# Recent Developments in the Field of the Metal-Insulator Transition in Two Dimensions

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## Abstract

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## 1. Introduction

## 2. Metal-Insulator Transition in Zero Magnetic Field

## 3. Influence of the Magnetic Field Parallel to the 2D Plane

## 4. Spin Susceptibility; g-Factor; the Effective Mass

## 5. Band Flattening: Possible Condensation of Fermions

- (a)
- data describing the electron system as a whole, like thermodynamic density of states, magnetization of the electron system, or the magnetic field required to polarize electron spins fully and
- (b)
- data related only to the electrons at the Fermi level, like the amplitude of the Shubnikov–de Haas oscillations. This yields the effective mass ${m}_{\mathrm{F}}$ and Landé g-factor ${g}_{\mathrm{F}}$ at the Fermi level.

## 6. Transport Evidence for a Sliding Quantum Electron Solid

## 7. Summary

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Temperature dependence of the resistivity of a Si MOSFET at different electron densities near the MIT in zero magnetic field (

**a**) and in a parallel magnetic field of 4 Tesla (

**b**). The electron densities are indicated in units of ${10}^{11}$ cm${}^{-2}$. Dashed curves correspond to ${n}_{\mathrm{s}}={n}_{\mathrm{c}}^{*}$, which is equal to $0.795\times {10}^{11}$ cm${}^{-2}$ in zero field and to $1.155\times {10}^{11}$ cm${}^{-2}$ in $B=4$ Tesla. Adapted from [17].

**Figure 4.**(

**a**) Dependence of the Pauli spin susceptibility on electron density in a silicon MOSFET obtained by three independent methods: thermodynamic measurements of the magnetization (dashed line), the magnetic field of the full spin polarization (circles), and magnetocapacitance (squares). The dotted line is a guide to the eye. Also shown by a solid line are the transport data of [29]. The inset shows the polarization field as a function of the electron density determined from the magnetization (circles) and magnetocapacitance (squares) data. The symbol size for the magnetization data reflects the experimental uncertainty. The dependence extrapolates linearly to zero field at a density ${n}_{\chi}$ just below ${n}_{\mathrm{c}}$. Adapted from [30]. (

**b**) The effective electron mass m (circles) and g-factor (squares) in a silicon MOSFET, determined from the analysis of the parallel field magnetoresistance and temperature-dependent conductivity, versus electron density. ${m}_{0}$ and ${g}_{0}$ are the electron mass and g-factor for non-interacting electrons in silicon. The dashed lines are guides to the eye. Adapted from [32].

**Figure 5.**Product of the Landé factor and effective mass as a function of electron density in a SiGe/Si/SiGe quantum well determined by measurements of the field of full spin polarization ${B}^{*}$ (squares) and Shubnikov–de Haas oscillations (circles) at $T\approx 30$ mK. The empty and filled symbols correspond to two samples. The experimental uncertainty corresponds to the data dispersion and is about 2% for the squares and about 4% for the circles (${g}_{0}=2$ and ${m}_{0}=0.19\phantom{\rule{0.166667em}{0ex}}{m}_{\mathrm{e}}$ are the values for non-interacting electrons; ${m}_{\mathrm{e}}$ is the free electron mass; and ${g}_{\mathrm{F}}\approx {g}_{0}$ is the g-factor at the Fermi level). The top inset shows schematically the single-particle spectrum of the electron system in a state preceding the band flattening at the Fermi level (solid black line). The dashed violet line corresponds to an ordinary parabolic spectrum. The occupied electron states at $T=0$ are indicated by the shaded area. Bottom inset: the effective mass ${m}_{\mathrm{F}}$ versus electron density determined by analysis of the temperature dependence of the amplitude of Shubnikov–de Haas oscillations, similar to [45]. The dashed line is a guide to the eye. From [43].

**Figure 6.**$V-I$ curves are shown for different electron densities in the insulating state of a silicon MOSFET at a temperature of 60 mK. The dashed lines are fits to the data using Equation (6). The top inset shows the $V-I$ curve for ${n}_{\mathrm{s}}=5.20\times {10}^{10}$ cm${}^{-2}$ on an expanded scale; also shown are the threshold voltages ${V}_{\mathrm{th}1}$ and ${V}_{\mathrm{th}2}$, the static threshold ${V}_{\mathrm{s}}={V}_{\mathrm{th}2}$, and the dynamic threshold ${V}_{\mathrm{d}}$ that is obtained by the extrapolation of the linear region of the $V-I$ curve to zero current. Bottom inset: activation energy ${U}_{\mathrm{c}}$ vs. electron density. Vertical error bars represent standard deviations in the determination of ${U}_{\mathrm{c}}$ from the fits to the data using Equation (6). The dashed line is a linear fit. From [67].

**Figure 7.**(

**a**) $V-I$ characteristics at ${n}_{\mathrm{s}}=5.36\times {10}^{10}$ cm${}^{-2}$ in a silicon MOSFET for different temperatures. The dashed lines are fits to the data using Equation (6). (

**b**) The broad-band noise as a function of voltage for the same electron density and temperatures. The three upper curves are shifted vertically for clarity. From [67].

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**MDPI and ACS Style**

Shashkin, A.A.; Kravchenko, S.V. Recent Developments in the Field of the Metal-Insulator Transition in Two Dimensions. *Appl. Sci.* **2019**, *9*, 1169.
https://doi.org/10.3390/app9061169

**AMA Style**

Shashkin AA, Kravchenko SV. Recent Developments in the Field of the Metal-Insulator Transition in Two Dimensions. *Applied Sciences*. 2019; 9(6):1169.
https://doi.org/10.3390/app9061169

**Chicago/Turabian Style**

Shashkin, Alexander A., and Sergey V. Kravchenko. 2019. "Recent Developments in the Field of the Metal-Insulator Transition in Two Dimensions" *Applied Sciences* 9, no. 6: 1169.
https://doi.org/10.3390/app9061169