Scattering from rough surfaces

The energy of charged particles in a quantum well is strongly affected by its thickness

Interface roughness scattering is most noticeable in confined systems, in which the energies for charge carriers are determined by the locations of interfaces. An example of such a system is a quantum well, which may be constructed from a sandwich of different layers of semiconductor. Variations in the thickness of these layers therefore causes the energy of particles to be dependent on their in-plane location in the layer.[2] Although the roughness ${\displaystyle \Delta _{z}(\mathbf {r} )}$ varies in a complicated way on a microscopic scale, it can be considered to exhibit a Gaussian distribution[3] characterised by a height ${\displaystyle \Delta }$ and a correlation length ${\displaystyle \Lambda }$ such that

${\displaystyle \langle \Delta _{z}(\mathbf {r} )\Delta _{z}(\mathbf {r'} )\rangle =\Delta ^{2}\exp \left(-{\frac {|\mathbf {r} -\mathbf {r'} |^{2}}{\Lambda ^{2}}}\right)}$