Three-dimensional electron gas with localization along one, two, and three directions R.M. Me Ândez-Moreno a , M.A. Ortiz a, * , S. Orozco a , M. Moreno a,b a Departamento de Fõ Âsica, Facultad de Ciencias, Universidad Nacional Auto Ânoma de Me Âxico, Apartado Postal 21-092, 04021 Mexico, DF, Mexico b Instituto de Fõ Âsica, Universidad Nacional Auto Ânoma de Me Âxico, Apartado Postal 20-364, 01000 Mexico, DF, Mexico Received 16 April 2001; accepted 29 October 2001 by E.E. Mendez Abstract A powerful non-perturbative technique, which allows a direct evaluation of the ground state properties of an interacting electron gas in three dimensions, has been developed. In a uni®ed approach, the low-, intermediate-, and high-density regions are considered. This technique is applied to three-dimensional (3D) systems with periodic electron density along one and two directions. The electronic and magnetic states of these systems are theoretically studied on the basis of the deformable jellium model, within a self-consistent Hartree±Fock approach. To determine the magnetic character of the 3D electron gas ground state, the paramagnetic and ferromagnetic energies are calculated and compared at low, intermediate, and high densities. As r s increases, several symmetry and/or magnetic transitions occur, in each system. The electronic and magnetic states obtained are compared with other theoretical results reported in the literature with different models. In addition to planar and linear periodic electron densities, cubic electron densities are considered in order to look for symmetry transitions among these systems. q 2002 Elsevier Science Ltd. All rights reserved. PACS: 71.10.2w; 71.30.1h; 71.10.Ca Keywords: D. Electronic states (localized); D. Phase transitions; D. Wigner crystal A large number of experimental and theoretical studies of the electronic and magnetic states for the electron gas have been carried out. Three-, two-, and one-dimensional electron gas models are of great interest because of theoretical [1±8] and technological [9,10] implications. Indeed, the electron gas is a cornerstone of many-body quantum theory applica- tions to condensed matter physics. Also, a lot of interesting studies of metallic thin ®lms have been carried out in recent years. The motion of electrons con®ned to move freely in three-dimensional (3D) thin ®lms, or even in one or two spatial dimensions gives rise to a variety of interesting phenomena [3,11,12]. Transport and optical properties are topics of investigation for our understanding and potential use in several devices [9,10], i.e. the results may be used to study heterostructures with ®nite thickness. On the other hand, most calculations of physical properties for low- dimensional systems have used the 3D results. As special cases of 3D systems, materials showing anisotropic properties have been studied, with different approaches [11±13]. The new technical developments should ensure that anisotropic materials remain a productive venue for pursuing the electronic properties of condensed matter for many years to come. The ground state of a 3D electron gas has been obtained with several methods. A wide variety of theoretical estimates were produced over the years, most of these were carried out within the Hartree±Fock (H±F) approxi- mation, or within the density-functional approach. At high densities, the system is in a ¯uid state, the competition between kinetic and potential energy favors a uniform charge distribution. As the density is reduced, the Coulomb interaction gains importance and, at intermediate and low densities, a symmetry breaking favors an inhomogeneous charge distribution. A periodic density becomes energeti- cally more favorable than the ¯uid state [1±8,14]. Finally, as Wigner suggested in the early thirties [15], at very low Solid State Communications 121 (2002) 223±227 0038-1098/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S0038-1098(01)00482-3 PERGAMON www.elsevier.com/locate/ssc * Corresponding author. Tel.: 152-56-22-48-55; fax: 152-56-22- 48-54. E-mail address: maof@hp.fciencias.unam.mx (M.A. Ortiz).