Reentrant insulating phases in the integer quantum Hall regime

Quantum Hall measurements are performed in a dilute high-quality GaAs two-dimensional hole system with carrier densities of (4.4–$5.3)\times {10}^{10}{\text{cm}}^{-2}$. With the $r_s$ value approximately 20, the interaction effect is already strong even in the absence of magnetic field. In the integer quantum Hall regime up to 2 T, a series of reentrant insulating phases (RIPs) are observed between fillings 1, 2, 3, and 4, with peak locations periodic in $1/B$. Moreover, the RIP peaks are also characterized by a striking inductance in response to ac excitations, which is likely due to the onset of a Wigner crystal state. The multiple RIP peaks indicate a more complex phase diagram than previously perceived.

Figure 1

(a) Diagram of quantum well. Distances are in nanometers. Band gap energies taken from Ref. [18] are in electron volts. (b) Shubnikov–de Haas oscillations. The $0^{\circ }$ and $+{90}^{\circ }$ quadrature components of the magnetoresistance, ${\rho }_{xx}$ and $Y_{xx}$, are shown in black and red, respectively. Phase is measured relative to the driving current. Blue is used for the Hall resistance ${\rho }_{xy}$.

Figure 2

(a) Hall measurement reveals two insulating peaks in the integer regime labeled P1 and P2. At P1, ${\rho }_{xx}$ reaches $10h/e^2$. For both peaks, $Y_{xx}$ rises substantially but elsewhere is less than 10% of the total signal. (b) Magnetoresistance shows flat frequency response in zero field up to the filter cutoff frequency. (c) Inductive behavior of P2 is clearly visible below the cutoff. (d) Frequency dependence on the shoulder of P1 $\left(B=1.62\text{T}\right)$ is similar to that at the maximum of P2. At larger sample impedance, the cutoff decreases, preventing similar measurement at the center of P1.

Figure 3

(a) A series of measurements for various temperatures (30 to 300 mK) showing both the real (solid) and imaginary (dashed) components of the magnetoresistance for fixed density at P2. A vertical dashed line appears at $B=0.89\text{T}$. (b) Real and imaginary components of the magnetoresistance at fixed $B=0.89\text{T}$ plotted as a function of $T^{-1/2}$ on a semilogarithmic scale. The dotted line represents fitting to Efros-Shklovskii [20] variable-range-hopping ${\rho }_{xx}\propto exp\left(\sqrt{T^{☆}/T}\right)$ with $T^{☆}=98\text{mK}$. The inset shows both ${\rho }_{xx}$ and $Y_{xx}$ as a function of $T$ with dotted lines as guides to the eye. (c) Hall measurement for the RIP near 1/3 at density $5.1\times {10}^{10}{\text{cm}}^{-2}$ (the color coding is the same as that in Fig. 2).

Figure 4

(a) Periodicity of $Y_{xx}$ in $1/B$. To the right of the vertical dashed line, $Y_{xx}$ has been amplified by a factor of 10 for clarity. Peaks locations are marked by upward arrows and indexed according to the closest integer filling factor. The inset shows a straight-line fit of the peak location in $1/B$ as a function of peak index. (b) Proposed phase diagram modified from Ref. [15]. Integer fillings are positioned to line up with the magnetoresistance minima in (a) as indicated by the vertical dashed arrows. A series of reentrant phases is proposed between integer fillings. Relative peak heights are qualitatively consistent with our data.