Tunneling Spectroscopy of a Two-Dimensionally Periodic Electron System

The tunneling current between an electron gas with a periodic potential in two dimensions and a plain two-dimensional electron system (2DES) has been studied. The strength of the periodic potential, the subband energy of the plain 2DES, and an applied in-plane magnetic field were varied, mapping the Fourier transform of the periodic wave function. Periodic peaks were observed and explained by translations in the reciprocal lattice. When the potential was strongly modulated to form an array of antidots, commensurability peaks were seen in lateral transport, but, as expected, not in tunneling.

Figure 1

Diagonal magnetoresistance of the antidot lattice (lower trace), and tunneling resistance between the antidots and the lower 2DES, with the series magnetoresistance of the antidot lattice removed (upper trace). The diagonal magnetoresistance shows a clear commensurability peak; the tunneling trace lacks it. Also shown is a $2\%$ band around the tunneling trace as a guide to the eye.

Figure 2

Tunneling measurements at zero magnetic field: (a) typical traces as a function of back-gate voltage $V_{\mathrm{b}\mathrm{g}}$. Electrons tunneling into the lower 2DES originate either from a plain upper 2DES ($V_{\mathrm{a}\mathrm{g}}=+0.02\mathrm{V}$), from a weakly modulated 2DES ($V_{\mathrm{a}\mathrm{g}}=-0.3\mathrm{V}$), or from an antidot lattice ($V_{\mathrm{a}\mathrm{g}}=-0.55\mathrm{V}$); (b) tunneling differential conductance as a function of $V_{\mathrm{b}\mathrm{g}}$ and $V_{\mathrm{a}\mathrm{g}}$. Black represents tunneling maxima. The peaks originate from 2D-2D tunneling (k and q peaks) or from the periodicity of the lattice (m and b peaks). Inset: Diagram of the device. $V_{\mathrm{s}\mathrm{g}}$ and $V_{\mathrm{s}\mathrm{b}\mathrm{g}}$ define independent contacts to each layer, $V_{\mathrm{a}\mathrm{g}}$ and $V_{\mathrm{b}\mathrm{g}}$ control the tunneling area.

Figure 3

Tunneling measurements in an in-plane magnetic field $B$. (a) Typical traces for various $V_{\mathrm{a}\mathrm{g}}$. $B$ is perpendicular to the Hall bar ($y$ direction). (b)–(e) tunneling maxima (black) as a function of $B$ and antidot-gate voltage $V_{\mathrm{a}\mathrm{g}}$. $B$ is applied at different angles $\alpha$ shown. We identify 2D-2D, k, q, m, and b peaks. Inset to (a): schematic diagram of the augmented spectral density function $B_1$ in k space. In horizontal and vertical directions, $B_1$ has peaks with periodicity close to $2\pi /p$, and, diagonally, peaks with periodicity $\sqrt{2}\pi /p$; in other directions there is no clear periodicity. [A limited set of circles was used for $A_1$ (it was assumed that those far from $k=0$ are blurred out due to deviations from perfect periodicity of the antidot lattice). The FT ${\tilde{\Phi }}_1^{\mathit{k}}({\mathit{k}}^{\prime })$ was taken to have broad peaks for $|\mathit{k}-{\mathit{k}}^{\prime }|$ near wave numbers corresponding to the carrier densities of the upper layer below and between the antidot gates (chosen to be 6 and 11 in units of $2\pi /p$, with $\mu$ set to the average of these).]

Figure 4

(a)–(d) Tunneling maxima (black) vs in-plane magnetic field $B$ and angle of the in-plane field $\alpha$ for the fixed antidot-gate voltages $V_{\mathrm{a}\mathrm{g}}$ shown at the right.