### 5.7 Interface Roughness Scattering

The roughness of interfaces in a heterostructure leads to spatial fluctuations of the well width, and
consequently to fluctuations of the energy levels. These fluctuations of the energy levels
act as a fluctuating potential for the motion of confined carriers [89]. A distribution of
terraces is present at the interfaces and the electrons are scattered elastically by them
[90].

The randomness of the interface is described by a correlation function at the in-plane position
x_{∥}= (x, y), which is usually taken to be Gaussian with a characteristic height of the roughness Δ, and
a correlation length Λ representing a length scale for fluctuations of the roughness along the interface
[91], such that

| (5.32) |

The perturbation in the potential V (z) due to a position shift Δ(x_{∥}) is given by

| (5.33) |

For the I-th interface, which is centered about the plane z_{I} and extends over the range [z_{L,I},z_{R,I}], the
scattering matrix element can be defined as

where |α^{′ } ⟩ and |α⟩ denote the final and initial wave functions, respectively. Here, φ_{α′α,I} is defined
as
| (5.35) |

and the rectangular function reads

The expectation value of the square of the matrix element is given by

| (5.36) |

Making use of eq. (5.32) and Fermi’s Golden Rule, the interface roughness induced scattering rates are
given by [92]

where k _{∥ }^{′ } and k_{∥} are the final and initial wave vectors, respectively, and θ is the scattering angle. The
integral is evaluated numerically by means of the MATLAB integration routines.