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Physical review. B, Solid state

Correlations in a two-dimensional electron system

A. K. Rajagopal, John C. Kimball

Published Mar 1, 1977• 183 citations
Publisher

Abstract

The correlation energy per particle (expressed in rydbergs) of a two-dimensional electron gas is calculated by summing the ring diagrams. For high densities, this is found to be of the form $C+D{r}_{s}\mathrm{ln}{r}_{s}+O({r}_{s})$, where ${r}_{s}$ is a dimensionless parameter such that ${r}_{s}^{\ensuremath{-}2}$ is proportional to the number of electrons per unit area. The value of $D$ is -0.172 and that of $C$ is -0.38 \ifmmode\pm\else\textpm\fi{} 0.04, the latter involves a numerical evaluation. By the same methods, we have also calculated the difference in total energy of the nonmagnetic and ferromagnetic states and we find for ${r}_{s}>2.3$ the system may be ferromagnetic. We have also obtained some exact relationships between the pair correlation function at zero separation and the asymptotic behavior of the structure factor and the momentum distribution function. These are of interest in relation to scattering experiments.

DOI
10.1103/physrevb.15.2819

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Journal of Physics Conference Series•Mar 1, 2016•4 citations
The correlation energy per particle (expressed in rydbergs) of a two-dimensional electron gas is calculated by summing the ring diagrams. For high densities, this is found to be of the form C+Drslnrs+O(rs)C+D{r}_{s}\mathrm{ln}{r}_{s}+O({r}_{s})C+Drs​lnrs​+O(rs​), where rs{r}_{s}rs​ is a dimensionless parameter such that rs\ensuremath−2{r}_{s}^{\ensuremath{-}2}rs\ensuremath−2​ is proportional to the number of electrons per unit area. The value of DDD is -0.172 and that of CCC is -0.38 \ifmmode\pm\else\textpm\fi{} 0.04, the latter involves a numerical evaluation. By the same methods, we have also calculated the difference in total energy of the nonmagnetic and ferromagnetic states and we find for rs>2.3{r}_{s}>2.3rs​>2.3 the system may be ferromagnetic. We have also obtained some exact relationships between the pair correlation function at zero separation and the asymptotic behavior of the structure factor and the momentum distribution function. These are of interest in relation to scattering experiments.
A discussion of the various properties of pair correlation functions of a magnetic electron gas is presented. The cusp conditions obeyed by the many-electron wave functions for a pair of electrons with parallel and antiparallel spins in close proximity are expressed as derivative conditions on the corresponding parallel-and antiparallel-spin pair correlation functions. In particular, it is shown that in a fully ferromagnetic gas the parallel spin correlations determine the correlation energy whereas in the paramagnetic case, this function is of less importance in comparison to the antiparallel-spin correlation function. In the high-momentum limit, it is found that the momentum distribution of electrons is dominated by antiparallel-spin correlations in the partially magnetic case whereas in the fully ferromagnetic case, it is dominated by parallel-spin correlations but is smaller by a factor of ${q}^{\ensuremath{-}2}$. A calculation of the parallel-spin correlation function for the ferromagnetic gas in perturbation theory is presented. A model for the exchange correlation energy of the system which takes into account the exact conditions given here is proposed. Corresponding results for the two-dimensional electron gas are also stated.
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.
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.
A methodology is introduced for the systematic treatment of electron-electron correlations in solids and other interacting quantum NNN-particle systems. The method is developed within the framework of electronic structure theory (band theory) but, in contrast to conventional approaches, which are based on the single-particle picture, it is formulated within a many-particle picture in which nnn particles in ddd-dimensional phase space are treated as a single particle in a phase space of nd\mathrm{nd}nd dimensions. In this phase space, interparticle interactions appear as external potentials allowing the treatment of the system of particles through the use of single-particle methods, while at the same time allowing a systematic, direct, and nonperturbative treatment of interparticle interactions. The method makes use of the invariance of the Hamiltonian describing an interacting-particle system under partitioning into subsystems of nnn particles. This treatment leads to exact results in the limit $n\ensuremath{\rightarrow}N$. Based on such partitioning, we propose a generalization of density functional theory and an appropriately defined local density approximation to treat the interactions between the nnn-particle units in a system of N> nN>~nN> n particles. This approach yields nnn-particle correlated densities and nnn-particle states which can be used in an analysis of the electronic properties of materials, such as total energy, phase stability, electronic transport, and others. We use the formal construct of multiple-scattering theory to develop the method for the calculation of the two-particle electronic structure of a solid and the corresponding total energy of the ground state. We also illustrate some of the properties of the method in terms of a Hubbard model Hamiltonian on a linear ring. Various features of the method and further possible applications are presented in a discussion section.
The ground state of a two-dimensional charged-boson system is investigated over the range of densities 1\ensuremath{\lesssim}Rs{R}_{s}Rs​\ensuremath{\lesssim}10 in the self-consistent-field approximation; Rs{R}_{s}Rs​=(a02{a}_{0}^{2}a02​\ensuremath{\pi}\ensuremath{\rho}${)}^{\mathrm{\ensuremath{-}}1/2}$, where a0{a}_{0}a0​ is the Bohr radius and \ensuremath{\rho} is the number density. Starting with numerical self-consistent calculations of the static structure factor, the elementary excitation, pair-correlation functions, pressure, and ground-state energy are evaluated. These results are compared with those of the two-dimensional and three-dimensional systems obtained from other methods. The ground-state energy is given as ${\mathit{E}}_{0=\mathrm{\ensuremath{-}}1.2918{\mathit{R}}_{\mathit{s}}}^{\mathrm{\ensuremath{-}}2/3}$+0.03, which improves that result from the ring-diagram approximation.
A trial wave function Ψ(1, 2,..., N ) of an N electron system can always be written as the product of an antisymmetric Fermion factor F { z ij } = Π i<j z ij , and a symmetric correlation factor G { z ij }. F results from the Pauli principle, and G is caused by Coulomb interactions. One can represent G diagrammatically [1] by distributing N points on the circumference of a circle, and drawing appropriate lines representing correlation factors (cfs) z ij between pairs. Here, of course, z ij = z i – z j , where z i is the complex coordinate of the i th electron. Laughlin correlations for the ν = 1/3 filled incompressible quantum liquid (IQL) state contain two cfs connecting each pair ( i,j ). For the Moore-Read state of the half-filled excited Landau level (LL), with ν = 2 + 1/2, the even value of N for the half-filled LL is partitioned into two subsets A and B , each containing N /2 electrons [2]. For any one partition ( A,B ), the contribution to G is given by G AB = Π i<j∈A z ij 2 Π k<ι∈B z kι 2 . The full G is equal to the symmetric sum of contributions G AB over all possible partitions of N into two subsets of equal size. For Jain states at filling factor ν = p/q < 1/2, the value of the single particle angular momentum ι satisfies the equation 2 ι = ν -1 N - C ν , with C ν = q + 1 - p . The values of (2 ι, N ) define the function space of G { z ij } , which must satisfy a number of conditions. For example, the highest power of any z i cannot exceed 2 ι + 1 - N . In addition, the value of the total angular momentum L of the lowest correlated state must satisfy the equation L = ( N /2)(2 ι +1 - N ) - K G , where K G is the degree of the homogeneous polynomial generated by G . Knowing the values of L for IQL states (and for states containing a few quasielectrons or a few quasiholes) from Jain's mean field CF picture allows one to determine K G . The dependence of the pair pseudopotential V(L 2 ) on pair angular momentum L 2 suggests a small number of correlation diagrams for a given value of the total angular momentum L . Correlation diagrams and correlation functions for the Jain state at ν = 2/5 and for the Moore-Read states will be presented as examples. The generalizations of the method of selecting G from small to larger systems will be discussed.