  Coulomb drag between parallel two-dimensional electronic layers is an
excellent tool for the study of electron-electron interactions. In actual
experiments, the layers display spatial charge density fluctuations due to
imperfections such as external charged impurities. However, at present a
systematic way of taking these inhomogeneities into account in drag
calculations has been lacking, making the interpretation of experimental data
problematic. On the other hand, there exists a highly successful and widely
accepted formalism describing transport within single inhomogeneous layers
known as effective medium theory. In this work, we generalize the standard
effective medium theory to the case of Coulomb drag between two inhomogeneous
sheets and demonstrate that inhomogeneity in the layers has a strong impact on
drag transport. In the case of exciton condensation between the layers, we show
that drag resistivity takes on a value determined by the amplitude of density
fluctuation. Next we consider drag between graphene sheets, in which the
existence of spatial charge density fluctuations is well-known. We show that
these inhomogeneities play a crucial role in explaining existing experimental
data. In particular, the temperature dependence of the experimentally observed
peaks in drag resistivity can only be explained by taking the layer density
fluctuations into account. We also propose a method of extracting information
on the correlations between the inhomogeneities of the layers. The effective
medium theory of Coulomb drag derived here is general and applies to all
two-dimensional materials.
