Valley Splitting of AlAs Two-Dimensional Electrons in a Perpendicular Magnetic Field

By measuring the angles at which the Landau levels overlap in tilted magnetic fields (the coincidence method), we determine the splitting of the conduction-band valleys in high-mobility two-dimensional electrons confined to AlAs quantum wells. The data reveal that, while the valleys are nearly degenerate in the absence of magnetic field, they split as a function of perpendicular magnetic field. The splitting appears to depend primarily on the magnitude of the perpendicular component of the magnetic field, suggesting electron-electron interaction as its origin.

Figure 1

Magnetoresistivity ${\rho }_{xx}$ as a function of perpendicular magnetic field $B_{\perp }$ for AlAs 2D electrons in sample S1. At low $B_{\perp }$, ${\rho }_{xx}$ in the inset shows strong minima at every fourth filling factor $\nu$, indicating a fourfold degeneracy of the Landau levels (two for spin and two for valley). At higher $B_{\perp }$, all minima become stronger than their ($\nu +4$) counterpart, implying that both spin and valley splitting increase with $B_{\perp }$.

Figure 2

Inset: An example of the Fourier transform of ${\rho }_{xx}$ vs $1/B_{\perp }$ for sample S2 at $n=7.9\times {10}^{11}{\mathrm{c}\mathrm{m}}^{-2}$. The presence of a peak at the frequency equal to one-quarter of frequency associated with the total density confirms the degeneracy of the spin and valleys at low $B_{\perp }$. Main: Fourier transform peak frequencies as a function of the front-gate bias.

Figure 3

(a) Magnetoresistivity ${\rho }_{xx}$ vs $1/\cos \theta$ and $\nu$ measured in sample S3 at $T=30\mathrm{m}\mathrm{K}$. Blue and red colors represent small and large values of ${\rho }_{xx}$, respectively. To increase the visibility of ${\rho }_{xx}$ oscillations over the whole field range, we scaled ${\rho }_{xx}$ by a Dingle-like factor, $\exp (1/6.4B_{\perp })$. A striking alternating diamond pattern emerges in the data. This pattern has an approximate right-left symmetry, has a periodicity of four in $\nu$, and changes little over the whole field range. (b) Fan diagram that qualitatively accounts for the data. In this diagram, $\Delta E_V$ scales linearly with $B_{\perp }$ and is independent of $B_{||}$.

Figure 4

Products $m^{*}\Delta E_V$ and $\Delta E_V$ (assuming $m^{*}=0.46$) as a function of $B_{\perp }$ with typical error bars. Solid line is the least squares linear fit for the high field valley splitting.