Semiclassical Boltzmann theory of spin Hall effects in giant Rashba systems

TL;DR

This paper develops a semiclassical Boltzmann theory for the spin Hall effect in systems with strong spin-orbit coupling, incorporating disorder effects like side-jump, and applies it to Rashba systems showing enhanced spin Hall angles with magnetic impurities.

Contribution

It introduces a novel semiclassical Boltzmann framework that includes the spin-current-side-jump velocity, addressing a gap in understanding disorder effects in spin Hall phenomena.

Findings

Enhanced spin Hall angle with magnetic impurities in Rashba systems.

Largely increased spin Hall effect when only the lower Rashba band is occupied.

Theoretical framework applicable to strong spin-orbit coupled materials.

Abstract

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 only the lower Rashba band is partially occupied.

Equations

$${\mathbf{j}}^z=\sum _l{f}_l{\mathbf{j}}_l^z,$$

$${\mathbf{j}}_l^z={\left({\mathbf{j}}_l^z\right)}^0+{\delta }^{in}{\mathbf{j}}_l^z+{\delta }^{ex}{\mathbf{j}}_l^z.$$

$${\delta }^{in}{\mathbf{j}}_l^z=\hslash e\sum _{{\eta }^{\prime }\ne \eta }\frac{2\mathrm{Im}⟨\eta \mathbf{k}|{\hat{ȷ}}^z|{\eta }^{\prime }\mathbf{k}⟩⟨u_{\mathbf{k}}^{{\eta }^{\prime }}|\hat{\mathbf{v}}\cdot \mathbf{E}|u_{\mathbf{k}}^{\eta }⟩}{{\left({ϵ}_{\mathbf{k}}^{\eta }-{ϵ}_{\mathbf{k}}^{{\eta }^{\prime }}\right)}^2}$$

$${\delta }^{ex}{\mathbf{j}}_l^z$$

$$\times \frac{{⟨⟨\eta \mathbf{k}|\hat{V}|{\eta }^{\prime }{\mathbf{k}}^{\prime }⟩⟨{\eta }^{\prime \prime }{\mathbf{k}}^{\prime }|\hat{V}|\eta \mathbf{k}⟩⟩}_c⟨{\eta }^{\prime }{\mathbf{k}}^{\prime }|{\hat{ȷ}}^z|{\eta }^{\prime \prime }{\mathbf{k}}^{\prime }⟩}{\left({ϵ}_{\mathbf{k}}^{\eta }-{ϵ}_{{\mathbf{k}}^{\prime }}^{{\eta }^{\prime }}-i{0}^{+}\right)\left({ϵ}_{\mathbf{k}}^{\eta }-{ϵ}_{{\mathbf{k}}^{\prime }}^{{\eta }^{\prime \prime }}+i{0}^{+}\right)}$$

$$+2\mathrm{Re}\sum _{{\eta }^{\prime }\ne \eta }\sum _{{\eta }^{\prime \prime }{\mathbf{k}}^{\prime \prime }}$$

$$\times \frac{{⟨⟨{\eta }^{\prime }\mathbf{k}|\hat{V}|{\eta }^{\prime \prime }{\mathbf{k}}^{\prime \prime }⟩⟨{\eta }^{\prime \prime }{\mathbf{k}}^{\prime \prime }|\hat{V}|\eta \mathbf{k}⟩⟩}_c⟨\eta \mathbf{k}|{\hat{ȷ}}^z|{\eta }^{\prime }\mathbf{k}⟩}{\left({ϵ}_{\mathbf{k}}^{\eta }-{ϵ}_{\mathbf{k}}^{{\eta }^{\prime }}+i{0}^{+}\right)\left({ϵ}_{\mathbf{k}}^{\eta }-{ϵ}_{{\mathbf{k}}^{\prime \prime }}^{{\eta }^{\prime \prime }}+i{0}^{+}\right)}.$$

$$e\mathbf{E}\cdot {\mathbf{v}}_l^0\frac{\partial f^0}{\partial {ϵ}_l}=-\sum _{l^{\prime }}{\omega }_{l,l^{\prime }}\left[f_l-f_{l^{\prime }}-\frac{\partial f^0}{\partial {ϵ}_l}e\mathbf{E}\cdot \delta {\mathbf{r}}_{l\prime ,l}\right],$$

$$f_l=f_l^0+g_l^n+g_l^a,$$

$${\mathbf{j}}^z=\sum _l{f}_l{\left({\mathbf{j}}_l^z\right)}^0+\sum _l{g}_l^{2s}{\delta }^{ex}{\mathbf{j}}_l^z+\sum _l{f}_l^0{\delta }^{in}{\mathbf{j}}_l^z,$$

$${\mathbf{j}}^z={\mathbf{j}}^{z,in}+{\mathbf{j}}^{z,sj}.$$

$$g_{\eta }^{2s}\left(ϵ\right)=\left(-{\partial }_{ϵ}{f}^0\right)e\mathbf{E}\cdot {\mathbf{v}}_{\eta }\left(ϵ,\varphi \right){\tau }_{\eta }^{tr}\left(ϵ\right),$$

$$1=\sum _{{\eta }^{\prime }}\int \frac{d{\varphi }^{\prime }}{2\pi }{\omega }_{\eta ,{\eta }^{\prime }}^{\varphi ,{\varphi }^{\prime }}\left[{\tau }_{\eta }^{tr}-\cos \left({\varphi }^{\prime }-\varphi \right){\tau }_{{\eta }^{\prime }}^{tr}\right].$$

$$g_{-\nu }^{2s}\left(ϵ\right)=\left(-{\partial }_{ϵ}{f}^0\right)e\mathbf{E}\cdot {\mathbf{v}}_{-\nu }\left(ϵ,\varphi \right){\tau }_{-\nu }^{tr}\left(ϵ\right),$$

$$1=\sum _{{\nu }^{\prime }}\int \frac{d{\varphi }^{\prime }}{2\pi }{\omega }_{-\nu ,-{\nu }^{\prime }}^{\varphi ,{\varphi }^{\prime }}\left[{\tau }_{-\nu }^{tr}-{\left(-1\right)}^{{\nu }^{\prime }-\nu }\cos \left({\varphi }^{\prime }-\varphi \right){\tau }_{-{\nu }^{\prime }}^{tr}\right].$$

$${\delta }^{ex}{\left({\mathbf{j}}_l^z\right)}_y^{nm}=-\frac{1}{{\tau }_0}\frac{\eta }{k_R}\frac{\hslash }{8}\cos \varphi ,$$

$$j_y^{z,sj}=\sum _l{g}_l^{2s}{\delta }^{ex}{\left({\mathbf{j}}_l^z\right)}_y^{nm}=\frac{e}{8\pi }{E}_x,$$

$${\delta }^{ex}{\left({\mathbf{j}}_l^z\right)}_y^{nm}=\frac{1}{{\tau }_0}\frac{1}{k_0\left(ϵ\right)}\frac{\hslash }{8}\cos \varphi .$$

$$j_y^{z,sj}=\sum _l{g}_l^{2s}{\delta }^{ex}{\left({\mathbf{j}}_l^z\right)}_y^{nm}=\frac{k_0\left({ϵ}_F\right)}{k_R}\frac{e}{8\pi }{E}_x$$

$${\delta }^{ex}{\left({\mathbf{j}}_l^z\right)}_y^m=-\frac{1}{3}\frac{n_m{V}_m^2}{n_0{V}_0^2}{\delta }^{ex}{\left({\mathbf{j}}_l^z\right)}_y^{nm}.$$

$$\frac{{\tau }_{\eta }^{tr}\left(ϵ\right)}{{\tau }_m}=\frac{8k_0\left(ϵ\right)-k_{\eta }\left(ϵ\right)}{7k_0\left(ϵ\right)}$$

$$\frac{{\tau }_{-\nu }^{tr}\left(ϵ\right)}{{\tau }_m}=\frac{k_0\left(ϵ\right)}{k_R}\frac{8k_R-k_{-\nu }\left(ϵ\right)}{7k_R}$$

$$j_y^{z,sj}=\sum _l{g}_l^{2s}{\delta }^{ex}{\left({\mathbf{j}}_l^z\right)}_y^m=\frac{1}{7}\frac{-e}{8\pi }{E}_x=\frac{1}{7}{j}_y^{z,in}$$

$${\alpha }_{sH}=\frac{e{j}_y^z/\left(\frac{\hslash }{2}\right)}{j_x}=\frac{-2\hslash }{{\tau }_m{ϵ}_R}\frac{1}{3+7\frac{{ϵ}_F}{{ϵ}_R}}.$$

$$j_y^{z,sj}=\frac{k_0\left({ϵ}_F\right)}{7k_R}\frac{-e}{8\pi }{E}_x=\frac{1}{7}{j}_y^{z,in}$$

$${\alpha }_{sH}=\frac{-2\hslash }{{\tau }_m{ϵ}_R}\frac{1}{\left(3-\frac{{ϵ}_F}{{ϵ}_R}\right)\sqrt{1+2\frac{{ϵ}_F}{{ϵ}_R}}}.$$

$${\alpha }_{sH}=\frac{-\frac{\hslash }{{\tau }_S{ϵ}_R}\frac{2}{3}\frac{{\tau }_0}{{\tau }_m}}{1+\frac{{\tau }_0}{{\tau }_m}+\left(1+\frac{7}{3}\frac{{\tau }_0}{{\tau }_m}\right)\frac{{ϵ}_F}{{ϵ}_R}}$$

$${\alpha }_{sH}=\frac{-\frac{\hslash }{{\tau }_S{ϵ}_R}\frac{2}{3}\frac{{\tau }_0}{{\tau }_m}}{1+\frac{{\tau }_0}{{\tau }_m}+\left[1-\frac{1}{3}\frac{{\tau }_0}{{\tau }_m}\right]\frac{{ϵ}_F}{{ϵ}_R}}\frac{1}{\sqrt{1+2\frac{{ϵ}_F}{{ϵ}_R}}},$$

Full text

I Introduction

It is now generally accepted that three mechanisms – intrinsic, side-jump and skew scattering – contribute to both the spin Hall effect (SHE) and anomalous Hall effect (AHE) Nagaosa2010 ; Sinova2015 . Among the three mechanisms, the side-jump mechanism is of special interest because it originates from scattering but can, in some simple cases Nagaosa2010 ; Sinova2015 ; Kovalev2010 ; Culcer2010 ; Yang2011 , be independent of both the disorder density and scattering strength. In particular, when the SHE or AHE arises from strong spin-orbit coupling in the band-structure, the side-jump belongs to the category of disorder-induced interband-coherence effect which has recently been an important topic in condensed matter physics Nagaosa2010 ; Sinova2015 ; Kovalev2010 ; Sinitsyn ; Hou2015 ; Culcer2017 ; Xiao2017SOT .

In investigating transport phenomena in solids, the semiclassical Boltzmann approach is appealing due to its conceptual intuition Ziman1960 . In the study of SHE-AHE, how to incorporate side-jump effects into the semiclassical formalism is an attracting theoretical issue Nagaosa2010 . In the study of AHE, the renewed semiclassical theory addressing this issue has proven useful in obtaining physical pictures Sinitsyn . In such a theory, the quantum mechanical information on side-jump is coded in the expressions of gauge-invariant classical concepts such as the coordinate-shift and side-jump velocity Sinitsyn . On the other hand, in the field of SHE when the spin is not conserved due to strong spin-orbit coupling in the band structure, such as in a Rashba two-dimensional electron gas (2DEG), a semiclassical description to side-jump SHE is still absent Zhang2005 ; noteZhang2005 . Although the modified Boltzmann equation Sinitsyn developed in studying the AHE can be directly applied to SHE, the spin–current-counterpart of the side-jump velocity in this case has not been addressed before.

In the present paper we formulate a semiclassical Boltzmann framework of SHE when the spin is not conserved due to strong band-structure spin-orbit coupling. This semiclassical theory takes into account interband-coherence effects induced by both the dc uniform electric field and weak static disorder. We work out the spin-current-counterpart of the side-jump velocity based on scattering-induced modifications of conduction-electron states. When the electric field turns on, this quantity contributes one part of the side-jump SHE.

As applications we consider the SHE in a 2DEG with giant Rashba spin-orbit coupling and short-range impurities. We focus on the enhancement of spin Hall angle when the Fermi energy is tuned down towards and below the band crossing point in giant Rashba 2DEGs with magnetic disorder. The spin Hall angle which measures the generation efficiency of the transverse spin current from the longitudinal electric current is the figure of merit of the SHE. Giant Rashba spin-orbit coupling energy comparable to or even larger than the Fermi energy is possible in the polar semiconductor BiTeX (X=Cl, Br and I) family and related surfaces and interfaces Eremeev2012 ; Sakano2013 ; Wu2014 . Thus these systems are promising to realize efficient conversion of charge current into spin current.

The paper is organized as follows. In Sec. II we outline the semiclassical formulation of SHE. Section III introduces the Rashba model and calculates the SHE. Section IV concludes the paper.

II Semiclassical formulation

Considering the linear response of the spin current polarized in one particular direction (z direction is chosen in the following) to a weak dc uniform electric field $\mathbf{E}$ in non-degenerate multiband electron systems in the weak disorder regime, one has the semiclassical formula

[TABLE]

where $\mathbf{j}_{l}^{z}$ is the amount of spin current carried by the conduction-electron state denoted by $l$, $f_{l}$ is the semiclassical distribution function.

The conduction-electron state may be modified by the electric field and static impurity scattering, $\mathbf{j}_{l}^{z}$ can thus deviate from the customary pure-band value Ziman1960 ; Zhang2005 : $\left(\mathbf{j}_{l}^{z}\right)^{0}\equiv\langle l|\mathbf{\hat{\jmath}}^{z}\mathbf{|}l\rangle$ with $\mathbf{\hat{\jmath}}^{z}$ the spin-current operator. Here $|l\rangle=|\eta\mathbf{k}\rangle=|\mathbf{k}\rangle|u_{\mathbf{k}}^{\eta}\rangle$ is the Bloch state, $\eta$ is the band index, $\mathbf{k}$ is the crystal momentum, $|\mathbf{k}\rangle$ and $|u_{\mathbf{k}}^{\eta}\rangle$ are the plane-wave and periodic parts of $|\eta\mathbf{k}\rangle$, respectively. Following the recipe based on the quantum-mechanical perturbation theory for the electric-field modified Bloch state and the Lippmann-Schwinger equation for the scattering modified conduction-electron state in Ref. Xiao2017SOT, , in the weak disorder regime nontrivial corrections caused by interband-coherence effects to $\left(\mathbf{j}_{l}^{z}\right)^{0}$ read:

[TABLE]

The intrinsic correction $\delta^{in}\mathbf{j}_{l}^{z}=2\operatorname{Re}\langle l|\mathbf{\hat{\jmath}}^{z}|\delta^{\mathbf{E}}l\rangle$ arises from the interband-virtual-transition (electron charge e) $|\delta^{\mathbf{E}}l\rangle=-i\hbar e\mathbf{E\cdot}\sum_{\eta^{\prime}\neq\eta}|\eta^{\prime}\mathbf{k}\rangle\langle u_{\mathbf{k}}^{\eta^{\prime}}|\mathbf{\hat{v}}|u_{\mathbf{k}}^{\eta}\rangle/\left(\epsilon_{\mathbf{k}}^{\eta}-\epsilon_{\mathbf{k}}^{\eta^{\prime}}\right)^{2}$ induced by the electric field Zhang2005 , with $\epsilon_{l}\equiv\epsilon_{\mathbf{k}}^{\eta}$ the energy of Bloch state $|\eta\mathbf{k}\rangle$ and $\mathbf{\hat{v}}$ the velocity operator. Thus

[TABLE]

is an electric-field-induced interband-coherence effect.

The extrinsic correction $\delta^{ex}\mathbf{j}_{l}^{z}$ originates from the interband-coherence during the elastic electron-impurity scattering process. The scattering-induced modification to conduction-electron states is captured by the Lippmann-Schwinger equation describing the scattering state $|l^{s}\rangle=|l\rangle+\left(\epsilon_{l}-\hat{H}_{0}+i\epsilon\right)^{-1}\hat{T}|l\rangle$ with the T-matrix $\hat{T}|l\rangle=\hat{V}|l^{s}\rangle$ related to the disorder potential $\hat{V}$ and disorder-free Hamiltonian $\hat{H}_{0}$. $|\delta l^{s}\rangle\equiv|l^{s}\rangle-|l\rangle$ denotes the scattering-induced modification to the Bloch state. Thus $\delta^{ex}\mathbf{j}_{l}^{z}$ is related to the values of $2\operatorname{Re}\left\langle\langle l|\mathbf{\hat{\jmath}}^{z}|\delta l^{s}\rangle\right\rangle_{c}$ and $\left\langle\langle\delta l^{s}|\mathbf{\hat{\jmath}}^{z}|\delta l^{s}\rangle\right\rangle_{c}$ in the lowest nonzero order in the disorder potential. Here $\left\langle..\right\rangle_{c}$ denotes the average over disorder configurations and we assume that the statistical average of the disorder potential is zero (nonzero value only shifts the origin of total energy) $\left\langle V\right\rangle_{c}=0$. Only the terms containing interband matrix elements of $\mathbf{\hat{\jmath}}^{z}$ represent the disorder-induced interband-coherence effects, therefore note-SO scattering

[TABLE]

It has been shown Xiao2017SOT that the side-jump velocity $\mathbf{v}_{l}^{sj}$ which is an important ingredient in the semiclassical theory of AHE Sinitsyn can also be obtained in this way ($\delta^{ex}\mathbf{v}_{l}=\mathbf{v}_{l}^{sj}$) and thus shares the same origin. $\delta^{ex}\mathbf{j}_{l}^{z}$ can therefore be deemed as the spin–current-counterpart of the side-jump velocity in the case of band-structure spin-orbit coupling.

The properly modified steady-state linearized Boltzmann equation in the presence of weak static disorder has been proposed as Sinitsyn :

[TABLE]

where $\mathbf{v}_{l}^{0}=\partial\epsilon_{l}/\hbar\partial\mathbf{k}$ is the band velocity, $f^{0}$ is the Fermi distribution function, $\delta\mathbf{r}_{l\prime,l}$ is the coordinate-shift in the scattering process ($l\rightarrow l^{\prime}$) Sinitsyn and $\omega_{l,l^{\prime}}$ the semiclassical scattering rate ($l^{\prime}\rightarrow l$). Up to the linear order of the electric field one has the decomposition Sinitsyn ; Xiao2017AHE

[TABLE]

with $g_{l}^{n}$ the normal part of the out-of-equilibrium distribution function satisfying the Boltzmann equation in the absence of $\delta\mathbf{r}_{l\prime,l}$ and $g_{l}^{a}$ the anomalous distribution function related to $\delta\mathbf{r}_{l\prime,l}$. It is now clear Sinitsyn that $\delta r_{l\prime,l}$ is a disorder-induced interband-coherence effect and so is $g_{l}^{a}$.

Given that the semiclassical formulation is relevant in the weak disorder regime, Eq. (1) reduces to Xiao2017SOT

[TABLE]

up to the zeroth order of total impurity density and scattering strength. $g_{l}^{2s}$ represents the value of $g_{l}^{n}$ in the lowest Born order Sinitsyn . In higher Born orders, some additional contributions to $g_{l}^{n}$ appear and are responsible for the transverse transport due to the breakdown of the principle of microscopic detailed balance. The analysis of these higher-Born-order contributions under the non-crossing approximation has been detailed in Ref. Xiao2017AHE, . Here we only mention that there is an interband-coherence scattering effect called “intrinsic-skew-scattering induced side-jump” appearing in the third Born order under the Gaussian disorder. Below we set $\mathbf{j}^{z,in}=\sum_{l}f_{l}^{0}\delta^{in}\mathbf{j}_{l}^{z}$ which is just the intrinsic contribution to the spin current independent of the disorder Zhang2005 , and $\mathbf{j}^{z,sj}=\sum_{l}g_{l}^{2s}\delta^{ex}\mathbf{j}_{l}^{z}$ because it is related to the spin–current-counterpart of the side-jump velocity. In general case of SHE induced by strong band-structure spin-orbit coupling, $\mathbf{j}^{z,sj}$ is just one part of the side-jump SHE arising from disorder-induced interband-coherence effects. Other two semiclassical contributions to the side-jump SHE (from the anomalous distribution function $g_{l}^{a}$ and the intrinsic-skew-scattering induced side-jump) note-sj and the skew scattering SHE arising from non-Gaussian disorder are all included in the first term of Eq. (7) Sinitsyn ; Xiao2017AHE .

To be more clear we can consider the case of randomly distributed scalar pointlike Gaussian disorder with density $n_{0}$ and average strength $V_{0}$. Then $g_{l}^{2s}\sim n_{0}^{-1}V_{0}^{-2}$, $\delta^{ex}j_{l}^{z}\sim n_{0}V_{0}^{2}$, $g_{l}^{a}\sim n_{0}^{0}V_{0}^{0}$, and the third-Born-order contribution to $g_{l}^{n}$ behaves as $\sim n_{0}^{0}V_{0}^{0}$ (thus is called the intrinsic-skew-scattering Sinitsyn ; Xiao2017AHE ). In this case the side-jump SHE may consist of three semiclassical contributions in the zeroth order of both the impurity density and scattering strength: $j^{z,sj}$, $\sum_{l}g_{l}^{a}\left(\mathbf{j}_{l}^{z}\right)^{0}$ and the intrinsic-skew-scattering induced side-jump.

III Model calculation

III.1 Model

The model Hamiltonian of a Rashba 2DEG is $H_{0}=\frac{\hbar^{2}\mathbf{k}^{2}}{2m}+\alpha_{R}\mathbf{\hat{\sigma}}\cdot\left(\mathbf{k}\times\mathbf{\hat{z}}\right)$, where $\mathbf{k}$ is the 2D wavevector, $m$ is the effective mass, $\mathbf{\hat{\sigma}}$ is the vector of Pauli matrices, $\alpha_{R}$ the Rashba coefficient. The internal eigenstates read $|u_{\mathbf{k}}^{\eta}\rangle=\frac{1}{\sqrt{2}}\left[1,-i\eta\exp\left(i\phi\right)\right]^{T}$, where $\eta=\pm$ label the two bands $\epsilon_{k}^{\eta}=\hbar^{2}k^{2}/\left(2m\right)+\eta\alpha_{R}k$, and $\tan\phi=k_{y}/k_{x}$.

For $\epsilon>0$ the corresponding wave number in $\eta$ band is given as $k_{\eta}\left(\epsilon\right)=-\eta k_{R}+k_{0}\left(\epsilon\right)$. Here $k_{R}=m\alpha_{R}/\hbar^{2}=\frac{1}{2}\left(k_{-}\left(\epsilon\right)-k_{+}\left(\epsilon\right)\right)$ measures the momentum splitting of two Rashba bands, whereas $k_{0}\left(\epsilon\right)\equiv\alpha_{R}^{-1}\sqrt{\epsilon_{R}^{2}+2\epsilon_{R}\epsilon}=\frac{1}{2}\sum_{\eta}k_{\eta}\left(\epsilon\right)$. The density of states of $\eta$ band takes the form $D_{\eta}\left(\epsilon\right)=D_{0}\frac{k_{\eta}\left(\epsilon\right)}{k_{0}\left(\epsilon\right)}$, with $D_{0}=m/2\pi\hbar^{2}$.

For $0>\epsilon>-\epsilon_{R}/2$, the iso-energy surface slices the spectrum two rings of radii $k_{-\nu}\left(\epsilon\right)=k_{R}+\left(-1\right)^{\nu-1}k_{0}\left(\epsilon\right)$, where $\nu=1,2$ denote the two monotonic segments (Fig. 1), $k_{0}\left(\epsilon\right)\equiv\alpha_{R}^{-1}\sqrt{\epsilon_{R}^{2}+2\epsilon_{R}\epsilon}=\frac{1}{2}\left(k_{-1}\left(\epsilon\right)-k_{-2}\left(\epsilon\right)\right)$. The density of states of $-\nu$ branch reads $D_{-\nu}\left(\epsilon\right)=D_{0}\frac{k_{-\nu}\left(\epsilon\right)}{k_{0}\left(\epsilon\right)}$.

The conventional definition of the spin current as an anti-commutator of velocity and spin is employed: $\mathbf{\hat{\jmath}}^{z}=\frac{\hbar}{2}\frac{1}{2}\left\{\hat{\sigma}_{z},\mathbf{\hat{v}}\right\}=\frac{\hbar}{2}\frac{\hbar\mathbf{k}}{m}\hat{\sigma}_{z}$. It is purely off-diagonal in band-index space in this model: $\left(\mathbf{j}_{l}^{z}\right)^{0}=0$, thus the SHE in Eq. (7) is determined only by

[TABLE]

The Boltzmann equation can be conveniently solved by using variables $l=\left(\epsilon,\eta,\phi\right)$ for $\epsilon>0$ and $l=\left(\epsilon,-\nu,\phi\right)$ for $0>\epsilon>-\epsilon_{R}/2$. Correspondingly, $\sum_{l}=\sum_{\eta\left(\nu\right)}\int d\epsilon D_{\eta\left(-\nu\right)}\left(\epsilon\right)\int\frac{d\phi}{2\pi}$ for $\epsilon>0$ $\left(0>\epsilon>-\epsilon_{R}/2\right)$. If $\epsilon>0$, in the lowest Born order the energy-integrated elastic scattering rate is $\omega_{\eta,\eta^{\prime}}^{\phi,\phi^{\prime}}\left(\epsilon\right)=\int d\epsilon^{\prime}D_{\eta^{\prime}}\left(\epsilon^{\prime}\right)\omega_{l\prime,l}^{2s}$. Whereas if $0>\epsilon>-\epsilon_{R}/2$ there exists elastic scattering between the two branches $\nu=1$ and $2$, and one has $\omega_{-\nu,-\nu^{\prime}}^{\phi,\phi^{\prime}}\left(\epsilon\right)=D_{-\nu^{\prime}}\left(\epsilon\right)\int d\epsilon_{l^{\prime}}\omega_{l\prime,l}^{2s}$. Assuming isotropic disorder potential, transport-time type solutions to $g_{l}^{2s}$ exist Xiao2016PRB . For $\epsilon>0$ we have

[TABLE]

where the transport time $\tau_{\eta}^{tr}\left(\epsilon\right)$ is determined by

[TABLE]

For $0>\epsilon>-\epsilon_{R}/2$ we have

[TABLE]

with the transport time $\tau_{-\nu}^{tr}\left(\epsilon\right)$ decided by

[TABLE]

III.2 Calculations

We consider the impurity potential is produced by randomly distributed short-range scatters at $\mathbf{R}_{i}$, i.e., $V\left(\mathbf{r}\right)=\sum_{i,\mu}V_{\mu}^{i}\sigma_{\mu}\delta\left(\mathbf{r}-\mathbf{R}_{i}\right)$ with $\mu=0,1,2,3$ and $\sigma_{0}$ the unity matrix in spin space Lu2013 . Here the short-range potential is approximated by the delta-potential. We assume Gaussian disorder approximation and isotropic magnetic scattering Lu2013 ; Inoue2006 . $n_{0}$ and $n_{m}$ are the concentrations of nonmagnetic and magnetic impurities, respectively. $V_{0}$ and $V_{m}$ are the average strengths for the nonmagnetic and magnetic scattering, respectively. The external electric field is applied in x direction.

III.2.1 Nonmagnetic impurities

When $\epsilon>0$, straightforward calculation leads to the spin-current-counterpart of the side-jump velocity

[TABLE]

with $\tau_{0}=\left(2\pi n_{0}V_{0}^{2}D_{0}/\hbar\right)^{-1}$. The transport time reads Xiao2016PRB $\tau_{\eta}^{tr}\left(\epsilon\right)=\tau_{0}D_{\eta}\left(\epsilon\right)/D_{0}$, and then the side-jump spin Hall current is

[TABLE]

which completely cancels out the intrinsic spin Hall current $j_{y}^{z,in}=\frac{-e}{8\pi}E_{x}$. This just reproduces the well-known Sinova2015 vanishing spin Hall current $j_{y}^{z}=0$ in the semiclassical Boltzmann theory for the first time.

When $0>\epsilon_{F}>-\epsilon_{R}/2$, the intrinsic spin Hall current is $j_{y}^{z,in}=\frac{k_{0}\left(\epsilon_{F}\right)}{k_{R}}\frac{-e}{8\pi}E_{x}$. Meanwhile the spin-current-counterpart of the side-jump velocity reads

[TABLE]

and thus the side-jump spin Hall current

[TABLE]

again cancels out the intrinsic one. This also coincides with the zero SHE obtained by the Kubo formula Grimaldi2006 .

III.2.2 Magnetic impurities

For isotropic delta-like magnetic impurity potential, since the contributions from $V_{x}^{i}$ and $V_{y}^{i}$ cancel out in Eq. (4), the spin–current-counterpart of the side-jump velocity is given by

[TABLE]

The transport time is given by

[TABLE]

for $\epsilon>0$, and

[TABLE]

for $0>\epsilon>-\epsilon_{R}/2$, with $\tau_{m}=\left(\frac{2\pi}{\hbar}n_{m}V_{m}^{2}D_{0}\right)^{-1}$.

When both Rashba bands are partially occupied, the side-jump spin Hall current

[TABLE]

enhances the total spin Hall current to $j_{y}^{z}=j_{y}^{z,in}+j_{y}^{z,sj}=\frac{8}{7}j_{y}^{z,in}$. This $j_{y}^{z}$ is the same as the weak-disorder-limit value of that obtained by Kubo diagrammatic calculations Inoue2006 ; Wang2007 . The longitudinal electric current is $j_{x}=\frac{e^{2}}{\pi\hbar^{2}}\tau_{m}\frac{3\epsilon_{R}+7\epsilon_{F}}{7}E_{x}$, the spin Hall angle is therefore

[TABLE]

When only the lower Rashba band is partially occupied, the side-jump and the total spin Hall currents are

[TABLE]

and $j_{y}^{z}=\frac{8}{7}j_{y}^{z,in}$, respectively. The longitudinal electric current is $j_{x}=\frac{e^{2}}{\pi\hbar^{2}}\tau_{m}\frac{3\epsilon_{R}-\epsilon_{F}}{7}\frac{k_{0}^{2}\left(\epsilon_{F}\right)}{k_{R}^{2}}E_{x}$, thus

[TABLE]

Although $\frac{\hbar}{\tau_{m}\epsilon_{R}}$ is a small quantity in giant Rashba systems, the factor $\sqrt{1+2\frac{\epsilon_{F}}{\epsilon_{R}}}$ can be very small leading to a large spin Hall angle when $\epsilon_{F}$ is located close to the band bottom of the lower Rashba band. For instance, if $\frac{\hbar}{\tau_{m}\epsilon_{R}}=0.02$, $1+2\frac{\epsilon_{F}}{\epsilon_{R}}=0.1$ leads to $\alpha_{sH}\simeq-4\%$, which is quite large Ebert2015 ; Fert2011 . Smaller $\tau_{m}$ and smaller $\sqrt{1+2\frac{\epsilon_{F}}{\epsilon_{R}}}$ may lead to larger $\alpha_{sH}$. However, the quantitative analysis of this possibility is beyond the scope of the semiclassical theory which is valid only in the weak disorder regime. From the above equation, $\alpha_{sH}$ goes to infinity as $\epsilon_{F}$ goes to the band bottom of the lower Rashba band. But this low carrier density limit is actually beyond the Boltzmann regime, and more rigorous microscopic treatments are called for.

III.2.3 Both nonmagnetic and magnetic impurities

The coexistence of nonmagnetic and magnetic impurities may be the more realistic case Inoue2006 ; Lu2013 . Only main results will be given in this case. Since there is no mixing between the nonmagnetic and magnetic scattering as pointed out by Inoue et al. Inoue2006 , the spin-current-counterpart of the side-jump velocity is $\delta^{ex}\left(\mathbf{j}_{l}^{z}\right)_{y}=\left[1-\frac{1}{3}\frac{\tau_{0}}{\tau_{m}}\right]\delta^{ex}\left(\mathbf{j}_{l}^{z}\right)_{y}^{nm}$. The total spin Hall current reads $j_{y}^{z}=\frac{\frac{8}{3}\frac{\tau_{0}}{\tau_{m}}}{1+\frac{7}{3}\frac{\tau_{0}}{\tau_{m}}}j_{y}^{z,in}$, depending on the relative weight of different types of scattering Yang2011 ; Lu2013 . The side-jump effect vanishes when $\tau_{0}=3\tau_{m}$, the same condition as that for the vanishing of the ladder vertex correction in the Kubo diagrammatic calculation in the spin-$\sigma_{z}$ basis for the case $\epsilon_{F}>0$ Inoue2006 ; note-vertex .

For Fermi energies above and below the band crossing point, the spin Hall angles are

[TABLE]

and

[TABLE]

respectively. Here we define $\tau_{S}^{-1}\equiv\tau_{0}^{-1}+\tau_{m}^{-1}$. Tuning the ratio $\tau_{0}/\tau_{m}$ one can find that $\alpha_{sH}$ changes monotonically and continuously from the scalar-disorder-dominated case to the magnetic-disorder-dominated regime.

IV Discussion and Summary

Before concluding this paper, we comment on some important issues not mentioned in above sections.

First, the simple form of the semiclassical Boltzmann equation (5) is exactly valid only for isotropic bands and isotropic scattering Nagaosa2010 ; Xiao2017AHE . In our opinion, in the presence of anisotropy a more generic and complicated form of the Boltzmann equation may be necessary, we refer the readers to Ref. Luttinger1957, for detailed discussions.

Second, the recently highlighted “coherent skew scattering” under the Gaussian disorder beyond the non-crossing approximation Ado2015 is also included in the first term of Eq. (7). This additional contribution is also in the zeroth order of both the impurity density and scattering strength in the weak disorder limit in the presence of only one type of disorder, like the side-jump contribution, but is not an interband-coherence scattering effect note-sj . Thus how to place this contribution into the classification of AHE-SHE mechanisms suggested in Refs. Nagaosa2010, ; Sinova2015, is still an open question. Therefore, in presenting our theory we avoid this issue. Fortunately, in the Rashba model considered in Sec. III the first term of Eq. (7) vanishes. Besides, we should remind the interested readers that this so-called “coherent skew scattering” has actually already been proposed sixty years ago by Kohn and Luttinger Luttinger1957 ; Luttinger1958 . We will provide a comprehensive description of a semiclassical Boltzmann theory going beyond the non-crossing approximation in a future publication.

Finally, in the presence of spin-orbit coupling the electron spin is not conserved thus the spin current is not uniquely defined. The conventionally defined spin current adopted in this study is not a conserved transport current. A physically attracting definition of the conserved spin current has been suggested by Shi et al. Shi2006 by introducing the torque dipole moment. However, disorder effects on the torque dipole spin current Nagaosa2006 in the Bloch representation are hard to deal with under the uniform external electric field in the Boltzmann theory. We reserve these for future studies.

In summary, we have formulated a semiclassical Boltzmann framework of spin Hall effects induced by strong band-structure spin-orbit coupling in non-degenerate multiband electron systems in the weak disorder regime. We worked out the absent ingredient in previous semiclassical theories, i.e., the spin–current-counterpart of the semiclassical side-jump velocity. This gauge-invariant quantity arises from the interband-coherence during the elastic electron-impurity scattering, and contributes one part of the side-jump spin Hall effect.

Applying this theory to a 2DEG with giant Rashba spin-orbit coupling, we showed an enhanced spin Hall angle when only the lower Rashba band is partially occupied in the presence of magnetic impurities. We note that this energy regime below the band crossing point in Rashba systems and similar systems is of intense theoretical interest also from the standpoint of enhanced efficiency of spin-orbit torque and of Edelstein effect Murakami2012 ; Xiao2016FOP ; Zhang2016 , as well as enhanced thermoelectric conversion efficiency Wu2014 ; Xiao2016PRB .

Acknowledgements.

The author is indebted to Qian Niu, Dingping Li and Zhongshui Ma for insightful discussions.