hours recording time. The record shown in Fig. 9 was obtained with about one-half an hour running time. The drift in our instrument could undoubtedly be reduced by properly shunting the more sensitive of the thermocouples in the detector and by a more careful isolation of the coaxial input cable so that the external grounded lead could not serve as a portion of an external thermoelectric circuit by being grounded in more than one place. We have found, however, that good records, can be obtained within 2 hours after opening the tank, changing the absorption cell, and re evacuating the tank. The merit of the recording system may be judged from the record shown in Fig. 6. This record was made using the Speedomax recorder with no feedback in the system. The r.m.s. value of the Brownian motion fluctuations of the gal vanometer mirror in the relay should correspond to about 1 X 10-9 volt in the galvanometer circuit. The calibration voltage applied was 2 X 10-8 volt. This record indicates that output fluctu ations are produced chiefly by the Brownian motion of the galvanometer. The sensitivity of the thermocouple is such that 3 X 10-4 microwatt of radiant energy will produce a galvanometer deflection equal to the r.m.s. value of the Brownian motion deflections. The record in Fig. 6 was obtained with the Cenco Hypervac pump used to evacuate the spectrograph running. This pump sits on the same floor as the relay and only about five feet away. With the pump not oper ating the fluctuations appear just as large as those seen in Fig. 6. This fact coupled with the fact that very little vibration insulation is used on the relay indicates that the Moll galvanometer is a good type to be used in such a relay. Since the photographs of the instrument were taken the control panel has been altered to include a counter and control for indicating and setting the fore prism position. It has been found necessary to slowly readjust the prism while recording across a spectral band at wave-lengths longer than 5 microns. It has been found con- This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 141.209.100.60 On: Sun, 21 Dec 2014 02:51:08R E COR DIN G V A C U U!VI G RAT I N G S PEe T R 0 !VI E T E R FOR I N F R A -RED 55 r I MINUTE (CALIBRATION 2XIO-e VOLTS FIG. 6. Speedomax record of noise and standard signal with no feedback. .,4 -L ~ '~ -.J U I :i U') M ..... CIl 0 ::r:: '-0 8 ::: b u ~ 0. (/J II) ...c:: .... .... 0 '1:l ... 0 u PE ...: 8 ~ ..... - ::i r- N ..... CIl ~ ::r:: .... OOtlllt-L '0 -r 8 a ~ ::: b u 8- ~ '" ~ II 0 ..., I " '0 i ~ '1:l ... § >:::: .. W:JO#lC; 00 8 ~ This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 141.209.100.60 On: Sun, 21 Dec 2014 02:51:0856 BELL, NOBLE, AND NIELSEN SLIT WIDTH • 0.20 eM" 2 <> .... '" '" '" FIG. 9. Record of the spectrum of CO2 at 4.3JL. venient also to change the calibration circuit to include a set of equal voltage steps. These steps can be adjusted with a potentiometer in the meter circuit and serve as an auxiliary zero adjustment to reset the recording meter on the central position of the scale after the instrument has drifted for some time. The reset steps are constant in size and do not involve the careful adjustment which is necessary when changing the zero adjustment potentiometers. IV. EXAMPLES OF SPECTRA RECORDED Records have been made with the new spec trometer of certain well-known spectra in order to investigate its performance. The fundamental absorption band of hydrogen chloride is shown in Fig. 7. The frequencies indicated on this record were obtain from the paper of Meyer and Levin,I° Figure 8 is a record of the 2.7 p. band in water vapor. The frequencies were taken from the manually recorded data of Nielsen,u Figure 9 is a record of the 4.3J.1. band in carbon dioxide 10 C. F. Meyer and A. A. Levin, Phys. Rev. 34, 44 (1929). 11 H. H. Nielsen, Phys. Rev. 62, 422 (1942). and the frequencies were taken from the paper of Nielsen and Yao.12 All of these spectra were obtained with a 7500 lines per inch replica grating made by R. W. Wood and used in the first order. Comparison of these records with those given in the references shows that the self recorded spectral resolution is equal to and in some cases better than the manually obtained resolution. In most parts of the spectrum, it is possible to resolve lines separated by intervals of less than 1 em-I. This spectrometer was constructed under the supervision of Mr. Carl McWhirt who presented many helpful suggestions relative to its design. Its construction was facilitated by grants-in-aid from the American Philosophical Society, Phila delphia, and The Research Corporation, New York. Work on the spectrometer was carried out by one of the authors (E.E.B.) under a pre doctoral fellowship of the National Research Council. For this helpfulness it is a pleasure for the authors to make acknowledgment. 12 A. H. Nielsen and Y. T. Yao, Phys. Rev. 68, 173 (1945). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 141.209.100.60 On: Sun, 21 Dec 2014 02:51:08
1.2177051.pdf	Asymmetric field variation of magnetoresistance in Permalloy honeycomb nanonetwork M. Tanaka, E. Saitoh, H. Miyajima, and T. Yamaoka Citation: Journal of Applied Physics 99, 08G314 (2006); doi: 10.1063/1.2177051 View online: http://dx.doi.org/10.1063/1.2177051 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 Investigation of Permalloy cross structure using magnetic force microscope and magnetoresistance measurement J. Appl. Phys. 99, 08B710 (2006); 10.1063/1.2177205 Domain structures and magnetic ice-order in NiFe nano-network with honeycomb structure J. Appl. Phys. 97, 10J710 (2005); 10.1063/1.1854572 Reversible and irreversible magnetoresistance of quasisingle domain permalloy microstructures J. Appl. Phys. 95, 6759 (2004); 10.1063/1.1688216 Stray fields of domains in permalloy microstructures—Measurements and simulations J. Appl. Phys. 95, 5641 (2004); 10.1063/1.1697642 Micromagnetics and magnetoresistance of a Permalloy point contact Appl. Phys. Lett. 74, 422 (1999); 10.1063/1.123048 [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: Sun, 30 Nov 2014 12:02:16Asymmetric ﬁeld variation of magnetoresistance in Permalloy honeycomb nanonetwork M. T anaka,a/H20850E. Saitoh, and H. Miyajima Department of Physics, Keio University, Hiyoshi 3-14-1, Yokohama 223-8522, Japan T . Yamaoka SII Nano Technology Inc., Takatsuka-Shinden 563, Matsudo 270-2222, Japan /H20849Presented on 1 November 2005; published online 27 April 2006 /H20850 The magnetic properties of two-dimensional network comprising a Permalloy wire-based honeycomb structure were investigated by magnetic force microscopy and magnetoresistancemeasurement. These results indicate that the magnetization of the wire behaves homogenously likea binary bit and that the magnetic interaction at the vertex governs this magnetization. This allowsus to achieve a magnetoelectronic device, based on the magnetic interaction among the wires.©2006 American Institute of Physics ./H20851DOI: 10.1063/1.2177051 /H20852 I. INTRODUCTION Recently, there has been extensive coverage of the study of laterally deﬁned nanoscale magnetic structures due to ad-vances in lithographic and magnetic measurementtechniques. 1–6The main motivation for studying nanoscale magnetic materials is the dramatic change in the magneticproperties that occurs when the magnetic length scale gov-erning certain phenomenon is comparable to the magneticelement size. We have already revealed that ferromagneticwire-based nanonetwork with honeycomb pattern showsfrustration due to the magnetic interaction among the wireswhen the magnetic moment in the wire behaves coherentlylike a spin. 7,8However, the frustration disappears in response to a decrease in the magnetic interaction at the vertices.9In this paper, we clarify that asymmetric ﬁeld variations ofmagnetoresistance in the Permalloy honeycomb system arederived from the domain wall location in each vertex. Theseresults imply a potential application for magnetoelectronicdevices, based on the frustration. II. EXPERIMENT Figure 1 shows a scanning electron microscope image of part of the Permalloy honeycomb nanonetwork system. Thesample was fabricated by the lift-off technique. A thin poly-methyl methacrylate resist /H20849ZEP-520 /H20850layer, 100 nm thick, was spin coated onto a thermally oxidized Si substrate. Afterprebaking, the desired pattern was drawn with electron beamlithography, followed by resist development. Subsequently,Permalloy ﬁlm was deposited at a rate of 0.1 nm/s by theelectron beam /H20849EB /H20850evaporator in a vacuum of 1 /H1100310 −8Torr. The sample was obtained after the resist mask was removed in solvent. The size of the honeycomb networkis as follows; width=50 nm, length=400 nm, and thickness=20 nm. The network system consists of 60 /H1100360 unit cells of the honeycomb structure. The magnetic domain structure of the sample was ob- served by means of magnetic force microscopy /H20849MFM,SPI4000/SPA300HV /H20850. A CoPtCr low moment probe was used in order to minimize the inﬂuence of the stray ﬁeldfrom the probe to the magnetic structure of the system. Ascanning probe microscope system, equipped with an evacu-ated /H208491.0/H1100310 −6Torr /H20850sample chamber, was used in dynamic force mode with an optimized quality factor of the probe of around 3000.10To measure the resistance of the system, two Cu electrodes were deposited at the edges of the network. Allmagnetoresistance /H20849MR /H20850measurements were performed at 77 K by applying a dc of 80 mA along the direction of J,a s shown in Fig. 1. III. RESULTS AND DISCUSSION Figure 2 /H20849a/H20850shows a MFM image for the remanent state after the application of an external magnetic ﬁeld /H2084910 kOe /H20850 perpendicular to the ﬁlm plane. In the MFM image, a leak- age ﬁeld signal caused by a domain wall is clearly observedat each vertex and no domain wall features are observed inthe wire parts. These results indicate that the magnetizationin the wire behaves coherently and that the magnetic prop-erty of the ferromagnetic network can be described in termsof the uniform magnetization in each wire and the magneticinteraction among the wires at the vertex. a/H20850Electronic mail: mtanaka@phys.keio.ac.jp FIG. 1. A scanning electron microscope image of a Permalloy honeycomb nanonetwork. The size of the wire system is as follows; wire width=50 nm, length=400 nm, and thickness=20 nm, respectively. Jdenotes the current direction for magnetoresistance measurements.JOURNAL OF APPLIED PHYSICS 99, 08G314 /H208492006 /H20850 0021-8979/2006/99 /H208498/H20850/08G314/3/$23.00 © 2006 American Institute of Physics 99, 08G314-1 [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: 128.114.34.22 On: Sun, 30 Nov 2014 12:02:16The intensity of each black or white contrast is almost equal, while there are four possible magnetic conﬁgurationsat the vertex, as shown in Figs. 2 /H20849b/H20850–2/H20849d/H20850,2/H20849d/H20850, and 2 /H20849e/H20850. Consider that the intensity of the contrast varies with theleakage ﬁeld corresponding to the magnetic conﬁguration atthe vertex. This indicates that the latter magnetic conﬁgura-tion is required to minimize the exchange energy. The mag-netization M i,i nt h e ith wire, is determined to be the vector sum ofMi, for the three wires jointed at the Nth vertex must not be zero vector. We describe the magnetic conﬁguration asa “one-in/two-out” or “two-in/one-out” magnetic conﬁgura-tion /H20851see Figs. 2 /H20849b/H20850and 2 /H20849c/H20850/H20852. The “three-in” or “three-out” magnetic conﬁgurations, in which the vector sum of M iat Nth vertex is zero vector, are unstable because of the large magnetic energy loss due to the abrupt magnetization rota-tion at the vertex /H20851see Figs. 2 /H20849d/H20850and 2 /H20849e/H20850/H20852. The micromag- netic simulation using OOMMF code was carried out in one vertex.11The cell size, the saturation magnetization MS, and the damping parameter /H9251are 5 nm, 1 T, and /H9251=0.01, re- spectively. Figures 3 /H20849a/H20850and 3 /H20849b/H20850show the results of the mi- cromagnetic simulations for the vertex of the system. Theconﬁgurations in Figs. 3 /H20849a/H20850and 3 /H20849b/H20850are the “one-in/two-out” and “two-in/one-out” magnetic conﬁgurations, respectively.The three-in or three-out magnetic conﬁgurations are unreal-izable in the simulation results. These results indicate that themagnetic interaction among the wires predominates over themagnetic conﬁguration. Recently we clariﬁed that thesemagnetic properties of the honeycomb system stably appearwhen the wire length of the honeycomb network is shorten. Figure 4 /H20849a/H20850shows a MR curve of the system at the ﬁeld angle /H9258=30°, where /H9258denotes the angle between the current Jand the projection of the ﬁeld Honto the ﬁlm plane /H20849see Fig. 1 /H20850. Before the MR measurement, the magnetic conﬁgu- rations were arranged by applying a magnetic ﬁeld Hini /H208495 kOe /H20850for/H9258=0°. The magnetic conﬁguration after the ap- plication of the ﬁeld Hiniis shown in Fig. 4 /H20849e/H20850. After apply- ing the magnetic ﬁeld H=−1.4 kOe, the resistance increases monotonically with increasing magnetic ﬁeld and reaches the value of 374.9 /H9024atH=0 kOe /H20849point A1 /H20850. The resistance is decreased with the ﬁeld variation from negative to positivesense, and it shows a steep jump at H=0.76 kOe. After reaching H=1.4 kOe, the ﬁeld is decreased with a peak of 375/H9024atH=0 kOe /H20849point A2 /H20850. The resistance also exhibits a jump at H=−0.86 kOe. This MR is interpreted by the aniso- tropic magnetoresistance effect which is produced by therapid reversals of the magnetization in the wires. 1 FIG. 2. /H20849a/H20850A magnetic force microscope image of a Permalloy honeycomb nanonetwork. The arrows denote the magnetization in the wires. /H20851/H20849b/H20850–/H20849e/H20850/H20852 The possible magnetic conﬁgurations at the vertex. FIG. 3. Magnetic conﬁgurations at the vertex part of the honeycomb systemfrom micromagnetic simulations /H20849 OOMMF code /H20850./H20849a/H20850The “one-in/two-out” conﬁguration. /H20849b/H20850The “two-in/one-out” conﬁguration. FIG. 4. /H20851/H20849a/H20850–/H20849d/H20850/H20852The magnetoresistance of the network system at the ﬁeld angle of /H9258=30° or −30°. Before the MR measurements, the magnetic con- ﬁgurations are arranged by applying magnetic ﬁeld Hinifor/H9258=0° or 180°. /H20849g/H20850The magnetic conﬁgurations at points A1 and A2. /H20849h/H20850The magnetic conﬁgurations at the points B1 and B2. The dashed lines in /H20849g/H20850and /H20849h/H20850 denote the magnetic domain walls. The magnetization characterized byblack arrows in /H20849g/H20850and /H20849h/H20850does not reverse in the MR measurement. J denotes the current for magnetoresistance measurements.08G314-2 T anaka et al. J. Appl. Phys. 99, 08G314 /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: 128.114.34.22 On: Sun, 30 Nov 2014 12:02:16This MR curve appears asymmetrical. This asymmetry is due to the position of the magnetic domain wall at the vertex.Due to the shape magnetic anisotropy, a high magnetic ﬁeldis needed to reverse the magnetization in the wire, whichcreates a right angle with the magnetic ﬁeld H/H20851black arrows in Fig. 4 /H20849g/H20850/H20852. Thus it would appear that this magnetization stayed unchanged in the MR measurement. Figure 4 /H20849g/H20850 shows the magnetic conﬁgurations at points A1 and A2 inFig. 4 /H20849a/H20850. The broken line in Fig. 4 /H20849g/H20850indicates the position of magnetic domain walls. At point A1, the magnetic domainwalls lie in the current ﬂow Jand strongly affect the resis- tance of the system. At point A2, the current ﬂow Jwas blockaged by the magnetic domain walls. Thus the magneticdomain walls contribute relatively little to the resistance. Theposition of the magnetic domain wall at the vertex bringsabout the difference of the resistance at H=0 kOe. The ori- gin of asymmetry is attributed to the minor loop while mea-suring the MR. Notable is that each magnetic conﬁgurationin Figs. 4 /H20849g/H20850and 4 /H20849h/H20850is stabilized. These asymmetric ﬁeld variations of the MR in a zigzag structure has been reported. 2 Figure 4 /H20849b/H20850shows the MR curve for /H9258=30° after apply- ing a magnetic ﬁeld Hini/H208495 kOe /H20850along /H9258=180°. Before the MR measurement, the magnetic conﬁgurations at the vertices are arranged, as shown in Fig. 4 /H20849f/H20850. The magnetic conﬁgura- tions at points B1 and B2 are shown in Fig. 4 /H20849h/H20850. As well as the case of Fig. 4 /H20849g/H20850, the magnetic domain walls lie on the different positions and the MR curve shows asymmetricvariation. The MR curve in Fig. 4 /H20849a/H20850has mirror symmetry with that in Fig. 4 /H20849b/H20850, while each of the MR measurements is performed by applying the magnetic ﬁeld at /H9258=30°. This is due to the difference in the position of the magnetic domainwall at the vertex. When comparing the magnetic conﬁgura-tion at A1 with that at B1, the positions of the magneticdomain walls at the vertices are different due to the magne-tization vector perpendicular to the magnetic ﬁeld H, indicat- ing that the domain walls act different effects on magnetore-sistance. As well as the MR measurements at /H9258=30°, part of thewires is perpendicular to the magnetic ﬁeld Hwhen the ﬁeld His applied along /H9258=−30°. Figures 4 /H20849c/H20850and 4 /H20849d/H20850show the MR for /H9258=−30° at the initial magnetic ﬁelds Hinialong /H9258 =0° and /H9258=180°, respectively. The MR curves appear asym- metrical. These MR curves can be explained by the positionof the magnetic domain walls without contradiction. IV. SUMMARY The magnetic properties of the Permalloy honeycomb nanonetwork were investigated by the MFM and MR mea-surement. These results reveal that the magnetization in thewire is a single domain and that it behaves like a binary bit.The magnetic energy at the vertex is dominant to the mag-netization in the wire and the magnetization in the wiresinteracts with each other. We revealed that the asymmetricMR curves for /H9258=30° and −30° are due to the position of the magnetic domain wall. ACKNOWLEDGMENT This work was supported by Grants-in-Aid for Scientiﬁc Research from the Ministry of Education, Culture, Sports,Science and Technology, Japan. 1J.-E. Wegrowe, D. Kelly, A. Franck, S. E. Gilbert, and J.-Ph. Ansermet, Phys. Rev. Lett. 82, 3681 /H208491999 /H20850. 2T. Taniyama, I. Nakatani, T. Yakabe, and Y. Yamazaki, Appl. Phys. Lett. 76,6 1 3 /H208492000 /H20850. 3D. A. Allwood, G. Xiong, M. D. Cooke, C. C. Faulkner, D. Atkinson, N. Vernier, and R. P. Cowburn, Science 296,2 0 0 3 /H208492002 /H20850. 4C. C. Faulkner, D. A. Allwood, M. D. Cooke, G. Xiong, D. Atkinson, and R. P. Cowburn, IEEE Trans. Magn. 39, 2860 /H208492003 /H20850. 5F. J. Castaño, K. Nielsch, C. A. Ross, J. W. A. Robinson, and R. Krishnan, Appl. Phys. Lett. 85, 2872 /H208492004 /H20850. 6T. Uhlig and J. Zweck, Phys. Rev. Lett. 93, 047203 /H208492004 /H20850. 7E. Saitoh, M. Tanaka, H. Miyajima, and T. Yamaoka, J. Appl. Phys. 93, 7444 /H208492003 /H20850. 8M. Tanaka, E. Saitoh, H. Miyajima, and T. Yamaoka, J. Magn. Magn. Mater. 282,2 2 /H208492004 /H20850. 9M. Tanaka, E. Saitoh, H. Miyajima, T. Yamaoka, and Y. Iye, J. Appl. Phys. 97, 10J710 /H208492005 /H20850. 10T. Yamaoka, K. Watanabe, Y. Shirakawabe, K. Chinone, E. Saitoh, M. Tanaka, and H. Miyajima, IEEE Trans. Magn. 41, 3733 /H208492005 /H20850. 11M. J. Donahue, http://math.nist.gov/oommf/08G314-3 T anaka et al. J. Appl. Phys. 99, 08G314 /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: 128.114.34.22 On: Sun, 30 Nov 2014 12:02:16
1.3491181.pdf	Analysis of passive scalar advection in parallel shear ﬂows: Sorting of modes at intermediate time scales Roberto Camassa, Richard M. McLaughlin, and Claudio Viotti Department of Mathematics, University of North Carolina, Chapel Hill, North Carolina 27599, USA /H20849Received 15 April 2010; accepted 9 August 2010; published online 4 November 2010 /H20850 The time evolution of a passive scalar advected by parallel shear ﬂows is studied for a class of rapidly varying initial data. Such situations are of practical importance in a wide range ofapplications from microﬂuidics to geophysics. In these contexts, it is well-known that the long-timeevolution of the tracer concentration is governed by Taylor’s asymptotic theory of dispersion. Incontrast, we focus here on the evolution of the tracer at intermediate time scales. We show how intermediate regimes can be identiﬁed before Taylor’s, and in particular, how the Taylor regime canbe delayed indeﬁnitely by properly manufactured initial data. A complete characterization of thesorting of these time scales and their associated spatial structures is presented. These analyticalpredictions are compared with highly resolved numerical simulations. Speciﬁcally, this comparisonis carried out for the case of periodic variations in the streamwise direction on the short scale withenvelope modulations on the long scales, and show how this structure can lead to “anomalously”diffusive transients in the evolution of the scalar onto the ultimate regime governed by Taylordispersion. Mathematically, the occurrence of these transients can be viewed as a competition in theasymptotic dominance between large Péclet /H20849Pe/H20850numbers and the long/short scale aspect ratios /H20849L Vel/LTracer /H11013k/H20850, two independent nondimensional parameters of the problem. We provide analytical predictions of the associated time scales by a modal analysis of the eigenvalue problem arising in the separation of variables of the governing advection-diffusion equation. The anomaloustime scale in the asymptotic limit of large kPe is derived for the short scale periodic structure of the scalar’s initial data, for both exactly solvable cases and in general with WKBJ analysis. Inparticular, the exactly solvable sawtooth ﬂow is especially important in that it provides a short cutto the exact solution to the eigenvalue problem for the physically relevant vanishing Neumannboundary conditions in linear-shear channel ﬂow. We show that the life of the corresponding modesat large Pe for this case is shorter than the ones arising from shear free zones in the ﬂuid’s interior.A WKBJ study of the latter modes provides a longer intermediate time evolution. This part of theanalysis is technical, as the corresponding spectrum is dominated by asymptotically coalescingturning points in the limit of large Pe numbers. When large scale initial data components are present,the transient regime of the WKBJ /H20849anomalous /H20850modes evolves into one governed by Taylor dispersion. This is studied by a regular perturbation expansion of the spectrum in the smallwavenumber regimes. © 2010 American Institute of Physics ./H20851doi:10.1063/1.3491181 /H20852 I. INTRODUCTION The advection-diffusion of a passive scalar is a pivotal problem in mathematical physics, the intense efforts spent onthe subject are witnessed by the large amount of literature/H20849an overview of theoretical developments with applications can be found, for instance, in Ref. 1, see also the recent survey 2on turbulent mixing /H20850. A number of factors can char- acterize the complexity of the problem /H20849e.g., dimensionality, structure of the velocity ﬁeld, and boundary conditions /H20850but much insight can be gained by focusing on simpliﬁed ﬂowconﬁgurations, where essential mechanisms can be isolatedand made amenable to complete mathematical analysis. Inthis work, we focus on simple steady parallel shear ﬂows,where we manage to characterize the interplay between arbi-trary tracer scales, advection, and diffusion. An example of ﬂows in this class is of course the one considered in the seminal work by Taylor. 3Since then, only some attention has been paid to the full evolution from initialdata to the long-time limiting behavior, which is governed by a one-dimensional renormalized diffusion equation and thusallows for a concise description of the evolution. The ﬁnite-time features of the problem have been considered by someauthors focusing on the identiﬁcation of transient stages act-ing on intermediate time scales. Transient dynamics can bephysically relevant in many situations. For example, the dis-persion of pollutants in rivers 1can occur at very large values of Péclet number, thus delaying the onset of the Taylor re-gime beyond those times that are physically relevant for, e.g.,monitoring purposes. Furthermore, passive scalar dynamicscan show up in more general contexts where intermediatetime evolution becomes the focus of interest. Examples in-clude the studies by Spiegel and Zalesky 4and Doering and Horsthemke5that recognize eigenmodes of the advection- diffusion problem to be a basic ingredient in the stabilityanalysis of an advection-diffusion-reaction system. By using a free-space solution introduced by Lighthill, 6PHYSICS OF FLUIDS 22, 117103 /H208492010 /H20850 1070-6631/2010/22 /H2084911/H20850/117103/16/$30.00 © 2010 American Institute of Physics 22, 117103-1Latini and Bernoff7have studied the complete evolution of /H9254-function initial data in axially symmetric parabolic shears, and compared this solution with short-time asymptotics andstochastic simulations. These authors have shown that thesolution exhibits two different time scales, marking the sepa-ration between three different regimes of dispersion: baremolecular diffusivity, anomalous superdiffusion, and Taylordispersion. The ﬁrst time scale is well-known to be verydependent on initial conditions, as Camassa et al. 8have rig- orously shown for pipe ﬂows; the second one is often closeto the cross-stream diffusive time r 2/D/H20849where ris the pipe diameter and Dthe diffusivity /H20850. Since one focus of our in- vestigation is the questioning of r2/Das the universal time scale that marks the transition into the Taylor regime, we willrefer to the time scale in which the homogenized equationbecomes a good approximation to the evolution as the“Taylor regime time scale,” to be distinguished from theabove diffusive time scale in the cross-stream direction. The behavior is very different if the passive scalar pos- sesses some intrinsic scale, such as that arising by imposingperiodic boundary conditions. The coupling between convec- tion and bare molecular diffusion in such setups can stillresult in overall anomalous diffusion, but distinguished fromthe classical Taylor regime due to the existence of long-livedmodes 9/H20849see also Ref. 10for scaling arguments /H20850. Camassa et al.9have recently considered this category of ﬂows in a study aimed at investigating the effect of shear on the statis-tical evolution of a random, Gaussian, and small scale distri-bution of dye. They observed that the probability distributionfunction /H20849PDF /H20850migrates toward an intermittent regime /H20849stretched-exponential tailed PDF /H20850. The physical picture that emerged was as follows. The scalar was ﬁrst seen to experi-ence an initial phase of stretching and ﬁlamentation, withﬂuctuations most efﬁciently suppressed within regions ofhigh shear. This is the “rapid expulsion” mechanism inRhines and Young. 10In a subsequent stage, the longest-lived concentration of dye was near shear-free regions, wheresome equilibrium between stretching and diffusion limits fur-ther distortion. This stage of the evolution was attributed tobe the collapse of the system onto a ground-state eigenmodeof an associated spectral problem. This suggests that such a“modal phase” of the evolution could play a role in nonpe-riodic problems on intermediate time scales, and we exploresuch scenario in the bulk of this paper. We remark that thisviewpoint is also used by Sukhatme and Pierrehumbert, 11 who described the scalar evolution with more complex ve-locity ﬁelds in terms of emerging self-similar eigenmodes. Our study of the modal evolution of an advected passive tracer at intermediate time scales is organized as follows. InSec. II, the formulation of the eigenvalue problem derivedfrom the advection-diffusion equation is presented. This willlead to a periodic second order nonself-adjoint operator.While the spectral theory for second order self-adjoint peri-odic equations /H20849Floquet theory /H20850is rather complete, 12for the more general case the full characterization of the spectrum isin general an open question /H20849which has recently been exam- ined in the context of the so-called PT-symmetry in quantum theory, where however most of the attention is focusedonly on the real part of the spectrum, see for example,Bender et al. , 13and some existence results for complex spec- trum have been obtained by Shin14/H20850. Here we derive simple bounds for the complex spectrum, while in Sec. III, we char-acterize the ordering of modes in three classes depending onthe interplay of advection with diffusion dictated by the lim-its /H9280→/H11009and/H9280→0, where /H9280=1//H20849kPe/H20850. The limit /H9280→0i s further classiﬁed into two categories depending on the rela- tive ordering with respect to the balance k=O/H20849Pe/H9251/H20850where the exponent /H9251is shown to depend on the smoothness properties of the velocity proﬁle. In the ﬁrst limit /H20849/H9280→/H11009/H20850, a straight- forward regular perturbation expansion sufﬁces to compute the spectrum, spanning the Taylor regime. In the second limit/H20849 /H9280→0/H20850, for the simplest cases such as piecewise linear shear layers, the analysis can be worked using exact techniques /H20849presented in Appendix B /H20850, while for more general cases, we use WKBJ asymptotics /H20849such method has proved to be useful in self adjoint problems, such as those arising in quantummechanics /H20850. We further discuss how this exactly solvable, piecewise-linear shear actually provides a shortcut to the ex-act solution for the physically relevant case involving van-ishing Neumann boundary conditions, and givesrise to thin boundary layers and decay rates scaling suchas /H92801/3as/H9280→0. In contrast, we establish the different scal- ings of /H92801/4and/H92801/2for the spatial internal layers and their decay rates, respectively, for generic locally quadratic shearﬂows. In Sec. IV, we present a study of the passive scalar evo- lution comparing the theoretical predictions with numericalsimulations. We test the predictive capabilities of the theoryon a set of numerical experiments focusing on both single-and multiscale initial data. In particular, the theory identiﬁesnew intermediate time scales, which are missed by classicalmoment analysis, and connects them to the spatial scales ofthe initial data. By manipulating the initial data, we can ex-tend the transient features associated with these intermediatetime scales beyond the Taylor time scale r 2/D. II. THE EIGENVALUE PROBLEM The advection-diffusion equation, assuming the velocity ﬁeld to be a parallel shear, is Tt+u/H20849y/H20850Tx=P e−1/H116122T, /H208491/H20850 with /H116122=/H20849/H115092//H11509x2,/H115092//H11509y2/H20850. We consider the problem to be periodic in the cross-ﬂow direction y, and it is understood a nondimensionalization based on the maximum velocity U and on a vertical length scale Lvelof the shear in such a way to ﬁx the yperiod as 2 /H9266. The Peclét number Pe is based on such scales and on the molecular diffusivity D, and it mea- sures the relative importance of advection and diffusion. Thevelocity ﬁeld is a parallel shear layer with velocity pointingalong xand dependent on y. We assume that the initial data T/H20849x,y,0/H20850admits a Fou- rier integral representation with respect to x, linearity and homogeneity in xguarantee the different Fourier components to be uncoupled. A solution of Eq. /H208491/H20850is expanded as117103-2 Camassa, McLaughlin, and Viotti Phys. Fluids 22, 117103 /H208492010 /H20850T/H20849x,y,t/H20850=/H20885 −/H11009/H11009 dk/H20858 n=0/H11009 an/H20849k/H20850/H9274n/H20849y,k/H20850e/H9275nt+ikx, /H208492/H20850 being /H9274n/H20849y/H20850, the eigenfunction basis associated with the wavenumber k, and /H9275n, the corresponding complex fre- quency. The freedom in kintroduces a second length scale LTracer, which is connected to the initial data on Eq. /H208491/H20850. Us- ing Eq. /H208492/H20850into the evolution equation, and projecting onto the adjoints, the eigenfunctions are found to satisfy /H20851/H9275+iku/H20849y/H20850/H20852/H9274=P e−1/H20849−k2/H9274+/H9274yy/H20850. Introducing /H9280=1//H20849kPe/H20850,/H9261=/H9275/k+k2/H9280, we ﬁnally write the periodic eigenvalue problem in normal- ized form /H20877/H9280/H9274yy=/H20851/H9261+iu/H20849y/H20850/H20852/H9274, /H9274/H20849−/H9266/H20850=/H9274/H20849/H9266/H20850,/H20878, /H208493/H20850 for the eigenfunction /H9274/H20849y/H20850and the eigenvalue /H9261. Notice that when the solution of Eq. /H208491/H20850is real, the eigenvalue-eigenfunction pairs satisfy /H9261/H20849/H9280/H20850=/H9261/H11569/H20849−/H9280/H20850,/H9274/H20849/H9280/H20850=/H9274/H11569/H20849−/H9280/H20850, /H20849here superscript/H11569indicates the complex conjugate /H20850. Note that without loss of generality /H9280can be regarded as a positive deﬁnite quantity. A. Exact estimates on /H9261 It is possible to show a priori that/H9261can lie only inside an horizontal strip of the complex plane. Writing separatelythe real and imaginary parts /H9261=/H9261 R+i/H9261I, we intend to estab- lish that /H9261R/H110210, − 1 /H11021/H9261 I/H110211. /H208494/H20850 Multiplying both sides of Eq. /H208493/H20850by/H9274/H11569, and summing each side of the resulting equation to the adjoint part is obtainedby the relation /H20849 /H9274/H11569/H9274y+/H9274/H9274y/H11569/H20850y−2/H20841/H9274y/H208412=2/H9280−2/H9261R/H20841/H9274/H208412. Integrating over the period P, we have −/H20885 P/H20841/H9274y/H208412dy=/H9280−2/H9261R/H20885 P/H20841/H9274/H208412dy, which implies the ﬁrst one of Eq. /H208494/H20850. If, otherwise, we re- peat the procedure subtracting the adjoint part, we obtain /H20849/H9274/H11569/H9274y−/H9274/H9274y/H11569/H20850y=2/H9280−2/H20849/H9261I+u/H20850/H20841/H9274/H208412, and the integration now yields /H20885 P/H20849/H9261I+u/H20850/H20841/H9274/H208412dy=0 . Such an expression implies that, in order for the integral to vanish, the quantity /H9261I+uhas to change sign within P. Hence /H9261I=u/H20849/H9264/H20850for some /H9264/H33528Pand the second one of Eq. /H208494/H20850 follows.III. THREE CLASSES OF MODES The problem /H208493/H20850can be studied in the two possible asymptotic limits /H9280→0 and /H9280→/H11009. Thinking in terms of Pe, large but ﬁxed, this represents a subdivision of the modesinto a high- and a low-wavenumber category with qualita-tively different properties. For ksmall enough we expect to ﬁnd a class of modes that behave in agreement with theTaylor renormalized-diffusivity theory, and that belongs tothe realm of homogenization theory. Within the solutions ofEq. /H208493/H20850in the /H9280→/H11009limit will be found a class referred as Taylor modes. In the opposite limit /H20849as we shall see later /H20850, the problem otherwise acquires a WKBJ structure, in thiscase we shall use the term WKBJ- or anomalous modes. Westress that the latter class of modes is more correctly to beconsidered as an intermediate-asymptotic category. Indeed,letting k→/H11009, diffusivity will, at some point, eventually dominate over the eigenvalue /H9261. This regime will be dis- cussed as the pure-diffusive mode. A. The limit /H9280\/H11557: Taylor modes In such limit, if the following expansions in /H9280are assumed: /H9274n=/H20858 j=0+/H11009 /H9280−j/H9274nj, /H208495/H20850 /H9261n=/H20858 j=0+/H11009 /H9280−j+1/H9261nj, /H208496/H20850 then a regular perturbation problem is found. The use of the above expansion inside the eigenvalue problem /H208493/H20850leads to a classical recursive system of equations O/H20849/H9280/H20850:L/H20851/H9261n0/H20852/H9274n0=0 , O/H208491/H20850:L/H20851/H9261n0/H20852/H9274n1=/H20851/H9261n1+iu/H20849y/H20850/H20852/H9274n0, ] O/H20849/H92801−m/H20850:L/H20851/H9261n0/H20852/H9274nm=/H20849/H9261n1+iu/H20849y/H20850/H20850/H9274nm−1 +/H20858 p=1m−1 /H9261np+1/H9274nm−p−1, where L/H20851/H9261n/H20852=d2/dy2−/H9261n. The same recursive problem was also derived by Mercer and Roberts,15whose starting point was a center manifold approach. At O/H20849/H9280/H20850, and normalizing to unitary L2-norm /H20648f/H206482=/H20848−/H9266/H9266f2dx, we have /H9274n0= cos ny//H20881/H9266,/H9261n0=−n2,/H20849n/H110220/H20850 =1//H208812/H9266, =0, /H20849n=0/H20850,/H208497/H20850 for symmetric modes, and /H9274n0= sin ny//H20881/H9266,/H9261n0=−/H20873n+1 2/H208742 ,/H20849n/H113500/H20850, /H208498/H20850 for the asymmetric ones. The longest-lived of all the above modes is the n=0 element in the symmetric class. This will be referred as the117103-3 Analysis of passive scalar advection in parallel shear ﬂows Phys. Fluids 22, 117103 /H208492010 /H20850Taylor mode . The corresponding eigenvalue is smaller than O/H20849/H9280/H20850, hence it requires us to proceed to higher orders to compute it. It turns out also to be the only eigenvalue with nontrivial regularly diffusive scaling. Corrections to the eigenvalue are gives by the solvability condition which the right-hand side is enforced to satisfy atany higher order. At O/H208491/H20850and O/H20849 /H9280−1/H20850, these are /H9261n1=− /H20851iu/H20849y/H20850,1/H20852, /H9261n2=− /H20853/H20851/H9261n1+iu/H20849y/H20850/H20852/H9274n1,1/H20854, where the notation /H20849·,·/H20850denotes the standard inner product. The ﬁrst equation expresses the physical fact that Taylor modes travel with the mean ﬂow speed. This corresponds topurely imaginary /H9261 n1that is the phase speed of the mode. The second equation yields a real decay rate. For a cosineproﬁle homogenization theory would yield the renormalizeddiffusivity D eff=/H208491/2/H20850U2Lvel2/D. Here this corresponds to n=0, which after some algebra yields /H926101=0 and /H926102=−1/2. To summarize, in the Taylor modes limit /H9280→/H11009eigenval- ues are given by /H9261n/H11011/H20902−/H9280n2+O/H20849/H9280−1/H20850,n/H110220 −1 2/H9280−1+O/H20849/H9280−2/H20850.n=0 ,/H20903/H208499/H20850 up to the ﬁrst nontrivial order. B. The limit /H9280\0: Anomalous modes As/H9280→0, we expect to ﬁnd a class of modes in which the ground-state element reproduces the long-lived structuresobserved in a periodic domain of O/H208491/H20850period, at large Pe by Camassa et al. 9We employ a different approach from the regular perturbation expansion adopted for the Taylor modes.Asymptotics are obtained via WKBJ method. At ﬁrst, this isdone to generalize the classical matched-asymptotics calcu-lation for real, self-adjoint operators /H20849in the classical litera- ture often referred as the two-turning-point problems as seenin Ref. 16/H20850. Here, however, the nonself-adjoint character of the problem presents additional complication and a reﬁne-ment of the technique is required. A second time, a regular-ized variant of the method is derived and the accuracy of thetwo approaches is compared. Before developing theasymptotic analysis, we introduce two particular exactlysolvable cases. 1. Exactly solvable linear case The ﬁrst exactly solvable special case consists in a nonanalytic “sawtooth” shear proﬁle /H20849with full details re- ported in Appendix B /H20850. Eigenfunctions in this case are con- structed using piecewise patched Airy functions, and eigen-values correspond to zeros of certain combinations of thesefunctions that can then be computed with systematic asymp-totics. The end result is that the decay rate scales as O/H20849 /H92801/3/H20850, a boost over the bare diffusivity O/H20849/H9280/H20850. Such scaling is also obtained by Childress and Gilbert17for a linear-shear chan- nel with homogeneous Dirichlet boundary conditions for thescalar /H20849which would correspond to the case of antisymmetric modes in our setup /H20850. We emphasize that this scaling differsfrom the generic case involving an analytic shear ﬂow, whose decay rate scales as O/H20849 /H92801/2/H20850as we show next. We remark that such differences are physically consistent. As mentioned in Sec. I, shear enhances diffusion and hence theabsence of shear-free regions yields, in fact, for large Pe, astronger damping of the modes. Nonetheless, perhaps sur-prisingly, even in this case a long-lived mode persists aroundthe corners, which has a counterpart in the analytic case inthe near shear-free regions as shown numerically in Ref. 9. However, as also shown in Appendix B, the eigenfunctionslocalize to a thinner region that scales such as O/H20849 /H92801/3/H20850as opposed to O/H20849/H92801/4/H20850for the analytic case, conﬁrming previous numerical ﬁndings in Ref. 9/H20849this comparison is depicted in Fig. 1, where the corresponding long-lived modes have the appearance of “chevrons” elongated in the streamwisedirection /H20850. While the sawtooth shear proﬁle is amenable to exact analysis, one may think that its physical signiﬁcance wouldbe per se limited. However, note that the symmetric subclassof the sawtooth eigenfunctions are exact solutions for theproblem involving a linear shear between two impermeablewalls /H20849T y=0 there /H20850, and hence the present analysis is physi- cally relevant. Moreover, the analysis of the sawtooth bringsforth the true essence of the boundary conditions’ effect.Generically, all ﬂows near a nonslip ﬂat wall will localize as/H20849weakly nonparallel /H20850linear shear and hence the sawtooth theory predicts the main structures of the scalar’s wall-boundary layer. This emerges clearly in the case of Poiseuilleﬂow in a channel, shown below in Fig. 2, following the asymptotic treatment below. 2. Exactly solvable cosine shear Our second example of an exactly solvable case is u/H20849y/H20850=cos y. The eigenvalue problem /H208493/H20850in this case reduces to the complex Mathieu equation FIG. 1. Ground-state mode for /H9280=10−3/H20849k=0/H20850, Pe=1000, comparison be- tween cosine u/H20849y/H20850=cos /H20849y/H20850and sawtooth u/H20849y/H20850=1− /H20841y/H20841shear proﬁles, respec- tively, top and bottom pictures. For the cosine ﬂow case, the eigenfunction isconstructed using Hermite uniform asymptotic approximation, for the saw-tooth it is computed exactly using Airy functions /H20849discussed in the text /H20850.117103-4 Camassa, McLaughlin, and Viotti Phys. Fluids 22, 117103 /H208492010 /H20850/H9280/H9274yy=/H20849/H9261+icosy/H20850/H9274, /H2084910/H20850 whose eigenfunctions can be written as /H9274/H20849y/H20850=S/H20849a,b,y/2/H20850, where Sis the /H9266-periodic Mathieu function and b=−2 i//H9280, a=−4/H9261//H9280. However asymptotics in the /H9280→0 limit are not immediately available for these functions, and it is more con-venient to resort to WKBJ methods. 3. WKBJ analysis We next examine analytic shear proﬁles which can be analyzed approximately with asymptotic WKBJ theory.Since the long-lived structures localize around the extremaofu/H20849y/H20850, we shall express the phase velocity of the WKBJ modes as a perturbation about y e, the location of a maximum/minimum of u/H20849y/H20850. Moreover, because for the cosine-ﬂow u/H20849ye/H20850=/H110061, we seek /H9261in the form /H9261/H11011a/H9280p+i/H20849b/H9280q−1/H20850,a s /H9280→0. /H2084911/H20850 In our derivation, we use a free-space approximation. Since we are interested in spatially localized eigenmodes, whichare rapidly vanishing away from the extrema of u/H20849y/H20850,w e shall drop the periodic boundary conditions in favor of a decay condition of the eigenfunctions. a. Eigenvalues from singular WKBJ solution. Turning points play a crucial role in this analysis. These are deﬁnedas those points in the complex plane where /H20851/H9261+icos/H20849y/H20850/H20852 =q/H20849y;/H9261/H20850=0. Since we are concerned with eigenvalues fol- lowing the scaling given in Eq. /H2084911/H20850,a s /H9280→0, we will have two turning points approaching y=0 symmetrically with re- spect to the origin, which correspond to simple roots of q, left and right turning points being denoted as yLand yR,respectively /H20849see Fig. 16/H20850. The eigenvalue condition for complex turning points given by the WKBJ method appears as a natural extension ofthe well-known result for self-adjoint problems, where theturning points lie on the real line. 16Since its derivation is quite involved, it is reported separately in Appendix A. TheWKBJ approximation of the eigenvalue /H9261 WKB is determined by the integral condition exp/H20873/H9280−1/2/H20885 /H9253q/H20849/H9256;/H9261WKB /H208501/2d/H9256/H20874=/H11006i, /H2084912/H20850 where /H9253is an arbitrary path in the complex plane connecting yLtoyRwithout looping around one of them /H20851because of the multivalueness of q/H20849/H9256;/H9261/H208501/2/H20852. Equation /H2084912/H20850can be rewritten as /H20885 /H9253q/H20849/H9256;/H9261WKB /H208501/2d/H9256=i/H92801/2/H9266/H20873n+1 2/H20874,n= 0,1,2 ... , /H2084913/H20850 which constitutes an implicit relation for a set of eigenvalues /H9261WKB, corresponding to even /H20849odd/H20850eigenmodes for neven /H20849odd/H20850. The left-hand side is a function of /H9261WKB that involves an integral of the elliptic kind, of which the limits of inte-gration contain themselves a dependence from /H9261 WKB. Such relation can be inverted only numerically. In order to make any analytical progress one could per- form a Taylor expansion of q/H20849/H9256;/H9261WKB /H20850, which once truncated would yield an explicitly integrable form. Such possibility will not be pursued here, but in Sec. II IB3b , i t will be related to the result obtained therein. We observe instead thatcondition /H2084913/H20850, even as it stands, unveils the scaling expo- 00.51 time=0 00.51 time=7 00.51 time=16 00.51 30 35 40 455 05 56 000.51 time=27 FIG. 2. /H20849Color online /H20850Snapshots of the time evolution governed by Eq. /H208491/H20850with u/H20849y/H20850=1−4 /H20849y−1/2/H208502and Pe=103from the initial condition T0/H20849x,y/H20850 =exp /H20849−/H20849x−Lx/2/H208502//H5129x2/H20850, with /H5129x=10−3/2Lx, and horizontal Fourier period Lx=20/H9266, with nx=1024 and ny=128, respectively, for horizontal and vertical Fourier modes. Neumann boundary conditions Ty/H208490/H20850=Ty/H208491/H20850=0 are enforced by even symmetry with respect to the y=0 and y=1 horizontal boundaries. The localization of the tracer near the walls and the center of the channel is evident as are the different speeds and decay rates for these two regions /H20849the peaks are normalized by the scalar’s maximum /H20850.117103-5 Analysis of passive scalar advection in parallel shear ﬂows Phys. Fluids 22, 117103 /H208492010 /H20850nents pandqin Eq. /H2084911/H20850. Both the integrand and the measure of the integration path /H9253are in fact O/H20849/H20881/H9261+i/H20850, which gives O/H20851/H9280−1/2/H20849/H9261+i/H20850/H20852=1, and it follows p=q=1/2. b. Eigenvalue condition from uniform approximation- .The fundamental drawback of WKBJ for eigenvalue prob- lems lies in the lack of asymptoticity for nﬁxed, although is typically necessary to consider just n=3 to 4 to obtain very accurate results and even for the ground state eigenvalueWKBJ is often a fairly good approximation /H20849see Ref. 16, which can also be a suitable reference for some facts used inthe rest of this section /H20850. The formal failure of WKBJ can be understood in terms of turning points approaching each othertoo quickly to allow the solution to be fast after a rescalingthat ﬁxes the distance between the turning points to be O/H208491/H20850. The cosine-shear ﬂow, thanks to its local quadratic be- havior, allows an alternative route that does not suffer of thelatter problem and has the value of giving a simple explicitexpression for the eigenvalues. As /H9280→0 Hermite functions /H20849or, equivalently, parabolic cylinder functions /H20850can be used to obtain a local inner-layer approximation in a region containing both turning points.Using Hermite functions we can express solutions of theequation /H9272/H11033+/H20849/H9263+1 2−1 4z2/H20850/H9272=0 , /H2084914/H20850 as /H9272=H e /H9263/H20849/H11006z//H208812/H20850e−z2/4. /H2084915/H20850 He/H9263represents the Hermite function of /H20849arbitrary and com- plex /H20850order /H9263/H20849see Ref. 18, Sec. 10.2 /H20850. This is exactly the equation one would ﬁnd expanding Eq. /H2084910/H20850up to second order about y=0 and applying the transformations /H92801/4z=21/4e−i/H9266/8y, /H2084916/H20850 /H9263=1 /H208812/H20849/H9261+i/H20850/H9280−1/2ei/H9266/4−1 2. /H2084917/H20850 We essentially can view Eq. /H2084910/H20850as a perturbation problem regularized by the variable rescaling /H9280−1/4yfor/H20841y/H20841/H112701, with leading-order solutions easily constructed from Eq. /H2084915/H20850. Such solutions can eventually be matched with outer WKBJ solutions to construct global approximants. However,the approximate eigenvalues /H9261 Hare determined at the level of the inner problem only, imposing asymptotic decay.Hence, the eigenvalues are just related to those of the Her-mite Eq. /H2084914/H20850through the transformation /H2084917/H20850. Since the ei-genvalues /H9263are determined by the condition that /H9274/H20849y/H20850be bounded for /H20841/H9280−1/2y/H20841→/H11009along the real axes, one has to ac- count for the phase shift involved in the coordinate transfor-mation /H2084916/H20850, and to understand the eigenvalues of Eq. /H2084914/H20850 as those values that allow /H9272to vanish for /H20841z/H20841→/H11009with arg/H20849z/H20850=/H9266/8. It is inferred from the large-argument expansion of Hermite functions that within a /H9266/8 phase shift of the argument from the real line the character at inﬁnity is notaltered, hence the eigenvalues of the Hermite equation wouldbe the same if the problem were posed on the real line. Sucheigenvalues are known to be just the integers /H9263=0,1,2..., it is then elementary to verify that /H9261H=−i−/H92801/2/H208491−i/H20850/H20849n+1 2/H20850,n= 0,1,2 ... . /H2084918/H20850 We point out that one would obtain the same result ex- panding up to second order the integrand in Eq. /H2084913/H20850and explicitly solving the integral via standard residue calcula-tion. This conﬁrms that also in case of complex parabolicpotential WKBJ provides exact eigenvalues, as well-knownfor the real self-adjoint Schrödinger equation. c. Comparison. A comparison of the two approaches de- scribed above is given in Table I. Generally, we obtain good accuracy even for moderately small /H9280, bare WKBJ being always more accurate than the uniform approximation. Theseresults, perhaps unexpected, are ultimately due to the boostof accuracy that WKBJ enjoys with locally parabolic poten-tials. Such accuracy boost absorbs the small- ndeﬁciency. We also observe that the error shows two opposite trends for n growing at /H9280ﬁxed, decreasing for /H9261WKB and increasing for /H9261H, respectively. This can be understood observing that while the WKBJ approximation is asymptotic for large n, the sec- ond approach is a completely local approximation, hencesuffering from the fact that the eigenfunctions widen as n grows. d. Poiseuille ﬂow in a channel: Intermediate time mode sorting. All the features in the analysis above come together in the classical case of Poiseuille ﬂow in a channel withwalls impermeable to the tracer T y=0. The different decay and propagation rates special to the locally linear and qua-dratic shear and captured exactly by the sawtooth and cosineﬂow result in a visible mode sorting during the evolution ofa generic initial condition. This is illustrated in Fig. 2, gen- erated by a numerical simulation /H20849details of the algorithm are described below in Sec. IV /H20850of the passive scalar evolution initially concentrated in a thin strip /H20849y- independent /H20850in an effectively inﬁnite long channel are advected by the ﬂowTABLE I. Approximate and exact eigenvalues, with percentage error on the quantity /H9261+i. −/H9261WKB /H20849err % /H20850−/H9261H /H20849err % /H20850 −/H9261Exact /H9280=10−1n=0 0.154 96+0.841 85 /H208492/H20850 0.158 11+0.841 89 i/H208494/H20850 0.151 73+0.841 75 i n=2 0.708 81+0.204 48 i/H208492/H20850 0.790 57+0.209 43 i/H208499/H20850 0.724 12+0.208 84 i /H9280=10−2n=0 0.049 69+0.949 99 i/H208490.6/H20850 0.050 00+0.950 00 i/H208491.2/H20850 0.049 37+0.949 99 i n=2 0.242 07+0.749 87 i/H208490.1/H20850 0.250 00+0.750 00 i/H208493/H20850 0.241 74+0.749 85 i /H9280=10−3n=0 0.015 78+0.984 19 i/H208490.2/H20850 0.015 81+0.984 19 i/H208490.4/H20850 0.015 75+0.984 19 i n=2 0.078 27+0.920 94 i/H208490.04 /H20850 0.079 06+0.920 94 i/H208491/H20850 0.078 24+0.920 94 i117103-6 Camassa, McLaughlin, and Viotti Phys. Fluids 22, 117103 /H208492010 /H20850u/H20849y/H20850=1−4 /H20849y−1/2/H208502with boundary conditions Ty/H208490/H20850=Ty/H208491/H20850 =0. By even-periodic extension of the ﬂow in y-direction, the ensuing periodic modes of Eq. /H208493/H20850may be separated be- tween symmetric and antisymmetric with respect to the walllocations, with the symmetric ones automatically satisfyingvanishing Neumann boundary conditions at the walls. Tracerinitial data symmetric with respect to the channel centerlineare spanned by these symmetric modes. This extension ofu/H20849y/H20850is schematically depicted in Fig. 3, which shows how the different asymptotic scalings of the /H20849imaginary /H20850compo- nent of /H9261−iu m=O/H20849/H92801/2/H20850and/H9261=O/H20849/H92801/3/H20850give rise to modes supported on regions of size O/H20849/H92801/4/H20850and O/H20849/H92801/3/H20850, respec- tively, for the interior and wall mode. The initial stage of the evolution shows the direct im- print of the shear proﬁle, with the initial distribution of tracerdeforming accordingly into a parabolic shape. While suchbehavior can be expected at startup, it soon evolves into amore interesting form of competition between advection anddiffusion. The initial /H20849purely advective /H20850mechanism that bends and stretches the tracer isolines is also acted on bydiffusion. This mechanism is progressively enhanced untilthe two effects equilibrate each other. When such nontrivialbalance is achieved the chevronlike structures predicted bythe asymptotic analysis can be clearly observed near thewall, as well as the fatter, longer-lived interior chevron asso-ciated with quadratic /H20849cosine /H20850shear proﬁle at the center. Ob- serve that the tracer distribution near the wall appears tomove, even as the local ﬂuid velocity approaches zero. Thisis predicted by the modal analysis, as the phase speed /H20849de- termined by imaginary component of the eigenvalue /H20850Im/H9261is nonzero. In Fig. 4, the position of the tracer distribution peak near the wall is shown for the simulation depicted in Fig. 2, and compared with an estimate based on the phase speed givenby the sawtooth theory. The tangency at short times of theprediction shows accurate comparison with the simulation,while the increase of velocity corresponds to the migrationtoward smaller k’s as the diffusion decay kills higher wave- numbers. This is in agreement with the dispersion relation,which shows an increasing phase speed of the wall modes askdecreases. /H20849A similar cascade occurs in the interior. /H20850This interplay of modal decay with the modal phase speed is aninteresting problem in its own right, which sheds muchneeded light on the intermediate scales of data evolution to- ward the Taylor regime. We will report on this in a separatestudy. C. The limit /H9280\0,kšPe1/3: Pure-diffusive modes The ordering of modes that sees straight diffusivity as the dominant effect at wavenumbers beyond the WKBJrange is a consequence of the large-Pe limit we are consid-ering. When this is not true, the time scale of streamwisemolecular diffusion /H20849Pe −1k2/H20850can overcome anomalous dif- fusion /H20849k/H9261/H20850over the whole k-spectrum, including Taylor scales. The threshold between WKBJ and pure-diffusive modes is simply found comparing the time scale Pe1/2k−1/2 from Sec. III B with the streamwise diffusion scale Pe k−2. Equating the two characteristic times we have k/H9261R/H11011Pe−1k2⇒k=O/H20849Pe1/3/H20850, implying that the condition for pure-diffusive modes is in- deed/H9280→0 with k/H11271Pe1/3. Diffusivity in the streamwise /H20849x/H20850direction is not present in the eigenvalue problem, implying that the spatial structure of pure-diffusive modes remains the same as for the WKBJmodes. However, the contraction of the streamwise wave-length eventually overcomes the effect of the gradient along y. IV. INITIAL VALUE PROBLEMS In this section, we shall present, guided by the analysis above, numerical simulations of initial-value problems forthe passive scalar evolution. We study the nondimensionaladvection-diffusion Eq. /H208491/H20850, employing the same numerical scheme as Ref. 9. This is a pseudospectral solver based on Fourier modes in both xand y, with an implicit-explicit third-order Runge–Kutta 19routine for time marching, that combines explicit treatment of the advective part with animplicit one for the diffusive stiff term. The scheme is anti-aliased by the standard 2/3 rule. By proceeding in successivereﬁnements, we have documented that all simulations pre-sented are well resolved. The computational solution en-forces doubly periodic boundary conditions. We ﬁrst exploreImλ=u+Ο(ε) Imλ=Ο(ε )u(y) y1/2 1/41/3 Ο(ε)1/3Ο(ε)m FIG. 3. Schematics of the periodic extension for channel ﬂow: support of interior and wall modes is determined by the scaling of the imaginary part ofthe eigenvalues of cosine and linear shear, respectively. Boundary sawtooth theory 0 1 02 03 04 032364044 timeMaximalocation FIG. 4. /H20849Color online /H20850Position of the tracer distribution peak near the wall for the simulation depicted in Fig. 2compared to the wall-mode theoretical prediction for the phase speed based on the characteristic wavenumber ofthe initial condition /H20849k/H11229 /H9266for the initial data in the simulation /H20850.117103-7 Analysis of passive scalar advection in parallel shear ﬂows Phys. Fluids 22, 117103 /H208492010 /H20850single-scale initial data and then examine nonperiodic /H20849lon- gitudinal /H20850evolutions using a period much larger than the horizontal extent of the initial data. A. Single-scale initial data Using the initial condition T0/H20849x,y/H20850=cos kx, we want to follow the smearing out of initial ﬂuctuations by the shear ﬂow, focusing in particular on the decay rate of the L2-norm, and how it depends on /H9280when this parameter spans the whole range of possible regimes. Let the decay rate /H9253/H20849t/H20850be deﬁned as /H9253/H20849t/H20850=−d dtlog/H20648T/H20849x,y,t/H20850/H206482, where /H20648·/H206482is the standard L2-norm over the period. For a pure exponential decay /H9253would be the constant in the exponent. First we observe in Fig. 5some snapshots of the time evolution when the wavenumber is chosen to have /H9280small. In the earlier stage the vertical bands are stretched and re-duced into thin ﬁlaments where the shear is stronger. Fromthe point of view of eigenmodes, this stage in which thevertical structure is built, corresponds to a collapse of theinitial superposition of many eigenfunctions on the groundstate mode. Owing to the complexity of the physical struc-ture of eigenfunctions, the modal approach is not very infor-mative at this stage. The temporal decay is faster than expo-nential: the mechanism of this transient enhanced diffusion isessentially the fast expulsion explained by Rhines and Young. 10Such process terminates when ﬂuctuations are com- pletely suppressed by shear, after which two long lived,chevron-shaped structures localize in thin layers around theshear-free regions. At this stage the decay rate settles on a constant value. The picture is analogous to the one considered in Ref. 9 for random initial data. The long-time behavior is in factcommon to a large class of initial condition with streamwisemodulation. In striking contrast on the other hand, are theresults obtained when a steady source is added /H20849see Ref. 20/H20850. In the latter case, the accumulation of the unmixed dye isseen to take place in the high shear region of the ﬂow. Thespectral analysis can provide further information on thesource problem. Besides being able to show details of thetransient evolution out of general initial data to the regimedictated by the source, the eigenvalues and eigenfunctionscan be assembled to produce an exact expression of theGreen’s function, which could then be analyzed byasymptotic methods. In this regard, we note that theasymptotic scaling of the inner layers in the source problemstudied in Refs. 20and21, while generically of order O/H20849 /H92801/3/H20850 would switch over to scalings of order O/H20849/H92801/4/H20850if the same source cos xwere made to move at a speed sufﬁciently close to the maximum ﬂuid velocity, an effect not reported bythese prior studies. Shown in Fig. 6is the opposite limit of /H9280large. The distribution sets on a Taylor mode with weak dependencefrom y, which is still visible /H20849right picture /H20850because /H9280is only moderately large. The decay rate /H9253is shown in Figs. 7and8as obtained from numerical simulations. These two ﬁgures report thesame data under different rescaling, to emphasize the /H9280de- pendence in the behavior. Also, in each ﬁgure a referencehorizontal line marks the asymptotic decay rates of theWKBJ and Taylor regimes respectively, obtained from thereal part of the ground-state eigenvalue given by Eqs. /H2084918/H20850 and /H208499/H20850. At large times, the data limits to one of these con- stant values depending on whether /H9280/H112701o r/H9280/H112711, and the different rescaling demonstrates the collapse. Note that therescaled /H9253approaches 1/2 in both limits /H20849this is only a coin- cidence happening for the shear proﬁle considered /H20850. When /H9280=O/H208491/H20850oscillations appear, particularly evident for/H9280=1. This phenomenon arises through interaction of the two nonorthogonal ground-state modes, with conjugate ei-genvalues corresponding to right- and left-traveling chevron-structures. As long as /H9280is small, the two trains of chevrons are each localized in the respective shear-free regions; this FIG. 5. Snapshots of the time evolution from an initial condition T0/H20849x,y/H20850 =cos kxfor/H9280=.001 /H20849k=1, Pe=103/H20850. Concentration ﬁeld is shown at tPe−1/2k1/2=0,.032,.095,1.89. While this is a single-mode computation in the streamwise direction, the number of Fourier modes used in the cross-ﬂow direction is ny=256. FIG. 6. Snapshots of the time evolution from an initial condition T0/H20849x,y/H20850 =cos kxfor/H9280=10 /H20849k=10−4,P e = 1 03/H20850. Concentration ﬁeld is shown at tPek2=0,.08. Resolution as for Fig. 5.117103-8 Camassa, McLaughlin, and Viotti Phys. Fluids 22, 117103 /H208492010 /H20850makes the eigenmodes almost orthogonal. As /H9280=O/H208491/H20850, one can roughly imagine the modes to be made of wide chevron- structures, now with non-negligible overlap. Here nonor-thogonality produces oscillations in the decay rate /H20849essen- tially a constructive-destructive interference depending onthe alignment of the two wave trains /H20850. The inset in Fig. 7 shows the exact reconstruction of /H9253using the two Mathieu ground-state functions, conﬁrming this assertion. The three regimes of modes are represented collectively in Fig. 9. This ﬁgure shows a time-averaged value of /H9253in which we have excluded the initial transient to better ap-proximate the inﬁnite time average. The averaging procedurecan be regarded as a device to obtain a measure of “effec-tive” decay rate even in the cases /H9280=O/H208491/H20850manifesting un- steadiness. This quantity can be considered essentially equivalent to Re /H20851k/H92610/H20852. B. Multiscale initial data In Sec. III, we have seen how streamwise variations of different scales behave under shear-distortion. The homog-enization of an initial condition with multiple scales is now discussed and illustrated with some numerical simulations.We consider initial distributions in the form of slowly modu-lated wave packets T 0=/H20849Asinkx+B/H20850e−x2//H5129x2, /H2084919/H20850 which perhaps provides the simplest setup to assess the in- terplay between two length scales with large separation. Weremark that such class of initial conditions captures the es-sential features of those realizable in simple experimentalsetups currently under study. Through Aand Bwe can tune the relative participation of high- versus low-frequency com-ponents. For simplicity we are only considering two longitu-dinal length scales, given by kand/H5129 x. We shall keep constant kand/H5129x, and to observe enough scale separation the latter are chosen such as kPe/H112701 and Pe //H5129x/H112711. In the following, we present four simulations, where all parameters are listedin Table II. Visualizations of the passive scalar ﬁelds at dif- ferent times are reported in Figs. 10–13. The general physical picture that emerges is as follows. At early times, the high-frequency components of the initialcondition govern the main features of evolution, similar tothex-periodic problem discussed in Sec. III /H20849chevron-shaped structures /H20850. The interplay between the two wide-separated length scales contained in the initial data adds further physi-cal features that lie in the subsequent phase of evolution; ingeneral, the small scales are wiped out during a global ho-mogenization stage that could not exist in the strictly peri-odic problem. The time scale of this wipe-out, the “cross-10-310-210-1100 1 10γPe1/2k-1/2 tP e-1/2k1/2.001 .01 .1 1 10 100 1-exact FIG. 7. Decay rates from numerical simulations as a function of time for different values of /H9280obtained setting Pe=1000 and k=10−p/H20849p=0:1:5 /H20850. Axes are rescaled on WKBJ time scale to show the collapse at /H9280/H112701 on the decay rate predicted by the WKBJ analysis /H20849marked by the horizontal line at 1/2/H20850. The inset contains the exact computation for /H9280=1/H20849intermediate value between WKBJ and Taylor regimes /H20850obtained using the two ground-state Mathieu functions. 10-610-510-410-310-210-1100 10-410-2100102104106γPe-1k-2 tP ek2 FIG. 8. Decay rates from numerical simulations as a function of time for different values of /H9280,a si nF i g . 7. Axes are rescaled on the Taylor time scale to show the collapse at /H9280/H112711 on the decay rate predicted by the regular perturbation analysis /H20849marked by the horizontal line at 1/2 /H20850. For the legend, see Fig. 7.10-810-610-410-2100102104 Pe-1Pe1/3γave k1/2 Pe k2 Pe-1k21/2 Pe-1/2k1/2 FIG. 9. Averaged decay rate /H20849see text /H20850vskfrom numerical simulations. The lines represents the three asymptotic behaviors of kR/H20851/H92610/H20852, including also three simulations from the pure diffusive regime. Results are for Pe=1000. TABLE II. Parameters used in numerical simulations. For all of them k=1, /H5129x=800, and Pe=50; the number of /H20849dealiased /H20850Fourier modes used is nx=12 288 ny=64. The fundamental wavenumbers are kx0=.001, ky0=1. The simulations are periodic in xwith a domain large enough to mimic an unbounded domain on the time scale of the simulations. AB Run 1 1 1 Run 2 0 1Run 3 1 0Run 4 1 0.001117103-9 Analysis of passive scalar advection in parallel shear ﬂows Phys. Fluids 22, 117103 /H208492010 /H208503500 3600 3700 3800 390 0036 time=0 3500 3600 3700 3800 390 0036 time=5 3500 3600 3700 3800 390 0036 time=15 3500 3600 3 700 3800 3900036 time=100 FIG. 10. /H20849Color online /H20850Snapshots showing the distribution of the passive scalar for run 1. Only a portion of the domain is shown. Colorbar as in Fig. 2,b u t ranging from /H110021t o1 . 3500 3600 3700 3800 390 0036 time=0 3500 3600 3700 3800 390 0036 time=5 3500 3600 3700 3800 390 0036 time=15 3500 3600 3 700 3800 3900036 time=100 FIG. 11. /H20849Color online /H20850Same as Fig. 10for run 2. 3500 3600 3700 3800 390 0036 time=0 3500 3600 3700 3800 390 0036 time=5 3500 3600 3700 3800 390 0036 time=15 3500 3600 3 700 3800 3900036 time=100 FIG. 12. /H20849Color online /H20850Same as Fig. 10for run 3.117103-10 Camassa, McLaughlin, and Viotti Phys. Fluids 22, 117103 /H208492010 /H20850over” time, which corresponds to the time when evolution starts to be governed by the homogenized equation /H20849the deﬁ- nition of Taylor time scale used in this paper /H20850, and can be different from the nondimensional cross-ﬂow diffusive timescale /H9270D=Pe. From a spectral perspective, the decay time of the high- frequency spectral bands is estimated by our WKBJ analysisto be k −1/2Pe1/2. While this time scale is in general shorter than/H9270D/H20849for large Pe /H20850, the initial data can be such that the relative energy of the high versus low frequency bands /H20851e.g., set by the parameters Aand Bin Eq. /H2084919/H20850/H20852can make the WKBJ modes observable well beyond /H9270D. Further, these dif- ferences can be seen by comparing the evolution of momentsversus norms, as we show below. Looking at the snapshots for run 1 one can see that by t=100 chevronlike structures are completely depleted. We inquire whether this major structural crossover is a bench-mark for the transition to the Taylor regime. Figure 15/H20849top panel /H20850depicts the crossover as the drop at t/H1101510 of the L 2-norm in run 1 to the nearly constant value given by the slowly evolving case in run 2. Note that the oscillations ob-served in the bottom image are induced by the nonorthogo-nality of anomalous modes as discussed in Sec. III. For further comparison, we consider the second moment of the tracer distribution, which sometimes is used as a di-agnostic to detect the onset of the Taylor regime. We displayin Fig. 14the variance-gap /H9268˜/H20849t/H20850ª/H9268T/H20849t/H20850−/H9268/H20849t/H20850, where /H9268is the variance in xof the distribution integrated in y, and /H9268T=2/H9266Petis the theoretical law for a Gaussian distribu- tion evolving according to Taylor-renormalized pure diffu-sion. This onset is clearly seen in Fig. 14. Notice that the variance does not capture differences in the structure of thedifferent runs, since fast ﬂuctuations belong to the high-frequency spectrum, and are thus missed by the variance /H20849but are accounted by the L 2-norm /H20850. We may conclude that in run 1, the crossover transition has occurred earlier /H20849t/H1101510/H20850than/H9270D, which at Pe=50 is well completed after t/H11015200. Perhaps more emphatic on this point is the comparison with the last two runs. In run 3, the initialcondition is chosen to have zero mean; thus the decay rate /H9253 /H20851reported in Fig. 15/H20849bottom /H20850/H20852settles on a constant value /H20849predictable from WKBJ eigenvalues as shown previously /H20850,and the crossover transition to the Taylor regime does not occur at all. In run 4, where the initial data are chosen with asmall mean, a clear crossover transition occurs at t/H1101550, i.e., deferred respect to run 1 /H20849notice how chevrons are still iden- tiﬁable in the latest time in Fig. 13/H20850. The difference that stands between the transition at time /H9270Dand the smearing out of fast scales is further illustrated by looking at run 2. Even at large scales, hence at low wave-numbers, the weak longitudinal variations O/H208491//H5129 x/H20850combine with the shear to build a weak vertical structure departing from the vertically homogeneous initial condition. The cross-stream diffusion sets this weak variation to a small amplitudeO/H20851/H20849Pe/H5129 x/H20850−1/H20852/H20849as given above in the analysis of Taylor modes /H20850once a time scale O/H20849Pe/H20850is reached. In the terminol- ogy of homogenization theory, the vertical structure correc- tion to the vertically independent leading-order is dominatedby the solution of the cell problem . V. CONCLUDING REMARKS A number of authors3,7–10have addressed the problem of passive scalar diffusion under simple ﬂow conditions, andnontrivial time scales have been identiﬁed and explained indifferent cases. The present study we believe contributes amore complete global understanding of the various scalingsthat such problems can exhibit. In particular, with the formu- 3500 3600 3700 3800 390 0036 time=0 3500 3600 3700 3800 390 0036 time=5 3500 3600 3700 3800 390 0036 time=15 3500 3600 3 700 3800 3900036 time=100 FIG. 13. /H20849Color online /H20850Same as Fig. 10for run 4. 1e31e41e5 1 10 100 1000σ~ tRun 4 Run 2 Run 1 σT FIG. 14. Gap of variance /H9268˜vs time. All distributions are normalized to unitary mass. The curves level off after the time scale /H9270D. Notice that the curves relative to runs 1, 2, and 4 result indistinguishable. Run 3 is notreported because by exact asymmetry the variance is identically zero at alltimes.117103-11 Analysis of passive scalar advection in parallel shear ﬂows Phys. Fluids 22, 117103 /H208492010 /H20850lation of the analysis as an eigenvalue problem we have identiﬁed and calculated the explicit long-lived slow modes,and further sorted these modes into two different categories.The ﬁrst is connected to the Taylor regime, governed by thehomogenized evolution equation. The second category isconnected to the intermediate- and short-time anomalousevolution. While ﬁrst computed for idealized periodic ﬂows,we have also shown how such modes provide insight formore physically relevant shears, such as the example of thePoiseuille channel ﬂow. Explicit analysis of a sawtooth shearﬂow provides the detailed structure and decay properties ofthe tracer’s boundary layer near ﬂat walls, while in the inte-rior shear-free regions /H20851near locations where u /H11032/H20849y/H20850=0/H20852more general WKBJ asymptotic analysis provides the longest lived anomalous modes, which persists well beyond the wallboundary layer modes in the limit /H9280→0. The analysis characterizes different stages of evolution, each one carrying the signature of a different spectral band.From the spectral point of view of the advection-diffusionproblem, the Taylor regime should be regarded as the limit-ing state in which any component of the spectrum has de-cayed and become negligible with respect to the n=0 modes in the k/H11270Pe −1range. The modes inside the range k/H11271Pe−1 characterize the structure of the solution in the super- Gaussian anomalous diffusive regime described, for instance,in the work by Latini and Bernoff. Considering the pointsource distributions discussed by these authors, the cross-section-averaged distribution is initialized from a ﬂat Fourierspectrum, which necessarily excites all three classes ofmodes and exhibits three regimes along the evolution. Weobserve how a simulation in a L x/H11003Lyperiodic domain would require a fundamental xwavenumber kx0=2/H9266/Lx/H112701/Pe in order to observe the Taylor regime. A smaller domain, withlower resolution in the wavenumber domain, leads to a cutoffof the Taylor modes, limiting the possible observable re-gimes up to the “anomalous diffusion” stage. By adjusting the initial relative energy in the bands we demonstrated howthe WKBJ may in principle be extended well beyond theclassical cross-stream-diffusion time scale r 2/D/H20849or/H9270Din nondimensional units /H20850. Additionally, compared to previous studies, the present investigation provides deeper insightinto the geometrical spatial structures arising during timeevolution. It is interesting to consider the implications of our analy- sis for the case of several superimposed passive scalars withdifferent diffusion coefﬁcients. For example, one can con-sider a setup consisting of two different chemical species thatare injected in the same point within a shear ﬂow. If themolecular diffusivities are different, our analysis indicatesthat the velocity ﬁeld acts as a separator for the two scalars,owing to their different interplay with advection. If thescalars are reactive, one could in turn expect the separationto affect reaction, possibly suppressing it, when the timescales of advection- diffusion are comparable to the timescale of reaction. A detailed investigation in this direction isinteresting and will be considered for further studies. Future studies will also include the extension of the con- cepts presented in this work to the more realistic setups ofﬂows in both two and three dimensions, with open andclosed streamlines and physical boundary conditions, wheresimilar phenomena including long-lived modes have beenobserved. In particular, the axially symmetric geometry natu-rally merits study for its relevance to pipe ﬂows. Furtherextensions of the methods presented here should also be di-rected to addressing time-dependent ﬂows possessing mul-tiple scales and even randomness. ACKNOWLEDGMENTS We thank Ray Pierrehumbert for helpful discussions and sharing notes from one of his presentations about strangeeigenmodes. We thank Neil Martinsen-Burrell for his initialwork on the subject at the end of his post-doctoral appoint-ment at UNC sponsored by NSF CMG Contract No. ATM-0327906. We also thank an anonymous referee for pointingout the consequences of our analysis in the case of reactionbetween two or more diffusing chemical species with differ-ent diffusivities. R.C. was partially supported by NSF Con-tract No. DMS-0509423 and NSF CMG Contract No. DMS-0620687. R.M.M. was partially supported by NSF CMGContract No. ATM-0327906, NSF Contract No. DMS-030868, and NSF RTG Contract No. DMS-0502266. C.V.has been partially supported by NSF CMG Contract No.DMS-0620687. APPENDIX A: DERIVATION OF WKBJ FORMULA The application of the concepts we are going to use can be traced back to the earlier attempts to solve theSchrödinger equation of quantum mechanics /H20849see, for in- stance, Ref. 22/H20850. In ﬂuid mechanics literature, similar ideas have also been applied to fourth-order operators in the theoryof hydrodynamic stability by Lin 23and others. Unlike the typical quantum mechanics turning-point problems, thepresent analysis demands additional effort, due to nonself-0102030405060708090100 1 10 100\|\|T\|\|2 tRun 4 Run 3 Run 2 Run 1 00.020.040.060.080.10.120.140.160.18 1 10 100γ t FIG. 15. Time evolution of L2norm /H20849top/H20850and decay rate /H20849bottom /H20850.R u n2i s not reported in the bottom picture because it would be too low to be visible.117103-12 Camassa, McLaughlin, and Viotti Phys. Fluids 22, 117103 /H208492010 /H20850adjointness and the existence of complex spectra. Some ac- knowledgment for complex WKBJ analysis should be attrib-uted to Wasow, 24in particular for the role of the Stokes phenomenon central to this problem. In general, we can write a WKBJ solution of Eq. /H2084910/H20850as a linear combination of two fundamental solutions /H92781,2=q/H20849y;/H9261/H20850−1/4exp/H20873/H11006/H9280−1/2/H20885 y0y q/H20849/H9256;/H9261/H208501/2d/H9256/H20874, /H20849A1/H20850 where so far y0is left unspeciﬁed. Not yet speciﬁed is also the domain of validity of such asymptotic solutions. Thebreakdown of Eq. /H20849A1/H20850close to turning points is a well- known fact, but moreover WKBJ solutions fail to hold in awhole annular region looping around a turning point 24 /H20849Stokes phenomenon /H20850. We point out that a given WKBJ so- lution has a full meaning only if the domain of deﬁnition isspeciﬁed. In fact, a single WKBJ solution could beasymptotic to two different exact solutions depending on theregion. Stokes lines are deﬁned by the property Re/H20851/H20848 y0yq/H20849/H9256/H208501/2d/H9256/H20852=0. The problem under consideration has simple turning points, hence we have three Stokes lines ema- nating from yLand yR, and the same number of anti-Stokes lines, where the latter are deﬁned by the specular conditionIm/H20851/H20848 y0yq/H20849/H9256/H208501/2d/H9256/H20852=0 with y0being yLoryR. In Fig. 16is reported a plot of the complex y-plane with such curves, showing the topology when the turning points yLand yRare collapsing at the bottom of the potential y=0 from the third and ﬁrst quadrant /H20849this follows from the ansatz assumed for WKBJ eigenvalues /H20850. On the Stokes and anti-Stokes lines WKBJ solutions exhibit limiting behaviors, on the ﬁrst theexponential is purely oscillatory while on the second it ispurely growing/decaying without oscillations. We introduce the four WKBJ solutions /H92781L,/H92782L,/H92781R, and/H92782R, where the subscript indicates the speciﬁc choice y0=yLory0=yR. The sector of deﬁnition of both /H92781Land/H92782L is the one contained in between /H9262L2and/H9262L3/H20849unshaded left region in Fig. 16/H20850. Similarly /H92781Rand/H92782Rare deﬁned in be- tween /H9262R2and/H9262R3/H20849unshaded right region in Fig. 16/H20850. Also,the branch choice ﬁxes /H92781Lto be the exponentially small component for ymoving to the left along the negative real axis, while /H92782Rwill be small for ymoving to the right along the positive real axis. 1. Connection formulas Since we are working under the assumption of free- space condition, in the far ﬁeld only the vanishing compo-nents of the solution /H20849 /H92781Land/H92782R/H20850are present. To determine the eigenvalues, one has to impose matching in the middleregion S/H20849shaded region in Fig. 16/H20850for the left- and right- hand side solutions, which come as /H92781Land/H92782Rfrom the lateral sectors /H20849blank regions in the same picture /H20850. Moving from the lateral sectors into Sthe two asymptotic solutions /H92781Land/H92782Rhave to be continued inside Saccounting for Stokes phenomenon. In other words, /H92781Land/H92782Rhave to be replaced by different expressions in order to be asymptotic tothe same solutions the two functions are asymptotic to out-side of S. If the four functions /H9278are extended inside Sby analytic continuation moving in the counterclockwise sensearound the turning points, the substitutions to perform are /H92781L→/H92781L+i/H92782L, /H20849A2a /H20850 /H92782R→/H92782R+i/H92781R. /H20849A2b /H20850 These are analogs of Jefferey’s connection formulas, gener- alized to connect asymptotic solutions valid in different sec-tors of the complex plane around a turning point, rather thanthe two parts of the real line divided by a turning point forreal self-adjoint problem. 2. Asymptotic matching After using the connection formulas, we enforce match- ing inside S. This can be performed in either a symmetric or antisymmetric manner /H92781L+i/H92782L=/H11006/H92782R/H11006i/H92781R. /H20849A3/H20850 For compactness of notation let now QL=/H9280−1/2/H20848yLyq/H20849/H9256/H208501/2d/H9256 and QR=/H9280−1/2/H20848yRyq/H20849/H9256/H208501/2d/H9256, Eq. /H20849A3/H20850/H20849dropping the prefactor q−1/4/H20850becomes eQL+ieQL=/H11006e−QR/H11006ieQR, where all the integrals now are now path-independent in S. Introducing also QLR=/H9280−1/2/H20848yLyRq1/2d/H9256, the above equation is equivalent to eQLR+QR+ie−QLR−QR=/H11006e−QR/H11006ieQR, which can be recombined as eQR/H20851eQLR/H11007i/H20852=e−QR/H20851−ie−QLR/H110061/H20852. /H20849A4/H20850 The last relation can be satisﬁed if the terms in brackets are equal to zero. This yields the eigenvalue condition /H2084912/H20850. yLyR ΜL1ΜL2 ΜL3ΜR1 ΜR2ΜR3 Π /MinusΠΠ /MinusΠ FIG. 16. /H20849Color online /H20850Stokes /H20849dashed /H20850an anti-Stokes /H20849continuous /H20850lines for turning points close to the origin. The shaded region is the /H20849open /H20850setS.117103-13 Analysis of passive scalar advection in parallel shear ﬂows Phys. Fluids 22, 117103 /H208492010 /H20850APPENDIX B: WKBJ MODES FOR SAWTOOTH SHEAR FLOW Here we consider the velocity proﬁle to be a piecewise linear shear ﬂow, namely u/H20849y/H20850=1− /H20841y/H20841with y/H33528/H20851−2,2 /H20852, and periodic boundary conditions apply. The eigenvalue problem /H208493/H20850, which now reads as /H9280/H9274yy=/H20849/H9261+i−i/H20841y/H20841/H20850/H9274, /H20849B1/H20850 can be solved in the /H9280→0 limit by a patching method. The starting point is again the assumption of the ansatz /H2084911/H20850. 1. Eigenvalue conditions Before proceeding, the derivation we simplify the prob- lem exploiting the symmetry respect to the origin. Sincesymmetry implies that all eigenfunctions have to be eithersymmetric or antisymmetric, we can work on the semi-interval, say /H208510,2/H20852, and impose symmetry /H20851 /H9274y/H208490/H20850=/H9274y/H208492/H20850=0/H20852 or antisymmetry /H20851/H9274/H208490/H20850=/H9274/H208492/H20850=0/H20852boundary conditions. The eigenfunction /H9274will be in general a linear combi- nation of two independent solutions of Eq. /H20849B1/H20850, say/H92741and /H92742. The boundary conditions require the existence of a non- trivial solution for the linear system /H9253/H92741y/H208490/H20850+/H9254/H92742y/H208490/H20850=0 , /H9253/H92741y/H208492/H20850+/H9254/H92742y/H208492/H20850=0 , in the symmetric case, or /H9253/H92741/H208490/H20850+/H9254/H92742/H208490/H20850=0 , /H9253/H92741/H208492/H20850+/H9254/H92742/H208492/H20850=0 , in the antisymmetric. Setting the determinant equal to zero we obtain the eigenvalue condition, but before writing it, wemake /H92741and/H92742explicit. We use the change of variable z=−/H9280−1/3i1/3y+/H9261+i /H92801/3i2/3, that transforms Eq. /H20849B1/H20850into an Airy equation in the variable z, hence two base solutions can be chosen as /H92741/H20849y/H20850=A1/H20851z/H20849y/H20850/H20852,/H92742/H20849y/H20850=A2/H20851z/H20849y/H20850/H20852, where A1/H20849z/H20850=Ai/H20849z/H20850and A2/H20849z/H20850=Ai/H20849/H9275z/H20850, with /H9275=ei2/H9266/3. The two boundary points y=0 and y=2 map, respectively, into thez-plane as z+=/H9261−i /H92801/3i2/3,z−=/H9261+i /H92801/3i2/3, so that the eigenvalue conditions for symmetric and antisym- metric eigenfunctions, respectively, read A1/H11032/H20849z+/H20850A2/H11032/H20849z−/H20850=A1/H11032/H20849z−/H20850A2/H11032/H20849z+/H20850, /H20849B2/H20850 A1/H20849z+/H20850A2/H20849z−/H20850=A1/H20849z−/H20850A2/H20849z+/H20850. /H20849B3/H20850 The ansatz for /H9261implies that we assume the form z+/H110112i−5/3/H9280−1/3, /H20849B4/H20850z−/H11011i−2/3/H20849a/H9280p−1/3+ib/H9280q−1/3/H20850, /H20849B5/H20850 forz+and z−. Observe that, while the orientation of z+in the complex plane is given by i−5/3, the orientation of z−is still to be determined. Representing it as i/H9251, we have the constraint 1 3/H11349/H9251/H113494 3. 2. Determination of pand q We ﬁrst show that assuming either p/H110211 3orq/H110211 3/H20849i.e., z−→/H11009/H20850, then Eqs. /H20849B2/H20850and /H20849B3/H20850cannot be satisﬁed, hence p/H113501 3and q/H113501 3. With little additional effort this condition will be turned in p=q=1 3. We rewrite the eigenvalue conditions separating z+from z−, namely, A1/H20849z+/H20850 A2/H20849z+/H20850=A1/H20849z−/H20850 A2/H20849z−/H20850, /H20849B6/H20850 A1/H11032/H20849z+/H20850 A2/H11032/H20849z+/H20850=A1/H11032/H20849z−/H20850 A2/H11032/H20849z−/H20850, /H20849B7/H20850 planning to obtain the /H9280→0 asymptotics for left- and right- hand sides of both expressions, and to show that no possibil-ity of matching exists if the above assumption is made. Westart from the antisymmetric case. 3. Antisymmetric case The leading order behavior of A1/H20849/H9267i/H9251/H20850and A2/H20849/H9267i/H9251/H20850as /H9267→+/H11009is known to be25 for/H9251/H33528/H20849−2 , 2 /H20850, /H20849B8/H20850 A1/H20849/H9267i/H9251/H20850/H11011/H9267−1/4exp/H20849−2 3/H92673/2i3/H9251/2/H20850, for/H9251/H33528/H20849−10 3,2 3/H20850, /H20849B9/H20850 A2/H20849/H9267i/H9251/H20850/H11011/H9267−1/4exp/H208492 3/H92673/2i3/H9251/2/H20850, where the actual constants that should appear in front of these expressions have been omitted, since they are irrel-evant for the present investigation. Such simpliﬁcation willbe also assumed in all analogous situations. The asymptotic behavior of the left-hand side of Eq. /H20849B6/H20850follows directly from the above formulas. In fact from Eq. /H20849B4/H20850, we are in the case /H9251=−5 3, in which both Eqs. /H20849B8/H20850 and /H20849B9/H20850are valid. Using Eq. /H20849B4/H20850and developing the ratio, we obtain A1/H20849z+/H20850 A2/H20849z+/H20850/H11011exp/H20873−8/H208812 3/H9280−1/2i−5/2/H20874. /H20849B10 /H20850 For the right-hand side, we notice that Eq. /H20849B5/H20850implies that, if z−is assumed to go to inﬁnity as /H9280→0, it has to belong to the sector Sgiven as /H9266/6/H11349arg/H20849z−/H20850/H113492/H9266/3/H20849corre- sponding to1 3/H11349/H9251/H113494 3/H20850. While the asymptotic expansion of A1holds in the whole extent of S, the one for A2needs to be split in two parts, respectively, valid in the two subsectors S1 and S2, deﬁned by /H9266/3/H11021arg/H20849z−/H20850/H113492/H9266/3 and /H9266/6/H11349arg/H20849z−/H20850 /H11021/H9266/3. In sector S1, it should be used Eq. /H20849B9/H20850with2 3/H11021/H9251 /H113494 3, which falls outside the range of validity for that117103-14 Camassa, McLaughlin, and Viotti Phys. Fluids 22, 117103 /H208492010 /H20850asymptotic expansion. Hence /H9251has to be taken to belong to −10 3/H11021/H9251/H11349−8 3.I n S2both expressions are valid with1 3/H11349/H9251 /H110212 3and no change of branch is needed. The case /H9251=2 3, the edge between S1and S2, will be treated separately at the end of this discussion. The resulting asymptotic expressions areas follows: /H208491/H20850Sector S 1,2 3/H11021/H9251/H113494 3 A1,2/H20849/H9267i/H9251/H20850/H11011/H9267−1/4exp/H20849−2 3/H92673/2i3/H9251/2/H20850. It follows A1/H20849z−/H20850 A2/H20849z−/H20850/H11011const., /H9280→0. /H20849B11 /H20850 /H208492/H20850Sector S2,1 3/H11349/H9251/H110212 3 A1/H20849/H9267i/H9251/H20850/H11011/H9267−1/4exp/H20849−2 3/H92673/2i3/H9251/2/H20850, A2/H20849/H9267i/H9251/H20850/H11011/H9267−1/4exp/H208492 3/H92673/2i3/H9251/2/H20850. It follows A1/H20849z−/H20850 A2/H20849z−/H20850/H11011exp/H20873−4 3/H92673/2i3/H9251/2/H20874, hence using Eq. /H20849B5/H20850 A1/H20849z−/H20850 A2/H20849z−/H20850/H11011exp/H20873−4 3/H20849a2+b2/H208503/4/H92803p−1/2i3/H9251/2/H20874,/H9280→0. /H20849B12 /H20850 Now it is possible to compare the asymptotics of left- and right-hand side of the eigenvalue condition /H20849B6/H20850.I ti s almost immediate to realize that no matching is possible . Indeed, from Eq. /H20849B10 /H20850, it is clear that left-hand side is as- ymptotically a growing exponential, while neither Eqs. /H20849B11 /H20850 and /H20849B12 /H20850are in the respective sectors. This fact excludes the possibility z−→/H11009, and we conclude that z−has to approach a constant value as /H9280→0/H20849which so far could be 0 /H20850. Using Eq. /H20849B5/H20850, this immediately implies p/H113501 3and q/H113501 3. 4. Symmetric case Now we have to consider the leading order behavior of A1/H11032/H20849/H9267i/H9251/H20850and A2/H11032/H20849/H9267i/H9251/H20850for large /H9267, which is25 for/H9251/H33528/H20849−2 , 2 /H20850, /H20849B13 /H20850 A1/H11032/H20849/H9267i/H9251/H20850/H11011/H92671/4exp/H20849−2 3/H92673/2i3/H9251/2/H20850, for/H9251/H33528/H20849−10 3,2 3/H20850, /H20849B14 /H20850 A2/H11032/H20849/H9267i/H9251/H20850/H11011/H92671/4exp/H208492 3/H92673/2i3/H9251/2/H20850. The same discussion of the sectors of validity for the antisymmetric case applies for the present case as well. Wecan also note that the structure of the asymptotic expressionsis still the same. In particular the exponential term, whichessentially determined our conclusions above, is not altered.Therefore the same result is also established for the symmet- ric case: the eigenvalue condition /H20849B7/H20850cannot hold unless p/H11350 1 3q/H113501 3. 5. Final step We ﬁrst come back to the case /H9251=2/3, which as already mentioned has to be discussed separately. Actually it requiresonly the observation that along this direction the exponentialbehavior of A 1and A2vanishes /H20849we are in fact on a Stokes line/H20850. Since both A2/A1and A2/H11032/A1/H11032can vanish only algebra- ically there are no chances, once again, for the eigenvalueconditions /H20849B6/H20850and /H20849B7/H20850to be satisﬁed. To turn the inequalities obtained for pand qinto equali- ties we can proceed by showing how the constant which z − approaches cannot be 0. For the antisymmetric mode it is proven that the ratio A1/H20849z−/H20850/A2/H20849z−/H20850goes to inﬁnity, we have also to conclude that z−has to approach a root of A2, being this the only possibility allowing A1/H20849z−/H20850/A2/H20849z−/H20850→/H11009, since Airy functions do not have any ﬁnite-range singularity. Moreover, A2does not have a root at the origin, and its roots are all aligned along the direction ei/H20849/H9266/3/H20850in the ﬁrst quadrant of the complex plane, so that the constant cannot be 0, hence p=q=1 3, and aand bare determined such that A2/H20849a+ib/H20850=0. The freedom left in the determination of aand byields a set of eigenvalues, with the one corresponding to the root closerto the origin being the ground state. The same is true for thesymmetric modes just replacing A 1and A2with A1/H11032and A2/H11032. We can further assess the accuracy of the asymptotic prediction from this analysis. Since A2/H20849z−/H20850/H20851and A2/H11032/H20849z−/H20850/H20852 have been shown to vanish exponentially fast as /H9280→0, O/H20851exp/H20849−const. /H9280−1/2/H20850/H20852, using Eq. /H20849B6/H20850and /H20849B10 /H20850and the fact that Airy functions /H20849and their derivatives /H20850have only simple roots, implies that z−has to approach exponentially fast a root of A2/H20849orA2/H11032/H20850. This in turn implies that the form of the remainder in the asymptotic equality /H20849B5/H20850is determined, and the remainder for the leading order expression of /H9261follows: /H9261=/H20849a+ib/H20850/H92801/3−i+O/H20851exp/H20849− const /H9280−1/2/H20850/H20852, as can be immediately obtained from the deﬁnition of z−. One may additionally be interested in the eigenvalue for ﬁnite/H9280, which may be found by using any root-ﬁnding nu- merical algorithm on Eqs. /H20849B2/H20850and /H20849B3/H20850. The above scaling results are conﬁrmed by a numerical approach. Moreover, anadditional behavior at ﬁxed value of /H9280merits mention, namely, that the imaginary part of the ground-state eigen-value vanishes at a critical ﬁnite value of /H9280, indicating nonanalytic bifurcations of the eigenvalues. This is essen-tially in agreement with previous ﬁndings by Doering andHorsthemke 5in the case of a linear-shear channel /H20849mostly equivalent to our sawtooth. /H20850 1W. R. Young and S. Jones, “Shear dispersion,” Phys. Fluids A 3, 1087 /H208491991 /H20850. 2A. J. Majda and P. R. Kramer, “Simpliﬁed models for turbulent diffusion: Theory, numerical modelling, and physical phenomena,” Phys. Rep. 314, 237 /H208491999 /H20850. 3G. I. Taylor, “Dispersion of soluble matter in solvent ﬂowing slowly through a tube,” Proc. R. Soc. London, Ser. A 219, 186 /H208491953 /H20850.117103-15 Analysis of passive scalar advection in parallel shear ﬂows Phys. Fluids 22, 117103 /H208492010 /H208504E. A. Spiegel and S. Zalesky, “Reaction-diffusion instability in a sheared medium,” Phys. Lett. A 106,3 3 5 /H208491984 /H20850. 5C. R. Doering and W. Horsthemke, “Stability of reaction-diffusion- convection systems: The case of linear shear ﬂow,” Phys. Lett. A 182, 227 /H208491993 /H20850. 6M. J. Lighthill, “Initial development of diffusion in Poiseuille ﬂow,” IMA J. Appl. Math. 2,9 7 /H208491966 /H20850. 7M. Latini and A. J. Bernoff, “Transient anomalous diffusion in Poiseuille ﬂow,” J. Fluid Mech. 441,3 9 9 /H208492001 /H20850. 8R. Camassa, Z. Lin, and R. M. McLaughlin, “The exact evolution of the scalar variance in pipe and channel ﬂow,” Commun. Math. Sci. 8, 601 /H208492010 /H20850. 9R. Camassa, N. Martinsen-Burrell, and R. M. McLaughlin, “Dynamics of probability density functions for decaying passive scalars in periodic ve-locity ﬁelds,” Phys. Fluids 19, 117104 /H208492007 /H20850. 10P. B. Rhines and W. R. Young, “How rapidly is a passive scalar mixed within closed streamlines?” J. Fluid Mech. 133, 133 /H208491983 /H20850. 11J. Sukhatme and R. T. Pierrehumbert, “Decay of passive scalars under the action of single scale smooth velocity ﬁelds in bounded two-dimensionaldomains: From non-self-similar probability distribution functions to self-similar eigenmodes,” Phys. Rev. E 66, 056302 /H208492002 /H20850. 12W. Magnus and S. Winkler, Hill’ s Equation /H20849Interscience-Wiley, New York, 1966 /H20850. 13C. M. Bender, G. V. Dunne, and P. N. Meisinger, “Complex periodic potentials with real band spectra,” Phys. Lett. A 252,2 7 2 /H208491999 /H20850. 14K. C. Shin, “On the shape of spectra for non-self-adjoint periodic Schrödinger operators,” J. Phys. A 37, 8287 /H208492004 /H20850.15G. N. Mercer and A. J. Roberts, “A centre manifold description of con- taminant dispersion in channels with varying ﬂow properties,” SIAM J. Appl. Math. 50, 1547 /H208491990 /H20850. 16C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers /H20849Springer, New York, 1999 /H20850. 17S. Childress and A. D. Gilbert, Stretch, Twist, Fold: The Fast Dynamo /H20849Springer-Verlag, Berlin, 1995 /H20850. 18N. N. Lebedev, Special Functions & Their Applications /H20849Dover, New York, 1972 /H20850. 19P. R. Spalart, R. D. Moser, and M. M. Roger, “Spectral methods for the Navier-Stokes equations with one inﬁnite and two periodic directions,” J. Comput. Phys. 96, 297 /H208491991 /H20850. 20T. A. Shaw, J. L. Thiffeault, and C. R. Doering, “Stirring up trouble: Multi-scale mixing measures for steady scalar sources,” Physica D 231, 143 /H208492007 /H20850. 21M. R. Turner, J. Thuburn, and A. D. Gilbert, “The inﬂuence of periodic islands in the ﬂow on a scalar tracer in the presence of a steady source,”Phys. Fluids 21, 067103 /H208492009 /H20850. 22J. L. Dunham, “The Wentzel-Brillouin-Kramers method of solving the wave equation,” Phys. Rev. 41, 713 /H208491932 /H20850. 23C. C. Lin, “On the stability of two-dimensional parallel ﬂows, part 1,” Q. Appl. Math. 3,1 1 7 /H208491945 /H20850. 24W. Wasow, “The complex asymptotic theory of a fourth order differential equation of hydrodynamics,” Ann. Math. 49, 852 /H208491948 /H20850. 25M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables /H20849Dover, New York, 1964 /H20850.117103-16 Camassa, McLaughlin, and Viotti Phys. Fluids 22, 117103 /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
1.357250.pdf	Currentinduced displacements of Bloch walls in NiFe films of thickness 120–740 nm E. Salhi and L. Berger Citation: Journal of Applied Physics 76, 4787 (1994); doi: 10.1063/1.357250 View online: http://dx.doi.org/10.1063/1.357250 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/76/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Thickness dependence of current-induced domain wall motion in a Co/Ni multi-layer with out-of-plane anisotropy Appl. Phys. Lett. 102, 152410 (2013); 10.1063/1.4802266 Currentinduced displacements and precession of a Bloch wall in NiFe thin films J. Appl. Phys. 73, 6405 (1993); 10.1063/1.352611 Observation of a currentinduced force on Bloch lines in NiFe thin films J. Appl. Phys. 67, 5941 (1990); 10.1063/1.346110 Domain Wall Structure and Magnetization Creep in Ni–Fe Films 1200–2000 Å Thick J. Appl. Phys. 41, 1338 (1970); 10.1063/1.1658930 Domain Walls in Thin Ni–Fe Films J. Appl. Phys. 34, 1054 (1963); 10.1063/1.1729367 [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, 01 Dec 2014 19:07:39Current-induced displacements of Bloch walls in Ni-Fe films of thickness 120-740 nm E. Salhi and L. Berger Physics Department, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 (Received 8 March 1994; accepted for publication 7 July 1994) Rectangular current pulses of duration 0.14 p, flowing across Bloch domain walls in N&Fe,, films, cause displacements Ax of these walls, observable by Kerr-contrast microscopy. In zero magnetic field, Ax reaches -14 @pulse at current densities ~30% above the value j, where wall motion starts. This critical current density is j,-1.2X 10” A/m2 for a film thickness w=263 nm. We have measured j, versus film thickness for w=120-740 nm, and find jCmw-‘.l. This suggests strongly that the observed wall motion is associated with an S-shaped distortion of the wall by the circumferential magnetic field of the current. This wall distortion is limited by the wall surface tension. The wall structure becomes that of the so-called asymmetric Neel wall. Through wall distortion, the current pulse pumps kinetic energy and momentum into the wall. This kinetic energy is then dissipated during ballistic wall motion happening largely after the end of the pulse. We also find j, to be independent of pulse duration. I. INTRODUCTION In N&Fe films of thickness w<35 nm, containing N&e1 walls, the dominant interaction between an electric current and a magnetic domain wall seems to arise from the s-d exchange and spin-orbit energies. Current-induced displace- ments of these walls are observed.’ Ni-Fe films with 35 nm <w<85 nm contain cross-tie walls, and a force arising from the resistivity gradient across the film thickness may be im- portant in that range.r On the other hand, Ni-Fe films of thickness w>85 nm contain Bloch walls. Short current pulses are also found to induce wall displacements in these films. In the case2 of a film of thickness 263 nm, the critical current density for in- cipient displacements was j,-1.35X10t” A/m’. The purpose of this article is to extend these measurements of j, to other values of the film thickness w. Here the effect of the current on the wall cannot be described by a force. Rather, the cur- rent pulse causes a variation of the angle $ between wall spins S and the original wall plane,’ as well as an S-shaped distortion of the wall shape. Through that process, the cur- rent pulse pumps kinetic energy and momentum into the wall. This kinetic energy is then dissipated during ballistic wall motion happening largely after the end of the pulse.” In Ref. 2, we assumed that the variation of $ mentioned above was caused by the so-called s-d exchange torque r,d. This current-induced torque on the wall arises3 from the fact that a conduction-electron spin fi/2 is flipped as the electron crosses the 180” wall. Since then, we have come to the real- ization that the circumferential field H,(y) generated by the current also produces a torque on the wall, similar in effect to +rsd. Here, z is the easy axis, and y is the. coordinate normal to the film. In Ni-Fe films of thickness -250 mn, this torque is 120 times larger than T,~, and is probably dominant. It will be discussed in Sec. III. Because of the larger torque, the wall probably moves faster than was assumed in Ref. 2, and the ballistic-overshoot duration is shorter. Il. MEASUREMENTS OF CRITICAL DENSITY], VERSUS FILM THICKNESS AND PULSE DURATION The experimental apparatus and technique were already described in Refs. 1 and 2. Walls are parallel to the induced easy axis of the N&Fe,, film, and normal to the current. They are observed by Kerr-contrast microscopy at a magni- fication of X20. In the present work, rectangular current pulses with a duration of 140 ns are used2 for most measure- ments. The films are prepared by evaporation in a vacuum of =10m6 Torr. A number of different film thicknesses, ranging between 120 and 740 nm, were obtained by varying the deposition time. The critical current density j, is determined by finding the smallest 1 j,] value for which a sequence of -600 pulses would produce a detectable total wall displace- ment. The smallest detectable displacement is about 20 nm/ pulse. Measured j, values are plotted versus thickness w in Fig. 1, on logarithmic scales. The data are consistent with jcocw-2,1. Tn the case of the sample of thickness w=263 nm, j, was also measured as a function of pulse duration +r between 50 and 300 ns. We see (Fig. 2) that j, is independent of r in that range. We also measured the displacements Ax, themselves, in the same sample. The zero-field Ax, values reach -14 /.D/ pulse at current densities ~30% above j,, in a sample of thickness 263 nm. These Ax, values are very sensitive to small easy-axis fields of order 10 ,uT. These results will be analyzed in a separate publication. Ill. THEORY OF WALL AT REST DURING THE PULSE A uniform dc current of density j, flowing along the x direction, through a thin film or ribbon located in the region -w/2<y=~ w/2, produces a circumferential field (some- times called global field by us), given inside the film by Hz= + js (Fig. 3). The magnetic easy axis is along the z direction. We consider a domain wall, originally in the yz plane. The atomic spins S shown on Fig. 3 are everywhere J. Appl. Phys. 76 (8), 15 October 1994 0021-8979/94/76(8)/4787/s/$6.00 Q 1994 American Institute of Physics 4787 [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, 01 Dec 2014 19:07:39100 1000 lo4 wind FIG. 1. Measured values of the critical current density j, of Nis,Fe,, films vs film thickness w. The best-fitted straight line has a slope -21, and gives j,=1.18X1010 A/m’ for w=263 nm. antiparallel to the local magnetization M, . The field H,(y) generates pressures of opposite sign iFig. 3) on the upper and lower halves of the wall. Hence, its effect is equivalent to a torque exerted on the wall. Under its influence, the wall un- dergoes an Z&shaped distortion’ (Fig.. 3). At the film surface, the wall displacement from its original position is called Xdis. The differential equations governing the equilibrium wall shape x(y) are:’ -2M,yj,=rr 2; dx -tan y=dy* (1) Here, M, is the saturation magnetization, CT the wall surface tension, y the angle between the local wall plane and the film normal, and ds the length element along the wall in the xy plane. Local minimizationof the wall surfaces energy leads to 1.5 m- 1 -E d 0 -b w 2 0.5 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 -‘c (10”s) FIG. 2 Measured values of j, vs current pulse duration T, for a NislFe19 film of thickness w=263 nm. FIG. 3. Distortion of a Bloch wall by the circumferential field H,(y) gen- erated by a uniform current density j, . the boundary condition475 y=O at the wall ends y= t w/2. Using the boundary condition and integrating Eqs. (1) twice, we find for X(Y = ~12) =xdis : dis = -Yf(uo); UO=jxljsat; jsat= 4i~llM,w”; u= l- - I 2Y 2 i 11 W ua; u du J-&“Ju,-u Here, the second form of the integral is obtained from the first by setting u = -sin s, uo= -sin so. This second form can be evaluated in terms of elliptic integrals of the first and second kinds. For Ij,l e js,r, X&s is proportional4 to j, . How- ever, as first pointed out in Ref. 5, f(uo) and Xd& diverge logarithmically when Ij,] approaches j,, (see Fig. 4). An- other significant feature of the equations is that j,,, varies like w-z x, is? . The wall momentum, appropriate for motion along 2M, -+w/2 p&V=-- J POYO -w/2 NY)-&. (3) Here yo=1.76X10u rad/s T is the gyromagnetic ratio, ~u0=1.25XlO-~ V s/A m, and pw is normalized to a unit length of wall along Z. Strictly speaking, (I, is3 the angle between the projection of S on the xy plane and the -y axis. We consider a simplified model of the wall, strictly valid only in bulk samples, where properties of the wall such as the surface tension (+ and the thickness A0 along the local wall normal are independent of local wall orientation, and the same in all parts of the wall. In that same model, we expect that, in order to minimize the demagnetizing-field en- 4788 J. Appl. Phys., Vol. 76, No. 8, 15 October 1994 E. Salhi and L. Berger [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, 01 Dec 2014 19:07:39FIG. 4. Average angle $ of wall spins S with the film normal y vs normal- ized current density, during the current pulse. Also, function f(j,/j,J de- scribing the wall displacement xdis for wall distortion during the pulse, ac- cording to Eqs. (2). Also, normalized amplitude Ao,,(7) of wall oscillations just after the pulse vs normalized current density. ergy, the spins S inside a distorted wall at rest tend to lie approximately in a direction parallel to the local wall plane (Fig. 3). This relation between wall shape and spin angle + is: -tan t/(y)= 5. Note that this structure of a distorted Bloch wall is rather similar to that of the so-called asymmetric Ndel wall. Also, we assume the current pulse to be long enough for the wall to come to rest. By combining Eqs. (4) and (l), we obtain after one in- tegration: q+(y)= y(y)=arcsin zf. (5) Finally, Eqs. (3) and (5) give, after setting t=2y/w (G(&J=ld arcsin[(l-t’) JJdt, 2M,w - PW=- E . @(ix ljaat). (6) Here, $ represents an average of $(y) over the wall. We have evaluated the function &j,/j,,J by numerical inte- gration of Eq. (6), and show the results in Fig. 4. The largest possible value of $, obtained when 1 j,] = js,, , is $=0.82 rad.=47.0”. Iv. THEORY OF WALL STEADY STATE AFTER THE PULSE After the end of the current pulse, the local pressure -2M,yj, vanishes. As a result, the wall curvature d ylds also vanishes for the steady state of the wall [Eq. (l)]. The angle I#= y is uniform over the whole wall [Fig. 5(a)], but not zero if the wall is moving with speed u,#O and momen- tum p,,,+O. Actually, when u,#O, + and y are only approxi- mately equal. But this approximation is quite good: Since the maximum wall demagnetizing field is large, being equal to -vw,pw a) Aosw H x& * I ?Xdir b> c==) c) -~ e d) FIG. 5. (a) Steady state of moving distorted wall, after the current pulse. (b) Oscillations of a moving distorted wall around the straight steady-state shape of (a), after the pulse. Oscillation amplitude is Aosc(7)=xdis-X& . (c) ‘IWO-dimensional Bloch wall in a Ni-Fe film, according to LaBonte. The spins S approximate a magnetic vortex with axis A. The solid curve marks the center of the wall. The wall is at rest. (d) Oscillations of a two- dimensional wall around the steady-state shape of (c). M,=l T, the wall spins wiIl stay close to the local wall plane. Note that 60, v,>O, pW>O, in the case shown in Fig. 5(a). If coercivity and Gilbert damping are ignored, the wall energy is independent of wall location and plays, therefore, the role of a kinetic energy. In our simplified model constant a; this energy is: For this wall, the momentum is obtained from Eq. (3): 2M,w pw= - - *. PO Yo _ Hence, the wall speed is dT -POYO dT -PoYoa sin ti --= “+‘=dpw= 2M,w de 2M, cos’ $ with (7) (8) (9) At the film surface, the wall displacement from its original position is W W x&=--tan y=-Ftan . (10) Here, we use the superscript > to differentiate this quantity from the displacement Xdis during the pulse. J. Appl. Phys., Vol. 76; No. 8, 15 October 1994 E. Salhi and L Berger 4789 [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, 01 Dec 2014 19:07:39V. ACTUAL WALL DYNAMlCS DURING AND AFTER THE PULSE We introduce the drift speed u, = - j,fn,e of the charge carriers, which are electron-like iu N&Fe,, . We consider the case u,>O (Fig. 3), i.e., j,<O. By looking at torques, one can show that the wall velocity u, is negative during the pulse, in that case. Therefore, the pinning force +2Mfi,w on the wall is actually positive, and equal to +2M@,w. By writing the equation dp,ldt= +2Mficw, and _using the second of Eqs. (6), one can then easily show that $, starting at zero at the beginning of the pulse, becomes negative. By writing the same equation, and using a reasonable value H,-100 A/m, one can see that the current-pulse duration 7= 140 ns, used in most cases, is quite long enough.for 1/1 to reach its steady- state value of Eq. (6) before the pulse end. We conclude that the curved wall shape of Sec. III, including the steady-state value of Xdis predicted by Eqs. (2), will be realized by the pulse end. Also, since the net torque vanishes, the wall will .~ come to rest. At the end of the pulse, the steady-state wall shape switches abruptly from the curved shape of Sec. III to the straight shape of Sec. IV. And u, switches from zero to a positive value. Of course, the actual wall shape does not change instantaneously at pulse end. Indeed, because of the small value a=lO-’ of the Gilbert damping parameter in Ni-Fe, a steady state will take a very long time to reestablish itself after the pulse. During that time, the wall will oscillate and flop around that new steady-state shape [Fig. 5(b)]. Therefore, the initial amplitude A.,,(T) of these oscillations will be the difference between the old value Xdis and the new value x& of the steady-state wall distortion. Note that, due to the internal nature of the surface-tension forces, and to the oscillatory nature of the damping forces involved in the wall oscillations, the wall momentum of the steady state at rest during the pulse and of the steady state after the pulse are equal. Hence, by Eqs. (6) and (8): - 9(i, lid7 01) where the left-hand side refers to the steady state after the pulse (Sec. IV). Therefore, by combining Eqs. (2), (lo), and (11): Aosc(~)EXdis-X&s= - F (2 f(jx/jsat) -tan[:i%ix/jsat>l~. w Values of A,,,(T) are plotted in Fig. 4. We see that A&T) differs appreciably from zero only when 1 j,l approaches j,, , and diverges at isat. Equations (1) constitute a nonlinear equation obeyed by the function x(y) for an -equilibrium wall at rest at j,#O. Nonlinear effects are important only when Idxldy 1 =ltau 4 =ltan 4 is large. The most obvious such nonlinear effect is a “softening” of the restoring forces which limit wall distor- tion. Due to this softening, Xdis and A,,, diverge when I&l reaches 47”, for 1 j,l =jsat (Fig. 4). But y=+=O before the pulse, and y and + grow at a finite rate towards equilibrium during the pulse, as discussed above. Hence, the wall is still stiff at the beginning of the pulse, and the leading edge of the I 0.8 5 ‘7 0.6 .2 Y o.40L--------1 0.2 0.4 0.6 0.8 XJW- FIG. 6. Ratio j,/jSat vs parameter x,/w, as predicted by Eq. (13). pulse only generates small oscillations. For this reason, we have-only considered the trailing edge in calculating A,,, 1% G91. ._ - VI. THEORY OF H, REDUCTION AFTER THE PULSE By using the work theorem T( +) - T(0) = 2M3,I Ax,,,] and a reasonable value ~~2x10~~ J/m2 in Eq. (7), it be- comes clear that the observed wall displacements Ax,-3-15 ,um are possible only if the coercive field of the film is reduced below its usual value H,-90 A/m, down to nearly zero. This reduction results from the wall oscillations [Fig. 5(b)] excited by the current pulse. According to the Baldwin model7 of coercivity, wall pin- ning is caused by potential-energy wells of radius x,. The value of x, is not well known but may* be of order 0.1-0.5 ,um. If the wall,oscillations have an amplitude A,,,(T) equal. to the well radius, the oscillating ,wall samples equally all parts of the pinning well, so that the pinning force is aver- aged to zero, and H, vanishes. Using Eq. (12), and identify- ing the current density where this happens with the critical current density j, , we obtain au implicit equation for j, : xc=5 12 f(j,lj,,)-tan[9(j,/js,,)l}l or 03) A solution of Eq. (13) is of the form jC/jsat=g(xClw), where g(x) is still another function. Using the definition of j,,, from Eqs. (2), this gives 4C L=m dxclw). s 114) The function &x,/w) has been found numerically from Eq. (13), and is plotted in Fig. 6. For x,Iw>O.2, g(x,/w) is close to unity, and nearly independent~of x,Iw. As shown in the next section, data iu the (j, ,H,) plane for the 263 um sample suggest x,/w=O.O2. By Fig. 6, this corresponds to g(x,/w)=O.715. Assuming x,/w and P to be independent of w, Eq. (14) predicts jCmwm2. This prediction is in good agreement with the w-‘.l dependence found experimentally (Fig. 1). By fitting Eq. (14) to the best experimental value j,-1.18X1010 A/ m2 for w=263 nm, we obtain the value 0-=2.85X10-~ J/m2, if x,/w=O.O2 and M,=l T are as- sumed. This is smaller than the value 0=10X10-~ J/m2 cal- 4790 J. Appl. Phys., Vol. 76, No. 8, 15 October 1994 E. Salhi and L. Berger [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, 01 Dec 2014 19:07:39% h 9 “0 c.. .2 j+H, (1 O-6 T) FIG. 7. The solid line represents the prediction of Eq. (15) for the boundary line between static walls and moving walls in the (j, ,H,) plane, for a value 0.02 of the parameter x&v. The circles and crosses represent the experi- mental boundary line for a sample with w=263 nm. Crosses and circles correspond to incipient wall displacements toward +x and -x, respectively. culated numerically9 for a two-dimensional Bloch wall in a N&Fe film of thickness 200 nm, but of the same order. The value j,=1.18X10r” A/m’ above differs slightly from the value 1.35X10r” quoted in Sec. I, because it results from averaging over several samples (Fig. 1). The fact that j, is found experimentally (Fig. 2) to be independent of pulse duration r is consistent with the idea (see Sec. V) that xdis reaches its final value much before the pulse end. VII. THEORY OF PHASE DIAGRAM IN THE (i,,H,) PLANE Assume that a constant and uniform easy-axis field Hz is present, in addition to the current pulse. If the wall-pinning potential wells have a parabolic shape for 1x1 GX, , the effect of Hz is to translate the wall by an amount x$lJH, along the x: direction, without changing the wall distortion. Hence, this displacement is simply added or subtracted from A,,,(T), at the film surface where the total displacement is the largest. The condition of Eqs. (13) for incipient wall motion be- comes: x,=?zxJYJH,+~ (2 f(i,ljsat)-tan[~(jx/jsat>l}, or Hz-H, k 17: bYjxOsat) - 3 tan[$(j,/j,,t)l) We show in Fig. 7 the phase boundary in the {j, ,H,) plane between regions corresponding to static and to moving walls, as predicted by Eq. (15) for ,~&~=llO pT, w=263 nm, M,=l T, u=3.75X10F4 Jlm2, and x,Iw=O.O2. We also show the experimental boundary line2 for the sample with w =263 mn. The predicted line has the same overall shape as the experimental line. The measured value j,=1.55X10” A/m2 at Hz=0 is somewhat larger (Fig. 7) than that quoted in Sec. VI, because’ the phase boundary .was obtained with pulses of exponential shape. Correspondingly, the IT value is larger. As mentioned in Sec. VI, the actual x, value is not well known for our samples. If x, is related to the size of crystal grains in Ni-Fe, it is expected to increase with increasing film thickness w. Then the x,/w parameter might be roughly independent of w. The value x,Iw=O.O2, used above for w =263 nm, leads to x, =5 nm. This is much smaller than the x,=0.1-0.5 ,um obtained’ for bulk samples. VIII. FREQUENCY OF DISTORTION OSCILLATIONS As discussed in Sec. III, wall distortion is caused by torques generated by the circumferential field H;(y) of the current. In the case of a. pulse, the time variation of the torque induces wall oscillations (Sec. V), which cause an observable reduction of H, (Sec. VI) for some time interval after the pulse. If, instead of having the form of a pulse, the current had a sinusoidal variation with time, we expect that the ampli- tude of the wall oscillations would exhibit a maximum when the frequency of the current is equal to the frequency f,,, of the wall oscillations. As far as we know, this experiment has not yet been performed. However, a similar resonance has been observedlO when an ac magnetic field H, is applied to a Bloch wall in a Ni-Fe t&u, along the in-plane hard-axis direction. It is clear that such a field exerts on the wall spins a torque similar to that H,(y) exerts on the wall itself. The product of f,,,, and w was found”. to be constant, and equal to 20 Hz m. In their interpretation of these experiments, the authors considered the magnetic vortex present in a two-dimensional Bloch wall in Ni-Fe films [Fig. 5(c)], and assumed that the resonance involved the oscillatory motion of the vortex axis along the y direction normal to the film. However, this motion of the vortex axis is associated [Fig. 5(d)] with motions of the wall center (i.e., surface where 8=~/2) along x, which have op- posite sign in the upper and lower part of the wall. We see that these motions of the wall center are very similar to the wall distortion of a one-dimensional wall considered in our Fig. 3, so that their picture of the wall oscillations is actually equivalent to ours. In order to describe the oscillations of a one-dimensional wall we userr the displacement x(3w/8) at the point of the = wall located at y = 3 w/8. This point is chosen because its displacement and speed are smaller and more typical than that of the point y= w/2 at the film surface. A linearized equation of motion like that for a harmonic oscillator can be written in the limit lx(3w/8)/ <w, for this wall coordinate. In it, the restoring-force term isr’ (32/39)( 12u/w”)x(3 w/8). After adding an inertial term containing the Doring mass m WP the equation of motion reads g(~)x(~)+mw$x(;)=o. Therefore, the sinusoidal solutions for x(3 w/8) will have a frequency J. Appl. Phys., Vol. 76, No. 8, 15 October 1994 E. Salhi and L. Berger 4791 [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, 01 Dec 2014 19:07:39fosez( g . ,y2 “L. w We use the same constant value (r=2.85X 10m4 J/m2 as before, Also, we use a constant Doring mass n~,,,-0.5XlO-~ kg/m2, as calculated numerically12 for a (two-dimensional) wall in Ni-Fe for w= 100-200 nm. Then, Eq. (16) predicts f osc +w=const.=38 Hz m. This prediction of a product f ,,sca w independent of w is in good agreement with the experimental finding&’ dis- cussed earlier. Also, the predicted value of the product is of the same order as the experimental value. This is as good an agreement as could be expected in view of the uncertain value of m, used. IX. CONCLUSIONS AND FINAL REMARKS Our main experimental finding is that the current density ’ j,, above which wall displacements appear after a current pulse, is proportional to the power -2.1 of the film thickness w (Fig. 1). This suggests strongly that these wall displace- ments are associated with an S-shape distortion (Fig. 3) of the Bloch wall during the pulse. Indeed, we.know from ear- lier worksVrl that the critical current density jsat at which this wall distortion becomes large is proportional to wm2. Wall structure resembles that of an asymmetric N&e1 wall. Wall motion happens mostly during a ballistic overshoot after the pulse. We also fmd j, to be independent of the duration r of the rectangular pulses (Fig. 2). The large size Ax,=~l4 pm of observed wall displace- ments suggests that the coercive~ field H, is considerably reduced below its normal value, during the ballistic over- shoot. This reduction is probably associated with oscillations of the distorted wall around its steady-state shape [Fig. 5(b)]. A simplified one-dimensional model of the wall, with constant surface tension o and Doring mass m,,,, explains satisfactorily our experimental results. It also explains the f osc~w-l dependence of the frequency f,,, of the wall reso- nance observed’ for Bloch walls exposed to high-frequency hard-axis fields. I The actual nature of Bloch walls in Ni-Fe films is two-dimensional’ pig. 5(c)]. Therefore, our one- dimensional model is not very realistic. However, it has the advantage of leading to simple analytical formulas [Eqs. (14)-(16)] which are also in quite good agreement with ex- periments. In particular, we can use the knowledge of diver- gences [Eq. (12) and Fig. 41 happening at a finite value jsat of the current density, which was developed in earlier work.‘,‘r ACKNOWLEDGMENTS This work was supported by NSF Grants No. DMR 88- 03632 and DMR 93-10460. We are thankful to the CMU Data Storage Systems Center, supported by NSF Grant No. ECD 89-07068, for the use of their facilities. We thank Al Thiele for useful discussions. t C.-Y. Hung and L. Berger, J. Appl. Phys. 63, 4276 (1988). aE. Salhi and L. Berger, J. Appl. Phys. 73,640s (1993). 3L Berger, Phys. Rev. B 33, 1572 (1986); J. Appl. Phys. 71, 2721 (1992). 4H. J. Williams and W. Shockley, Phys. Rev. 75, 178 (1949); R. Aleonard, P. Brissonneau, and L. Neel, J. Appl. Phys. 34, 1321 (1963). ‘L. Niel, C. R. Acad. Sci. (Paris) 254, 2891 (1963); Y. Hsu’and L. Berger, J. Appl. Phys. 53; 7873 (1982). A factor of 4 should be inserted in the denominator of Eq. (2) of this last paper. 6A. P. Malozemoff and J. C. Slonczewski, Magnetic Domain WaZZs in Bubble Materials’ (Academic, New York, 1979), pp. 126, 154. 7J. A. Baldwin and-G. J. Culler, J. Appl. Phys. 40, 2828 (1969). “L. Berger, J. Appl. Phys. 50, 2137 (1979). See Eq. (8), and text under Bq. (10). ‘A. E. LaBonte, J. Appl. Phys. 40, 2450 (1969). ‘ON. Smith, T. Jagielinski, J. Freeman, and P. Koeppe, J. Appl. Phys. 73, 6013 (1993). A talk. was delivered at the Annual Conference on Magne- tism and Magnetic Materials, Houston, Dec. 1992. “A. K. Agarwala and L. Berger, J. Appl. Phys. 57,3505 (1985). See Eq. (5). “S. W. Yuan and H. N. Bertram, Phys. Rev. B 44, 12395 (1991). 4792 J. Appl. Phys., Vol. 76, No. 8, 15 October 1994 E. Salhi and L. Berger [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, 01 Dec 2014 19:07:39
1.2024855.pdf	(PRM) need this correction at higher frequencies and shows that taking this measure can eliminate a system error of 10- 2 to 10- 3 orders of magni- tude. 9:35 W3. Viscoelastic effects on solid amplitude and phase transformers. L. M. B.C. Campos (Instituto Superior Tcnico, 1096 Lisboa Codex, Portugal) The longitudinal oscillations of nonuniform bars are used as displace- ment amplifiers in power tools and other devices. These often operate near resonance [E. Eisner, J. Acoust. Soc. Am. 35, 1367-1372 (1963) ], so that the elastic model should be replaced by a viscoelastic one, which includes damping. In the present paper, the longitudinal vibrations of a tapered viscoelastic bar are discussed generally from the wave equations for the displacement and strain. Exact solutions are obtained for the exponential, catenoidal, sinusoidal, and inverse shapes, and also for the Gaussian and power-law shapes. These solutions generalize earlier results for elastic bars; e.g., the elastic Gaussian bar [D. A. Bies, J. Acoust. Soc. Am. 34, 1567-1569 (1962)] is generalized to a viscoelastic one. Diagrams of wavenumber and damping ratio versus frequency and viscous relaxation time are presented for several shapes of tapered bar; they describe the propagation and dissipation of oscillations and their effects on amplitude and phase. 9:50 W4. Relationship between the impedances and the reflection and transmission coefficients of a junction. L. J. Maga and G. Maidanik (David Taylor Naval Ship Research and Development Center, Bethesda, MD 20084) A junction constitutes the coupling between dynamic systems. In this case, the one-dimensional dynamic systems so coupled are characterized by their wavenumbers and mass densities. The interface impedances of the dynamic systems at the junction may then be defined. The junction may be defined in terms of a junction impedance matrix. The nature of this matrix will be discussed. This matrix, together with the interface impedances of the dynamic systems, may then be employed to derive the junction inter- action matrix, which consists of reflection and transmission coefficients. The manner and results of this derivation will be discussed and a few examples will be cited. 10:05 WS. Damping and vibration analysis of bonded beams with a lap joint. Mohan D. Rao and Malcolm J. Crocker (Mechanical Engineering Department, Auburn University, Auburn, AL 36849) In this paper, a theoretical model to calculate the loss factors and the resonance frequencies of flexural vibrations of a system of two parallel beams bonded with a single lap joint by a viscoelastic material has been developed. First, equations of motion of the joint region are derived using a differential element approach considering the transverse displacements of the upper and the lower beam to be different. The normal force between each beam and the adhesive layer is represented by a Pasternak base mod- el, which consists of closely spaced linear springs. The shear force at the interface is modeled using a viscous model for friction. The resulting equa- tions of motion, together with equations of transverse vibrations of the beams away from the joint, are solved using motion continuity conditions and boundary conditions at the free ends of the beams. Equations for calculating the resonance frequency and the loss factor for the case of clamped-clamped boundary conditions are then derived. Numerical re- sults are generated and are compared with experiments for a system of two beams with a simple lap joint made of graphite epoxy composite material. [Work supported by NASA-MSFC. ] 10:20 W6. Vibration analysis of mass-loaded beams. Dhanesh N. Manikanahally and Malcolm J. Crocker (Department of Mechanical Engineering, Auburn University, Auburn, AL 36849) A general procedure for determining the dynamic response of a mass- loaded free free beam subjected to a harmonic and transient force is given. Though free free beams are considered for analysis, the same procedure could be extended for other end conditions also. The beam is assumed to have structural damping for determining the steady-state response due to harmonic force excitation. The mode shapes for free vibration, dynamic response, and dynamic strain due to forced excitation are presented in a graphical form. The analysis is used to study a space structure, modeled as a mass-loaded free free beam, by making an exhaustive optimization search for minimum dynamic response due to harmonic and transient excitation forces. The computer program developed for the analysis is used to check some simple beam problems [G. B. Warburton, The Dy- namical Behaviour of Structures (Pergamon, New York, 1976) ]. [Work supported by SDIO/DNA, Contract No. DNA-001-85-C-0183. ] 10:35 W7. Finite element analysis of a large vibrating space structure. B. S. Sridhara and Malcolm J. Crocker (Department of Mechanical Engineering, Auburn University, Auburn, AL 36849) The large flexible space structure to be used in the Strategic Defense Initiative Project is modeled as a long flexible beam with three point masses. Using the Galerkin method, finite element equations are formu- lated that take advantage of the fact that the natural boundary conditions come out as a result of integration by parts. Hermite polynomials have been chosen for the shape or interpolation function. A tubular beam made of graphite epoxy material, 100m in length with 2.0-m o.d. and 1.95-m i.d. is considered for the purpose of this analysis. Masses of 500, l0 000, and 1000 kg are placed at the left extreme end, at a distance of 20 m from the left extreme end, and at the fight extreme end of the beam, respectively. Natural frequencies of the long flexible beam are calculated using stan- dard methods and the mode shapes are also plotted. Results are also ob- tained for a different location and increased values of the largest mass. Computer codes are being prepared to obtain the dynamic and steady- state response of the beam subjected to an impulse and a sinusoidal force. 10:50 W8. Procedure for the measurement of wavenumber/frequency admittance of structures. Karl Grosh, W. Jack Hughes, and Courtney B. Burroughs (Applied Research Laboratory, The Pennsylvania State University, P.O. Box 30, State College, PA 16804) A procedure for measuring the wavenumber/frequency admittance of structures using an array of drives in a standing wave pattern was devel- oped in a paper presented in the ASA meetings in Indianapolis, IN on 14 May 1987 [ K. Grosh, J. H. Hughes, and C. B. Burroughs, J. Acoust. Soc. Am. Suppl. 1 81, S73 (1987) ]. In this paper, data are presented to verify the validity of the measurement procedure. The wavenumber/frequency spectra are presented for arrays of point drives and the vibration responses of lightly damped beams and long beams with heavily damped ends. The effect of steering the array to different wavenumbers and varying the num- ber of drives is examined. 11:05 W9. Influence of various parameters on the transmission of vibrational power. T. Gilbert, J. M. Cuschieri, M. McCollurn, and J.L. Rassineux (Center for Acoustics and Vibrations, Department of Ocean Engineering, Florida Atlantic University, Boca Raton, FL 33431 ) The transmission of vibrational power between two thin plates in an L- shape configuration is investigated using an SEA model. Only bending waves are considered in the model. Expressions are developed for both the ratio of energy levels in the two plates and the ratio of the transmitted power to the input power. It is found that these ratios are only dependent upon three parameters: frequency, dampings of the plates, and coupling loss factors between the two plates. Therefore, a way to decrease both the $51 J. Acoust. Soc. Am. Suppl. 1, Vol. 82, Fall 1987 114th Meeting: Acoustical Society of America $51 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 121.205.113.172 On: Tue, 18 Aug 2015 22:49:01