Hopping conductivity for localised electronic states

Abstract

The form of the current density operator, involving a velocity operator and suitable for use in band-conduction calculations, is used here to calculate the conductivity in a system of localised electronic states. In the site representation, it has zero diagonal matrix elements. Using a relationship between diagonal and off-diagonal elements of position and electron-phonon interaction valid for this system, the authors show that their result is equivalent to a similar Green function calculation of Manucharyants and Zvyagin (1974), where the time derivative of the dipole moment operator was used as the current operator. In doing the calculation, they are able to emphasise the equivalent role played by diagonal and off-diagonal elements of position and electron-phonon interaction. They confirm that the same result is attained when a canonical transformation is performed to remove the diagonal elements of electron-phonon interaction from the Hamiltonian.