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arXiv:cond-mat/0509159 · Physical Review B

Construction of model dielectric functions for two- and three-dimensional electron liquids from density functionals

Ana Paula Favaro, Klaus Capelle, João Vítor Batista Ferreira

Published Jan 31, 2006• 6 citations
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Abstract

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.

DOI
10.1103/physrevb.73.045133

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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→0)(k\ensuremath{\rightarrow}0)(k\ensuremath→0) limit of the random-phase approximation is not fully correct.
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.
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.
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 problem of the Friedel oscillations in two- and three-dimensional electron Fermi liquids is examined by means of the many-body local fields theory as aided by the most recent results of accurate numerical studies based on the quantum Monte Carlo method. Within linear response, an exact answer is obtained for the amplitude of the electron density distortion due to both short- and long-ranged impurity potentials. It is discussed how a measurement of the local environment of an impurity embedded in an otherwise homogeneous electron liquid can be used to characterize the systems as a Fermi liquid as well as to extract the magnitude of the static many-body corrections beyond the random phase approximation for wave vector equal to 2kF2{k}_{F}2kF​.
Spin correlations in an interacting electron liquid are studied in the high-frequency limit and in both two and three dimensions. The third-moment sum rule is evaluated and is used to derive exact limiting forms (at both long and short wavelengths) for the spin-antisymmetric local-field factor, ${limop}_{\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\omega}}\ensuremath{\infty}}{G}_{\ensuremath{-}}(\mathbf{q},\ensuremath{\omega}).$ In two dimensions ${limop}_{\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\omega}}\ensuremath{\infty}}{G}_{\ensuremath{-}}(\mathbf{q},\ensuremath{\omega})$ is found to diverge as 1/q1/q1/q at long wavelengths, and the spin-antisymmetric exchange-correlation kernel of time-dependent spin density functional theory diverges as 1/q2{1/q}^{2}1/q2 in both two and three dimensions. These signal a failure of the local-density approximation, one that can be redressed by alternative approaches.
In a recent paper [B. P. van Zyl et al., Phys. Rev. A 89, 022503 (2014)], the average density approximation (ADA) was implemented to develop a parameter-free, nonlocal kinetic energy functional to be used in the orbital-free density functional theory of an inhomogeneous, two-dimensional (2D) Fermi gas. In this work, we provide a detailed comparison of self-consistent calculations within the ADA with the exact results of the Kohn-Sham density functional theory and the elementary Thomas-Fermi (TF) approximation. We demonstrate that the ADA for the 2D kinetic energy functional works very well under a wide variety of confinement potentials, even for relatively small particle numbers. Remarkably, the TF approximation for the kinetic energy functional, without any gradient corrections, also yields good agreement with the exact kinetic energy for all confining potentials considered, although at the expense of the spatial and kinetic energy densities exhibiting poor pointwise agreement, particularly near the TF radius. Our findings illustrate that the ADA kinetic energy functional yields accurate results for both the local and global equilibrium properties of an inhomogeneous 2D Fermi gas, without the need for any fitting parameters.