A general expression for the spin Hall conductivity (SHC) in the s-wave superconducting state at finite temperatures is derived. Based on the expression, we study the SHC in a two-dimensional electron gas model in the presence of Rashba spin-orbit interaction (SOI). The SHC is zero in the normal state, whereas it takes a large negative value as soon as the superconductivity occurs, due to the change in the quasiparticle contributions. Since this remarkable behavior is independent of the strength of the SOI, it will be widely observed in thin films of superconductors with surface-induced Rashba SOI, or in various non-centrosymmetric superconductors.
The Rashba superconductor, in which spin-splitting bands become superconducting, is fascinating as a novel superconducting system in low dimensional systems. Here, we present the results of $\textit{in-situ}$ transport measurements on a Rashba-type surface state of the striped incommensurate (SIC) phase of a Pb atomic layer on Ge(111) surface with additional Pb islands/clusters on it. We found that two-step superconducting transitions at around 7 K and 3 K occurred. The latter superconducting transition is suggested to be induced at the non-superconducting Rashba SIC area because of the latera…
We investigate the properties of a weak link between two Rashba-based superconducting nanowires with geometric misalignment. By applying an external magnetic field the system can be driven into a topological non-trivial regime. We demonstrate that the Josephson current can be modulated in amplitude and sign through the variation of the applied field and, remarkably, via the angle controlling the spin-orbit locking mismatch at the interface of the nanowires. The proposed setup with misaligned coplanar nanowires provides the building block configuration for the manipulation of coherent transport…
Spin Hall effect due to random Rashba spin-orbit coupling in the two-dimensional honeycomb lattice of carbon atoms (graphene) is considered theoretically. Using the Green function method and diagrammatic technique we show that fluctuations of the Rashba interaction around zero average value give rise to nonzero spin Hall conductivity. Generally, the conductivity is not universal, but depends on the ratio of the total momentum and spin-flip relaxation rates.
We study the effect of anisotropy of the Rashba coupling on the extrinsic spin Hall effect due to spin-orbit active adatoms on graphene. In addition to the intrinsic spin-orbit coupling, a generalized anisotropic Rashba coupling arising from the breakdown of both mirror and hexagonal symmetries of pristine graphene is considered. We find that Rashba anisotropy can strongly modify the dependence of the spin Hall angle on carrier concentration. Our model provides a simple and general description of the skew scattering mechanism due to the spin-orbit coupling that is induced by proximity to large…
For the spin Hall effect arising from strong band-structure spin-orbit coupling, a semiclassical Boltzmann theory reasonably addressing the intriguing disorder effect called side-jump is still absent. In this paper we describe such a theory of which the key ingredient is the spin-current-counterpart of the semiclassical side-jump velocity (introduced in the context of the anomalous Hall effect). Applying this theory to spin Hall effects in a two-dimensional electron gas with giant Rashba spin-orbit coupling, we find largely enhanced spin Hall angle in the presence of magnetic impurities when o…
The critical current of a superconductor can depend on the direction of current flow due to magnetochiral anisotropy when both inversion and time-reversal symmetry are broken, an effect known as the superconducting (SC) diode effect. Here, we consider one-dimensional (1D) systems in which superconductivity is induced via the proximity effect. In both topological insulator and Rashba nanowires, the SC diode effect due to a magnetic field applied along the spin-polarization axis and perpendicular to the nanowire provides a measure of inversion symmetry breaking in the presence of a superconducto…
In a two dimensional electron gas with Rashba spin-orbit coupling, the external electric field may cause a spin Hall current in the direction perpendicular to the electric field. This effect was called the intrinsic spin Hall effect. In this paper, we investigate the influences of spin accumulation on this intrinsic spin Hall effect. We show that due to the existence of boundaries in a real sample, the spin Hall current generated by the intrinsic spin Hall effect will cause spin accumulation near the edges of the sample, and in the presence of spin accumulation, the spin Hall conductivity will…
We present a theoretical analysis of two-dimensional Dirac-Rashba systems in the presence of disorder and external perturbations. We unveil a set of exact symmetry relations (Ward identities) that impose strong constraints on the spin dynamics of Dirac fermions subject to proximity-induced interactions. This allows us to demonstrate that an arbitrary dilute concentration of scalar impurities results in the total suppression of nonequilibrium spin Hall currents when only Rashba spin-orbit coupling is present. Remarkably, a finite spin Hall conductivity is restored when the minimal Dirac-Rashba …
We predict two spin-dependent transport phenomena in two-dimensional electron systems, which are induced by spatially fluctuating Rashba spin-orbit interaction. When the electron gas is magnetized, the random Rashba interaction leads to the anomalous Hall effect. An example of such a system is a narrow-gap magnetic semiconductor-based symmetric quantum well. We show that the anomalous Hall conductivity reveals a strongly nonlinear dependence on the magnetization, decreasing exponentially at large spin density. We also show that electron scattering from a fluctuating Rashba field in a two-dimen…
We analyze influence of a dimerization on the topological phases of the Rashba nanowire proximitized to a superconducting substrate. We find that periodic alternations of the hopping integral and spin-orbit coupling can lead to band inversion, inducing transition to topologically nontrivial superconducting phase that hosts the Majorana zero-energy modes. This "dimerization-induced topological superconductivity" completely repels the topological phase of the uniform nanowire, whenever they happen to overlap. We provide an analytical justification for this puzzling behavior based on the symmetry…
Two-dimensional electron systems with spin-orbit coupling in the proximity of a superconductor and a magnetic insulator have recently been considered as promising candidates to realize topological superconducting phases. Here we discuss electromagnetic response properties of these systems. Breaking of time-reversal symmetry leads to an anomalous Hall effect with a characteristic non-monotonic gate voltage dependence and a Hall conductivity that can change a sign as temperature is varied. The imaginary part of the Hall conductivity at finite frequency, which shows up for example in the Kerr rot…
Changes dopant ion concentrations in the sides of a symmetric quantum well are known to create a random Rashba-type spin-orbit coupling. Here we demonstrate that, as a consequence, a finite size spin-Hall effect is also present. Our numerical algorithm estimates the result of the Kubo formula for the spin-Hall conductivity, by using a tight-binding approximation of the Hamiltonian in the framework of a time-dependent Green's function formalism, well suited for very large systems.
We report an intrinsic form of the inverse spin Hall effect (ISHE) in ferromagnetic (FM) metal with Rashba spin orbit coupling (RSOC), which is driven by a normal charge current. Unlike the conventional form, the ISHE can be induced without the need for spin current injection from an external source. Our theoretical results show that Hall voltage is generated when the FM moment is perpendicular to the ferromagnetic layer. The polarity of the Hall voltage is reversed upon switching the FM moment to the opposite direction, thus promising a useful readback mechanism for memory or logic applicatio…
We study theoretically the spin and orbital angular momentum (OAM) Hall effect in a high mobility two-dimensional electron system with Rashba and Dresselhuas spin-orbit coupling by introducing both the spin and OAM torque corrections, respectively, to the spin and OAM currents. We find that when both bands are occupied, the spin Hall conductivity is still a constant (i.e., independent of the carrier density) which, however, has an opposite sign to the previous value. The spin Hall conductivity in general would not be cancelled by the OAM Hall conductivity. The OAM Hall conductivity is also ind…
For the two-dimensional ideal electron gas with the Rashba spin-orbit interaction and in the presence of non-magnetic short-ranged potential impurities the spin-Hall conductivity $σ_{sH}$ is found by direct microscopic calculation. Within the semiclassical approximation $\hbar/τ\ll ε_F$ and $Δ\ll ε_F$ the value of $σ_{sH}$ is zero for arbitrary ratio of the spin-orbit splitting $Δ$ and the inverse elastic scattering time $1/τ$.
Quasi--one dimensional edge channels are formed at the boundary of a two-dimensional electron system subject to a strong perpendicular magnetic field. We consider the effect of Rashba spin--orbit coupling, induced by structural inversion asymmetry, on their electronic and transport properties. Both our analytical and numerical results show that spin--split quantum--Hall edge channels exhibit properties analogous to that of Rashba--split quantum wires. Suppressed backscattering and a long spin life time render these edge channels an ideal system for observing voltage--controlled spin precession…
We study the effect of disorder on the intrinsic anomalous Hall (AH) conductivity in a spin-polarized two-dimensional electron gas with a Rashba-type spin-orbit interaction. We find that AH conductivity vanishes unless the lifetime is spin-dependent, similar to the spin Hall (SH) conductivity in the non-magnetic system. In addition, we find that the SH conductivity does not vanish in the presence of magnetic scatterers. We show that the SH conductivity can be controlled by changing the amount of the magnetic impurities.
We calculate the intrinsic spin Hall conductivity σ^{\mathrm{sH}} of a two-dimensional electron system within a generalized Rashba model, showing that it is, in general, finite and model-dependent. Considering arbitrary band dispersion, we find that σ^{\mathrm{sH}} in the presence of the linear-in-momentum spin-orbit coupling of the Rashba form does not vanish in the presence of impurities except for the precisely parabolic spectrum. We show, using the linear response Kubo formalism, how the exact cancellation happens for the quadratic dispersion, and why it does not occur in general. We deriv…
Recent experiments in semiconductor nanowires with a spin-orbit coupling and proximity-induced superconductivity exhibit signatures of Majorana bound states predicted to exist in the topological phase. In this work we predict that these nanowire systems exhibit unconventional magnetoelectric effects showing a sharp crossover behavior at the topological phase transition. We find that magnetic fields with a component parallel to the spin-orbit field can give rise to currents in equilibrium. Surprisingly, also fields perpendicular to the spin-orbit field may induce currents and can be employed in…
The spin Hall effect in a finite ballistic two-dimensional system with Rashba and Dresselhaus spin-orbit interaction is studied numerically. We find that the spin Hall conductance is very sensitive to the transverse measuring location, the shape and size of the device, and the strength of the spin-orbit interaction. Not only the amplitude of spin Hall conductance but also its sign can change. This non-universal behavior of the spin Hall effect is essentially different from that of the charge Hall effect, in which the Hall voltage is almost invariant with the transverse measuring site and is a …
Using the four-terminal Landauer-Büttiker formula and Green's function approach, we calculate numerically the spin-Hall conductance in a two-dimensional junction system with the Rashba spin-orbit (SO) coupling and disorder. We find that the spin-Hall conductance can be much greater or smaller than the universal value $e/8π$, depending on the magnitude of the SO coupling, the electron Fermi energy and the disorder strength. The spin-Hall conductance does not vanish with increasing sample size for a wide range of disorder strength. Our numerical calculation reveals that a nonzero SO coupling can…
We study transport properties of a two-dimensional electron system with Rashba spin-orbit coupling in a perpendicular magnetic field. The spin orbit coupling competes with Zeeman splitting to introduce additional degeneracies between different Landau levels at certain magnetic fields. This degeneracy, if occuring at the Fermi level, gives rise to a resonant spin Hall conductance, whose height is divergent as 1/T and whose weight is divergent as $-\ln T$ at low temperatures. The Hall conductance is unaffected by the Rashba coupling.
We consider spin Hall effect in a system of massless Dirac fermions in a graphene lattice. Two types of spin-orbit interaction, pertinent to the graphene lattice, are taken into account - the intrinsic and Rashba terms. Assuming perfect crystal lattice, we calculate the topological contribution to spin Hall conductivity. When both interactions are present, their interplay is shown to lead to some peculiarities in the dependence of spin Hall conductivity on the Fermi level.
We calculate the longitudinal and spin-Hall conductances in four-lead bridges with Rashba - Dresselhaus spin-orbit interactions. Numerical results are obtained both within Landauer-Buttiker formalism and by the direct evaluation of the Kubo formula. The microscopic Hamiltonian is obtained in the tight-binding approximation in terms of the neareast-neighbor hopping integral $t$, the Rashba spin-orbit coupling $V_R$, the Dresselhaus spin-orbit coupling $V_D$ and an Anderson-like, on-site disorder energy strength $W$. We reconfirm that below a critical disorder threshold, the spin-Hall effect is …
In this work we show that Rashba-based topological superconductor nanowires, where the spin-orbit coupling may change its sign, support three topological phases protected by chiral symmetry. When a superconducting phase gradient is applied over the interface of the two nontrivial phases, the Andreev spectrum is qualitatively phase shifted by πcompared to usual Majorana weak links. The topological π-junction has the striking property of exhibiting maximum supercurrent in the vicinity of vanishing phase difference. Qualitative features of the junction are robust against disorder and magnetic fie…
Anomalous and spin Hall effects are investigated theoretically for a magnetic tunnel junction where the applied voltage produces a Rashba spin-or bit coupling within the tunneling barrier layer. The ferromagnetic electrodes are the source of the spin-polarized current. The tunneling electrons experience a spin-orbit coupling inside the barrier due to the applied electrical field. Charge and spin Hall currents are calculated as functions of the position inside the barrier and the angle between the magnetizations of the electrodes. We find that both charge and spin Hall currents are located insi…
We propose an all-electrical nanoscopic structure where {\em pure} spin current is induced in the transverse voltage probes attached to {\em quantum-coherent} one-dimensional ring when conventional unpolarized charge current is injected through its longitudinal leads. Tuning of the Rashba spin-orbit coupling in semiconductor heterostructure hosting the ring generates quasi-periodic oscillations of the predicted spin Hall current due to {\em spin-sensitive quantum-interference effects} caused by the difference in Aharonov-Casher phase acquired by opposite spins states traveling clockwise and co…
We consider normal-superconductor (NS) and superconductor-normal-superconductor (SNS) junctions based on one-dimensional nanowires with Rashba spin-orbit coupling and proximity-induced $s$-wave spin-singlet superconductivity and analytically demonstrate how both even- and odd-frequency and spin-singlet and -triplet superconducting pair correlations are always present. In particular, by using a fully quantum mechanical scattering approach, we show that Andreev reflection induces mixing of spatial parities at interfaces, thus being the unique process which generates odd-frequency pairing; on the…
We study a one-dimensional helical system with random Rashba spin-orbit coupling. Using renor- malization group methods, we derive a consistent set of flow equations governing the important con- trol parameters of the backscattering process. Thereby, we prove the existence of disorder-induced two-particle backscattering that can even be non-local in space. This analysis allows us to derive the scaling form of the conductance at low temperatures. We find that two-particle backscattering due to random spin-orbit coupling differs from the one off a single Rashba impurity by both the scaling of th…
We report on a comparative numerical study of the spin Hall conductivity in two-dimensions for three different spin-orbit interaction models; the standard k-linear Rashba model, the k-cubic Rashba model that describes two-dimensional hole systems, and a modified k-linear Rashba model in which the spin-orbit coupling strength is energy dependent. Numerical finite-size Kubo formula results indicate that the spin Hall conductivity of the k-linear Rashba model vanishes for frequency $ω$ much smaller than the scattering rate $τ^{-1}$, with order one relative fluctuations surviving out to large syst…
We study the band structure modulation and spin-Hall effect of a two-dimensional heavy-hole system with $k^3$-Rashba spin-orbit coupling (RSOC), irradiated by linearly polarized light. We find that the band structure becomes anisotropic under the illumination by the light. Most remarkably, a pair of additional spin-degeneracy points (apart from $Γ$ point) emerge in the energy dispersion, the locations of which are solely determined by the strength of the amplitude of the incident light. If this degeneracy occurs around the Fermi level, the spin-Hall conductivity (SHC) exhibits a resonance. Awa…
We consider the spin-Hall current in a disordered two-dimensional electron gas in the presence of Rashba spin-orbit interaction. We derive a generalized Kubo-Greenwood formula for the spin-Hall conductivity $σ$ and evaluate it in an systematic way using standard diagrammatic techniques for disordered systems. We find that in the diffusive regime both Boltzmann and the weak localization contributions to $σ$ are of the same order and vanish in the zero frequency limit. We show that the uniform spin current is given by the total time derivative of the magnetization from which we can conclude that…
A nonequilibrium Green's functions approach to spin-Hall effect is developed in a diffusive two-dimensional electron system with Rashba spin-orbit interaction. In the presence of long-range impurities, the coupled quantum kinetic equations are solved analytically in the self-consistent Born approximation. It is shown that the intrinsic spin-Hall effect stems from the dc-field-induced perturbation of the density of states. In addition, there is an additional disorder-mediated process, which involves the transition of nonequilibrium electrons between two spin-orbit-coupled bands. It results in a…
Current-driven magnetization dynamics of single ferromagnets in heavy metal/ferromagnet bilayers has been recently realized. In this work, spin torque induced by so-called Rashba spin-orbit coupling and spin Hall effect are considered within a diffusive model. The dependence of the resulting torque as a function of the thicknesses of the ferromagnet and heavy metal is analyzed. We show that (i) both torques are on the form ${\bf T}=T_{||}{\bf m}\times{\bf y}+T_\bot{\bf m}\times({\bf y}\times{\bf m})$, (ii) the ratio $T_{||}/T_\bot$ strongly depends on the thickness of the layers and (iii) the …
In a two-dimensional electron gas with Rashba and Dresselhaus spin-orbit couplings, there are two spin-split energy surfaces connected with a degenerate point. Both the energy surfaces and the topology of the Fermi surfaces can be varied by an in-plane magnetic field. We find that, if the chemical potential falls between the bottom of the upper band and the degenerate point, then simply by changing the direction of the magnetic field, the magnitude of the spin Hall conductivity can be varied by about 100 percent. Once the chemical potential is above the degenerate point, the spin Hall conducti…
The anomalous Hall and Nernst effects are considered theoretically within Matsubara-Green's function formalism. The effective Hamiltonian of a magnetized two-dimensional electron gas with cubic Rashba spin-orbit interaction may describe transport properties of electronic states at the interfaces or surfaces of perovskite oxides or another type of heterostructures that, due to symmetry, may be described by the same effective model. In the quasi-ballistic limit, both effects are determined by the topological (Fermi sea) contribution whereas the states at the Fermi level gives a negligibly small …
The spin-Hall effect in the two-dimensional electron gas (2DEG) generates symmetric out-of-plane spin Sz accumulation about the current axis in the absence of external magnetic field. Here we employ the real space Landauer-Keldysh formalism [B. K. Nikolic et al., Phys. Rev. Lett. 95, 046601 (2005); Phys. Rev. B 73, 075303 (2006)] by considering a four-terminal setup to investigate the circumstances in which this symmetry is broken. For the absence of Dresselhaus interaction, starting from the applied out-of-plane B corresponding to Zeeman splitting energy 0 - 0.5 times the Rashba hopping energ…
We propose an experimental realization of the Spin Hall effect in graphene by illuminating a graphene sheet on top of a substrate with circularly polarized monochromatic light. The substrate induces a controllable Rashba type spin-orbit coupling which breaks the spin-degeneracy of the Dirac cones but it is gapless. The circularly polarized light induces a gap in the spectrum and turns graphene into a Floquet topological insulator with spin dependent edge states. By analyzing the high and intermediate frequency regimes, we find that in both parameter limits, the spin-Chern number can be tuned b…
Using the Matsubara Green function formalism we calculate the temperature dependence of spin Hall and spin Nernst conductivities of a two-dimensional electron gas with Rashba spin-orbit interaction in the linear response regime. In the case of spin Nernst effect we also include the contribution from spin-resolved orbital magnetization, which assures correct behavior of the spin Nernst conductivity in the zero-temperature limit. Analytical formulas for the spin Hall and spin Nernst conductivities are derived in some specific situations. Using the Ioffe-Regel localization criterion, we have also…
We analyze the dynamics of the helical edge modes of a quantum spin Hall state in the presence of a spatially non-uniform Rashba spin-orbit (SO) interaction. A randomly fluctuating Rashba SO coupling is found to open a scattering channel which causes localization of the edge modes for a weakly screened electron-electron (e-e) interaction. A periodic modulation of the SO coupling, with a wave number commensurate with the Fermi momentum, makes the edge insulating already at intermediate strengths of the e-e interaction. We discuss implications for experiments on edge state transport in a HgTe qu…
Skew scattering on rare impurity configurations is shown to dominate the anomalous Hall effect in a 2D Rashba ferromagnet. The mechanism originates in scattering on rare impurity pairs separated by distances of the order of the Fermi wave length. Corresponding theoretical description goes beyond the conventional non-crossing approximation. The mechanism provides the only contribution to the anomalous Hall conductivity in the most relevant metallic regime and strongly modifies previously obtained results for lower energies in the leading order with respect to impurity strength.
We investigate the spin-current linear response conductivity tensor to an electric field in a paramagnetic two dimensonal electron gas with both Rashba and Dresselhaus spin-orbit coupling in the weak scattering regime. In the ususal case where both spin-orbit-split bands are occupied, we find that the spin-Hall conductivity depends only on the sign of the difference in magnitude of the Rashba and Dresselhaus coupling strengths except within a narrow window where both coupling strengths are equal. We also find a new effect in which a spin current is generated in the direction of the driving fie…
We present a quantum Boltzmann equation analysis of the spin-Hall effect in a diffusive Rashba two-dimensional electron system. Within the framework of the self-consistent Born approximation, we consider the roles of disorder-induced quasiclassical relaxation, collisional broadening of the quasiparticles, and the intracollisional field effect in regard to spin-Hall dynamics. We present an analytical proof that the spin-Hall current vanishes, independently of the coupling strength, of the quasiparticle broadening, of temperature and of the specific form of the isotropic scattering potential. A …
Pure spin currents in semiconductors are essential for implementation in the next generation of spintronic elements. Heterostructures of III- nitride semiconductors are currently employed as central building-blocks for lighting and high-power devices. Moreover, the long relaxation times and the spin-orbit coupling (SOC) in these materials indicate them as privileged hosts for spin currents and related phenomena. Spin pumping is an efficient mechanism for the inception of spin current and its conversion into charge current in non-magnetic metals and semiconductors with Rashba SOC $via$ spin Hal…
We calculate the spin-Hall conductivity of a disordered two-dimensional Rashba electron gas within the self-consistent Born approximation and for arbitrary values of the electron density, parametrized by the ratio $E_F /E_0$, where $E_F$ is the Fermi level and $E_0$ is the spin-orbit energy. We confirm earlier results indicating that in the limit $E_F/E_0 \gg 1$ the vertex corrections suppress the spin-Hall conductivity. However, for sufficiently low electron density such that $E_F\lesssim E_0$, we find that the vertex corrections no longer cancel contributions from the Fermi surface, and they…
We discuss the transport properties of a quantum spin-Hall insulator with sizable Rashba spin-orbit coupling in a disk geometry. The presence of topologically protected helical edge states allows for the control and manipulation of spin polarized currents: when ferromagnetic leads are coupled to the quantum spin-Hall device, the ballistic conductance is modulated by the Rashba strength. Therefore, by tuning the Rashba interaction via an all-electric gating, it is possible to control the spin polarization of injected electrons.
An intriguing magneto-transport property is demonstrated by tight-binding lattice electrons with Rashba spin-orbit coupling (SOC) in a magnetic field. With the flux strength $ϕ={2π/N}$ ($N$ is an integer) and the Zeeman splitting fixed, when increasing the Rashba SOC $λ$, the spin-Hall and charge-Hall conductances (SHC and CHC) undergo four-step evolutions: the SHC shows size-dependent resonances and jumps at three critical $λ_{c}$'s, and changes its sign at $λ_{c1}$ and $λ_{c3}$; while the CHC exhibits three quantum jumps by $-Ne^2/h$, $+2Ne^2/h$ and $-Ne^2/h$. Such four-step evolutions are a…
We study the Magnus transport in a gapped 2D electron gas with Rashba spin-orbit coupling using semiclassical Boltzmann transport formalism. Apart from its signature in the charge transport coefficients, the inclusion of Magnus velocity in the spin current operator enables us to study Magnus spin transport in the system. In particular, we study the roles of mass gap and Fermi surface topology on the behavior of Magnus Hall and Nernst conductivities and their spin counterparts. We find that the Magnus spin Hall conductivity vanishes in the limit of zero gap, unlike the universal spin Hall condu…
We have investigated the complex two-dimensional electron system with unequal Rashba and Dresselhaus spin-orbit interactions in the presence of a perpendicular magnetic field. The spin polarizations are obtained in a wide range of magnetic fields. It is shown that such a system is hard to be magnetized. We also find that the resonant charge and spin Hall conductances occurs simultaneously at a certain magnetic field, at which two (nearly) degenerate Landau levels are filled partly. The resonant Hall effects are universal in this type of semiconductor materials, and could have potential applica…
