  We present a theory of local electric polarization in crystalline solids and
apply it to study the case of wurtzite group-III nitrides. We show that a local
value of the electric polarization, evaluated at the atomic sites, can be cast
in terms of a summation over nearest-neighbor distances and Born effective
charges. Within this model, the local polarization shows a direct relation to
internal strain and can be expressed in terms of internal strain parameters.
The predictions of the present theory show excellent agreement with a formal
Berry phase calculation for random distortions of a test-case CuPt-like InGaN
alloy and InGaN supercells with randomly placed cations. While the present
level of theory is appropriate for highly ionic compounds, we show that a more
complex model is needed for less ionic materials, in which the strain
dependence of Born effective charges has to be taken into account. Moreover, we
provide ab initio parameters for GaN, InN and AlN, including hybrid functional
values for the piezoelectric coefficients and the spontaneous polarization,
which we use to accurately implement the local theory expressions. In order to
calculate the local polarization potential, we also present a point dipole
method. This method overcomes several limitations related to discretization and
resolution which arise when obtaining the local potential by solving Poisson's
equation on an atomic grid. Finally, we perform tight-binding supercell
calculations to assess the impact of the local polarization potential arising
from alloy fluctuations on the electronic properties of InGaN alloys. In
particular, we find that the large upward bowing with composition of the InGaN
valence band edge is strongly influenced by local polarization effects.
Furthermore, our analysis allows us to extract composition-dependent bowing
parameters for the energy gap and valence and conduction band edges.
