Electrical Control of the Zeeman Spin Splitting in Two-Dimensional Hole Systems

Semiconductor holes with strong spin-orbit coupling allow all-electrical spin control, with broad applications ranging from spintronics to quantum computation. Using a two-dimensional hole system in a gallium arsenide quantum well, we demonstrate a new mechanism of electrically controlling the Zeeman splitting, which is achieved through altering the hole wave vector $k$. We find a threefold enhancement of the in-plane $g$-factor $g_{\parallel }(k)$. We introduce a new method for quantifying the Zeeman splitting from magnetoresistance measurements, since the conventional tilted field approach fails for two-dimensional systems with strong spin-orbit coupling. Finally, we show that the Rashba spin-orbit interaction suppresses the in-plane Zeeman interaction at low magnetic fields. The ability to control the Zeeman splitting with electric fields opens up new possibilities for future quantum spin-based devices, manipulating non-Abelian geometric phases, and realizing Majorana systems in $p$-type superconductor systems.

Figure 1

(a) Schematic of the GaAs hole quantum well used in this Letter, where ${\psi }_{H1}$ is the heavy hole wave function and $V(z)$ is the confinement potential controlled via electric fields ${\overrightarrow{F}}_{\text{front}}$ and ${\overrightarrow{F}}_{\mathrm{back}}$ applied using the front and back gates, respectively. Here, the total electric field $\overrightarrow{F}=F_z\widehat{z}\equiv ({\overrightarrow{F}}_{\text{front}}+{\overrightarrow{F}}_{\mathrm{back}})/2=0$. (b) The device is tilted at an angle $\theta$ with respect to the applied magnetic field $\mathbf{B}$. (c)–(e) Magnetoresistance oscillations ${\rho }_{xx}$ of the 2D holes in a symmetric quantum well at various tilt angles for three hole densities $p$. Because of the in-plane magnetic field, a resistance minimum gradually evolves into a maximum or vice versa, for example, at $p=1\times {10}^{11}{\mathrm{cm}}^{-2}$ and $\nu =15$ for angles $\theta =0°$, 63.5°, 78.9° (see dashed box). (d)–(e) The effect of the in-plane magnetic field on the resistance becomes more pronounced as the 2D hole density increases: the transition between a resistance minimum and maximum occurs at smaller angles at higher densities (see dashed box). The traces have been offset for clarity.

Figure 2

(a)–(c) Resistance oscillations $\mathrm{\Delta }{\overline{\rho }}_{xx}=\mathrm{\Delta }{\overline{\rho }}_{xx}(B_{\parallel })\equiv ({\rho }_{xx}^{\nu }-{\rho }_{xx}^{\nu +1})/({\rho }_{xx}^{\nu }+{\rho }_{xx}^{\nu +1})$ at various filling factors $\nu$ for different $p$. (d)–(f) Schematic of the energy band dispersion $E(k_x,k_y)$, showing how the $k$-dependent Zeeman splitting, where $k\equiv \sqrt{k_x^2+k_y^2}$, changes the effective masses $m_{+}$ and $m_{-}$ in the two bands. (g)–(i) Schematic of the Landau levels corresponding to the band structure in (d)–(f) for an even $\nu$. The shaded regions refer to the filled Landau levels. As $B_{\parallel }$ is increased from 0 T to $B_{\parallel }=B_1>0\mathrm{T}$, the Landau level spacing in the two spin-split bands changes due to the different masses. (j) The in-plane magnetoresistance $\mathrm{\Delta }{\overline{\rho }}_{xx}$ arises from the change in the density of states at the Fermi energy as illustrated in (d)–(f). (k) The difference in the hole density of the two spin-split bands $\mathrm{\Delta }p$ as a function of $B_{\parallel }$ for different $p$, as indicated by the three different colors. (Inset) Shows $g_{\parallel }{m}^{*}$ (solid black line with solid circles) extracted using Eq. (1). Using the effective masses obtained from $6\times 6$ $\mathit{k}·\mathit{p}$ calculations [43], we find $g_{\parallel }$ (solid red line with open circles), which increases linearly with density.

Figure 3

(a) In the absence of Rashba splitting ($F_z=0\mathrm{MV}/\mathrm{m}$), the magnetoresistance minima change into maxima (or vice versa) when $\theta$ is increased from 0° to 62°, as highlighted in the dashed box for $\nu =40$. (b) At $F_z=0.43\mathrm{MV}/\mathrm{m}$, the Shubnikov–de Haas oscillations are far less affected by $\theta$: when $\theta$ is increased from 0° to 62° the positions of the minima and maxima remain almost unchanged, and only when $\theta$ is large ($\theta >80°$) do the resistance minima evolve into maxima (the dashed box highlights $\nu =36$). Here, $p=2\times {10}^{11}{\mathrm{cm}}^{-2}$. The shape of the potential profile and ground state envelope wave function are shown in green and red. The traces are offset for clarity.