Current-induced spin polarization in spin-orbit-coupled electron systems

Current-induced spin polarization (CISP) is rederived in ballistic spin-orbit-coupled electron systems, based on equilibrium statistical mechanics. A simple and useful picture is correspondingly proposed to help understand the CISP and predict the polarization direction. Nonequilibrium Landauer-Keldysh formalism is applied to demonstrate the validity of the statistical picture, taking the linear Rashba-Dresselhaus [001] two-dimensional system as a specific example. Spin densities induced by the CISP in semiconductor heterostructures and in metallic surface states are compared, showing that the CISP increases with the spin-splitting strength, and hence suggesting that the CISP should be more observable on metal and semimetal surfaces due to the discovered strong Rashba splitting. An application of the CISP designed to generate a spin-Hall pattern in the in-plane, instead of the out-of-plane, component is also proposed.

Figure 1

(Color online) Statistical picture of the current-induced spin polarization phenomenon.

Figure 2

(Color online) Effective magnetic field of a $100\text{−}\text{Å}$-thick [001] InGaAs quantum well with $⟨k_z^2⟩=3.6\times {10}^{-4}{\text{Å}}^{-2}$.

Figure 3

(Color online) Spin orientation in a $30a\times 10a$ channel with $a=1\text{nm}$. Channels with linear Rashba model are considered in (a) and (c) while those with linear Dresselhaus [001] model are in (b) and (d). The direction of each sharp triangle represents the in-plane spin vector ${⟨\mathbf{S}⟩}_{\parallel }=\left(⟨S_x⟩,⟨S_y⟩\right)$ of the local spin density. The size of the triangle depicts the magnitude of ${⟨\mathbf{S}⟩}_{\parallel }$. Effective magnetic fields due to individually the Rashba and the Dresselhaus [001] fields are shown in the insets.

Figure 4

(Color online) Local spin densities by LKF in a square Rashba-Dresselhaus [001] channel with (a) left to right, (b) left bottom to right top, (c) bottom to top, and (d) right bottom to left top bias configurations. Bias regime belongs to low: $e{V}_0=2\text{meV}$. Inset: ${\left[\mathbf{S}\right]}_{\mathbf{k}}^{\text{LKF}}$ vs ${\mathbf{B}}_{RD}^{\left[001\right]}\left(\mathbf{k}\right)$ in the $k_x\text{−}{k}_y$ coordinate.

Figure 5

(Color online) Bias dependence of spin densities induced by the CISP in Rashba 2DESs.

Figure 6

(Color online) Mapping of the (a) local charge current density, and local spin densities (b) $⟨S_z⟩$, (c) $⟨S_x⟩$, and (d) $⟨S_y⟩$ in a four-terminal square channel with a special bias arrangement. Unit in (b)–(d) is $\hslash /2$.