# New Reentrant Insulating Phases in Strongly Interacting 2D Systems with Low Disorder

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## Abstract

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## Featured Application

**high mobility transistors.**

## Abstract

_{s}, (~20–40) and a high purity. A new complex phase diagram was proposed, which includes zero-field MIT, low magnetic field RIPs, integer QH states, fractional QH states, high field RIPs and insulating phases (HFIPs) with υ < 1 in which the insulating phases are explained by the formation of a Wigner crystal. Furthermore, evidence of new intermediate phases was reported. This review article serves the purpose of summarizing those recent experimental findings and theoretical endeavors to foster future research efforts.

## 1. Introduction

_{s}(the ratio between the Coulomb energy and the kinetic energy, ${r}_{s}\equiv 1/\left[{a}_{B}^{\ast}\sqrt{\pi p}\right],{a}_{B}^{\ast}={\hslash}^{2}\u03f5/{m}^{\ast}{e}^{2}$ is the effective Bohr radius) is much greater than one.

_{s}. Secondly, experimental observations of a possible Wigner crystal melting and new intermediate phases are shown and examined. Finally, relevant theoretical models that try to explain the MIT are briefly discussed.

## 2. New Reentrant Insulating Phases at Low Magnetic Fields

_{s}~ 30–40 [17,18], which had not been shown in experimental measurements until the study in Reference [10]. Therefore, the observation of the new RIP in Reference [10] provided a phase diagram consistent with the theory [18], which further suggested that the zero-field MIT is a liquid-WC transition.

#### 2.1. Resistivity

^{2}(Figure 4a) and the data do not have a high dynamic range to warrant a reliable fitting to the model. Therefore, the mechanism (thermal activation vs. variable-range-hopping) of the temperature dependent resistivity in the RIPs remains to be seen.

#### 2.2. Capacitance Measurement

#### 2.3. Inductance

_{s}in the density-perpendicular magnetic field plane is consistent with the liquid-WC transition phase diagram in clean 2D systems.

## 3. Possible Transport Evidence for Intermediate Phases and Wigner Crystal Melting

_{s}, then an important question arises: What type of phase transitions are experimentally observed for a 2D metallic liquid to insulator transition (whereas the insulator is either the zero-field insulator or the RIPs at a low magnetic fields). Given the well-known theoretical results that there is no long-range order in a 2D solid and the possible existence of various phases intermediate between a 2D WC and a liquid [20,21,22,23,24,25,26,27,28,29,30,31,32,33], is there any experimental evidence for intermediate phases when a WC melts into liquid?

## 4. Discussion and Outlook

_{s}2D systems is examined beyond the zero magnetic field to finite perpendicular magnetic fields. In contrast to weak interaction theories, many other theories emphasize the importance of strong correlations and relevance of WC physics in the systems showing a 2D MIT [9,26,27,33] and therefore, may be further compared with the experimental results. These theories are based on a number of different approaches: Analytical mean-field models [26,27], quantum Monte Carlo simulations [30], or dynamic mean field theory (DMFT) [32,33]. In the mean-field theories by Kivelson and Spivak [26,27], the various spectacular transport behavior in the correlated 2D systems showing an MIT are attributed to the Fermi liquid to WC transition where intermediate states are unavoidable [31]. In the intermediate ‘micro-emulsion’ states (e.g., WC bubbles in a Fermi liquid background), it is the interplay or transformation between a Fermi liquid and the WC components tuned by the temperature or a magnetic field that dictates the transport behavior and gives rise to the resistivity change of the system. It seems that the most relevant micro-emulsion phase to the experimentally observed RIPs and intermediate phases is the scenario where WC bubbles co-exist in a Fermi liquid. Whether other micro-emulsion states (Fermi liquid bubbles in WC, 1D ordered stripes) exist in experiments requires further research and more theoretical developments are desired to establish more quantitative predictions on the experimental systems. New theoretical approaches, based on hydrodynamics, seem quite promising and are currently being developed [41,42]. In addition to the mean-field models by Kivelson and Spivak, a strong interaction and Wigner-Mott transition based theoretical studies, led by Dobrosavljevic and collaborators [32,33], may also be relevant to the experimental findings. Modern DMFT was applied to study the MIT in 2D carriers with a high r

_{s}and the early approaches of Wigner and Mott were reconciled. Based on this ‘Wigner-Mott’ transition scenario, DMFT calculations are able to explain many detailed behaviors in the electrical transport and charge ordered intermediate phases similar to a charge density wave (CDW) predicted to form before the system enters a WC. It is worth noting that in the DMFT theory, both metallic CDW and insulating CDW are found [33]. It will be very interesting to see whether such CDW states exist in experiments.

_{F}are limited and worth exploring further [46]. In addition to transport, new techniques such as thermopower measurements are strongly desired to shed new light on the RIPs. For instance, diverging thermopower observed for strongly interacting 2D electrons in Si-MOSFET [47] revealed the critical nature of the zero-field MIT, complementing the views from critical behavior in resistance or effective mass [48]. Such critical behavior in thermopower or effective mass has generated theoretical interest and analogy to other strongly correlated materials like high T

_{c}cuprate may be made [49]. Finally, although this review is focused on strongly interacting 2D systems with an ultra-high purity/mobility and the effect of the impurities or disorder is not discussed much, understanding the effect of disorder on a 2D MIT has elicited rich physics [50] and thus we expect the disorder effect on the RIPs and RIP to liquid transition to be an interesting outstanding issue.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Longitudinal resistivity map plotted for the density (p)—magnetic field (B) plane at T = 150 (

**a**), 80 (

**b**) and 50 (

**c**) mK for dilute 2D holes in a 10 nm wide high mobility GaAs quantum well. The reentrant insulating phases (RIP) phase (${\rho}_{xx}>h/{e}^{2}$) becomes more prominent at a lower temperature. (

**d**) The proposed phase diagram includes the metal-insulator transition (MIT), RIP, integer and fractional quantum Hall (QH) states. Figure was adapted from Reference [10].

**Figure 2.**(

**a**) Observation of multiple RIPs between Landau-level filling factors 1, 2, 3 and 4 in a dilute 2D hole system in GaAs with r

_{s}~20. (

**b**) The modified phase diagram with multiple RIPs. Figure is taken from Reference [12].

**Figure 3.**(

**a**) 2D MIT at a zero magnetic field in a dilute 2D hole system in a 10 nm wide GaAs quantum well, ${p}_{c}~0.8\times {10}^{10}/{\mathrm{cm}}^{2}$. (

**b**) ${\rho}_{xx}\left(B\right)$ with $p=0.86\times {10}^{10}/{\mathrm{cm}}^{2}$. (

**c**) Arrhenius plot of the RIP peak resistance at various hole densities. (

**d**) The fitted thermal activation gap. Figure from Reference [10].

**Figure 4.**(

**a**) Real (solid) and imaginary (dashed) parts of the magnetoresistance for a RIP between υ = 2 and 3 for 2D holes in a 20 nm wide GaAs quantum well. (

**b**) Real and imaginary components of the magnetoresistance plotted on a semi-log scale. Figure from Reference [12].

**Figure 5.**(

**a**) Capacitance (symbol) and resistance (line) vs. perpendicular magnetic field at several hole densities in a 10 nm wide GaAs quantum well system. (

**b**) The phase diagram viewed in the longitudinal resistance map at 70 mK. (

**c**) The phase diagram viewed in the capacitance map at 70 mK. Figure from Reference [10].

**Figure 6.**(

**a**) Magnetoresistance of 2D holes in GaAs showing two RIPs at ν > 1 (labeled as P1 and P2). (

**b**) mechanism (thermal activation vs. variable-range-hopping) (

**d**) The longitudinal resistivity ρ

_{xx}and inductive signal Y

_{xx}vs. frequency at various magnetic fields, showing the clear inductive effect of the RIPs. Figure taken from Reference [12].

**Figure 7.**Voltage-current characteristics of 2D electrons in a Si-metal-oxide-semiconductor field-effect transistor (Si-MOSFET) in a possible Wigner crystal (WC) state. V-I curves of different electron densities are shown to show the depinning of the WC. Figure taken from Reference [35].

**Figure 8.**Voltage-current characteristics of 2D holes in GaAs in a possible WC state to show the two-stage melting of the WC. (

**a**) dc IV at 28 mK. (

**b**) IVs at different temperatures. (

**c**) Temperature dependence of ${r}_{d\left(T\right)}{|}_{V\to 0}.$ (

**d**) Suggested phase diagram. Figure from Reference [36].

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**MDPI and ACS Style**

Qiu, R.L.J.; Liu, C.-W.; Liu, S.; Gao, X.P.A.
New Reentrant Insulating Phases in Strongly Interacting 2D Systems with Low Disorder. *Appl. Sci.* **2018**, *8*, 1909.
https://doi.org/10.3390/app8101909

**AMA Style**

Qiu RLJ, Liu C-W, Liu S, Gao XPA.
New Reentrant Insulating Phases in Strongly Interacting 2D Systems with Low Disorder. *Applied Sciences*. 2018; 8(10):1909.
https://doi.org/10.3390/app8101909

**Chicago/Turabian Style**

Qiu, Richard L. J., Chieh-Wen Liu, Shuhao Liu, and Xuan P. A. Gao.
2018. "New Reentrant Insulating Phases in Strongly Interacting 2D Systems with Low Disorder" *Applied Sciences* 8, no. 10: 1909.
https://doi.org/10.3390/app8101909