Determinant approach to the scattering matrix elements in quasi-one-dimensional and two-dimensional disordered systems

Abstract

We have developed an approach based on the characteristic determinant ͑the Green function poles͒ to solve the Dyson equation in quasi-one-dimensional ͑Q1D͒ and two-dimensional disordered systems without any restriction on the numbers of impurities and modes. We consider two different models for a disordered Q1D wire: a set of two-dimensional ␦ potentials with signs and strengths determined randomly, and a tight-binding Hamiltonian with several modes and on-site disorder. We calculate analytically the scattering matrix elements for particles coming both from the left and from the right without actually determining the eigenfunctions of the electrons. It is shown that the poles of the Green functions for these models can be deduced from a determinant of rank N ϫ N ͑N is the number of scatterers͒ instead of the rank NM ϫ NM ͑M is the number of modes͒. We calculate the inverse localization lengths for the two models. They are exactly on the order of w 2 for the weak disorder regime and are valid for an arbitrary number of channels, M.

Localization length in the quasi one-dimensional disordered system revisited

“…Particularly, LLs calculated in Refs. [4,5,7,8] result in an incorrect dependence on M , while LLs calculated [6,9] correctly predicted the M dependence, but failed to provide the exact magnitude of LL.…”

“…A non-perturbative analytical approach, based on the Green's function formalism to solve the Dyson equation in Q1D and two-dimensional (2D) disordered systems without any restriction on the numbers of impurities and modes, was developed in Refs. [7][8][9]. For a TB Hamiltonian with several modes and on-site disorder the electron's scattering matrix elements T nm (hence the wire conductance G = nm T nm T * nm (in units of e 2 /h)) were analytically calculated for an arbitrary impurity profile without actually determining the eigenfunctions.…”

“…For a TB Hamiltonian with several modes and on-site disorder the electron's scattering matrix elements T nm (hence the wire conductance G = nm T nm T * nm (in units of e 2 /h)) were analytically calculated for an arbitrary impurity profile without actually determining the eigenfunctions. In these papers [7][8][9] only HW conditions were discussed, which correspond to arranging the parallel equidistant chains on a plane.…”

“…Once this is established, motivated by our doubts about the correctness of LLs results of Refs. [4][5][6][7][8][9] and to overcome the difficulty of the discrepancy, we have reconsidered the calculation of LL for the Q1D TB anisotropic Anderson model, using the Greens function approach, developed in Refs. [7][8][9].…”

“…Next, closely following Refs. [7][8][9], we evaluate the scattering matrix elements T nm , in the weak disordered regime. The result for the electron transmission amplitude T nm is…”