Experimental Observation of the Spin-Hall Effect in a Two-Dimensional Spin-Orbit Coupled Semiconductor System

We report the experimental observation of the spin-Hall effect in a 2D hole system with spin-orbit coupling. The 2D hole layer is a part of a $p\mathrm{\text{−}}n$ junction light-emitting diode with a specially designed coplanar geometry which allows an angle-resolved polarization detection at opposite edges of the 2D hole system. In equilibrium the angular momenta of the spin-orbit split heavy-hole states lie in the plane of the 2D layer. When an electric field is applied across the hole channel, a nonzero out-of-plane component of the angular momentum is detected whose sign depends on the sign of the electric field and is opposite for the two edges. Microscopic quantum transport calculations show only a weak effect of disorder, suggesting that the clean limit spin-Hall conductance description (intrinsic spin-Hall effect) might apply to our system.

Figure 1

Basic characteristics of the LED. (a) The schematic cross section of the coplanar $p\mathrm{\text{−}}n$ junction LED device. At forward bias of order of the GaAs band gap, electrons move from the 2DEG to the 2DHG where they recombine. The highest intensity of the emitted light is in the $p$ region near the junction step edge. (b) Numerical simulations of the conduction and valence band profiles along the $⟨001⟩$ growth direction ($z$ axis) in the unetched part of the wafer near the step edge. Black lines correspond to an unbiased $p\mathrm{\text{−}}n$ junction and red lines to a forward bias of 1.5 V across the junction. In the detailed image of the upper $(\mathrm{A}\mathrm{l}\mathrm{G}\mathrm{a})\mathrm{A}\mathrm{s}/\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}$ interface (with the conduction band shifted down in energy for clarity), we indicated possible sub-GaAs-gap radiative recombinations with the 2DHG involving either 3D electrons in the conduction band (red arrow) or impurity states (black arrow). The black line corresponds to unbiased wafer 1 and the blue line to wafer 2. (c),(d) Photoluminescence spectra (black lines) and electroluminescence spectra (red lines) at low (lower curves) and high (higher curves) bias measured in wafers 1 and 2 at 4.2 K. Impurity (I), excitonic (X), and 3D electron to 2D hole transitions (A, B, C) are identified. We focus on peak B, which has small overlap with the I and X lines.

Figure 2

Spin-polarization in the absence of the SHE driving current. (a) Theoretical energy spectrum of the 2DHG along the $k_y$ component of the wave vector (parallel to $I_{\mathrm{L}\mathrm{E}\mathrm{D}}$) of the heavy-hole (HH) and the light-hole (LH) subband, spin-split due to the SO coupling. Unbiased wafer 1 was assumed in the calculations. A small shift of the subband energies towards the top of the valence band is expected for wafer 2 and under bias. Schematic color plot demonstrates circularly polarized light emission induced by transitions from the asymmetrically occupied conduction band to the $\mathrm{H}\mathrm{H}+$ subband. (b) Theoretical spin-polarization components along $k_y$ in the $\mathrm{H}\mathrm{H}+$ and $\mathrm{H}\mathrm{H}-$ subbands. (An infinitesimal magnetic field along $z$ direction was added to break the degeneracy at $\mathbf{k}=0$). Except for a small region near $k_y=0$, spins are oriented in plane and perpendicular to the $k$ vector. (c) Circular polarization, defined as the relative difference between intensities of left and right circularly polarized light, plotted as a function of the in-plane magnetic field along the $x$ axis, measured in wafer 1. The observation angle is $\alpha =85°$, which is the maximum detection angle allowed in our experimental setup. The experiment measures, to high accuracy, the $x$ component of the light polarization vector which is proportional to $⟨s_x⟩$. (d) Spectral plot of the circular polarization near peak B for field $\pm 3\mathrm{T}$. (e) Same as (c) for magnetic and light detection axis along the normal to the 2DHG plane. (f) Same as (d) for field 0 and $\pm 3\mathrm{T}$. At zero magnetic field the $z$ component of the polarization vanishes. All measurements are done at 4.2 K.

Figure 3

The SHE experiment. (a) Scanning electron microscopy image of the SHE LED device. The top (LED 1) or bottom (LED 2) $n$ contacts are used to measure the EL at opposite edges of the 2DHG $p$ channel parallel to the SHE driving current $I_p$. (b) Polarization along $z$ axis measured with active LED 1 for two opposite $I_p$ current orientations. Spectral region of peak B of the high bias EL curve of wafer 1 is shown. Nonzero and opposite out-of-plane polarization for the two $I_p$ orientations demonstrates the SHE. (c) Polarization along $z$ axis measured with fixed $I_p$ current and for biased LED 1 or LED 2. The data show opposite polarizations at opposite edges of the 2DHG channel confirming the SHE origin of the measured signal. (d) Theoretical intrinsic SHE conductivity in units of $e/8\pi$ versus quasiparticle lifetime broadening and 2D hole density. Parameters corresponding to our 2DHG, indicated by a white dot, fall into the strong intrinsic SHE part of the theoretical diagram.