Computing the full spectrum of large sparse palindromic quadratic eigenvalue problems arising from surface Green's function calculations

Abstract

Full spectrum of a large sparse inverted perpendicular-palindromic quadratic eigenvalue problem inverted perpendicular-PQEP) is considered arguably for the first time in this article. Such a problem is posed by calculation of surface Green's functions SGFs) of mesoscopic transistors with a tremendous non-periodic cross-section. For this problem, general purpose eigensolvers are not efficient, nor is advisable to resort to the decimation method etc. to obtain the Wiener-Hopf factorization. After reviewing some rigorous understanding of SGF calculation from the perspective of inverted perpendicular-PQEP and nonlinear matrix equation, we present our new approach to this problem. In a nutshell, the unit disk where the spectrum of interest lies is broken down adaptively into pieces small enough that they each can be locally tackled by the generalized inverted perpendicular-skew-Hamiltonian implicitly restarted shift-and-invert Arnoldi G inverted perpendicular SHIRA) algorithm with suitable shifts and other parameters, and the eigenvalues missed by this divide-and-conquer strategy can be recovered thanks to the accurate estimation provided by our newly developed scheme. Notably the novel non-equivalence deflation is proposed to avoid as much as possible duplication of nearby known eigenvalues when a new shift of G inverted perpendicular SHIRA is determined. We demonstrate our new approach by calculating the SGF of a realistic nanowire whose unit cell is described by a matrix of size 4000 x 4000 at the density functional tight binding level, corresponding to a 8 x 8 nm(2) cross-section. We believe that quantum transport simulation of realistic nano-devices in the mesoscopic regime will greatly benefit from this work. (c) 2017 Elsevier Inc. All rights reserved.