Figure 1

(a) Schematic drawing of the sample with voltage probes for measuring modulated (ULSL) and unmodulated (reference) part. (b),(c) Scanning electron micrographs of the EB-resist gratings that introduce strain-induced potential modulation. Darker areas correspond to the resist. A standard line-and-space pattern (b) was utilized for samples 1, 3, and 4. Sample 2 employed a patterned grating (c) designed to partially relax the strain.Reuse & Permissions

Figure 2

(Color online) Sample parameters as a function of the electron density $n_e$, varied either by LED illumination (open symbols) or by back-gate voltage (solid symbols). (a) Modulation amplitude $V_0$. (b) Mobility $\mu$. (c) Damping parameter ${\mu }_{\mathrm{W}}$ of CO. Quantum mobility ${\mu }_{\mathrm{s}}$ for samples 1 and 2 are also plotted by $\times$ and $+$, respectively. Inset in (a) shows $\Delta {\rho }_{xx}^{\mathrm{osc}}∕{\rho }_0$ experimentally obtained by subtracting a slowly varying background from the magnetoresistance trace (for sample 2 at $n_e=2.20\times {10}^{15}{\mathrm{m}}^{-2}$, shown by solid trace) and calculated by Eq. (2) using $V_0$ and ${\mu }_{\mathrm{W}}$ as fitting parameters (dotted trace, showing almost perfect overlap with the experimental trace).Reuse & Permissions

Figure 3

(Color online) Fermi surface in the $x\text{−}{p}_x\text{−}{p}_y$ space for (a) $\beta =0$, (b) $0<\beta <1$, and (c) $\beta =1$. Each electron orbit is specified by the cross section of the Fermi surface by a constant-$p_y$ plane. Streaming orbits are present in the shaded area.Reuse & Permissions

Figure 4

(Color online) Functions $F\left(\beta \right),G\left(\beta \right),\varphi \left(\beta \right)$ (thin dotted, solid, and dash-dotted lines, respectively, left axis) and ${\beta }^{2}G\left(\beta \right)$ (thick solid line, right axis).Reuse & Permissions

Figure 5

(Color online) Magnetoresistance traces for various values of $n_e$. Selected values of $n_e$ are noted in the figure (in ${10}^{15}{\mathrm{m}}^{-2}$). Dotted traces indicate that the $n_e$ is attained by LED illumination. Note that the vertical scale is expanded by five times for sample 4.Reuse & Permissions

Figure 6

(Color online) Replot of Fig. 5 with abscissa normalized by the extinction field $B_{\mathrm{e}}$ and ordinate by the sample-parameter-dependent prefactor in Eq. (12), $\alpha {\mu }_{\mathrm{W}}\mu V_0^{5∕2}{a}^{-2}{n}_e^{-3∕2}$, with the coefficient $\alpha \equiv 4\sqrt{2}{\pi }^{-1}{\Phi }_0^{-1∕2}{\mu }_{\mathrm{B}}^{*-5∕2}\simeq 4.04\times {10}^7{\mathrm{T}}^2{\mathrm{meV}}^{-5∕2}{\mathrm{m}}^{-1}$. Vertical scale is expanded twice for sample 4. The function ${\beta }^{2}G\left(\beta \right)$ is also plotted for comparison.Reuse & Permissions

Figure 7

(Color online) (a) Magnetoresistance traces for sample 4 with the inflection point $B_{\inf }$ marked by downward open triangles. Traces are offset proportionally to the change in $n_e$. Selected values of $n_e$ in ${10}^{15}{\mathrm{m}}^{-2}$ are noted in the figure. (b) Illustration of the procedure to pick up $B_{\inf }$ (an example for $n_e=2.33\times {10}^{15}{\mathrm{m}}^{-2}$). The point at which the second derivative $(d^2∕d{B}^2)(\Delta {\rho }_{xx}∕{\rho }_0)$ (solid curve, right axis) crosses zero upward (marked by open downward triangle) is identified as $B_{\inf }$. $B_{\inf }$ is marked also on $\Delta {\rho }_{xx}∕{\rho }_0$ (dotted curve, left axis). Shaded area indicates the contribution from SO. (c) Plot of $B_{\inf }$ versus $B_{\mathrm{e}}$ calculated by Eq. (1) using experimentally obtained $V_0$. The line represents $B_{\inf }=B_{\mathrm{e}}$.Reuse & Permissions

Figure 8

(Color online) Illustration of $\mathbf{E}\times \mathbf{B}$ drift velocity $v_{\mathrm{d}}$ affecting the electrons during the cyclotron motion. Orbits are depicted neglecting the modification due to the modulation $V\left(x\right)=V_0\cos \left(qx\right)$ (drifting movement and slight variation of the velocity depending on $x$) for simplicity. Top diagrams represent slightly larger $B$ than bottom ones for both (a) and (b). On averaging $v_{\mathrm{d},y}$ along an orbit, most contribution comes from minimum- and maximum-$x$ edges, as shown by solid arrows in the figure. Open arrows indicate the direction of $E_x=(q{V}_0∕e)\sin \left(qx\right)$ at the edges. (a) For $B$ large enough so that electrons can travel cycles before being scattered. Depending on $B,v_{\mathrm{d}}$ at both edges are constructive (top diagram, $2R_{\mathrm{c}}∕a=n+1∕4$ with $n$ integer) or destructive (bottom diagram, $2R_{\mathrm{c}}∕a=n-1∕4$), resulting in maxima and minima in the magnetoresistance, respectively. (b) For small $B$ so that electrons are scattered before completing a cycle. The interrelation of $v_{\mathrm{d},y}$ at both edges is not simply determined by $B$. The edges affect the magnetoresistance independently.Reuse & Permissions

Figure 9

(Color online) Replot of magnetoresistance traces normalized by the prefactor in Eq. (22), ${\alpha }^{\prime }{\mu }^2{V}_0^2{a}^{-1}{n}_e^{-3∕2}$, with ${\alpha }^{\prime }={(\pi ∕2)}^{1∕2}{\Phi }_0^{-1}{\mu }_{\mathrm{B}}^{*-2}=4.06\times {10}^{14}\mathrm{T}{\mathrm{meV}}^{-2}{\mathrm{m}}^{-2}$, after subtracting the contribution from SO, $\delta {\rho }_{xx}^{\mathrm{SO}}∕{\rho }_0$ in Eq. (12). Contribution attributable to drift velocity of incompleted cyclotron orbits is given by $\mid B\mid$, which is also plotted by dash-dotted line.Reuse & Permissions