Theoretical aspects of the Edelstein effect for anisotropic two-dimensional electron gas and topological insulators

A charge current driven through a two-dimensional electron gas (2DEG) with Rashba spin-orbit coupling generates a spatially homogeneous spin polarization perpendicular to the applied electric field. This phenomenon is the Aronov–Lyanda-Geller–Edelstein (ALGE) effect. For selected model systems, we consider the ALGE effect within the semiclassical Boltzmann transport theory. Its energy dependence is investigated, in particular the regime below the Dirac point of the 2DEG. In addition to an isotropic 2DEG, we analyze systems with anisotropic Fermi contours. We predict that the current-induced spin polarization vanishes if the Fermi contour passes through a Lifshitz transition. Further, we corroborate that topological insulators (TI) provide a very efficient charge-to-spin conversion.

Figure 1

Energy spectrum of a 2DEG with Rashba SOC (solid lines) for one direction in $\vec{k}$ space. Blue and red represent the $+$ and $-$ branches in Eq. (2), respectively. For comparison, the free-electron parabola without SOC is shown (dashed line). The arrows indicate the spin expectation values with respect to a quantization axis perpendicular to $\vec{k}$.

Figure 2

Fermi lines $({\mathcal{E}}_{\text{F}}>0$) with corresponding spin expectation values (arrows). Blue and red indicate the branch of the energy dispersion, $+$ or $-$. (a) In equilibrium, the total spin polarization vanishes. (b) If an external electric field $\vec{E}\parallel {\vec{e}}_x$ is applied, the Fermi lines are shifted opposite to the field direction, and a nonvanishing spin polarization perpendicular to $\vec{E}$ results.

Figure 3

Expectation value of the total spin ${\langle \sigma \rangle }^{\mathrm{total}}$ (gray) versus Fermi energy ${\mathcal{E}}_{\mathrm{F}}$. The contributions of the $+$ and $-$ branches are shown in blue and red, respectively. Above the Dirac point (region II), the expectation value of the total spin is constant with respect to ${\mathcal{E}}_{\mathrm{F}},{\langle \sigma \rangle }_0=\frac{\left|e\right|{\alpha }_{\mathrm{R}}N}{\pi c_{\mathrm{i}}{\left|U\right|}^2}\left|\vec{E}\right|$. For Fermi energies between the band edge and the Dirac point, ${\mathcal{E}}_{min}<{\mathcal{E}}_{\mathrm{F}}<0$ (region I), the total spin increases linearly.

Figure 4

(a) Energy dispersion of a Rashba system with $C_{2\mathrm{v}}$ symmetry ($m_x=0.5m_y,{\alpha }_{\mathrm{R}x}={\alpha }_{\mathrm{R}y}$) in two-dimensional $\vec{k}$ space (schematic). (b) In addition, the energy spectrum is shown for the $k_x$ and $k_y$ directions. At ${\mathcal{E}}_{\mathrm{S}}$ the band minimum in one direction (here, the $k_x$ direction) is reached, and a saddle point occurs. (c)–(f) The Fermi lines for selected energies; the arrows represent the direction of the spin expectation value ${\langle \vec{\sigma }\rangle }_{\vec{k}}^{\pm }$.

Figure 5

Total spin expectation value of a system with $C_{2\mathrm{v}}$ symmetry ($m_x=0.5m_y,{\alpha }_{\mathrm{R}x}={\alpha }_{\mathrm{R}y}$). The external electric field is applied in the $x$ direction. The contributions of the $+$ and $-$ branches are shown in blue and red, respectively. The selected energies for which the Fermi lines are sketched in Figs. 4–4 are marked by arrows.

Figure 6

Fermi lines of a system with $C_{3\mathrm{v}}$ symmetry and ${\mathcal{E}}_{\mathrm{F}}>0$ ($m=m_{\mathrm{e}},{\alpha }_{\mathrm{R}}=1\mathrm{eV}\AA$ and $\lambda =5.8\mathrm{eV}{\AA }^3$). The inner line corresponds to the $+$ branch; the outer corresponds to the $-$ branch. The arrows illustrate the in-plane component of the spin expectation value; the color scale indicates the out-of-plane component.

Figure 7

Adaptive triangle method for the determination of the Fermi lines. (a) The $\vec{k}$ space is divided by a rectangular mesh (gray dots). The mesh points are connected in such a way that triangles are formed (blue lines). The red line is a complex Fermi line. (b) Detail of (a). To determine the $\vec{k}$ points on the Fermi line (red dots) an adaptive procedure is used. Triangles A and B are not intersected by the Fermi line; therefore, they are not considered further. The Fermi line passes through triangles C and D; they are divided iteratively into smaller triangles until the required precision is reached.