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Physical review. B, Condensed matter

Dynamical local-field factors and effective interactions in the two-dimensional electron liquid

Gurinder S. Atwal, Iya G. Khalil, Neil William Ashcroft

Published Mar 12, 2003• 22 citations
Publisher

Abstract

We present an analytical study of the dynamical local-field factors associated with the response of a homogeneous two-dimensional interacting electron liquid as functions of momentum, frequency, and density. We derive sum rules that constrain their asymptotic forms (in momentum and frequency) for both the spin-symmetric and spin-antisymmetric cases. Parametrized expressions for the local-field factors are proposed, based on all available sum rules and on many-body perturbation theory, and these are found to be in good agreement with quantum Monte Carlo calculations. Finally, these expressions are used to evaluate the effective electron-electron interaction in a local approximation for two-dimensional systems. It is shown that both the quantitative and qualitative behaviors of the interaction are sensitive to the inclusion of dynamical correlations.

DOI
10.1103/physrevb.67.115107

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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.
We analyze the effective electron-electron interaction in a two dimensional polarized paramagnetic system. The spin degree of freedom, s, is manifestly present in the expressions of spin dependent local field factors that describe the short range exchange (x) and correlation (c) effects. Starting from the exact asymptotic values of the local field correction functions for large and small momentum at zero frequency we obtain self-consistent expressions across the whole spectrum of momenta. Then, the effective interaction between two electrons with spins s and s' is calculated. We find that the four effective interactions, up-up, up-down, down-up and down-down, are different. We also obtain their qualitative dependence on the electronic density and polarization and note that these results are independent of the approximation used for the local field correction functions at intermediate momenta.
We present a quantitative study of the quasiparticle properties of a three-dimensional electron Fermi liquid. Our approach is based on the theory of the many-body local field factors which we use to include vertex corrections associated with charge and spin fluctuations. Extensive use is made of the results of recent quantum Monte Carlo calculations. Several models for the wave-vector dependence of the spin-antisymmetric many-body local field factor ${G}_{\ensuremath{-}}$, for which no numerical results are currently available, are discussed and compared. Both the real and imaginary parts of the self-energy as well as the quasiparticle renormalization constant and the enhancement of the effective mass are calculated in the experimentally accessible range of electron densities. The results obtained by means of the on-shell approximation are critically compared with those given by a self-consistent solution of the Dyson equation. An ultraviolet catastrophe in the Coulomb-hole part of the self-energy is identified and a satisfactory resolution of this impasse is presented. The same many-body local field factors are also used to obtain the Landau interaction function. This allows us to obtain both the spin susceptibility and the proper compressibility of the system.
We present numerical calculations of the effect of electron-electron interactions on the quasiparticle properties such as the effective mass and the Land\'e ggg-factor in a GaAs∕AlGaAs\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{Ga}\mathrm{As}GaAs∕AlGaAs triangular quantum well from which the spin susceptibility is obtained. For this purpose, we consider quantum many-body effects associated with charge- and spin-density fluctuations induced many-body vertex corrections. The approach is based on the many-body local-field factors which are extracted from Fermi hypernetted-chain self-consistent calculation through the fluctuation-dissipation theorem. We find the spin susceptibility in good agreement with the recent experimental measurements and quantum Monte Carlo simulation data for such a system in the weak and intermediate coupling limits.
Motivated by a number of recent experimental studies we have revisited the problem of the microscopic calculation of the quasiparticle self-energy and many-body effective mass enhancement in an unpolarized two-dimensional electron liquid. Our systematic study is based on the many-body local field theory and takes advantage of the results of the most recent diffusion Monte Carlo calculations of the static charge and spin response of the electron liquid. We report extensive calculations of both the real and imaginary parts of the quasiparticle self-energy. We also present results for the many-body effective mass enhancement and the renormalization constant over a broad range of electron densities. In this respect we critically examine the relative merits of the on-shell approximation, commonly used in weak coupling situations versus the actual self-consistent solution of the Dyson equation. We show that already for ${r}_{s}\ensuremath{\simeq}3$ and higher, a solution of the Dyson equation proves necessary in order to obtain a well-behaved effective mass. Finally we find confirmation that the inclusion of both charge- and spin-density fluctuations beyond the random phase approximation is indeed crucial to get reasonable agreement with recent measurements.