Commensurability Oscillations of Composite Fermions Induced by the Periodic Potential of a Wigner Crystal

When the kinetic energy of a collection of interacting two-dimensional (2D) electrons is quenched at very high magnetic fields so that the Coulomb repulsion dominates, the electrons are expected to condense into an ordered array, forming a quantum Wigner crystal (WC). Although this exotic state has long been suspected in high-mobility 2D electron systems at very low Landau level fillings ($\nu \ll 1$), its direct observation has been elusive. Here we present a new technique and experimental results directly probing the magnetic-field-induced WC. We measure the magnetoresistance of a bilayer electron system where one layer has a very low density and is in the WC regime ($\nu \ll 1$), while the other (“probe”) layer is near $\nu =1/2$ and hosts a sea of composite fermions (CFs). The data exhibit commensurability oscillations in the magnetoresistance of the CF layer, induced by the periodic potential of WC electrons in the other layer, and provide a unique, direct glimpse at the symmetry of the WC, its lattice constant, and melting. They also demonstrate a striking example of how one can probe an exotic many-body state of 2D electrons using equally exotic quasiparticles of another many-body state.

Figure 1

Overview of our measurements. (a) Our bilayer system has a high-density top layer which hosts a CF Fermi sea at high magnetic fields near its filling factor ${\nu }_T=1/2$. The bottom layer has a much lower density so that it is at a very small filling factor (${\nu }_B\ll 1$), thus allowing a WC to form with lattice constant $a=\sqrt{2/\sqrt{3}{n}_B}$ ($n_B$ is the bottom-layer density). The top layer feels a periodic potential modulation from the bottom WC layer’s charge density; $l_B$ is the magnetic length. (b) As the magnetic field is swept near the top layer’s $1/2$ filling, the CFs in the top layer execute cyclotron motion, leading to commensurability maxima in the magnetoresistance of the top layer when the CF cyclotron orbit encircles $1,3,7,\dots$ lattice points.

Figure 2

(a) $R_{xx}$ vs $1/{\nu }_T$ traces for the bilayer sample, measured as the bottom-layer density is reduced via applying negative back-gate voltage. The bottom-layer densities ($n_B$) are 19, 1.7, and zero (in units of ${10}^9{\mathrm{cm}}^{-2}$) for $V_{\mathrm{BG}}=0$, $-45$, and $-90\mathrm{V}$, respectively. The top-layer density ($n_T$) for these traces varies slightly (see Fig. S1 in Supplemental Material [24]); the top $x$ axis is normalized to represent $1/{\nu }_T$, where ${\nu }_T$ is the top-layer filling factor [the lower axis (field) scale is only for the middle trace]. The triangles mark the expected positions for the $i=1$, 3, 7 CF cyclotron orbit commensurability with a triangular WC potential [Fig. 1]. These positions are based on the measured $n_T$ and $n_B$. For the lowest trace, the triangles fall far from ${\nu }_T=1/2$, making the expected commensurability-induced resistance maxima indiscernible because of the strong FQHE features. (b) Temperature dependence of $R_{xx}$ traces near ${\nu }_T=1/2$. All traces were taken for $n_B=1.7\times 10^9{\mathrm{cm}}^{-2}$ ($V_{\mathrm{BG}}=-45\mathrm{V}$). For the $T=114\mathrm{mK}$ trace, we also show the expected ($i=1$, 2, 4, 9) commensurability positions if the WC had square symmetry. (c) $R_{xx}$ vs $1/{\nu }_T$ traces are shown near ${\nu }_T=1/2$ for different $n_B$, corresponding to $-60\le V_{\mathrm{BG}}\le -20\mathrm{V}$. In all traces for $0.6\le n_B\le 1.7\times 10^9{\mathrm{cm}}^{-2}$ (corresponding to $440\ge a\ge 260\mathrm{nm}$), $R_{xx}$ exhibits a strong maximum at the expected positions for the $i=1$ commensurability for ${\nu }_T>1/2$. Similar to (a), the lower axis (field) scale is only for the $n_B=1.7\times 10^9$ trace. In (a)–(c), two traces are shown for each condition: The black is for the up sweep of $B$ and the red is for the down sweep. The origin of the slight hysteresis seen in some of the traces is unknown. Traces for different conditions are shifted vertically for clarity.