Algorithm for the construction of self-energies for electronic transport calculations based on singularity elimination and singular value decomposition

Abstract

We present a complete prescription for the numerical calculation of surface Green's functions and self-energies of semi-infinite quasi-one-dimensional systems. Our work extends previous results generating a robust algorithm to be used in conjunction with ab initio electronic structure methods. We perform a detailed error analysis of the scheme and find that the highest accuracy is found if no inversion of the usually ill conditioned hopping matrix is involved. Even in this case however a transformation of the hopping matrix that decreases its condition number is needed in order to limit the size of the imaginary part of the wave vectors. This is done in two different ways: either by applying a singular value decomposition and setting a lowest bound for the smallest singular value or by adding a random matrix of small amplitude. By using the first scheme the size of the Hamiltonian matrix is reduced, making the computation considerably faster for large systems. For most energies the method gives high accuracy, however in the presence of surface states the error diverges due to the singularity in the self-energy. A surface state is found at a particular energy if the set of solution eigenvectors of the infinite system is linearly dependent. This is then used as a criterion to detect surface states, and the error is limited by adding a small imaginary part to the energy.