The frequency dependence of nonlinear conductivity in non-homogeneous systems: An analytically solvable model

For a single-particle hopping model of arbitrary dimension we determine analytically the linear and nonlinear parts of the response to a periodic external field. Despite its simplicity the model contains the effects of localized double-well potential dynamics as well as long-range transport, i.e. reflecting key elements of the dynamics in truly disordered systems. The model parameters reflect typical symmetries between adjacent sites as well as the degree of barrier disorder. It is shown that in the 1D case the dc-limit of the nonlinear conductivity behaves differently than that in higher dimensions. Only for low dimensions the nonlinear conductivity displays a minimum. The scaling of this minimum with system parameters is derived. It is shown that for a broad range of frequencies the nonlinear conductivity can be expressed as a superposition of one contribution related to the linear conductivity, and another one related to the nonlinearity in a double-well potential. The present results are also discussed in the context of recent measurements of the non-linear conductivity for inorganic ion conductors as well as ionic liquids.