Large-scale linear system solver using secondary storage: Self-energy in hybrid nanostructures

Abstract

We present a Fortran library which can be used to solve large-scale dense linear systems, A x = b . The library is based on the LU decomposition included in the parallel linear algebra library PLAPACK and on its out-of-core extension POOCLAPACK. The library is complemented with a code which calculates the self-polarization charges and self-energy potential of axially symmetric nanostructures, following an induced charge computation method. Illustrative calculations are provided for hybrid semiconductor–quasi-metal zero-dimensional nanostructures. In these systems, the numerical integration of the self-polarization equations requires using a very fine mesh. This translates into very large and dense linear systems, which we solve for ranks up to 3 × 10 5 . It is shown that the self-energy potential on the semiconductor–metal interface has important effects on the electronic wavefunction. Program summary Program title: HDSS (Huge Dense System Solver) Catalogue identifier: AEHU_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEHU_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 98 889 No. of bytes in distributed program, including test data, etc.: 1 009 622 Distribution format: tar.gz Programming language: Fortran 90, C Computer: Parallel architectures: multiprocessors, computer clusters Operating system: Linux/Unix Has the code been vectorized or parallelized?: Yes. 4 processors used in the sample tests; tested from 1 to 288 processors RAM: 2 GB for the sample tests; tested for up to 80 GB Classification: 7.3 External routines: MPI, BLAS, PLAPACK, POOCLAPACK. PLAPACK and POOCLAPACK are included in the distribution file. Nature of problem: Huge scale dense systems of linear equations, A x = B , beyond standard LAPACK capabilities. Application to calculations of self-energy potential in dielectrically mismatched semiconductor quantum dots. Solution method: The linear systems are solved by means of parallelized routines based on the LU factorization, using efficient secondary storage algorithms when the available main memory is insufficient. The self-energy solver relies on an induced charge computation method. The differential equation is discretized to yield linear systems of equations, which we then solve by calling the HDSS library. Restrictions: Simple precision. For the self-energy solver, axially symmetric systems must be considered. Running time: About 32 minutes to solve a system with approximately 100 000 equations and more than 6000 right-hand side vectors using a four-node commodity cluster with a total of 32 Intel cores.