  In realistic nanoelectronics, disordered impurities/defects are inevitable
and play important roles in electron transport. However, due to the lack of
effective quantum transport method to do disorder average, the important
effects of disorders remain largely un-explored or poorly understood. Here, we
report a generalized non-equilibrium vertex correction method with coherent
potential approximation for the non-equilibrium quantum transport simulation of
disordered nanoelectronics. In this method, the disorder average of various
Green's functions are computed by a generalized coherent potential
approximation. A generalized non-equilibrium vertex correction algorithm is
then developed to calculate disorder average of the product of any two real
time single-particle Green's functions. We obtain nine non-equilibrium vertex
corrections and find they can be solved by a set of simple linear equations. As
a result, the averaged non-equilibrium density matrix and various important
transport properties, including averaged current, disordered induced current
fluctuation and the averaged shot noise, can all be efficiently computed in a
unified simple scheme. Moreover, the relationship between the non-equilibrium
vertex correction method and the non-equilibrium coherent potential
approximation theory is clarified, and we prove the non-equilibrium coherent
potential equals the non-equilibrium vertex correction and this equivalence is
guaranteed by the Keldysh's formulas. In addition, a generalized form of
conditionally averaged non-equilibrium Green's function is derived to
incorporate with density functional theory to enable first-principles quantum
transport simulation. Our approach provides a unified, efficient and
self-consistent method for simulating non-equilibrium quantum transport through
disordered nanoelectronics.
