Critical scaling of the insulating phase at the two-dimensional metal-insulator transition: which critical point governs Efros-Shklovskii hopping, and why is it unchanged by full spin polarization?

In dilute, strongly interacting two-dimensional (2D) electron systems in silicon MOSFETs, the zero-field metal-insulator transition (2D MIT) occurs at a critical electron density $n_c$, and the insulating phase conducts by Efros-Shklovskii variable-range hopping with a characteristic temperature $T_{ES}$ that vanishes on approach to the transition as $T_{ES} = A[(n_c - n_s)/n_c]^{\beta}$. Li and Sarachik [Phys. Rev. B 95, 041301(R) (2017)] showed that fully spin polarizing the electrons with an in-plane magnetic field merely shifts the critical density while leaving the hopping law and, when densities are referenced to the respective critical densities, the scaling of $T_{ES}$ and of $\sigma_0 = 1/\rho_0$ unchanged; they extracted $\beta = 2.5 \pm 0.1$ far from the transition with two-parameter scaling, $\beta \approx 1.7$ near the transition with one-parameter scaling, against $\beta \approx 1.6$ from earlier one-parameter analyses. The problem, posed by the source paper itself, is to establish the critical behavior of the insulating phase approaching the 2D MIT: which critical point (the disorder-driven $n_c$ or the putative disorder-independent, interaction-driven $n_i$) the vanishing of $T_{ES}$ refers to, or whether several closely spaced transitions or a mixed/intermediate phase intervene just below the transition; whether the scaling is one-parameter or two-parameter and what universality class applies; and why full spin polarization leaves the hopping form and scaling parameters invariant apart from the shift of $n_c$.

Background

A silicon MOSFET (metal-oxide-semiconductor field-effect transistor) confines electrons to a layer at a semiconductor-oxide interface, with the electron density $n_s$ tunable by a gate voltage. At very low densities the Coulomb interaction energy greatly exceeds the kinetic (Fermi) energy, and such strongly interacting 2D systems exhibit a sharp change, at a critical density $n_c$, between metallic-like low-temperature behavior (resistivity decreasing as $T$ is lowered) and insulating behavior. This 2D metal-insulator transition (2D MIT) was unexpected: the scaling theory of localization of Abrahams, Anderson, Licciardello and Ramakrishnan (1979) predicts that non-interacting 2D carriers always localize at zero temperature, so no 2D metallic phase should exist. The 2D MIT is widely regarded as one of the major unresolved problems of condensed matter physics (reviews: Abrahams, Kravchenko and Sarachik, Rev. Mod. Phys. 73, 251 (2001); Spivak, Kravchenko, Kivelson and Gao, Rev. Mod. Phys. 82, 1743 (2010)).

On the insulating side ($n_s < n_c$) conduction occurs by thermally activated hopping between localized states. In variable-range hopping (VRH) an electron hops not just to the nearest site but to the site optimizing distance against energy cost; when electron-electron interactions open a Coulomb gap — a suppression of the single-particle density of states near the Fermi level caused by the long-range Coulomb repulsion — the Efros-Shklovskii (ES) law holds, $\rho(T) = \rho_0 \exp[(T_{ES}/T)^{1/2}]$, replacing Mott's interaction-free 2D law with exponent 1/3. In ES theory $T_{ES} \propto e^2/(\epsilon\xi)$, where $\epsilon$ is the (density-dependent) dielectric constant and $\xi$ the localization length. If $\epsilon \propto [n_c/(n_c - n_s)]^{\zeta}$ and $\xi \propto [n_c/(n_c - n_s)]^{\nu}$ as $n_s$ approaches $n_c$ from below, then $T_{ES} = A[(n_c - n_s)/n_c]^{\beta}$ with $\beta = \zeta + \nu$; the prefactor $\rho_0$ (equivalently $\sigma_0 = 1/\rho_0$) may obey its own critical scaling. 'One-parameter' scaling analyses of the 2D MIT use a single crossover temperature $T_0$ to collapse data on both sides of the transition, while 'two-parameter' analyses let $T_{ES}$ and $\rho_0$ scale independently; earlier one-parameter experiments near the transition reported exponents near 1.6.

Because the electron layer is thin, a magnetic field $B$ applied in the plane of the 2D system couples almost only to the electron spin. The strong positive in-plane magnetoresistance saturates at a field $B_{sat}$ that Shubnikov-de Haas measurements — quantum oscillations of the resistivity in a magnetic field, which resolve the spin splitting of the electronic energy levels — show corresponds to full spin polarization, and applying $B > B_{sat}$ is known to shift the critical density $n_c$ upward. Mokashi, Li, Wen, Kravchenko, Shashkin, Dolgopolov and Sarachik [Phys. Rev. Lett. 109, 096405 (2012)] proposed that two quantum critical points may be in play near the 2D MIT in low-disorder samples: a disorder-driven critical density $n_c$ and a disorder-independent, interaction-driven critical density $n_i$, too close to each other to be separated experimentally.

In this context Li and Sarachik [Phys. Rev. B 95, 041301(R) (2017)] measured the insulating phase of a low-disorder Si MOSFET (0.25-2 K) in zero field (unpolarized electrons) and in a 5 T in-plane field (fully polarized electrons). In both spin configurations the resistivity obeys the ES law, and the parameters $T_{ES}$ and $\sigma_0$ of the two configurations map onto each other when the density is referenced to the respective critical densities, $n_c(0) = (0.782 \pm 0.014)\times10^{11}\,\mathrm{cm}^{-2}$ and $n_c(B_{sat}) = (0.985 \pm 0.014)\times10^{11}\,\mathrm{cm}^{-2}$: the sole effect of full spin polarization is a shift of the critical density. Their data far from the transition yield $\beta = 2.5 \pm 0.1$ with two-parameter scaling, while the portion of the data near the transition is consistent with one-parameter scaling and $\beta \approx 1.7$; they note that 2.5 is close to the classical percolation exponent 2.3 but argue against that interpretation because the disorder potential in Si is short-ranged.

Problem Statement

Establish the critical behavior of the insulating phase approaching the two-dimensional metal-insulator transition in strongly interacting 2D electron systems, and the role of spin in it, as posed by Li and Sarachik (2017). A complete answer determines: (1) which critical point governs the observed scaling — whether the vanishing of the Efros-Shklovskii parameter $T_{ES}$ (and the criticality of $\sigma_0$) as $n_s$ approaches $n_c$ from below refers to the disorder-driven critical density $n_c$, to the putative disorder-independent interaction-driven critical density $n_i$, or to several closely spaced transitions; (2) the correct scaling structure — whether the critical behavior is described by one-parameter or two-parameter scaling, with which exponent(s) $\beta$ (reconciling $\beta \approx 1.6$-$1.7$ extracted near the transition with $\beta \approx 2.5$ extracted farther away) and which universality class; (3) whether an intermediate phase or mixed-phase region exists in the narrow density range just below the transition, where the resistivity is a very weak function of temperature and cannot be reliably fit to the Efros-Shklovskii form, or whether the apparent drift of the exponent is a crossover effect of a single transition; and (4) the physical explanation of why full spin polarization of the electron system shifts the critical density but leaves the Efros-Shklovskii hopping form and the scaling parameters $T_{ES}$ and $\sigma_0$ unchanged when densities are referenced to their respective critical densities. These four questions jointly fix one objective — the transition's insulating-side criticality and its spin dependence — and progress on them requires experimental data closer to the transition (lower temperatures, lower and systematically varied disorder to separate $n_c$ from $n_i$) and/or microscopic theory of hopping transport in a strongly interacting, disordered 2D electron system near criticality, including its spin dependence.

References

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