Contour integral method for obtaining the self-energy matrices of electrodes in electron transport calculations

Abstract

We propose an efficient computational method for evaluating the self-energy matrices of electrodes to study ballistic electron transport properties in nanoscale systems. To reduce the high computational cost incurred in large systems, a contour integral eigensolver based on the Sakurai-Sugiura method combined with the shifted biconjugate gradient method is developed to solve exponential-type eigenvalue problem for complex wave vectors. A remarkable feature of the proposed algorithm is that the numerical procedure is very similar to that of conventional band structure calculations. We implement the developed method in the framework of the real-space higher-order finite difference scheme with nonlocal pseudopotentials. Numerical tests for a wide variety of materials validate the robustness, accuracy, and efficiency of the proposed method. As an illustration of the method, we present the electron transport property of the free-standing silicene with the line defect originating from the reversed buckled phases.

Complex band structure calculations based on the overbridging boundary matching method without using Green's functions

“…(11) at different energies ε = ε. As a consequence of exploiting the general property of the Krylov subspace, we can omit expensive operations of matrix-vector products in iterative solvers for second and subsequent reference energies, and greatly reduce the computational cost for executing the Sakurai-Sugiura projection method [21,27]. This allows us to draw band structures denser in energy.…”

“…, it is seen that a BiCG solver solves Eq. (11) for two sampling points z a and z b = 1/z * a at once [21]. Note that the superscript * denotes the complex conjugate of a scalar.…”

“…To further reduce the computational costs, it should be noted that the integral along an outer (inner) radius of an annular domain can be reused when calculating the integral along the inner (outer) radius of the next annular domain. Thus we can successively reduce the total number of the BiCG iterations thanks to the convergence property of the BiCG method and the reuse of the integral although the product N rhs N div is larger than N rhs of the single domain on k-plane [21] in some cases. Now let us compare the accuracy of λ and ψ obtained by solving Eq.…”

“…In this section, we compare the computational time for solving the eigenvalue problems for generalized Bloch states in an annular domain with the present method with/without the two computational technique as well as with the methods proposed in Refs. [16,21]. The coefficient matrices of Eq.…”

“…The Sakurai-Sugiura projection method reduces the dimension of an eigenvalue problem to the number of the eigenvalues located in a specific domain of the complex plane. Recently, we applied it to a quadratic eigenvalue problem (3) represented in the real-space finite-difference formalism by making use of the advantage of the Sakurai-Sugiura projection method on the contour integral for sparse matrices [17,21,22]. The method for solving the quadratic eigenvalue problem is clearly advantageous over the conventional OBM method in terms of avoiding expensive computation of the Green's function matrix and treating sparse coefficient matrices.…”