Experimental evidence for predicted magnetotransport anomalies in rectangular superlattices

We investigate commensurability oscillations in the magnetoresistances of unstressed ungated rectangular two-dimensional superlattices of different periods. The amplitude of the commensurability oscillations in these systems exhibits a nonmonotonic dependence on the applied magnetic field, which is not present in 1D or square superlattices. Furthermore, the high and low resistance directions switch between the two axial directions of the superlattices depending on the magnetic field. Our observations are explained by the drift of the cyclotron motion guiding center along contours of a magnetic-field-dependent effective potential as put forward in a recent theory. Comparison of the data with the theoretical predictions shows good agreement. For a larger modulation amplitude, we observe a flattening of commensurability oscillation minima, which is also predicted by the calculation.

Figure 1

(a) Sketch of the L-shaped Hall bar geometry showing the extent and orientation of the superlattices. The $x$- and $y$-directions coincide with the cleavage directions of the (100) GaAs wafer. (b) Normal and (c) inverted views of a representative AFM measurement on an etched $136\mathrm{nm}\times 103\mathrm{nm}$ period superlattice after removal of the resist.

Figure 2

(Color) Left panel, middle: Magnetoresistances measured at $4\mathrm{K}$ for transport along the $136\mathrm{nm}$ ($x$-direction, thick blue) and the $103\mathrm{nm}$ ($y$-direction, thin red) period directions of a rectangular superlattice. Bottom: Magnetoresistances ${\rho }_{xx}$ (blue) and ${\rho }_{yy}$ (red) calculated using Ref. 1 for the parameters $V_x=5.6\%$ $E_F$, $V_y=3.9\%$ $E_F$, a ratio $a:b=1.32$ and $q\lambda =26$. The vertical lines indicate the calculated $B$ values of the CO minima of ${\rho }_{xx}$ for indices $k_a$ (blue dashed) and of ${\rho }_{yy}$ for indices $k_b$ (red dotted). They coincide with the zeroes of the corresponding effective potential amplitudes ${\left[V_x^{\mathrm{eff}}\right]}^2$ (blue line) and ${\left[V_y^{\mathrm{eff}}\right]}^2$ (red line) shown, in arbitrary units, at the top as function of $1∕\left(q{R}_c\right)$. Right panels: A, B, C, D, E, and F show the contours of the effective potential $V_x^{\mathrm{eff}}\cos (2\pi X∕a)+V_y^{\mathrm{eff}}\cos (2\pi Y∕b)$ at the peaks marked in the left graph by A, B, C, D, E, and F, respectively. The contours in these panels correspond to guiding center trajectories in real space.

Figure 3

Magnetoresistances measured at $4\mathrm{K}$ for transport in rectangular superlattices of three different pairs of lattice constants having the ratio 1.3: thick lines for transport along the longer periods $\left({\rho }_{xx}\right)$, thin lines for transport along the shorter periods $\left({\rho }_{yy}\right)$; periods are indicated. For clarity, the middle and the upper pair of traces have been offset by $30\Omega$ and by $60\Omega$, respectively. The three CO minima (in ${\rho }_{xx}$) for $k_a=5$ occur at the same dashed vertical line. The CO peaks in ${\rho }_{xx}$ between the $k_a=3$ and 4, indicated by downward arrows, are with increasing modulation period increasingly suppressed.

Figure 4

(Top) Magnetoresistances measured at $4\mathrm{K}$ for transport along the $136\mathrm{nm}$ (thick line) and $103\mathrm{nm}$ (thin line) period of a rectangular superlattice on heterostructure B. The vertical lines mark the calculated magnetic field values of the CO minima for indices $k_a$ and $k_b$. Note that the broad CO minima at $k_a=2$ and $k_b=3$ are flattened. Bottom: Magnetotransport calculation using Ref. 1 with $V_x=9\%$ $E_F$, $V_y=7\%$ $E_F$, the ratio $a:b=1.32$ and $q\lambda =37$.

Figure 5

The Magnetoresistances of Fig. 2 (thin) and Fig. 4 (thick) plotted versus $1∕\left(q{R}_c\right)$, so CO minima are aligned. Note that the SdH oscillations for the CO minima at $k_a=2$ and $k_b=2,3$ are suppressed for the thick lines as compared to the thin lines.