Controlled incorporation of soft Wigner-crystal collective charge modes into the self-doping theory of the two-dimensional metal-insulator transition

Experiments on very clean semiconductor two-dimensional electron gases (2DEGs) show a sharp zero-field metal-insulator transition (2D-MIT) at interaction parameters $r_s\gtrsim 10$, with an effective mass that grows strongly as the transition is approached. Pankov and Dobrosavljević proposed an interaction-driven mechanism in which the Wigner crystal melts through vacancy-interstitial (self-doping) excitations, mapped onto a two-band Hubbard model solved in static slave-boson mean field. That treatment, like all of its later descendants, keeps the lattice rigid: the very soft collective charge-density modes (Wigner-crystal phonons) generated by the long-range Coulomb interaction enter only through semi-phenomenological electrostatic parameters. The open problem is to build a controlled microscopic theory that treats local strong correlations and the long-wavelength Coulomb collective fluctuations together, so that the renormalized charge-transfer gap and the transition densities $n_c$ and $n_1$ follow from microscopic inputs rather than from fitted parameters. No such synthesis exists in the subsequent Wigner-Mott literature, so the problem remains open.

Background

In a two-dimensional electron gas (2DEG) confined at a semiconductor interface, the dimensionless interaction parameter $r_s$ (the ratio of the mean inter-electron spacing to the effective Bohr radius) measures Coulomb repulsion against kinetic energy. At sufficiently large $r_s$ the ground state of jellium — the idealized electron gas neutralized by a uniform, rigid positive background — is a Wigner crystal: electrons localize on a triangular lattice to minimize repulsion, at the cost of kinetic energy. A Mott insulator is instead a lattice system localized by short-range repulsion at commensurate filling; at low density these pictures merge, since the self-formed Wigner lattice plays the role of the crystal lattice, and the resulting state is often called a Wigner-Mott insulator. Since the mid-1990s, experiments on ultra-clean silicon metal-oxide-semiconductor field-effect transistors (Si MOSFETs, gated semiconductor devices hosting an interfacial 2DEG) and comparable heterostructures have shown a sharp metal-insulator transition (the 2D-MIT) near $r_s\gtrsim 10$, far below the accepted $r_s\sim 37$ for Wigner crystallization of the unpolarized jellium, accompanied by a dramatic resistivity drop and an effective mass $m^*$ that grows roughly as $(n-n_c)^{-1}$ while the Landé $g$-factor stays nearly unrenormalized. Whether this transition is driven primarily by disorder (as in the two-parameter scaling theory of Finkel'shtein and Punnoose, which resums interaction corrections to Anderson localization in the diffusive regime) or by interactions in the clean limit remains contested.

Pankov and Dobrosavljević (arXiv:0704.0250, published as Physica B 403, 1440 (2008); and the companion paper arXiv:0705.3428, Phys. Rev. B 77, 085104 (2008)) developed the clean-interaction scenario. They approached the transition from the insulating side, describing the Wigner crystal's dominant quantum defects — vacancy-interstitial pairs — by a two-band Hubbard (charge-transfer) model: 'site' electrons pinned to the Wigner lattice and 'interstitial' electrons on the interstices, hybridized and with on-site repulsion treated by the slave-boson mean-field method (a technique that enforces the no-double-occupancy constraint by auxiliary bosons and yields a quasiparticle weight $Z$). Delocalizing the interstitials lowers the free energy at nonzero 'self-doping' $\delta$ (deviation from half filling, in analogy to the solid-to-normal-liquid transition of $^3$He), so the insulator becomes unstable to self-doping above a critical density $n_c$, producing a heavy Fermi liquid; in a large parallel magnetic field (spins fully polarized) the model reduces to a band insulator whose gap closes linearly at a higher density $n_1$, matching magneto-transport phenomenology.

The acknowledged deficiency of this and all subsequent treatments in the same program is the static lattice. Because the Coulomb interaction is long-ranged, the charge-density collective modes of the Wigner crystal (its phonons, whose shear branches are gapless and whose longitudinal branch is anomalously soft at long wavelength) are expected to strongly renormalize the microscopic parameters, in particular the charge-transfer gap $e_c-e_f$ between the site and interstitial bands, thereby shifting the transition to the experimentally observed low densities ($r_s\gg 1$), in close analogy to Coulomb-gap physics in disordered insulators. In the 2007 work these effects enter only through the choice of electrostatic model parameters. Extended dynamical mean-field theory (EDMFT) — the extension of dynamical mean-field theory in which the local charge susceptibility is screened by a self-consistently determined bosonic bath representing nonlocal Coulomb interactions — was explicitly named by the authors as a candidate controlled framework for incorporating such collective fluctuations, alongside Coulomb-glass techniques (methods developed for the Coulomb glass, a disordered insulator with long-range Coulomb interactions whose low-energy excitation spectrum develops a soft gap) and other schemes exploiting the special properties of the Coulomb potential. Building that synthesis is the objective of this task.

Problem Statement

Construct a controlled microscopic theory of the clean, interaction-driven two-dimensional metal-insulator transition in the dilute 2DEG in which both ingredients that the static slave-boson self-doping treatment of Pankov and Dobrosavljević kept separate are treated together and systematically: (i) the local strong correlations that drive self-doping of the Wigner-Mott insulator via vacancy-interstitial (two-band Hubbard) excitations, and (ii) the long-wavelength collective charge-density modes — the soft Wigner-crystal phonons generated by the long-range Coulomb interaction — whose renormalization of the charge-transfer gap $e_c-e_f$ is expected to place the transition at the experimentally observed low densities $r_s\gg 1$.

The single overarching objective is a parameter-free account of the transition: the microscopic inputs (carrier density, band masses, dielectric environment) must enter without fitting to the measured transition. Within the same framework, the theory must deliver all of these required outputs — the renormalized charge-transfer gap, together with the resulting predictions of both the critical density $n_c$ of the zero-field transition and the density $n_1$ at which the activation gap of the spin-polarized system closes in a large parallel magnetic field — from the same microscopic inputs, instead of reproducing these quantities through separately chosen electrostatic parameters. A microscopic input is admissible only if its value is fixed by the material and the independently measured carrier density, without reference to the transition; a quantity whose value is instead chosen so that the predicted $n_c$ or $n_1$ matches the measured value is a semi-phenomenological parameter and is not admissible. Candidate frameworks include extended dynamical mean-field theory (EDMFT) with self-consistent dynamical screening, Coulomb-glass-inspired techniques, or any many-body scheme that can justify its approximations in the strong-coupling regime; demonstrating that the soft collective modes shift the transition to $r_s\gtrsim 10$ without semi-phenomenological parameter choices is the central requirement.

References

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