Source: https://www.groundai.com/project/spin-and-impurity-effects-on-flux-periodic-oscillations-in-core-shell-nanowires/
Timestamp: 2019-04-25 21:46:24+00:00

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We study the quantum mechanical states of electrons situated on a cylindrical surface of finite axial length to model a semiconductor core-shell nanowire. We calculate the conductance in the presence of a longitudinal magnetic field by weakly coupling the cylinder to semi-infinite leads. Spin effects are accounted for through Zeeman coupling and Rashba spin-orbit interaction (SOI). Emphasis is on manifestations of flux-periodic (FP) oscillations and we show how factors such as impurities, contact geometry and spin affect them. Oscillations survive and remain periodic in the presence of impurities, noncircular contacts and SOI, while Zeeman splitting results in aperiodicity, beating patterns and additional background fluctuations. Our results are in qualitative agreement with recent magnetotransport experiments performed on GaAs/InAs core-shell nanowires. Lastly, we propose methods of data analysis for detecting the presence of Rashba SOI in core-shell systems and for estimating the electron g-factor in the shell.
Recent years have seen significant progress in the development and fabrication of semiconductor nanostructures. Nanowires of diameters of the order of 10−100 nm can now be grown. Bakkers and Verheijen (2003); Li et al. (2006); Thelander et al. (2006); Yang et al. (2010) Core-shell nanowires are composed of a thin layer (shell) surrounding a core in a tubular geometry. The cross section may be circular, but also polygonal (e. g. hexagonal) reflecting the lattice structure of the materials. If both shell and core are semiconductors, they may be chosen such that the difference in conduction band energies forms a potential barrier confining carriers to either the core van Tilburg et al. (2010); Popovitz-Biro et al. (2011) or the shell. Rieger et al. (2012); Jung et al. (2008) Recent examples include nanowires composed of an InAs shell grown on a GaAs core resulting in the formation of a conductive electron gas in the shell which may be further augmented by modulation doping the core. Gül et al. (2014); Blömers et al. (2013) Such systems provide a fascinating means for studying fundamental quantum effects such as interference.
In this paper, we analyze the energy spectrum, charge and current densities of electrons confined to a closed cylindrical surface of finite length pierced by a longitudinal magnetic flux. We calculate the magnetoconductance of the finite system by coupling it to leads. In particular, we focus on FP oscillations in both conductance and spectrum and discuss the effects of core donor impurities and electron spin, which is included through Zeeman splitting and Rashba SOI and may thus affect transport nontrivially. The experimentally-relevant effects of nonuniform coupling to leads are also discussed. While donor impurities dampen conductance oscillations, they remain resolvable after extensive averaging over multiple random impurity configurations, assuming realistic donor densities. Using parameters comparable to those reported in Ref. Gül et al., 2014 , we attribute background conductance oscillations, which are superimposed on the FP oscillations, to an interplay between the finite system length and Zeeman splitting, propose means for detecting the presence of Rashba SOI and discuss a method for estimating the shell electron g-factor based on transport data.
In Sec. II we describe the closed-system model and transport formalism and discuss results in Sec. III. The case with impurities is treated in Sec. IV and in Sec. V we give a comparison with recent experimental data. Finally, we offer concluding remarks in Sec. VI.
We consider a cylindrical core-shell nanowire of radius r0 and length L0 where the shell and core are composed of different semiconductors such that the difference in conduction band energies confines conduction electrons to the shell with negligible wavefunction leakage into the core. We assume that the shell thickness is small compared to r0 and L0 such that only the lowest radial mode is occupied and approximate the shell as infinitely thin. In principle r0 corresponds to the mean radius of the shell and thus we model the nanowire as a two-dimensional cylindrical surface.
where α determines the SOI strength and σφ=cos(φ)σy−sin(φ)σx describes the tangential spin projection. A Dresselhaus type SOI, arising due to inversion asymmetry in the crystal structure of the shell, may also be included. Such effects are typically minor in InAs compared to those of the Rashba term, and we have therefore chosen to neglect them in the present context.
Here n∈Z is the orbital angular momentum quantum number, s=±1 describes the spin projection along z, χs are eigenspinors of σz and p∈Z+ arises due to the longitudinal quantization.
In order to calculate the conductance of the finite cylindrical system we couple it to leads. The leads are taken as cylindrical continuations of the finite central system with the same radius r0, extending from the junctions to the left (L) and right (R) along the z-axis over z<0 and L0<z, respectively. Their purpose is to supply phase-coherent electrons to the now open central system (S) from two reservoirs (or contacts) maintained at chemical potentials μL and μR. Datta (1995) Electrons propagate through the leads to the junctions where, as with the central system, hard-wall boundary conditions are imposed, but injections into the central system are made possible through a geometry-dependent coupling kernel in the form of an overlap integral between each lead and the central cylinder. Gudmundsson et al. (2009) Aside from backscattering due to hard-wall boundary conditions, we assume that all scattering takes place in the central system. Since HS in Eq. (5) is time-independent, only elastic scattering is considered.
which will be used to calculate conductance in this paper. Energy-dependent quantities are evaluated at the chemical potential μ which is uniform throughout the system. From Eq. (12) it is clear that GS and hence G are primarily determined by the geometry and properties of the central system through HS. The leads enter through the self-energies and provide level-broadening for the central system as discussed before.
and invert it numerically. Evaluating the self-energy matrix elements ⟨a|Σi|b⟩ with i=L,R also yields the matrix representations of Γi in Eq. (16) and is thus sufficient to calculate G using Eq. (17).
where (z,φ) and (z′,φ′) are coordinates of the central cylinder and lead i, respectively. The kernel is real and due to the δ-function conserves the angular coordinate between lead and central system producing a circularly symmetric coupling. gi0 is a parameter with the dimension energy/length which governs the overall strength of the coupling and can be used to control level-broadening in the central system. The parameter diz determines how rapidly the coupling decreases along the cylinder axis. To be consistent with the assumption of only indirect coupling between leads via the central system, diz is chosen such that the exponential coupling of a given lead vanishes in the vicinity of the other lead. The Ki-modulated overlap integral in Eq. (22) incorporates the geometry and properties of both central system and leads giving state-dependent level-broadening.
where nL∈Z, kL∈R+, sL=±1 and χsL is an eigenspinor of σz. Right lead states ΨRnRsRkR(r) are obtained by switching the index L→R and setting z→z−L0 such that they vanish at z=L0.
Using the eigenstates of HS [Eq. (5)] and the isolated leads [Eq. (24)] along with the kernel of Eq. (22), the self-energy matrix elements [Eq. (19)] are evaluated. Note that each self-energy matrix element contains two overlap integrals. The sum over the lead quantum number nL is truncated at the same value as the central system angular modes n [Eq. (7)]. For each nL, both spin projections are included and the integral over the continuous lead quantum number kL is done analytically by extension into the complex plane. From the self-energy matrix elements of both leads at E=μ, the matrix representations of the operators GS(μ) [Eq. (12)] and Γi(μ) [Eq. (16)] follow and then G(μ) is calculated using Eq. (17).
To conclude this section, we mention that alternative, grid-based transport methods exist to calculate the conductance of nanowires. Datta (1995) An example is the scattering matrix formalism, implemented for tubular nanowires in Ref. Serra and Choi, 2009 .
We consider a cylindrical shell using material parameters for InAs. Bringer and Schäpers (2011); Manolescu et al. (2013) The effective electron g-factor is ge=−14.9 and we use the Rashba SOI parameter α=20 meVnm which corresponds to a strong confining radial field. As the effective mass of conduction electrons at the Γ-point we use me=0.023m0. The dielectric constant is taken as ϵr=14.6. Unless otherwise specified, we assume a shell radius r0=16.8 nm and nanowire length L0=50.4 nm corresponding to an aspect ratio η=L0/r0=3 which ensures that angular and axial quantization result in approximately equal level spacing. Growth of nanowires of comparable radius has been reported in Refs. Jung et al., 2008; Richter et al., 2008; Bakkers and Verheijen, 2003 .
In the following subsections we discuss FP oscillations and spin effects on the magnetoconductance in our model. We furthermore consider cylindrical symmetry breaking due to Coulomb impurities and a broken circular symmetry of the coupling scheme.
Figure 1: (Color online) Conductance of a cylinder with aspect ratio η=3 for varying values of μ and Φ with: (a) α=ge=0. (b) α=0, ge=−14.9. (c) α=20 meVnm, ge=0. (d) α=20 meVnm, ge=−14.9. Conductance peaks correspond to broadened chemical potential intersections with the spectrum resulting in periodic conductance oscillations provided ge=0 (compare with Fig. 2). Their shape and phase depends on the value of μ considered.
Figure 2: (Color online) Spectrum of a cylinder with aspect ratio η=3 as a function of Φ with: (a) α=ge=0. (b) α=0, ge=−14.9. (c) α=20 meVnm, ge=0. (d) α=20 meVnm, ge=−14.9. Provided ge=0, the spectrum exhibits periodic oscillations even in the presence of Rashba SOI.
Figure 2 shows the calculated conductance of a finite cylinder as a function of Φ and μ. Different subfigures demonstrate the effects of the spin-dependent terms Eqs. (3) and (4) on G. For reference, the flux-dependence of the closed cylinder spectrum is given over the corresponding energy range in Fig. 2. A detailed analysis of the spectra of closed cylinders of different aspect ratios is given in Ref. Gladilin et al., 2013 .
Roughly, conductance peaks correspond to μ intersections with the spectrum, so G(Φ) manifests as the broadened and slightly shifted spectrum ϵSa(Φ) of HS. This is due to the self-energy operators of the leads in GS [Eq. (12)]. We have deliberately chosen coupling parameters such that the induced shift and level-broadening are both of the order 1 meV, such that the close correspondence between spectrum and conductance in Figs. 2 and 2 becomes evident. Our intention is thus to minimize the effects of the leads and the particular form of the coupling kernel [Eq. (23)] on G, which should be governed by the physics of the central system, i. e. by HS.
In the absence of Rashba SOI, the central system Hamiltonian has the eigenstates Eq. (7). At Φ=0 each level is quadruply degenerate, except states with n=0 which are only doubly degenerate. At Φ=0 states with successively higher orbital angular momentum Lz=±ℏn pile into a given axial mode forming a ring-like spectrum until a new axial mode sets in. Hence, the spectrum can be thought of as a superposition of the ring-like spectra of different axial modes. When ge=0 the spectrum is periodic in Φ with period Φ/Φ0=1, i. e. increasing Φ/Φ0 by 1 is equivalent to reducing Lz by ℏ at a fixed energy. Lorke et al. (2000) Hence, the oscillations are similar to Aharonov-Bohm oscillations which have been studied extensively in ring-like Sheng and Chang (2006); Takai and Ohta (1993, 1994); Washburn et al. (1987); Nowak and Szafran (2009); Alexeev and Portnoi (2012); van Oudenaarden et al. (1998); Daday et al. (2011); Gefen et al. (1984); Büttiker et al. (1984) and cylinder-like Holloway et al. (2013); Jung et al. (2008); Gladilin et al. (2013); Tserkovnyak and Halperin (2006); Ferrari et al. (2009) geometries, i. e. without and with a longitudinal degree of freedom, respectively. There is no coupling between Φ and longitudinal electron motion, so FP oscillations on cylinders manifest due to the same principles as those observed in quantum rings, but with an added degree of freedom through the length-dependent p2/L20-term.
Figure 3: (Color online) Equilibrium current density j on the cylinder surface uncoupled to leads with α=0, pierced by a longitudinal magnetic flux Φ/Φ0≈0.4. The current forms concentric circles and is circularly symmetric.
Figure 3 shows the equilibrium current density j [Eq. (8)] on the surface of the cylinder uncoupled to leads at Φ/Φ0≈0.4 with α=0. The current density is obtained by summing up the contributions from the N=8 lowest states, a realistic number of electrons given the density reported in Ref. Bringer and Schäpers, 2011 . Setting ge=0 does not change j in this case, since the Zeeman term does not affect the velocity operator v [Eq. (10)]. However, spin-splitting increases with Φ, changing the orbital characteristics of the energetically lowest states as they become spin-polarized, which affects j. The system is invariant under rotations around the z-axis since [HS,Dz(θ,^z)]=0 where Dz(θ,^z)=exp(iθJz/ℏ) is the rotation operator around the cylinder axis by the finite angle θ. As a result, the electron density ρ on the cylinder surface is circularly symmetric. Since the velocity operator v commutes with Dz, j is rotationally invariant and because ⟨a|δ(r−r′)vz|a⟩∝i⟨p|δ(z−z′)∂z|p⟩ is purely imaginary for all p, the axial component jz vanishes [Eqs. (7) and (9)]. Thus j is composed of concentric circles, each of constant current density. In the closed system, ρ and j thus reflect that electrons enclose a magnetic flux resulting in the FP oscillations observed in the spectrum.
In Figs. 2 (a) and (b) we show the calculated conductance of a cylinder coupled to leads as a function of μ and Φ without (a) and with (b) the Zeeman term. The flux-dependence of the spectrum in the corresponding energy range is given in Figs. 2 (a) and (b). When ge=0 the conductance evidently retains the periodicity of the spectrum and hence exhibits oscillations with period Φ/Φ0=1. While the conductance oscillations are periodic for all values of μ, their phase and shape in a single period is sensitive to the value of μ considered. Including the Zeeman term breaks the periodicity of conductance oscillations as it does to the spectrum. Tserkovnyak and Halperin (2006) The resulting spin-splitting of states can produce magnetoconductance curves which are gradually increasing, decreasing or relatively stable at low values of Φ/Φ0 depending on the value of μ considered, as may be seen in Fig. 2 (b). This point will be further discussed in Sec. V. Our numerical results show the same overall trends in the density of states (DOS) depending on μ as the flux is varied.
Including Rashba SOI, we obtain the eigenstates of HS given by Eq. (5) by numerical diagonalization in the basis (7). Examples of the resulting energy spectrum are shown in Figs. 2 (c) and (d) for ge=0 and ge≠0, respectively. Compared to the spectrum with α=0 and ge=0 in Fig. 2 (a) the Rashba term generally lifts spin-degeneracy at Φ≠0, but crossings still appear at integer values of Φ due to the fact that HS commutes with Jz.
where θ∈R. Analogous to the finite cylinder, one then obtains vanishing tangential spin density ⟨χ±n(k)|σϕδ(r−r′)|χ±n(k)⟩=0 which reconciles the two cases.
Figure 2 (c) shows the conductance of the finite cylinder coupled to leads with Rashba SOI included and ge=0. Compared with the case α=ge=0 shown in Fig. 2 (a), the Rashba term causes a split and shift of conductance curves. Generally, this results in the appearance of more peaks of smaller amplitude within a given period at fixed μ. The splitting and shift is analogous to that which occurs in the closed-cylinder spectrum [Figs. 2 (a) and (c)], further demonstrating the close correspondence between spectrum and conductance in this formalism. As with the spectrum, including Rashba SOI alone is insufficient to break the periodic oscillations in conductance with Φ at a fixed μ. Tserkovnyak and Halperin (2006) Instead, it modifies the shape and phase of conduction curves within a single period. Including the Zeeman term also breaks the periodicity of the spectrum as in the case when α=0, see Fig. 2 (d). Again, this is reflected in the conductance as Fig. 2 (d) shows.
Figure 4: (Color online) Magnetoconductance evaluated at μ=29 meV of the cylinder with spin neglected and unrestricted coupling (solid), compared to the case with coupling restricted to (dashed): (a) (φLmin,φLmax)=(0,π) and (φRmin,φRmax)=(π,2π). (b) (φLmin,φLmax)=(π/2,3π/2) and (φRmin,φRmax)=(π/2,2π). Restricting the coupling alters the shape of the conductance oscillations, but they remain flux-periodic.
Figure 4 compares the magnetoconductance of the cylinder with restricted and unrestricted coupling. Spin is neglected for simplicity. We see that the oscillations indeed remain flux-periodic. However, the overall conductance is reduced and the shape of the oscillations within a given period may change significantly depending on the intervals considered.
In realistic core-shell nanowires the number of shell conduction electrons may be increased by modulation doping the core with donors. Blömers et al. (2013); Gül et al. (2014) This produces ionized Coulomb impurities in the core, i. e. attractive potentials to shell conduction electrons. In this section we discuss the effects of such donor-like impurities on both closed and open cylindrical systems.
where Qm−1/2 are associated Legendre functions of the second kind of zeroth order and half-integer degree and γ=[r20+r2i+(z−zi)2]/2r0ri. The Legendre functions are obtained using the code provided in Ref. Segura and Gil, 2000 .
which we solve numerically as a system of equations.
Figure 5: (Color online) Electron (top) and current (bottom) densities of 8 electrons on the cylinder with aspect ratio η=3 pierced by a longitudinal flux Φ/Φ0≈0.4 with two, separate impurity configurations (left and right). Due to the impurities (filled dots) the rotational and parity symmetries are broken (compare with Fig. 3). Bright and dark regions correspond to regions of high and low charge density, respectively.
Impurity potentials of the form Eq. (29) break the circular symmetry of the system, except in the special case when the impurities lie on the cylinder axis ri=0. Gladilin et al. (2013) Small deviations in impurity location from the cylinder axis introduce in the spectrum avoided crossings for states with low Lz. The gaps are small if only few impurities are present and located close to the cylinder axis. Avoided crossings in rings due to disorder are for example discussed in Ref. Büttiker et al., 1984 . The impurities also couple to longitudinal electron motion and so their location on the z-axis can strongly affect densities, as the impurity potentials generally ruin also the longitudinal parity symmetry of the system. For example, if the impurities are concentrated close to the upper end z=L0 of the cylinder, the longitudinal symmetry is manifest broken as ρ and j increase in the upper half but decrease in the lower half. Placing impurities close to the cylinder center (ri=0, zi=L0/2) will however produce densities that are nearly indistinguishable from the case without impurities.
Impurities are generally not located solely around the center of the cylinder axis in realistic core-shell nanowires and densities may differ significantly for more generalized distributions. In Fig. 5 we show the densities for two distributions, where impurity coordinates (φi,zi) are marked with large dots. The calculations are done with spin suppressed. The impurities in Figs. (a) and (c) (configuration 1) are uniformly distributed along the radial direction with coordinates ranging between 0.15≤ri/r0≤0.82, more concentrated in the upper half of the cylinder. In Figs. (b) and (d) (configuration 2) the impurities are condensed into a narrow angular interval around φi≈π/2 close to the cylinder surface with 0.55≤ri/r0≤0.76. Both configurations strongly break the rotational and parity symmetries in the closed system as the densities show. Configuration 2 is composed of impurities that are evenly distributed along the cylinder length at comparable distances from the surface in a narrow angular interval. As a result, they form a potential well around φ≈π/2 which traps states of low orbital angular momentum Lz. This “flattens” the corresponding energy levels as functions of Φ and suppresses their FP oscillations, similar to a transverse electric field. Barticevic et al. (2002); Alexeev and Portnoi (2012) This is reflected in j [Fig. 5 (d)] which shows the formation of a vortex circulating the impurity cluster, greatly deforming the circular motion. As Fig. 5 (c) shows, configuration 1 affects j more modestly by for example introducing nonvanishing jz.
Figure 6: (Color online) Magnetoconductance of a cylinder with aspect ratio η=3 with impurity configuration 1 [Figs. 5 (a) and (c)] without (a) and with (b) spin included. The figures are qualitatively similar to the case without impurities given in Figs. 2 (a) and (d), but with damped oscillations. Impurities alone are insufficient to break the periodicity of the oscillations.
to shift the central system spectrum for a given configuration so that ground state energies match with and without impurities, in order to make possible a comparison between different impurity configurations at the same chemical potential. To demonstrate the effects of impurities on magnetoconductance, let us consider impurity configuration 1 used in Figs. 5 (a) and (c). Realigning the spectrum requires a gate voltage VG=19.1 mV. Figure 6 shows the resulting conductance of the finite cylinder coupled to leads without (a) and with (b) spin included as a function of Φ/Φ0 and μ. Provided the oscillations were periodic prior to the inclusion of impurities (i. e. if ge=0) they remain so when impurities are included. This is because the impurity potentials do not couple to Φ. The conductance curves with and without impurities [Figs. 2 (a) and (d)] are qualitatively similar, but the impurities dampen oscillation amplitudes by reducing maxima and increasing minima.
Figure 7: (Color online) Magnetoconductance evaluated at μ=29 meV of the cylinder with spin neglected, averaged over Nc=250 (dashed) random impurity configurations containing (a) N=4 and (b) N=8 impurities each. Further averaging does not affect the results significantly. The solid lines show G(Φ) without impurities. Impurity averaging reduces conductance oscillations, but even with a highly doped core (N=8) they are still clearly visible.
Figure 7 (a) compares G(Φ) at μ=29 meV without impurities and ⟨G(Φ)⟩ averaged over Nc=250 configurations of N=4 impurities. We neglect spin for simplicity. Further averaging does not affect the results significantly. The applied gate voltage VG is obtained by averaging the shift of the ground state over multiple N-impurity configurations. The oscillations are indeed damped, but present. Increasing the number of impurities to N=8 [Fig. 7 (b)] the amplitude drops, but the oscillations still survive. Actually, for N=8 impurities, even averaging over Nc=10 configurations already yields qualitatively the same ⟨G(Φ)⟩ as observed in Fig. 7 (b) after extensive averaging, which implies that at such high core donor concentrations the exact impurity configuration is not paramount. The damping suggests that conductance oscillations may be reduced in amplitude beyond achievable experimental resolution in extremely disordered samples. However, our simulations indicate that even in the presence of a large but realistic core-donor density, the oscillations are clearly resolvable. Finally, we mention that our model donor impurities [Eq. (29)] do not account for screening. Screening of donor impurities in the core would reduce their effects on conduction electrons and hence on both closed and open system properties. Similarly, electron-electron interaction would oppose impurity-induced localizations in the system, e. g. as in Figs. 5 (b) and (d), and hence weaken impurity effects. Gladilin et al. (2013) By ignoring screening effects in the core and electron-electron interaction, our simulations thus describe “a worst-case scenario” of the electron-impurity interactions.
In this section we compare simulations using realistic parameters with recently reported measurements performed on GaAs/InAs core-shell nanowires. Results in Ref. Gül et al., 2014 show FP conductance oscillations superimposed on slowly-varying background oscillations in hexagonal GaAs/InAs core-shell nanowires. The background oscillations are attributed to universal conductance fluctuations. Our cylindrical model represents an idealized core-shell nanowire and neglects some aspects present in experiment, notably the hexagonal structure, shell thickness and electron-electron interaction. Out of the effects considered in this paper, we have shown that only Zeeman splitting can break the periodicity of oscillations in cylinders, e. g. Fig. 2.
Figure 8: (Color online) Spectrum of a cylinder with r0=55 nm and L0=100 nm with: (a) ge=−14.9, α=0. (b) ge=−29.8, α=0. (c) ge=−29.8, α=20 meVnm. Due to Zeeman splitting, axial-sublevel minima produce sloped linear “traces” of parabola minima, marked with dots, resulting in large-scale DOS variations at a fixed energy. Increasing ge amplifies this effect and reveals crossings between traces. With Rashba SOI included the crossings become avoided. The values of μ used to calculate G(Φ) in Fig. 9 are marked with horizontal dashed lines.
Consider a cylinder with r0=55 nm and L0=100 nm. Figure 8 (a) shows the energy spectrum with ge=−14.9 and α=0. Each axial mode has an energy minimum, the spin-degeneracy of which is lifted by the Zeeman term for Φ≠0, producing sloped “traces” of the corresponding parabolic bottoms marked with filled circles, yielding a flux-modulated DOS at a fixed energy. The energies 15 and 21 meV (dashed) are located between two such traces approaching the former and distancing from the latter, resulting in a monotonically increasing and decreasing DOS, respectively. This is reflected in G(Φ), which Fig. 9 (a) shows evaluated at the corresponding values μ=15 and 21 meV, decreasing and increasing gradually on average, comparable to the experimental results of Ref. Gül et al., 2014 . It follows that the measured background oscillations could be explained as an interplay between the finite system length and spin. To further illustrate this effect, Fig. 8 (b) shows the cylinder spectrum with double Zeeman interaction, ge=−29.8, and α=0 still. The slopes of the parabola-minima traces increase revealing crossings and the DOS modulation is amplified. The DOS is maximum when two such bottom-band traces cross, i. e. for Φ/Φ0≈15 just below 21 meV, and minimum at the largest energy separation between them. Figure 9 (b) shows the corresponding conductance for μ=15 and 21 meV and reveals that the crossings manifest as peaks in background conductance oscillations.
as s1−s2=±2 because only traces of opposite spin may intersect. Here, (Φ/Φ0)c is the magnetic flux at which the lines intersect. It may be estimated from Fig. 9 as the flux at which the background conductance oscillations peak. Applying this to G at μ=21 meV in Fig. 9 (b), we find (Φ/Φ0)c≈15. If the chemical potential is known, the axial modes follow from the condition p21<2meL20μ/(ℏπ)2<p22, which must hold at Φ=0. For the present example, we find p2=4 and p1=3 which yields ge≈−30 compared to the input value ge=−29.8. If the chemical potential is not known, a guess of the relevant axial modes is needed.
Figure 9: (Color online) A cylinder with r0=55 nm and L0=100 nm. (a), (b) and (c): G(Φ) evaluated at μ=15 meV (solid) and μ=21 meV (dashed) with ge=−14.9, α=0 (a), ge=−29.8, α=0 (b) and ge=−29.8, α=20 meVnm (c). Due to Zeeman splitting, conductance oscillations are superimposed on background fluctuations, the form of which depends on μ as is reflected in the spectrum [compare with Figs. 8 (a), (b) and (c)]. (d) Flux-averaged conductance ⟨G(N)⟩ relative to ⟨G(N=1)⟩ at μ=21 meV plotted against flux number N with ge=−29.8 for different values of α. As α increases, the amplitude of the peak around Φ/Φ0=N=15 is reduced, reflected in the Rashba-induced avoided crossings of “traces” in Fig. 8 (c).
Finally, Fig. 8 (c) shows the spectrum with ge=−29.8 and α=20 meVnm. Interestingly, due to the SOI the crossings of the traces become avoided, their energy separation increasing with α. The resulting energy “gap” dampens the background-oscillation peaks of G(Φ), as is shown for μ=21 meV at Φ/Φ0≈15 in Fig. 9 (c). Rashba SOI also dampens the FP oscillations themselves as discussed in Sec. III.4. To understand how the amplitude of the background conductance oscillations varies with α, Fig. 9 (d) shows how the conductance ⟨G(N)⟩ averaged over the N-th flux N−1≤Φ/Φ0≤N with N∈Z+ varies with N relative to ⟨G(N=1)⟩ for different values of α. By averaging over the intervals between integer fluxes we exclude the Φ-periodic part of G and isolate the Zeeman-induced background oscillations. In analogy with Figs. 9 (b) and (c), ⟨G(N)⟩ peaks around N=15 and as α increases the peak is reduced in amplitude relative to ⟨G(1)⟩. It has been shown Engels et al. (1997); Nitta et al. (1997); Liang and Gao (2012) that α is controllable by applying a gate voltage and therefore measurements on peaks in the background oscillations of magnetoconductance in GaAs/InAs core-shell nanowires may allude to the existence of Rashba SOI in such tubular systems. Importantly, for α≠0 the background conductance oscillations flatten, but the peaks do not shift much compared to α=0, and so Eq. (35) may still be applied to estimate ge.
We performed transport calculations of electrons situated on a cylindrical surface in the presence of a longitudinal magnetic flux and obtained flux-periodic oscillations at different chemical potentials. Varying μ shifts the chemical potential of the system relative to the fixed spectra of the central part and the leads, similar to the experimental setup in Ref. Gül et al., 2014 , where both the nanowire and the contacts are placed on a substrate used as a back gate. An alternative model is to shift only the central system spectrum relative to the leads and some fixed chemical potential, but our calculations (not shown here) reveal that these two methods are essentially identical, with only a minor difference in level-broadening. The oscillations survive and remain periodic in the presence of impurities and occur even if the contacts do not have a uniform angular coverage of the cylindrical surface. Hence, they are robust to deviations from the ideal circular and parity symmetries in the nanowire. Furthermore, the oscillations remain flux-periodic when Rashba SOI is included. The oscillations are also still present when Zeeman interaction is included, although they cease to be flux-periodic. Instead, a rich structure of beating patterns and background oscillations is identified, the latter of which also relates to the finite system length. By analyzing these oscillations, it is possible to estimate the g-factor of the electrons in the shell and detect the presence of Rashba SOI, provided the SOI strength can be varied. Our results are in qualitative agreement with recent measurements on GaAs/InAs core-shell nanowires.
We would like to thank Thomas Schäpers, Fabian Haas, Sigurdur Ingi Erlingsson, Llorens Serra and Thorsten Arnold for enlightening discussions. This work was supported by the Research Fund of the University of Iceland and the Icelandic Research and Instruments Funds.
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