Electron transport in double quantum wells under the longitudinal size-effect regime

We study electron transport in contacted double quantum wells (DQW's) under conditions when a competition between the longitudinal (Ohmic) currents and transverse (tunneling) current leads to inhomogeneous distributions of the currents in the wells. If the characteristic length ${\mathrm{\kappa }}^{\mathrm{-}1}$ (estimated as a square root from the ratio of the averaged in-plane conductivity to tunneling conductance) of such inhomogeneities is comparable with the length L of the DQW's, the conductivity of the system depends on the size factor κL, i.e., the longitudinal size effect takes place. Basic equations describing in-plane distribution of the potentials in the wells are derived for a linear response regime in a phenomenological approach, while transport coefficients appearing in these equations are calculated from a microscopic theory. General form of boundary conditions (relations between the currents and potentials at the contacts) for the basic equations is presented. Different contacting schemes containing independent and common contacts are under consideration. When the frequency of the applied voltage becomes comparable with the ratio of the tunneling conductance to the transverse polarizability of DQW's, the appearance of retardation effects gives rise to a nontrivial size dependence of the ac response and transient characteristics of the DQW's under the longitudinal size-effect regime.