Probing the Melting of a Two-Dimensional Quantum Wigner Crystal via its Screening Efficiency

One of the most fundamental and yet elusive collective phases of an interacting electron system is the quantum Wigner crystal (WC), an ordered array of electrons expected to form when the electrons’ Coulomb repulsion energy eclipses their kinetic (Fermi) energy. In low-disorder, two-dimensional (2D) electron systems, the quantum WC is known to be favored at very low temperatures ($T$) and small Landau level filling factors ($\nu$), near the termination of the fractional quantum Hall states. This WC phase exhibits an insulating behavior, reflecting its pinning by the small but finite disorder potential. An experimental determination of a $T$ vs $\nu$ phase diagram for the melting of the WC, however, has proved to be challenging. Here we use capacitance measurements to probe the 2D WC through its effective screening as a function of $T$ and $\nu$. We find that, as expected, the screening efficiency of the pinned WC is very poor at very low $T$ and improves at higher $T$ once the WC melts. Surprisingly, however, rather than monotonically changing with increasing $T$, the screening efficiency shows a well-defined maximum at a $T$ that is close to the previously reported melting temperature of the WC. Our experimental results suggest a new method to map out a $T$ vs $\nu$ phase diagram of the magnetic-field-induced WC precisely.

Figure 1

Overview of transport and penetration current results. The inset shows a schematic of measurement configuration. The yellow layers are the top and bottom gates, and the blue layer is the 2DES. An ac voltage ($V_{\mathrm{ac}}$) applied to the bottom gate generates the source electric field $E_0$ between this gate and the 2DES. The penetration electric field $E_P$ reaches the top gate and induces $I_P$, which is measured by a lock-in amplifier in ammeter mode (A). The black trace is the longitudinal magnetoresistance ($R_{xx}$); the gray trace shows $R_{xx}$ reduced by a factor of 30 at high fields. The blue trace is the Hall resistance ($R_{xy}$). The red trace is the penetration current ($I_P$). The $I_P$ maximum observed at the filling factor between $2/9$ and $1/5$, where the reentrant WC is present, is marked by a green arrow. The minima between the high-field WC and the FQHSs at $\nu =2/9$ and $1/5$ are indicated by the red triangles.

Figure 2

(a) Evolution of $I_P$ with magnetic field ($B$) and temperature ($T$). (b)–(e) $T$ dependence of $I_P$ for various phases of the 2DES: (b) compressible, electron liquid states at $\nu =1/2,1/4$, and $3/10$; (c) incompressible FQHSs at $\nu =1/3$ and $1/5$; (d) reentrant Wigner crystal (RWC) phase at $\nu =0.21$; and (e) WC phase at $\nu =0.17$. In (d) and (e), the $T$ dependence of $R_{xx}$ (gray trace) at the same $\nu$ is also plotted. (b) and (c) share the same $I_P$ scale on the left. (d) and (e) have the same $I_P$ scale on the left, and $R_{xx}$ scale on the right (in gray). The arrows from panels (b), (c), (d) and (e) to panel (a) indicate the $B$ positions for different phases.

Figure 3

The measured critical temperature ($T_C$) vs $\nu$. The red and blue circles are the data measured at $n=4.2$ and $6.0\times {10}^{10}{\mathrm{cm}}^{-2}$, respectively. The data represented by solid circles are measured by sweeping $B$ at fixed $T$ [Fig. 2], while the empty circles are measured by sweeping $T$ at fixed $B$ [Figs. 2]. The gray zones indicate the regime of nearby FQHSs. The curves connecting data points serve as guides to the eye.