  Motivated by the observation and prediction of clustering behavior for
impurities substituted into the host lattice of a real material, and the
dramatic impact this can have on electronic properties, we develop a simple
approach to describe such an effect via the electron self-energy. We employ a
disorder averaged T-matrix expansion taken to second order, which we modify to
include a clustering probability parameter. This approach circumvents the need
for specific cluster probability distributions, simplifying greatly the
analysis of clustered impurities. To gain analytical insights, we study a
nearest-neighbor square lattice tight-binding Hamiltonian with clustered
impurity substitutions to investigate clustering of off-diagonal hopping
impurities. We find that our T-matrix approach is in excellent agreement with
exact numerical results from a tight-binding computation performed with the
KWANT package. We observe a variety of interesting impurity clustering-induced
effects in the self-energy such as the suppression of quasi-particle lifetimes
at certain momenta and an increase in localization, as indicated by the inverse
participation ratio. The KWANT results are reproduced in our modified T-matrix
approach. In addition, our method allows for a full analytical treatment of
clustering effects which can aid in physical insight.
