Disorder dependence of phase transitions in a Coulomb glass

Abstract

We have performed a Monte Carlo study of a three dimensional system of classical electrons with Coulomb interactions at half filling. We systematically increase the positional disorder by starting from a completely ordered system and gradually transitioning to a Coulomb glass. The phase transition as a function of temperature is second order for all values of disorder. We use finite size scaling to determine the transition temperature T C and the critical exponent n. We find that T C decreases and that n increases with increasing disorder.Electrons with long range Coulomb interactions in three dimensions display a rich and complex behavior. If there is translational invariance and a background of compensating positive charge, the system forms a Wigner crystal at low densities where the potential energy dominates the kinetic energy [1,2]. In the presence of quenched disorder the competition between interactions and disorder produces a Coulomb glass. Comparing these two extremes reveals similarities and differences. For example both undergo a phase transition when the temperature is lowered. In one case an ordered arrangement of electrons is formed while in the case of the Coulomb glass a highly disordered arrangement is frozen into place. Yet both low temperature phases have a gap in their single particle density of states.In this paper we study the effect of gradually introducing disorder into a three dimensional system of electrons with long range Coulomb interactions. The system is discrete in the sense that the electrons sit on half of the available sites. In the ordered case the sites form a cubic lattice. The disorder is introduced in the positions of the sites and their deviation from a cubic lattice. The Hamiltonian iswhere we set the charge e ¼ 1, n i is the number operator for site i, r ij ¼ jr i À r j j, and K is a compensating background charge making the whole system charge neutral. n i ¼ 1 (À1) for an occupied (unoccupied) site. We consider half-filling with K ¼ 1=2.We have simulated three dimensional systems of linear size L ¼ 4, 6, and 8. We place N ¼ L 3 sites in the system. We have only considered the case of half filling in order to take advantage of the particle-hole symmetry. For the ordered case the sites form a cubic lattice. In the ground state, every other site is occupied; the occupied sites form a face centered cubic (FCC) lattice. We can gradually introduce disorder by allowing the deviation of a site from its position in a cubic lattice to be chosen from a Gaussian distribution with a standard deviation of s. This gives the radial distance from the cubic lattice site. The angular coordinates of the site are chosen randomly using a uniform distribution. The ordered case corresponds to s ¼ 0. s ¼ 1 corresponds to a very disordered case with a standard deviation equal to the cubic lattice constant a. For all values of the disorder, the system

1/fnoise in electron glasses

“…The exact form of g(ε, T ) is not known, but some have argued [30,31,32] that its low temperature asymptotic behavior is described by g(ε = 0, T ) ∼ T d−1 . We have done Monte Carlo simulations of a three dimensional Coulomb glass with off-diagonal disorder and we find that g(ε = 0, T ) cannot be described by a simple power law [28,33]. The results of such simulations do not produce a density of states that is suitable for use in our noise integrals due to finite size effects.…”

On the application of the Edwards–Anderson order parameter to the Coulomb glass

“…This state of a disordered system possesses many intriguing features, such as, for instance, the energy gap in the density of single-particle excitations called a Coulomb gap [1]. It has never been clarified however, whether this correlated distribution of electrons really evidences a kind of a glasslike order, although several studies have been addressed to this particular question [2][3][4][5].…”

“…Further study was performed by Grannan and Yu [4], as well as by Overlin et al [5] using the three-dimensional lattice model with a weak spatial disorder introduced by some displacement of lattice sites with respect to their ideal positions. The authors recognized the glass transition in the Coulomb glass as the phase transition of the second order.…”