Hopping transport equation for electrons in superlattices with vertical disorder

We develop a theory of vertical hopping transport in doped superlattices with intentional uncorrelated vertical disorder introduced by controlled random variations of well widths. For structures with sufficiently large disorder, the vertical conductance (in the direction of the growth axis) is limited by phonon-assisted hopping between the wells. It is shown that due to quasi-equilibrium situation within the wells, the master rate equation for transitions between the electronic states of the structure can be reduced to a truncated rate equation for inter-well transitions only. At low bias, the solution of this rate equation is shown to be equivalent to finding total resistance of a quasi-one-dimensional network of resistances expressed in terms of integral transition rates between the wells. Using this approach, we estimate the temperature dependence of the vertical resistance of superlattices with intentional disorder and show that it can be non-activated for not too low doping levels and temperatures.