Two-dimensional Metal-insulator Transition Research Articles

Disorder-induced two-dimensional metal-insulator transition in moiré transition metal dichalcogenide multilayers

We develop a minimal theory for the recently observed metal-insulator transition (MIT) in two-dimensional (2D) moir\'e multilayer transition metal dichalcogenides (mTMD) using Coulomb disorder in the environment as the underlying mechanism. In particular, carrier scattering by random charged impurities leads to an effective 2D MIT approximately controlled by the Ioffe-Regel criterion, which is qualitatively consistent with the experiments. We find the necessary disorder to be around $5$-$10\times10^{10}$cm$^{-2}$ random charged impurities in order to quantitatively explain much, but not all, of the observed MIT phenomenology as reported by two different experimental groups. Our estimate is consistent with the known disorder content in TMDs.

Resistivity of the insulating phase approaching the two-dimensional metal-insulator transition: The effect of spin polarization

The resistivities of the dilute, strongly-interacting 2D electron systems in the insulating phase of a silicon MOSFET are the same for unpolarized electrons in the absence of magnetic field and for electrons that are fully spin polarized by the presence of an in-plane magnetic field. In both cases the resistivity obeys Efros-Shklovskii variable range hopping $\rho(T) = \rho_0 \mbox{exp}[(T_{ES}/T)^{1/2}]$, with $T_{ES}$ and $1/\rho_0$ mapping onto each other if one applies a shift of the critical density $n_c$ reported earlier. With and withoug magnetic field, the parameters $T_{ES}$ and $1/\rho_0 = \sigma_0$ exhibit scaling consistent with critical behavior approaching a metal-insulator transition.

Functionalized graphene as a model system for the two-dimensional metal-insulator transition.

Reports of metallic behavior in two-dimensional (2D) systems such as high mobility metal-oxide field effect transistors, insulating oxide interfaces, graphene, and MoS2 have challenged the well-known prediction of Abrahams, et al. that all 2D systems must be insulating. The existence of a metallic state for such a wide range of 2D systems thus reveals a wide gap in our understanding of 2D transport that has become more important as research in 2D systems expands. A key to understanding the 2D metallic state is the metal-insulator transition (MIT). In this report, we explore the nature of a disorder induced MIT in functionalized graphene, a model 2D system. Magneto-transport measurements show that weak-localization overwhelmingly drives the transition, in contradiction to theoretical assumptions that enhanced electron-electron interactions dominate. These results provide the first detailed picture of the nature of the transition from the metallic to insulating states of a 2D system.

Signatures of localization in the effective metallic regime of high-mobility Si MOSFETs

Combining experimental data, numerical transport calculations, and theoretical analysis, we study the temperature-dependent resistivity of high-mobility 2D Si MOSFETs to search for signatures of weak localization induced quantum corrections in the effective metallic regime above the critical density of the so-called two-dimensional metal-insulator transition (2D MIT). The goal is to look for the effect of logarithmic insulating localization correction to the metallic temperature dependence in the 2D conductivity so as to distinguish between the 2D MIT being a true quantum phase transition versus being a finite-temperature crossover. We use the Boltzmann theory of resistivity including the temperature dependent screening effect on charged impurities in the system to fit the data. We analyze weak perpendicluar field magnetoresistance data taken in the vicinity of the transition and show that they are consistent with weak localization behavior in the strongly disordered regime $k_F\ell\gtrsim1$. Therefore we supplement the Botzmann transport theory with a logarithmic in temperature quantum weak localization correction and analyze the competition of the insulating temperature dependence of this correction with the metallic temperature dependence of the Boltzmann conductivity. Using this minimal theoretical model we find that the logarithmic insulating correction is masked by the metallic temperature dependence of the Botzmann resistivity and therefore the insulating $\log T$ behavior may be apparent only at very low temperatures which are often beyond the range of temperatures accessible experimentally. Analyzing the low-$T$ experimental Si MOSFET transport data we identify signatures of the putative insulating behavior at low temperature and density in the effective metallic phase.

Two-dimensional metal-insulator transition as a strong localization induced crossover phenomenon

Low-disorder and high-mobility 2D electron (or hole) systems undergo an apparent metal-insulator-transition (MIT) at low temperatures as the carrier density (n) is varied. In some situations, the 2D MIT can be caused at a fixed low carrier density by changing an externally applied in-plane magnetic field parallel to the 2D layer. d\rho/dT changes its sign at some nonuniversal sample-dependent critical carrier density n_c separating an effective 2D metal (d\rho/dT >0) for n>n_c from an effective 2D insulator (d\rho/dT<0) for n<n_c. We study the 2D MIT phenomenon as a possible strong localization induced crossover process controlled by the Ioffe-Regel criterion, k_F l=1. Calculating the quantum mean free path (l) in the effective metallic phase from a realistic transport theory including disorder scattering effects, we solve the integral equation defined by the Ioffe-Regel criterion to obtain the nonuniversal critical density n_c as a function of the applicable physical experimental parameters including disorder strength, in-plane magnetic field, spin and valley degeneracy, background dielectric constant and carrier effective mass, and temperature. The key physics underlying the nonuniversal parameter dependence of the n_c is the temperature and density dependence of the Coulomb disorder. Our calculated results for the crossover n_c appear to be in qualitative and semi-quantitative agreement with the available experimental data in different 2D semiconductor systems lending credence to the possibility that the apparent 2D MIT signals the onset of the strong localization crossover in disordered 2D systems. We provide some results for graphene where a low-temperature 2D MIT becomes possible in the presence of intervalley scattering. We also provide an extensive comparison with the theoretical results obtained on the basis of 2D MIT being considered as a percolation transition.

Two-dimensional metal-insulator transition as a potential fluctuation driven semiclassical transport phenomenon

We theoretically consider the carrier density tuned (apparent) two-dimensional (2D) metal-insulator-transition (MIT) in semiconductor heterostructure-based 2D carrier systems as arising from a classical percolation phenomenon in the inhomogeneous density landscape created by the long-range potential fluctuations induced by random quenched charged impurities in the environment. The long-range Coulomb disorder inherent in semiconductors produces strong potential fluctuations in the 2D system where a fraction of the carriers gets trapped or classically localized, leading to a mixed 2-component semiclassical transport behavior at intermediate densities where a fraction of the carriers is mobile and another fraction immobile. At high carrier density, all the carriers are essentially mobile whereas at low carrier density all the carriers are essentially trapped since there is no possible percolating transport path through the lake-and-mountain inhomogeneous potential landscape. The low-density situation with no percolation would mimic an insulator whereas the high-density situation with allowed percolating paths through the lake-and-mountain energy landscape would mimic a metal with the system manifesting an apparent MIT in between. We calculate the transport properties as a function of carrier density, impurity density, impurity location, and temperature using a 2-component (trapped and mobile carriers) effective medium theory. Our theoretically calculated transport properties are in good qualitative agreement with the experimentally observed 2D MIT phenomenology in 2D electron and hole systems. We find a high (low) density metallic (insulating) temperature-dependence of the 2D resistivity, and an intermediate-density crossover behavior which could be identified with the experimentally observed 2D MIT.

Wigner-Mott scaling of transport near the two-dimensional metal-insulator transition

Electron-electron scattering usually dominates the transport in strongly correlated materials. It typically leads to pronounced resistivity maxima in the incoherent regime around the coherence temperature $T^{*}$, reflecting the tendency of carriers to undergo Mott localization following the demise of the Fermi liquid. This behavior is best pronounced in the vicinity of interaction-driven (Mott-like) metal-insulator transitions, where the $T^{*}$ decreases, while the resistivity maximum $\rho_{max}$ increases. Here we show that, in this regime, the entire family of resistivity curves displays a characteristic scaling behavior $\rho(T)/\rho_{max}\approx F(T/T_{max}),$ while the $\rho_{max}$ and $T_{max}\sim T^{*}$ assume a powerlaw dependence on the quasi-particle effective mass $m^{*}$. Remarkably, precisely such trends are found from an appropriate scaling analysis of experimental data obtained from diluted two-dimensional electron gases in zero magnetic fields. Our analysis provides strong evidence that inelastic electron-electron scattering -- and not disorder effects -- dominates finite temperature transport in these systems, validating the Wigner-Mott picture of the two-dimensional metal-insulator transition.

Energy scale behind the metallic behaviors in low-density Si MOSFETs

We show that the unexpected metallic behavior (the so-called two-dimensional metal-insulator transition) observed in low-density Silicon metal-oxide-semiconductor field-effect transistors (Si-MOSFETs) is controlled by a unique characteristic energy scale, the polarization energy. On one hand, we perform Quantum Monte Carlo calculations of the energy needed to polarize the two dimensional electron gas at zero temperature, taking into account Coulomb interactions, valley degeneracy and electronic mobility (disorder). On the other hand, we identify the characteristic energy scale controlling the physics in eight different sets of experiments. We find that our {\it ab-initio} polarization energies (obtained without any adjustable parameters) are in perfect agreement with the observed characteristic energies for all available data, both for the magnetic field and temperature dependence of the resistivities. Our results put strong constraints on possible mechanisms responsible for the metallic behavior. In particular, there are strong indications that the system would eventually become insulating at low enough temperature.

Observation of percolation-induced two-dimensional metal-insulator transition in a Si MOSFET

By analyzing the temperature $(T)$ and density $(n)$ dependence of the measured conductivity $(\ensuremath{\sigma})$ of two-dimensional (2D) electrons in the low-density $(\ensuremath{\sim}{10}^{11}\text{ }{\text{cm}}^{\ensuremath{-}2})$ and temperature (0.02--10 K) regimes of high-mobility (1.0 and $1.5\ifmmode\times\else\texttimes\fi{}{10}^{4}\text{ }{\text{cm}}^{2}/\text{Vs}$) Si metal-oxide-semiconductor field-effect transistors, we establish that the putative 2D metal-insulator transition is a density-inhomogeneity-driven percolation transition where the density-dependent conductivity vanishes as $\ensuremath{\sigma}(n)\ensuremath{\propto}{(n\ensuremath{-}{n}_{p})}^{p}$, with the exponent $p\ensuremath{\sim}1.2$ being consistent with a percolation transition. The ``metallic'' behavior of $\ensuremath{\sigma}(T)$ for $n&gt;{n}_{p}$ is shown to be well described by a semiclassical Boltzmann theory, and we observe the standard weak localization-induced negative magnetoresistance behavior, as expected in a normal Fermi liquid, in the metallic phase.

Self-doping instability of the Wigner-Mott insulator

We present a theory describing the mechanism for the two-dimensional (2D) metal-insulator transition (MIT) in the absence of disorder. A two-band Hubbard model is introduced, describing vacancy-interstitial pair excitations within the Wigner crystal. Kinetic energy gained by delocalizing such excitations is found to lead to an instability of the insulator to self-doping above a critical carrier concentration $n={n}_{c}$, mapping the problem to a density-driven Mott MIT. This mechanism provides a natural explanation of several puzzling experimental features, including the large effective mass enhancement, the large resistivity drop, and the large positive magnetoresistance on the metallic side of the transition. We also present a global phase diagram for the clean 2D electron gas as a function of $n$ and parallel magnetic field ${B}_{\ensuremath{\parallel}}$, which agrees well with experimental findings in ultraclean samples.

Quantum critical behavior in the insulating region of the two-dimensional metal-insulator transition

We show that the quantum critical point associated with the metal-insulator transition phenomenon in two dimensions controls an extended critical region encompassing not only the usual quantum critical sector but also a range of the low-temperature insulator region. The extended range of criticality permits a unified analysis of data from the insulating region and quantum critical sector, allowing us to determine both the dynamical critical exponent $z$ and the correlation length critical exponent $\ensuremath{\nu}$ from published data from a single experiment in the insulator critical region. We show that the critical exponents determined from the insulator sector consistently describe the temperature dependence of the resistance data from the same experiment in the quantum critical sector. This provides evidence for the presence of a quantum critical point in these systems.

2D-MIT as self-doping of a Wigner–Mott insulator

We consider an interaction-driven scenario for the two-dimensional metal–insulator transition in zero magnetic field (2D-MIT), based on melting the Wigner crystal through vacancy–interstitial pair formation. We show that the transition from the Wigner–Mott insulator to a heavy Fermi liquid emerges as an instability to self-doping, resembling conceptually the solid to normal liquid transition in He3. The resulting physical picture naturally explains many puzzling features of the 2D-MIT.

Capacitively induced high mobility two-dimensional electron gas in undoped Si∕Si1−xGex heterostructures with atomic-layer-deposited dielectric

The authors demonstrate that a high mobility two-dimensional electron gas can be capacitively induced in an undoped Si∕Si1−xGex heterostructure using atomic-layer-deposited Al2O3 as the dielectric. The density is tuned up to 4.2×1011∕cm2, limited by the gate leakage current. The mobility increases with the density rapidly and reaches 5.5×104cm2∕Vs at the highest density. The observation of well developed quantum Hall states and two-dimensional metal-insulator transition shows that the devices are suitable for two-dimensional electron physics studies.

Spin–valley phase diagram of the two-dimensional metal–insulator transition

Using symmetry breaking strain to tune the valley occupation of a two-dimensional (2D) electron system in an AlAs quantum well, together with an applied in-plane magnetic field to tune the spin polarization, we independently control the system's valley and spin degrees of freedom and map out a spin-valley phase diagram for the 2D metal-insulator transition. The insulating phase occurs in the quadrant where the system is both spin- and valley-polarized. This observation establishes the equivalent roles of spin and valley degrees of freedom in the 2D metal-insulator transition.

Two-dimensional metal-insulator transition and in-plane magnetoresistance in a high-mobility strained Si quantum well

The apparent metal-insulator transition is observed in a high quality two-dimensional electron system (2DES) in the strained Si quantum well of a Si/Si_{1-x}Ge_x heterostructure with mobility \mu=1.9 x 10^5 cm^2/Vs at density n=1.45 x 10^{11} cm^{-2}. The critical density, at which the thermal coefficient of low T resistivity changes sign, is ~ 0.32 x 10^{11} cm^{-2}, so far the lowest observed in the Si 2D systems. In-plane magnetoresistance study was carried out in the higher density range where the 2DES shows the metallic-like behavior. It is observed that the in-plane magnetoresistance first increases as ~ B_{ip}^2 and then saturates to a finite value \rho(B_c) for B_{ip} > B_c. The full spin-polarization field B_c decreases monotonically with n but appears to saturate to a finite value as n approaches zero. Furthermore, \rho(B_c)/\rho(0) ~ 1.8 for all the densities ranging from 0.35 x 10^{11} to 1.45 x 10^{11} cm^{-2} and, when plotted versus B_{ip}/B_c, collapses onto a single curve.

Two-Dimensional Metal-Insulator Transition as a Percolation Transition in a High-Mobility Electron System

By carefully analyzing the low temperature density dependence of 2D conductivity in undoped high-mobility n-GaAs heterostructures, we conclude that the 2D metal-insulator transition in this 2D electron system is a density inhomogeneity driven percolation transition due to the breakdown of screening in the random charged impurity disorder background. In particular, our measured conductivity exponent of approximately 1.4 approaches the 2D percolation exponent value of 4/3 at low temperatures and our experimental data are inconsistent with there being a zero-temperature quantum critical point in our system.

The limitimg behaviour of the well width of a quantum two-dimensional metal–insulator transition

Metal–Insulator transition using an exact two-dimensional (2D) dielectric function is investigated for a shallow donor in an isolated well of a GaAs/Ga1−xAlsAs superlattice system within the effective mass approximation. Vanishing of the donor ionization energy as a function of well width and the donor concentration suggests that a phase transition is not possible even below a well width of 10Å, supporting the scaling theory of localization. The effects of Anderson localization, exchange and correlation in the Hubbard model are included in a simple way. The relationship between the present model and the Mott criterion in terms of Hubbard model is also brought out. The critical concentration appears to be enhanced when a random distribution of impurities is considered. The limiting behaviour of the well width for a quantum 2D well is brought out. A simple expression is derived for a Mott constant in 2D, a*Nc1/2 exp (9.86 exp (−L/a*))=0.123, where Nc is the critical concentration per area. Results are compared with the existing data available and discussed in the light of existing literature.

Mott constant in two-dimensional metal–insulator transition

The recent developments of scaling theory of metal–insulator transition in two dimensions are elicited in a good manner. Metal–Insulator transition using exact two-dimensional dielectric function is investigated for a shallow donor in an isolated well of GaAs/Ga1−xAlsAs superlattice system within the effective mass approximation. Vanishing of the donor ionization energy as a function of well width and the donor concentration suggests that the phase transition is not possible even below a well width of 10Å supporting the scaling theory of localization. The effects of Anderson localization, exchange and correlation in the Hubbard model are included in a simple way. The relationship between the present model and the Mott criterion in terms of Hubbard model is also brought out. A simple expression for a Mott constant in 2D, a*Nc1/2 exp (−9.86exp(−L/a*))=0.123, where Nc is the critical concentration per area, is derived. The critical concentration is enhanced when a random distribution of impurities is considered. The limiting behaviour of well width for a quantum 2D is brought out. The results are compared with the existing data available.

Extended self-similarity in the two-dimensional metal-insulator transition.

We show that extended self-similarity, a scaling phenomenon first observed in classical turbulent flows, holds for a two-dimensional metal-insulator transition that belongs to the universality class of random Dirac fermions. Deviations from multifractality, which in turbulence are due to the dominance of diffusive processes at small scales, appear in the condensed-matter context as a large-scale, finite-size effect related to the imposition of an infrared cutoff in the field theory formulation. We propose a phenomenological interpretation of extended self-similarity in the metal-insulator transition within the framework of the random beta-model description of multifractal sets. As a natural step, our discussion is bridged to the analysis of strange attractors, where crossovers between multifractal and nonmultifractal regimes are found and extended self-similarity turns out to be verified as well.

Coherent backscattering near the two-dimensional metal-insulator transition.

We have studied corrections to conductivity due to the coherent backscattering in low-disordered two-dimensional electron systems in silicon for a range of electron densities including the vicinity of the metal-insulator transition, where the dramatic increase of the spin susceptibility has been observed earlier. We show that the corrections, which exist deeper in the metallic phase, weaken upon approaching the transition and practically vanish at the critical density, thus suggesting that the localization is suppressed near and at the transition even in zero field.