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Physical review. B, Condensed matter

Static local-field corrections of two-dimensional electron liquids

Naoki Iwamoto

Published Jan 15, 1991• 61 citations
Publisher

Abstract

The parametrized form of local-field corrections is introduced to construct the dielectric function and the nonlocal spin susceptibility that describe the dielectric properties of a paramagnetic degenerate electron liquid in two dimensions beyond the random-phase approximation (RPA). The parameters that take into account the spin-parallel and spin-antiparallel electron-electron interactions at short distances are determined in such a way that the response functions correctly give the Monte Carlo values for the compressibility and the spin susceptibility, respectively, in the long-wavelength limit. The local-field correction thus obtained is compared with that in the Hubbard approximation; the static dielectric function and the spin susceptibility are also compared with those in the RPA. It is found that the sign of the static dielectric function depends on both the dimensionless coupling constant and the wave number, which is a feature markedly different from the RPA results.

DOI
10.1103/physrevb.43.2174

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The parametrized form of local-field corrections is introduced to construct the dielectric function and the nonlocal spin susceptibility that describe the dielectric properties of a paramagnetic degenerate electron liquid in two dimensions beyond the random-phase approximation (RPA). The parameters that take into account the spin-parallel and spin-antiparallel electron-electron interactions at short distances are determined in such a way that the response functions correctly give the Monte Carlo values for the compressibility and the spin susceptibility, respectively, in the long-wavelength limit. The local-field correction thus obtained is compared with that in the Hubbard approximation; the static dielectric function and the spin susceptibility are also compared with those in the RPA. It is found that the sign of the static dielectric function depends on both the dimensionless coupling constant and the wave number, which is a feature markedly different from the RPA results.
A deeper understanding of low-dimensional and small-scale electronic systems requires an accurate characterization of the many-electron effects. Based on the electron liquid (EL) model, these many-body effects in the three-dimensional (3D) and two-dimensional (2D) electronic systems are dealt going beyond the random phase approximation within the zero-temperature framework. The dielectric formulation of the many-body problem is recapitulated and the longitudinal dielectric function of the EL is obtained using the self-consistent local-field correction scheme, known as STLS, under a general spin-valley degeneracy. Extensive explicit expressions are provided for the use of other researchers. The performance of this formalism is compared in both 3D and 2D by the quantum Monte Carlo data and the pseudopotential approach introduced by Pines and co-workers. STLS is observed to be a highly satisfactory technique, with some reservations on the long-wavelength behaviour. Ground-state energy, correlation energy and the isothermal compressibility of the 2D EL are computed using a variety of approaches. Negative compressibility and overscreening are discussed in the light of the recent experiments.
We use the Ceperley and Alder Green's-function Monte Carlo data for the static form factor, S(q)S(q)S(q), of an electron liquid to calculate the static local-field correction, G(q)G(q)G(q), to the dielectric function which takes into account short-range electron correlations. The resulting local-field correction in the long-wavelength limit is obtained by smoothly fitting to the results of Iwamoto and Pines, which are exact in this limit. We use the G(q)G(q)G(q) thus obtained to calculate the static and dynamic form factors, the contribution to the correlation energy from different momentum transfers, plasmon dispersion, and the plasmon-dispersion coefficient. We compare these theoretical quantities with the available experimental data, and use the ${\ensuremath{\omega}}^{3}$-sum rule (which we also calculate from the Monte Carlo data) to discuss the limitations arising from the use of the static local-field correction.
The Thomas-Fermi (TF) approximation for the static dielectric function of a three-dimensional (3D) electron liquid can be derived by minimizing the TF local-density approximation for the kinetic-energy functional. Here we show that this connection between energy functionals and model dielectric functions is not an artifact, but a general paradigm. Four examples are worked out in detail, by calculating the model dielectric functions that follow, respectively, from (i) exchange corrections to TF theory in 3D, i.e., TF-Dirac theory, (ii) further correlation corrections to TF-Dirac theory in 3D, (iii) TF theory in two dimensions (2D), and (iv) exchange corrections to TF theory in 2D. Each of these cases has certain interesting features, revealing connections between independent many-body methods, that are discussed in some detail. As a by-product of these investigations we also find that a common textbook statement about the long-wavelength $(k\ensuremath{\rightarrow}0)$ limit of the random-phase approximation is not fully correct.
We discuss interaction effects for the one-dimensional electron gas with a repulsive \ensuremath{\delta}-function interaction potential by using the random-phase approximation and a local-field correction. Analytical results for the local-field correction of charge-density fluctuations and spin-density fluctuations are obtained. The ground-state energy is found to be in better agreement with the exact result than the ground-state energy calculated within the random-phase approximation. We calculate the pair-correlation functions for electrons with parallel and antiparallel spins and the paramagnetic susceptibility. The energies of the collective density modes ${\mathrm{\ensuremath{\omega}}}_{\mathrm{d}}$ (q) and the collective spin modes ${\mathrm{\ensuremath{\omega}}}_{\mathrm{s}}$ (q) are calculated and compared with the energy of the electron-hole excitations ${\mathrm{\ensuremath{\omega}}}_{\mathrm{eh}\ifmmode\pm\else\textpm\fi{}}$ (q): ${\mathrm{\ensuremath{\omega}}}_{\mathrm{d}}$ (q\ensuremath{\rightarrow}0)>${\mathrm{\ensuremath{\omega}}}_{\mathrm{eh}\ifmmode\pm\else\textpm\fi{}}$ (q\ensuremath{\rightarrow}0)>${\mathrm{\ensuremath{\omega}}}_{\mathrm{s}}$ (q\ensuremath{\rightarrow}0). The critical exponents for the long-distance behavior of correlation functions are found to be described by local-field corrections. We compare our results for the critical exponents with recent results obtained by bosonization techniques and the conformal field theory.
The problem of the many-body enhancement of the static spin susceptibility at long wavelengths and its relation to the quasiparticle effective mass is investigated for a normal electron gas in two-dimensional space as a function of the electronic density. We start from a discussion of the results of the simple Hartree-Fock approximation for various interaction potentials and proceed to develop a complete theory. We find that the effects of the electron-electron interaction are significantly larger than in the familiar three-dimensional case. Our approach is based on a new self-consistent scheme which goes beyond the simple random-phase approximation by explicitly allowing for charge- and spin-fluctuation-induced vertex corrections of the Hubbard type. We show that when the latter are neglected, the many-body enhancement of the spin susceptibility can be cast in a remarkably simple and elegant analytic form.
We have evaluated the density-density response of the two-dimensional electron gas at zero temperature by solving the Dyson equation for the particle-hole Green's function, including exchange Coulomb matrix elements and short-range contributions in the ladder approximation. We study the effect of these correlations on the total energy, compressibility per particle, local field factor G ( q ), static structure factor and pair-correlation function. Results are compared with the normal random-phase approximation, local field theories and quantum Monte Carlo calculations.