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Physical review. B, Condensed matter

Ground state of the two-dimensional electron gas

B. Tanatar, David M. Ceperley

Published Mar 15, 1989• 1,034 citations
Publisher

Abstract

Variational and fixed-node Green's-function Monte Carlo calculations have been performed to find the ground-state properties of the two-dimensional electron gas in the density range 1\ensuremath{\le}${r}_{s}$\ensuremath{\le}100. Our calculations predict a Wigner crystallization at the density ${r}_{s}$\ensuremath{\simeq}37\ifmmode\pm\else\textpm\fi{}5. The electron system is found to be in the normal- (paramagnetic) fluid state below the transition density, but the fully polarized state is very close in energy. We have tabulated the values of pair distribution function g(r), the static structure factor S(k), and the momentum distribution n(k) at several densities of interest both in the normal and the polarized phases. An estimate of the spin susceptibility \ensuremath{\chi} is also given.

DOI
10.1103/physrevb.39.5005

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Variational and fixed-node Green's-function Monte Carlo calculations have been performed to find the ground-state properties of the two-dimensional electron gas in the density range 1\ensuremath{\le}rs{r}_{s}rs​\ensuremath{\le}100. Our calculations predict a Wigner crystallization at the density rs{r}_{s}rs​\ensuremath{\simeq}37\ifmmode\pm\else\textpm\fi{}5. The electron system is found to be in the normal- (paramagnetic) fluid state below the transition density, but the fully polarized state is very close in energy. We have tabulated the values of pair distribution function g(r), the static structure factor S(k), and the momentum distribution n(k) at several densities of interest both in the normal and the polarized phases. An estimate of the spin susceptibility \ensuremath{\chi} is also given.
Quantum Monte Carlo methods are used to calculate various ground-state properties of charged bosons in two dimensions, throughout the whole density range where the fluid phase is stable. Wigner crystallization is predicted at ${r}_{s}\ensuremath{\simeq}60.$ Results for the ground-state energy and the momentum distribution are summarized in analytic interpolation formulas embodying known asymptotic behaviors. Near freezing, the condensate fraction is less than 1%. The static structure factor S(k)S(k)S(k) and susceptibility $\ensuremath{\chi}(k)$ are obtained from the density-density correlation function in imaginary time, $F(\mathbf{k},\ensuremath{\tau}).$ An estimate of the energy of elementary excitations, given in terms of an upper bound involving S(k)S(k)S(k) and $\ensuremath{\chi}(k),$ is compared with the result obtained via analytic continuation from $F(\mathbf{k},\ensuremath{\tau}).$
We have used the variational and diffusion quantum Monte Carlo methods to calculate the energy, pair-correlation function, static structure factor, and momentum density of the ground state of the two-dimensional (2D) homogeneous electron gas. We have used highly accurate Slater-Jastrow-backflow trial wave functions and twist averaging to reduce finite-size effects where applicable. We compare our results with others in the literature and construct a local-density-approximation exchange-correlation functional for 2D systems.
We use quantum Monte Carlo methods to calculate the zero-temperature phase diagram of the two-dimensional homogeneous electron gas. We find a transition from a paramagnetic fluid to an antiferromagnetic triangular Wigner crystal at density parameter r(s)=31(1) a.u. and a transition to a ferromagnetic crystal at r(s)=38(5) a.u. The fully spin-polarized fluid is never stable. We search for, but do not find, the ferromagnetic "hybrid" phase proposed by H. Falakshahi and X. Waintal [Phys. Rev. Lett. 94, 046801 (2005)10.1103/PhysRevLett.94.046801].
Accurate diffusion Monte Carlo technique within the fixed-node approximation has been used to determine the equation of state of the two-dimensional electron gas in three phases: the normal paramagnetic liquid, the ferromagnetic (spin-polarized) liquid, and the Wigner crystal. Our backflow fixed-node calculation, which is a significant improvement over two previous conflicting Slater-Jastrow calculations, shows that there is only one clear phase transition from the paramagnetic liquid to the Wigner crystal as the density decreases. Unlike the three-dimensional case, we do not see the stability of the polarized liquid phase at intermediate densities before the crystallization.
We present a numerical study of magnetic phases of the 2D electron gas near freezing. The calculations are performed by diffusion Monte Carlo in the fixed-node approximation. At variance with the 3D case we find no evidence for the stability of a partially polarized phase. With plane wave nodes in the trial function, the polarization transition takes place at r s = 20, whereas the best available estimates locate Wigner crystallization around r s = 35. Using an improved nodal structure, featuring optimized backflow correlations, we confirm the existence of a stability range for the polarized phase, although somewhat shrunk, at densities achievable nowadays in 2-dimensional hole gases in semiconductor heterostructures. The spin susceptibility of the unpolarized phase at the magnetic transition is approximately 30 times the Pauli susceptibility.
Variational and diffusion quantum Monte Carlo methods are employed to investigate the zero-temperature phase diagram of the three-dimensional homogeneous electron gas at very low density. Fermi fluid and body-centered cubic Wigner crystal ground-state energies are determined using Slater-Jastrow-backflow and Slater-Jastrow many-body wave functions at different densities and spin polarizations in finite simulation cells. Finite-size errors are removed using twist-averaged boundary conditions and extrapolation of the energy per particle to the thermodynamic limit of infinite system size. Unlike previous studies, our results show that the electron gas undergoes a first-order quantum phase transition directly from a paramagnetic fluid to a body-centered cubic crystal at density parameter rs=86.6(7){r}_{\text{s}}=86.6(7)rs​=86.6(7), with no region of stability for an itinerant ferromagnetic fluid.