Spin Hall Effect

Abstract

Since electrons have an internal degree of freedom, spin, they are characterized not only by charge density and electric current, but also by spin density and spin current. The spin current is described by a tensor $q_{ij}$, where the first index indicates the direction of flow, while the second one says which component of the spin is flowing.

Appendix: The Generalized Kinetic Equation

The Boltzmann equation, generalized for the existence of spin and taking account of the spin-orbit interaction is probably the best, most simple, and economic theoretical tool for studying spin phenomena in conditions when the orbital motion can be considered classically (when, for example, the Landau quantization is not important). It can be derived from the quantum equation for the one-particle density matrix $\hat{\rho }_{\varvec{p}_1,\varvec{p}_2} (t)$. We use the momentum representation, hats indicate matrices in spin indices.

$$\begin{aligned} i\hbar \frac{\partial \hat{\rho }}{\partial t}=[\hat{H},\hat{\rho }]. \end{aligned}$$

(8.58)

To approach classical physics, one indtroduces the Wigner density matrix $\hat{\rho }(\varvec{r},\varvec{p},t)$, by putting $\varvec{p}=(\varvec{p}_1+\varvec{p}_2)/2$, $\varvec{\kappa }= \varvec{p}_1-\varvec{p}_2$ and doing the Fourier transform over $\varvec{\kappa }$:

$$\begin{aligned} \hat{\rho }(\varvec{r},\varvec{p},t)= \int \exp (i\varvec{\kappa }\varvec{r}/\hbar ) \hat{\rho }_{\varvec{p}+\varvec{\kappa }/2, \varvec{p}-\varvec{\kappa }/2} (t) \frac{d^3\varvec{\kappa }}{(2\pi \hbar )^3}. \end{aligned}$$

(8.59)

Consider a Hamiltonian $\hat{H}(\varvec{p})$ describing the electron spin band splitting and the interaction with an external electric field $\varvec{E}$ (e is the absolute value of the electron charge):

$$\begin{aligned} \hat{H}(\varvec{p})= \frac{p^2}{2m}+\hbar \varvec{\varOmega }(\varvec{p})\hat{\varvec{s}}+e\varvec{E}\varvec{r}. \end{aligned}$$

(8.60)

Using (8.58) and (8.60) one can derive the kinetic equation for $\hat{\rho }(\varvec{r},\varvec{p},t)$:

$$\begin{aligned} \frac{\partial \hat{\rho }}{\partial t} +\{\hat{\varvec{v}},\nabla \hat{\rho }\}- e\varvec{E}\frac{\partial \hat{\rho }}{\partial \varvec{p}} + i[\varvec{\varOmega }(\varvec{p})\hat{\varvec{s}}, \hat{\rho }] = \hat{I}\{\hat{\rho }\}, \end{aligned}$$

(8.61)

where $\{A,B\}=(AB+BA)/2$ and $\hat{\varvec{v}}=\partial \hat{H}(\varvec{p})/\partial \varvec{p}$ is the electron velocity, which is a matrix in spin indices. The left-hand side of (8.61) follows exactly from (8.58). The right-hand side is the collision integral, which is added “by hand” (like in the conventional Boltzmann equation).²² One has to evaluate the change of the density matrix, $\delta \hat{\rho }$, during one collision and make a sum over all collisions per unit time.

Spin-orbit interaction (8.36) during the act of an individual collision makes the integral operator $\hat{I}$ a matrix with 4 spin indices. For elastic collisions with impurities and if the effect of band splitting can be neglected while considering individual collisions, we have [76]:

$$\begin{aligned} \hat{I}\{\hat{\rho }(\varvec{p})\}_{\mu \mu '}=\int d\varOmega _{\varvec{p'}}\{ W^{\mu \mu '}_{\mu _1\mu '_1}\cdot \rho _{\mu _1\mu '_1}(\varvec{p'})- \frac{1}{2} [W^{\mu \mu _1}_{\mu _2 \mu _2} \cdot \rho _{\mu _1\mu '}({\varvec{p}}) + \rho _{\mu \mu _1}({\varvec{p}}) \cdot W^{\mu _1 \mu '}_{\mu _2 \mu _2} ] \}. \end{aligned}$$

(8.62)

Here the summation over repeated indices is implied. This expression is the generalization of the Boltzmann collision integral taking account of spin-orbit effects. If spin-orbit interaction during collisions is neglected, it reduces to the conventional Boltzmann term. The transition probability matrix $\hat{W}$ can be expressed through the scattering amplitude $\hat{F}_{\varvec{p'}}^{\varvec{p}}$ given by (8.28):

$$\begin{aligned} W^{\mu \mu '}_{\mu _1\mu '_1}=N v F^{\mu \varvec{p}}_{\mu _1 \varvec{p'}} (F^{\mu ' \varvec{p}}_{\mu _1' \varvec{p'}}) ^{\star }, \end{aligned}$$

(8.63)

where N is the impurity concentration, $v=p/m=p'/m$ is the electron velocity, and $F^{\mu \varvec{p}}_{\mu _1 \varvec{p'}}$ is the scattering amplitude for the transition from the initial state $\mu _1 \varvec{p'}$ to the final state $\mu \varvec{p}$.

The question of whether or not the spin band splitting can be neglected, while considering collisions, is a subtle one, especially when dealing with non-diagonal in band indices elements of the density matrix.²³

Equation 8.61 resembles the usual Boltzmann equation, the main differences being the form of the collision integral and the additional commutator term due to the spin band splitting. It can be separated into two coupled equations by putting $\hat{\rho }=(1/2)f\hat{\mathcal{I}}+ 2\varvec{S}\hat{\varvec{s}}$, where the particle and spin distributions in phase space are related to $\hat{\rho }$ by the relations:

$$\begin{aligned} f(\varvec{r},\varvec{p},t)=\text {Tr}(\hat{\rho }), \qquad \varvec{S}(\varvec{r},\varvec{p},t)=\text {Tr}(\hat{\varvec{s}} \hat{\rho }). \end{aligned}$$

(8.64)

The spin polarization density $\varvec{P}(\varvec{r})$ used in this chapter is related to the distribution $\varvec{S}$ by $\varvec{P}(\varvec{r})=2\int \varvec{S}(\varvec{r},\varvec{p})d^3 \varvec{p}$. The equations for f and $\varvec{S}$ can be derived from (8.61):

$$\begin{aligned} \frac{\partial f}{\partial t} + \frac{p_i}{m}\frac{\partial f}{\partial x_i}- eE_i\frac{\partial f}{\partial p_i} + \frac{\partial \varOmega _j}{\partial p_i} \frac{\partial S_j}{\partial x_i}= \text {Tr}(\hat{I}), \end{aligned}$$

(8.65)

$$\begin{aligned} \frac{\partial S_k}{\partial t} + \frac{p_i}{m}\frac{\partial S_k}{\partial x_i} - eE_i\frac{\partial S_k}{\partial p_i} + \frac{1}{4}\frac{\partial \varOmega _k}{\partial p_i} \frac{\partial f}{\partial x_i} - [\varvec{\varOmega }(\varvec{p}) \times \varvec{S}]_k = \text {Tr}(\hat{s}_k \hat{I}). \end{aligned}$$

(8.66)

These equations are further simplified in the spatially homogeneous situation.

They contain most of the relevant spin physics in semiconductors: spin relaxation, spin diffusion, coupling between spin and charge currents (due to skew scattering, as well as intrinsic), etc. After including magnetic field in the usual manner, they can be used to study magnetic effects in spin relaxation and spin transport. Similar, but more complicated, equations were derived and analysed for $J=3/2$ holes in the valence band and carriers in a gapless semiconductor [26]. Compared to other, more sophisticated techniques, the approach based on the kinetic equation, has the advantage of being much more transparent and of allowing to use the physical intuition accumulated in dealing with the Boltzmann equation. It also avoids invoking quantum mechanics where it is not really necessary.