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Chapter 6 Microscopic Theory of a Strongly Correlated Two-Dimensional Electron Gas

MIKHAIL V. ZVEREV, V. A. Khodel

Published May 25, 2017• 0 citations
Publisher

Abstract

This chapter presents the microscopic theory of a two-dimensional (2D) electron gas based on an ab initio functional approach to the many-body problem developed by V. A. Khodel et al. and V. R. Shaginyan. It discusses the detailed structure of the theory and scheme of the numerical solution of relevant equations. The chapter also discusses the method of evaluation of the single-particle spectrum of a 2D electron gas. It examines the behavior of a 2D electron gas in external magnetic fields. The chapter focuses on the disappearance of magnetic oscillations in electron systems of metal-oxide-semiconductor field-effect transistors due to merging of spin- and valley-split Landau levels in silicon, being the topological rearrangement of the Landau state, analogous to the formation of the fermion condensate. It explores the impact of the violation of the equidistance of electron spectra in strong magnetic fields on the structure of magnetic oscillations near the critical density.

DOI
10.4324/9781315364575-6

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Physical review. B, Condensed matter•Oct 15, 1983•133 citations
This chapter presents the microscopic theory of a two-dimensional (2D) electron gas based on an ab initio functional approach to the many-body problem developed by V. A. Khodel et al. and V. R. Shaginyan. It discusses the detailed structure of the theory and scheme of the numerical solution of relevant equations. The chapter also discusses the method of evaluation of the single-particle spectrum of a 2D electron gas. It examines the behavior of a 2D electron gas in external magnetic fields. The chapter focuses on the disappearance of magnetic oscillations in electron systems of metal-oxide-semiconductor field-effect transistors due to merging of spin- and valley-split Landau levels in silicon, being the topological rearrangement of the Landau state, analogous to the formation of the fermion condensate. It explores the impact of the violation of the equidistance of electron spectra in strong magnetic fields on the structure of magnetic oscillations near the critical density.
The rearrangement of the Fermi surface in a diluted two-dimensional electron gas beyond the topological quantum critical point has been examined within an approach based on the Landau theory of Fermi liquid and a nonperturbative functional method. The possibility of a transition of the first order in the coupling constant at zero temperature between the states with a three-sheet Fermi surface and a transition of the first order in temperature between these states at a fixed coupling constant has been shown. It has also been shown that a topological crossover, which is associated with the joining of two sheets of the Fermi surface and is characterized by the maxima of the density of states N(T) and ratio C(T)/T of the specific heat to the temperature, occurs at a very low temperature T ⋄ determined by the structure of a state with the three-sheet Fermi surface. A momentum region where the distribution n(p, T) depends slightly on the temperature, which is manifested in the maximum of the specific heat C(T) near T *, appears through a crossover at temperatures T ∼ T * > T ⋄. It has been shown that the flattening of the single-particle spectrum of the strongly correlated two-dimensional electron gas results in the crossover from the Fermi liquid behavior to a non-Fermi liquid one with the density of states N(T) ∝ T −α with the exponent α }~ 2/3.
We study two interacting electrons in a two-dimensional system under a strong magnetic field and show that their numerically exact solutions organize into a set of {\em sub-Landau levels} characterized by relative angular momentum quantum number mmm. These sub-levels define correlation-resolved subspaces of the Landau-level Hilbert space, while retaining the full degeneracy associated with center-of-mass motion. Within this structure, the accessible states in each correlation channel are effectively reduced, leading to a natural organization of guiding-center states consistent with a fractional occupancy. We further analyze the role of electron correlation, Zeeman splitting, and disorder in stabilizing spin-polarized electron-pair states. Building on the two-electron states, we construct a class of many-electron trial wavefunctions based on correlated electron pairs with fixed mmm, which encode short-range correlations through the vanishing of the pair wavefunction at small separation. Our results establish a direct connection between exact two-body physics and the organization of correlated many-electron states in the lowest Landau level, providing a microscopic perspective on how relative angular momentum structures can underpin the emergence of correlated phases in quantum Hall systems.
The nature of the ground state of degenerate two dimensional (2D) fermions at zero magnetic field is an outstanding open problem, which has not been deciphered despite decades of research. In the absence of disorder, the ground state is believed to be determined by the interplay between the kinetic energy, E f, and the inter-particle interaction energy, Ec = e2/ica, where a = (nn)~1^ 2 is the typical inter-particle distance, n is the areal particle density, and k is the host dielectric constant. The relative importance of the two energy scales is characterized by rs = a/ao, with ao being the Bohr radius. For electrons in a single band, rs = Ec/ E f while for the (100) surface of silicon rs = Ec/2E p , due to the two-fold valley degeneracy. At very high densities (rs <£ 1) a 2D system approaches the paramagnetic limit of a non-interacting degenerate gas characterized by the Pauli susceptibility, xo • As the density is reduced, the growing ferromagnetic correlations lead to substantial enhancement of the spin susceptibility, The system is predicted to remain paramagnetic up to rs « 20, where numerical calculations [1] find a quantum phase transition to a ferromagnetic liquid phase [2]. At lower density, rs & 37 [3], the Coulomb correlations are predicted to lead to a second phase transition to an anti-ferromagnetic quantum Wigner crystal. The energy balance between the ferro and paramagnetic states is very subtle, therefore the density window where ferromagnetism can take place is small [1].
The rearrangement of single-particle degrees of freedom of a dilute two-dimensional electron gas in the vicinity of the quantum critical point is examined within a microscopic approach. It is shown that just beyond the critical point, the Landau state undergoes self-consistent rearrangement of the quasiparticle spectrum and momentum distribution. At very low temperatures, there emerges a multi-connected quasiparticle momentum distribution. With increasing temperature, two crossovers occur: a fermion condensate appears in the first and disappears in the second, giving way to universal non-Fermi-liquid behavior. Manifestations of these crossovers in thermodynamic properties of the electron gas are studied and characterized. The four quasiparticle phases predicted to exist in the vicinity of the critical point are collected in a schematic phase diagram.
This chapter focuses on experimental results obtained in low-disordered strongly interacting two-dimensional (2D) electron systems; in particular in (100)-silicon metal-oxide-semiconductor field-effect transistors (MOSFET). It describes the additional confirmation in favor of zero-temperature zero-field metal-insulator transition is provided by magnetic and thermopower measurements. The chapter argues that the metal-insulator transition in silicon samples with very low level of disorder is driven by interactions. In a low-disordered 2D electron system in silicon MOSFETs, the resistivity at a certain "critical" electron density shows virtually no temperature dependence over a wide range of temperatures. In 2000, it was experimentally found that the ratio between the spin and the cyclotron splittings in silicon MOSFETs strongly increases at low electron densities. The strong enhancement of the spin susceptibility has indicated that at low electron densities, the system behaves well beyond the weakly interacting Fermi liquid.
We study the two-dimensional electron gas in a quantizing magnetic field for the cases of Coulomb and harmonic interactions among the electrons. Numerical solutions for the quantum states of clusters of up to five electrons show that the strength of the excitation gap is a strongly oscillating function of the density not unlike what is observed in the anomalous quantum Hall effect. We present analytic results for the case of harmonic interactions and show that the variational wave function recently proposed by Laughlin for the Coulomb problem is in fact an exact eigenstate of the harmonic problem.