Transport of a sliding Wigner crystal in the four flux composite fermion regime

In two-dimensional (2D) electron systems, Wigner crystals (WC) and fractional quantum Hall effect (FQHE) liquids are competing ground states under low temperatures $\left(T\right)$ and high magnetic fields $\left(B\right)$. Here we report differential conductivity results demonstrating the reentrant insulating phase around $\nu =1$/5 in a 2D hole system in AlGaAs/GaAs quantum wells and unexpected features in the solid-liquid phase transition between WC and FQHE liquids in ultrahigh magnetic fields up to 45 T. Remarkably, the electric field $\left(E\right)$ plays an equivalent role as the temperature does in our phase diagram. From the $E-T$ “duality” analysis, a characteristic length of 450 nm is derived, which can be understood as the phase-coherent domain size of WC. Moreover, evidence shows that with weak disorder the insulating phase and composite fermion liquid could be coexisting around $\nu =$ 1/5, pointing to the possibility that the insulating phase is the four flux quantum Wigner crystal, as proposed by theories.

Figure 1

The results of the derivative conductivity $\left(dI/dV\right)$ measurement. (a) For some magnetic fields (certain filling factors), the derivative conductivity shows the large threshold $E$ field $\left(E_{\text{th}}\right)$ and dramatic slope. $B=34$- and 39-T data exhibit complex features, which come from the (2/9 and 1/5) FQHE liquid order in the mixing of WC-melted liquid. Inset: Schematic of the conductivity measurements on the Corbino sample. (b) The threshold for the electric field under magnetic fields and the corresponding 1/$\nu$. (c) The linear or linearlike $dI/dV$ features at $B=37.7$, 37.8, and 38.1 T.

Figure 2

$E$-dependent conductivity measurement results. (a) The 3D plot for the longitudinal conductivity ${\sigma }_{xx}$ under high $B$: 18–45 T, with $E\sim 0$–690 mV/cm. The red traces highlight the filling factors. Also we highlight the divided ranges by the dotted lines, “A” and “B”; between those the robust $\nu =1$/5 FQHE liquid shows a strong effect. (b) The black solid square curves are the boundary for totally pinned WC, and blue open circles indicate the robust influence by the $\nu =1/5$ FQHE state. (c) The peak positions for the conductivity in panel (a), which can be clearly seen as the “mountain chain” in the 3D plot. (d) The phase diagram comes from the superposition of (b) and (c). In (b)–(d), the data are all shown under $B$ and 1/$\nu$.

Figure 3

$T$-dependent conductivity results. (a) ${\sigma }_{xx}$ under a $T$ range of 20–370 mK; the top axis shows 1/$\nu$. (b) The Arrhenius plot $({\sigma }_{xx}$ vs 1/$T)$ for 36, 38, and 40 T. (c) Energy gaps $E_g$ and $\mathrm{\Delta }$ under $B$ fields: 34–40 T. The data follow ${\sigma }_{xx}\propto \text{exp}(E_g/k_{B}T),{\sigma }_{xx}\propto \text{exp}(-\mathrm{\Delta }/k_{B}T)$. (d) The boundary between the melted liquid and WC. The boundary $T_b$ comes from the peak position in the $T$-dependent ${\sigma }_{xx}$, and $T_{C1}$ is the transition temperature in the $\mathrm{\Lambda }$-shape feature for fixed $\nu$ $\left(B\right)$. (e) $\mathrm{\Delta }$ vs $T_{C1}$ presents a linear relation: $\mathrm{\Delta }=10.5T_{C1}-0.8$ K. (f) Our obtained $E-T$ phase diagram is rather similar to the $E-B$ diagram. (g) In the $E_g$ vs $T_{C2}$ panel, the gap decreases with the critical $T$, with a gradient of $-10.5$.

Figure 4

The phase diagram under $B-T_B$ (boundary temperature) or $B-E$ coordinates. (a) This panel combines the data from Fig. 3 and the $E-T$ duality, with the $E$ axis (left axis) and $T_B$ axis (right axis) in it. (b) The data fittings are carried out for the phase boundaries. The red fitting curves for the black circle and the blue square data follow the relations $T_B\propto {(B-B_0)}^{1/2}$. In the range of $1/5<\nu <2/9$ and $\nu <1/5$, there are dramatic protuberances (outlined by the black short dashed line). The inverse of filling factor $\nu$ and the CF fillings ${\nu }^{*}$ axis are shown too.