First introduced by M. Pollak, the Coulomb gap is a soft gap in the single-particle density of states (DOS) of a system of interacting localized electrons. Due to the long-range Coulomb interactions, the single-particle DOS vanishes at the chemical potential, at low enough temperatures, such that thermal excitations do not wash out the gap.

Theory

At zero temperature, a classical treatment of a system gives an upper bound for the DOS near the Fermi energy, first suggested by Efros and Shklovskii. The argument is as follows: Let us look at the ground state as the energy of an electron is given by the expression:

1. .

. This is an upper bound for the Coulomb gap. Efros considered single electron excitations, and obtained an integro-differential equation for the DOS, showing the Coulomb gap in fact follows the above equation (i.e., the upper bound is a tight bound).

Other treatments of the problem include a mean-field numerical approach, as well as more recent treatments such as, also verifying the upper bound suggested above is a tight bound. Many Monte Carlo simulations were also performed, some of them in disagreement with the result quoted above. Few works deal with the quantum aspect of the problem. Classical Coulomb gap in clean system without disorder is well captured within Extended Dynamical Mean Field Theory (EDMFT) supported by Metropolis Monte Carlo simulations.

Experimental observations

Direct experimental confirmation of the gap has been done via tunneling experiments, which probed the single-particle DOS in two and three dimensions. The experiments clearly showed a linear gap in two dimensions, and a parabolic gap in three dimensions. Another experimental consequence of the Coulomb gap is found in the conductivity of samples in the localized regime. The existence of a gap in the spectrum of excitations would result in a lowered conductivity than that predicted by Mott variable-range hopping dependence is obtained, for any dimension. The observation of this is expected to occur below a certain temperature, such that the optimal energy of hopping would be smaller than the width of the Coulomb gap. The transition from Mott to so-called Efros–Shklovskii variable-range hopping has been observed experimentally for various systems. that fits neither the Mott nor the Efros–Shklovskii theories.

See also

• Coulomb's law

References