Figure 1

(a) In-plane strain breaks the valley energy degeneracy; e.g., tension along [100] results in a transfer of electrons from the [100] valley to the [010] valley. (b) Single-particle fan diagram for the AlAs 2DES using band parameters. (c) Single-particle (dashed lines) and measured (circles) $\nu =1$ QHS gap, ${}^1\Delta$, as a function of the single-particle valley splitting, $\Delta E_v$, for $n=2.55\times {10}^{11}{\mathrm{cm}}^{-2}$. ${}^1\Delta$ and $\Delta E_v$ are given both in meV (top and right axes) and also in units of the Coulomb energy $e^2/4\pi ϵl_B$, where $ϵ=10{ϵ}_0$ is the dielectric constant of AlAs and $l_B=(\hslash /eB{)}^{1/2}$ is the magnetic length; in our experiments, $B=10.5\mathrm{T}$ and $e^2/4\pi ϵl_B=18.2\mathrm{meV}$. (d) ${}^1\Delta$ vs $\Delta E_Z$ for spin Skyrmions in a single-valley 2DES. The dashed line is the single-particle Zeeman splitting. The theoretical prediction for the Skyrmion excitation energy is shown by the solid curve and the solid symbols show the experimentally determined gaps at $\nu =1$ (see text).Reuse & Permissions

Figure 2

Dependence of ${}^1\Delta$ on $\Delta E_v$ for an AlAs 2DES with $n=2.55\times {10}^{11}{\mathrm{cm}}^{-2}$ at different tilt angles. Single-particle $\Delta E_Z$ at $\nu =1$ is 1.22 meV and 1.75 meV for $\theta =0$ and $\theta =46°$, respectively. Inset shows a comparison between $\hslash {\omega }_c$ (dashed line) and ${}^1\Delta$ (squares) measured at $\theta =0$ as a function of perpendicular magnetic field (density) for a 2DES with only the [010] valley occupied ($\Delta E_v=3.4\mathrm{meV}$).Reuse & Permissions

Figure 3

Magnetoresistance of the AlAs 2DES for different values of applied strain at $T=0.3\mathrm{K}$ (main figure) and 2 K (inset). The value of $\Delta E_v$ is indicated for each trace; the bold traces correspond to the smallest magnitude of $\Delta E_v$. Traces are vertically offset from each other for clarity.Reuse & Permissions