AlAs two-dimensional electrons in an antidot lattice: Electron pinball with elliptical Fermi contours

We report ballistic transport measurements in a two-dimensional electron system confined to an AlAs quantum well and patterned with square antidot lattices of period $a=0.6$, 0.8, 1.0, and $1.5\mu \mathrm{m}$. In this system two in-plane conduction-band valleys with elliptical Fermi contours are occupied. The low-field magnetoresistance traces exhibit peaks corresponding to the commensurability of the cyclotron orbits and the antidot lattice. From the dependence of the position of the peak associated with the smallest commensurate orbit on electron density and $a$, we deduce the ratio of the longitudinal and transverse effective masses $m_l∕m_t=5.2\pm 0.4$, a fundamental parameter for the anisotropic conduction bands in AlAs.

Figure 1

(a) Micrograph of the AD lattice region with period $a=0.8\mu \mathrm{m}$. (b) The Fermi contours of AlAs in-plane valleys $X$ and $Y$ in $k$ space. The Fermi wave vectors $k_{F,X}$ and $k_{F,Y}$ are indicated. (c) The first four commensurate orbits for the $X$ and $Y$ valleys that give rise to peaks in magnetoresistance for current along the $x$ direction; these orbits have diameters in the $y$ direction that are equal to a multiple integer of the AD lattice period (see text); this integer is given by the index $i$ in $Xi$ and $Yi$.

Figure 2

Low-field MR traces for all four AD regions with periods (from top to bottom) equal to $a=0.6$, 0.8, 1.0, and $1.5\mu \mathrm{m}$. Traces are offset vertically for clarity, except for the bottom one. The Hall bar with the different AD regions is schematically shown on the right.

Figure 3

Low-field MR traces for the AD region with period $a=0.8\mu \mathrm{m}$ for $V_G=-0.1\text{to}0.15\mathrm{V}$ (from top to bottom), corresponding to a linear variation of the density $n_T$ from $2.27\times {10}^{11}\text{to}3.53\times {10}^{11}∕{\mathrm{cm}}^2$. Traces are offset for clarity, except for the bottom one.

Figure 4

MR obtained from numerical simulations (smooth curves are guides to the eye). Vertical lines indicate the expected positions of the peaks for orbits $X1$, $X2$, $X3$, and $Y1$, based on Fig. 1c. We assumed equal densities for the two valleys and a current along the $x$ direction. $B_0$ is the magnetic field of the first commensurate orbit if the Fermi contour were circular. Inset: Simulation snapshot showing various types of trajectories, chaotic, pinned, and skipping orbits, for $X$-valley electrons when the $X1$ peak occurs $(B=\sqrt[4]{m_l∕m_t}{B}_0)$.

Figure 5

Summary of the density dependence of $B_{P,A}^2$ for all four AD regions; $B_{P,A}$ is the position of peak $\mathrm{A}$ observed in MR traces. The straight lines are linear fits using Eq. (2). The error bars reflect the uncertainty in the peak positions which were determined by subtracting second-order polynomial backgrounds from the MR traces. Inset: Slope $\left(\beta \right)$ of the $B_{P,A}^2$ vs $n_T$ lines of the main figure are plotted as a function of $a^{-2}$. The dashed line is a linear fit to the data.

Figure 6

Summary of the density dependence of $B_P^2$ of peaks $\mathrm{A}$ and $\mathrm{B}$ for the AD regions: $a=\left(\mathrm{a}\right)$ 0.6 and (b) $0.8\mu \mathrm{m}$. The lines marked $X1$ are linear fits to peak $\mathrm{A}$ using Eq. (2). Lines $X2$, $X3$, and $Y1$ are the predicted peak positions calculated using equations similar to Eq. (2) (see text).