Universality of the two-parameter scaling description of the zero-field two-dimensional metal–insulator transition: critical exponents, critical resistivity, and the scaling function R(X,Y)

High-mobility silicon metal–oxide–semiconductor field-effect transistor (Si-MOS) samples show a zero-field metal–insulator transition (MIT) in the two-dimensional electron system whose transport data near the critical density can be collapsed with a two-parameter scaling construction, but the extracted critical exponents and critical resistivity vary from sample to sample and the scaling function itself is only an empirical fit. The open problem is to determine whether universal two-parameter scaling genuinely governs this transition: (i) establish whether the sample-to-sample variation of the exponents and critical resistivity is extrinsic (long-range disorder, finite disorder correlation length, density inhomogeneity) or intrinsic non-universality, as tested by cleaner samples with much shorter disorder correlation length; and (ii) determine the universal scaling function from microscopic theory or validate/extend the empirical form beyond the measured range.

Background

In a two-dimensional electron system (2DES) confined in a semiconductor interface, the scaling theory of localization for noninteracting electrons predicts that all states are localized at zero temperature, so no true metallic phase should exist. Experiments by Kravchenko et al. in 1994 (Phys. Rev. B 50, 8039, DOI: 10.1103/PhysRevB.50.8039) on high-mobility silicon metal–oxide–semiconductor field-effect transistors (Si-MOSFETs, in which a gate controls the electron density $n$ in an interfacial 2D layer) nevertheless showed a sharp crossover between metallic and insulating behavior as $n$ is lowered through a critical density $n_c$, at zero magnetic field. The leading theoretical framework, developed by Finkelstein, treats disorder and electron–electron interaction together through two coupled renormalization-group (RG) equations: one for the dimensionless conductance $\sigma$ (in units of $e^2/h$) and one for the interaction amplitude $\gamma_2$ (related to the Fermi-liquid triplet-channel parameter $F_0^\sigma$). The transition corresponds to a fixed point of these equations, and this framework is called two-parameter scaling, in contrast to the earlier one-parameter scaling in which density is the only driving parameter and the separatrix resistivity between metallic and insulating behavior is temperature-independent (contrary to the data). Linearizing the RG equations near the fixed point, the measured resistivity takes the scaling form $\rho = \mathcal{R}(X,Y)$ with two scaling variables $X = (T/T_0)^{-\kappa}(n-n_c)/n_c$ and $Y = (T/T_1)^{\zeta}$, where $T_{0,1}$ are sample-dependent energy scales and the exponents are fixed by the eigenvalues of the linearized equations: $\kappa = p/(2\nu)$ with $\nu$ the correlation-length exponent and $p$ the temperature exponent set by the anomalous dimension of the quasiparticle-weight factor $z$ at the fixed point ($p = 2/(2+\gamma_z)$, where $d\ln z/d\eta = \gamma_z$ and $\eta$ is the logarithm of the running length scale), and $\zeta = -py/2$ with $y$ the (negative) irrelevant exponent. The theory predicts that $\kappa$, $\zeta$, and the zero-temperature critical resistivity $\rho_c^0$ (in units of $h/e^2$) are universal, as is the scaling function $\mathcal{R}(X,Y)$ itself. In the Phys. Rev. Lett. 100, 046405 (2008) experiment, Knyazev, Omel'yanovskii, Pudalov, and Burmistrov measured $\rho(T,n)$ on five Si-MOS samples and performed this two-parameter analysis self-consistently: 64000 scaled data points from all samples collapse close to a single surface in $(X,Y,\rho/\rho_c^0)$ space, modeled empirically as $\mathcal{R}(X,Y) = \rho_c^0 \exp[f_1(X)] f_2(Y)$ with $f_1 = -X + 0.07X^2 + 0.001X^3$ and $f_2 = (1-Y+1.48Y^2)/(1+1.88Y^2+1.72Y^3)$, agreeing with the data within about 4% rms over $|X| \le 5$ and $Y \le 3$. However, the extracted parameters vary appreciably across the five samples: $\kappa = 0.61$–0.82, $\zeta = 0.35$–0.86, and $\rho_c^0 = 2.35$–5.27 (in $h/e^2$), with a trend of $\kappa$ and $\zeta^{-1}$ toward smaller values at lower mobility. The authors attribute this trend plausibly to long-range components of the random potential (favoring percolation-type behavior over true scaling) and to electron-density inhomogeneity, and they note that in theory the explicit form of $\mathcal{R}(X,Y)$ is unknown. They conclude that settling universality requires new samples in which the random potential has a much shorter correlation length and the density inhomogeneities are reduced.

Problem Statement

Determine whether the zero-field two-dimensional metal–insulator transition in high-mobility Si-MOS structures is governed by universal two-parameter scaling, in the sense required by the renormalization-group theory tested by Knyazev et al. (Phys. Rev. Lett. 100, 046405 (2008), DOI: 10.1103/PhysRevLett.100.046405). This comprises two jointly linked questions: (i) Universality of the critical parameters — establish whether the sample-to-sample variation of the critical exponents $\kappa$ and $\zeta$ and of the critical resistivity $\rho_c^0$ observed across Si-MOS samples is extrinsic, i.e. caused by long-range components of the random potential, the finite correlation length of the disorder, and electron-density inhomogeneities that modify or mask the true renormalization-group scaling, or intrinsic, i.e. a genuine departure from universality; in particular, test whether in samples with much shorter disorder correlation length and reduced density inhomogeneity a single universal set $\{\kappa, \zeta, \rho_c^0\}$ emerges from the two-parameter analysis. (ii) The scaling function — determine the universal scaling function $\mathcal{R}(X,Y)$ relating resistivity to the scaling variables $X = (T/T_0)^{-\kappa}(n-n_c)/n_c$ and $Y = (T/T_1)^{\zeta}$, either by deriving it from the microscopic two-parameter (Finkelstein) renormalization-group theory, or by extending and validating the empirical reconstruction (currently accurate to ~4% rms only within $|X| \le 5$ and $Y \le 3$) to wider parameter regimes. These two parts support one objective: deciding whether the observed criticality is a genuine interaction-driven quantum phase transition in the two-parameter-scaling universality class, with universal exponents, universal critical resistivity, and a universal scaling function that any theory of the transition must reproduce.

References

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