Full spin-orbit coefficient in III-V semiconductor wires based on the anisotropy of weak localization under in-plane magnetic field

Because of the one-dimensional confinement of electron momentum in narrow semiconductor wires, spin relaxation is suppressed irrespective of the presence of spin-orbit (SO) interaction. In quantum transport, weak localization corrections to conductivity are reflected as suppressed spin relaxation, which makes quantification of the SO strength difficult because quantum correction theory requires weak antilocalization when evaluating SO coefficients. Narrow wires with strong SO interaction are potential platform for Majorana particles and parafermions for topological electronics and quantum computation, so revealing the SO strength in semiconductor wire structures is beneficial. Herein, we present quantification of the full SO coefficient under weak localization in InGaAs-based narrow wires. Using anisotropic weak localization observed under an in-plane external magnetic field with various orientations, one can ascertain the relative ratio between Rashba ($\alpha$) and linear Dresselhaus (${\beta }_1$) SO coefficients with no fitting. Furthermore, we find that widely tuning the potential profile of the quantum well through the top gate can expose a Rashba-predominant region in magnetoconductance, where the $\alpha$ value can be extracted reliably from two-dimensional quantum correction theory. Finally, we quantify the full SO coefficients including Rashba, linear Dresselhaus, and cubic Dresselhaus terms in the wire.

Figure 1

(a) Conduction and valence bands as well as electron probability density in a (001) ${\mathrm{In}}_{0.53}{\mathrm{Ga}}_{0.47}\mathrm{As}/{\mathrm{In}}_{0.52}{\mathrm{Al}}_{0.48}\mathrm{As}$ two-dimensional electron gas calculated using the Poisson-Schrödinger equation. (b) Modulation of the conduction band profile in various sheet carrier densities modulated by the top gate. (c) Top-gate voltage dependence of sheet carrier density at $T=1.6\mathrm{K}$. (d) Device configuration for parallel wire structures along [−110], [010], and [110] crystal orientations with applied perpendicular and in-plane external magnetic fields. Inset shows a microscope image for the parallel wires. (e) Schematic illustration of Rashba and Dresselhaus spin-orbit induced effective magnetic fields in the studied crystal orientations. Panels (f) and (g) show the orientations of Rashba (red), Dresselhaus (blue), and in-plane (purple) magnetic fields for [−110] wire. The in-plane field is either (f) parallel or (g) perpendicular to the spin-orbit field, inducing different configurations for the total magnetic field direction (green arrows).

Figure 2

Color-coded magnetoconductance as functions of perpendicular magnetic field $B_z$ and in-plane magnetic field angle ${\theta }_{\mathrm{in}}$ for (a) [−110], (b) [010], and (c) [110] wires at $B_{\mathrm{in}}=1.5$ T. (d)–(f) Schematic illustration of Rashba (red), Dresselhaus (blue), and total spin–orbit fields (pink, light green, and light blue arrows) for (d) [−110], (e) [010], and (f) [110] wires. (g) Magnetoconductance of [010] wire with maximum (red) and minimum (blue) weak localization amplitude, respectively, at ${\theta }_{\mathrm{in}}={35}^{\circ }$ and ${125}^{\circ }$. (h) Conductance amplitude $\mathrm{\Delta }\sigma$ at fixed $B_z=20$ mT as a function of ${\theta }_{\mathrm{in}}$ for [010] wire. Dashed lines correspond to the maximum and minimum conductance amplitude $\mathrm{\Delta }{\sigma }_{\max }$ and $\mathrm{\Delta }{\sigma }_{\min }$. (i) Polar plot of normalized conductance $\mathrm{\Delta }\tilde{\sigma }$ as a function of in-plane field angle for [−110] (red), [010] (green), and [110] (blue) wires.

Figure 3

(a) Polar plot of normalized conductance $\mathrm{\Delta }\tilde{\sigma }$ as a function of in-plane field angle under different top-gate voltage: $V_g=-0.55$, −0.45, and +0.10 V under $B_{\mathrm{in}}=2.0$ T. (b) Polar plot of numerically computed conductance for different $\alpha /{\beta }_1$ ratio: $\alpha /{\beta }_1=-0.23$, −1, and −1.73. (c) Gate voltage dependence of $\alpha /{\beta }_1$ ratio for [010] wire under $B_{\mathrm{in}}=2.0$ T.

Figure 4

(a) Magnetoconductance measured in a Hall bar at $T=0.3\mathrm{K}$ in a Rashba-predominant region by setting the sheet carrier density $N_s$ between $2.37\times {10}^{16}$ and $2.58\times {10}^{16}{\mathrm{m}}^{-2}$. Solid colored lines correspond to fits using universal modeling of WAL corrections based on the real-space simulation developed by Sawada et al. [27]. (b) Extracted spin-orbit length $L_{\mathrm{SO}}$ and mean free path $L_{\mathrm{tr}}$ as a function of carrier density. The $L_{\mathrm{tr}}>L_{\mathrm{SO}}$ corresponds to the ballistic regime for WAL corrections. (c) Rashba spin-orbit coefficient $\left|\alpha \right|$ as a function of carrier density.

Figure 5

(a) Carrier density dependence of Rashba $\left|\alpha \right|$ parameters (open blue squares) in a Hall bar and $\alpha /{\beta }_1$ ratio (open red circles) obtained from [010] wire. Full spin-orbit coefficients $\left|\alpha \right|, |{\beta }_1|$, and $|{\beta }_3|$, in wires are shown as blue, red, and green squares. (b) Carrier density dependence of spin-orbit induced effective field angle ${\theta }_{\mathrm{eff}}$ for [010] wire and fit (solid green line) using Eq. (1). (c) Magnetoconductance originating from the Dresselhaus spin-orbit interaction in a Hall bar at $T=0.3\mathrm{K}$. The carrier density at which the Rashba SO coefficient is vanished $N_s=1.38\times {10}^{16}{\mathrm{m}}^{-2}$. The solid blue line corresponds to fitting using the Sawada model.