ELECTRON TRANSPORT IN NANO DEVICES MATHEMATICAL INTRODUCTION AND PRECONDITIONING Olga Trichtchenko Master of Science Department of Mathematics and Statistics McGill University Montr´eal,Qu´ebec June 15, 2009 A thesis submitted to McGill University in partial fulfilment of the requirements of the degree of Master of Science c Olga Trichtchenko, 2009 ACKNOWLEDGEMENTS I thank my supervisor and the students at both McGill and my temporary home, SFU. I also thank my family and my friends for their support. ii ABSTRACT In this thesis we outline the mathematical principles behind density func- tional theory, and describe the iterative Kohn-Sham formulation for compu- tation of electronic density calculations in nano devices. The model for com- putation of the density of electrons in such device is a non-linear eigenvalue problem that is solved iteratively using the resolvent formalism. There are several bottlenecks to this approach and we propose methods to resolve them. This iterative method involves a matrix inversion. This matrix inversion is called upon when calculating the Green's function for a particular system, the two-probe device. A method to speed up this calculation is to use a precondi- tioning technique to accelerate the convergence of the iterative method. Tests the existing algorithm for a one-dimensional system are presented. The results indicate that these preconditioning methods reduce the condition number of the matrices. iii RESUM´ E´ Dans cette th`ese,nous pr´esentons les principes math´ematiques`ala base de la th´eoriede la fonctionnelle de la densit´e,et nous d´ecrivons la formule Kohn- Sham it´erative pour le calcul des densit´esd'´electrondans les composants nano- ´electroniques. Le mod`elede densit´e´electroniqueest un probl`eme de valeur- propre non-lin´eaireque l'on r´esoutde mani`ereit´erative. Il y a plusieurs compli- cations li´ees`acette technique et nous proposons des m´ethodes pour y rem´edier. On formule le syst`eme`al'aide du calcul de l'op´erateurhamiltonien dans une base particuli`ere. Cette inversion de matrice est n´ecessairelors du calcul de la fonction de Green pour le syst`eme en question: l'appareil `adeux sondes. Afin d'acc´el´ererce calcul, nous utilisons une technique de pr´econditionnement bas´eesur la nature it´erative du probl`eme. Nous pr´esentons les r´esultatsde nos essais avec diff´erents pr´econditionneurs. Ceux-ci indiquent que ces m´ethodes r´eduisent le nombre de conditionnement de notre matrice. Ce pr´econditionnement est donc appliqu´e`ades algorithmes d'inversion it´eratives classiques tels que la m´ethode de Gauss-Seidel et la m´ethode du r´esiduminimal g´en´eralis´ee.En effet, nous observons une r´eductiondu nombre d'it´erationsn´ecessairespour le calcul de la matrice inverse. iv TABLE OF CONTENTS ACKNOWLEDGEMENTS . ii ABSTRACT . iii RESUM´ E..................................´ iv LIST OF TABLES . vii LIST OF FIGURES . viii 1 Introduction . 1 1.1 Overview . 2 2 Density Functional Theory . 5 2.1 Overview . 5 2.2 The Hamiltonian . 5 2.3 Density Functional Theory . 6 2.4 Variational Principle . 9 2.5 Full Energy Functional . 10 2.5.1 Hartree Potential, UH . 11 2.5.2 Exchange-Correlation Potential, UXC . 12 2.6 Summary . 13 3 Green's Functions and Density . 14 3.1 Overview . 14 3.2 Green's Functions . 14 3.3 Density . 17 3.3.1 Contour Integration . 17 3.3.2 Integration Along Real Axis . 19 3.4 General Formalism . 21 3.5 Approximations . 24 3.6 Summary . 24 4 Two Probe Device . 26 4.1 Overview . 26 4.2 Infinite System . 26 4.3 Non-Equilibrium Eigenvalues . 29 v 4.4 Total Density . 31 4.4.1 Fermi Distribution . 31 4.4.2 Fermi-Dirac Statistics . 32 4.5 Self-Energies . 34 4.5.1 Inverse of Block Tridiagonal Matrix . 34 4.5.2 Inverse of a Hamiltonian for a Periodic Potential . 37 4.5.3 Bloch Theorem . 38 4.5.4 Boundary Condition . 39 4.6 Spectra . 41 4.7 Discussion of Basis Functions . 44 4.8 Summary . 45 5 Numerical Methods and Results . 46 5.1 Overview . 46 5.2 Kohn-Sham Equations . 46 5.2.1 Schematic of the Solution Scheme . 47 5.2.2 Bottlenecks . 48 5.3 Broyden's Method . 48 5.3.1 Convergence . 50 5.4 Gaussian Quadrature . 51 5.5 Pseudo-Code . 51 5.6 Conditioning . 53 5.7 Sample One-Dimensional System . 54 5.7.1 Iterative Matrix Inversion Schemes . 55 5.7.2 Some Notes on Inverses . 57 5.7.3 Preconditioning . 58 5.7.4 Results . 59 5.8 Comments and Future Directions . 67 5.9 Summary . 67 6 Conclusion . 68 APPENDIX: Condition Numbers for Three-Dimensional System . 70 REFERENCES . 73 vi LIST OF TABLES Table page 5{1 Given energy values, Ei for a sample one-dimensional system. 54 vii LIST OF FIGURES Figure page 1{1 This is a scheme of the algorithm used. The two sets of grey arrows represent the two iterative steps. One iterative step is inside the full algorithm over the iterative matrix inversion step and the other showing that the full algorithm is repeated until self-consistency. 4 3{1 The poles for GR lie in the lower half plane and are enclosed by A the contour Γ2 whereas the poles for G lie in the upper half complex plane and are enclosed by Γ1. 21 4{1 A diagram of a two probe device representing the atoms and their arrangement. 27 4{2 A schematic diagram of the different regions which represent a device with two leads of infinite length labelled LL and RR. 27 4{3 This figure represents the allowable energy states for a sample crystal. The diagram on the left shows the energy bands for particular points in Fourier space and the diagram on the right shows where these points are located in a crystal lattice. 31 4{4 This figure illustrates that depending on where the Fermi- energy level is, the inorganic material will have different properties. 34 5{1 The sparsity of the Hamiltonian for a two-probe, one-dimensional device where the size of the matrix is 696 by 696. 55 5{2 This figure illustrates the difference in the number of iterations it takes to solve the linear system using Gauss-Seidel and GMRES methods with and without a preconditioner. The Gauss-Seidel is in red in the upper half of the figure, whereas GMRES is in the lower half in blue. The preconditioner used (1) (k−1) (k) was G(E1) and the stopping criteria was kx −x kL2 ≤ 10−6 with preconditioned results shown using a solid line. For comparison, the dotted line is what happens when no preconditioner is applied. Preconditioned inside means preconditioned inside the GMRES calculation. 61 viii 5{3 This figure illustrates the difference in the number of iterations it takes to solve the linear system using Gauss-Seidel and GMRES methods with and without a preconditioner. The Gauss-Seidel is in red in the upper half of the figure, whereas GMRES is in the lower half in blue. The preconditioner used (1) (k−1) (k) was G(E2) and the stopping criteria was kx −x kL2 ≤ 10−6 with preconditioned results shown using a solid line. For comparison, the dotted line is what happens when no preconditioner is applied. Preconditioned inside means preconditioned inside the GMRES calculation. 62 5{4 This figure illustrates the difference in the number of iterations it takes to solve the linear system using Gauss-Seidel and GMRES methods with and without a preconditioner. The Gauss-Seidel is in red in the upper half of the figure, whereas GMRES is in the lower half in blue. The preconditioner used (1) (k−1) (k) was G(Ei) and the stopping criteria was kx −x kL2 ≤ 10−6 with preconditioned results shown using a solid line. For comparison, the dotted line is what happens when no preconditioner is applied. Preconditioned inside means preconditioned inside the GMRES calculation. 63 5{5 This figure illustrates the difference in the number of iterations it takes to solve the linear system using Gauss-Seidel and GMRES methods with and without a preconditioner. The Gauss-Seidel is in red in the upper half of the figure, whereas GMRES is in the lower half in blue. The preconditioner (k−1) (k−1) used was G(Ei) and the stopping criteria was kx − (k) −6 x kL2 ≤ 10 with preconditioned results are shown using a solid line. For comparison, the dotted line is what happens when no preconditioner is applied. Preconditioned inside means preconditioned inside the GMRES calculation. 64 5{6 This figure illustrates the different condition numbers of ma- (k) (k) trices [EiS − H ]. Notice that the most successful preconditioning algorithms are the ones where the energy values of preconditioners coincide with the energy values of (k) (k) the iterates of [EiS − H ]. 65 5{7 This figure illustrates the different condition numbers for the two most successful preconditioning schemes. 66 6{1 Chosen Gaussian quadrature points for the integration of the system in three dimensions. 72 ix 6{2 Condition numbers of a sample two-probe device for each computed Hamiltonian. Recall that H is a function of energy value in the complex plane, as well as the wave vector. The line divides two successive iterations of the Kohn-Sham schemes. 72 x CHAPTER 1 Introduction One learns early in an undergraduate program how to compute the en- ergetic states of the hydrogen atom, but the generalization even to helium is already time-consuming without approximations.