Diagonal and off-diagonal disorder in doped quantum wells

We consider the formation of an impurity band in the electronic density of states (DOS) of a quantum well of ${\mathrm{Ga}}_{1\mathrm{-}\mathrm{x}}$ ${\mathrm{Al}}_{\mathrm{x}}$As/GaAs. We consider both the bandwidth coming out of a fluctuation in the binding energy (diagonal disorder) and a fluctuation in the transfer matrix ($V_{\mathrm{ij}}$) due to randomness on the impurity locations in the plane parallel to the interfaces. We calculate the DOS for an impurity layer thickness half the well length, and show that in this case the off-diagonal disorder is dominant. A comparison with experimental results is made.