We present the analytical description of the anomalous Hall effect (AHE) in a 2DEG ferromagnet within the Keldysh formalism. These results unify all three linear response approaches to anomalous Hall transport and close a long standing debate. We are able to identify a new extrinsic AHE regime dominated by a hybrid skew scattering mechanism. This new contribution is inversely proportional to the impurity concentration, resembling the normal skew scattering, {\em but} independent of the impurity-strength, resembling the side-jump mechanism. Within the Kubo formalism this regime is captured by higher order diagrams which, although weak, dominate when both subbands are occupied; this regime can be detected by variable remote doping experiments that we describe.
We show that the density matrix for a finite conductor attached to reservoirs obtained by Keldysh formalism is of MacLennan-Zubarev form. On the basis of the fact that the density matrix is the invariant part proposed by Zubarev, it is shown that Keldysh formalism may describe the irreversible processes and steady-state feature of the system. An important consequence of the MacLennan-Zubarev form of the density matrix is a generalization of the Kubo formula in a nonequilibrium case. On the basis of the result, we propose the formula of shot noise and a nonequilibrium identity between different…
Reconciling Kubo and Keldysh Approaches to Fermi-Sea-Dependent Nonequilibrium Observables: Application to Spin Hall Current and Spin-Orbit Torque in Spintronics
We revisit a theory of heat transport in the light of a gauge theory of gravity and find the proper heat current with a corresponding gauge field, which yields the natural definitions of the heat magnetization and the Kubo-formula contribution to the thermal conductivity as torsional responses. We also develop a general framework for calculating gravitational responses by combining the Keldysh and Cartan formalisms. By using this framework, we explicitly calculate these two quantities and reproduce the Wiedemann-Franz law for the thermal Hall conductivity in the clean and non-interacting case.…
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.
Rashba spin-orbit coupling appears in 2D systems lacking inversion symmetry, and causes the spin-splitting of otherwise degenerate energy bands into an upper and lower helicity band. In this paper, we explore how impurity scattering affects transport in the ultra-low-density regime where electrons are confined to the lower helicity band. A previous study has investigated the conductivity in this regime using a treatment in the first Born approximation. In this work, we use the full T-matrix to uncover new features of the conductivity. We first compute the conductivity within a semiclassical Bo…
Using the Kubo formalism, the magnetic properties of the system in the linear regime have been investigated. Mainly the effect of non-magnetic substrate on the spin susceptibility is calculated. Results show that the Rashba coupling interaction has a central role in the magnetic response function of the system and it is really remarkable since this type of spin orbit coupling can be effectively controlled by an external gate voltage. Most importantly it was shown that, in the presence of the Rashba interaction a magnetic phase transition could be observed. This magnetic phase corresponds to a …
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…
Spin electromagnetic fields driven by the Rashba spin-orbit interaction, or Rashba-induced spin Berry's phase, in ferromagnetic metals is theoretically studied based on the Keldysh Green's function method. Considering a limit of strong sd coupling without spin relaxation (adiabatic limit), the spin electric and magnetic fields are determined by calculating transport properties. The spin electromagnetic fields turn out to be expressed in terms of a Rashba-induced effective vector potential, and thus they satisfy the Maxwell's equation. In contrast to the conventional spin Berry's phase, the Ras…
Within the Kubo formalism, we estimate the spin-Hall conductivity in a two-dimensional electron gas with Rashba spin-orbit interaction and study its variation as a function of disorder strength and system size. The numerical algorithm employed in the calculation is based on the direct numerical integration of the time-dependent Schrodinger equation in a spin-dependent variant of the particle source method. We find that the spin-precession length, L_s controlled by the strength of the Rashba coupling, establishes the critical lengthscale that marks the significant reduction of the spin-Hall con…
In model studies of the spin/anomalous Hall effect, effective Hamiltonians often serve as the starting point. However, a complete effective quantum theory contains not only the effective Hamiltonian but also the relation linking the physical observables to the canonical ones. We construct the semiclassical Boltzmann (SB) transport framework in the weak disorder-potential regime directly in the level of the effective quantum theory, and confirm this construction by formulating a generalized Kohn-Luttinger density matrix transport theory also in this level. The link and difference between the pr…
We investigate the effect of Rashba spin-orbit coupling (SOC) on the optoelectronic properties of n- and p-type monolayer MoS2. The optical conductivity is calculated within the Kubo formalism. We find that the spin-flip transitions enabled by the Rashba SOC result in a wide absorption window in the optical spectrum. Furthermore, we evaluate the effects of the polarization direction of the radiation, temperature, carrier density, and the strength of the Rashba spin-orbit parameter on the optical conductivity. We find that the position, width, and shape of the absorption peak or absorption wind…
We study the spatial distribution of spin torque induced by a strong Rashba spin-orbit coupling (RSOC) in a ferromagnetic (FM) metal layer, using the Keldysh non-equilibrium Green's function method. In the presence of the s-d interaction between the non-equilibrium conduction electrons and the local magnetic moments, the RSOC effect induces a torque on the moments, which we term as the Rashba spin torque. A correlation between the Rashba spin torque and the spatial spin current is presented in this work, clearly mapping the spatial distribution of Rashba Spin torque in a nano-sized ferromagnet…
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 …
Spin Hall effect (SHE) has been discussed in the context of Kubo formulation, geometric physics, spin orbit force, and numerous semi-classical treatments. It can be confusing if the different pictures have partial or overlapping claims of contribution to the SHE. In this article, we present a gauge-theoretic, time-momentum elucidation, which provides a general SHE equation of motion, that unifies under one theoretical framework, all contributions of SHE conductivity due to the kinetic, the spin orbit force (Yang-Mills), and the geometric (Murakami-Fujita) effects. Our work puts right an ambigu…
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…
We study spin torques induced by Rashba spin-orbit coupling in two-dimensional ferromagnets under the good-metal condition $ε_{F}τ/\hbar\gg1$ ($ε_{F}$ the Fermi energy, $τ$ the electron lifetime) by employing the Kubo formula. We find that, in the presence of spin-dependent disorder the Rashba torque changes greatly as the system evolves out of the weak disorder limit where $\hbar/τ$ is much smaller than any intrinsic energy scale characterizing the multiband structure. The antidamping-like component of Rashba torque can be comparable to and larger than the field-like one out of the weak disor…
The traditional Kubo formula is generalized to describe the linear response with respect to non-Abelian fields. To fulfil the demand for studying spin transport, the SU(2) Kubo formulae are derived by two conventional approaches with different gauge fixings. Those two approaches are shown to be equivalent where the non-conservation of the SU(2) current plays an essential role in guaranteeing the consistency. Some concrete examples relating Spin Hall Effect are considered. The dc spin conductivity vanishes in the system with parabolic unperturbed dispersion relation. By applying a time-dependen…
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 calculate the optical conductivity of an anisotropic two-dimensional system with Rashba spin-flip excitation within the Kubo formalism. We show that the anisotropic Rashba effect caused by an external field changes significantly the magnitude of the spin splitting. Furthermore, we obtain an analytical expression for the longitudinal optical conductivity associated with inter-band transitions as a function of the frequency for an arbitrary polarization angle. We find that the diagonal components of the optical conductivity tensor are direction-dependent and the spectrum of optical absorption…
Recently we have developed a theory of Keldysh formalism for mesoscopic systems. The resulting nonequilibrium Kubo formula for differential conductance makes it possible to propose the new formula of shot noise $S_h$, and thus to study shot noise in correlated systems at any temperature and any bias voltage. Employing this new approach, we analyze shot noise in the Kondo regime through a quantum dot for the symmetric case at zero temperature. Using the renormalized perturbation theory, we prove that in the leading order of bias voltage $S_h$ equal to noise power at zero temperature conventiona…
The Kubo formula is used to calculate the spin-Hall conductivity in a spin-polarized two-dimensional electron system with Rashba-type spin-orbit interaction. As in the case of the unpolarized electron system, spin Hall conductivity is entirely determined by states at the Fermi level, a property that persists in the presence of magnetic impurities. In the clean limit, the spin-Hall conductivity decreases monotonically with the Zeeman splitting, a result of the ordering effect on the electron spins produced by the magnetic field. In the presence of magnetic impurities, the spin-dependent scatter…
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…
Edelstein and spin Hall effect response functions for a two-dimensional (2D) system of pseudospin-1 particles is investigated. These two response functions denoted by $ σ_{SH} $ and $ σ_{EE} $ have been analyzed in a pseudospin-1 particle system in the presence of the Rashba-type spin-orbit interaction using the Kubo-Streda technique and vertex corrections with non-magnetic impurities. Then, for a given range of the Rashba coupling, response functions of the spin Hall effect (SHE) and Edelstein effect (EE) have been estimated at various energy gaps and Fermi energies. Results indicate that in …
We investigate the influence of magnons on the temperature-dependence and the anisotropy of the spin-orbit torque (SOT). For this purpose we use 3rd order perturbation theory in the framework of the Keldysh formalism in order to derive suitable equations to compute the magnonic SOT. We find several contributions to the magnonic SOT, which depend differently on the spin-wave stiffness $\mathcal{A}$ and on the temperature $T$, with the dominating contribution scaling like $T^{2}/\mathcal{A}^{2}$. Based on this formalism we compute the magnonic SOT in the ferromagnetic Rashba model. For large Ras…
Second-order dc conductivity in the velocity-gauge Keldysh formalism: gauge-invariant decomposition into nonlinear Drude, Berry-curvature-dipole, and quantum-metric responses
Stacking switching between correlation-protected radial Rashba field and persistent spin textures in graphene encapsulated by 1T-TaS$_2$ monolayers
Institute of Physics, Pavol Jozef Šafárik University in Košice（帕沃尔·约瑟夫·沙法里克大学科希策物理研究所）；Institute of Informatics, Slovak Academy of Sciences（斯洛伐克科学院信息学研究所）；Faculty of Physics, University of Belgrade（贝尔格莱德大学物理学院）；Institute of Experimental Physics, Slovak Academy of Sciences（斯洛伐克科学院实验物理研究所）；New Technologies Research Centre, University of West Bohemia（西波希米亚大学新技术研究中心）
We study theoretically the quantum transport properties of the Dirac fermions with spin-orbit interactions (SOIs) in graphene by using the method of Schwinger proper time together with decomposition over Landau level poles and Kubo formula. The analytical expressions for both longitudinal and Hall conductivities are derived explicitly. It is found that, from some numerical examples, when the Rashba SOI is taken into account the Shubnikov-de Haas (SdH) oscillation peaks of the longitudinal conductivity versus the chemical potential are split, while the SdH oscillation of the longitudinal conduc…
We investigate the electric and thermal transport properties in a disordered Weyl ferromagnet on an equal footing by using the Keldysh formalism in curved spacetime. In particular, we calculate the anomalous thermal Hall conductivity, which consists of the Kubo formula and the heat magnetization, without relying on the Wiedemann-Franz law. We take nonmagnetic impurities into account within the self-consistent $T$-matrix approximation and reproduce the Wiedemann-Franz law for the extrinsic Fermi-surface and intrinsic Fermi-sea terms, respectively. This is the first step towards a unified theory…
The spin Hall and longitudinal conductivity of a 2D heavy-hole gas with {\it k}-cubic Rashba and Dresselhaus spin-orbit interaction is studied in the ac frequency domain. Using Kubo linear-response theory and a recently proposed definition for the (conserved) spin current operator suitable for spin-3/2 holes, it is shown that the spin conductivity tensor exhibit very distinguishable features from those obtained with the standard definition of the spin current. This is due to a significant contribution of the spin-torque term arisen from the alternative definition of spin current which strongly…
In this paper we study the current-induced spin polarization in a two-dimensional electron gas, known also as the Edelstein effect. Compared to previous treatments, we consider both the Rashba and Dresselhaus spin-orbit interaction as well as the spin-orbit interaction from impurity scattering. In evaluating the Kubo formula for the spin polarization response to an applied electric field, we explicitly take into account the side-jump and skew-scattering effects. We show that the inclusion of side-jump and skew-scattering modifies the expression of the current-induced spin polarization.
We propose a mechanism of spin Hall effect in two-dimensional electron gas with spatially random Rashba spin-orbit interaction. The calculations based on the Kubo formalism and kinetic equation show that in contrast to the constant spin-orbit coupling, spin Hall conductivity in the random spin-orbit field is not totally suppressed by the potential impurity scattering. Even if the regular contribution is removed by the vertex corrections, the terms we consider, remain. Therefore, the intrinsic spin-Hall effect exists being, however, non-universal.
We study theoretically the spin-Hall effect as well as its reciprocal phenomenon (a transverse charge current driven by a spin-dependent chemical potential gradient) in electron and hole finite size mesoscopic systems. The Landauer-Buttiker-Keldysh formalism is used to model samples with mobilities and Rashba coupling strengths which are experimentally accessible and to demonstrate the appearance of measurable charge currents induced by the spin-dependent chemical potential gradient in the reciprocal spin-Hall effect. We also demonstrate that within the mesoscopic coherent transport regime the…
The intrinsic spin-Hall effect in hexagon-shaped samples is investigated. To take into account the spin-orbit couplings and to fit the hexagon edges, we derive the triangular version of the tight-binding model for the linear Rashba [Sov. Phys. Solid State 2, 1109 (1960)] and Dresselhaus [Phys. Rev. 100, 580 (1955)] [001] Hamiltonians, which allow direct application of the Landauer-Keldysh non-equilibrium Green function formalism to calculating the local spin density within the hexagonal sample. Focusing on the out-of-plane component of spin, we obtain the geometry-dependent spin-Hall accumulat…
Spin-charge interconversion is currently the focus of intensive experimental and theoretical research both for its intrinsic interest and for its potential exploitation in the realization of new spintronic functionalities. Spin-orbit coupling is one of the key microscopic mechanisms to couple charge currents and spin polarizations. The Rashba spin-orbit coupling in a two-dimensional electron gas has been shown to give rise to the inverse spin galvanic effect, i.e. the generation of a non-equilibrium spin polarization by a charge current. Whereas the Rashba model may be applied to the interpret…
Recent experiments demonstrate that antiferromagnets exhibit the spin Hall effect. We study a tight-binding model of an antiferromagnet on a square lattice with Rashba spin-orbit coupling and disorder. By exact diagonalization of a finite system connected to reservoirs within the Landauer-B$\ddot{\text{u}}$ttiker formalism, we compute the transverse spin Hall current in response to a longitudinal voltage difference. Surprisingly, the spin Hall conductance can be considerably larger in antiferromagnets than in normal metals. We compare our results to the Berry-phase-induced spin Hall effect gov…
We calculate the spin Hall conductivity driven by Rashba spin-orbit interaction in $p$-type two-dimensional semiconductors in the presence of a perpendicular magnetic field. For a highly confined quantum well, the system is described by a $k$-cubic Rashba term for two-dimensional heavy holes. The eigenstates of the system can be described by Landau spinor states. First we consider the conventional spin Hall conductivity. The contribution of the interband transitions to the Kubo-Greenwood formula gives the density dependent intrinsic spin Hall conductivity, which approaches its universal value …
It is well known that the Dresselhaus spin-orbit coupling (SOC) in semiconductor two dimensional electron gases (2DEGs) possesses both linear and cubic in momentum contributions. Nevertheless, the latter is usually neglected in most theoretical studies. However, recent Kerr rotation experiments have revealed a significant enhancement of the cubic Dresselhaus interaction by increasing the drift velocities in 2DEGs hosted in GaAs quantum wells. Here, we present a study of the optical spin Hall conductivity in 2DEGs under the simultaneous presence of Rashba and (linear plus cubic) Dresselhaus SOC…
In these lecture notes we study the disordered two-dimensional electron gas in the presence of Rashba spin-orbit coupling, by using the Keldysh non-equilibrium Green function technique. We describe the effects of the spin-orbit coupling in terms of a SU(2) gauge field and derive a generalized Boltzmann equation for the charge and spin distribution functions. We then apply the formalism to discuss the spin Hall and the inverse spin galvanic (Edelstein) effects. Successively we show how to include, within the generalized Boltzmann equation, the side jump, the skew scattering and the spin current…
We present a study of transport in multiple-band non-interacting Fermi metallic systems based on the Keldysh formalism, taking into account the effects of Berry curvature due to spin-orbit coupling. We apply this formalism to a Rashba 2DEG ferromagnet and calculate the anomalous Hall effect (AHE) and anisotropic magnetoresistance (AMR). The numerical calculations reproduce analytical results in the metallic regime revealing the crossover between the skew scattering mechanism dominating in the clean systems and intrinsic mechanism dominating in the moderately dirty systems. As we increase the d…
We consider a two-dimensional electron system subjected to a short-ranged nonmagnetic disorder potential, Coulomb interactions, and Rashba spin-orbit coupling. The path-integral approach incorporated within the Keldysh formalism is used to derive the kinetic equation for the semiclassical Green's function and applied to compute the spin current within the linear response theory. We discuss the frequency dependence of the spin Hall conductivity and further elucidate the role of electron interactions at finite temperatures for both the ballistic and diffusive regimes of transport. We argue that …
In this work, we study the spin Hall effect and Rashba-Edelstein effect of a 2D Weyl fermion system in the clean limit using the Kubo formalism. Spin transport is solely due to the spin-torque current in this strongly spin-orbit coupled (SOC) system, and chiral spin-flip scattering off non-SOC scalar impurities, with potential strength $V$ and size $a$, gives rise to a skew-scattering mechanism for the spin Hall effect. The key result is that the resultant spin-Hall angle has a fixed sign, with $θ^{SH} \sim O \left(\tfrac{V^2}{v_F^2/a^2} (k_F a)^4 \right)$ being a strongly-dependent function o…
Local spin and charge densities on a two-dimensional honeycomb lattice are calculated by the Landauer-Keldysh formalism (LKF). Through the empirical tight-binding method, we show how the realistic band structure can be brought into the LKF. Taking the Bi(111) surface, on which strong surface states and Rashba spin-orbit coupling are present [Phys. Rev. Lett. 93, 046403 (2004)], as a numeric example, we show typical intrinsic spin-Hall accumulation (ISHA) patterns thereon. The Fermi-energy-dependence of the spin and charge transport in two-terminal nanostructure samples is subsequently analyzed…
We present a review of experimental and theoretical studies of the anomalous Hall effect (AHE), focusing on recent developments that have provided a more complete framework for understanding this subtle phenomenon and have, in many instances, replaced controversy by clarity. Synergy between experimental and theoretical work, both playing a crucial role, has been at the heart of these advances. On the theoretical front, the adoption of Berry-phase concepts has established a link between the AHE and the topological nature of the Hall currents which originate from spin-orbit coupling. On the expe…
The intrinsic contribution to the spin Hall effect in a 2DEG with non-magnetic impurities is studied in a quantum Boltzmann approach. It is shown that if the steady state response is perturbative in the spin-orbit coupling parameter $λ$, then the precession term--vital for Dyakonov-Perel relaxation and the key to the spin Hall effect in previous similar Boltzmann studies--must be left out to first order in spin-orbit coupling. In such a case one would have that to lowest order in the parameters electric field, spin-orbit coupling, impurity strength and impurity concentration there is no intrin…
Using the non-equilibrium Green\noindent 's function method and the Keldysh formalism, we study the effects of spin-orbit interactions and time-reversal symmetry breaking exchange fields on non-equilibrium quantum transport in graphene armchair nanoribbons. We identify signatures of the quantum spin Hall (QSH) and the quantum anomalous Hall (QAH) phases in non-equilibrium edge transport by calculating the spin-resolved real space charge density and local currents at the nanoribbon edges. We find that the QSH phase, which is realized in a system with intrinsic spin-orbit coupling, is characteri…
The coupled kinetic equation for density and spin Wigner functions are derived including spin-orbit coupling, electric and magnetic field as well as selfconsistent Hartree meanfields suited for SU(2) transport. The interactions are assumed to be with scalar and magnetic impurities as well as scalar and spin-flip potentials among the particles. The spin-orbit interaction is used in a form suitable to solid state physics with Rashba or Dresselhaus coupling, graphene, extrinsic spin-orbit coupling as well as effective nuclear matter coupling. The deficiencies of the two-fluid model are worked out…
