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Physical review. B, Condensed matter

Spin susceptibility in a two-dimensional electron gas

Sudhakar Yarlagadda, Gabriele F. Giuliani

Published Sep 15, 1989• 45 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.40.5432

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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.
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.
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.
The spin susceptibility of the electron gas is studied using a functional integral formulation. A Hubbard-Stratonovich transformation is used to decouple the electron-electron interaction. A uniform Fock mean field is chosen, and effects of small fluctuations about this mean field are retained up to second order. The term that gives the usual spin random-phase approximation susceptibility is identified, with the ladder sum contained in the propagator for the field fluctuations. Computations of the static spin susceptibility outline the region of stability as a function of electron-gas coupling parameter and wave vector. These computations parallel those of Hamann and Overhauser [Phys. Rev. 143, 183 (1966)] for the Yukawa interaction. We use the analytic continuation method of Staunton et al. [Phys. Rev. Lett. 82, 3340 (1999)] to extend the calculations to nonzero frequency. The resulting spectral density shows the increase in low-energy spin fluctuations as the spin-density-wave transition is approached, and the development of a pseudogap in the spin fluctuation spectrum near the wave vector of the transition.
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.
A simple calculation of the wave-vector- and frequency-dependent paramagnetic spin susceptibility of an interacting electron gas is presented. The equation of motion of the double-time Green's function is solved using a moment-conserving decoupling approximation. For the short-range interaction, our expression for the dynamic spin susceptibility reduces to a previous result of Wolff. In the static and long-wavelength limit, our result for the Stonerenhanced spin susceptibility coincides with that obtained from consideration of the groundstate energy in the Hartree-Fock approximation.
The main topic of the present thesis is represented by the many-body effects which characterize the physical behavior of an electron liquid in various realizations. We begin by studying the problem of the response of an otherwise homogeneous electron liquid to the potential of an impurity embedded in its bulk. The most dramatic consequence of this perturbation is the existence of so called Friedel density oscillations. We present calculations of their amplitude valid in two as well as in three dimensions. The second problem we will discuss is that of the correlation effects in a three dimensional electron liquid in the metallic density regime. A number of quasiparticle properties are evaluated: the electron self-energy, the quasiparticle effective mass and the renormalization constant. We also present an analysis of the effective Landé g-factor as well as the compressibility. The effects of the Coulomb interactions beyond the random phase approximation have been treated by means of an approach based on the many-body local field factors theory and by utilizing the latest numerical results of Quantum Monte Carlo numerical simulations. The final chapter includes the results of our extensive work on various aspects regarding the two dimensional Fermi liquid in the presence of linear Rashba spin-orbit coupling. By using a number of many-body techniques, we have studied the interplay between spin-orbit coupling and electron-electron interaction. After proving an extension to the famous Overhauser Hartree-Fock instability theorem, a considerable amount of work will be presented on the problem of the density and spin response functions. For the study of the spin response, we will present the results of extensive numerical calculations based on the time dependent mean field theory approach.