Optimal block-tridiagonalization of matrices for coherent charge transport

• Published in: Journal of computational physics Vol. 228; no. 23; pp. 8548 - 8565
• Main Authors: Wimmer, Michael, Richter, Klaus
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Inc 10.12.2009; Elsevier
• Subjects: ALGORITHMS; Block-tridiagonal matrices; Blocking; CHARGE TRANSPORT; CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Coherent quantum transport; Computational techniques; ELECTRON GAS; Exact sciences and technology; GRAPH THEORY; Graphene; GREEN FUNCTION; HAMILTONIANS; Mathematical analysis; Mathematical methods in physics; MATRICES; Matrix reordering; Optimization; Partitioning; Physics; QUANTUM DECOHERENCE; Recursive Green’s function algorithm; SEMICONDUCTOR MATERIALS; Semiconductors; Transport; TRANSPORT THEORY; 72.10.Bg; 05C78; 05C50; Recursive Green’s function algorithm; 02.70.−c; Coherent quantum transport; Block-tridiagonal matrices; Matrix reordering; Graph theory; 02.10.Ox; Semiconductor materials; Green function; Calculation methods; Algorithms; 02.70.-c; Hamiltonians; Recursive Green's function algorithm; Calculation; Tridiagonal matrix; Performance; Mesoscopic systems; Quantum transport; Electron gas
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2009.08.001

Numerical quantum transport calculations are commonly based on a tight-binding formulation. A wide class of quantum transport algorithms require the tight-binding Hamiltonian to be in the form of a block-tridiagonal matrix. Here, we develop a matrix reordering algorithm based on graph partitioning techniques that yields the optimal block-tridiagonal form for quantum transport. The reordered Hamiltonian can lead to significant performance gains in transport calculations, and allows to apply conventional two-terminal algorithms to arbitrarily complex geometries, including multi-terminal structures. The block-tridiagonalization algorithm can thus be the foundation for a generic quantum transport code, applicable to arbitrary tight-binding systems. We demonstrate the power of this approach by applying the block-tridiagonalization algorithm together with the recursive Green’s function algorithm to various examples of mesoscopic transport in two-dimensional electron gases in semiconductors and graphene.