Hartree-Fock energy of a density wave in a spin polarized two-dimensional electron gas

We calculate the Hartree-Fock energy of a density-wave in a spin polarized twodimensional electron gas using a short-range repulsive interaction. We find that the stable ground state for a short-range potential is always either the paramagnetic state or the uniform ferromagnetic state. The energy of a density-wave state is, however, reduced by a factor proportional to (1 − ζ), where ζ is the polarization of the electron gas. Since this situation occurs in the most unfavorable conditions (shortrange repulsive interaction) it is therefore conceivable that by including higher order many-body corrections to the interaction a density-wave ground state is indeed found to be stable. Hartree-Fock energy of a density wave in a spin polarized two-dimensional electron gas2 The possible existence of a non-uniform ground state for a two-dimensional (2D) electron system in the presence of a magnetic field has been proposed in explaining the results of a number of recent experiments. Several examples of such puzzling data are the unidirectional charge-density-wave states that appear in partially filled Landau levels [1, 2], the incompressible, inhomogeneous insulating phase in p-GaAs/AlGaAs heterostructures displaying metal-insulator transition [3, 4], and the unusual magnetooptical properties of the ferromagnetic phase of p-type Cd1−xMnxTe quantum wells [5]. Numerical calculations[6] have definitely proved that at low densities an unpolarized electron gas forms a Wigner crystal. However, a study of point defects in the twodimensional Wigner crystal suggests that the quantum melting could be continuous rather than first order[7], leaving open the possibility of inhomogeneous intermediate phases. The idea of a non-uniform ground state in a paramagnetic 2D electron system has surfaced before. Prior to the discovery of the quantum Hall effect, it was argued that in 2D GaAs-type structures the ground state was a charge-density-wave [8]. More recent experimental[1, 2] and theoretical[9, 10] results point indeed to the existence of charge-density-wave states in partially filled higher Landau levels on account of the quasi-one-dimensional electron motion. In this work we are interested in the possible formation of density waves in a 2D electron system that exhibits a very large Zeeman splitting. This situation occurs in II-VI dilute-magnetic semiconductor structures where the effective Landé factor is up to thousands of times the band value. This is exactly opposite to the case in GaAs, where the cyclotron energy is dominant. For example, in GaAs the ratio of the cyclotron frequency to the Zeeman splitting is around 14, while the same ratio is close to 1/20 in Zn1−xCdxSe. Quantum Monte Carlo results show that the exchange-correlation hole is larger in the polarized electron gas[11, 12] suggesting that a highly polarized system is prone to density instabilities. Whether or not the instability develops before the Wigner crystallization and its dependence on the degree of polarization has not been studied yet. Here, we calculate the Hartree-Fock (HF) energy of a spin density wave state in a spin polarized system for a delta-function repulsive interaction, the most unfavorable situation for the development of a density-wave instability in an isotropic system. It is well known that in three dimensions, within the HF approximation and for an unscreened Coulomb interaction, the paramagnetic state is unstable with respect to formation of a spin density wave with wave vector near 2kF [13]. However, for a short-range potential the stable HF solution is always either the usual paramagnetic state or the uniform ferromagnetic state [14]. We find that this result holds also in the 2D polarized system. In this case, however, the difference in energy between a density wave state with momentum q, E(q)/N , and the ferromagnetic state, E(q = 0))/N , is reduced by a factor proportional to (1− ζ), where ζ = (n↑ − n↓)/n is the polarization Hartree-Fock energy of a density wave in a spin polarized two-dimensional electron gas3 of the electron gas, E(q)− E(q = 0) N = q 1− ζ 4 . (1) If the background charge is allowed to relax, as in the deformable jellium model, our result applies to both spinand charge-density waves [15]. Since the effective electronic interaction in real systems is somewhere in between the short-range delta-function and the unscreened Coulomb potential our result suggest the possibility of an inhomogeneous density wave ground state in highly polarized twodimensional electron systems. If this inhomogeneous state exists it will be most probably a charge density wave, since electronic correlations in real systems favor charge-density over spin-density wave instabilities [15]. Below we present the details of our analysis. A density wave develops when the correlation function between an electron with momentum k and spin σ and another electron with momentum k+q and spin σ, 〈ψ kσψk+qσ′〉 becomes finite. When σ = σ , a charge-density wave is formed, while σ 6= σ corresponds to a spin-density wave. The interacting electron system Hamiltonian is diagonalized by a canonical transformation that introduces a new set of operators: ψ k = cos (θk/2)ψk−2σ + sin (θk/2)ψk+ q 2 σ (2) ψ k = − sin (θk/2)ψk−2σ + cos (θk/2)ψk+ q 2 σ , (3) where θk is the coupling parameter and ψ lower k (ψ upper k ) refers to the new lower (upper) band excitations. The Hartree-Fock ground state energy is a function of q and the parameters θk [14]. For a system with N electrons, volume V and polarization ζ the total ground state energy is: