Parallel magnetic-field tuning of valley splitting in AlAs two-dimensional electrons

We demonstrate that, in a quasi-two-dimensional electron system confined to an AlAs quantum well and occupying two conduction-band minima (valleys), a parallel magnetic field can couple to the electrons’ orbital motion and tune the energies of the two valleys by different amounts. The measured density imbalance between the two valleys, which is a measure of the valley susceptibility with respect to parallel magnetic field, is enhanced compared to the predictions of noninteracting calculations, reflecting the role of electron-electron interaction.

Figure 1

(Color online) Diagrams showing: (a) Experimental setup; (b) and (c) valley occupations at zero (light green/gray) and finite (dark green/gray) parallel magnetic fields; (d) and (f) Fermi contour distortions due to parallel magnetic field for the [100] and [010] valleys; (e) and (g) density of states (DOS) for the [100] and [010] valleys before (light green/gray) and after (dark green/gray) the application of parallel magnetic field.

Figure 2

(Color online) (a) Resistance vs strain traces for an 11-nm-wide AlAs QW taken at $\nu =3$ for different parallel magnetic fields. The “balanced point” for a given $B_{\parallel }$ is marked by a closed circle on the corresponding trace. (b) Landau-level fan diagram for $B_{\parallel }=3.4\text{T}$ as a function of $ϵ$.

Figure 3

(Color online) (a) Resistance vs strain traces for the same sample as in Fig. 2 but at a higher density, taken at different fillings for a fixed tilt angle, $\theta =74.5°$. (b) Landau-level fan diagram as a function of $ϵ$ for $\nu =6$. (c) Valley polarization $\left(P_V\right)$ as a function of $ϵ$ at different fillings. The intercepts of the lines with the $y$ axis give $P_V$ (or alternatively, $\Delta n$) for the corresponding values of $B_{\parallel }$. Inset shows $\Delta n$ as a function of $B_{\parallel }^2$.

Figure 4

(Color online) (a) Summary of density imbalance as function of $ϵ$ for three densities for the 11-nm-wide QW. (b) and (c) Summary of density imbalance as function of $B_{\parallel }^2$ for three densities for the 11-nm-wide and 15-nm-wide AlAs QWs. Insets show the valley susceptibilities, normalized to their band values, with respect to two variables, $ϵ$ and $B_{\parallel }^2$.