Electron Transport in Nano Devices Mathematical Introduction and Preconditioning

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 functional theory, and describe the iterative Kohn-Sham formulation for computation of electronic density calculations in nano devices. The model for computation 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 preconditioning 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 thèse,nous présentons les principes mathématiquesàla base de la théoriede la fonctionnelle de la densité,et nous décrivons la formule Kohn- Sham itérative pour le calcul des densitésd’électrondans les composants nano- électroniques. Le modèlede densitéélectroniqueest un problème de valeur- propre non-linéaireque l’on résoutde manièreitérative. Il y a plusieurs compli- cations liéesàcette technique et nous proposons des méthodes pour y remédier. On formule le systèmeàl’aide du calcul de l’opérateurhamiltonien dans une base particulière. Cette inversion de matrice est nécessairelors du calcul de la fonction de Green pour le système en question: l’appareil àdeux sondes. Afin d’accélérerce calcul, nous utilisons une technique de préconditionnement baséesur la nature itérative du problème. Nous présentons les résultatsde nos essais avec différents préconditionneurs. Ceux-ci indiquent que ces méthodes réduisent le nombre de conditionnement de notre matrice. Ce préconditionnement est donc appliquéàdes algorithmes d’inversion itératives classiques tels que la méthode de Gauss-Seidel et la méthode du résiduminimal généralisée.En effet, nous observons une réductiondu nombre d’itérationsnécessairespour 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 energetic states of the hydrogen atom, but the generalization even to helium is already time-consuming without approximations. How can this be generalized even further to small scale devices in which there are many individual molecules, atoms and electrons that each play important roles in the properties of the system? Well, starting from density functional theory, this goal can be achieved. Density functional theory is used to describe a many-body system, accounting for every interaction and every contributing particle, by correctly computing the electronic density [14]. From this, the electronic transport through a nano device can be computed. In turn, the understanding of these electronic properties leads to new applications for the devices for purposes like early cancer detection and faster quantum computing that comprise the general growing ﬁeld of nanotechnology [1]. However, calculating the electronic transport properties is no small feat. Starting from the work by Hohenberg and Kohn in the 60s [9],[14] with additions by Sham a few years later [13], [12] and currently, the ongoing work done by Prof. Hong Guo’s group at McGill [19], [20] which motivated this thesis, a more complete theory and computational scheme for electronic density calculations was formulated. This thesis provides a review of some key physical and mathematical concepts behind density functional theory and derivations of the Kohn-Sham scheme to compute electronic density. This review includes physical intuition as well as numerical algorithms involved in accomplishing the goal. Math- ematically, density functional theory is a very rich ﬁeld. Emphasizing the

1 techniques from numerical analysis in particular, density functional calculations use algorithms such as ﬁxed point iteration, numerical eigenvalue solvers, quadrature, weak formulation of partial diﬀerential equations, direct and iterative matrix inversion schemes and last but not least, preconditioning. These are only some of the techniques used in this ”ab-initio” method of modelling nano-electronic devices. The main focus of this present work was to speed up the matrix inversion algorithm currently invoked by introducing preconditioning schemes based on the iterative nature of the problem. These prove to be successful in lowering the condition number of the matrix to be inverted and reducing the iteration number in classical matrix iterative inversion schemes. 1.1 Overview

The layout of the thesis is as follows. Chapter 2 will introduce the reader to basic concepts such as the Hamiltonian, the Schrödingerequation and variational principle to compute ground state density. It will outline the computational difficulties associated with the many-particle Hamiltonian and end with the Kohn-Sham Hamiltonian based on the electronic density including terms such as the Padéapproximation of the exchange-correlation energy and the Hartree potential computed by solving a Poisson equation for the particular set-up as seen in [19]. The next chapter, will familiarize the reader with the method used to compute the electronic density given a Hamiltonian. This is done by the Green’s functional approach and contour integration. Some basic complex analysis will be outlined to make the derivation self-contained. Special attention should be paid to the fact that in matrix notation, the Green’s function is the inverse of the Hamiltonian and the density is the projection onto the eigenbasis of the Hamiltonian. This chapter contains a discussion on what happens when the device gets more complicated.

2 The following chapter proceeds with the deﬁnition of the Hamiltonian matrix for a device of particular interest, the two-probe device. Two algorithms involving the Fourier transform and an iterative block tri-diagonal inverse are also discussed in this chapter. The two-probe device has particular boundary conditions which introduce an asymmetry in the Hamiltonian. The discussion of the spectrum of the Hamiltonian and the physical intuition behind it, follows. The bottleneck of the scheme are shown to be in the change of basis and the matrix inversion steps of the Kohn-Sham algorithm. Focussing on the matrix inversion bottleneck, the thesis is concluded by Chapter 5 with a discussion of the types of condition numbers seen in the particular examples provided by Prof. Hong Guo’s group. Matrix inversion algorithms are discussed with focus on their utility for the current model of electron transport. Some tables which illustrate the speed of Gauss-Seidel and GMRES are included. Concluding remarks follow in Chapter 6. Note, since there are two iterative steps in the algorithm, a schematic diagram representing them is shown in Figure 1–1. One iterative step is inside the full algorithm and is invoked when a linear system is solved as represented by a smaller set of elliptical arrows in the ﬁgure. The other shows that the full algorithm to compute the electronic density is iterated over until self- consistency is reached and this can be seen as the bigger set of grey arrows.

3 !"#$%&'(%&)*)'%+&,*& 6%72%&84%))&9".&5&,*& 34%))&5& -'%./0$%&+/'.-1& :"2'"4.&-2'%8./0"2& -2$%.)-"2&

Figure 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 CHAPTER 2 Density Functional Theory 2.1 Overview

The main goal of this chapter is to illustrate how, given a system with many atoms and therefore many electrons, the total energy state of the system represented by a Hamiltonian can be written in terms of the electronic density. This is done by considering the Hohenberg-Kohn theorems [12], [9], that result in a statement which says there exists a unique map between the Hamiltonian and the electronic density. This Hamiltonian is written as a combination of diﬀerent potentials which account for the interactions between electrons and the system, these being the Hartree, exchange-correlation and an external contribution, as well as the kinetic energy. A discussion of the closed form of these potentials then follows with concluding remarks about the diﬃculty of the task of computing them. 2.2 The Hamiltonian

Before embarking on the discussion about the fundamental theories for the model discussed throughout the thesis, it is important to review the concept of the Hamiltonian, as used in quantum mechanics. The Hamiltonian, often denoted as H, is an operator from L2 to L2 . For our purposes, its eigenbasis corresponds to the wave functions of electrons, denoted as Ψ(x). By construction, this Hamiltonian is also Hermitian in the L2 inner product. That is, it can be diagonalized, i.e. has an orthonormal basis with the real eigenvalues representing the quantized energy, , associated with the system. This section will describe how to ﬁnd a suitable form for the Hamiltonian H.

5 Since the Hamiltonian represents the total energy state of the system, it has to contain the following information: • kinetic energies of the electrons T , • interaction with an external potential V and • Coulomb interactions U. The many-electron Schr¨odinger equation summarizes the above in a form shown in Equation 2.1, where x can be interpreted as a vector in R3 and N refers to the number of electrons [14]

" N N N # X 2 X X HΨ(x) = − ~ ∇2 + V (x ) + U(x , x ) Ψ(x) = Ψ(x). (2.1) 2m i i i j i i i

Equation 2.1 includes

N N N X 2 X X T ≡ − ~ ∇2,V ≡ V (x ) and U ≡ U(x , x ). 2m i i i j i i i

Note that summing over N number of pairwise interactions can be an in- tractable problem when the number of electrons in the system is close to Avogadro’s number, N = 1023. This is where it is useful to introduce a few simpliﬁcations and approximations to the problem and the concepts of Kohn, Sham and Hohenberg can become handy in actually computing the state of a many-electron system [14], [11], [12], [9]. 2.3 Density Functional Theory

The main use of density functional theory is to simplify the interactions encountered in the Schr¨odingermany-particle equation, Equation 2.1, by using density to write down the expression for the Hamiltonian, H. In the past, it has not been easy to predict the total state of a system which contains many electrons because accounting for pairwise short and long range interactions is a time and memory consuming task. However, another approach to take is to obtain all the forces acting on a system from the electronic structure instead

6 of using the many-body system of equations. Density functional theory states that if a density for a system is known (in this case the electronic density), there exists a functional which will describe the total energy state of the system in terms of that density. Once this total energy state of the system is computed, there are a number of other interesting properties which can be derived from this density such as the current passing through the system [3]. The basis of density functional theory relies on work by Hohenberg and Kohn with later additions by Kohn and ﬁnally Sham [12]. The fundamental theorem is given by Theorem 2.3.1. Once we know the density of the system, the potential V can be calculated, or vice versa. In order to state and prove this theorem, an energy functional must be introduced as

Z H[ρ] = ρ(x)v[ρ](x)dx + T [ρ] + Vext[ρ]. (2.2) 3 R Here, the interaction term, U was replaced by a term, v[ρ](x), that depends on electronic density, ρ

Z U(xi, xj) → ρ(x)v[ρ](x)dx. 3 R The other terms represent the kinetic energy , T , which remains the same and a potential energy due to interactions with an external potential the system is in, Vext[ρ]. Note that the electronic density has yet to be deﬁned. Theorem 2.3.1 (Hohenberg-Kohn Theorem 1). The electronic ground state density, ρ, of an N electron system, in the presence of an external potential, v[ρ], uniquely determines this potential [9].

Proof of Hohenberg-Kohn Theorem 1. Let ρ be a nondegenerate ground-state density of N electrons in two diﬀerent potentials, v1 and v2. Corresponding to these potentials are two eigenvalue equations where Ψ1 and Ψ2 are two sets

7 of eigenstates and 1, 2 are two diﬀerent sets of eigenvalues of energies.

H1Ψ1 = 1Ψ1 and

H2Ψ2 = 2Ψ2.

Using Equation 2.2, two separate equations involving the density ρ are derived

Z Z 2 ∗ ∇ H1[ρ] = v1[ρ]ρdx − Ψ1(x) Ψ1(x)dx + Vext[ρ], R3 R3 2

Z Z 2 ∗ ∇ H2[ρ] = v2[ρ]ρdx − Ψ2(x) Ψ2(x)dx + Vext[ρ]. R3 R3 2

Since Ψ2 is the ground state for H2 and since it was assumed to be nondegenerate, that means any other states have a diﬀerent energy associated with them, which will be higher than the ground state energy,

Z Z 2 ∗ ∇ 1 < v2[ρ]ρdx − Ψ2(x) Ψ2(x)dx + Vext[ρ] R3 2 Z 1 <2 + (v1[ρ] − v2[ρ])ρdx (2.3) R3 and similarly, for a ground state Ψ2 (not necessarily non-degenerate, hence the ≤),

Z 2 ≤ 1 + (v2[ρ] − v1[ρ])ρdx (2.4) R3

Adding Equations 2.3 and 2.4, this leads to a contradiction,

1 + 2 < 1 + 2.

Thus, v1 ≡ v2 and therefore there exists only one potential which is deﬁned by the density ρ [12].

8 2.4 Variational Principle

If we are interested in computing the total energy of the N-electron system, there are two ways of accomplishing the task. One is diagonalizing the Hamiltonian in its matrix form, however this is complicated if the eigenbasis is not known, and the other is by introducing a variational principle as done by the second theorem of Hohenberg and Kohn, Theorem 2.4.1 which is based on the Rayleigh-Ritz minimal principle [12]. Theorem 2.4.1 (Hohenberg-Kohn Theorem 2). Let H be a Hamiltonian and R ρ˜ any electronic density, such that ρ˜ ≥ 0 and 3 ρ˜ dx = N. Then the energy R of the ground state density, ρ, given by H[ρ], will be lower or equal to that of H[˜ρ]

H[ρ] ≤ H[˜ρ]. (2.5)

Outline of proof of Hohenberg-Kohn Theorem 2. The proof follows from The- orem 2.3.1 since each densityρ ˜ determines its own potentialv ˜[˜ρ].

Taking into consideration the above theorem, the solution to the Schr¨odinger equation, Equation 2.1, can now be written in variational form by using the Rayleigh quotient which is also referred to as the Rayleigh-Ritz formula. The smallest eigenvalue of H is given by Q[ρ]

R Ψ∗(x)HΨ(x)dx Q[ρ] = min . (2.6) ρ R Ψ∗(x)Ψ(x)dx

Minimization can be achieved by introducing the quantity , considered as a Lagrange multiplier with units of energy, and ﬁnding the minimum of

Z E[ρ] ≡ [Ψ∗(x)Ψ(x) − Ψ∗(x)HΨ(x)] dx. (2.7) R3

9 2.5 Full Energy Functional

The underlying concept of the formulation is to assume there exist orthonormal wavefunctions ψi(x) which represent the eigenstates of the operator. These deﬁne the total energetic state of the Hamiltonian system, H. The Schr¨odingerequation, Equation 2.8, can be interpreted as an eigenvalue equation where H has N orthogonal eigenvectors,

Hψi = Eiψi. (2.8)

The Kohn-Sham scheme is the following. The density, ρ, can be deﬁned in terms of those wavefunctions ψi and their inner product as shown in Equation 2.9

N X ∗ ρ = ψi(x) ψi(x). (2.9) i=1 The Hamiltonian operator in the Schr¨odinger equation, Equation 2.8, can now be written in terms of the density, ρ instead of an operator acting on ψi(x) with Equation 2.9 deﬁning a map between ψi and ρ. Consider separating the contributions to the energy state of the Hamilto- nian into the kinetic energy, TKS and the potential energy Utot functionals.

H[ρ] = TKS[ρ] + Utot[ρ] (2.10)

The kinetic energy term, TKS is deﬁned as

1 Z T [ρ] = − (Ψ(x))∗∇2Ψ(x)dx. (2.11) KS 2

The kinetic energy, TKS of electrons is large where the density of electrons is large. This follows by the Pauli exclusion principle since these electrons must arrange themselves into momentum states by pairs of up and down spin.

10 The other condition for a high kinetic energy is where the gradient of the wavefunction is large, by definition shown in Equation 2.11. The beauty of the Kohn-Sham formalism is that it defines the electronic density in terms of wavefunctions as shown in Equation 2.9, so that density and wavefunctions have a direct relationship and the effective potential energy functional, Utot can be written in terms of the electronic density, ρ. The effective potential energy can be further decomposed into the contribution due to the presence of ions (sometimes referred to as the external potential), the Hartree potential and the exchange-correlation potential which is mainly responsible for chemical reactions and bonds

Utot[ρ] = UH [ρ] + UXC [ρ] + Vext[ρ]. (2.12)

The individual contributions will be discussed further, except for the contribution Vext. This latter potential energy will group all the energy terms which have to do with the presence of an external potential and interactions with the ions assuming that once they are calculated for a particular system, they will not change depending on the density and eigenstates of the electrons, only on their total number, N. Since the Kohn-Sham equations deﬁne unique maps between potentials and density, the diﬀerent energy functional contributions will be discussed in terms of the potentials from which they are calculated as done by Hong Guo’s group at McGill [19].

2.5.1 Hartree Potential, UH The Hartree potential can be thought of as the Coulombic interaction which is usually computed using Gauss’ Law. The Hartree electric potential

11 equation is obtained from solving the Poisson problem, Equation 2.13.   2 ∇ VH (x) = −4πρ in Ω, (2.13)   VH (x) = Vbulk(x) on ∂Ω where Vbulk(x) is some already known function available from a previous computation and Ω is the region of the electronic system, in other words, the device region. Equation 2.13 can then be related to a potential energy through integration of the density as

Z UH = ρVH [ρ]dx. (2.14)

2.5.2 Exchange-Correlation Potential, UXC The potential for the exchange correlation contribution can be written as shown in Equation 2.15, in terms of a variation of the exchange-correlation energy, UXC

δU [ρ] V = XC . (2.15) XC δρ

The exchange correlation functional is obtained by using Padé’sapproxima- tion, or in other words, a quotient of polynomials with coefficients determined from fitting to experimental data.

2 3 a0 + a1rs + a2rs + a3rs UXC [ρ0] = −ρ0 2 3 4 , (2.16) b1rs + b2rs + b3rs + b4rs where 3 1/3 r = s 4πρ and coeﬃcients are given as: a0 = 0.458165293, a1 = 2.217058676, a2 =

0.74055173, a3 = 0.019682278, b1 = 1.0, b2 = 4.504130959, b3 = 1.110667363, b4 = 0.23592917 [19]. This contribution to the potential is phenomenologically derived.

12 2.6 Summary

To summarize, given an exact form for the potential, v[ρ], there exists a unique electronic density, ρ corresponding to it. Using this concept, the Hamiltonian, H, can then be written in the form of a ﬁeld equation in terms of the electronic density, ρ. As shown in this chapter, the total energy functional then looks like

Z Z 1 ∗ 2 E[ρ] = − (ψi(x)) ∇ ψi(x)dx + ρVH [ρ]dx ... 2 3 3 R R 2 3 a0 + a1rs + a2rs + a3rs −ρ0 2 3 4 + Vext. (2.17) b1rs + b2rs + b3rs + b4rs

The two contributions to note are the Hartree potential which is the solution to a Poisson problem as shown in Equation 2.13, and the exchange correlation contribution as shown in Equation 2.16. The Hartree potential correponds to Coulombic interaction of electrons and is hard to calculate since it involves the solution of a partial diﬀerential equation. The exchange-correlation is the potential which explains bonding and is derived empirically. Using the variational of the energy functional, the Hamiltonian in the Kohn-Sham scheme can then be written as

2 − ~ ∇2ψ (x) + v[ρ]ψ (x) = ψ (x), (2.18) 2m i i i i

δE[ρ] with v[ρ] ≡ ρ . From this, electron transport in a particular device is a more tractable problem.

13 CHAPTER 3 Green’s Functions and Density 3.1 Overview

We are interested in a sum of all the occupied eigenstates of a Hamilto- nian and the associated eigenvalues for further analysis of a speciﬁc device. Instead of solving an eigenvalue equation, this chapter provides a summary of work on the Green’s functional approach [19], [26]. These Green’s functions, G, can be written down in matrix notation in a particular basis such as the linear combination of atomic orbitals (LCAO), [19], and for diﬀerent devices [3]. In order to compute the electronic density, ρ, calculus of residues is used, [25] [23], and a contour integral of the Green’s function is performed in the complex energy plane. This is reminiscent of the resolvent formalism and pro- jections into the eigenbasis [2], which is mentioned in the chapter. Equilibrium and non-equilibrium situations are examined and the causal Green’s function is discussed [16], [10]. Starting from the concept of the Fermi distribution and chemical potential as well as bound and excited states [14], [11], bounds on equilibrium and non-equilibrium contributions are applied as in [20], complet- ing the closed form of the electronic density. 3.2 Green’s Functions

In the previous chapter, we saw the introduction of the energy functional as in Equation 3.1

Z E[ρ] = Ψ∗(x)Ψ(x)dx − H[ρ]. (3.1) 3 R Equation 3.1 will be equivalent to the Schr¨odingerequation if E[ρ] ≡ 0, that is, it is minimized. Equation 3.1 can be interpreted as a matrix equation so

14 that all the quantities are discrete. This can be done by writing the operator

R ∗ R ∗ H[ρ] as a matrix in some basis, H = 3 Ψ HΨ. Deﬁning S ≡ 3 Ψ Ψdr the R R overlap matrix, the matrix equation would be

E[ρ] = S − H.

It achieves a minimum, E[ρ] = 0, when the eigenvectors, Ψ of the Hamiltonian are known, and the energy, is equivalent to the eigenvalues, i. That also means that S ≡ I. Noting that H is unitarily diagonalizable with the elements of the diagonal matrix given by

∗ ∗ ψl Hψi = iψl ψi = iδli,

∗ where ψi represent an element of the eigenbasis with ψi ψi = 1 and i represents the corresponding eigenvalue. Also, δli is the Kronicker delta and will be 1 if l = i and 0 otherwise. In order to solve the Schr¨odingerequation, Equation 2.1 in matrix notation, the eigenvalues, i and eigenvectors, ψi need to be computed simultane- ously. Recalling that neither the eigenvalues or the eigenvectors are known, a matrix representation of the Hamiltonian written in another basis can be derived. Introducing a change of coordinates, C which is a linear map from Ψ to Φ as

X ∗ X ∗ ∗ ψi = cijφj and ψl = φk(cjk) , (3.2) j k the Hamiltonian can be written in a new, complete basis where S is no longer the identity matrix I. However, it is important to keep in mind that this map C, is unknown. Substituting for ψi and its complex conjugate in Equation 3.2, Equation 3.3 is obtained where it is important to note that

15 P P ∗ ∗ k j(clk) φkφjcij = 1 ⇐⇒ l = i

X X ∗ ∗ ∗ ∗ {(clk) φkφjcij − (clk) φkHφjcij} = ( − i)δli. (3.3) k j

Introducing ˜ ∗ Hkj = φkHφj and ˜ ∗ Skj = φkφj, as well as bringing ( − i) to the left-hand side of the equation, will yield

∗ X n ˜ ˜ o (clk) cij Skj − Hkj = δli. (3.4) − i kj

i We deﬁne, ∀i, j, k, the causal Green’s function, Gkj(), where for each eigenenergy, i its components are given by [23]

∗ i (cik) cij Gkj() = . (3.5) − i

This leads to writing down the following

n o X ˜ ˜ i Skj − Hkj Gkj() = 1. (3.6) kj

Rewriting the above as a matrix equation instead of a component equation leads to

n o ES˜ − H˜ G(E) = I (3.7)

The Green’s function has a singularity when E (a matrix with on the diagonals) is comprised of eigenvalues. Looking for where the Green’s function

i diverges to ∞ (i.e. lim→i Gkj() → δ(i − )), will give an indication that the Schr¨odingerequation has an eigenvalue at = i. In the future, the tilde

16 notation will be dropped and note that if the basis functions were diagonal, then S ≡ I. 3.3 Density

As done before for the Hamiltonian, the density, ρ, can be written in the P new basis, Φ, as well. Using the coordinate transformations, ψi = j cijφj, set

i ∗ X ∗ ∗ ρ = ψi ψi = φkcikcijφj jk and denote,ρ ˜, which would be the density in the new basis as

i ∗ ρ˜kj = cikcij. (3.8)

Thus, the density in the old basis will simply be given by

ρ = Φ∗ρ˜Φ.

Going back to component notation, the relationship between the Green’s function and the density can be derived using the calculus of residues and by applying the Sokhotsky-Weierstrass theorem [26], [22]. Thus, since the Green’s function can be computed in the new basis, then even without knowing the eigenbasis or the linear map C, an expression for the density can be derived. 3.3.1 Contour Integration

We can use the Laurent expansion of G about i in order to approximate the Green’s function near the singularity at i

∞ i X n Gkj() = cn( − i) . n=−∞

i |c−1| As → i, G () ≈ . Since the density, ρ is deﬁned as kj |(−i)|

i i ρ˜ = lim( − i)G (), →i

17 we see that

i ρ˜ = |c−1|.

i where c−1 is the coeﬃcient of the Laurent series which represents Gkj(). From

i this, a deﬁnition for ρkj as a residue follows. In general terms, let g(z) be analytic on a simple closed contour, C and at all points interior to C except for a simple pole at z0, then the residue is given by

1 I Res[g(z), z0] = g(z)dz. (3.9) 2πi C

Another property to note is that if g(z) is an analytic function except for a singularity at z0 then the principle of deformation of contours, states that the residue will not be dependent on the precise shape of C, i.e. all the simple closed curves that contain the singularity of f(z) will lead to the same value of the integral. If there exist many poles, zk for k = 1..n of an analytic function (ie the function is meromorphic), then the residue theorem shown in Equation 3.10 applies [25]

N 1 I X g(z)dz = Res[g(z), z ]. (3.10) 2πi k C k=1 It is then clear that the density can be computed by Equation 3.11 which

i i formulates ρkj as a contour integral of Gkj() over a contour C enclosing the i-th eigenvalue, i

I i 1 i ρ˜kj = Gkj()d. (3.11) 2πi C

In matrix notation, using the fact that the Green’s function contains more than one pole (or the Hamiltonian has more that one eigenvalue), i, the

18 deﬁnition for the density given a Green’s function then becomes

1 I ρ˜ = G(E)dE, (3.12) 2πi C where the contour, C encloses all the poles [23]. 3.3.2 Integration Along Real Axis

Implementing Equation 3.11 in a numerical scheme is still not that easy. It is important to introduce another idea for evaluating functions at singularities. Let z ∈ Ω, where the domain Ω is the ball B(0,R) with radius R centered about 0. Let z be a radial coordinate. Then, for any integrable function f(z) ∈ B(0,R) which is zero outside of the ball, i.e. f(z) = 0 for |z| > R, the Sokhotsky-Weierstrass Formula is as seen in [26],[22]

Z R f(z) Z R f(z) limη→0+ dz = P dz ∓ iπf(0). (3.13) −R z ± iη −R z

R f(z) In Equation 3.13, P z dz, stands for Cauchy’s principal value de- ﬁned by Equation 3.14 [23]

Z R f(z) Z ∞ f(z) Z −η f(z) Z ∞ f(z) P dz = P dz = limη→0+ dz + dz . −R z −∞ z −∞ z η z (3.14)

Suppose that all the eigenvalues, i, are in the interval [Emin,Emax]. Then, the Green’s function, G() will have poles along the real axis in this interval. Limits can be set for the deﬁnition of density in Equation 3.15. These have a physical interpretation involving chemical potentials and scattering states which will be discussed in a later section. Applying Sokhotsky-Weierstrass will give the formula for an element of the density matrix shown in Equation

19 3.15

Z Emax ∗ Z Emax ∗ i 1 (clk) cij (clk) cij ρ˜kj = lim d − limη→0+ d (3.15) η→0+ 2i Emin − i − iη Emin − i + iη

To perform the above integrals, two more Green’s functions can be deﬁned (in matrix notation) as [3]

[ES − H − iη]GR(E) = I and the other [ES − H + iη]GA(E) = I

Writing everything in matrix notation gives

Z Emax 1 R A ρ = limη→0+ G (E) − G (E) dE. (3.16) 2i Emin

Numerically, integrating Equation 3.16 along the real line is not advan- tageous. This is because the values of G along the axis, E, will be close to singular since they are close to the values of GR/A(E) at the poles. This will lead to computing a GR/A(E) which varies greatly depending on the value of E and can lead to numerical errors. However, to compute the integrals in a more stable fashion, recall that the poles of G are on the real axis, therefore GA has poles in the upper half plane and GR has poles in the lower half plane [19]. If GR/A is analytic in the upper/lower complex plane respectively, then

I Z Emax Z GR/A(E)dE = GR/A(E)dE + GR/A(E)dΓ = 0, (3.17) C Emin Γ where the above holds for any closed path Γ lying in the upper/lower complex planes where GR/A are analytic and once again invokes Cauchy-Gorsat

20 1.1

Γ1

Γ2

−1.1 −1.1 0 1.1 Figure 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. theorem. Therefore

Z Emax Z GR/A(E)dE = − GR/A(E)dΓ. (3.18) Emin Γ

Thus, Equation 3.18 gives the result that choosing a contour, Γ in the domain where the respective Green’s functions are analytic, will allow the computation of the integral over the real axis. 3.4 General Formalism

Hermitian matrices are normal. Normal matrices can be diagonalized, i.e. a unitary matrix, X, containing eigenvectors, xi as columns and a matrix, Λ, containing the associated eigenvalues, λi on the diagonals, can be found such that

A = XΛX∗. (3.19)

21 Let Xi be a unitary matrix formed by i columns of X where

∗ Xi Xi = Ii.

The eigenprojection, Pi, associated with the eigenvalues, λi, of a matrix A can be deﬁned by

∗ Pi = XiXi and the matrix diagonalization scheme as in Equation 3.19, can now be written in terms of a matrix projection as

X A = λiPi. (3.20) i It can now also be shown that the eigenprojection is given by the residues of the resolvent, R(A, z) = (A − zI)−1, as shown in Equation 3.21 over the simple, closed contour Γ which encloses the singularities of the resolvent [2]

1 I P = − R(A, z)dz. (3.21) 2πi Γ

Proof. Proof of Equation 3.21 This is a special case of the proof for the contour integral shown in Equa- tion 3.21. For the general case, see [2]. Let the algebraic multiplicity, the number of repetitions of a certain eigenvalue λi, be mi, and the number of distinct eigenvalues of a matrix A be d. To start oﬀ, recall that the resolvent

−1 set of A, z ∈ res(A), is a set of all points for which (A − zI) exists. Let Xi be a matrix which is formed by mi column vectors associated with a speciﬁc eigenvalue, λi. Also, deﬁning a projection Pi onto the eigenspace associated with λi, as

∗ Pi = XiXi ,

22 then by the spectral theorem, the Hermitian matrix A can be decomposed into

d X A = λiPi. i Recalling that if A = XDX∗ where D = diag{λ}, then A−1 = XD−1X∗ and also for a complex number, z

(A − zI)−1 = X(D − zI)−1X∗.

−1 Thus, Pi as deﬁned above, is also a spectral projection of (A − zI) with eigenvalues given by

˜ −1 λi = (λi − z) . (3.22)

Using Equation 3.22, the spectral decomposition of the resolvent, R(A, z), is given by Equation 3.23

d X 1 (A − zI)−1 = P (3.23) λ − z i i=1 i

Let Ck be a simple closed contour which isolates λk, then by the residue theorem there are two cases such that   I 1 −2πi ⇐⇒ i = k = (3.24) λ − z Ck i  0 else.

Then,

I 1 −1 − (A − zI) dz = Pk (3.25) 2πi Ck

23 In our case, the Green’s function, G, can be thought of as a resolvent of the Hamiltonian operator H and the density, ρ, as the projection onto the eigenspace of the operator. 3.5 Approximations

For the purposes of our work, the specific form of the Hamiltonian will not be of concern. The only thing to note will be that since electronic configura- tions and systems have many forces and therefore many energy contributions, the form of the Hamiltonian will not usually be exact. That means there is no guarantee of having ”nice” eigenvalues, or in other words, it might not be diagonalizable. This is where the resolvent formalism becomes useful. In the system particular to the density functional theory and the Kohn- Sham formulation, the Hamiltonian is actually a function of the density. Since the real density of the system is not known until the eigenstates are computed, the Hamiltonian cannot be written down in the basis of eigenfunctions from the start (chicken or egg scenario). Instead, it is written down in an approximate basis and then refined with each iteration of a self-consistent loop until some criteria is met. This can be viewed as a fixed point method. How to compute a Hamiltonian given a density of the system is known. That means given any guess for this density, the Hamiltonian can be written down in that basis. This is where the matrix representation of the operator will be used since it is easier to visualize. 3.6 Summary

This section has outlined how to go between the Hamiltonian, to the computation of electronic density. This is done by the Green’s functional approach. Since this will be used further, it is most useful to write the Green’s function in matrix notation as

[ES − H]G = I.

24 From this, in order to calculate the electronic density, recall that it is given by a sum over the inner product of the system’s N electrons with wavefunctions

ψi, N X ∗ ρ = ψi ψi. i It can be shown that a density given by an integral over a contour around the poles of the Green’s function

1 I ρ = G(E)dE. 2πi

It was also shown that the above holds even when the eigenbasis is not known by using any complete basis that has a linear map to the eigenbasis. The corresponding Green’s function and density in that basis can then be calculated. Introducing proper extensions to the Green’s function into the complex plane deﬁned as GR = G − iη and GA = (GR)∗ = G + iη, the contour integral can then be rewritten as an integral in the complex plane, where the density will be given by

1 Z Z ρ = GR(E)dE − GA(E)dE . 2πi Γ1 Γ2

This contour, Γ1 is then a contour in the upper half plane where we are assured

R that there are no poles of G , the retarded Green’s function, and Γ2 is in the lower half plane where the advanced Green’s function, GA, will not have poles. This helps in stabilizing the integration algorithm since we are now away from singularities.

25 CHAPTER 4 Two Probe Device 4.1 Overview

Thus far, we know how to write down a Hamiltonian in terms of electronic density in some given basis. We also know how to calculate a density given a Hamiltonian via a Green’s functional approach. However, we have yet to discuss what the devices are for which we are interested in calculating the electronic density. The device of interest in our case, is the two probe device. By considering interactions in different regions of the device, a Green’s function in the form of a sparse matrix arises [19]. These two probe devices can be used for circuitry as gate switches and molecular wires, [3]. Due to their usefulness, they are discussed in detail in this chapter with an explicit derivation for the form for a Green’s function motivated by Bloch theorem [14], [16], and a recursive approach as seen in [24], [7], and [4]. The difference from that work, will be that in our case, these Green’s functional calculations are done for a device where the boundary conditions for the bulk potential are known as in [19]. Doing this, will complete the picture of how to compute the electronic density for two probe nano-device. 4.2 Infinite System

Suppose that we have the system that consists of atoms as shown in Figure

4–1. There is a molecule in the center (in this case C60, but could be others) that connects on the left and right to some bulk material. This bulk material, usually a metal, extends to inﬁnity on either side and will be referred to as the left and right lead. The region of interest will be the molecule with a ﬁnite part of the bulk material on the left and on the right. For a schematic

26 Figure 4–1: A diagram of a two probe device representing the atoms and their arrangement.

Figure 4–2: A schematic diagram of the different regions which represent a device with two leads of infinite length labelled LL and RR. diagram of the region of interest, see Figure 4–2 and the sections CL, CC and CR. The left and right infinite leads are labelled by LL and RR respectively.

The full matrix for the Hamiltonian represented here as HF ull, will consist of submatrices representing the diﬀerent regions in the device. If only the neighbouring regions interact, the it will have the tridiagonal form. Following the convention from Figure 4–2, the full matrix will lead to the expression as   HLL HLC 0     HF ull =  H H H  . (4.1)  CL CC CR    0 HRC HRR

The matrices describing the interaction with the leads are HCL and HCR and

∗ ∗ since the device is symmetric, HLC = HCL and HRC = HCR.

27 Working in the orthonormal (S ≡ I), then the matrix equation for the full Green’s function GF ull is of the form

(EF ull − HF ull)GF ull = I, and in the expanded matrix notation, it is shown by Equation 4.2 as

      EF ull − HLL HLC 0 GLL GLC GLR I 0 0              H E − H H  G G G  = 0 I 0 .  CL F ull CC CR   CL CC CR         0 HRC EF ull − HRR GRL GRC GRR 0 0 I (4.2)

The properties of interest are associated with the central device, so that the only Green’s function needed is GCC . It can be calculated from only three equations listed below

(EF ull − HLL)GLC = 0, (4.3)

HCLGLC + (EF ull − HCC )GCC + HCRGRC = 0, (4.4)

(EF ull − HRR)GRC = 0. (4.5)

Solving for GRC and GLC using Equation 4.3 and 4.5 respectively and substi- tuing into Equation 4.4, GCC is obtained

(EF ull − HCC − ΣRR − ΣLL)GCC = I, (4.6)

where the deﬁnitions for ΣLL and ΣRR which are referred to as self-energies of the left and right lead respectively, are given by

−1 ΣLL = HCL[Efull − HLL] HLC , (4.7)

28 −1 ΣRR = HCR[Efull − HRR] HRC . (4.8)

The eﬀect of the leads can be seen as adding an extra energy contribution ¯ to the overall Hamiltonian. Representing H = HCC +Σ where Σ = ΣRR +ΣLL, then the eﬀective Hamiltonian, H¯ can once again be represented in a diagonal form with shifted eigenvalues as written in Equation 4.9 [15]

¯ Hψi = ¯iψi. (4.9)

4.3 Non-Equilibrium Eigenvalues

Now that we have described the Green’s function for the device of interest which contains left and right infinite leads, an expression for density can be written in the case where the leads affect the central device. Recall that the effects of the leads is essentially an addition of an extra term, Σ to the Hamitonian, H. In order to calculate the projection onto the eigenbasis, a similar procedure as in the equilibrium case without the leads can be followed. Defining a retarded Green’s function, GR to be

[ES − H − ΣR]GR = I, (4.10) and its complex conjugate, the advanced Green’s function, GA, to be

[ES − H − ΣA]GA = I, (4.11) with (ΣA)∗ = ΣR. The electronic density can be calculated from the spectral function, A, which is deﬁned as A = i(GR − GA).

29 Note that A can be re-written

A = i(GR{(GA)−1GA} − {GR(GR)−1}GA).

Grouping terms, get

A = iGR((GA)−1 − (GR)−1)GA.

Rewriting Equations 4.10 and 4.11 as

ES − H − ΣR = (GR)−1, (4.12)

ES − H − ΣA = (GA)−1, (4.13) and then subtracting Equation 4.12 from Equation 4.13, we obtain

ΣR − ΣA = (GA)−1 − (GR)−1 = −iΓ, where we introduced a new variable, Γ, which represents the diﬀerent in the energies between the leads [3]. The spectral function now becomes

A = GRΓGA ≡ −iG<.

In this case, G< is the Green’s function for an injection of charge. From this, the density of states can be deﬁned as

1 I ρ = G<(E)dE, 2π where the integral only has poles on the real line as shown before. It can be computed by using Cauchy-Gorsat theorem and integrating along a path in the complex plane as done for the equilibrium case when there are no semi-inﬁnite leads as in Chapter 3.

30 Figure 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.

4.4 Total Density

4.4.1 Fermi Distribution

To better understand what the eigenstates of the Hamiltonian represent, it is important to know the concept of Fermi-Dirac statistics. A solid can be described in terms of the allowed and forbidden energy levels, or energy bands that its electrons can be in. This band structure is due to the wave- like behaviour of electrons which diffract from the crystal lattice structure of a solid. It defines a solid’s electronic and optical properties and is shown on the left in Figure 4–3 that depicts the bands of energy (in eV) as different lines that depend on the position in the solid with positions labelled as Greek letters on the x-axis. These positions are wave vectors (that is, they are Fourier transforms of real space variables) and are shown for a simple crystal on the right of the figure. Statistically speaking, the energy bands of a solid at a given temperature are filled according to the Fermi-Dirac distribution. This gives a way to average the behaviour of electrons to get an idea of the macroscopic properties [14], [11].

31 4.4.2 Fermi-Dirac Statistics

Fermi-Dirac distributions are used to describe fermions (or particles with half integer spin) such as electrons, (spin 1/2), that obey the Pauli exclusion principle. It states that no more than one electron can exist in a speciﬁc energy conﬁguration or quantum state. The Fermi-Dirac distribution, labelled as f(i), describes the probability of state i with energy i is

1 f(i) = . (4.14) e(i−µ)/kT + 1

Here, the chemical potential is µ, the temperature T and k is the Boltzmann constant. The statistical distribution is valid only if the number of fermions in the system is large enough so that the chemical potential µ does not change if more electrons are added to the system. At temperature, T = 0 and for energies less than the Fermi energy, all the states are occupied, f(i) = 1, while for energies greater than the Fermi energy, there are no electrons in those states, f(i) = 0. The notion of an electronic chemical potential, Equation 4.15, is also important. It is the functional derivative of the density functional with respect to the electron density

δE[ρ] µ(r) = . (4.15) δρ(r) ρ=ρref Considering the particular form of the energy functional, the chemical potential is eﬀectively the electrostatic potential experienced by the negative charges present in the system. At the ground state density, the chemical potential is constant and the electron density is at a steady state so the forces are all balanced. The use of a chemical potential will become more evident when examining the device with semi-inﬁnite leads that we are interested in.

32 There are three categories of solids which can be identified based on their energy band structure. These are classified according to the location of the valence band and conduction band. The valence band represents the highest ranges of energy at the zero temperature, which can be occupied by electrons without being ejected. In organic compounds, this is equivalent to highest occupied molecular orbital. This concept is equivalent to finding all bound eigenstates of a Hamiltonian below the Fermi energy. On the other hand, the conduction band, or the lowest unoccupied molecular orbital is the lowest energy level that free electrons can achieve which is above the Fermi- level. For inorganic compounds, the different energy levels are illustrated in Figure 4–4. Three types of inorganic solids can be identified 1. Metals: where the conduction band and valence band overlap. 2. Semi-conductors: where the conduction band and valence bands are separated by a small gap. 3. Insulators: where the conduction band and valence bands are separated by a large gap which is rarely traversed so that electrons cannot be ejected from the material and therefore the material cannot have induced current. Having this knowledge, the full computation for an electronic density will now require the idea of the Fermi energy and chemical potential. Recall that

I ρneq = G<(E)dE (4.16) C where now, the energy for left and right leads will incorporate the Fermi-Dirac distribution

G<(E) = GRΣ<[f LL(E), f RR(E)]GA (4.17)

33 Figure 4–4: This ﬁgure illustrates that depending on where the Fermi-energy level is, the inorganic material will have diﬀerent properties. with Σ<[f LL(E), f RR(E)] = −2iIm[f LLΣLL + f RRΣRR].

The Fermi distribution function, f(E) and it’s representation will not be discussed in great detail and mentioned for completeness. For a reference, see [19] or [20]. 4.5 Self-Energies

in this section, we will outline how to compute the contributions to the Hamiltonian which arise from the semi-infinite leads. At first, the discussion will outline how to compute a Green’s function, also known as the fundamental solution, for an infinite domain. We then show how to construct the Green’s function for the semi-infinite domain. We derive a series representation for the Green’s function for the semi-infinite leads.. This will be done in order to write a closed form for the Green’s function of the central region. 4.5.1 Inverse of Block Tridiagonal Matrix

We will show a derivation for a formula for computing the inverse of a tridiagonal matrix using a recursion relation. We start from an example from

34 [7]. Suppose we are interested in solving for Gfull in

¯ HfullGfull = I,

¯ where Hfull is a block tridiagonal matrix. Moreover, let the blocks satisfy ¯ ¯ ∗ Hij = Hji, then       ¯ ¯ H11 H12 0 0 G11 G12 G13 G14 I 0 0 0        ¯ ∗ ¯ ¯      H H22 H23 0  G21 G22 G23 G24 0 I 0 0  12          =   . (4.18)  0 H¯ ∗ H¯ H¯  G G G G  0 0 I 0  23 33 34  31 32 33 34         ¯ ∗ ¯ 0 0 H34 H44 G41 G42 G43 G44 0 0 0 I

Equation 4.18 represents a system of linear equations, four of which are shown in Equation 4.19.

¯ ¯ H11G11 + H12G21 = I

¯ ∗ ¯ ¯ H12G11 + H22G21 + H23G31 = 0

¯ ∗ ¯ ¯ H23G21 + H33G31 + H34G41 = 0

¯ ∗ ¯ H34G31 + H44G41 = 0. (4.19)

Introducing γ, a matrix deﬁned by

+ γ4 = 0

+ ¯ ¯ + −1 ¯ ∗ γ3 = H34[H44 − γ4 ] H34

+ ¯ ¯ + −1 ¯ ∗ γ2 = H23[H33 − γ3 ] H23

+ ¯ ¯ + −1 ¯ ∗ γ1 = H12[H22 − γ2 ] H12, (4.20)

35 − γ1 = 0

− ¯ ∗ ¯ − −1 ¯ γ2 = H12[H11 − γ1 ] H12

− ¯ ∗ ¯ − −1 ¯ γ3 = H23[H22 − γ2 ] H23

− ¯ ∗ ¯ − −1 ¯ γ4 = H34[H33 − γ3 ] H34. (4.21)

Solving the systems of equations in Equation 4.19 as follows,

¯ + −1 ¯ ∗ G41 = − [H44 − γ4 ] H34G31

¯ + −1 ¯ ∗ G31 = − [H33 − γ3 ] H23G21

¯ + −1 ¯ ∗ G21 = − [H22 − γ2 ] H12G11

¯ + − −1 G11 = [H11 − γ1 − γ1 ] . (4.22)

Notice that the Gij block is deﬁned recursively. Following the logic of a system with 4 by 4 blocks, the formula for a block tridiagonal matrix of any size can be derived

¯ + − −1 Gii = [Hii − γi − γi ] ,

¯ + −1 ¯ ∗ Gij = − [Hii − γi ] Hi−1,iGi−1,j, ∀ i > j

¯ − −1 ¯ Gij = − [Hii − γi ] Hi+1,iGi+1,j, ∀ i < j (4.23) where

+ ¯ −1 ∗ γi = Hi,i+1[Hii − γi+1] Hi,i+1 and

− ∗ ¯ −1 γi = Hi−1,i[Hii − γi−1] Hi−1,i. (4.24)

The set of Equations 4.23 is a recursive relation which provides the blocks

− comprising the full Green’s function, Gfull. It should be noted that γi =

+ ∗ (γi ) . Also, in order to solve the above recursive form, a boundary condition ± for γi should be assigned in order to compute them uniquely. For example,

36 in the system of 4 by 4 matrix blocks, the conditions were set to 0, i.e.

+ − γ4 = 0 and γ1 = 0.

From this comes the formula which states the dependence of the Green’s function at i + 1 (or i − 1) on the one at i as shown in Equation 4.25

+ + + Gi,i = γi + γi Gi−1,i−1γi

− − − Gi,i = γi + γi Gi+1,i+1γi (4.25)

4.5.2 Inverse of a Hamiltonian for a Periodic Potential

We now consider a bulk, periodic system. The Hamiltonian is a block- tridiagonal, inﬁnitely sized matrix with repeating rows as shown below   ...    ∗  ...Hint H0 Hint 0 0 ...    ∗  Hfull = ... 0 H H H 0 ... . (4.26)  int 0 int     ∗  ... 0 0 Hint H0 Hint ...  .  ..

The above matrix is a simpliﬁed form of the general block tridiagonal matrix considered in section 4.4.1. In other words

Hii = H0, ∀ i = 1, ..., ∞ and

∗ Hi,i+1 = Hi+1,i = Hint, ∀ i = 1, ..., ∞.

37 That means the recursive relations outlined in Equation 4.23 and Equa- tion 4.25 are now simpliﬁed to look like

+ ∗ + −1 γ = Hint[H0 − γ ] Hint

+ + + Gi+1,i+1 = γ + γ Gi,iγ , (4.27) where it is still important to note that a boundary condition needs to be given to solve for γ+. 4.5.3 Bloch Theorem

Instead of going about solving the system by introducing the recursive relations, they can be uncoupled by taking note of the Bloch theorem which is essentially described by performing a Fourier transform. Note that the matrix

Hfull is a Hermitian matrix. The Schr¨odingerequation

HfullΨfull = EΨfull, can also be written in matrix form as shown in Equation 4.28.

 .   .  .. .      ∗    ...Hint E − H0 Hint 0 0 ... Ψl−1         ... 0 H∗ E − H H 0 ...  Ψ  = 0. (4.28)  int 0 int   l       ∗    ... 0 0 Hint E − H0 Hint ... Ψl+1  .   .  .. .

Equation 4.28 consists of inﬁnitely many coupled linear systems, one of which is shown in Equation 4.29

∗ HintΨl−1 + (E − H0)Ψl + HintΨl+1 = 0. (4.29)

It would be difficult to solve for eigenstates Ψl, since there are infinitely many coupled systems, however, the process can be simplified by a convenient ansatz

38 [4]. Assume

X inl Ψl = e φn, n then

X in(l−1) Ψl−1 = e φn, n

X in(l+1) Ψl+1 = e φn. n We use this to uncouple Equation 4.30 to get

X ∗ −in in inl Hinte + (E − H0) + Hinte e φn = 0. (4.30) n Equation 4.30 is now the eigenvalue problem written in a diﬀerent basis. For ease, denote the Hamiltonian in brackets as

∗ −in in Heff (n) = Hinte + (E − H0) + Hinte .

As before, we deﬁne the Green’s function

¯ ink ∗ −1 inl Gkl(n) = (e ) [H(n)eff ] e . (4.31)

¯ We now need to represent Gkl in the original basis via

∞ X ∗ ink ∗ −1 inl Gkl = φn(e ) [H(n)eff ] e φn (4.32) n Equation 4.32 provides an explicit representation for the periodic inﬁnite system. In Equation 4.23, the Green’s function is computed recursively. These two ideas can be combined to get a closed form for the Green’s function of a periodic system for a semi-inﬁnite lead which in turn will allow us to have a closed form for GCC , the Green’s function for the central device. 4.5.4 Boundary Condition

Since we are interested in semi-inﬁnite leads, we know that the Green’s function must be zero at some z0 which is where the leads starts. In other

39 words, the fundamental solution is known and it is equivalent to the Green’s function for an infinite matrix. However, the Green’s function for the semi- infinite leads has to satisfy the zero boundary condition at one end of the lead where it is connected to the central molecule. Finding this Green’s function, is similar to the usual techniques in PDEs. Once again working in operator notation, we are now looking for a function such that it satisfies Equation 4.33.   0 0 0 H(z, z )GLL(z, z ) = δ(z − z ) in Ω (4.33)  0  GLL(z, z ) = 0 on ∂Ω, where Ω is in the region of the full device. That is, we are looking to modify the fundamental solution derived in Equation 4.32 to make sure that it is zero on the boundary. In other words, the Green’s function for the left semi-infinite lead, GLL will look like

0 ∞ 0 LL GLL(z, z ) = G (z, z ) − γ (z, z0), (4.34) where γLL needs to be determined. Using the recursion relation as in Equation 4.27, a Green’s function for the semi-inﬁnite lead, GLL can be computed from the general form of the Green’s function for a block tridiagonal matrix, by realizing that the contribution from the term which is on the boundary has to be zero. Let this term be the n + 1 term at the boundary, z0, then

LL ∞ 0 0 γ (z, z0) =γ(z, z0)Gn,n(z, z )γ(z0, z )

X 0 γLL(z, z ) ≡ φ∗ (einl(z−z0))∗φ∗ [H(n ) ]−1φ einm(z0−z )φ . (4.35) 0 nm nl l eff nl nm nl,nm

40 The full Green’s function for a semi-inﬁnite left lead will then be given by Equation 4.36

X 0 G (z, z0) = φ∗ (einlz )∗[H(n ) ]−1einlzφ LL nl l eff nl nl X 0 − φ∗ (einl(z −z0))∗φ∗ [H(n ) ]−1φ einm(z0−z)φ . (4.36) nm nl l eff nl nm nl,nm The Green’s function can be veriﬁed to satisfy the correct boundary conditions

0 for the left lead. Let z = z = z0 to see that

X G (z , z ) = φ∗ (einlz0 )∗[H(n ) ]−1einlz0 φ LL 0 0 nl l eff nl nl X − φ∗ (einl(z0−z0))∗φ∗ [H(n ) ]−1φ einm(z0−z0)φ . nm nl l eff nl nm nl,nm Using

inm(z0−z0) φnl e φnm = φnl φnm = δml where γml is the Kronecker delta. We then arrive at

GLL(z0, z0) ≡ 0

A similar process is used for the right lead [19], [16]. Once we have a closed form for GRR and GLL, these can now be substituted into Equations 4.7 and

4.8 for ΣLL and ΣRR. In turn, these can now be used to ﬁnd GCC using Equation 4.6. 4.6 Spectra

Let X be an operator in a Banach space with elements x and the identity operator I. The resolvent, R(λ) is deﬁned in Equation 4.37 where λ is a complex number.

R(λ) = (λI − x)−1 (4.37)

41 The resolvent set is then the set of all λ in the resolvent. The spectrum, σ(X) is the deﬁned as the set of all λ in the complement of the resolvent set as shown in Equation 4.38

σ(λ) ≡ C \ R(λ) = {λ ∈ C | λ∈ / R(λ)}. (4.38) as in [5]. The Hamiltonian has a spectrum that can be grouped into three categories [18]: 1. The point spectrum: the spectrum of the operator, G that consists of eigenvalues, λ, such that λI − H is not injective, i.e. the set of all the eigenvalues is the point, or discrete spectrum. The eigenstates associated with this spectrum correspond to bound states. 2. The continuous spectrum: the set of all numbers λ which are not in the discrete eigenvalue set and those whose Range(λI − H) is dense. 3. The singular spectrum: the set of λ for which Range(λI − H) is not dense. For example, the Schr¨odinger equation for the hydrogen atom has both a point and a continuous spectrum. The form of the Hamiltonian for this particular system is c −∆ψ(r) + ψ(r) = λψ(r), r where c is a constant. It can be shown that λn = 1/n for integers n such that

0 < n < N with discrete eigenvalues λN ≤ ... ≤ λ1 ≤ 0. This s referred to as the discrete spectrum. However, it can be seen that as r → ∞, then the Schr¨odingercan be simpliﬁed to

−∆ψ(r) = λψ(r).

42 This has the solutions ψ(r) = e−ikr for |k|2 = λ. These solutions are valid for all λ ∈ [0, ∞) and the result is a continuous spectrum. To emphasize, the point spectrum of a Hamiltonian has associated states which are bound whereas the continuous spectrum corresponds to free states. The electrons which contribute to the current in an electronic device, are those electrons which are allowed to move around the device and thus have a continuous spectrum. The energy for which bound states become free states, corresponds to the energy which is above the highest eigenvalue in the point spectrum of the Hamiltonian. This is eigenvalue is the Fermi energy. Intuitively, we can now state that the total electron density, ρTOT , is the sum of the equilibrium and non-equilibrium contributions,

ρTOT = ρeq + ρneq.

The non-equilibrium contribution to the density, ρneq, arises from having a flow of electrons. If the leads that are part of the device of interest, have potential difference ∆V , the electrons will flow in the device to try to compensate and bring the system to equilibrium. This concept introduces natural bounds for the integral which is used to compute ρneq as shown in Equation 4.39

1 Z Emax ρneq = G<(E)dE, (4.39) 2π Emin where

Emin = min(µLL + VLL, µRR + VRR) and Emax = max(µLL + VLL, µRR + VRR)

and Emin and Emax correspond to Fermi levels of the right and left leads, RR and LL. The rest of the contribution to the electronic density comes from the

43 eq equilibrium density, ρ where the bounds are set by 0 and Emin [20].

1 Z Emin 1 Z Emin ρeq = GR(E) − GA(E) dE ≡ − Im[GR(E)]dE. (4.40) 2πi 0 π 0

4.7 Discussion of Basis Functions

As previously seen in Chapter 3, in order to compute the Green’s function, ˜ a Hamiltonian H, must be written in some complete basis φi

˜ ∗ Hij = φi Hφj.

These basis functions should ideally be chosen such that the new Hamiltonian, ˜ H, is a sparse matrix. This can be done by choosing those φi as the linear combination of atomic orbital basis functions (LCAO). These are derived from solving isolated atom equations and cutting off the functions at some artificial barrier. However, there is one problem with writing down the full problem in the LCAO basis. That comes up when one is trying to compute the contributions to the Hamiltonian as outlined in Equation 5.3 and solving the Poisson equation for the Hartree potential, Equation 2.13. The potential has a representation in the real basis, but the representation for the PDE and the associated boundary conditions would be unclear in the LCAO basis. That means once there is a guess for the electronic density, it must be converted from real space to LCAO basis for the computations of the Green’s function and thus the electronic density. After this, the form of the potential must be refined by converting the electronic density to a real space density and then solving the Hartree contribution to the potential. This is summarized in Equation 4.41

H˜ → ρ˜ → ρ → V → V˜ → H.˜ (4.41)

44 where ˜ represents the computations done in the LCAO basis and the ones without it are the quantitites computed in the real basis, R3. 4.8 Summary

In this chapter, the Green’s function for a two-probe device is now given in terms of self-energies of the leads, Σ as

R R −1 A ∗ G = [Efull − H − Σ ] = (G )

The Green’s function due to injection of charge, G<, was introduced as

G< = GA − GR.

Also, deﬁnitions of the Fermi distributions and chemical potentials which allowed the calculations of Emin and Emax as bounds for the integral used to compute the non-equilibrum contribution to the density, ρneq where

1 Z Emax ρneq = G<(E)dE. 2π Emin

Using several identities, ρeq can also be calculated as

1 Z Emin ρeq = − Im[GR(E)]dE. π 0

After this, a closed form for the Green’s function was introduced by the use of Bloch’s theorem and a recursive formula. This led to the deﬁnition seen in

Equation 4.36. Using that form of GLL and a similar form for GRR, these can be substituted into the block tridiagonal matrix for the Green’s function as seen in Equation 4.2. From this, the electronic density for the central region, based on GCC can now be computed.

45 CHAPTER 5 Numerical Methods and Results 5.1 Overview

So far, a complete overview of how to model a two probe nano-electronic device has been laid out. This chapter will discuss how to compute the electronic density numerically. The method and code to compute electronic density was formulated by Dr. Hong Guo’s group at McGill university [19], [20] and is described here. The diﬃculty of the computation is shown to be the computation of the Green’s function at diﬀerent energy values [19] for each iteration of the Kohn-Sham scheme. To alleviate this computational bottleneck, a preconditioning method is presented. This method is based on the iterative nature of the problem. As it is shown here, successful preconditioning lowers the condition number. In turn, solving a linear system iteratively as described in [21], [8], with preconditioning, is shown to lower the number of iterations required. 5.2 Kohn-Sham Equations

Recall from Chapter 2 that the total energy functional of the system can be uniquely deﬁned in terms of density as

Z E[ρ] = v(x)ρ(x)dx + T [ρ] + Vext[ρ]. (5.1) Ω

46 Also recall that the density, ρ, is

N X 2 ρ(x) = |ψi(x)| with i Z ρ(x)dx =N. (5.2) 3 R Thus, the Schr¨odinger was written as an eigenvalue problem

2 − ~ ∇2ψ (x) + v (x)ψ (x) = ψ (x), (5.3) 2m i eff i i i with veff calculated using

δE [ρ(x)] v (x) = V [ρ(x)] + xc + V [ρ(x)]. (5.4) eff H δρ(x) ext

It should be emphasized that knowing the correct exchange-correlation energy,

EXC , and thus the eﬀective potential, veff , would lead to computing the exact ground state density. However, since EXC is phenomenologically derived, we can only ever have approximate ρ.

Equation 5.2 - 5.4 are the Kohn-Sham scheme.We emphasize that ψi and ρ are interdependent and the Kohn-Sham scheme is a non-linear eigenvalue problem. It can be solved iteratively, starting with an initial guess for the electronic density and updating it as shown in the scheme in section 5.2.1. 5.2.1 Schematic of the Solution Scheme

1. Computation of density in LCAO basis via the Green’s functions:

1 Z Emin 1 Z Emax ρ˜TOT = − Im[G˜R(E)]dE + G˜<(E)dE π 0 2π Emin

2. Conversion of density into a basis in R3,

ρ˜ = Φ∗ρΦ

47 3. Computation of the potential

V = VH + VXC

from

2 ∇ VH ≡ −4πρ in Ω, subject to VH = Vbulk on ∂Ω

and 2/3 1/3 4/3 a1ρ + a2ρ + a3ρ + a4 VXC = δ ρ 2/3 1/3 /δρ b1ρ + b2ρ + b3ρ + b4 4. Conversion of the above potential to LCAO basis

V˜ = C∗VC

5. Finally, computation of H˜ from V˜ . 5.2.2 Bottlenecks

In the scheme outlined, there are two bottlenecks. One of them occurs at step 1 and the other at step 2 and equivalently step 4. The ﬁrst calculation involves a matrix inversion in order to compute G and the other two steps involve a conversion from one basis to another. This chapter deals with what can be done to speed up the numerical inversion of the Hamiltonian, H, to compute the Green’s function, G, in order to avoid that bottleneck. 5.3 Broyden’s Method

Broyden’s method is used to compute the solution to the self-consistent set of Kohn-Sham equations. This method is a fixed point algorithm and is sometimes referred to as a generalization of the secant method to a system of equations from the general category of ”quasi Newton -Raphson” methods. Broadly, to find a fixed point x for f(x), define a g(x) such that

g(x) ≡ f(x) − x.

48 At a ﬁxed point, g(x) = 0. The derivation of the method starts from expanding the given function, g(x) as a Taylor series up to the ﬁrst derivative aboutx ¯

g(x) ≈ g(¯x) + g0(¯x)(x − x¯) = 0, and rearranging to ﬁndx ¯ g(¯x) x ≈ x¯ − . g0(¯x)

This yields to an iterative method. Let x0 be an initial guess, then for k = 0, 1, ..., n, g(x(k)) x(k+1) ≈ x(k) − . g0(x(k)) In one variable, perhaps the simplest way to approximate the derivative ap- pearing on the right hand side is to use the secant method. In other words, we use a ﬁnite diﬀerence approximation

g(x(k)) − g(x(k−1) g0(x(k)) ≈ . x(k) − x(k−1)

Suppose we wish to solve a system of equations, G(x) = 0, where x(k) is a vector quantity. The derivative in the approximation now becomes a Jacobian, J(x(k)), leading to Newton’s method

−1 x(k+1) ≈ x(k) − J(x(k)) G(x(k)). (5.5)

When the system is nonlinear, the Jacobian is relatively costly to compute. Broyden’s method introduces an approximation to the Jacobian and therefore, the partial derivatives which appear in the Jacobian as shown in Equation 5.6

∂Gi Jij(x) = (x). ∂xj

49 The Jacobian required to update the initial guess, or J(x(0)) = J (0) can be computed either analytically, or according to the secant method.

G (x(0) + e h) − G (x(0)) J (x(0)) ≈ i j i , (5.6) ij h where ej is the unit vector in the j-th direction and h is a small real number. All the Jacobians for the successive iterates, or J (k), k ≥ 1 will be computed according to the ”least-change secant approximation”. Thus, Broyden’s method becomes

G(x(k) − J (k−1)(x(k) − x(k−1)) J (k) ≈ J (k−1) − ||x(k) − x(k−1)||T , (5.7) ||x(k) − x(k−1)||2

T where ||.|| indicates the L2 norm and indicates a transpose [17]. For our problem, there is an operator A : ρ → H[ρ] and B : H[ρ] → ρ. We want to calculate the ﬁxed point of B ◦ A. 5.3.1 Convergence

The criteria for terminating the self consistent loop, i.e. the criteria for determining how good the guess for the electronic density, ρ, depends on calculating three quantities 1. Density : the diﬀerence between the values of electronic density ρ be-

tween diﬀerent iterations is measured until the L2-norm smaller than 10−3,

(k−1) (k) −3 ||ρ − ρ ||2 < 10 .

2. Effective Potential : since the Hamiltonian is dependent on the electronic density and the electronic density in turn depends on the Hamiltonian, the difference between different iterates is also measured. Only the ef-

fective potential changes with each iteration, so only the Veff term is of

50 interest. The criteria for its convergence is also 10−3,

(k−1) (k) −3 ||Veff − Veff ||2 < 10 .

3. Band Structure Energy : Deﬁning EBS ≡ T r[ρH], the band structure energy diﬀerence between successive iterates can also be compared and used as a stopping criteria once its below 10−3. 5.4 Gaussian Quadrature

Since the numerical problem at hand is how to speed up the matrix inversion that occurs when computing the Green’s function

G(E) = [ES − H − Σ]−1, it is important to mention how many times this matrix inversion occurs. The function that invokes the computation of G(E) calls the matrix inversion for each energy, E because I ρ = G(E)dE.

We use Gaussian quadrature to compute the contour integral with a summation of the function at speciﬁc values of E multiplied by a weight, w(Ei) as shown in Equation 5.8

I X G(E)dE ≈ wiG(Ei). (5.8) i

5.5 Pseudo-Code

To understand how relevant it is to ﬁnd a faster way of inverting the matrices to compute G, it will be shown how frequently these matrices are

1 From now on, for ease, the tilde notation will be dropped. For example, G˜ will now be written as G.

51 called within the numerical algorithm. As shown before, G(Ei) depends on the Gaussian quadrature points, so it is computed n times. The density, ρ(k) is then updated using the Broyden’s method which depends on the new Green’s

(k) (k−1) function and the previously calculated density, F(G (Ei), ρ ). This is outlined in Algorithm 1. This will be referred to as the one-dimensional case since there was no computation which involved the Fourier transforms of the inﬁnite leads. The number of Gaussian quadrature points examined in this system is 40.

Algorithm 1 One dimensional system while convergence is not reached do for i from 1 to n, the number of quadrature points do (k) (k) (k) −1 G (Ei) = [EiS − H ] end for (k) Pn (k) (k−1) ρ = F( i=1 G (Ei), ρ ) k = k + 1 end while

The three-dimensional algorithm involves a summation over wavevectors since Bloch theorem for their computation was invoked. This means there exists an extra for-loop as shown in Algorithm 2. The number of wavevectors used in this model is 64. In total, there are then 2560 matrix inversions for the computation of the electronic density per iteration of the self-consistent scheme. Algorithm 2 Three dimensional system while convergence is not reached do for j from 1 to number of wavevectors do for i from 1 to n, the number of quadrature points do (k) (k) (k) −1 Gj (Ei) = [EiSj − Hj ] end for (k) Pn (k) (k−1) ρj = F( i=1 Gj (Ei), ρj ) end for (k) Pm (k) ρ = ( j=1 ρj ) k = k + 1 end while

52 5.6 Conditioning

Recall that a well-conditioned function f(x), is one with the property that all small perturbations of the data, x, lead to small changes of f(x). An ill-conditioned problem is one where small perturbation of x leads to a large change in f(x). If δx denotes a small perturbation of x and the corresponding change in the δf where

δf = f(x + δx) − f(x), then the absolute condition number κˆ is deﬁned as

kδfk κˆ = lim sup , (5.9) δ→0 kδxk≤δ kδxk where the norm k.k is taken to be the L2 norm for the duration of the chapter, unless otherwise speciﬁed. Similarly, the relative condition number can be deﬁned as

kδfk kxk κ = sup . (5.10) δx kf(x)k kδxk

In the context of linear systems, consider computing

Ax = b.

The relative condition number of the problem will be

kxk κ = kAk (5.11) kAxk

Moreover, if A is an invertible matrix, then the condition number of A satisﬁes

κ ≤ kAkkA−1k (5.12)

53 Table 5–1: Given energy values, Ei for a sample one-dimensional system.

i Ei 1 −0.000149697867065979 + 0.0173024080601708i 2 −0.0983151054208785 + 0.432393745199718i 3 −0.734490875450673 + 0.964108347013472i 4 −1.63487570292988 + 0.772614290464068i 5 −1.98264415420533 + 0.185500582764808i

The notion of conditioning is important in order to see if the Kohn-Sham scheme is accurate at each step of the algorithm where the guess for electronic density is reﬁned. Since the density depends on the integral of an inverse of the Hamiltonian matrix, the condition number for that should be relatively low so that we know the inverse is well-deﬁned. 5.7 Sample One-Dimensional System

As previously described, the one-dimensional system does not involve sampling over wavevectors, but does depends on the energy, Ei of the system. Sample data with 5 given Hamiltonians, these being the 16th, 19th, 22nd, 25th and 28th iterates of the self-consistent Kohn-Sham scheme, were given. Also given, were 5 values of E on the complex plane for which the Green’s function was computed as shown in Table 5–1. The size of the matrices was 696 by 696 with the sparsity pattern can be seen in Figure 5.7. The methods considered for speeding up the computation of the direct inverse are the following: 1. Using basic iterative methods 2. Using Krylov subspace methods 3. Preconditioning A preconditioner was also applied in methods 1 and 2 to speed up their convergence. In testing the algorithms, only the ﬁrst column of the inverse was computed.

54 Figure 5–1: The sparsity of the Hamiltonian for a two-probe, one-dimensional device where the size of the matrix is 696 by 696.

5.7.1 Iterative Matrix Inversion Schemes

The current algorithm to invert matrices involved in computations of electronic properties of the nano-device, uses Matlab’s ”inv” function. This function is part of a package called ”LAPACK” which has a lengthy user’s guide and for a summary, the description can be found in [21], so it will not be outlined here. In this work, alternative methods for ﬁnding an inverse were attempted. These were iterative schemes.The most common include 1. Gauss-Seidel and successive overrelaxation - for matrices that are either diagonally dominant, or symmetric and positive deﬁnite. 2. GMRES - is used for non-Hermitian matrices. Consider the system Ax = b, with A is an n by n complex valued square matrix and x, b ∈ Cn The matrix A can be split up into an upper triangular matrix U, a lower triangular matrix, L and a diagonal matrix, D.

A = D + U + L

55 then, Dx + (L + U) x = b.

Solving for x in the above leads to

x = D−1 [b − (L + U) x] .

If the successive iterates of the elements of x are x(k), then the scheme can be written as

x(k+1) = D−1 b − (L + U) x(k) , (5.13)

The result is known as the Jacobi method and it is iterated until self- consistency is reached. The Gauss-Seidel method takes advantage of the fact that in order to compute x(k+1), some of the elements of x(k+1) are already known. This leads to a modiﬁed scheme given by Equation 5.14 known as the Gauss-Seidel method.

x(k+1) = (D + L)−1 b − Ux(k) . (5.14)

The above method can be improved by introducing a relaxation factor, ω, that governs how much weight is put on the terms involving the previous iterate and the current iterate as shown in Equation 5.15 as seen in [21],

(D + ωL)x(k+1) = (−ωU + (1 − ω)D)x(k) + ωb. (5.15)

Equation 5.15 is referred to as successive overrelaxation. However, it must be noted that the parameter ω is hard to predetermine. Thus far, the methods outlined are some of the simplest and their use is limited. The more common way to iteratively solve a linear system, is to use the Generalized Minimal Residual method (GMRES). To explain the

56 method, the deﬁnition for the residual is needed

r = b − Ax. (5.16)

The idea behind the method is thus minimizing the residual at every iteration. This method is more lengthy to describe, so instead for its derivation, the book by Trefethen [21] is an excellent resource. Trying out the basic iterative methods proved to be successful. Gauss-Seidel converged without a preconditioner, but in applying a preconditioner, it converged faster. This is discussed in detail later on. Successive overrelaxation also converged, but the optimal relaxation parameter is hard to ﬁnd and in the tests performed, never improved the convergence rate of Gauss-Seidel, these results are not shown. 5.7.2 Some Notes on Inverses

The criteria for the conjugate gradient method to converge, is that the matrix is symmetric and positive deﬁnite. This is not met by the Hamilto- nian matrices. MINRES also can’t be applied since the Hamiltonian with the additions of self-energies is no longer Hermitian. However, the generalized minimal residual algorithm, GMRES, does work on the system. We used GMRES without restarting. In our case, if we want the full inverse, A−1, this can be computed by knowing that if b is set to be a vector where only a particular element is 1 and

T the rest are zero, for example b = (1, 0,..., 0) ≡ e1, then the resulting x will be the ﬁrst column on the inverse. Then if only the second element of b is 1 and the rest are zero, the second column of A−1 is computed. Thus, the full inverse can be obtained by varying the entries in b. This is derived by using the fact that AA−1 = I

57 and letting b be successive columns of the identity matrix. However, some of the electron transport properties do not rely on the full inverse, but instead on traces of the inverse (for details, see [3]). Thus, we want to iteratively solve a linear system

(k) (k) (k) [EiS − H ]x = ei (5.17)

(k) with x being a vector we want to compute and ei being a column vector with only the i-th entry as 1 and the rest zero. 5.7.3 Preconditioning

Suppose we are trying to ﬁnd a solution, x to the linear system

Ax = b.

If A is ill-conditioned, the inverse will be hard to compute. Introducing now a matrix, M, also called a preconditioner, and multiplying both sides of the original equation M −1Ax = M −1b = c, would lead to x = [M −1A]−1c.

If the condition number of M −1A was now lower, then the system would be faster to solve iteratively. The hard part is ﬁnding this matrix M, a preconditioner to lower the condition number without many extra computational steps. For the particular system at hand, it is desired to ﬁnd the Green’s function for each iteration (k) of the self consistent loop of the Kohn-Sham scheme, given by

G(k) = (ES − H(k) − Σ)−1. (5.18)

58 Label H˜ (k) = ES −H(k) −Σ where this H˜ (k) is previously calculated. In trying to calculate the next iterate of the Green’s function, G(k+1), the previous iterate G(k) can be used to make the matrix have a lower condition number. Since the self-consistent iteration over the Green’s functions ﬁnds an inverse of a matrix that is similar to the matrix in the previous step, their inverses at each step (since we are interested in reﬁning the electronic density at each step until convergence) should also be similar. Thus,

G(k+1)H˜ (k) ≈ I (5.19)

H˜ (k+1)G(k) ≈ I. (5.20)

One method of preconditioning is then

G(k+1) = G(k)(H˜ (k+1)G(k))−1. (5.21)

This means the matrix to be inverted when preconditioned with the previous iterate’s inverse, will have a lower condition number. For iterative inversion algorithms which are useful for large sparse matrices, this helps the schemes to converge faster. Also, if we are only interested in computing speciﬁc columns of the inverse, this will be even faster with an iterative linear solve for x(k) as in Equation 5.17. 5.7.4 Results

The ﬁgures below summarize the number of iterations is takes to solve a linear system for diﬀerent preconditioning methods. There are four methods for each scheme and the dotted line always represents the non-preconditioned Gauss-Seidel (in red on the upper part of the graphs) and GMRES methods (in blue on the lower part of the graphs) for comparison. The Gauss-Seidel algorithm was written by us and Matlab’s ”GMRES” code with and without a preconditioner was used. The dashed line for GMRES represents what happens

59 when the preconditioner was applied at each iteration of the iterative solver and the solid line represents inverting a previously preconditioned matrix. Only computation of the ﬁrst column of the inverse, x(k) was done, as in

−1 (k) (k) (k) −1 M [EiS − H ]x = M e1.

Tolerance for when the algorithm was assumed to have converged was set to

−6 10 . It represents the L2 norm of the diﬀerence between successive iterates of the inverses generated by the iterative inverse algorithm

tolerance = kx(k−1) − x(k)k ≤ 10−6.

The preconditioner varied depending on the method and was always a previously calculated Green’s function, M −1 ≡ G.

60 List of methods to solve the linear system:

(1) 1. Using G (E1) as a preconditioner.

2500 preconditioned 2000 no preconditioner

1500

1000

500

Number of Iterations to Solve Linear System 0 1 6 11 16 21 25 Iterate of [E S(k) − H(k)] i

500

400

300 preconditioned before preconditioned inside 200 no preconditioner

100

Number of Iterations to Solve Linear System 0 1 6 11 16 21 25 Iterate of [E S(k) − H(k)] i

Figure 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

(1) (k−1) (k) −6 used was G(E1) and the stopping criteria was kx − x kL2 ≤ 10 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 (1) 2. Using G (E2) as a preconditioner.

2500 preconditioned 2000 no preconditioner 1500

1000

500

Number of Iterations to Solve Linear System 0 1 6 11 16 21 25 Iterate of [E S(k) − H(k)] i

500 preconditioned before 400 preconditioned inside no preconditioner 300

200

100

Number of Iterations to Solve Linear System 0 1 6 11 16 21 25 Iterate of [E S(k) − H(k)] i

Figure 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

(1) (k−1) (k) −6 used was G(E2) and the stopping criteria was kx − x kL2 ≤ 10 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 (1) 3. Using G(Ei) as a preconditioner with energy values corresponding to the same energy as the matrix to be inverted.

2500 preconditioned 2000 no preconditioner 1500

1000

500

Number of Iterations to Solve Linear System 0 1 6 11 16 21 25 Iterate of [E S(k) − H(k)] i

400 preconditioned before 300 preconditioned inside no preconditioner 200

100

Number of Iterations to Solve Linear System 0 1 6 11 16 21 25 Iterate of [E S(k) − H(k)] i

Figure 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

(1) (k−1) (k) −6 used was G(Ei) and the stopping criteria was kx − x kL2 ≤ 10 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 4. Using the previously computed Green’s function from an iteration of

(k−1) the self-consistent Kohn-Sham algorithm loop, G(Ei) as a preconditioner energy values corresponding to the same energy as the matrix to be inverted.

2500 preconditioned 2000 no preconditioner 1500

1000

500

Number of Iterations to Solve Linear System 0 1 6 11 16 20 Iterate of [E S(k) − H(k)] i

400 preconditioned before 300 preconditioned inside no preconditioner 200

100

Number of Iterations to Solve Linear System 0 1 6 11 16 20 Iterate of [E S(k) − H(k)] i

Figure 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) (k) −6 used was G(Ei) and the stopping criteria was kx − 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 The condition numbers for the diﬀerent methods are shown in Figure 5–6. It is important to note that the condition numbers for matrices which had no

(1) preconditioning and the matrices where the preconditioner was G (E2) had the same result. Also, these do not represent an absolute measure of how long it takes to invert a matrix, however, they gives an indication nevertheless.

120 G(1)(E ) 1 G(1)(E ) 2 100 G(2)(E ) 3 G(1)(E ) i G(k−1)(E ) 80 i no precond.

Condition Number 40

0 1 6 11 16 21 25 Iterate of [ES(k)− H(k)] i

Figure 5–6: This ﬁgure illustrates the diﬀerent condition numbers of matrices

(k) (k) [EiS −H ]. Notice that the most successful preconditioning algorithms are the ones where the energy values of preconditioners coincide with the energy

(k) (k) values of the iterates of [EiS − H ].

(1) (k−1) Also, the preconditioners G (Ei) and G (Ei) produce really small condition numbers and a new ﬁgure to show just how small these numbers are, is shown in Figure 5–7. This ﬁgure shows that preconditioning is most successful when the preconditioners have the same energy value as the linear

65 system since the energy is the greatest contribution to the condition number. This is because depending on the energy, we can be closer to the eigenvalues of the Hamiltonian and therefore closer to poles of the Green’s function. If we were computing a Green’s function at the pole, the matrix would then be singular.

1.035 G(1)(E ) i 1.03 G(k−1)(E ) i

1.025

1.02

1.015 Condition Number

1.01

1.005

1 1 6 11 16 21 25 Iterate of [E S(k) − H(k)] i

Figure 5–7: This ﬁgure illustrates the diﬀerent condition numbers for the two most successful preconditioning schemes.

2 In these ﬁgures, the vertical dotted lines represent the diﬀerent iterations of the self-consisten Kohn-Sham algorithm. The superscript k in H(k) and G(k) represents the iteration number of the Kohn-Sham loop and corresponds to iterates 6, 11, 16 and 21. In between the dotted lines, the integration variable, Ei, is varied for i = 1, ..., 5.

66 5.8 Comments and Future Directions

It was shown that for a one-dimensional system, diﬀerent preconditioning schemes reduce the number of iterations required to solve a linear system and these can be compared with the known solutions for the example shown. However, the interesting application of this would be for three-dimensional systems since in that case, it is not necessarily possible to compute a direct inverse for the larger matrices. In the appendix, the condition numbers for these larger systems can be seen. Unfortunately due to lack of computing power and no sample data, the preconditioning strategies were not tested on these matrices. This will be a subject for future work. 5.9 Summary

In this section, the numerical computation of the electronic density was shown. In one-dimension, this is given by Algorithm 1 and in three-dimensions, this is given by Algorithm 2. Both of these algorithms contain at least one ”for” loop and a ”while” loop. That means a matrix inversion is done inside at least two loops and this is time-consuming. Since not all entries of the electronic density are needed, an iterative linear solver is suggested such as Gauss-Seidel or GMRES. Successful preconditioners to speed up the iterative solvers are

(1) (k−1) also suggested such as using G (Ei) and G (Ei). This lowers the number of iterations required to solve the system, speeding up the computation of electronic density.

67 CHAPTER 6 Conclusion In this thesis, we have outlined the mathematical principles behind electronic density calculations for two-probe devices. At ﬁrst, the principles of density functional theory based on theorems by Hohenberg-Kohn [12], [9] were explained, which led to the formulation of the problem using the Kohn-Sham equations [13]. They were shown to have solutions that are iteratively computed by the Green’s functional formalism. Using matrix notation, an explicit equation for electronic density was derived, which required a contour integration of the Green’s function [23]. Numerically, this is a time-consuming step which invokes a matrix inversion for each point on the contour integral. To speed up the matrix inversion, preconditioning strategies were introduced for iterative methods such as Gauss-Seidel and GMRES. These were tested on supplied sample calculations and several iterates of the Kohn-Sham scheme for a one-dimensional system. The result of the preconditioning strategies can be summarizes as follows

• The condition number varies greatly depending on the energy, Ei.

(k) (k) • Preconditioning the current iterate, k, of the Hamiltonian, EiS − H

(k−1) by the Green’s function from a previous, (k − 1) iteration, G (Ei),

(k) (k) k−1 works to reduce the condition of the matrix [EiS − H ]G (Ei) to almost unity. • Preconditioning lowers the number of iterations that a GMRES or Gauss- Seidel needs to go through to converge. Using the results of this work can lead to faster algorithms for the calculation of electron transport. If the end goal is to calculate the electronic density,

68 then classic iterative matrix inversions of preconditioned matrices instead of the currently used direct inversion currently emplyed [19] is strongly recom- mended. For larger systems, this method of preconditioning can be coupled to the already existing block diagonal inversion algorithms as in [15], [4] [24] and [7] and used on three-dimensional systems. However, as it stands, this is still a subject of future work.

69 APPENDIX: Condition Numbers for Three-Dimensional System

70 The condition numbers for a sample two-probe system were also examined. Once again, a plot of quadrature points is included in Figure 6–1. In total there are 40 points with values ranging from 0 to −2.5 along the real axis and with imaginary values ranging from 0 to 1.2. Figure 6–2 illustrates the condition numbers of the Hamiltonian given an energy value, E and a wave vector, k for the two-probe system. The wave vectors are 64 equally spaced points on a square grid ranging from values of −3 to 3 on each side. The condition numbers of the Hamiltonian range from 10 to just below 75. They are seen to depend great only the energy value and the k-point, but not as much on the iteration of the Kohn-Sham scheme. The line in Figure 6–2 represents the end of one iteration of the Kohn-Sham scheme while each local lowest point represents the end of each inner ”for” loop over the energy values (for details refer to Algorithm 2). The greatest inﬂuence on the condition numbers were the energy values, with the one closest to 0 making the Hamiltonian the most ill-conditioned. The symmetric nature of the condition numbers due to the regularity of wave vector sampling points can be seen by examining the iterate of H near the 1280 value. The condition numbers of the Hamiltonian about that axis are symmetric. Also, the condition numbers for energy values away from E = 0 are very similar and can be considered low.

71 Figure 6–1: Chosen Gaussian quadrature points for the integration of the system in three dimensions.

Figure 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.

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