Microscopic foundation of the compressibility-driven scaling scenario for the two-dimensional metal-insulator transition: metallic-phase nature, stabilization against weak localization, and untested scaling predictions

Belitz and Kirkpatrick (Phys. Rev. Lett. 117, 236803 (2016)) proposed a scaling description of the two-dimensional metal-insulator transition (2D MIT) in which the compressibility $\chi_n=\partial n/\partial\mu$, rather than the diffusion coefficient, vanishes at the critical point. The theory is phenomenologically consistent but leaves its own microscopic basis open: the nature of the metallic phase (Fermi liquid vs. non-Fermi liquid), the mechanism that suppresses weak localization in an ostensibly two-dimensional system, the physical meaning of the observed negative electronic compressibility, the role of quenched disorder and possible lifting of degenerate dynamical exponents, the existence of an order parameter, and the experimental verification of the predicted scaling laws for the tunneling density of states and the specific heat.

Background

In high-mobility silicon metal-oxide-semiconductor field-effect transistors (Si MOSFETs) and related dilute two-dimensional electron systems (2DESs), the resistivity at low temperature was found in the mid-1990s to change sharply from metallic to insulating behavior as the carrier density is reduced through a critical value, apparently contradicting the scaling theory of localization, which holds that noninteracting electrons in two dimensions are always localized by disorder at sufficiently low temperature (Abrahams, Kravchenko and Sarachik, Rev. Mod. Phys. 73, 251 (2001)). Whether this two-dimensional metal-insulator transition (2D MIT) is a genuine continuous quantum phase transition, and if so what drives it, has been contested ever since. Candidate frameworks include disorder-based extensions of scaling theory with interactions, Wigner-Mott quantum criticality (crystallization of the electron fluid), and spatially mixed 'microemulsion' phases; the diversity of these proposals is reviewed in Spivak, Kravchenko, Kivelson and Gao, Rev. Mod. Phys. 82, 1743 (2010).

A central thermodynamic quantity is the electronic compressibility $\chi_n=\partial n/\partial\mu$, the response of the density $n$ to the electrochemical potential $\mu$. Quantum Monte Carlo calculations of the clean two-dimensional electron gas (Tanatar and Ceperley, Phys. Rev. B 39, 5005 (1989)) show that beyond mean-field (Hartree) contributions, exchange and correlation lower the energy at low density strongly enough that $\chi_n$ becomes negative, and capacitance measurements confirmed negative compressibility in interacting 2DESs (Eisenstein, Pfeiffer and West, Phys. Rev. Lett. 68, 674 (1992)). Scanning single-electron-transistor (SET) potentiometry — a scanning probe that maps the local thermodynamic compressibility — found strongly inhomogeneous compressibility landscapes near the zero-field 2D MIT (Ilani, Yacoby, Mahalu and Shtrikman, Phys. Rev. Lett. 84, 3133 (2000)), and a separate thermodynamic study found a sharp feature in $\chi_n$ at the transition (Dultz and Jiang, Phys. Rev. Lett. 84, 4689 (2000)).

Belitz and Kirkpatrick (Phys. Rev. Lett. 117, 236803 (2016)) built a scaling theory of the 2D MIT on the assumption that the transition is continuous and driven by $\chi_n\to 0$ while the diffusion coefficient $D$ stays noncritical, so that the density response remains diffusive but with a vanishing prefactor; the insulator is accordingly an incompressible quantum fluid, echoing Mott-transition physics (a Mott transition is an interaction-driven transition between a metal and an insulator; for a two-dimensional continuous-Mott scenario with $z=2$ gauge fluctuations see Senthil, Phys. Rev. B 78, 045109 (2008)). Here $z$ is the dynamical critical exponent, relating the characteristic frequency of critical modes to their wave number, $\omega\sim k^{z}$; $z=1$ is characteristic of ballistic (Fermi-liquid-like) soft modes and $z=2$ of diffusive ones. With the correlation-length exponent $\nu$ (describing divergence of a length scale as the critical density is approached) fixed to $\nu=1$ by the thermopower data of Mokashi et al., Phys. Rev. Lett. 109, 096405 (2012) — a value that holds provided there either is only one dynamical exponent, or the thermopower depends on a $z$ that is equal to 1 — and $z\approx\nu\approx1$ from transport scaling within substantial error bars, the framework predicts homogeneity (power-law scaling) relations for the electrical conductivity, the tunneling density of states (DOS), and — sensitive to the largest dynamical exponent if several are degenerate — the specific heat. A Fermi liquid (FL) is the standard Landau theory of interacting metals with long-lived quasiparticles; a 'strange metal' is a metallic state without well-defined quasiparticles, often with resistivity linear in temperature. Belitz and Kirkpatrick note that since switching off the electron-electron interaction would render the system insulating by weak localization (the quantum-interference tendency of a disordered 2D electron gas to localize), and since $k_{\mathrm{F}}\ell$ (the product of Fermi wave number and mean free path) is not large even in high-mobility samples, the metallic phase is unlikely to be a conventional FL, and they conjecture that the observed negative compressibility $K_T<0$ — where $K_T$ is the electronic compressibility probed in these experiments, related to the density response by $\chi_n=\partial n/\partial\mu=n^{2}K_{T}$ — renders the system mechanically unstable and drives a spontaneous rearrangement into a spatially inhomogeneous state, introducing a new length scale that cuts off the weak-localization singularities — a suggestion related to earlier microemulsion transport ideas (Spivak and Kivelson, Ann. Phys. 321, 1231 (2006)).

Problem Statement

As posed at the close of Belitz and Kirkpatrick's scaling theory, the problem is to supply — or definitively refute — the microscopic foundation of the compressibility-driven scenario for the two-dimensional metal-insulator transition. Its interlinked components, which jointly serve this single objective, are: (1) identify the microscopic nature of the metallic phase of the 2D MIT — Fermi liquid or non-Fermi-liquid 'strange' metal — and thereby the origin of the observed dynamical exponent $z=1$; (2) determine the physical mechanism that stabilizes a metallic phase in a nominally two-dimensional disordered system where $k_{\mathrm{F}}\ell$ is not large, in particular establish or falsify the conjecture that the mechanical instability associated with negative electronic compressibility drives a spontaneous spatially inhomogeneous state whose new length scale cuts off the weak-localization singularities; (3) establish the physical meaning of the experimentally observed negative compressibility in the metallic phase (spatial reorganization, phase separation, or otherwise) and its relation to the assumed vanishing of $\chi_n=\partial n/\partial\mu$ at the transition; (4) determine the role of quenched disorder, in particular whether disorder lifts degeneracies among dynamical exponents (e.g. a crossover from $z=1$ to $z=2$) and whether treating the diffusion coefficient as noncritical is self-consistent; (5) decide whether this MIT admits an order-parameter description and, if so, whether $\chi_n$ or the single-particle density of states is the order parameter; and (6) assess how the predicted homogeneity laws and exponent values (with $\nu=1$, $z=1$ fixed by existing thermopower and transport data) could be tested by tunneling-density-of-states and specific-heat measurements on the same samples used for transport and compressibility studies, and what existing data already imply. All six components are mandatory: the task is resolved only when each is addressed — supplied where a component asks for an account, or shown by a definitive argument to fail — and engaging with a proper subset does not resolve it.

References

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