Quantum Wigner solid in two-dimensional electron systems in semiconductors

TL;DR

The paper addresses whether a low-density 2D electron system in semiconductors forms a quantum electron solid (Wigner crystal) and how to distinguish it from conventional insulating mechanisms. It combines transport measurements in Si MOSFETs and ultra-high mobility SiGe/Si/SiGe wells to reveal robust two-threshold $V$-$I$ characteristics accompanied by a broadband noise peak, interpreted via a collective depinning model with $U(V)=U_{ ext{c}}(1-V/V_{ ext{s}})$ and a thermally activated current form. The results show that double-threshold behavior persists in zero field and becomes more pronounced with perpendicular magnetic fields, where onset voltages and densities shift in a way consistent with magnetic stabilization of the solid, including a characteristic filling factor $ u o ext{~0.27}$ at high $B$. Importantly, quantum Hall insulating states near integer fillings do not exhibit the same double-threshold signatures, arguing against a quasi-particle quantum Hall Wigner solid in this system. Overall, the findings support the existence of a quantum electron solid in these 2D systems and demonstrate the generality of the double-threshold/depinning phenomenology across material platforms, with magnetic fields enhancing the solid’s stability.

Abstract

We review recent transport experiments that reveal two-threshold voltage-current characteristics, marked by a significant increase in noise between the two threshold voltages, at low electron densities in the insulating regime in two-dimensional (2D) electron systems, specifically in silicon metal-oxide-semiconductor field-effect transistors (MOSFETs) and SiGe/Si/SiGe heterostructures. The double-threshold voltage-current characteristics closely resemble those observed in the collective depinning of the vortex lattice in type-II superconductors. By adapting the model used for vortices to the case of an electron solid, good agreement with the experimental results is achieved, which supports a quantum electron solid forming in the low electron density state. When a perpendicular magnetic field is applied, the double-threshold behavior occurs at voltages an order of magnitude lower and at significantly higher electron densities than the zero-field case. This indicates the stabilization of the quantum electron solid, aligning with theoretical predictions. Interestingly, the double-threshold voltage-current curves, indicative of electron solid formation at low densities, are not observed in the quantum Hall regime. This lack of observation does not confirm the existence of a quasi-particle quantum Hall Wigner solid and indicates that quasi-particles near integer filling do not form an independent subsystem.

Figures (11)

• Figure 1: $V-I$ characteristics measured at different electron densities in the insulating state at a temperature of 60 mK. The dashed lines are fits to the data using Eq. (\ref{['I']}). In the top inset, the $V-I$ curve for $n_{\text{s}}=5.20\times 10^{10}$ cm$^{-2}$ is shown on an expanded scale; the threshold voltages $V_{\text{th1}}$ and $V_{\text{th2}}$, the static threshold $V_{\text{s}}=V_{\text{th2}}$, and the dynamic threshold $V_{\text{d}}$ (which is obtained by extrapolating the linear portion of the curves to $I=0$) are indicated. In the bottom inset, the activation energy $U_{\text{c}}$vs. electron density is plotted. The dashed line is a linear fit. Adapted from Ref. brussarski2018transport.
• Figure 2: (a) $V-I$ characteristics measured at $n_{\text{s}}=5.36\times 10^{10}$ cm$^{-2}$ and different temperatures. The dashed lines are fits to the data using Eq. (\ref{['I']}). (b) The broadband noise as a function of voltage for the same electron density and temperatures. Adapted from Ref. brussarski2018transport.
• Figure 3: Noise as a function of frequency at $n_{\text{s}}=5.36\times 10^{10}$ cm$^{-2}$, $T=60$ mK, and $V=4.26$ mV with resolution/bandwidth 2.5 Hz. The broad maxima at $f\sim10$ and 60 Hz on the order of 1 pA/$\sqrt{{\text{Hz}}}$ are within the experimental uncertainty. The dashed line shows the $1/f^{0.6}$ dependence. Adapted from Ref. brussarski2018transport.
• Figure 4: (a) The $V$-$I$ characteristics measured in double-gate samples at various electron densities, displayed from left to right as follows: 7.95, 6.98, 6.21, and 5.82 $\times 10^9$ cm$^{-2}$. The temperature is $T = 30$ mK. In panel (b), the $V$-$I$ characteristics for triple-gate samples are shown at different electron densities, listed from left to right as: 6.37, 6.19, 6.01, 5.92, 5.83, 5.74, 5.65, and 5.56 $\times 10^9$ cm$^{-2}$, also at $T = 30$ mK. The dashed lines represent fits to the data using Equation (I). The inset displays the $V$-$I$ characteristics for electron densities of $n_{\text{s}} = 6.01 \times 10^9$ cm$^{-2}$ and $5.92 \times 10^9$ cm$^{-2}$ on an expanded scale. Arrows indicate the dynamic threshold voltage $V_{\text{d}}$, which is obtained by extrapolating (as shown by the dotted line) the linear portion of the $V$-$I$ curves to zero current, along with the static threshold voltage $V_{\text{s}}$. From Ref. melnikov2024triple.
• Figure 5: (a) The $V$-$I$ characteristics are shown in the upper panel and the broadband noise is shown in the lower panel at a temperature of approximately 30 mK and various electron densities (from left to right): 5.92, 5.83, 5.74, 5.65, 5.56, 5.47, and $5.29\times10^9$ cm$^{-2}$. Panels (b) and (c) display the $V$-$I$ characteristics for two electron densities on an expanded scale. Also indicated are the threshold voltages, $V_{\text{th1}}$ and $V_{\text{th2}}$, the dynamic threshold $V_{\text{d}}$, which is obtained by extrapolating the linear portion of the $V$-$I$ curves to zero current, and the static threshold $V_{\text{s}}=V_{\text{th2}}$. The dashed lines are fits to the data using Eq. (\ref{['I']}). From Ref. melnikov2024collective.