Non-Linear and Non-Equilibrium Properties of Two-Dimensional Disordered Systems. I: Conductivity

Abstract

The non-linear dependences of the non-metallic part of conductivity σ′ on uniform electric and magnetic fields, E and H, are investigated in two-dimensional disordered systems. We calculate (i) σ′(T, E), (ii) σ′(T, H) and (iii) σ′(T, E, H) in the crossing fields E⊥H. There exist two kinds of the characteristic fields of non-linearity: E0=√6ℏ/eτl and H0=cℏ/el2, which are independent of temperature T, and E0(T)=E0η3/2 and H0(T)=H0η, which depend on T. Here η=τ/τε(T) and τ and l are the relaxation time and mean free path due to the elastic impurity scattering, and τε is the energy relaxation time. We show that (1) σ′(T, E) behaves like ln(E/E0) in Ec(T)≪E≪E0, (2)σ′(T, H) behaves like ln(H/H0) in Hc(T)≪H≪H0, (3) both approach lnη in E≪Ec(T) and H≪Hc(T) respectively, (4) σ′(T, E, H) depends still logarithmically, though a little weakened, on the respective fields in their ranges of intermediate strengths, and (5) σ′(T, E, H) crosses over to σ′(T, H) or σ′(T, E) as E or H is weakened. Recent experiments on σ′(T, E) are explained by our theory qualitatively. The possible log E dependence of the excess conductivity of a superconducting film is studied in the Appendix.