Quantum Transport Calculations Using Wave Function Propagation and the Kubo Formula

Master Thesis

Quantum transport calculations using wave function propagation and the Kubo formula

Troels Markussen (s001477) [email protected]

Supervisors: Mads Brandbyge and Antti-Pekka Jauho

MIC – Department of Micro and Nanotechnology Technical University of Denmark

9th March 2006

Preface

This thesis is submitted in candidacy for the Master of Science degree in Engi- neering Physics from the Technical University of Denmark (DTU). The project has been carried out at MIC - Department of Micro and Nanotechnology in the period from February 1st 2005 to February 1st 2006 with Mads Brandbyge and Antti-Pekka Jauho as supervisors. I would like to thank my supervisors for their enthusiasm and constructive criticism. I have enjoyed the many fruitful discussions and in general working in the Theoretical Nanotechnology group. In particular I thank Magnus Paulson for help with obscure Linux problems and Python programming. I would like to thank Oticon Fonden for their generous ﬁnancial support. This enabled me to visit N. Lorente and R. Rurali in Toulouse, France, to discuss collaborative work on the metallic silicon nanowires and learn about the siesta- program. The grant also made it possible to participate in the conference Elec- Mol’05 in Grenoble, France, where some of the results from the project was presented on a poster. I thank the people at MIC for making it a pleasant place to spend a lot of time. I also thank the guys in room 119 for fruitful and indispensable discussions concerning anything but my project. Finally, I thank my girlfriend Wiebke for her patience and support during the project, and for reminding me that there are other things in life than physics.

Troels Markussen MIC – Department of Micro and Nanotechnology Technical University of Denmark 9th March 2006

Abstract

This project is concerned with modelling the electrical properties of silicon nanowires. The fabrication of nanowires can be controlled with high accuracy, making them very promising candidates for future electronic components with the capability of exceeding conventional technologies. Also, it has been demonstrated that nanowires as chemical and biological sensors can be very sensitive with capability of single virus detection. Calculations on nanowires with randomly placed dopants or defects, require an atomistic model since quantum effects are important due to the small diameter, which can be down to 2 nm. The length of the wires can, however, be several µm and the numerical method should be able to treat more than 104 atoms, implying that (N) scaling is required. O Two (N) methods, both using a tight-binding model based on ab initio calculations,O have been applied in this project. The first approach, derived from the Kubo-Greenwood formula, uses time propagation of wave packets to estimate the conductance. This method has until now only been applied by a single group and primary to model carbon nanotubes. One of the main objectives in this work has been to understand, implement and test this relatively new method. The second approach is based on the Landauer formula and the conductance is found by recursive calculations of Green’s functions. Comparison of the two approaches revealed that the Green’s function method is the preferred choice for modelling thin nanowires. It is more rigorous than the Kubo method and does not suffer from having many convergence parameters. The transport properties in both pure silicon wires and wires passivated with hydrogen is shown to be strongly affected by surface disorder. Randomly missing H atoms causes the electrons to localize, and both the elastic mean free paths and localization lengths scales linearly with the average distance between H vacancies. Comparison with estimated electron-phonon interactions indicates that impurities might be the dominant scattering mechanism for thin wires even at room temperature.

Resum´e

Dette projekt omhandler modellering af elektriske egenskaber i silicium nanotr˚ade. Fremstillingen af nanotr˚ade kan kontrolleres med stor nøjagtighed, hvilket gør nanotr˚ade til lovende kandidater for fremtidige elektroniske komponenter med mulighed for at overg˚akonventionelle teknologier. Det er ligeledes blevet demonstreret, at nanotr˚ade som kemiske og biologiske sensorer kan være meget følsomme med mulighed for at detektere enkelte virusmolekyler. Modellering af nanotr˚ade med tilfældigt fordelte doteringer eller defekter kræver en atomar model, da kvanteeffekter er væsentlige p˚agrund af den lille diameter, som kan være ned til 2 nm. Længden af nanotr˚adene kan til gengæld være adskillige µm, og den numeriske metode skal derfor være i stand til a behandle mere end 104 atomer, hvilket betyder, at (N) skalering er nødvendigt. O Der er i projektet blevet anvendt to (N) metoder, der begge benytter en tight-binding model baseret p˚a ab initioO beregninger. Den første metode, der er udledt fra Kubo-Greenwood formlen, bruger tidsudvikling af bølgepakker til at estimere konduktansen. Denne metode er indtil nu kun blevet anvendt af en enkelt gruppe til at studere kulstof nanorør, og et af hovedform˚alene i dette arbejde har derfor været at forst˚a, implementere og teste denne relativt nye metode. Den anden fremgangsm˚ade er baseret p˚aLandauer formlen og konduktansen findes ved rekursive beregninger af Greens funktioner. En sammenligning af de to metoder gav, at Greens funktions metoden er det foretrukne valg til at modellere tynde nanotr˚ade. Den er mere stringent end Kubo-metoden, og har ingen konvergens parametre. Det er vist, at transportegenskaberne i b˚ade rene siliciumtr˚ade og hydrogen passiverede tr˚ade, er stærkt p˚avirket af overfladedefekter. Tilfældigt fjernede H atomer bevirker, at elektronerne lokaliserer, og b˚ade den elastiske fri mid- delvejlængde og lokaliseringslængden skalerer lineært med den gennemsnitlige afstand mellem H vakancerne. Sammenligning med estimerede elektron-phonon vekselvirkninger indikerer, at defekter kan være den væsentligste sprednings- mekanisme selv ved stuetemperatur.

Contents

List of symbols xi

1 Introduction 1 1.1 Semiconducting nanowires ...... 1 1.1.1 Growth of silicon nanowires ...... 2 1.1.2 Electronic applications ...... 3 1.1.3 SiNWsassensors...... 4 1.1.4 Critical issues and possible theoretical help ...... 5 1.2 Theoretical techniques ...... 6 1.2.1 Atomic and electronic structure ...... 6 1.2.2 Transport calculations ...... 7 1.2.3 Real-space Kubo-method ...... 7 1.2.4 Applications of the real-space Kubo method ...... 8 1.3 Motivation and outline ...... 10 1.3.1 Outline ...... 11

2 The real-space Kubo formalism 13 2.1 Derivation of the real-space Kubo formula ...... 13 2.1.1 Preliminary notations ...... 13 2.1.2 Rewriting the Kubo-Greenwood formula ...... 15 2.1.3 Time evolution ...... 17 viii CONTENTS

2.2 Three regimes of transport ...... 17 2.2.1 Ballistic regime ...... 18 2.2.2 Diﬀusiveregime ...... 18 2.2.3 Localization...... 19 2.2.4 Simple model for diﬀusion and localization ...... 20

3 Numerical methods 22 3.1 Exact solution to the time-dependent Schrödinger equation ...... 22 3.2 The Chebyshev method ...... 23 3.2.1 Basic properties of Chebyshev polynomials ...... 23 3.2.2 Expansion of Uˆ(t) in Chebyshev polynomials ...... 23 3.2.3 Convergence properties ...... 24 3.2.4 Convergence of the coefficients ...... 25 3.2.5 Comparison of Chebyshev and Taylor ...... 26 3.3 Continued fraction technique ...... 26 3.4 Overview of the computer codes ...... 28 3.4.1 Important parameters ...... 28 3.4.2 Dataflow ...... 29

4 The one dimensional chain: Testing of the method 31 4.1 Local Density of States ...... 31 4.1.1 The infinite chain ...... 32 4.1.2 Comparison of numerical and analytical LDOS ...... 32 4.1.3 Comparing the velocities ...... 34 4.2 Disorderedchain ...... 35 4.2.1 Random initial states ...... 36 4.2.2 Time dependent diffusion coefficient (E,t)...... 36 D 4.2.3 Meanfreepath ...... 39 4.2.4 Conductance of disordered wire ...... 40 4.3 Carbon nanotubes: Comparison with published results ...... 43 4.4 ParallelChains ...... 46 CONTENTS ix

4.4.1 Calculation of the conductance ...... 47

5 The Landauer formula and recursive Green’s function method 50 5.1 Recursive Green’s function method ...... 50 5.1.1 Introduction to the Landauer formalism ...... 50 5.1.2 Recursive (N)growthprocess...... 52 O

6 Building a Tight-binding model 55 6.1 Introduction and ideas behind the model ...... 55 6.1.1 Finding the tight-binding parameters ...... 56 6.2 From atomic basis set to orthogonal tight-binding ...... 57 6.2.1 Hamiltonian for the full wire ...... 58 6.2.2 L¨owdin transformation ...... 59 6.2.3 Approximate orthogonalization ...... 60 6.2.4 Example - The one-dimensional chain ...... 61

7 Modelling of silicon nanowires 64 7.1 Metallic silicon nanowires ...... 64 7.1.1 Truncating the Hamiltonian ...... 65 7.1.2 Conductance of a pristine wire ...... 66 7.1.3 Anderson disorder ...... 67 7.1.4 Sub-conclusion ...... 69 7.2 Hydrogen passivated Si-wire ...... 71 7.2.1 Setting up the model ...... 71 7.2.2 Length and energy dependent conductance ...... 73 7.2.3 Distribution of conductances ...... 74 7.2.4 Resistance vs. length ...... 76

7.2.5 Mean free path and peaks of σG(L)...... 77 7.2.6 Comparison with analytical models ...... 78

7.2.7 Scaling of le and ξ ...... 80 7.2.8 Sub-conclusion ...... 81 x CONTENTS

8 Summary and outlook 82 8.1 Evaluation of the real-space Kubo method ...... 82 8.2 Modelling of silicon nanowires ...... 85 8.3 Outlook ...... 86

Appendix 87

A L¨owdin transformation 88 A.1 The generalized eigenvalue problem ...... 88 A.2 Formal orthogonalization procedure ...... 90 A.3 How to do the orthogonalization ...... 91

B Numerical Methods 93 B.1 The Continued Fraction Technique ...... 93 B.1.1 Truncation of the continued fraction - self-energy . . . . . 94 B.2 Tridiagonalization procedure ...... 96 B.3 Exact solution to the time-dependent Schr¨odinger equation . . . 97 B.3.1 Time propagation using Taylor expansion ...... 98 B.4 Recursive calculation of self-energies ...... 99

C Miscellaneous 101 C.1 Total and Local Density of States ...... 101 C.1.1 Green’s function and local density of states ...... 102 C.2 Ohm’s law and mean free paths ...... 104 List of common symbols

Symbol Description Unit

2 1 (E,t) Time- and energy depentent diﬀusion m s− D coeﬃcient 1 di(E) Local density of state eV− E Energy eV 1 G Green’s function matrix eV− 1 Gˆ Green’s function operator eV− 1 G Conductance Ω− H Hamiltonian matrix eV Hˆ Hamiltonian operator eV h On-site hamiltonian matrix eV 1 k Wavevector nm− L Length nm le Elastic mean free path nm N Number of conducting channels Ni Number of non-zero elements in a random phase state Ntri Size of tri-diagonal matrix, Htri Ntot Size of Hamiltonian matrix, H 1 n(E) Total density of state eV− R Position m R Resistance Ω R0 Contact resistance Ω S Overlap matrix xii LIST OF SYMBOLS

Symbol Description Unit

T (E) Transmission Tn(x) Chebyshev polynomial of order n t Time fs Uˆ(t) Time evolution operator 1 Vˆx x-component of velocity operator, m s− Schrödinger picture 1 Vˆx(t) x-component of velocity operator, Heis- m s− senberg picture V Coupling matrix eV v Coupling matrix eV 1 v(E) Velocity at energy E m s− Xˆ Position operator, Schrödinger picture m Xˆ(t) Position operator, Heissenberg picture m 2(E,t) Squared, average displacement of wave m2 X packets with energy E γ Hopping energy eV δ Cutoff energy eV δi Random on-site noise eV ∆ε Noise amplitude (disorder strength) eV ε0 On-site energy eV η Infinitesimal energy eV 1 ρ Resistivity Ω m− 1 σDC DC conductivity Ω− m σ Overlap integral 1 σG Standard deviation of conductance Ω− Σ Self-energy eV ξ Localization length nm ψ Random phase state | ri ψi General state |n,iR n’th orbital located at position R |Φ i n’th Chebysheb state | ni

Constant Description Value

34 h Planch’s constant 6.63 10− J s × 34 · ~ Planch’s constant h/2π 1.05 10− J s × 19 · e Electron charge 1.602 10− C × Chapter 1

Introduction

In this chapter we begin with an introduction to the ﬁeld of nanowires focusing on the fabrication process and the demonstrated applications. Next, we discuss some of the theoretical techniques that have been applied to model nanowires, and pay special attention to a new real-space method, which we will apply in the project. We end up discussing the motivations for this project and give an outline of the report.

1.1 Semiconducting nanowires

Over the last decade increasing effort has been put into the research of semiconducting nanowires, experimentally as well as theoretically. The experimental work is pioneered by the groups of C. M. Lieber (Harvard) and L. Samuelson (Lund). The growth of the nanowires can be controlled to a very high degree: The length of the wires can be several µm and is controlled by the growth time, while the location and diameter are determined by the position and size of a seed particle [1, 2, 3]. Various heterostructures with atomically sharp transitions have been produced, opening up for a wide range of device applications such as single electron transistors (SET), resonant tunneling diodes and field effect transistors (FET) [2, 3, 4, 5]. Silicon nanowires (SiNWs) have also been applied as chemical and biological sensors with very high sensitivity and capability of single virus detection [6, 7]. In the following subsections we will go into more details about the fabrication techniques and applications. Since the calculations in this work have been concerned with silicon nanowires we will focus on these, and pay less attention to the III-V nanowires such as InAs, InP and GaAs. We end up discussing critical issues where modeling might be a helpful tool to gain understanding. 2 1.1 Semiconducting nanowires

1.1.1 Growth of silicon nanowires

Most of the SiNWs reported in the literature are produced by a vapuor-liquid- solid (VLS) growth method [8]. The basic principles of the method are the following. Gold nanoclusters are placed on an oxidized silicon wafer inside a reaction chamber where pressure, temperature and chemical environment can o be controlled. The chamber is heated to 400 C and SiH4 is added [1, 9]. The growth initiates at the Au nanocluster where∼ Si crystallizes and pushes the gold particle upwards forming a wire underneath. The diameter of the grown SiNWs is controlled within a few nm by the size of the Au nanoparticle [1] and ranges from 1.3 nm [10] to 100 nm [11]. Most typical sizes are, however, 5 30 nm [1, 4, 6, 12, 13]. Note,∼ that the very thin SiNWs reported in Ref. [10] were− made in a diﬀerent way using an oxide-assisted growth method, where a powder of SiO is heated in an alumina tube to form wires. The growth directions are primarily [111] for larger wires while smaller wires tend to grow in the [110] direction [1]. The length of the SiNWs is controlled by the growth time.

Figure 1.1: Left: SiNW with a Si core and a relative thick sheath of SiOx. Scale bar, 10 nm. The image is from ref. [9]. Right: STM image of a hydrogen passivated SiNW. The SiOx has been removed by a HF etch. The image is from ref. [10].

After the growth process the SiNWs are covered by a thin sheath ( 1 5 nm) of ∼ − amorphous SiOx [1]. This is clearly seen in figure 1.1 (left), showing a diffraction contrast TEM image of a SiNW with a relative thick SiOx sheath [9]. The scale bar is 10 nm. The SiOx can be removed using a hydroflouric (HF) etch resulting in a hydrogen passivated surface [10]. An STM image of such a wire is shown to the right [10]. It is possible during growth to add dopants to the SiNW thereby making them either n- or p-type semiconductors. Adding B2H6 to the reaction chamber results in boron-doped SiNWs (p-type), while phosphorus doping (n-type) can be obtained by shining high intensity laser pulses on an Au-P nanocluster during 1.1 Semiconducting nanowires 3 growth [11]. Importantly, the doping can be altered during growth thereby creating heterostructures such as p-n junctions[13]. Recently metallic/semiconducting heterostructures with nm sharp transitions have been fabricated using NiSi as the metallic part [12]. First a normal, doped SiNW is grown and subsequently lithographically defined regions of the wire are covered by Ni. Heating the wire to 550oC and afterwards removing the residual Ni produces a metallic NiSi wire with a slightly larger diameter than the original SiNW. The junctions between the metallic NiSi part and the semiconducting Si part forms Ohmic contacts, and FETs were produced with the metallic NiSi regions working as source and drain electrodes [12].

1.1.2 Electronic applications

The ability to create nanowire heterostructures with very high accuracy makes them very promising candidates for future electronic devices. Moreover, the excellent growth control is a major advantage compared to carbon nanotubes, since it is not (yet) possible to control whether a single wall nanotube is metallic or semiconducting. High performance ﬁeld eﬀect transistors (FETs) have been demonstrated using boron doped SiNWs contacted to Ti/Gold electrodes. Measured key parameters such as transconductance and mobility showed that the SiNWs have the potential to exceed conventional MOSFET devices [5]. The principal setup of the

Figure 1.2: Left: Schematic setup of a SiNW FET. The inset shows a TEM image of a SiNW. Notice the amorphous SiOx at the edges. Right: Conductance vs. gate voltage before (green) and after (red) passivation of surface defects. The polarized surface defects screen the gate ﬁeld and eﬀectively lower the gate capacitance. Both images are from ref. [5].

SiNW FET is shown in ﬁgure 1.2 (left). To the right is shown the conductance 4 1.1 Semiconducting nanowires vs. gate voltages before (green) and after (red) passivation of surface defects. We shall return to this below. Diodes have been fabricated using modulation doped SiNWs forming p-n junctions [13] and more complex structures such as complementary inverters and bipolar transistors have been produced using systems of crossed p- and n-type wires [4]. Nanowires of diﬀerent III-V semiconductor materials, such as InAs-InP heterostructures have been applied to make diodes, single electron transistors (SET) and resonant tunneling diodes [2, 3]. The various devices demonstrated and the metal-semiconductor wire-junctions all together make it possible to integrate very dense nanosystems that include both active device areas and high-performance interconnects [14].

1.1.3 SiNWs as sensors

In the recent years new applications of SiNWs as chemical and biological sensors have been demonstrated [6, 7, 15]. Zhou et al. [15] showed that a bundle of SiNWs, where the SiOx sheath was removed by a HF etch, was highly sensitive to NH3 but not to N2. With the SiOx sheath still present no significant response were observed. The sensitivity of the resistance to exposure of NH3 could either be due to changes in the surface resistance of individual wires or a changed inter-wire connection. Lieber et al. [6, 7] have produced several sensors all based on a field effect: A boron-doped SiNW is coated with anti-bodies specific for a certain species, e.g. a virus. Since the virus is electrically charged it will work as a local gate when it binds to the anti-body and the resistance of the wire changes. Figure 1.3 (left) shows the principle in such a sensor. The one-dimensional nature of the nanowires is important for the sensitivity, since the binding of a molecule to a wire surface will alter the carrier concentration in the whole cross section of the wire, whereas only the surface region of a planar device would be changed [6]. The right part of figure 1.3 shows simultaneous electrical and optical data. The upper panel shows the conductance of the wire as function of time. The numbers correspond to the optical images of the flourescently labeled influenza virus in the lower panel. Remarkably, the conductance drops precisely when the virus binds to the wire (points 2 and 4) and returns to the base level when the virus unbinds. These data show that SiNW sensors are very sensitive and that single virus detection is possible. Using different antibody receptors on different wires Patolsky et al. [7] were able to detect and distinguish two different viruses, thereby opening up for massive parallel detection of various species. 1.1 Semiconducting nanowires 5

Figure 1.3: A: Principle sketch of a sensor. As a virus binds to the surface of a SiNW device modiﬁed with antibody receptors, the conductance drops. When the virus unbinds, the conductance returns to the base value. B: Simultaneous conductance and optical data recorded for a SiNW device. The images correspond to the two binding/unbinding events marked by points 1-3 and 4-6 in the conductance data, with the ﬂourescently labeled virus appearing as a red dot in the images. The images are from Ref. [6].

1.1.4 Critical issues and possible theoretical help

Several issues concerning SiNWs are still unanswered, and theoretical work and computer modelling might be helpful together with experiments to gain better understanding. Here we shall discuss some questions that might be addressed using transport calculations. It is evident from the above-mentioned that doping of SiNWs is a crucial point when making devices. Since the nanowires are quasi one-dimensional structures with diameters down to 10 atoms, the dopant atoms act as strong scattering centers that might affect∼ the performance significantly. From experiments it is difficult to determine the nature of the scattering events, i.e. whether defects, vacancies, dopants or maybe phonons govern the resistance. Transport calculations, on the other hand, can address the different scattering events selectively. A realistic model should preferably be fully ab initio or at least based on parameters obtained from ab initio calculations. Interesting studies could for instance be to find out whether some dopants scatter less than others and thereby lead to better performance. Such a knowledge could possibly help experimentalist to make more efficient devices. 6 1.2 Theoretical techniques

In bulk materials, electron-phonon interactions are often the dominant scattering mechanism at room temperature. However, the phonon scattering is suppressed in small diameter nanowires [16], and recent experiments indicated ballistic transport in undoped Si/Ge core-shell wires at room temperature with an estimated phonon scattering mean free path lph = 540 nm [17]. This might imply that even at room temperature, defects could be the most important scattering source. Moreover, as shown in ref. [5] the conductance of a SiNW increased by approximately one order of magnitude after thermal annealing. The reason to the enhanced conductance is believed to be better metal-SiNW contacts and passivation of defects in the Si-SiOx interface. Furthermore, surface modification by reaction with 4-nitrophenyl octadecanoate lead to an increase in transconductance and mobility by approximately one order of magnitude, see figure 1.2. The explanation is probably that polar surface sites are passivated and therefore no longer screen the gate field [5]. Modelling the effect of defects and vacancies could possibly support and elaborate on the present explanations. It might also be possible to model some of the sensor applications. Calculations on e.g. NH3 and N2 on SiNW surfaces would be an interesting study that could help to understand why NH3 affects the resistance significantly while N2 does not [15]. It might also be possible to model more complex molecules and their influence on the conductance.

1.2 Theoretical techniques

Theoretical modelling of nanowires can be divided into two ﬁelds: First, deter- mination of the atomic and electronic structure, and second, calculation of the transport properties of the wire. The transport properties will often depend critically on the structure of the wire and both aspects are therefore important for a complete description.

1.2.1 Atomic and electronic structure

The atomic and electronic structure is primarily determined from density functional theory (DFT) calculations [18, 19, 20, 21] or from tight binding (TB) models [20, 22]. While DFT calculations can be very accurate they suffer from an (N 3) scaling, where N is the number of atoms. This means that an increase in computerO power by a factor of 1000 only allows to study systems increased by a factor of 10. Today, the system sizes are limited to 1000 atoms. TB calculations can, on the other hand be of (N) [23]. The drawback∼ of TB methods is that they rely on accurate input parametersO that are often specific for a given system. An example of structure calculations on pure silicon wires (i.e. with no surface passivation of O or H) is given in ref. [20, 21], where both DFT and TB 1.2 Theoretical techniques 7 methods were applied to study wires grown in the [100]-direction. It was shown that small changes in the surface reconstruction had pronounced effects on the electronic structure, changing the wire from being semi-metallic to metallic. In chapter 7 we will apply this specific structure in conductance calculations, thus highlighting the important connection between structural and transport properties.

1.2.2 Transport calculations

Transport calculations in nanoscale systems range from full ab initio methods based on non-equilibrium Green’s functions [24, 25, 26] over tight-binding models [27] to the Boltzmann equation and effective mass theory [28]. As the accuracy goes down (from ab initio to Boltzmann) the possible system sizes go up. Das et al. [28] used the Boltzmann equation to calculate the carrier mobility in relative thick (d = 10 90 nm) GaAs wires focusing on the diameter dependence. Sundaram et al. [29]− also used the Boltzmann equation to study surface effects on the transport in large diameter wires. Y. Zheng et al. [27] applied a tight-binding model of a hydrogen passivated wire and studied the effect of wire thickness on the bandgap, effective masses and transmission. The band gap was found to increase for decreasing diameters, in agreement with experiments [10] and ab initio calculations [30]. In chapter 7 we will use the same wire structure as ref. [27] to calculate transport properties of the H-passivated wires with randomly removed H atoms. Very recently X. Blase and coworkers [26] used DFT calculations to study the effect of dopants on the transmission through both passivated an un-passivates SiNWs. It was shown that B- and P-doping lead to significantly different conductance properties in the wires as compared to bulk material. To our knowledge, no theoretical works concerning nanowires, based on ab initio methods and including many scattering events, have been published.

1.2.3 Real-space Kubo-method

One of the primary objectives in this work has been to implement and test a relatively new real-space method to calculate electronic transport properties such as conductance and mean free paths. The method is developed by the group of S. Roche and D. Mayou and has been applied in a number of publications discussed below [31, 32, 33, 34, 35, 36, 37, 38, 39, 40]. The method is based on the Kubo-Greenwood formalism [41, 42] rewritten in a real-space framework, and we will therefore refer to it as the ’real-space Kubo-method’. The fundamental philosophy behind the method is that the transport properties are governed by the movement of electrons in a given sample as time evolves. If the sample 8 1.2 Theoretical techniques is a perfect crystal, the electrons can travel through the sample without any backscattering and the resistance of the sample is zero (except for the contact resistance, of course). This is called the ballistic regime since the motion resembles classical, ballistic particles. If defects are present, e.g. in the form of different atoms (dopants) or missing atoms (vacancies), the electrons will eventually hit a defect and scatter, and the motion through the sample resembles more a random walk, leading to a diffusive behaviour and Ohmic resistance. In the real-space Kubo method one calculates the time-dependent broadening of initially localized wave packets. This measures the ease with which electrons can propagate in the sample and using this, the conductivity, mean free path or resistance can be calculated. Importantly, the method scales as (N) which means the computation time doubles if the system size doubles. O In the following subsection we briefly review the applications and results obtained with the method. A more throughout discussion of the details is given in chapter 2.

1.2.4 Applications of the real-space Kubo method

Most of the published work using the real-space Kubo method have been concerned with carbon nanotubes (CNTs). S. Roche and R. Saito showed that the magnetoresistance of small diameter CNTs could be tuned from positive to negative by changing the chemical potential and the orientation of the magnetic ﬁeld [35].

F. Triozon et al. [36] calculated the elastic mean free path, le, in diﬀerent single walled carbon nanotubes (SWNTs) as function of energy and disorder strength. Figure 1.4 shows large variations of le of three diﬀerent armchair SWNTs as

Figure 1.4: Energy dependent mean free path in diﬀerent armchair SWNTs with random on-site disorder. The inset shows the scaling of the mean free path with disorder strength. The ﬁgure is from ref. [36]. 1.2 Theoretical techniques 9

function of energy. The inset shows the scaling of le vs. disorder strength in a (5,5) SWNT. In chapter 4 we validate our numerical implementation by reproducing the scaling of le and compare with analytical calculations. S. Latil et al. [37] calculated transport properties of chemically doped CNTs. The electronic structure of a graphene layer with one boron or nitrogen substitution was calculated using DFT, and renormalized tight-binding parameters were obtained through ﬁtting of the band structures. Figure 1.5 (left) shows the super cell of the graphene layer, with a boron atom located in the corner. As an eﬀect of the doping atom, the on-site energies were changed up to third nearest neighbours, while the hopping energies were unchanged. The changed tight-binding parameters calculated for graphene were assumed to be valid for a CNT as well. Figure 1.5 (right) shows the energy dependent conductance for

Figure 1.5: Left: A DFT calculation on a graphene layer with a single boron substitution (in the corner) is used to obtain renormalized on-site energies up to third nearest neighbours. Right: Energy dependent conductance for a (10,10) CNT with 0.1% boron atoms calculated for diﬀerent lengths.

different tube lengths. The boron concentration is 0.1%, and the boron atoms are randomly distributed on the tube. For short tubes the conductance at positive energies is ballistic with well defined conductance plateaus, whereas for longer tubes the conductance is strongly affected by the dopants. Notice that the largest tube is more than 1 µm long, thus containing more than 105 atoms. In a related study S. Latil et al. [40] recently calculated transport properties of CNTs with random coverage of physisorbed molecules. Tight-binding parameters were again determined from band structure fitting, and the analysis showed that the impact of physisorption on the transport strongly depends on the HOMO-LUMO gap of the attached molecules. 10 1.3 Motivation and outline

1.3 Motivation and outline

The motivation for the present work is that computer modelling is becoming increasingly important in understanding physical properties of nanoscale systems. As discussed above, there are several issues concerning nanowires where a combination of theoretical and experimental work could benefit our knowledge. A calculation of the conductance of, say a SiNW with randomly positioned doping atoms or defects, puts strong requirements on the method. The quasi one-dimensional nature of the nanowires requires on the one hand an atomistic model taking quantum effects into account and, on the other hand the method should be able to treat 104 atoms and include many scattering events due to the µm length of the wires.∼ In this work, two (N) methods based on DFT calculations are used to study the effect of manyO random defects in long nanowires. Since the real-space Kubo method has been successfully applied to CNT systems and since it scales as (N), it is appealing to understand the method in detail, implement it and use itO to study SiNWs. Because only one group has used it in the literature and applied it primary on carbon nanotubes, there are many unanswered questions that will be addressed in the following chapters:

How does one derive the formulas used in the articles, and are the deriva- • tions rigorous?

How should the method be implemented? •

Does the method yield the same results as other methods? •

Is it applicable to other, more diﬃcult systems than carbon nanotubes? •

The second method we apply is more well-known. It is based on the Landauer formula, and the transmission through a sample is found from recursive calculations of Green’s functions (GFs). The aim of the work is not to make very accurate calculations reproducing experimental data exactly. To that the model is too simple: We use a single electron model, neglecting electron-electron interactions as well as electron-phonon interactions. Also, any leads connected to the samples will be assumed to be perfectly conducting with reﬂectionless contacts to two reservoirs [43]. Any real contact resistance will thus not be included. The objective is therefore rather to make qualitative estimates and answer questions like: What kind of defects are important? Is surface- or bulk disorder the most important? Does some dopants scatter more than other? Is phonon or defect scattering the most important? 1.3 Motivation and outline 11

1.3.1 Outline

The outline of the report is the following: In chapter 2 we review the details in the real-space Kubo formula, and show that the transport properties are governed by the time dependent diffusion coefficient. We end the chapter by discussing three regimes of transport: ballistic, diffusive and localization. A key element in the numerical procedure is the time evolution of wave packets. In chapter 3 we show how this efficiently can be calculated using Chebyshev polynomials. The chapter also includes an introduction to the continued fraction technique which is used to calculate densities of states, together with a brief description of the numerical implementation. In chapter 4 we test the numerical implementation of the Kubo method on a simple one-dimensional chain. We focus on the parameters governing the accuracy and compare numerical result with analytical expressions. We further validate our implementation by reproducing previously published results. We end the chapter with a discussion of an inherent problem in the method leading to incorrect conductances near band edges. The recursive Green’s function method is explained in chapter 5. Starting with a semi-infinite lead, a device area is ’grown’ by adding unit cells one at a time. At each growth step the transmission through the wire is calculated thus giving a length dependent resistance. As discussed above, a reliable model of e.g. a SiNW should be based on first principles calculations. In chapter 6 we show how to construct a tight-binding Hamiltonian from a DFT calculation. In order to perform the time evolution in the Kubo method, it is preferable to have an orthogonal basis set. However, the DFT program siesta uses a non-orthogonal atomic basis set, and it is therefore necessary to orthogonalize the output from siesta by a so-called Löwdin transformation. In chapter 7 we apply both the real-space Kubo method and the recursive GF method to study silicon nanowires. First, an un-passivated, metallic wire with random on-site disorder is considered. Is is shown that disorder in the bulk has little influence on the conductance at the Fermi level, whereas disorder on the surface atoms affects the transport significantly. Comparison of the two methods show that they yield approximately the same results. The GF method is, however, found to be faster than the Kubo method, and to some extent also more reliable, and the rest of the results are obtained using the GF approach. All wires reported in the literature are passivated by either SiOx or hydrogen. We therefore model an H passivated SiNW with randomly missing H atoms. Resistance scalings and mean free paths are calculated together with localization lengths. In chapter 8 we summarize the comparison of the two numerical methods. The real-space Kubo method suffers from inherent problems near band edges, which 12 1.3 Motivation and outline has significant influences for the studied SiNWs. Another drawback is that different accuracy parameters need to be determined manually. The GF method is more rigorous and does not contain any adjustable parameters. Also, it is possible to use a non-orthogonal basis and thus avoid the Löwdin transformation. On the other hand, the Kubo method is able to find the conductance at all energies in one calculation, whereas a full calculation is needed for each energy when using the recursive GF method. We finish the report with an outlook. Chapter 2

The real-space Kubo formalism

In this chapter the real-space Kubo formulas for conductivity and conductance are derived. This is not a standard textbook procedure and has only partly been presented in the literature [33]. We will therefore present the derivations in detail. In section 2.2 we will discuss three diﬀerent regimes of transport: Ballistic, diﬀusive and localization, all of which are captured by the method.

2.1 Derivation of the real-space Kubo formula

We start by introducing the notation and continue to rewrite the Kubo-Greenwood formula. The following derivations are primarily based on the (French) Ph.D. thesis of F. Triozon [44].

2.1.1 Preliminary notations

The mean value of an arbitrary operator Aˆ over states having energy E is written as:

1 N Tr[δ(E Hˆ )Aˆ] Aˆ = ψ(i) Aˆ ψ(i) = − , (2.1) h iE N h E | | E i ˆ i Tr[δ(E H)] X − 14 2.1 Derivation of the real-space Kubo formula where ψ(i) are N degenerate eigenstates of Hˆ , all having energy E. The last | E i equality sign can be veriﬁed by using the eigenstates, ψ of Hˆ in the trace: | En i Tr[δ(E Hˆ )Aˆ] ψ δ(E Hˆ ) Aˆ ψ − = nh En | − | En i Tr[δ(E Hˆ )] ψ δ(E Hˆ ) ψ − P nh En | − | En i ψ δ(E E )Aˆ ψ = Pnh En | − n | En i ψ δ(E E ) ψ P nh En | − n | En i 1 = P ψ(i) Aˆ ψ(i) N h E | | E i i X The mean value of the spreading in the x-direction of states having energy E is:

2 2(E,t) = Xˆ(t) Xˆ(0) , (2.2) X − E i Htˆ i Htˆ where Xˆ(t) = e ~ Xˆ e− ~ is the x-component of the position operator in the Heisenberg picture. The velocity autocorrelation function of states with energy E is in the same way deﬁned as:

C(E,t) = Vˆx(t), Vˆx(0) = Vˆx(t) Vˆx(0) + Vˆx(0) Vˆx(t) . (2.3) { } E E D E D E Diﬀerentiation of (2.2) leads to

d 2 (E,t) = Vˆx(t) Xˆ(t) Xˆ(0) + Xˆ(t) Xˆ(0) Vˆx(t) , (2.4) dtX − − E D E where the velocity operator is deﬁned as d i Vˆ (t) = Xˆ(t) = H,ˆ Xˆ(t) . (2.5) x dt ~ h i Changing the time arguments in the trace (2.4) allows us to write

d 2 (E,t) = Vˆx(0) Xˆ(0) Xˆ( t) + Xˆ(0) Xˆ( t) Vˆx(0) , dtX − − − − E D E and the second derivative of 2(E,t) can thus be written as: X 2 d 2 (E,t) = Vˆx(0) Vˆx( t) + Vˆx( t)Vˆx(0) . dt2 X − − E D E Changing the time arguments again and using (2.3) we ﬁnally get the relation between the velocity autocorrelation function and the average spread:

d2 2(E,t) = C(E,t). (2.6) dt2 X 2.1 Derivation of the real-space Kubo formula 15

2.1.2 Rewriting the Kubo-Greenwood formula

The Kubo-Greenwood formula for the DC-conductivity, σDC , at zero temperature can be written as [33]

2 ~ e2 π σ (E) = Tr Vˆ δ(E Hˆ ) Vˆ δ(E Hˆ ) , (2.7) DC Ω x − x − h i where Ω is the volume of the system. The last delta-function is rewritten as an integral

1 ∞ i(E Hˆ )t/~ δ(E Hˆ ) = dt e − − 2π~ Z−∞ and inserted into (2.7):

2 1 ∞ i(E Hˆ )t/~ σ (E) = 2 ~ e π dt Tr Vˆ δ(E Hˆ ) Vˆ e − DC 2π~ x − x Z−∞ h i 2 ∞ iEt/~ iHt/ˆ ~ = e dt Tr Vˆ δ(E Hˆ ) e Vˆ e− . x − x Z−∞ h i We have incorporated Ω in the states in the trace, thus normalizing them to the volume. Due to the delta-function we can write eiEt/~ = ei Hˆ t/~ and thereby get ∞ σ (E) = e2 dt Tr Vˆ (0) δ(E Hˆ ) Vˆ (t) , (2.8) DC x − x Z−∞ h i where Vˆx(t) is the x-component of the velocity operator, written in the Heisen- berg notation. Using the notation from (2.1) of an energy average we get

Tr Vˆx(0) δ(E Hˆ ) Vˆx(t) = Tr δ(E Hˆ ) Vˆx(t) Vˆx(0) , − − E h i h i D E which is inserted into (2.8):

2 ∞ σDC (E) = e dt Tr [δ(E Hˆ )] Vˆx(t) Vˆx(0) . (2.9) − E Z−∞ D E Using (2.3) and (2.6) we can rewrite the integral and evaluate it:

2 ∞ σDC (E) = e dt Tr [ δ(E Hˆ ) ] C(E,t) 0 − Z t= d ∞ = e2 Tr [ δ(E Hˆ ) ] 2(E,t) − dtX t=0 d = e2 Tr[ δ(E Hˆ ) ] lim 2(E,t), (2.10) t − →∞ dtX 16 2.1 Derivation of the real-space Kubo formula

d 2 2 since dt (E,t) t=0 = 0. Using the deﬁnition of (E,t) in equation (2.2) and the generalX equation (2.1) of the energy mean, weX ﬁnally get:

2 2 d σDC (E) = e Tr[ δ(E Hˆ ) ] lim Xˆ(t) Xˆ(0) t − dt − E →∞ ˆ ˆ ˆ 2 d Tr δ(E H) (X(t) X(0)) = e2 Tr[ δ(E Hˆ ) ] lim − − − t dt  h Tr[ δ(E Hˆ )] i →∞ − d   = e2 Tr[ δ(E Hˆ ) ] lim (t (E,t)) , (2.11) t − →∞ dt D where we have defined the time and energy dependent diffusion coefficient, (E,t), by D 2 Tr Xˆ(t) Xˆ(0) δ(E Hˆ ) 2(E,t) 1 − − (E,t) = X = . (2.12) D t t Tr δ(E Hˆ ) − h i In the diffusive regime, (E,t) = (E) and (2.11) reduces to D D0 2 σDC (E) = e n(E) lim (E,t). (2.13) t →∞ D where the total density of states Tr[ δ(E Hˆ ) ] = n(E). − The conductance of a wire of length L is found from (2.13) as [35] e2 G(E, L) = n(E) (E,τ) (diffusive), (2.14) L D where the time τ is defined by the condition L = 2(E,τ), (2.15) X i.e. the time for the electrons at energyp E to spread out by an amount equal to L. This relation between τ and L seems rather arbitrary and could just as well have been L = a 2(E,τ) with a being of the order of 1. X Strictly speaking,p the formula (2.14) is only correct in the diffusive regime. It yields, however the correct physics in the ballistic regime also, since there 2 d (E,t) = v(E) t and dt (t (E,t)) = 2 (E,t), resulting in a factor 2 differ- enceD compared to (2.14): D D e2 G(E, L) = 2 n(E) (E,τ) (ballistic). (2.16) L D

In spite of the difference between the diffusive and ballistic formula, equation (2.14) is used in ref. [35, 36, 37, 33, 32] and applied to both regimes, not commenting on the factor 2 difference. In the section 2.2 we will look into three regimes of transport using (2.14). The discussion is mostly qualitative and we shall not care about the factor 2 difference. 2.2 Three regimes of transport 17

2.1.3 Time evolution

In order to calculate (E,t) we need to move the time-dependence from the operator Xˆ(t) to the statesD in the trace. Consider therefore the numerator in (2.12) which we name nx(E,t) and rewrite as

2 n (E,t) = Tr Xˆ(t) Xˆ(0) δ(E Hˆ ) x − − iHtˆ iHtˆ iHtˆ iHtˆ = Tr e Xeˆ − Xˆ δ(E Hˆ ) e Xeˆ − Xˆ , − − − h i where we have used that the trace is permutation invariant. Inserting Iˆ = iHtˆ iHtˆ iHtˆ e e− and using that [H,ˆ e ] = 0 leads to

iHtˆ iHtˆ iHtˆ iHtˆ n (E,t) = Tr e Xˆ Xeˆ − δ(E Hˆ ) Xeˆ − e− Xˆ x − − − h i = Tr [X,ˆ Uˆ(t)]† δ(E Hˆ ) [X,ˆ Uˆ(t)] , (2.17) − h i iHtˆ where Uˆ(t) = e− is the time evolution operator (see chapter 3). The eﬀect of operating with the commutator, [X,ˆ Uˆ(t)], on one of the states, ψi , in the trace is: | i [X,ˆ Uˆ(t)] ψ = Xˆ ψ (t) Uˆ(t) Xˆ ψ . (2.18) | ii | i i − | ii ˆ (x) ˆ X operates on the time propagated state ψi(t) , and the state ψi = X ψi is (x) | i | i (x| ) i propagated to time t, ψi (t) . Denoting the state Ψi(t) = ψi(t) ψi (t) , the numerator in (2.12)| is compactlyi written as a sum| of locali | densitiesi − | of statei terms: n (E,t) = Ψ (t) δ(E Hˆ ) Ψ (t) . (2.19) x h i | − | i i i X In chapter 3 we show how to calculate Uˆ(t) using Chebyshev polynomials and explain how the local densities of states are calculated with a continued fraction method.

2.2 Three regimes of transport

It is evident from the above mentioned that the transport properties to a large extent are governed by the time and energy dependent diffusion coefficient (E,t). Figure 2.1 illustrates a typical time evolution of (E,t). The bal- Dlistic regime is characterized by an initial linear increase, withD a slope given by the velocity. This is followed by the diffusive regime with constant diffusion coefficient (E,t) = 0(E). Finally the electrons start to localize resulting in a decreasingD diffusionD coefficient. In the following we shall briefly discuss some of the properties characterizing the three regimes. 18 2.2 Three regimes of transport

D(t)

Diffusive D0

Localization Ballistic

Figure 2.1: Illustration of the time development of the diffusion coefficient (E,t). From an initial ballistic regime the transport becomes diffusive before enteringD the localization regime.

2.2.1 Ballistic regime

For ballistic propagation 2(E,t) = v(E)2 t2 the diﬀusion coeﬃcient grows linearly with time (E,t)X = v(E)2 t. Inserting this into (4.14) and writing L = v(E) τ we get: D

G(E) = e2 n(E) v(E). (2.20)

The energy dependence of the one-dimensional density of states and the velocity cancels, thus resulting in a length independent conductance, proportional to the number of conducting channels, as expected from the Landauer formula [43]. The elastic mean free path can be estimated from the diﬀusion coeﬃcient as

max (E,t), t > 0 (E) l (E) = {D } = D0 , (2.21) e v(E) v(E) where the velocity v(E) is determined from the initial slope, α(E), of the diffusion coefficient, as v(E) = α(E) - see figure 2.1. p 2.2.2 Diffusive regime

In the diﬀusive regime, where (E,t) is constant the resistance increases linearly with length: D 1 L R(E, L) = = , (2.22) G(E) e2 n(E) (E) D0 2.2 Three regimes of transport 19

2 1 in agreement with Ohm’s law. Using (2.20) to write R0 = (e n(E) v(E))− (2.22) can be rewritten as

R(E, L) = R0 L/le(E). (2.23)

The Ohmic slope is therefore determined by the mean free path, a relation we shall use in chapter 7, when analyzing numerical data. The conductivity (2.13) becomes

σ (E) = e2 n(E) (E), (2.24) DC D0 which is the Einstein relation for conductivity [43].

2.2.3 Localization

Localization in disordered systems has been studied during the last 50 years since Andersons work in 1958 [45]. The ﬁeld is extensively reviewed by Lee and Ramakrishnan [46] and by Kramer and MacKinnon [47], while Al’tshuler and Lee [48] give a more introducing review. A consequence of disorder in one- dimensional systems is that electrons always localize with a resistance increasing exponentially with length [46]

R(L) = R (eL/ξ 1), (2.25) 0 − where ξ is called the localization length. The formula was originally derived by Anderson et al. [49] using a subtle argumentation: One seeks a quantity, f(L) that has an additive mean which implies that the averaged resistance should not depend on how you add pieces of the sample together, i.e. if a piece of length L0 results in a factor f(L0) another piece with length 2L0 should give a factor 2 f(L0). Anderson et al. showed that the quantity ln(1+r) (r is the resistance in units of h/2e2) fulﬁlled this requirement meaningh that oni averaging over the distribution of two scatters with resistances r1 and r2 one have

ln(1 + r) = ln(1 + r ) + ln(1 + r ) . h i h 1 i h 2 i 1 This leads to a scaling behaviour ln(1+r) = αL, where α = ξ− is interpreted as the inverse localization length,h and (2.25)i follows. Notice, that for L ξ ≪ (2.25) gives R(L ξ) R0L/ξ. Comparing with (2.23) we see that for the one- dimensional system≪ the≈ mean free path is of the same order as the localization length, ξ le, in accordance with the general relation ξ Nle, where N is the number of∼ conducting channels [46, 50]. ∼ Although the study of disordered one-dimensional systems dates back many years it is still an active field of research. The emergence of new quasi one- dimensional structures such as CNTs and nanowires has opened up for a variety of experimental systems where localization can be studied. Very recently Gómez-Navarro et al. [51] were able to tune the localization length in SWNT by 20 2.2 Three regimes of transport ion irradiation. An exponentially increasing resistance was observed in excellent agreement with the theory (2.25). In the framework of the real-space Kubo method, localization is seen for long time propagations, where the spreading of the wave packet reaches a constant level 2(E,t) L for t . X → 0 → ∞ It follows from (2.12) that the diffusion coefficient will decrease for increasing times leading to a decreasing conductance.

2.2.4 Simple model for diﬀusion and localization

We now introduce a simple, classical model for diﬀusion and localization. The model closely follows Todorov [50] and the reason for using it, is that it ﬁts numerical calculations very well (see chapter 7.2).

Left Device Right jin jout jref

0 L x

Figure 2.2: A device is coupled to two perfect conducting leads (Left and Right). Within the device area the propagation is diﬀusive. The incoming, reﬂected and outgoing currents (jin, jref , jout) constitute the boundary conditions.

We consider the model in figure 2.2 with left and right leads being perfect conductors coupled to a device area. We assume N conducting channels each having transmission probability T , and wish to calculate T = jout , where j jin in and jout is the incoming and outgoing currents, respectively. First, we assume that the electrons diffuse through the device area. The number of conducting electrons per unit length is given by the steady-state diffusion equation:

n′′(x) = 0 n(x) = A + Bx. (2.26) ⇒ The constants A and B are determined by the boundary conditions. To the left, n(0)v = j + j = 2j j yielding A = jin (2 jout ), where v is the x in ref in out vx jin x − − 1 velocity in the x-direction. Using that j = Dn′(L), where D = v l is the out − 2 x e diﬀusion coeﬃcient and le is the mean free path, the right boundary condition 2.2 Three regimes of transport 21

2A n(L)vx = jout yields B = . The transmission can now be found as − le+2L j v n(L) l T = out = x = e (2.27) jin jin le + L

2 2e 1 and the resistance R = ( h NT )− becomes

R0 R = R0 + L, (2.28) le h where R0 = 2e2N . This is simply Ohms law, where the resistance increases linearly with the length L of the sample, and the Ohmic slope is determined by the contact resistance, R0, and the mean free path, le (cf. equation (2.23). This result can also be found by calculating the transmission probability of a series of scatterers, as shown in appendix C.2. In order to treat localization we extend the model by adding a random distribution of traps in the device area, which occasionally captures the electrons. The traps are modeled by an eﬀective sink, η2n(x), term added to the diﬀusion equation (2.26): −

2 ηx ηx n′′(x) η n(x) = 0 n(X) = Ae + Be− . (2.29) − ⇒ Again, the constants are found from the boundary conditions and the resistance becomes [50]: h R(L) = [c sinh(L/ξ) + cosh(L/ξ)] , (2.30) 2e2N

le 2 where c = ξ(1 + ( 2L ) )/le and ξ = 1/η is the localization length. For L ξ (2.30) reduces to Ohms law (2.28). Furthermore, for systems with one conduct-≪ L/ξ ing channel where le ξ, the resistance increases for L ξ as R(L) R0e , in accordance with (2.25).∼ With a few conducting channels≫ (2.25) gives≈ a faster deviation from the linear region than (2.30). Equation (2.30) fits numerical data remarkably well, as will be evident in chapter 7.2. Studies of the localization regime should, however, be done with great care. Since localization is caused by interference effects, phase randomization due to inelastic scattering will ’destroy’ the localization. Any inelastic scattering event such as electron-phonon or electron-electron scattering will result in a finite phase coherence length, lφ. If lφ < ξ localization will not be seen, which is the reason why everyday life metallic wires are conducting. Normally, we should therefore not expect to see localization effects at room temperature, and the localization length in the numerical data should probably mostly be an indicator of the length range in which we can apply the model, without inelastic effects. However, Lu et al. [17] recently reported ballistic transport in nanowires at room temperature, with estimated lφ = 540 nm. By introducing defects purposely as done in ref. [51] it might be possible to see localization effects in nanowires. In any case, the quantum mechanical nature of localization is interesting and we will return to it in chapter 7. Chapter 3

Numerical methods

In the previous chapter we saw that the transport properties of a system is determined by the time dependent diffusion coefficient. In order to calculate this we need to know the time evolution of the states involved in the traces (2.12). In section 3.1 and 3.2 we discuss time evolution and present in detail the efficient Chebyshev method, where the time-evolution operator is expanded in a set of orthogonal polynomials - the Chebyshev polynomials. In section 3.3 we show how local densities of states are calculated using a continued fraction technique. We finish the chapter by a short description of the numerical implementation focusing on important parameters and the data flow.

3.1 Exact solution to the time-dependent Schr¨odinger equation

The time evolution of a quantum mechanical state is governed by the time- dependent Schrödinger equation (TDSE), which, together with a boundary condition, constitutes a first order differential equation

∂ Hˆ ψ(t) = i~ ψ(t) , ψ(t = 0) = ψ , (3.1) | i ∂t| i | i | 0i with Hˆ being the Hamiltonian operator, which we shall assume to be time independent. The solution to (3.1) is found by simple integration:

i Htˆ ψ(t) = Uˆ(t) ψ = e− ~ ψ , (3.2) | i | 0i | 0i i Htˆ where we have introduced the time-evolution operator Uˆ(t) = e− ~ . Although the formal solution in (3.2) is very simple, it does not help us, actually ﬁnding the state ψ(t) at time t. In order to proceed, we project the Hamiltonian | i 3.2 The Chebyshev method 23 operator Hˆ onto a known basis set yielding a Hamiltonian matrix, H, and assume to know the eigenstates: H E = E E . (3.3) | ni n | ni As shown in appendix B.3 the time-evolution matrix can be written as

iE1t iE2t iEnt U(t) = V diag e− , e− ,...,e− ,... V†, (3.4) where the matrix V with the eigenvectors E as columns, diagonalizes H: | ni V† H V = diag(E1, E2,...,En,...). However, in general we do not know the eigenstates of the system, and ﬁnding them involves the diagonalization of H, an operation which scales as (N 3). For a long Si-wire containing 100,000 atoms this is an impossible taskO. We therefore have to ﬁnd the time-evolution operator in an other way.

3.2 The Chebyshev method

In this section we describe in detail how the time-evolution operator Uˆ(t) can be expanded in the orthogonal set of Chebyshev polynomials. We ﬁrst discuss some basic properties of the Chebyshev polynomials and then describe how the expansion is implemented and discuss the convergence properties. Finally, we compare the Chebyshev method with a simple Taylor expansion.

3.2.1 Basic properties of Chebyshev polynomials

The Chebyshev polynomials, Tn(x) are deﬁned by the weight function w(x) = 1 through the orthogonality relation[52]: √1 x2 − 1 1 π π Tn(x) Tm(x) δnm , n, m > 0 Tn(x) w(x) Tm(x)dx = dx = 2 2 1 2 1 √1 x 2 , n = m = 0 Z− Z− − with the ﬁrst Chebyshev polynomial T0(t) = 1. The other polynomials can be determined from the orthogonality relation through a Gram-Schmidt procedure. The Chebyshev polynomials obey the simple and very useful recurrence relation:

Tn+1(x) = 2xTn(x) Tn 1(x) , n 1, (3.5) − − ≥ which we shall use extensively in the following.

3.2.2 Expansion of Uˆ(t) in Chebyshev polynomials

We now show how to expand the time-evolution operator, Uˆ(t), in the Cheby- shev polynomials. If the spectrum of the Hamiltonian does not lie within the 24 3.2 The Chebyshev method interval [ 1; 1], the Hamiltonian is mapped to the new, normalized operator [53]: −

2Hˆ (Emax + Emin)Iˆ Hˆ EminIˆ Hˆ ′ = − = 2 − I,ˆ (3.6) E E E E − max − min max − min with eigenvalues in the range [ 1; 1]. In the following we shall assume that the Hamiltonian is properly scaled.− Given an initial state, ψ = ψ(t = 0) and a final time, t, we have: | 0i | i iHtˆ ψ(t) = Uˆ(t) ψ = e− ψ | i | 0i | 0i ∞ ∞ = c (t) T (Hˆ ) ψ = c (t) Φ , (3.7) n n | 0i n | ni n=0 n=0 X X where we have defined the Chebyshev states Φ T (Hˆ ) ψ . The coefficients | ni ≡ n | 0i c(t) are calculated in section 3.2.4. Using the recurrence relation (3.5) the Chebyshev states can by calculated recursively as

Φ = T (Hˆ ) ψ = ψ | 0i 0 | 0i | 0i Φ = T (Hˆ ) ψ = Hˆ ψ | 1i 1 | 0i | 0i Φ = T (Hˆ ) ψ = 2HTˆ (Hˆ ) ψ T (Hˆ ) ψ | 2i 2 | 0i 1 | 0i − 0 | 0i = 2Hˆ Φ Φ | 1i − | 0i . .

Φn+1 = Tn+1(Hˆ ) ψ0 = 2Hˆ Φn Φn 1 . | i | i | i − | − i The (n + 1)’th Chebyshev state, is simply found from the two previous and one operation with the Hamiltonian. This utilization of the recurrence relation makes it computationally very eﬃcient to calculate the terms in the sum (3.7).

3.2.3 Convergence properties

In order to investigate the convergence properties of the expansion, we ﬁrst recall the operator norm:

Let Oˆ : H H , Oˆ B(H, H) be a bounded operator from H to H. We deﬁne→ the operator∈ norm by:

Oˆ B = sup Oxˆ H , x H 1 k k {k k k k ≤ }

Since the Chebyshev polynomials constitute an orthogonal set, we have

N ∞ 1 = Uˆ(H,tˆ ) = c T c T . k k k n nk ≥ k n nk n=0 n=0 X X 3.2 The Chebyshev method 25

This means that the expansion is converged, when it holds that N cn(t) Φn 1 < ǫ, (3.8) k | ik − n=0 X for some predefined tolerance ǫ. For any ǫ we can always find a N such that the inequality (3.8) is satisfied. This has the great advantage, that one does not need to have any knowledge about the system (besides an estimate of the spectrum of the Hamiltonian) before doing the expansion. One simply keeps adding terms to the sum (3.7) until the approximated operator becomes unitary, i.e. (3.8) is satisfied.

3.2.4 Convergence of the coeﬃcients

We have just seen that the expansion (3.7) converges, but we have not yet said anything about how fast this convergence is. This is determined by the behavior of the coefficients, cn(t). These appear as the scalar product (defined by the weight function w(t)) between the Chebyshev polynomials and the operator to expand: ′ 1 ix t 2 e− Tn(x′) cn(t) = dx′. 2 π 1 √1 x′ Z− − Applying the scaling procedure (3.6) and defining W = E E , β = max − min Emax + Emin leads to W ′ β 1 i( 2 x + 2 )t 2 e− Tn(x′) cn(t) = dx′ 2 π 1 √1 x′ − Z −W ′ 1 i 2 x t 2 i β t e− Tn(x′) = e− 2 dx′ 2 π 1 √1 x′ Z− − i β t n W = 2 e− 2 ( i) J t , (3.9) − n 2~ where Jn(x) is the Bessel function of the first kind and order n, and where we have reinserted ~. For large n the Bessel function behaves as 1 x n J (x) . n ≈ n! 2 n+1 n and by using Stirlings formula, n! √2πn 2 e− the asymptotic behavior of the coefficients become ≈ exp(1) W t n c (t) 0 for n . | n | ∼ 4 ~ n → → ∞ exp(1) W t Thus, when n > 4 ~ the coefficients go to zero exponentially. This decay is especially fast when t is large, because in that case n also has to be large, exp(1) W t and and when finally n > 4 ~ the coefficients drop to almost zero since (1 δ)n 0 for n 1, even for small δ. We therefore expect a very sudden decay− of→ the coefficients,≫ when a certain expansion order is reached, and the expansion (3.7) will converge fast. 26 3.3 Continued fraction technique

3.2.5 Comparison of Chebyshev and Taylor

In order to test the Chebyshev method, we consider a tight-binding description of a simple one-dimensional chain with only nearest neighbour (n.n.) interactions. The non-zero elements of the Hamiltonian matrix are:

Hii = ε0 , Hi,i+1 = Hi+1,i = γ. (3.10) The electron is initially located at one site and the time evolution is solved in three different ways: Exact solution by diagonalizing the Hamiltonian and using (3.4), by Taylor expansion of Uˆ(t) for different expansion orders and different time steps, ∆t (see appendix B.3), and finally ψ(t) is found with the Chebyshev method. As a measure of error we use the maximum| i (or sup) norm: exact Error = max ψi (t) ψi(t) , i {| − |} where ψ (t) is the i’th element in the vector ψ(t) . i | i Figure 3.1 shows the error of both the Taylor and Chebyshev method as function of operations with the Hamiltonian. The hopping energy γ = 3 eV while the on- site energies were set to zero, ε0 = 0, and the total propagation time T = 100 fs. The size of the chain were N = 4000 sites. It is evident that the Chebyshev

0 10

−5 10

Error 3rd order Taylor −10 10 4th order Taylor 5th order Taylor Chebyshev

−15 10 2 3 4 5 6 10 10 10 10 10 Operations with H

Figure 3.1: Comparison between Taylor and Chebyshev expansions method is much more eﬃcient giving a very high accuracy at more than two orders of magnitude fewer Hamiltonian operations. Also, the very abrupt change in error for the Chebyshev method shows the point when the coeﬃcients in the expansion suddenly decay, as discussed in the previous section.

3.3 Continued fraction technique

The local density of states (LDOS) terms like φ δ(E H) φ , where φ is an arbitrary vector, are evaluated using the continuedh | fraction− | i technique.| Thei 3.3 Continued fraction technique 27 numerical details are described in appendix B.1 and B.2. Here we summarize the basic steps in the computation:

1. The Hamiltonian H is mapped to a smaller tridiagonal matrix called Htri:

M ( φ , N ) : H H . (3.11) | i tri → tri The mapping depends on the vector φ and on the number N which | i tri determines the size of Htri. The energy resolution is partly determined by Ntri but also depends on the size of H. The technical details in how the mapping is performed are described in appendix B.2.

2. The local density of the state φ , can be calculated as (see appendix C.1.1): | i

1 dφ(E) = φ δ(E H) φ lim Im G11(E iη) , (3.12) h | − | i ≈ η 0 −π − → 1 where the Greens function G(E) = (E Htri)− . The small imaginary energy η acts as a self-energy term and is− needed to make a smooth LDOS. For η 0 the LDOS for any ﬁnite system will be a series of delta peaks → 1 separated by an energy ∆E N − . In all calculations we therefore use a 1 ∝ η N − . ∝ tri 3. The ﬁrst diagonal element of the Greens function is calculated by the continued fraction:

1 G11(E) = , (3.13) β2 E α 1 − 1 − β2 E α 2 − 2− . .. 2 E αN β Σ(E) − tri − Ntri

where αi and βi are the diagonal and oﬀ-diagonal elements of Htri. Usu- ally, the coeﬃcients converge and the remaining terms of the continued fraction can therefore be analytically summed up to give a self-energy, Σ(E).

The most time-consuming step in the calculation is the construction of Htri, since it involves Ntri operations with H. This means that the time it takes to calculate dφ(E) at many energies does not diﬀer much from the time it takes to calculate it for a single energy. This is a major advantage of the Kubo method, since e.g. the conductance can be found for the whole energy spectrum in one calculation. 28 3.4 Overview of the computer codes

3.4 Overview of the computer codes

All computer code used in calculations with the Kubo method is written by myself. Much initial testing and prototyping was carried out in matlab but all larger calculations was performed using the program language Python. This is a relatively easy language, since many standard libraries with build-in functions are readily imported. Also, one should not care about allocating memory for the variables. Moreover, with the extension MPI Python, it is relatively simple to make the programs run in parallel, a feature we have used extensively. In this section we brieﬂy run through the program focusing on the data ﬂow and numerically important parameters.

3.4.1 Important parameters

The important parameters determining the speed and accuracy of the calculations are the following:

Size of the Hamiltonian, Ntot: The total system size determines the maximum propagation time, since no reﬂection from the end of the wire is allowed. Also, it partly determines the energy resolution (together with Ntri). A large Ntot results in a long computation time, since the matrix operations with the Hamil- tonian depend on the total system size.

Size of the tridiagonal matrix, Ntri: Partly determines the energy resolution. The computation time depends strongly on Ntri, since the construction of Htri requires Ntri operations with the full Hamiltonian, and the subsequent evaluation of the continued fractions also involves Ntri steps. For systems with closely lying bands such as the SiNWs considered in chapter 7, a large value of Ntri & 3000 is needed. Width of the delta peaks in the density of states, η: The imaginary energy broadens the delta peaks in the density of states making the LDOS smooth, which is intended in order to simulate an infinite system. But it also broadens the van-Hove peaks, and should be as small as possible, still making the LDOS smooth. As mentioned above, we use η Ntri, which generally works satisfying. The computation time does not depend∝η. Random phase state: We use a limited number of random phase states to estimate the traces in (2.12), and they are discussed further in the next chapter. Briefly, they are normalized vectors of length Ntot, with Ni random complex numbers (of the same absolute value) in the middle of the vector. The ends of the vector consist of zeros. Generally Ni should be chosen large to ensure fast convergence of the LDOS, but a large Ni requires also a large Ntot in order for the electron not to ’hit’ the edge as time evolves. The number of random phase states is another important parameter. It is prior to a calculation difficult to say, how many random phase states are needed to give a good estimate of the 3.4 Overview of the computer codes 29 trace, and we have checked for convergence manually. Note, that the estimation of traces is ideal for parallel computation, since each node can calculate parts of the trace individually.

Time vector: A vector tvec = [t1, t2,..., tn] determines the times at which to evaluate the spread of the initial state. The time evolution is performed in steps such that φ(t ) = Uˆ(t t ) φ(t ) . | i+1 i i+1 − i | i i File ID: In order to subsequently reload the results for analysis, each calculation is characterized by a file ID. At the end on the calculations, each node in the parallel program saves the results (with a filename specific for each node) using the python package ’pickle’.

3.4.2 Data ﬂow

The data flow in the program in summarized in figure 3.2. The program uses MPI Python which enables parallel computation on N nodes (node0, node1 ,..., nodeN 1). The input parameters described above, are specified in a main file. − The Hamiltonian is constructed by node0 and broadcasted to the other nodes. We use the package ’spmatrix’ to represent the Hamiltonian in a sparse format speeding up the calculations dramatically and saving a lot of memory. Each of the N nodes constructs a random phase state φ and calculates the | ni local density of state φn δ(E Hˆ ) φn . In a loop over the times t1,..., tN the time evolution of theh | initial− state| togetheri with the local density of states (x) 2 dn (E,ti) = φn(ti) (Xˆ(t) Xˆ(0)) δ(E Hˆ ) φn(ti) is calculated for each time. At the end ofh the calculation,| − each node− saves| thei results in a file specific for that node. If specified, each node can construct several random phase states and repeat the calculation. The results are subsequently reloaded and the total densities of states are estimated as the sum of all the local densities, calculated by each node:

n(E) = φ δ(E Hˆ ) φ h n| − | ni n X (x) and in the same way, nx(E,ti) is found as the sum of diﬀerent dn (E,ti):

n (E,t ) = φ (t ) (Xˆ(t) Xˆ(0))2 δ(E Hˆ ) φ (t ) . x i h n i | − − | n i i n X The time dependent diﬀusion coeﬃcient is then

n (E,t ) (E,t ) = x i , D i n(E, ) and the results can be extracted from this. 30 3.4 Overview of the computer codes

node_0

Construct Hamilton, H Make ID−files Distribute H and ID−file names to the other nodes Receive H and ID−file names node_1 node_2 node_3

φ Make initial state i φ δ φ d i = ι (E−H) i

Propagate φι(t) (x) φ 2 δ φ d i = i [X(t)−X(0)] (E−H) i Save data (using ID−file names) ...... Load saved data and analyze results

Σ φι δ(E−H)φ Calculate traces n(E) = ι i Σ φ 2 δ φ nx (E) = ι i [X(t)−X(0)] (E−H) i

n (E) x Calculate diffusion coefficient, D(E,t) D(E,t) = n(E) D

Plot results

Figure 3.2: Principal data ﬂow in a parallel computation. Chapter 4

The one dimensional chain: Testing of the method

In this chapter we evaluate the numerical implementation, primary using the inﬁnite one-dimensional chain as a test case. There are several reasons for studying this system: First of all it is the simplest one-dimensional system and therefore a natural starting point for a theoretical as well as a numerical analysis. Due to the simplicity, analytical expressions can be found, and thus serve to validate the numerical methods. Secondly, many properties of the one- dimensional chain are also found in more complicated systems. The chapter is organized as follows: In section 4.1 we calculate the local density of state (LDOS) using the continued fraction technique and compare with analytical results. This is followed by section 4.2, where we discuss how to estimate traces using random phase states and show how to calculate various transport properties of the one-dimensional chain with Anderson disorder. To further validate the numerical method, we calculate the mean free path in a disordered carbon nanotube and compare with published results. In section 4.4 we use a model system consisting of two parallel chains to explain some apparent problems with the real-space Kubo-formula. The problems arise for degenerate energies near the band edges.

4.1 Local Density of States

In this section we derive an analytical result for the local density of state in an inﬁnite chain. We use this to test the numerical result, which we obtain using the continued fraction technique (section 3.3), focusing on the parameter Ntri, determining the accuracy. 32 4.1 Local Density of States

4.1.1 The inﬁnite chain

We consider the inﬁnite one-dimensional chain shown in ﬁgure 4.1. If we assume only nearest neighbour interaction, the non-zero elements of the Hamiltonian matrix are: Hii = ε0 , Hi,i+1 = Hi+1,i = γ, (4.1) where N /2

Figure 4.1: Model system. We consider only nearest neighbour interaction with the hopping matrix element γ and the on-site energy ε0. The lattice constant is a. assumed to be inﬁnitely long it is translational invariant, and Bloch’s theorem thus applies. Using this, the eigenenergies are readily found [54]:

E(k) = ε 2 γ cos(ka) (4.2) 0 − | |

4.1.2 Comparison of numerical and analytical LDOS

From the bandstructure (4.2) we obtain the total density of states per unit length as [54]

1 1 1 ∂E(k ) − 1 n(k) = ′ = L 2π ∂k ′ 4π γ a sin(k a) ′ k =k | | 1 1 n(E) = θ(E Emin) θ(Emax E), (4.3) L 2 − − E ε0 4π γ a 1 − | | − 2γ r where we have defined Emin = ε0 2 γ and Emax = ε0 + 2 γ . Since the total density of states can be expressed− as| a| sum of the local densities| | of states (see appendix C.1), and since all the local densities of states for the infinite chain must contribute with the same weight, it follows that the local density of states for the infinite wire is given by: 1 1 di(E) = n(E) = , E [Emin; Emax] (4.4) N 2 ∈ tot E ε0 4π γ 1 − | | − 2γ r 4.1 Local Density of States 33

where we have used that the length L = Ntot a. Figure 4.2 shows the local density of states calculated with Ntri = 500 (solid blue) and Ntri = 3000 (dash- dotted black) compared with the analytical result (dashed red). Both numerical calculations are performed on a chain containing Ntot = 5000 sites in total. The left graph shows the whole spectrum demonstrating a general good agreement between the analytical and numerical results. The right graph, however, reveals deviations around the van-Hove singularity at E = 6 eV. It is also evident that the curve corresponding to Ntri = 3000 resembles the analytical result best, as expected. To quantify the diﬀerence between the analytical result and the

0.5 1 N =500 N =500 tri tri N =3000 N =3000 0.4 tri 0.8 tri Analytical Analytical

0.3 0.6 LDOS LDOS 0.2 0.4

0.1 0.2

0 0 −6 −4 −2 0 2 4 6 5.6 5.8 6 6.2 6.4 Energy [eV] Energy [eV]

Figure 4.2: Left:LDOS for the inﬁnite chain calculated analyticaly (dashed, red) and with the continued fraction technique (Ntri = 500 and Ntri = 3000). Right: Zoom-in on the van-Hove sigularity at the energy band-edge. The numerical results are improved for larger Ntri.

different numerical values corresponding to different values of Ntri we define an error function as

∞ error(N ) = d (E) d˜(E, N ) dE, (4.5) tri i − tri Z−∞

where di(E) is the analytical expression (4.4) and d˜(E, Ntri) is the numerical result. Figure 4.3 shows the error as function of Ntri for three different sizes of the total chain. The error clearly decays for increasing Ntri, but approaches a certain minimal error. The reason why the error does not go to zero, is that for very large Ntri the peaks in the density of state begin to show, revealing the finiteness of the system. In order to minimize the error further, the total system size, Ntot, should be enlarged, as the figure also illustrates. 34 4.1 Local Density of States

−1 10 LDOS

E −2 10 LDOS Error

E N=4000 N=7000 N=10000 −3 10 2 3 4 10 10 10 N tri

Figure 4.3: The integrated error of the LDOS curves versus Ntri for three different sizes of the total chain (Ntot = 4000, 7000, 10000). The error decays as 0.5 Ntri− . For small Ntri the error is caused at the band edge as illustrated in the∼ left inset. The van-Hove peak is not very sharp and there is a signiﬁcant tail, both due to a relatively large imaginary energy, iη, (see equation (3.12)). When Ntri Ntot the error reaches a plateau, the reason being an oscillating behaviour of≈ the numerical LDOS (right inset), revealing the ﬁniteness of the system.

4.1.3 Comparing the velocities

The analytical group velocity is found from the band structure as

1 ∂E (k) 2 γ a v(k) = ± = | | sin(k a) ~ ∂k ~ 2 γ a E ε 2 v(E) = | | 1 − 0 θ(E E ) θ(E E). (4.6) ~ − 2γ − min max − s The numerical velocity is found from the time-dependent diffusion coefficient, (E,t), through the relation v(E) = α(E), where α is the slope of the diffu- Dsion curve. As discussed in section 2.1, the initial behaviour of (E,t) is always p linear corresponding to ballistic propagation until the scatteringD mean free time. For a perfect system (as the unperturbed linear chain), the transport is ballistic for all times and (E,t) will just be a straight line. Figure 4.4 (left) shows (E,t) for differentD energies clearly showing an energy dependent velocity. On Dthe right graph the numerically calculated velocities are compared with the an- 4.2 Disordered chain 35 alytical result. In general the results agree, however, the numerically calculated velocities oscillate around the analytical. This oscillation is again due to the finiteness of the system which also resulted in the oscillating density of states.

2500 10 E =0 eV F E =3 eV 2000 F 8

/t E =5 eV

〉 F (E,t)

2 1500 6 X 〈

1000 4 Velocity [a/fs] Diffusion = 500 2 Numerical Analytical 0 0 0 5 10 15 20 25 30 −6 −4 −2 0 2 4 6 time [fs] Energy [eV]

Figure 4.4: Left: Time dependent diffusion coefficient at three different energies. The velocity is found from the slope, α(E) of the diffusion curve as: v(E) = α(E). Right: Numerical (solid red) and analytical (dashed blue) velocities for the whole band. The oscillating behaviour is due to the finiteness of the system. p

4.2 Disordered chain

We continue our discussion of the numerical procedures by adding random on- site noise to the one-dimensional chain, such that the on-site energies become

Hii = ε0 + δi, (4.7)

∆ε ∆ε where δi is a random number in the interval 2 ; 2 . This, so-called An- derson disorder, was first introduced by P.W. Anderson− in 1958 [45] and has been applied to various systems in order to model defects. The physical reason for adding on-site disorder could be to model a system, say a carbon nanotube, with different kinds of doping atoms. Such a system is considered by S. Latil et al. [37], where the dopant atoms result in different on-site energies around the dopant, but where the hopping parameters are unchanged - see the Introduc- tion, sectionKubo-applications. On the average it might therefore be reasonable to model the influence of dopants by random disorder. The disorder strength, ∆ε, could be thought of as a measure of the dopant concentration. In the presence of disorder we expect a transition from ballistic to diffusive (i.e. Ohmic) propagation after a certain time, giving rise to a length dependent resistance. In the following we calculate the resistance using the real-space Kubo approach and we compare the results with the Landauer formula, described in chapter 5. But first we introduce the random phase states. 36 4.2 Disordered chain

4.2.1 Random initial states

As shown in section 2.1 the transport is determined by the average squared spreading 2(E,t) X

Tr [X,ˆ Uˆ(t)]† δ(E Hˆ ) [X,ˆ Uˆ(t)] 2(E,t) = − , (4.8) X n Tr δ(E Hˆ ) o − which involves the calculation of twon traces. Wheno disorder is present, the system is no longer translationally invariant and sites are in principle no longer equivalent. Estimating the trace using the ’site-basis’ is therefore ineﬃcient. On a larger scale, however, the chain can still be considered as homogeneous: One sub-chain of, say, S sites presumably has the same properties as another sub-chain of the same size. Following this line of arguments (and inspired by the work of Triozon et al.[36]) we instead approximate the traces by summing over a limited number (< 50) of random phase states, ψr . The random phase states are placed in the middle of the system and extend| overi a (possibly large) number of orbitals, Ni:

Ni/2 1 2 i π α(j) ψr = e j , (4.9) | i √Ni | i j= Ni/2 X− where α(j) is a random number in the interval [0; 1[ and j = [0,..., 1, 0,...]T | i is the j’th basis state with the ′1′ on the j’th element. In the numerical implementation, ψr is a vector of length Ntot (size of the Hamiltonian matrix), with non-zero elements| i in the middlemost part given by (4.9). Figure 4.5 illustrates the advantage of random states compared to single sites. The figure shows the error of the density of states vs. the number of random states, when the initial states extend over 1 (solid blue) and 100 (dashed red) sites. The case Ni = 1 corresponds to tracing over the site-basis. The error defined similarly to (4.5), and as reference we use a DOS obtained using 200 different random states with Ni = 500. The figure shows that the DOS obtained using the random states (Ni = 100) approximately is converged for 20 different random states, whereas 60 initial states are needed in the case where N = 1. This is a general trend ∼ i and one should therefore use random phase states with a large Ni. The total system size, Ntot must always be larger than Ni, since there must be space for the initial state to propagate in without reaching the boundary. Therefore, the total propagation time and Ntot set an upper limit for Ni. The inset in figure 4.5 shows the estimated DOS. The oscillations are caused by the disorder and are not a finite-size effect as discussed above.

4.2.2 Time dependent diffusion coefficient (E,t) D Knowing how to calculate the local densities of states and what parameters are important, we are ready to calculate the time and energy dependent diffusion 4.2 Disordered chain 37

N =1 i 0.2 N =100 i

DOS 0.1

0 −5 0 5 Energy Error [arb. unit]

0 20 40 60 80 100 # Initial states

Figure 4.5: Error of the estimated density of states vs. number of random phase states. Solid blue and dashed red correspond to Ni = 1 and Ni = 100, respectively. The case Ni = 1 is equivalent to using the site-basis in the trace. The inset shows the converged density of state.

coefficient, (E,t), defined in equation (2.12). We estimate the traces in (4.8) using randomD phase states and calculate the time evolution using the Chebyshev expansion, as explained in section 3.2. Figure 4.6 illustrates the time evolution of an initial state. The plots show the absolute squared wave function ψ(x,t) 2 as function of the position for different times. | |

t=0 t t 1 2 2 (t)| ψ |

Figure 4.6: Probability distribution along the chain for diﬀerent times.

Figure 4.7 (left) shows the mean value of the diffusion coefficient as calculated from 50 initial random states with different values of Ni. Red, blue and black 38 4.2 Disordered chain

correspond to Ni = 10, 100, 800 respectively. The error bars show the standard deviation among the different random states. The right figure shows the summed distribution function for Ni = 100 and t = 140 fs plotted together with the error-function with the statistical mean and variance as parameters. Since the two curves are very close to each other, it is concluded that the different random states yield a normal distribution of diffusion coefficients (since the error function is the integral of a normal distribution). We can use the statistical

3000 1 Numerical Data Error function 2500 0.8

2000 0.6 1500 P(x) dx ∞ ε − Diffusion

∫ 0.4 1000 N =10 i N =100 0.2 500 i N =800 i 0 0 0 50 100 150 −1 −0.5 0 0.5 1 (D − µ − ε)/µ Time [fs] i

Figure 4.7: Left: Average diffusion curves for different initial states for a disordered chain with ∆ε = 0.2 γ . Red, blue and black correspond to Ni = 10, 100, 800 respectively. In all three| | cases the average is obtained from 50 different states. The error bars mark the standard deviations. Right: The summed distribution function for the numerical data is plotted together with the integrated distribution function for a normal distribution (the error-function) with the statistical mean and variance as parameters. data to estimate how many random states are needed to estimate the mean value within an uncertainty of 90%. We use the Student’s t-distribution given as [55] x¯ µ t = − √n, (4.10) s σ where µ is the population mean (calculated from the 50 different random states), σ is the estimator for population standard deviation (also calculated from the 50 different random states).x ¯ is the mean value from a sample with n data points, i.e. from n different random phase states. Rearranging (4.10) we get an inequality expressing a relative difference between the true mean value, µ, and the estimated,x ¯, of 0.1: x¯ µ t σ − = s < 0.1. (4.11) µ µ √n This leads to a minimal number of data-points n:

t σ 2 n > s . (4.12) 0.1µ 4.2 Disordered chain 39

Looking up ts for different values of n in a table gives the estimate n & 10. This estimate will of course change for different physical systems, e.g. a different noise amplitude, and we have not performed the analysis for every system considered. However, it gives us a rule of thumb that only one or two random states are insufficient, whereas a number between 10 and 20 would probably suffice. This number of random states roughly corresponds to the number needed for the density of states to converge, as shown in figure 4.5.

4.2.3 Mean free path

Figure 4.8 (left) shows the diﬀusion curves for diﬀerent disorder strengths, ∆ε (in units of γ ). Note that all curves show an initial linear behaviour with the same initial| slope| determined by the velocity. The mean free path is found as [36]:

max (E,t), t > 0 l (E) = {D }. (4.13) e v(E)

2 The right plot in ﬁgure 4.8 shows le plotted against (∆ε)− showing a quadratic

250 Numerical data 2000 Linear fit

] 200 ∆ε γ

−1 = 0.2| |

fs 1500 ∆ε = 0.3|γ| 2 150 ∆ε = 0.4|γ| ∆ε = 0.5|γ| 1000 ∆ε = 1.0|γ| 100 Diffusion [a 500 Mean−free path [a] 50

0 0 0 5 10 15 20 25 30 0 200 400 600 ∆ε γ −2 Time [fs] ( / )

Figure 4.8: Left: Diffusion coefficient vs. time for different disorder strengths, 2 ∆ε (. Right: Mean free path as function of (∆ε)− . The quadratic dependence on the disorder strength is in qualitative agreement with Fermi’s golden rule. dependence. This is in qualitative agreement with a Fermi’s golden rule estimate, since the scattering rate between an initial state i and a final state f is proportional to Γ f V (∆ε) i 2, where the perturbation| i Hamiltonian| i fi ∼ |h | | i| V (∆ε) = diag(δ1, . . . , δi . . .) is the random on-site disorder, which is propor- 2 tional to ∆ε. The scattering time τe is thus inversely proportional to (∆ε) and so is the mean free path since le = vτe. Focusing on the strongest and weakest disordered systems, we consider the squared displacement, 2(E,t), as shown in figure 4.9. The solid blue line (left X 40 4.2 Disordered chain axis) and dashed black line (right axis) correspond to ∆ε = 0.2 γ and ∆ε = 1.0 γ respectively (note the different scales). The inset focuses| on| the initial behaviour| | of the ∆ε = 0.2 γ -case. The circles mark the numerical data points, while the solid red line is a| quadratic| fit, showing that the initial propagation is ballistic. For times 100 . t . 400 fs the curve is approximately linear and the propagation is therefore diffusive. For larger times the electron slowly starts to localize. For the strongly disordered system, the localization starts already at t 100 fs, as revealed by the approximately constant value of 2(E,t). ≈ X

6 x 10 2 4000

∆ε = 1.0|γ| ] 2 ∆ε γ

[a = 0.2| | 〉 1 2000 (E,t)

2 4 x 10

X 10 〈

0 0 20 40 60 0 0 0 200 400 600 800 Time [fs]

Figure 4.9: Squared displacement, 2(E,t) as function of time. The solid blue line (left axis) and dashed black lineX (right axis) correspond to ∆ε = 0.2 γ and ∆ε = 1.0 γ respectively. The inset focuses on the initial behaviour of the| | ∆ε = 0.2 γ -case.| | The circles mark the numerical data points, while the solid red line is| a| quadratic ﬁt.

4.2.4 Conductance of disordered wire

As discussed in section 2.1, the conductance of a chain of length L is calculated as [36]: e2 G(E, L) = α n(E) (E,τ), (4.14) L D where α = 2 in the ballistic regime and α = 1 in the diffusive regime. The time τ is defined by the condition L = 2(E,τ), i.e. the time for the electrons at energy E to spread out by an amountX equal to L. Figure 4.10 illustrates how p to find (E,τ). The solid black curve (left axis) is the average displacement, 2(E,tD ). The intersection between 2(E,t) and L (marked by ’1’) defines X X the time τ. In practice we only calculate 2(E,t) for a discrete set of times, p p and the intersection point, and hence τ, isX found by linear interpolation of p 2(E,t) between the two points surrounding τ. Knowing τ we make another X p 4.2 Disordered chain 41 linear interpolation of the diffusion coefficient curve (dashed red, right axis) to find (E,τ) (marked by ’2’). D

1000 4000

[a] 800 1/2

) 3000 〉 2 X 〈 600

2 2000 400

L 1000 Diffusion coefficient D 200 1

Mean displacement ( τ 0 0 0 50 100 150 200 250 300 Time

Figure 4.10: Illustration of how to ﬁnd (E,τ). The intersection between 2(E,t)(solid black) and L (horizontal, dashedD blue) gives the time τ (marked byX ’1’). The precise intersection is found by linear interpolation. (E,τ) is af- p terwards found from another linear interpolation. The present curvesD correspond to a disorder strength ∆ε = 0.4 γ . | |

The calculated conductances for different lengths are shown in figure 4.11. Dash- dotted blue and solid black correspond to the ballistic and diffusive regimes respectively. The dashed red curve is obtained using the Green’s function (GF) approach, described in chapter 5. The error bars show the standard deviation of 10 different samples. For each different sample, i.e. different Hamiltonian, the mean value is estimated using 12 random phase states. The 10 mean values are subsequently used to calculate the sample-averaged conductance and standard deviation (shown in the figure). Clearly, the general trend is the same for the two approaches: A decreasing conductance for increasing length, as expected. 2 e For small values of L the conductance approaches one conductance unit ( h ) for the ballistic Kubo approaches as well as for the Landauer method. The ballistic Kubo-method, however, gives a clearly larger conductance than the GF approach for larger values of L, whereas the diffusive Kubo-method gives a significantly smaller value throughout the length range. Figure 4.12 shows the resistivity for different disorder strengths calculated with the three different methods. Interestingly, the diffusive Kubo-method gives almost the same resistivity as the GF method, whereas the ballistic formula yields lower values. The resistivity is obtained as the slope of the resistance vs. length curve. An example of such is shown in the inset. 42 4.2 Disordered chain

1.2 Landauer Kubo (ballistic) 1 Kubo (diffusive)

0.8 /h] 2 0.6 [e 〉 G 〈 0.4

0.2

0 50 100 150 200 250 300 Chain length [a]

Figure 4.11: Conductance vs. length for a chain with disorder strength ∆ε = 0.4 t . The ballistic Kubo-method (dash-dotted blue) yields in general a higher conductance| | than the GF approach (dashed red) whereas the diffusive Kubo- method (solid black) gives significantly lower values. The errorbars mark the standard deviation between 10 different samples.

0.02 Landauer 1.1

] Kubo, ballistic 2 1.05 Kubo, diffusive

a] 0.015 2

R [h/2e 1

0 10 20 0.01 Length [a]

Resistivity [h/2e 0.005

0 0.2 0.4 0.6 0.8 Noise amplitude, ∆ε [eV]

Figure 4.12: Resistivity vs. disorder strength calculated with both the Kubo- and the GF method. The resistivity is found as the slope of the resistance vs. length curve, shown in the inset. The GF method gives approximately the same resistivities as the diﬀusive Kubo approach. 4.3 Carbon nanotubes: Comparison with published results 43

The resistance is simply calculated as R = 1/ G 1, where denotes the sample mean. It should be noted that the resistivityh musti be foundh·i with some care, since it is only properly defined, when when the resistance increases linearly with the length. This is, however, not the case for very small lengths, where the resistance is constant, nor for very large lengths where the electron becomes localized and the resistance increases exponentially. The values in figure 4.12 suffer therefore from the uncertainty associated with a subjective estimate of the linear regime. With this in mind, it is still a clear trend that the diffusive version of the real-space Kubo approach yields almost the same resistivity as the GF method whereas the ballistic version yields a too low resistivity (and hence resistance). It is not surprising that the diffusive version of the Kubo method resembles the GF results the most, since the resistivity is found from the linear region of the resistance vs. length curve, i.e. from the diffusive regime.

4.3 Carbon nanotubes: Comparison with published results

Most of the work published by the group of S. Roche concerns carbon nanotubes (CNTs). It is therefore natural to test the implemented programs on carbon nanotube systems and compare the results with published data material. Following Triozon et al. [36], we model the CNT with the simplest tight-binding model treating only the coupling between π electrons and only between nearest neighbours. Setting the on-site energy to zero but adding on-site noise, the Hamiltonian for the system reads

H = δ π π + γ π π , (4.15)  i| iih i| | iih j | i j X X   where the i-summation runs over all the sites and the j-summation runs over the nearest neighbours for each i. The hopping energy is γ = 3 eV. As for the one-dimensional chain, each δ is chosen randomly in the interval− ∆ε ; ∆ε . i − 2 2 The mean free path in a weakly disordered armchair CNT can be estimated using Fermi’s golden rule (FGR). White and Todorov [56] deriveed the expression

2 3 γ Nuc l = 2 d0, (4.16) 8 σε where Nuc is the number of atoms in a unit cell (for a (5, 5) CNT Nuc = 20), 2 d0 = 0.25 nm is the length of the unit cell in the tube direction and σε is the variance of the noise distribution. We use a uniform noise distribution with the

1Note that in general this not be the same as R′ = h1/Gi. Especially in the localized regime the diﬀerence is pronounced. 44 4.3 Carbon nanotubes: Comparison with published results probability density function given by

1 for x ∆ε , P (x) = ∆ε | | ≤ 2 (4.17) 0 for x > ∆ε . ( | | 2 and with the variance

2 ∞ ∆ε σ2 = P (x) x2 dx = (4.18) ε 12 Z−∞

Figure 4.13 (left) shows the mean free path in an armchair (5, 5) CNT with diﬀerent disorder strengths calculated using the FGR result (4.16) (dashed blue) together with the numerical result obtained with the Kubo method. The red circles mark the numerical data points and the solid black line is a linear ﬁt. Both methods yield a quadratic dependence on the disorder strength, but the values are not equal. This is not surprising, since the FGR only takes into account the lowest order of the impurity potential, whereas the Kubo method in principle treats the disorder exactly. To the right is shown the result of

Numerical results 500 Linear fit FGR 400

300

200 Mean free path [nm] 100

0 0 5 10 15 20 25 1/∆ε2

Figure 4.13: Left: Mean free path in an armchair (5, 5) CNT with diﬀerent disorder strengths (in units of γ ). The red circles mark the numerical data points and the solid black line is| a| linear ﬁt. Right: Result from Triozon et al. [36]. The parameter W is equal to ∆ε.

Triozon et al. [36]. The parameter W is equal to ∆ε in our calculations. There is generally a good agreement between the two results, which validates that our numerical implementation (and understanding of the theory) is correct. 4.3 Carbon nanotubes: Comparison with published results 45

60 ∆ε = 0.7|γ| (10,10) 55 (9,9) (8,8) 50 (7,7) 45

(6,6) 40 Mean−free path [nm] (5,5) 35 6 8 10 12 14 Diameter [Å]

Figure 4.14: Left: Mean free path plotted against tube diameter for diﬀerent armchair CNTs all with a disorder strength of ∆ε = 0.7 γ . Right: Mean free path vs. tube diameter, for boron doped CNTs [37]. | |

In figure 4.14 (left) we show the mean free path plotted against tube diameter for different armchair CNTs all with a disorder strength of ∆ε = 0.7 γ . To the right is also shown the mean free path vs. tube diameter, for boron doped| | CNTs [37]. Even though these two types of disorder are different, the two results still show that we capture some of the relevant physics: The mean free path increases linearly with tube diameter implying that the transport in larger tubes is less sensitive to disorder and defects.

10 /h] 2 8

4 Conductance [2e 2

0 −10 −8 −6 −4 −2 0 2 4 6 8 10 Energy [eV]

Figure 4.15: Left: Conductance of a (5, 5) CNT. There are well deﬁned levels, but peaks in the conductance are observed around the band edges. Right: The same peaks are observed by S. Roche et al. (unpublished results) for a (10, 10) CNT.

Figure 4.15 (left) shows the conductance of a pristine (5, 5) CNT. There are well defined levels, but surprisingly the levels are not flat as they should be according to the Landauer formula. The overshooting at the band edge is also 46 4.4 Parallel Chains observed by Roche et al. as seen to the left (private correspondence), where the calculated conductance (dashed line) of a (10,10) CNT is shown together with the Landauer results (solid line). Both figures show the same peaks in the conductance near the band edges. We believe that the deviations near the band edges is a general failure in the real-space Kubo method, and in the following section we further investigate this problem.

4.4 Parallel Chains

We consider a model consisting of two inﬁnite parallel chains as shown in ﬁgure 4.16. Only nearest neighbour interactions are taken into account. The tight- binding parameters are γ1 and γ2 for hopping between and along the chains respectively . The distance between two atoms in the same chain is called a. In

γ 2 γ 1 a

Figure 4.16: Model system. We consider only nearest neighbour interaction with the tight-binding parameters γ1 and γ2 corresponding to hopping in the chain- direction and in perpendicular direction, respectively. The on-site energy is ε0. the Fourier domain the Hamiltonian is ε + 2 γ cos(k a) γ H = 0 1 2 , (4.19) γ ε + 2 γ cos(k a) 2 0 1 where ε0 is the on-site energy. The eigenvalues are

E (k) = ε0 + 2 γ1 cos(k a) γ2 (4.20) ± ±

The band structure is shown in ﬁgure 4.17 (left) where also the energies E1 - E4 are indicated. The density of states for the two bands is 1 n (E) = θ(E E1,2) θ(E3,4 E), (4.21) ± 2 − − E ε0 γ2 4π γ a 1 − ∓ | 1| − 2γ1 r and is plotted in ﬁgure 4.17 (right). The corresponding group velocities read

2 2 γ1 a E ε0 γ2 v (E) = | | 1 − ∓ θ(E E1,2) θ(E3,4 E).(4.22) ± ~ − 2γ − − s 1 4.4 Parallel Chains 47

E 4

E 3

E Density of states 2

E 1 −π/a k π/a E E E E 1 2 Energy 3 4

Figure 4.17: Banddiagram (left) and density of states (right) for the two bands separately.

4.4.1 Calculation of the conductance

We continue to calculate the diﬀusion coeﬃcient Tr [Xˆ(t) Xˆ(0)]2δ(E Hˆ ) 1 2 1 − − (E,t) = X (t) E = . (4.23) D t t Tr δ(E Hˆ ) − In the energy interval E [E2; E3] there are two bands and the δ(E Hˆ ) projects the states in the trace∈ into a linear combination of the eigenstates− from the two bands. We can therefore rewrite (4.23) as

2 2 1 n+(E)X+(E) + n (E)X (E) (E,t) = − − , (4.24) D t n+(E) + n (E) − where X2 (E) = (v (E)t)2 is the spread of wave packets belonging to each band. Defining± an effective± velocity as (for clarity we omit the explicit energy dependence) 2 2 (1) n+v+ + n v veff = − − (4.25) s n+ + n − the diffusion coefficient becomes

(1) 2 (veff ) (E,t) = . (4.26) D t Following the derivation of the ballistic conductance in chapter 2.1 and deﬁning (1) the length L = veff t, leads to a wrong conductance

(1) 2 (1) G (E) = e n(E) veff

2 2 2 n+v+ + n v = e (n+ + n ) − − (wrong) (4.27) − s n+ + n − 48 4.4 Parallel Chains

The conductance should rather be

(2) 2 G (E) = e [n+ v+ + n v ] , (4.28) − − which is easily verified by inserting (4.21) and (4.22). This result could be (1) obtained if we instead of the effective velocity, veff in (4.25) use the average velocity (2) n+v+ + n v veff = − − (4.29) n+ + n − (1) (2) Figure 4.18 (left) shows the effective velocities veff (dashed red) and veff (solid blue) together with the numerically calculated velocity (dash-dotted black). The numerical result is obtained by using the site-basis in the trace. Due to symmetry it suffices to only include one basis state in the trace. The on-site energy is zero and γ1 = 3 eV and γ2 = 2 eV. At first sight there does not seem to be any significant− difference between− the two analytical results. This is, however, not the case. The right graph shows the difference between the numerical result (1) (2) and the two analytical expressions, i.e. vnum veff (dashed red) and vnum veff | − | (1) | − | (solid blue). It is evident that the effective velocity veff (4.25) resembles the (2) numerical results better than the average velocity veff (4.29) in the interval E [ 4; 4] eV, which exactly is the interval where the two bands overlap. The deviations∈ − at E 4 are due to finite size effects and can be made smaller by enlarging the system.≥ | |

10 2 |v −v(2)| num eff 8 |v −v(1)| 1.5 num eff |

6 (1,2) eff

− v 1 num

Velocity 4 Numerical | v v(2) 0.5 2 eff v(1) eff 0 0 −5 0 5 −6 −4 −2 0 2 4 6 Energy Energy

Figure 4.18: Left: Velocity vs. energy calculated numerically (dash-dotted black) (1) (2) and analytically by veff (dashed red) and veff (solid blue). Right: Difference between the numerical and the two analytical velocities. It is evident that the (1) velocity veff (4.25) resembles the numerical result the most.

Figure 4.19 (left) shows the two analytical expressions for the conductances G(1)(E) (dashed red) and G(2)(E) (solid blue) together with the numerical result (dash-dotted black). To the right are shown segments of the same curves around E = 4 eV. 4.4 Parallel Chains 49

5 5 G(2)(E) G(2)(E) (1) (1) 4 G (E) 4 G (E)

/h] Numerical /h] Numerical 2 2

3 3

2 2

Conductance [e 1 Conductance [e 1

0 0 −10 −5 0 5 10 3 3.5 4 4.5 5 Energy [eV] Energy [eV]

Figure 4.19: Left: The numerically calculated conductance (dash-dotted black) is plotted together with the two analytical results G(1)(E) (4.27) (dashed red) and G(2)(E) (4.28) (solid blue). Right: Zoom in around E = 4 eV, showing that G(1)(E) resembles the numerical result very much.

It is evident that G(1)(E) (4.27) resembles the numerical result very much. Especially we find peaks in the conductance around the band edges which are similar to the peaks we found for the CNT. Rewriting the effective velocity (4.25) as 2 2 (n+v+ + n v ) + n+n (v+ + v ) (1) − − − − veff = , (4.30) q n+ + n − we see that the reason why we observe peaks in the conductance is a mixing of the two density of states given by the second term in the numerator. This mixing is an inherent failure in the numerical calculation of the diffusion coefficient. If instead of tracing over the site basis, the eigenstates of the Hamiltonian (4.19) 1 1 are used, i.e. ψ+ = ( 0 + 1 ) and ψ = ( 0 1 ), where 0 and | i √2 | i | i | −i √2 | i − | i | i 1 are the basis states in the middle of the chain, the correct conductance is| i obtained. The reason being that the δ(E Hˆ ) in this case only projects the initial state ψ onto the corresponding− band n (E) and the mixing is thus avoided. In| the±i general case, where disorder is present± we can no longer talk about bands since the system is no longer translational invariant, and it is difficult to find appropriate initial states that avoid the density of states mixing. In the following calculations we will continue to estimate the traces using random phase states, but bear in mind that in the vicinity of band edges care should be taken since the calculated conductance is too large. Chapter 5

The Landauer formula and recursive Green’s function method

5.1 Recursive Green’s function method

In this section we show how to calculate the conductance of a long wire using a recursive Green’s function (GF) method. Inspired by T. N. Todorov [50] we start out with a pristine semi-inﬁnite wire acting as the left lead. A device region is grown by recursively adding pieces of wire and calculating a certain GF and self-energy. In each growth step we calculate the transmission through the disordered wire, giving a length dependent resistance. We start out by introducing the Landauer formalism.

5.1.1 Introduction to the Landauer formalism

In many transport problems the Landauer formalism is the standard approach and we shall use it to compare with the results obtained with the Kubo formula and to model the properties of SiNWs in chapter 7. We consider a general system consisting of device region, coupled to two leads (left and right) as illustrated in ﬁgure 5.1. 5.1 Recursive Green’s function method 51

VL VR

HL H HR D Left lead (L) Device (D) Right lead (R)

VL VR

Figure 5.1: Principal model of a device area (D) coupled to two perfect conducting semi-inﬁnite leads (L) and (R).

The Green’s function equation for the whole system reads [43]

EI HL VL 0 G G G I 0 0 − − L LD LR VL† EI HD VR GDL GD GDR = 0 I 0 ,  − − − · G G G   I  0 VR† EI HR RL RD R 0 0  − −     (5.1)  from where we get

1 G = (EI H Σ (E) Σ (E))− , (5.2) D − D − L − R where we have deﬁned the self-energies ΣL(E) and ΣR(E) as

1 0 Σ (E) = V† (EI H )− V = V† G V (5.3) L L − L L L L L and

1 0 Σ (E) = V (EI H )− V† = V G V† . (5.4) R R − R R R R R 0 1 GL = (EI HL)− is the Green’s function for the isolated left lead, and like- 0− wise for GR. A recursive method to calculate the self-energies is explained in appendix B.4, and for now we shall assume that we know them1. The elements of the self-energy matrix have nonzero imaginary parts, corresponding to a ﬁnite lifetime [43]. This can be understood qualitatively as follows: Suppose an electron is placed in the device region. Since there is a ﬁnite coupling to the leads, there is a certain probability that the electron escapes into the leads. When it does so, it propagates in a perfect conductor and is never scattered back, and eventually the whole wave function has escaped the device region. The transmission through the device can be found as [43]:

T (E) = Tr GD† ΓR GD ΓL , (5.5) h i 1The Python program used to calculate the self-energies was provided by M. Brandbyge. 52 5.1 Recursive Green’s function method where the trace runs over states in the device region and where

0 Γ (E) = 2 Im [Σ (E)] = 2V† Im G V (5.6) L − L − L L L 0 ΓR(E) = 2 Im [ΣR(E)] = 2VR ImGR V† . (5.7) − − R Let us try to look into the physics behind the transmission formula (5.5). We focus on one of the states in the trace, say ψi , and calculate the contribution from this state to the transmission: | i

t (E) = ψ G† Γ G Γ ψ . (5.8) i h i| D R D L | ii

The eﬀect of operating with ΓL on ψi is ﬁrst to ’move’ it to the left lead by operating with V . Suppose ψ is| a localizedi state to the far right, it would L | ii not contribute to the trace. On the other hand, if ψi is close to the left lead |0 i it will give a contribution. The next term, Im GL , can be thought of as a measure of the local density of the states in the left lead that couple to ψi . This is due to the fact that the local density of any state i can be written| asi [54] | i

1 di(E) = lim Im [Gii(E + iη)] , (5.9) η 0 −π → 1 where the Green´s function G(E + iη)=[(E + iη)I H]− - see appendix C.1. −

VL† now brings the state back to the device area. The total eﬀect of operating with Γ is thus to measure the coupling strength between ψ and the lead, L | ii weighted with the local density of the involved lead-states. GD brings the state to the other end of the device, where ΓR works analog to ΓL, before GD† brings the state back. In short: The Γ’s measure the coupling strength to the leads weighted by the local densities of states of the lead-states, while the GD’s measure the ease with which the electrons can travel through the sample. Knowing the transmission we get the conductance using the Landauer formula, which at low temperatures simply reads [43]

2e2 G(E) = T (E), (5.10) h where the factor 2 comes from spin degeneracy.

5.1.2 Recursive (N) growth process O The zero temperature conductance of any device can, in principle, be found using equation (5.5) and (5.10). The calculation, however, involves the inversion of 3 HD (equation (5.2)), which in general scales as (N ), N being the number of states in the device region. However, it is possibleO to calculate the conductance with an recursive (N) algorithm, which is the topic of this subsection. O 5.1 Recursive Green’s function method 53

VL Vi−1,i VR

H H(1) (i−1)(i) (M) H L D HDHD HD R

VL Vi−1,i VR

Figure 5.2: The device region is divided into M sub-cells. The sub-cells are so large that they only couple to the nearest neighbour cells. Also, the left and right leads couple only to the ﬁrst and last cell respectively.

As illustrated in figure 5.2 we divide the device region into M cells, and make the cells so large that each cell only couples to the nearest neighbour cells. We start out with the semi-infinite left lead to which we add the first sub-cell (1) of the device, HD , and consider the equation ˜ EI HL VL X GL1 I 0 − − (1) = . (5.11) V˜ † EI H · GiL G11 0 I − L − D !

The matrix V˜ L couples the left lead only to the ﬁrst sub-cell in the device. (1) ˜ We assume HD to be large enough that VL and VL have the same non-zero ˜ ˜ 0 ˜ elements. Solving for G11 and denoting ΣL(E) = VL† GLVL yields 1 G = EI H(1) Σ˜ (E) − . (5.12) 11 − D − L Notice that this operation only involves the inversion of a matrix with the size of the sub-cell. Adding the next sub-cell results in the equation ˜ EI HL VL 0 X G G I 0 0 − − L1 L2 ˜ (1) G Y G I  VL† EI HD V12  1L 12 = 0 0 , − − − (2) ·    0 V† EI H G2L G21 G22 0 0 I  − 12 − D       (5.13)  from where we get 1 G = EI H(2) Σ (E) − , (5.14) 22 − D − 1 where the new self-energy, Σ1(E) is given by

(1) 1 Σ (E) = V† EI H Σ˜ (E) − V = V† G V . (5.15) 1 12 − D − L 12 12 11 12 54 5.1 Recursive Green’s function method

The eﬀect of the lead on the Green’s function matrix G22 thus only enters through the self-energy, Σ˜ L(E) in the expression for G11, which we already know. We can therefore continue the growth process by only considering the (i) (i 1) new cell, generally described by HD , the previous cell described by HD− minus the self-energy, Σi 1(E), together with the coupling matrix Vi,i 1. In each growth step we solve− the equation −

(i 1) EI HD− Σi 1(E) Vi 1,i X Gi 1,1 I 0 − − − − − − = . (i) G G 0 I Vi† 1,i EI HD ! · i,i 1 ii − − − − (5.16) yielding

1 G = EI H(i) Σ (E) − (5.17) ii − D − i Σi(E) = Vi† 1,iGi 1,i 1 Vi 1,i. (5.18) − − − −

Since Gi 1,i 1 is known from the previous growth step, each step involves two matrix multiplications− − and one matrix inversion. All the calculations, however, involve only matrices of the size of the new sub-cell. For a long wire, the whole process thus only scales as (N). O In each growth step we calculate the transmission through the truncated device region consisting of i sub-cells as

(i) † (i) i Ti(E) = Tr GD ΓR GD ΓL , (5.19) where 1 G(i) = EI H(i) Σ (E) Σ˜ (E) − (5.20) D − D − i − R Γi (E) = 2Im[Σ (E)] (5.21) L − i Γ (E) = 2 Im Σ˜ (E) (5.22) R − R h i and Σ˜ R(E) is found as Σ˜ L(E) described above. In this way we calculate the conductance (and hence resistance) as function of device length.

Non-orthogonal basis

In the case of a non-orthogonal basis, the time independent Schr¨odinger equation reads H ψ = E S ψ , | i | i where S is the overlap matrix (see chapter 6). The calculation of the Green’s functions and self-energies changes according to [24, 57]

1 Gs = E S(i) H(i) Σ (E) − (5.23) ii D − D − i s Σi (E) = (E Si† 1,i Vi† 1,i)Gi 1,i 1 (E Si 1,i Vi 1,i). (5.24) − − − − − − − −

For an orthogonal basis Sorth = I, and Si 1,i contains only zeros. The compli- cations caused by the non-orthogonal basis− set are very limited. Chapter 6

Building a Tight-binding model

6.1 Introduction and ideas behind the model

The tight-binding method, originally introduced by Slater and Koster (1953) [58], has been used to study a great variety of material properties. The works include modelling of the atomic and electronic structure, elasticity constants, electronic transport properties etc. [22]. The tight-binding method is less accurate but faster than ab initio methods and therefore useful for situations where quantum mechanical effects are significant, but the large system sizes make ab initio calculations impractical. Nanowires seem to be perfect candidates for modelling with tight-binding, since quantum mechanical effects are important due to the small diameter but ab initio calculations of long wires are impossible. The tight-binding method has indeed been used in the field of nanowires: recently R. Rurali et al. [20, 21] used both DFT and tight-binding to study the atomic and electronic structure of small diameter SiNW, and Zheng et al. [27] studied the effect of wire thickness on the bandgap, effective masses and transmission using a tight-binding model. Also, all the work of S. Roche et al. discussed in the Introduction (chapter 1) is based on a tight-binding model. In many situations electrons can be thought of as being localized in space. Chemical bonds or hybridization orbitals in many semiconductors are more easily understood in terms of localized electrons as compared to extended Bloch states. These are the physical pictures that underlie the tight-binding model. The ideal tight-binding model involves an orthogonal and short ranged basis set, with only nearest neighbour (n.n.) interactions. It is, however, not possible 56 6.1 Introduction and ideas behind the model to construct a physically rigorous basis set satisfying both demands [22]. Nev- ertheless, it is often assumed that such a basis set exists (e.g. the π-model used by Roche and many others for CNTs). The Hamiltonian describing the system is a matrix with the elements, called 1 tight-binding parameters , HmR′nR = m, R′ Hˆ n, R , where Hˆ is the Hamil- tonian operator entering the Schrödingerh equation| | andi n, R is the n’th orbital | i of the atom localized at position R. Due to the localized basis states, HmR′nR is nonzero only when R′ R < d . An accurate description will often require | − | 0 up to third nearest neighbour interaction, such that d0 1 - 5 A˚ [22, 59]. In the following we discuss how to find the tight-binding parameters.∼

6.1.1 Finding the tight-binding parameters

Any tight-binding model relies on suitable input parameters (i.e. the Hamil- tonian matrix elements). This is the primary drawback of the method since these parameters are not transferable. A model for, say Si in the diamond structure is unlikely to work for amorphous Si. Different methods have been applied to obtain the parameters, here two of them will be briefly discussed and a third one - the method we use - is explained in more detail. All the approaches are based on ab initio calculations, either by fitting to DFT results or by directly using the output from a DFT calculation.

Fitting of band structure

An often used method involves ﬁtting of the band structure to the result from a DFT calculation. Given a basis set n, R , the band structure is found by considering the Fourier transformed states{| i}

1 ik R n, k = n, R e · , (6.1) | i √N R | i X and calculating the Hamiltonian matrix elements Hmn(k) = m, k H n, k . One can ﬁt the eigenvalues of H(k) to the DFT band structure ath certain| | highi symmetry k-points by adjusting the tight-binding parameters, i.e. m, R′ Hˆ n, R . h | | i This method was introduced by Vogl et al. (1983) [60] and is still being used, e.g. by S. Latil et al. [37] to ﬁnd renormalized energies in doped CNTs (see Introduction, section 1.2.3).

1The diagonal elements of H are often referred to as on-site energies while the oﬀ-diagonal elements are called hopping energies. 6.2 From atomic basis set to orthogonal tight-binding 57

Wannier functions

The energy eigenstates in any periodic system are known to be Bloch states, ψm,k extending throughout the space. However, it is possible to construct a |set ofi localized Wannier functions (WF) from the Bloch states. The general formula for a Wannier function located around the lattice point R is [61]:

V (k) ik R w R = U ψ k e− · dk, (6.2) | n, i (2π)3 nm | m, i BZ m Z X where V is the volume of the unit cell and U is a unitary matrix describing a rotation in the sub-space of Bloch orbitals with wave vector k. Usually, the sum in (6.2) involves only occupied Bloch states. Importantly, the WFs form a complete and orthonormal basis set [61]. Since U could be any unitary matrix the WFs given by (6.2) are not unique. In order to reduce the arbitrariness in U, one can impose constrictions on the constructed WFs. Thygesen et al. [62] used a constriction of maximal localization of the WFs and showed that using both occupied and unoccupied Bloch states in the sum (6.2) leads to more localized WFs with better symmetry properties. The Wannier functions represent a localized, real-space representation ideal for calculating tight-binding parameters. These are simply the matrix elements of the Hamiltonian with the new WF basis. However, it can be a diﬃcult task to actually calculate the WFs for a large system since they require a plane- wave based DFT calculation with many bands in order to obtain the maximal localization.

6.2 From atomic basis set to orthogonal tight- binding

Much work has been put into finding suitable tight-binding parameters for silicon, primarily pioneered by Chadi and coworkers [22]. In spite of this we have chosen to build our own tight-binding model of a SiNW from ab initio calculations using the DFT program siesta [63]. There are several reasons for doing this: First, siesta uses an atomic basis set, inherently localized in space which is the basic requirement for a tight-binding basis. Second, we want to model defects, which give rise to changed tight-binding parameters: both vacancies, dopants and surface reconstructions locally change the atomic and electronic structure and thereby also the hopping energies. This complicates the use of standard tight-binding parameters. Finally, siesta is a relatively fast DFT program as compared to the plane wave based dacapo [64], making it much easier to make a siesta calculation than to construct a set of Wannier functions. The major drawback with siesta is that the atomic orbitals are nonorthogonal. The real-space Kubo method requires an orthgonal basis set in order to per- 58 6.2 From atomic basis set to orthogonal tight-binding form the time evolution. It is therefore necessary to make a so-called Löwdin transformation of the siesta output, as explained below. Very briefly, a siesta calculation is performed by supplying an initial set of atomic coordinates. These are relaxed with a molecular dynamics (MD) calculation and the electronic structure is calculated self consistently in the Kohn- Sham approximation [18]. If the forces on the atoms are larger than a certain threshold another MD calculation is performed and the procedure is repeated until the forces are sufficiently small. The speed and accuracy, of course, depend on how many basis functions are used. We use a minimal basis set (called single- ζ), where only the valence states of the free atoms are included. So for hydrogen only the 1s state is included and for Si the basis states are one 3s and three 3p states (3px, 3py, 3pz). The minimal basis set is used primarily to make the subsequent calculations with the real-space Kubo method faster. Consequently, we can not expect the conduction bands to be accurately described. The relevant output from siesta in this context is a coordinate vector, r, the s s Hamiltonian matrix H and an overlap matrix SmR′nR = m, R′ n, R , where n, R is the n’th basis state of the atom positioned at R. h | i | i s ˆ The elements of the Hamiltonian is HmR′nR = m, R′ H n, R and (in atomic units) h | | i 1 n(r ) δE [n(r)] Hˆ [n] = 2 + ′ dr + v(r) + xc , (6.3) −2∇ r r δn(r) Z | − ′| where v(r) is the ionic potential and Exc is the exchange-correlation energy, approximated by the generalized gradient approximation (GGA)[18]. Hˆ depends on the electron density n(r), which is found by solving the Kohn-Sham equations

Hˆ [n]ψi(r) = ǫi ψi(r) (6.4) n(r) = ψ (r) 2 (6.5) | i | i X where the sum runs over all occupied states. Since the states ψi(r) depend on the density through Hˆ [n], the equations (6.3) - (6.5) must be solved self- consistently.

6.2.1 Hamiltonian for the full wire

When modelling a wire with randomly placed defects (see chapter 7) several siesta calculations are performed: one for the pristine wire and one for each defect. The calculations on SiNWs are performed on super cells containing four unit cells, with the defect in unit cell two, close to cell number three. This ensures that the eﬀect of the defect is localized in the middle of the super cell. We neglect interactions between unit cells that are not nearest neighbours. This is justiﬁed by the fact that the unit cells considered are 5.5 A˚ long in the wire direction. The siesta Hamiltonian then has the form (neglecting∼ the interaction 6.2 From atomic basis set to orthogonal tight-binding 59 between the ends of the wire due to periodicity)

h1 v1 0 0 v† h2 v2 0 Hs =  1  , (6.6) 0 v2† h3 v3  v† h   0 0 3 4    where e.g. h1 is the ’on-site’ Hamiltonian for the ﬁrst unit cell and v1 is the coupling matrix between the ﬁrst and the second unit cells2. From Hs we pick out the sub-matrices

h = h2 , v = v2 (pristine) (6.7) in the case of a pristine wire and

h v 0 h = 2 2 , v = (defect) (6.8) v† h v 2 3 3 for a wire with a defect. The reason for the diﬀerence is that the sub-matrix h3 can ’feel’ the defect, that is located in h2 close to h3. Before picking out h and v, the Hamiltonians are shifted in order to align the Fermi energies and thereby ensure charge conservation. The defect-Hamiltonian is shifted by

Hs Hs (E E0 )Ss, → − F − F 0 where EF (EF ) is the Fermi energy in the pristine (defective) wire. An exception for this procedure is made when the systems are semiconducting and a defect state forms in the bandgap. This will be discussed in chapter 7.2. The Hamiltonian for the full wire containing N unit cells becomes h(1) v(1) (1) (2) (2) (v )† h v H =  . .  , (6.9) (v(2)) .. ..  †   .   .. h(N)      where h(i) and v(i) are either from the pristine wire (6.7) or from one of the wires with a defect (6.8). An equivalent procedure is performed for the overlap matrices to set up the total overlap matrix S. The matrices H and S can be used directly in the recursive Greens function method described in chapter 5.

6.2.2 L¨owdin transformation

As mentioned above, the real-space Kubo method requires an orthogonal basis to perform the time evolution. We now show how to transform the Hamiltonian

2The Python code that extracts the Hamiltonian from the siesta output was provided by M. Brandbyge. 60 6.2 From atomic basis set to orthogonal tight-binding

H, with a non-orthgonal basis, to a new one H, with an orthogonal basis. Here we explain the main steps in the transformation, while a more detailed description is given in appendix A. e The Schr¨odinger equation for the wire constitutes a so-called generalized eigenvalue problem:

H φn = EnS φn, (6.10)

1/2 1/2 1/2 1/2 1 By writing S = S S and deﬁning S− = (S )− we rewrite (6.10) as

1/2 1/2 1/2 1/2 H S− S φn = EnS S φn 1/2 1/2 H S− φñ = EnS φñ 1/2 1/2 1/2 1/2 S− H S− φñ = En S− S φñ

H˜ φ˜n = En φ˜n, (6.11)

1/2 where we have deﬁned the orthogonalized vectors φ˜n = S φn. We have thus transformed our generalized eigenvalue problem (6.10) to a normal eigenvalue problem by mapping the original Hamiltonian onto the new one:

1/2 1/2 H H = S− H S− . (6.12) → 1/2 In appendix A is it shown howe to actually calculate S− . The mapping in equation (6.12) is formally a great step forward, since it tells us how to ’orthogonalize’ the Hamiltonian. However, the advances are mostly formal, since we have to invert the matrix S, which has the same size as the Hamiltonian for the full wire. In practical calculations we therefor have to make some assumptions 1/2 about the shape of S− .

6.2.3 Approximate orthogonalization

We wish to construct a unit cell Hamiltonian, h0 described in an orthogonal basis. To this end, we consider a small sub-matrix H(i) in the interior of the full Hamiltonian, describing a few unit cells, including unit cell number i, i.e.:

hi 1 vi 1 0 0 − − (i) vi† 1 hi vi 0 H =  −  (6.13) 0 vi† hi+1 vi+1  v† h   0 0 i+1 i+2    The corresponding overlap matrix is denoted S(i). We now perform the orthogonalization on this sub-system and get:

1/2 1/2 H(i) = S(i) − H(i) S(i) − . (6.14) e 6.2 From atomic basis set to orthogonal tight-binding 61

The basic assumption is that the following equalities hold

(i) H22 = Hii (i) H = Hi,i+1 (6.15) e 23 e where H is the full, orthogonalizede Hamiltoniane obtained from (6.12). Denoting (i) (i) h0 = H22 and v = H23 the Hamiltonian for a homogeneous wire becomes e e e h0 v v† h0 v   H = . . . (6.16) v .. ..  †   .   .. h   0    The validity of the approximations in (6.15) can be tested by enlarging H(i), such that it contains more unit cells. Such a test is performed for the one- dimensional chain and presented in the following subsection.

6.2.4 Example - The one-dimensional chain

Consider a one-dimensional chain with M sites in total described by the Hamil- tonian

H = ǫ , H = H = γ , for i = M/2 ...M/2 (6.17) ii 0 i,i+1 i+1,i − and a nearest neighbour overlap

Sj,j+1 = σ. (6.18) From the center of the Hamiltonian matrix we take out a square matrix H(N), (N) such that Hij = Hij for ( i , j ) N/2, and likewise for the overlap matrix. This sub-system is orthogonalized| | | | ≤ as described by equation (6.14). Figure 6.1 (left) shows absolute values of the elements of the zero’th row in the Löwdin transformed Hamiltonian, i.e. the elements that couple to the middlemost site, for the case N = 19. The elements are calculated for three different values of the overlap: σ = 0.01, 0.05, 0.1. The elements of the original Hamiltonian are ǫ0 = 0 and γ = 3. In the figure, the values at i = 0 are the on-site energies, i = 1, 1 correspond− to the nearest neighbour interaction etc. It is evident that site− ’0’ now couples to all other sites, whereas it only coupled to the nearest neighbours (’-1’ and ’1’) before the orthgonalization. The price of an orthogonal basis is a longer range (cf. the discussion above), and is a well known fact [22, 58]. However, the neighbour interaction decays exponentially with distance, and it therefore suffice to include only a few neighbours. The interaction decays faster for smaller σ, as would be expected. Figure 6.2 illustrates how the real-space orbitals transform according to equation (A.14), appendix A. The original orbitals of the one-dimensional chain (dashed 62 6.2 From atomic basis set to orthogonal tight-binding

0 10 0 10 σ = 0.01 σ = 0.05 σ = 0.1 −5 −5 10 10 Error H(0,:) −10 −10 10 10 σ = 0.01 σ=0.05 −15 σ=0.1 10 −15 10 −10 −5 0 5 10 0 5 10 15 20 25 Site number, i System size, N

Figure 6.1: Left: Absolute values of the elements of the Löwdin transformed Hamiltonian concerning the middlemost site, calculated for three different values of the overlap element γ. Blue, red and black correspond to σ = 0.01, 0.05, 0.1 respectively. Right: The error, defined in (6.19), vs. size of the sub-matrix, N. Color codes are the same as in the left graph.

2 (x µi) blue) are assumed to have a Gauss shape, ψ (x) exp( − −2 ), centered i ∝ 2σ around the atom at x = µi, with a nearest neighbour overlap σ = 0.1. The solid red curve shows the L¨owdin transformed orbital of the middlemost atom. Notice, the oscillating behaviour and the longer range of the transformed orbital.

1.2 1 0.8 (x)

ψ 0.6 0.4 0.2 0 x

Figure 6.2: Original orbital of the one-dimensional chain (dashed blue) and one L¨owdin transformed orbital (solid red). The circles inticate the positions of the nuclei in the chain. The nearest neighbour overlap between the original orbitals is σ = 0.1.

In order to see the importance of the size of the sub-matrix H(N) we perform the Löwdin transformation for different values of N and compare the results to 6.2 From atomic basis set to orthogonal tight-binding 63 the exact result, where N = M. As a measure of the error made by only having a small N, we define an error function by

N/2 Error(N) = H(N) H (6.19) | 0i − 0i| i= N/2 X− e e The right graph in figure 6.1 shows the error as function of N for three different values of γ. The error decays exponentially, again with the fastest decay for the smallest σ. This shows that in order to construct an orthogonal basis, it suffices to invert the small sub-matrix S(N) (e.g. with N & 5 for σ = 0.01) and follow the procedure described above. Chapter 7

Modelling of silicon nanowires

In this chapter the real-space Kubo method used together with the recursive Green’s function (GF) method to study silicon nanowires. There are two main objectives: First, using both methods on the same system we can compare them and validate, whether the Kubo or the GF approach is the most suitable to study SiNWs. Second, by using an ab initio based Hamiltonian, we can make a fairly realistic model of long SiNWs, which hopefully can be compared with experiments and thereby help to understand the physics.

7.1 Metallic silicon nanowires

As mentioned previously, R. Rurali et al. [20, 21] showed that nanowires grown in the [1 0 0] direction could be metallic due to a certain surface reconstruction. Fortunately, we were allowed to use this structure and the first study of SiNWs were performed on these. Figure 7.1 shows an end-view (left) and a side-view (right) of a wire containing four unit cells. There are 57 atoms in one unit cell, the diameter is 1.5 nm, and the length of the unit cell (in the growth direction) is 0.55 nm. Notice∼ that the structure at the surface differs significantly from the bulk of the wire. This particular surface structure makes the wire metallic, which is evident in figure 7.2 showing the band structure with the Fermi energy marked by the dashed red line. Four bands are crossing the Fermi level (two being degenerate). 7.1 Metallic silicon nanowires 65

Figure 7.1: End-view (left) and side-view (right) of a pure SiNW with four unit cells. Each unit cell contains 57 atoms, the diameter is 1.5 nm and the length of each unit cell is 0.55 nm. ∼

0.5 [eV]

F 0 E − −0.5

−1 Γ k X

Figure 7.2: Band structure of the wire in ﬁgure 7.1. The wire is metallic with four bands (two degenerate) crossing the Fermi energy, marked by the dashed red line.

7.1.1 Truncating the Hamiltonian

In order to speed up the real-space Kubo method, we truncate the Hamiltonian such that if Hij < δ, the element is set to zero. The cutoff energy, δ, is chosen so small that| the| band structure calculated from the truncated Hamiltonian does not deviate significantly from the un-truncated. Figure 7.3 (left) shows, on a double logarithmic scale, the maximum deviation between the truncated and the un-truncated band structure for different values of δ. The error is only calculated from the four bands around the Fermi level. To the right is shown the fraction of non-zero Hamiltonian elements after the truncation as function of δ. There is a trade off of having few matrix elements and thus making the subsequent computations fast, while still making accurate calculations. We have 66 7.1 Metallic silicon nanowires

0 10 70

60 −1 10 50

40 −2 10 30

max error [eV] −3 20 10 10

−4

10 Fraction of non−zero H elements 0 −6 −5 −4 −3 −2 −6 −5 −4 −3 −2 10 10 10 10 10 10 10 10 10 10 Cutoff [eV] Cutoff [eV]

Figure 7.3: Left: Maximum deviation between the truncated and the un- truncated band structure around the Fermi energy for different values of the cutoff energy δ. Right: Fraction (in %) of non-zero elements in the truncated Hamiltonian matrix as function of cutoff energy δ.

3 chosen a cutoﬀ of δ = 10− eV yielding a maximum error of 0.023 eV while 87% of the matrix elements are set to zero. ∼

7.1.2 Conductance of a pristine wire

Figure 7.4 shows the conductance of a pristine wire calculated with the real- space Kubo method (solid blue) together with the analytical conductance obtained by simple band counting (dashed red). The two results agree qualitatively: the band gaps are approximately the same, the first conductance step from E = 0.3 eV to 0.1 eV is reproduced and the level at G= 6 e2/h around E = 0 eV− is clearly seen.− However, there are quantitative differences, and especially near the band edges there is a clear tendency that the Kubo method overshoots. This is believed to be due to the mixing of the different bands, as discussed in section 4.4. The smaller fluctuation in the Kubo conductance is presumably due to a (too) small number of random phase states to estimate the traces. As expected, the calculated conductance is length independent (not seen in the figure). The specific band structure for the wire complicates the use of the Kubo method in two ways: The rather flat bands around the Fermi energy imply low group velocities and it is therefore necessary to propagate for relatively long times. At other energies the propagation is faster, and to ensure that the wave packets (which have components of all energies) do not reach the end of the wire, very long wires are needed. Generally the wires should be 50 times longer than the propagation length of the electrons around the Fermi∼ energy. In many calculations the wire consists of up to 1000 unit cells (Lwire 550 nm) while a typical propagation length around the Fermi energy is 10 nm.≈ ∼ 7.1 Metallic silicon nanowires 67

10 Kubo Bandstucture 8 /h] 2 6

4 Conductance [e 2

0 −0.4 −0.2 0 0.2 0.4 Energy [eV]

Figure 7.4: Conductance vs. energy of a pristine wire calculated with the Kubo method (solid blue) and from the band structure (dashed red).

7.1.3 Anderson disorder

As a ﬁrst and simple model of physical defects (vacancies, adatoms, dopants) random noise is added to the on-site energies (Anderson disorder). As in chapter 4.2 the on-site enrgies are shifted according to

H H + δ , ii → ii i where δi is a random number in the interval [ ∆ε/2; ∆ε/2]. Figure 7.5 shows the estimated total density of states before (dashed− black) and after (solid red) addition of noise. The two densities of states are qualitatively equal, but the sharp van-Hove peaks are blurred by the noise. The positions of the band gaps are almost unchanged.

Noise Pristine Density of states

−0.4 −0.2 0 0.2 0.4 Energy [eV]

Figure 7.5: Total density of states of a pristine wire (dashed black) and of a wire with random on-site noise (solid red). The sharp van-Hove peaks are blurred by the noise. 68 7.1 Metallic silicon nanowires

As mentioned above, the special surface reconstruction leads to metallic wires, and the conducting channels are believed to be located in the surface layers [20]. It is therefore interesting to study the effect of spatially localized disorder. We consider two types: Surface (or edge) disorder, where only the on-site energies of the surface atoms are changed, and bulk disorder, where the bulk atoms (i.e. all but the surface atoms) are changed. Figure 7.6 shows that the two types of disorder (both having ∆ε = 0.2 eV) have very different effects. The system with

400 400 350 [Å] 1/2 )

〉 200

] 300 (t) 2 X −1 〈 250 ( 0 fs 0 100 200 300 2 Time [fs] 200 Bulk disorder 150

D(t) [Å 100

50 Edge disorder

0 0 50 100 150 200 250 300 Time [fs]

Figure 7.6: Time dependent diffusion coefficient at E = 0.1 eV. Inset shows the mean displacement. The bulk disorder has little effect and the transport is ballistic (linearly increasing displacement). The edge disordered system shows Ohmic behaviour with a constant diffusion coefficient. bulk disorder (dashed black) acts almost as a ballistic conductor resulting in a nearly linearly increasing diffusion coefficient. In the case of surface disorder, the diffusion coefficient reaches a constant level around t = 150 fs, and the transport becomes diffusive. The inset shows the corresponding displacements 2(t). The bulk disordered system results in a linearly increasing displacementX with p the slope given by the velocity, while the edge disordered wire deviates from the ballistic line around t = 50 fs. To avoid the wrong conductance peaks around the Fermi energy, the curves in the figure are obtained at E = 0.1 eV. The results in figure 7.6 are mean values from 10 different samples, and for each sample 10 random phase states are used to estimate the traces. The qualitatively different conductance properties of the surface- and bulk disordered systems show that the conduction around the Fermi energy almost entirely takes place at the surface atoms, qualitatively in agreement with the conclusions drawn in ref. [20]. This conclusion is perhaps not surprising, but it still implies that the used tight-binding model with the Löwdin transformed and truncated Hamiltonian together with the real-space Kubo method gives physically reasonable results. 7.1 Metallic silicon nanowires 69

4 Kubo Landauer 3 /nm] Ω

Resistivity [k 1

0 0.2 0.3 0.4 0.5 0.6 0.7 Noise amplitude, ∆ε [eV]

Figure 7.7: Resistivity vs. noise amplitude, ∆ε, for edge disorder calculated with both the real-space Kubo method and with the recursive Green’s function/Landauer method. E = 0.2 eV. −

To further test the Kubo method, the resistivity of surface disordered wires is calculated for a number of different disorder strengths, ∆ε, using both the real-space Kubo method (in the diffusive regime, i.e. the conductance formula (2.14)) and the recursive Green’s function method. For comparison, the Hamil- tonian used in the GF method was truncated in the same way as for the Kubo method. The Kubo results are, again, average results of 10 different samples with 10 random phase states in each. The GF results are calculated from 50 different samples. Figure 7.7 shows the results obtained at energy E = 0.2 eV, revealing that both the Kubo method (red circles) and the GF method− (blue squares) approximately fit the same quadratic dependence of ∆ε (solid black) obtained for the GF approach. The quadratic dependence on ∆ε corresponds 2 to a mean free path le ∆ε− as expected from Fermi’s golden rule and observed for carbon nanotubes∼ - see chapter 4. The resistivity values are subject to uncertainties since they (as discussed in section 4.2) are found by hand from the linear region of the resistance vs. length curve. The deviations between the Kubo and the GF method are believed to lie within these uncertainties, thus validating the real-space Kubo method.

7.1.4 Sub-conclusion

The analysis of the metallic SiNWs showed, in agreement with [20], that the conduction around the Fermi level takes place at the surface of the wires and is almost unaﬀected by bulk disorder. We conclude that the real-space Kubo method can be applied to SiNWs and gives approximately the same results as the recursive Green’s function method. 70 7.1 Metallic silicon nanowires

Furthermore, the orthogonal tight-binding model obtained with the Löwdin transformation and truncation of the Hamiltonian is usable. However, the Kubo method is, for the considered systems, slower and less reliable than the GF method. This is partly due to the very flat and closely lying bands, which give rise to overestimates of the conductance due to ’band mixing’. Moreover, the Kubo method suffers from having several convergence parameters that need to be determined by hand. It was in general difficult to obtain fully converged results with the Kubo method. To minimize the tails of the van-Hove peaks and avoid overlapping of the tails in the closely lying bands, high values of Ntri > 3000 were needed, resulting in relatively long computation times. Also, long time propagations were needed due to the small velocities around the Fermi energy. This results in much waste of work since the wires must be very long to avoid high velocity states to hit the boundary. A possible improvement might be to introduce artificial absorbing potentials at the ends of the wire. Tran [65] showed that this efficiently suppressed backreflection from the boundaries. If successfully implemented in the Kubo method, this would enable the use of much shorter wires thus reducing the computation time significantly. Whether or not this is possible has not been pursued further. Altogether, the recursive GF method is the preferred choice for modelling very thin SiNWs, and the following results are obtained using this method. For wires with a larger diameter and hence larger unit cells the GF method will be significantly slower since the inversion of the unit cell Hamiltonian scales as (N 3), and the Kubo approach might become the method of choice. O 7.2 Hydrogen passivated Si-wire 71

7.2 Hydrogen passivated Si-wire

The metallic SiNWs are both physically and technologically exciting, but probably also very fragile objects, since small changes in the surface such as defects or adatoms presumably will change the performance drastically. Moreover, the pure silicon wires have not been realized experimentally. Instead the wires are surface passivated by either SiOx or hydrogen, as discussed in the Introduction (section 1.1), and the focus in this section will be on such wires. The passivated wires are semiconducting, often with a direct bandgap that increases for small diameters [10, 27, 30]. The simplest way to model surface passivated wires is to add hydrogen atoms to the surface such that each Si atom has four nearest neighbours. Such wires resemble those reported by Ma et al. [10].

7.2.1 Setting up the model

Following ref. [27] we model an H passivated wire grown in the [1 0 0]-direction. Figure 7.8 shows a top-view (left) and side-view (right) of the wire unit cell containing 81 Si atoms and 36 H atoms. The side length is 1.8 nm (from H to H atom) and the length of the unit cell in the wire direction is 0.54 nm. The wire- direction is perpendicular to the paper plan in the left image and horizontally in the right image, so the wire has a square cross section. The Hamiltonian for

Figure 7.8: Top-view (left) and side-view (right) of a single unit cell containing 81 Si atoms and 36 H atoms. The side length is 1.8 nm and the length in the wire direction is 0.54 nm. the wire is constructed as described in chapter 6 using a single-ζ basis set. Even though the structure is relaxed in siesta, other conﬁgurations such as rounding of the corners making the cross section octagonal, might be more stable. This 72 7.2 Hydrogen passivated Si-wire has not been further investigated. Due to the minimal basis set, we can not expect the conduction band to be accurately described, and we therefore focus on the valence band. Figure 7.9 (right) shows the calculated band structure together with the result from ref. [27] obtained with a tight-binding model. The two results are similar with the most notable diﬀerence being that the two highest lying bands, which are degenerate in our calculations, split up in the result from [27]. This is due to spin-orbit coupling included in [27] but not in siesta.

−2.6

−2.7

−2.8

−2.9 Energy [eV] −3 0 Wavevector pi/a

Figure 7.9: Band structure calculated in ref. [27] (left) and in siesta (right) with a single-ζ basis set. The two highest lying, degenerate bands (right) split up in the left ﬁgure due to spin-orbit coupling.

It is likely that a passivated wire will have surface defects e.g. in the form of randomly missing H atoms. This situation is difficult to model with a full DFT calculation since the number of atoms (> 10, 000) becomes too large. The recursive GF method is ideal for this purpose, since the unit cells are relatively small and the Hamiltonian can be readily inverted. To simulate a random distribution of vacancies we consider four spatially different vacancy sites: removal of the middlemost H atom on each side of the unit cell. This is not an ideal model, and one could argue that the rotational symmetry of the vacancy sites affects the randomness. We will return to this below. When ’growing’ the wire with the recursive Green’s function method, care must be taken. Usually when joining two pieces of wire their Fermi energies should be aligned. However, this is not the case for the unit cells with a H vacancy. The reason is that the vacancy leads to a spatially localized Si dangling bond (DB) with an energy within the band gap as shown in figure 7.10. This causes the Fermi level to shift by 0.76 eV from approximately in the middle of the band gap (dashed red) to the energy level of the DB (horizontally, solid blue). Alignment of the Fermi levels would therefore lead to an unphysical shift in energies for sites far away (& 1 nm) from the vacancy, which cannot ’feel’ the defect. We therefore align the Si on-site energies instead, taking as reference a Si atom in the unit cell next to the vacancy cell. 7.2 Hydrogen passivated Si-wire 73

0.5

−0.5

−1 E for pristine wire F

−1.5 0.76 eV Energy [eV] −2 0.75 eV Defect state −2.5

−3 0 Wavevector π/a

Figure 7.10: Band structure of a super cell containing four unit cells with one H vacancy in the second unit cell. The localized dangling bond (DB) state has an energy within the band gap, which causes the Fermi level for the pristine wire (dashed red line) to shift by 0.76 eV to the DB energy.

7.2.2 Length and energy dependent conductance

The conduction of a wire containing a single H vacancy, connected to two infinite leads of the same (pristine) wire is shown in figure 7.11 (dashed blue). The conductance of a pristine wire (solid blue) is shown as reference. Solid red, black and green are average conductances, G , of wires having different lengths (L = 27, 54, 108 nm respectively) with ah randomi distribution of the four H vacancy types. The average distance between two vacancies is dH = 5.4 nm corresponding to 10% of the unit cells missing an H atom. Theh meani value of the conductance is obtained by averaging over 50 different configurations. There is evidently a strong energy dependence: At some energies the transmission goes almost to zero for the long wires, while at other energies the transport is much less affected. This is more quantitatively seen in figure 7.12 (left) showing the conductance at E = 2.74 (dashed blue) and E = 2.84 eV (solid red). The two energies are marked− by the two vertical dashed− lines in figure 7.11. The error bars mark the standard deviations, σG, among the 50 configurations. The right graph shows σG(L) vs. wire length, L, for the same two energies (same color code as to the left), with the maximum positions marked by the vertical lines. Notably, the curve corresponding to E = 2.74 eV has a pronounced maximum. Similar peaks are observed for all E .− 2.8 eV, which also are the energies at which the conductance is most affected− by the vacancies. 74 7.2 Hydrogen passivated Si-wire

5 Pristine 1 defect 4 L = 27 nm L = 54 nm L = 108 nm 3

2 Transmission

0 −3 −2.9 −2.8 −2.7 −2.6 Energy [eV]

Figure 7.11: Conductance vs. energy. Pristine wire (solid blue), through 1 defect (dashed) blue and sample averages for diﬀerent lengths with random distribu- tions of vacancies: red, black and green correspond to L = 27, 54 and 108 nm respectively. The average distance between vacancies is d = 5.4 nm. h H i

2.5 E = −2.84 eV E = −2.74 eV 2 0.5 /h] 2 0.4 /h]

1.5 2

0.3 1 (L) [2e

σ 0.2 0.5 Conductance [2e 0.1 0 0 0 20 40 60 80 100 0 20 40 60 80 100 Length [nm] Length [nm]

Figure 7.12: Left: Average conductance vs. length for E = 2.84 eV (solid red) and E = 2.74 eV (dashed blue). The error bars mark the standard− deviations, − σG of 50 diﬀerent samples. Right: Standard deviation vs. length. The peak positions are indicated with dashed vertical lines.

7.2.3 Distribution of conductances

From figure 7.12 (left), it appears that around L = 40 nm the standard deviations corresponding to E = 2.74 eV become larger than the mean value, since the bottom of the error bars− is below zero. The conductance is then no longer a well determined quantity. Interestingly, this lengths roughly corresponds to 7.2 Hydrogen passivated Si-wire 75 when the resistance starts to deviate from the initial Ohmic behaviour and becomes localized (see figure 7.14). It is characteristic for the localization regime that the distribution of conductances is very broad [46, 49] resulting in large relative standard deviations (σG/ G ). Figure 7.13 (left) shows, on a logarithmic scale, the conductances at Eh=i 2.74 eV of 50 wires (listed by increasing values) with length L = 32 nm (red− triangles) and L = 108 nm (blue circles). Red and blue dashed lines indicate the mean values for the two lengths. The short wire, which has a resistance in the Ohmic regime (see figure 7.14), shows a relatively narrow distribution, with many data points around the mean value. The long wire, which is deep within the localization regime (see figure 7.14), has a very broad distribution of conductances ranging over 6 orders of magnitude and few data points around the mean value. This is a quite remarkable result, since all the long wires have the same number of vacancies, only the positions differ. The right plot in figure 7.13 shows the conductance during two different growth processes, i.e. two different vacancy configurations. The solid red curve decreases every time a vacancy unit cell is added, resulting in a final 3 2 conductance of G = 2 10− e /h. The dashed black curve, on the other hand, sometimes increases yielding· a final conductance G = 0.6 e2/h.

4 0 10 /h] 2

/h] 3 2 −2 10

2 −4 10

Conductance [e 1

Conductance [e −6 10 L = 32 nm L = 108 nm 0 0 10 20 30 40 50 0 20 40 60 80 100 120 # sample number (sorted) Length [nm]

Figure 7.13: Left: Conductances of 50 different samples with length L = 32 nm (red triangles) and L = 108 nm (blue circles). The short wire is in the Ohmic range with a relatively narrow distribution, whereas the long wire is in the localization regime with conductances ranging over six orders of magnitude. Right: Conductance during the growth of two different samples. For some vacancy configurations, the addition of a new vacancy yields an enhanced conductance, highlighting the quantum mechanical nature of the scattering mechanism.

Altogether, these results highlight the quantum mechanical wave nature of the electrons, since the huge differences among the samples only can be understood in terms of different interference patterns in the different configurations. We emphasize, that these results could not have been obtained with a single transmission calculation through one vacancy, and that averaging over many impurity configurations is absolutely essential in the localization regime. 76 7.2 Hydrogen passivated Si-wire

7.2.4 Resistance vs. length

The resistances corresponding to figure 7.12 are shown in figure 7.14 with linear- and semi-logarithmic scales to the left and right respectively. At both energies, the resistance shows an initial linear increasing (Ohmic) behaviour characterized by the mean free path, le through the relation R(L) = R0 + R0L/le. While the resistance for E = 2.84 eV (solid red) is Ohmic throughout the range, the curve corresponding− to E = 2.74 eV (dashed blue) rises above the Ohmic line around L = 35 nm where the− localization sets in. The exponentially increasing resistance, characteristic for the localization regime, is clearly seen in the right graph. The two black, dash-dotted lines iindicate the two regimes. From the left figure we estimate the mean free path le 20 nm at E = 2.74 eV while the slope, α, of the black line to the right gives≈ the localization− length ξ = 1/α 45 nm. These estimates are, as previously discussed, subject to uncertainty,≈ since the linear regions are manually determined.

12 1 E = −2.84 eV 10 E = −2.84 eV 10 E = −2.74 eV E = −2.74 eV ] ] 2 2 8

4 0 10 Resistance [h/2e Resistance [h/2e 2

0 0 20 40 60 80 100 120 0 20 40 60 80 100 Length [nm] Length [nm]

Figure 7.14: Resistance vs. length. The right graph shows the same curves in a semi-logarithmic plot. We estimate the mean free path as le = R0/ρ, where R0 is the intersection with the y-axis and ρ is the resistivity given by the initial (Ohmic) slope. The localization length is determined from the ﬁnal slope, α, of the right curves as ξ = 1/α.

In order to see the significance of having different positions of the vacancies, the resistance at E = 2.74 eV is calculated for wires with different configurations and shown in figure− 7.15 (left). For a single vacancy position (solid red), i.e. always on the same side of the wire but randomly positioned along the wire, the resistance reaches a constant level corresponding to one conducting channel. For two possible vacancy positions opposite to each other (dash-dotted black), the situation is the same - one channel is blocked completely, while the other is unaffected. Placing the vacancies on adjacent sides, both channels are affected and the resistance increases exponentially throughout the length range following the curve corresponding to four possible vacancy positions (dotted green). The reason for this behaviour is probably that the two degenerate bands at E = 2.74 eV have a symmetry as illustrated to the right in figure 7.15. The − 7.2 Hydrogen passivated Si-wire 77

y 7 Single site 6 Two sites, adjacent

] x 2 Two sites, opposite 5 Four sites 4

3 y 2

Resistance [h/2e 1 x 0 0 20 40 60 80 100 Length [nm]

Figure 7.15: Left: Resistance vs. length for different configurations of vacancy positions. Right: Scematic profile in the xy-plane of the two degenerate states at E = 2.74 eV. − states have a node in the xz- or the yz-plane, with the z-axis in the wire direction. When a vacancy is positioned along the y-axis, the upper state will be unaffected, while the lower state will ’feel’ the defects at both sides equally well. The symmetric positioning of the vacancy sites is a drawback of the model, and a more realistic calculation ought to include many more possible vacancy positions. Due to lack of time, this has not been done, and the following results are obtained with the vacancies positioned on all four sides.

7.2.5 Mean free path and peaks of σG(L)

The mean free path is calculated for diﬀerent energies and plotted in ﬁgure 7.16

(blue circles) together with the peak positions of the standard deviations, σG(L) (marked by the vertical lines in figure 7.12 (right)). Remarkably, the maximum of σG almost exactly coincide with l when the latter is less than 50 nm. The e ∼ same correspondence between le and σG was also observed by Todorov [50], but was left almost uncommented. It is interesting that the conductance fluctuations bear clear information about the transition from ballistic to Ohmic transport, analog to a thermodynamic phase transition. The correlation can be understood from a simple model: Consider an ensemble of wires with a single vacancy, and suppose the probability of being reflected as function of propagation length has a Gaussian like distribution1 centered around the mean free path as illustrated in figure 7.17. For a wire of length L = le, 50% of the electrons will be reflected and 50% will not, and the standard deviation of the conductance has its maximum for this particular length. This simple model seems to work well for energies

1For randomly placed scatterers the probability would actually be be a Poisson distribution, but the qualitative argument remains the same. 78 7.2 Hydrogen passivated Si-wire

400 l = R /ρ e 0 Peak of σ (L) G 300

200

100 Mean free path [nm]

0 −3 −2.9 −2.8 −2.7 Energy [eV]

Figure 7.16: Elastic mean free path, le (solid blue). Remarkably, the peak positions of the σG(L) (dashed red) coincide with le determined from the ohmic slope.

0.2 1

0.1 0.5 P(L) D(L)

0 0 l l L e L e

Figure 7.17: Left: Probability function P (L) of being reﬂected. It is most likely to scatter and be reﬂected at L = le. Right: Probability distribution D(L) = L P (x)dx. At L = le, 50% of the electrons will scatter and 50% will not. The standard−∞ diviation of the conductance σ (L) has a similar form as P (L) with R G the maximun at L = le.

E . 2.85 eV where only a single (degenerate) band is present, while for lower energies,− where more bands excist, the situation is more complicated.

7.2.6 Comparison with analytical models

We now focus on the resistance at energy E = 2.74 eV (blue dashed line in previous ﬁgures). Figure 7.18 shows the numerically− calculated resistance (solid blue) together with the two analytical models discussed in chapter 2.2.3: The expression for localization in a one-dimensional chain (dashed-dotted red):

R(L) = c(eL/ξ 1) + R (7.1) − 0 7.2 Hydrogen passivated Si-wire 79 and the result obtained with the continuum model (dashed black):

R(L) = R0 [c sinh(L/ξ) + cosh(L/ξ)] , (7.2)

h where c = ρ ξ, R0 = 2e2N is the contact resistance and N = 2 is the number of conducting channels. The Ohmic resistivity, ρ, and localization length, ξ, is determined from the black dash-dotted lines in ﬁgure 7.14 as explained above. The simple continuum model (7.2) resembles the numerical results very well

15 Numerical data c*sinh(L/ξ) + cosh(L/ξ)/2

] L/ξ 2 c(e −1) + R 0 10

5 Resistance [h/2e

0 0 20 40 60 80 100 Length [nm]

Figure 7.18: Resistance vs length. The numerical data (solid blue) is compared with the two analytical expressions, equations (7.1) (dashed black) and (7.2) (dash-dotted red). The continuum model (7.2) ﬁts the numerical result throughout the length range. throughout the length range, whereas the 1D expression (7.1) deviates when entering the localization regime. The same trend is observed for other vacancy concentrations as shown in ﬁgure 7.19 and was also observed in ref. [50]. It

] 3 2 15 30

2 10 20

1 5 10 Resistance [h/2e 0 0 0 0 100 200 0 50 100 0 40 80 Length [nm]

Figure 7.19: Resistance vs. length for three concentrations: d = 28 nm, 5.5 h H i nm, 2.8 nm (left, middle, right) with dH being the average distance between two vacancies. The color code is the sameh i as in ﬁgure 7.18. 80 7.2 Hydrogen passivated Si-wire is expected that the 1D model gives a faster rise from the Ohmic line than the numerical data since there are two conducting channels in the SiNW, and therefore also a possibility of scattering between the conducting channels, thus delaying the localization. More surprisingly, the simple continuum model seems to capture the correct physics, ﬁtting the numerical data accurately for all concentrations. We emphasize, that any comparison between the numerical data and the two models only makes sense for an ensemble average. The resistance of a single, long wire is unpredictable with both models.

7.2.7 Scaling of le and ξ

The mean free path and localization length is calculated for a number of different vacancy concentrations and plotted in figure 7.20 as function of mean distance between the vacancies, d . Both l and ξ increase approximately linearly h H i e for increasing dH as would be expected from simple arguments: As discussed in appendix C.2,h i the mean free path should roughly be the average distance between scatterers divided by the probability of scattering l N d /(T T ), e ∼ h H i 0− 1 where N is the number of channels, T0 is the transmission through a pristine wire and T1 through a wire with one defect. From figure 7.11 we get T0 T1 0.5 and therefore l 4 d , in good agreement with the data in figure 7.20.− ≈ e ∼ h H i

Mean free path, l e 150 Localization length, ξ

100 [nm] ξ , e l 50

0 0 10 20 30 〈d 〉 [nm] H

Figure 7.20: Mean free path, le, (red squares) and localization lengths, ξ, (black circles) vs. average distance between two vacancies, d . The two dashed lines h H i are linear ﬁts to the numerical data. The linear dependence on dH is consistent with theoretical models. h i

The localization length is longer than the mean free path by a factor of 1.5 in reasonable agreement with the rule of thumb ξ Nl [46, 50]. The linear∼ ∼ e 7.2 Hydrogen passivated Si-wire 81 increase of ξ vs. distance between defects corresponds with recent calculations and experiments in ref. [51].

7.2.8 Sub-conclusion

In this section we have considered a fairly realistic model, based on ab initio calculations, of a hydrogen passivated SiNW with randomly distributed H vacancies. The mean free path was found to increase linearly with the average distance between vacancies, in agreement with theoretical models. For two conducting channels, the standard deviation of conductance showed clear maxima at lengths equal to the mean free paths, which could be explained using simple arguments. The localization regime is characterized by an average resistance increasing exponentially with length, and by a very broad distribution of resistances. It is emphasized, that averaging over many different sample configurations is crucial. A simple continuum model fitted the average resistance throughout the length range for different vacancy concentrations. Finally, the localization length increased linearly with the average distance between vacancies with values approximately a factor of two higher than the mean free path. Results, like those presented here, might help experimentalists to estimate real defect concentrations and clear out the important scattering mechanisms. Chapter 8

Summary and outlook

8.1 Evaluation of the real-space Kubo method

One of the main objectives in this work has been to understand, implement and test the real-space Kubo method. Since only one group has used it in the literature and applied it primary on carbon nanotubes, there were many unanswered questions:

How does one derive the formulas used in the articles, and are the deriva- • tions rigorous?

How should the method be implemented? • Does the method yield the same results as other methods? • Is it applicable for other, more diﬃcult systems than carbon nanotubes? •

The first question was addressed in chapter 2 where the the real-space Kubo formula was derived from the Kubo-Greenwood formula. It was shown that the conductance formulas in the ballistic and diffusive regimes differed by a factor of 2, a point not mentioned in the literature. Apart from that, the approach seems rigorous and yields well known results, such as length independent conductances in the ballistic regime as well as the conductivity reduces to the Einstein relation in the diffusive regime. Chapter 3 dealt with the question of implementation. Expansion of the time evolution operator in Chebychev polynomials was shown to be a very efficient way to numerically solve the time dependent Schrödinger equation. Local densities of states were calculated with a continued fraction technique, which suffers from having different adjustable convergence parameters. 8.1 Evaluation of the real-space Kubo method 83

The implementation was tested in chapter 4 primary using the one-dimensional chain. Several analytical results were reproduced, and it was shown that estimation of traces was efficiently done using few random phase states. Calculations of mean free paths in carbon nanotubes with random disorder reproduced published results and agreed with theoretical estimates. However, near band edges the conductance had incorrect peaks, which was shown to be caused by a mixing of the densities of states corresponding to the different bands. The resistivities in both the one-dimensional chain and in the metallic SiNW (chapter 7.1) were calculated with the real-space Kubo method and with the recursive Green’s function method, showing reasonable agreement between the two approaches. This result justified the construction of the orthogonal (Löwdin transformed) basis set used in the Kubo approach. However, the analysis of the SiNWs revealed that the Kubo method is slower and less reliable than the GF method. The latter is partly due to the flat and closely lying bands, which give rise to overestimates of the conductance due to the band mixing. Moreover, due to small velocities around the Fermi energy, long time propagations and long wires were needed, making the computations slow. Table 8.1 summarizes the pros (X) and cons ( ) of the two methods. Among † the advantages of the Kubo method is that it scales as (Nuc), where Nuc is the number of orbitals in the unit cells, whereas the GFO method scales as 3 (Nuc) due the inversion of the unit cell Hamiltonians. For the thin wires consideredO here, this has not been a limitation of the GF method since the unit cells are relatively small. For wires with a larger diameter and hence larger unit cells the GF method will be significantly slower. The Kubo method is parallel in energy, i.e. the time it takes to calculate the diffusion coefficient for all energies is not much longer than for a single energy. The GF method, on the other hand, requires a full calculation for each energy. However, this is a task ideal for parallel computation, since each node can calculate the transmission at different energies. The Kubo method is numerically more difficult than the GF method for a number of reasons: First, the traces are estimated with a number of random phase states. How many random states are needed to obtain convergence is unknown prior to a calculation. Second, the full Hamiltonian has to be stored thus requiring a lot of memory. In the recursive GF method, the wire is grown on the fly and there is no need for storing the full Hamiltonian matrix. Third, there are a number of convergence parameters, which complicate the Kubo method. The GF method has no such adjustable parameters, which ease the implementation and use of the method. As discussed in chapter 4.4, the Kubo approach leads to overestimates of the conductance near band edges. For systems, like the metallic SiNWs, where the bands lie close together, this leads to significant errors. The recursive GF method is more rigorous yielding the correct conductances at all energies. 84 8.1 Evaluation of the real-space Kubo method

Real-space Kubo Recursive Green’s function

X (N ) scaling (N : number of (N 3 ) scaling uc uc † O uc Oorbitals in unit cell)

X Incommensurate systems Requires periodic leads † (e.g. MWNT)

X ’Parallel’ in energy Full calculation for each energy † X Possibility of time-dependent Static model † potential (e.g. real space lattice vibrations)

Many convergence parametersa X No adjustable parameters † Numerically more diﬃcult X Numerically simple † Overshoot conductance near X Correct conductance † band edges

Slower for the thin wires X Faster for the thin wires † Requires orthogonal basis, X No L¨owdin transformation † L¨owdin transformation

aNtot: total system size, Ntri: size of the tridiagonal matrix, η: imaginary energy, Ni: size of random phase states, and the number of random phase states.

Table 8.1: Pros (X) and cons ( ) of the real-space Kubo and recursive Green’s function methods. †

The need for an orthogonal basis is another drawback of the Kubo method. The L¨owdin orthogonalization procedure results in longer ranging basis states and more non-zero Hamiltonian matrix elements making the computations slower. Altogether, we conclude that the recursive Green’s function method is the preferred choice for modelling thin SiNWs. For wires with a larger diameter and hence larger unit cells the GF method will be slower, and the Kubo approach might become the method of choice. Also, for incommensurate systems such as certain multi walled carbon nanotubes, it is possible to use the Kubo method as done in ref. [36], while the GF method can not be applied, since it requires periodic leads. 8.2 Modelling of silicon nanowires 85

8.2 Modelling of silicon nanowires

The models of SiNWs used in this work are all based on ab initio calculations. We believe this is crucial in modelling small diameter nanowires, since quantum effects are important. Moreover, using DFT calculations it is possible, and relatively straight forward, to model defects - in this work vacancies, but adatoms or dopants could also be included. This would complicate the use of standard tight-binding parameters. The results concerning the un-passivated, metallic wires revealed that the conduction at the Fermi energy is strongly affected by on-site noise at the surface atoms, whereas noise at the bulk atoms had little influence. From this we conclude that the conducting states are radially located at the surface atoms, in agreement with ref. [20]. This probably implies that such wires, if fabricated, would be very sensitive to any disturbances at the surface. This could possibly be used in sensor applications, if some molecule binds to the surface easier than others. However, it is probably more likely that the fragile surface construction is a drawback, since reproducible wires could be difficult to fabricate. Still, the metallic SiNWs are interesting from a scientific point of view. Very thin hydrogen passivated SiNWs have been fabricated by Ma et al. [10] and the second part of chapter 7 was concerned with similar wires. The effect of removing a single H atom was modelled using the DFT program siesta. By extracting Hamiltonians from both a pristine wire and from wires with different H vacancy, a long wire with randomly spaced vacancies is constructed and the transmission through it was calculated using the recursive Green’s function method. It was shown that the mean free path and the localization length both scale linearly with the average distance between two vacancies, in accordance with theoretical estimates. Resent results indicated ballistic transport in 10 nm Si/Ge core-shell nanowires at room temperature [17]. The acoustic phonon scattering rate was estimated using Fermi’s golden rule [17]

1 π k T Ξ2 = B n(E ), (8.1) ~ 2 F τap ρm vs where Ξ is the deformation potential, vs is the sound velocity, ρm is the mass density, and the density of states for the ﬁrst subband was calculated as

1 2 m n(E ) = ∗ , (8.2) F π2 ~ r2 E r F where m∗ is the eﬀective heavy hole mass and r is the wire radius. For Ge the mean free path is lap = vF τac = 540 nm [17], and a similar value holds for Si. Since the electron-phonon scattering rate decreases for wires with smaller diameters, even longer mean free paths might be expected in the very thin wires considered in this work. For carbon nantubes, the impurity scattering rate, on 86 8.3 Outlook the other hand, increases for decreasing radii as shown in chapter 4. If the same is true for silicon wires, and if similar values for the electron-phonon interaction apply to the wires considered here, defects could be the most dominant scattering source for small concentrations even at room temperature. A good understanding of the defect or dopant scattering will therefore be important for device engineering.

8.3 Outlook

The continously growing computer power increasingly enables theoretical physicists to model materials and explain or predict experiments. One might argue that the faster computers support the use of very accurate ab initio calculations while less accurate models such as tight-binding become irrelevant. However, the contrary might also be true, since an increase in computer power by a factor of 1000 allows to study systems only 10 times larger with (N 3) ab initio methods. On the other hand, with an (N) tight-binding modelO the system sizes can be 1000 times larger than today,O thus enabling atomistic models of real devices. Using the methods presented in this work, future studies could be to examine the diameter dependence of surface defect scattering. Another interesting investi- gation concerns dopants and the scattering caused by them. Relevant questions are: Where will the dopants be positioned? Does the scattering depend on the dopant location? What is the scaling of the mean free path with wire diameter for constant dopant concentrations? It might also be possible to model some of the chemical and biological sensors demonstrated in the literature. The charged molecules that bind to the surface give rise to a local field effect, which probably could be modelled by locally altered tight-binding parameters. Such models might be used to predict e.g. the diameter dependence of the sensitivity. The main focus in all cases will be to explain or predict qualitative behaviours and give estimates of quantities such as the mean free path or the mobility. Very accurate predictions and comparisons with experiments should probably not be expected. A very interesting extension of the methods would be to include electron-phonon interactions. Lattice vibrations have already been included in the real-space Kubo method by Roche et al. [39, 38], where the electron-phonon coupling is semi-classical and incorporated through distant depended hopping parameters. Inclusion of phonons in the recursive Green’s function method is a task we will be concerned with in future works. Another desirable extension would be to include a finite source-drain voltage in the calculations. To conclude, there is a great variety of interesting studies of nanowires using tight-binding like models based on ab initio calculations, and the future for a constructive interplay between experiments and theoretical modelling looks indeed very bright. Appendix Appendix A

L¨owdin transformation

The following pages describe how to construct an orthogonal basis set from a non-orthogonal one by means of a so-called L¨owdin transformation. We start by deriving a generalized eigenvalue equation from the Schr¨odinger equation, and continue by formally mapping the problem to a normal eigenvalue problem.

A.1 The generalized eigenvalue problem

The starting point for the analysis is the time independent Schrödinger equation Hˆ ψ = ǫ ψ , (A.1) | ni n| ni which together with some boundary conditions constitutes an eigenvalue equation. The space spanned by the eigenvectors, ψn is in general infinite dimensional. In order to perform calculations it is necessary| i to limit the problem to a finite dimensional space. We therefore assume to have a basis set α consist- {| ii} ing of N basis vectors. The basis set is in general non-orthogonal, αi αj = Sij . Notice that the overlap matrix S is real and symmetric and henceh Hermitian.| i This basis set could e.g. be an atomic, real-space basis set used in a density functional theory calculation. The limitation of the problem from an infinite to a finite dimensional space is a major simplification, and care must be taken. The validity of the approximation can be tested by including more basis states and check for convergence of the results. We now expand the eigenvector ψ in the atomic basis set: | ni N ψ φ (i) α , (A.2) | ni ≈ n | ii i X where φn(i) = αi ψn . The equality is only approximate since we only have a limited numberh of basis| i states. Inserting (A.2) in (A.1) yields another eigenvalue A.1 The generalized eigenvalue problem 89 equation: Hˆ φ (i) α = E φ (i) α , (A.3) n | ii n n | ii i i X X where in general En = ǫn and n N. In order for the basis set αn to be a good approximation6 there should≤ be reasonable correspondence{| betweeni} the eigenenergies En and ǫn in the relevant energy interval - usually around the Fermi energy. Operating with α from the left and denoting H = α Hˆ α yields h j | ij h j | | ii

Hji φn(i) = En Sji φn(i), i i X X which can be written as a matrix equation

H φn = EnS φn, (A.4)

T where the i’th element of the vector φn is φn(i): φn = [φn(1),...,φn(i),...] . Equation (A.4) constitutes a so-called generalized eigenvalue problem. Notice, in the special case where the basis set αi is orthogonal, the overlap matrix S I and we have a normal eigenvalue{| problem.i} → The matrix S is symmetric and positive deﬁnite, and as such it deﬁnes a metric T vector space [66] by the scalar product (y, x)s = y Sx. We shall now show that the eigenvectors φn are orthogonal with respect to this scalar product. From (A.4) we can write:

φm† H φn = Enφm† S φn (A.5)

φn† H φm = Emφn† S φm. (A.6)

Taking the hermitian conjugate of (A.6) yields

φm† H† φn = Em∗ φm† S† φn,

φm† H φn = Em∗ φm† S φn, (A.7) where we have used that both H and S are hermitian. Subtracting (A.7) from (A.5) gives

(E E∗ ) φ† S φ = 0, (A.8) n − m m n from which we deduce that En is real, since En = En∗, and that

φm† S φn = δnm. (A.9)

Equation (A.9) is the generalized version of the usual orthogonality relation for eigenvectors of a (generalized) eigenvalue problem. The eigenvectors in equation (A.4) are orthogonal in the metric space deﬁned by S. 90 A.2 Formal orthogonalization procedure

A.2 Formal orthogonalization procedure

We now proceed to show how to construct a basis set that is orthogonal in the usual metric. By writing S = S1/2 S1/2 we get from (A.4):

1/2 1/2 φn† S S φm = I 1/2 1/2 (S φn)† S φm = I ˜ ˜ φn† φm = I, (A.10)

1/2 where we have deﬁned the orthogonalized vectors φ˜n = S φn. In order to get from line 1 to 2 we have used that S1/2 is real and symmetric (this will be 1/2 1/2 shown below) wherefore it holds that (S )† = S . We now return to equation (A.4), which we rewrite using the new basis states 1/2 1/2 together with the relation I = S− S :

1/2 1/2 1/2 1/2 H S− S φn = EnS S φn 1/2 1/2 H S− φñ = EnS φñ 1/2 1/2 1/2 1/2 S− H S− φñ = En S− S φñ

H˜ φ˜n = En φ˜n. (A.11)

We have thus transformed our generalized eigenvalue problem (A.4) to a normal eigenvalue problem by mapping the original Hamiltonian onto the new one:

1/2 1/2 H H = S− H S− . (A.12) →

Before we show how to performe the orthogonalization (A.12) in practice, we brieﬂy return to the atomic real-space basis set and show how to construct a new orthogonal basis. From (A.12) we get

1/2 1/2 Hij = S−il Hlk S−kj Xkl 1/2 1/2 e = S− α Hˆ α S− il h l| | ki kj Xkl = α Hˆ α , (A.13) h i | | j i where we have deﬁned the new, orthogonale e basis α as {| j i}

1/2 e α = S− α . (A.14) | j i kj | k i Xk e A.3 How to do the orthogonalization 91

To show that α is an orthogonal basis, we simply calculate the overlap: {| j i} 1/2 1/2 α α = α S− S− α eh i | i i h k | ik lj | l i Xkl 1/2 1/2 e e = S− α α S− ik h k | l i lj Xkl 1/2 1/2 = S−ik Skl S−lj Xkl 1/2 1/2 = S− S S− = Iij = δij, (A.15) ij which shows that α indeed is an orthogonal basis. {| j i} e A.3 How to do the orthogonalization

1/2 We now show how to ﬁnd the matrix S− . We consider the overlap matrix S and assume to know the eigenvalues λ1,...,λn and eigenvectors v1,...,vn, which fulﬁll: S vi = λi vi.

Let V be the eigenmatrix where the i’th column is the eigenvector vi, such that

S V = V diag(λ1,...,λn). (A.16)

Since S is real and symmetric, the eigenvectors are real and orthogonal, and it thus holds that VT V = I. (A.17) The eigenvectors furthermore constitutes a complete set and it therefor also holds that V VT = I. (A.18) ˆ This is just the matrix version of the more familiar relation I = i ψi ψi . Using the orthogonality relation it follows from (A.16) that | ih | P T S = V diag(λ1,...,λn) V . (A.19)

Writing diag(λ1,...,λn) = diag(√λ1,..., √λn) diag(√λ1,..., √λn) and using the orthogonality condition we rewrite (A.19) as

T S = V diag( λ1,..., λn) diag( λ1,..., λn) V T T S = V diag(pλ1,..., pλn) V Vpdiag( λp1,..., λn) V , from where it follows thatp p p p

1/2 T S = V diag( λ1,..., λn) V , (A.20) p p 92 A.3 How to do the orthogonalization which shows that S1/2 indeed is real and symmetric, as claimed above. Since 1 1 1 1/2 diag(√λ1,..., √λn) − = diag( ,..., ), we get S− from (A.20) by √λ1 √λn multiplying from the right with V, diag( 1 ,..., 1 ) and VT as follows: √λ1 √λn

1/2 1/2 S− S = I 1/2 T S− V diag( λ1,..., λn) V = I

1/2 1 1 T p p S− = V diag( ,..., ) V ,(A.21) √λ1 √λn where we again have made use of the orthogonality- and completeness relations (A.17) and (A.18). Appendix B

Numerical Methods

B.1 The Continued Fraction Technique

In this appendix we show how to calculate diagonal elements of Green’s functions by means of the continued fraction technique (CFT) and the recursion method. This method is very useful when calculating the local density of states (LDOS).

Consider a tridiagonal, hermitian Hamiltonian H with diagonal elements αi and oﬀ diagonal elements βi:

α1 β1 0 ...... β1 α2 β2 0 ......   0 β2 α3 β3 0 ......  . .. ..   . 0 β3 . .  H =   . (B.1)  . . . . .   . . 0 ......     ......   ......     . . . . .   ......      For a general Hamiltonian that is not tridiagonal it is always possible to bring it to the tridiagonal form using a mapping described in appendix B.2. We which to ﬁnd the ﬁrst diagonal elements of the Green’s matrix 1 G = (E I H)− , (B.2) − 1 i.e. we should calculate G = (E I H)− . 11 − 11 From linear algebra we know that this inversion problem can be solved by use of Cramers Theorem [66]:

1 i+j Dji A− = ( 1) , (B.3) ij − det A 94 B.1 The Continued Fraction Technique

where A is a quadratic matrix and Dji is the (j, i)’th sub-determinant belonging to A. It is the determinant of the matrix obtained by removing the j’th row and the i’th collumn from A. Consider now the matrix E α β 0 ...... − 1 − 1 β1 E α2 β2 0 ......  −0 −β E− α β 0 ......  − 2 − 3 − 3  . .. ..   . 0 β3 . .  E I H =  −  . (B.4) −  . . . . .   . . 0 ......     ......   ......     . . . . .   ......      Using that a determinant in general can be written as

det A = ai1Ai1 + ai2Ai2 + ai1Ai1 . . .

i+j where aij is the (i, j)’th element in A and Aij = ( i) Dij , it is readily seen that we have − det (E I H) = (E α ) det (H ) β2 det (H ) − − 1 1 − 1 2 , where Hn is the matrix obtained by removing the ﬁrst n rows and columns from E I H. Since Hn has the same tridiagonal structure as H we have in general − det (H ) = (E α ) det (H ) β det (H ) n − n n+1 − n+1 n+2 K = (E α )K β2 K (B.5) n − n n+1 − n+1 n+2 , where we have introduced the notion Kn = det (Hn). From (B.3) we get: K K 1 G (E) = 1 = 1 = . (B.6) 11 2 2 K2 det (E I H) (E α1)K1 β K2 (E α ) β − − − 1 − 1 − 1 K1 By recursive use of (B.5) and (B.6) we obtain the continued fraction 1 G11(E) = (B.7) β2 E α 1 − 1 − β2 E α 2 − 2− β2 E α − 3 3 . − ..

B.1.1 Truncation of the continued fraction - self-energy

In many problems the considered Hamiltonian might be of infinite dimensions, wherefore the continued fraction contains infinitely many terms. Fortunately it B.1 The Continued Fraction Technique 95 often suffices to include a limited number of terms without affecting the result. We shall therefor assume that for a given N, all the coefficients are converged, such that αn = α and βn = β for n N. The infinitely many remaining terms of the continued∞ fraction can∞ then≥ be analytically summed up to give a self-energy, Σ(E). We thus have 1 G11(E) = (B.8) β2 E α 1 − 1 − β2 E α 2 − 2− . .. E α β2 Σ(E) − N − N The self energy is found from 1 1 Σ(E) = = . (B.9) β2 E α β2 Σ(E) ∞ E α ∞ 2 − − ∞ − ∞ − β E α ∞ − ∞ − . ..

Solving this second-degree equation leads to an expression for the self-energy

E α i 4β2 (E α )2 Σ(E) = − ∞ − ∞ − − ∞ , E E E , (B.10) 2β2 min ≤ ≤ max p ∞ where Emin and Emax deﬁnes the spectrum of the Hamiltonian.

How many terms are needed in the continued fraction in order for G11(E) to converge depends on the speciﬁc problem, and in general it should be tested whether or not the values of αn and βn have converged. 96 B.2 Tridiagonalization procedure

B.2 Tridiagonalization procedure

Most often the Hamiltonian has not the tridiagonal form as in (B.1). It is, however, always possible to perform a change of basis such that the Hamiltonian in the new basis is tridiagonal. In other words, there always exist a mapping from the original problem to a generalized one-dimensional chain, described by a tridiagonal matrix. By ’generalized’ we understand a chain, where the onside elements εi = εj and the hopping terms γij = γi′j′ . We shall now demonstrate how the tridiagonalization6 procedure is carried6 out. Given a Hamiltonian matrix, H, described by an initial orthogonal basis n . We seek a new orthogonal basis, denoted m , that tridagonalizes the Hamiltonian.| i Usually we wish to perform the tridiagonalization| } to calculate the local density of states at a given site i . We therefore choose the ﬁrst element in the new basis to be equal to the considered| i site:

1 i . | } ≡ | i In the new basis all the matrix elements m H 1 must vanish except 1 H 1 and 2 H 1 . Operating with H on 1 we{ thus| | get} { | | } { | | } | } H 1 = α 1 + β 2 . | } 1| } 1| }

Since the new basis states must be orthogonal, we find the coefficient α1 = 1 H 1 . The second basis vector 2 is found from { | | } | } β 2 = H 1 α 1 = 2˜ 1| } | } − 1| } | } and the coefficient β is determined from the normalization β = 2˜ 2˜ . 1 1 { | } This procedure can be iterated and in general, for n > 1 we have q

H n = βn 1 n 1 + αn n + βn n + 1 . (B.11) | } − | − } | } | } Again, the coeﬃcient α = n H n , while n + 1 and β are determined n { | | } | } n from the vector H n βn 1 n 1 αn n and the normalization condition n + 1 n + 1 = 1.| } − − | − } − | } { | } B.3 Exact solution to the time-dependent Schr¨odinger equation 97

B.3 Exact solution to the time-dependent Schr¨odinger equation

The time evolution of a quantum mechanical state is governed by the time- dependent Schrödinger equation (TDSE), which together with a boundary condition constitutes a first order differential equation:

∂ Hˆ ψ(t) = i~ ψ(t) , ψ(t = 0) = ψ , (B.12) | i ∂t| i | i | 0i with Hˆ being the Hamiltonian operator, which we shall assume to be time independent. The solution to (B.12) is found by simple integration:

i Htˆ ψ(t) = Uˆ(t) ψ = e− ~ ψ , (B.13) | i | 0i | 0i i Htˆ where we have introduced the time-evolution operator Uˆ(t) = e− ~ . Although the formal solution in (B.13) is very simple, it does not help us, actually ﬁnding the state ψ(t) at time t. In order to proceed, we assume to know the eigenstates of the Hamiltonian:| i Hˆ E = E E . (B.14) | ni n | ni Since Hˆ is a hermitian operator, the eigenstates constitutes a complete, orthonormal basis, and the following completeness condition holds

E E = I. (B.15) | nih n| n X

Using (B.15) we get from (B.14) , by operating with En from the right and summing over the states: h |

=I ˆ k iHtˆ ∞ ( iHt) Uˆ(t) = e− = − E E k! | nih n| n kX=0 Xz }| { ∞ ( iEt)k = − E E k! | nih n| n X kX=0 iEn t = E E e− (B.17) | nih n| n X 98 B.3 Exact solution to the time-dependent Schr¨odinger equation

Equation (B.17) simply uses the superposition principle: A given state is projected into the energy-eigenstates which evolves with a simple phase oscillation. We shall now rewrite equation (B.17) to make it easier to implement numerically. In all numerical problems in this work, the Hamiltonian operator, Hˆ , is replaced by a Hamiltonian matrix H. Let V be the unitary matrix, with the eigenvectors E as columns that diagonalizes H: | ni

V† H V = diag(E1, E2,...,En,...)

Since V V† = V† V = I, we rewrite equation (B.17) as

iHt iEn t e− = V V† E E V† V e− . | nih n| n X T Note that V† En = [0,... 0, 1, 0,...] = 1n , where the 1 is at the n’th position. We thus get | i | i

iHt iEn t e− = V 1 1 V e− | nih n| n X iE1t iE2t iEnt = V diag e− , e− ,...,e− ,... V†. (B.18)

This expression for the time-evolution operator, in terms of matrices is easy to implement numerically once the eigenstates and eigenenergies are known.

B.3.1 Time propagation using Taylor expansion

Although we formally have found the exact solution to the TDSE, this approach is in general very time consuming since we need to diagonalize Hˆ . To avoid the diagonalization, approximations have to made. The most natural way to approximate the time evolution operator is probably to expand it in a Taylor series:

2 N i Hˆ ∆t 1 1 Uˆ(∆t) = e− 1 iHˆ ∆t Hˆ ∆t + . . . + iHˆ ∆t = Uˆ (∆t). ≈ − − 2 N! − N (B.19) The Taylor expansion is, however, only eﬃcient for short time durations, that is small ∆t. The total propagation is therefor obtained by k successive operations with the approximated:

Uˆ(t ) Uˆ (∆t)Uˆ (∆t) . . . Uˆ (∆t), 1 ≈ N N N such that t1 = k∆t. B.4 Recursive calculation of self-energies 99

B.4 Recursive calculation of self-energies

In this appendix we show how to calculate the self-energy of the ﬁrst cell in a semi-inﬁnite wire. The procedure follows ref. [67]. The Green’s function, G, for the full wire is the solution to the equation

(w H) G = I, (B.20) − where H is the full Hamiltonian for the wire, and w should, where ever it appear, be understood as wI, with the dimension of I given by the context. The wire is illustrated in ﬁgure B.1 and consists of inﬁnitely many equal unit cells described (0) (0) (0) (0) with the Hamiltonian h and coupling matrices α and β = (α )†. The super-scripts are included for later convenience.

β(0) β(0) β(0)

(0) (0) (0) (0) h h h h0

α(0) α(0) α(0)

Figure B.1: Semi-inﬁnite wire with equal unit cells described by the on-site Hamiltonian, h(0), and coupling matrices α(0) and β(0).

We seek the element G00 of the full Green’s function, describing the interaction of the zeroth cell with the rest of the wire. We therefore consider the zeroth column of (B.20), and for clarity we denote the unit cell Hamiltonian for the (0) zeroth unit cell (at the end of the wire) h0 :

(0) (0) w h0 α 0 . . . G00 I − (0) − (0) (0) β w h α 0 G01 0  − − − (0) (0)    =   . (B.21) 0 β w h α G02 0  . − − −.   .   . (0) ..   .   0   . 0 β       −      From all odd-numbered rows in (B.21) we isolate G0n, n = 1, 3, 5,... , yielding

(0) (0) (0) (0) G0 n = g β G0 n 1 + g α G0 n+1, − (0) (0) 1 where g = (w h )− . This is inserted into the even numbered rows yielding, for n = 0 − 6 (1) (1) (1) β G0 2(n 1) + (w h )G0 2n α G0 2(n+1) = 0, − − − − 100 B.4 Recursive calculation of self-energies and for n = 0 (w h(1))G α(1) G = I, − 0 00 − 02 where we have deﬁned

β(1) = β(0) g(0) β(0) α(1) = α(0) g(0) α(0) h(1) = h(0) + β(0) g(0) α(0) + α(0) g(0) β(0) (1) (0) (0) (0) (0) h0 = h0 + α g β .

This gives another, renormalized semi-inﬁnite chain, and we can continue the same procedure by removing every second row. After j iterations, the Green’s function equation for the zeroth cell becomes

(j) (j) w h0 α 0 . . . G0 0 I − (j) − (j) (j) β w h α 0 G0 1 2j 0  − − − (j) (j)   ·  = , (B.22) 0 β w h α G0 2 2j   · 0  . − − −.   .   . (j) ..   .   0   . 0 β       −      where

β(j+1) = β(j) g(j) β(j) α(j+1) = α(j) g(j) α(j) h(j+1) = h(j) + β(j) g(j) α(j) + α(j) g(j) β(j) (j+1) (j) (j) (j) (j) h0 = h0 + α g β 1 g(j) = w h(j) − . − The point is now, that α(j) 0 and β(j) 0 for j , such that we end up with a block diagonal matrix→ with cells that→ do not→ couple ∞ to each other, and therefore we can compute G00 as

1 1 (j) − (0) − G00 = lim w h0 = w h0 Σ , j − − − →∞ where the last equality deﬁnes the self-energy, Σ, which is computed as

(j) (0) Σ = lim h0 h0 . j →∞ − Appendix C

Miscellaneous

C.1 Total and Local Density of States

In the numerical calculations we shall repeatedly calculate both local and total densities of states. The density of state for a system described by the Hamil- tonian matrix H is defined by: n(E) = Tr δ(E H). (C.1) − The physics in this definition is more clearly seen if the trace is performed over the eigenstates, ψ , of the Hamiltonian: | i i n(E) = ψ δ(E H) ψ h i | − | i i i X = δ(E ǫ ), − i i X i.e. for a discrete set of eigenenergies, ǫi, the density of states is a set of delta- peaks. The local density of states of an arbitrary vector ν is defined as | i d (E) = ν δ(E H) ν . (C.2) ν h | − | i ˆ By inserting I = i ψi ψi , where the sum is over the eigenstates, ψi , of H, we rewrite (C.2) as| ih | | i P d (E) = ν ψ ψ δ(E H) ν ν h | | i ih i | − | i i ! X = ν ψ δ(E ǫ ) ψ ν h | i i − i h i | i i X = ν ψ 2 δ(E ǫ ), (C.3) |h | i i| − i i X 102 C.1 Total and Local Density of States i.e. the local density of the state ν is a sum of delta-peaks positioned at the eigenenergies each weighted by the| i squared magnitude of the corresponding eigenfunction projected onto ν . | i If we sum over all the local densities of states of the vectors in an arbitrary, complete basis, µ , we get the total density of states: {| j i} d (E) = µ ψ 2 δ(E ǫ ) µj |h j | i i| − i j j i X X X = δ(E ǫ ) = n(E), (C.4) − i i X where we have used that the eigenstates are normalized, i.e. µ ψ 2 = 1. j |h j | i i| So the local densities of states can be obtained from (C.3)P if we know the eigenstates. However, this is in general not the case, so it is advantageous to express the local density of state in terms of a Green’s function.

C.1.1 Green’s function and local density of states

We deﬁne the retarded Green’s function matrix, G(E), in the energy domain as the solution the equation

[(E + iη)I H] G(E) = I − 1 G(E) = [(E + iη)I H]− , (C.5) − where the infinitisimal η is needed for convergence. It corresponds to a finite with of the delta peaks in the density of states, and in real systems coupled to e.g. leads it is replaced by a finite self-energy. In the time-domain the Green’s function is given by i H t G(t) = iθ(t) e− , (C.6) − which is seen by Fourier transformation. We can formally write the Hamiltonian in terms of the eigenstates and eigenenergies

H = ǫ ψ ψ = Ψ diag(ǫ , ǫ ,...)Ψ†, (C.7) i| i ih i | 1 2 i X where the i’th column of Ψ is the eigenvector ψi . Using this we proceed to ﬁnd an arbitrary element of the Green´s function| matrix:i

iHt G (t) = iθ(t) e− lm − lm ∞ 1 n = iθ(t) i Ψ diag(ǫ , ǫ ,...)Ψ†t . − n − 1 2 n=0 !lm X C.1 Total and Local Density of States 103

Using that ΨΨ† = I we get

∞ 1 n G (t) = iθ(t) Ψ ( i diag(ǫ , ǫ ,...)t) Ψ† lm − n − 1 2 n=0 ! X lm iǫit = iθ(t) ψ (l)e− ψ∗(m), (C.8) − i i i X where ψi(k) is the k’th element in the i’th eigenvector. Fourier transformation of (C.8) gives the (l,m)’th element of the Green´s function matrix in the energy domain:

∞ i(E+iη)t iǫit G (E + iη) = dt e ( i)θ(t) ψ (l) e− ψ∗(m) lm − i i i Z−∞ X ψ (l) ψ (m) = i i . (C.9) E + iη ǫ i i X − 1 Using the relation Im limη 0 = π δ(x) [68] we ﬁnally arrive at → x+iη − h i 1 Im lim Gll(E + iη) = ψi(l) ψi∗(l)δ(E + ǫi) −π η 0 → i X = ψ (l) 2 δ(E + ǫ ) = d (E). (C.10) | i | i l i X The local density of of site l can therefore be found from the imaginary part of the l´th diagonal element of the Green´s function. This is the way we in general shall use to calculate the local densities of states, applying the continued fraction technique (see appendix B.1) to evaluate the diagonal elements of the Green’s function. 104 C.2 Ohm’s law and mean free paths

C.2 Ohm’s law and mean free paths

In this appendix we will use a simpliﬁed model to obtain expressions for the length scaling of the resistance. The derivation which follows the procedure described Datta [43] is semi-classical neglecting the phase of the electrons. We consider a sample with a single mode and two successive scatterers with transmission probabilities T1 and T2 and reﬂection probabilities R1 and R2. The combined transmission probability is given by [43] T T T = 1 2 . (C.11) 1 + R1R2

Current conservation implies R1 = 1 T1, R2 = 1 T2 which we use to rewrite (C.11) − − 1 T 1 T 1 T − = − 1 + − 2 , (C.12) T T1 T2 1 T showing that when combining scatterers the quantity −T has an additive prop- erty. Combining N scatterers in series, each with transmission probability T we get 1 T (N) 1 T T − = N − T (N) = . T (N) T ⇒ N(1 T ) + T − Deﬁning the density of scatterers as ν = N/L we get

T T ν(1 T ) L0 − T (N) = = T = , (C.13) νL(1 T ) + T L + ν(1 T ) L + L0 − − where we have defined L = T . If we assume T 1, L is of the order of 0 ν(1 T ) ≈ 0 the mean free path, l , since the− latter fulfills (1 T )ν l 1. For a wire with e − e ∼ N modes, the mean free path is le N/ν(T0 T1), where T0 and T1 are the transmissions of a pristine and defected∼ wire respectively.− The resistance given by the Landauer formula (5.10) can be written as h h 1 T R = + − , (C.14) 2e2 2e2 T where the first term is the contact resistance and the second term the ’scattering resistance’. Inserting (C.13) in (C.14) yields Ohms law: h h L h h L R(L) = 2 + 2 2 + 2 . (C.15) 2e 2e L0 ≈ 2e 2e le In the Ohmic regime, where the resistance increases linearly with length (R(L) = β αL + β) we can extract the mean free path as le = α . In chapter 7 we will use this to estimate the mean free path in silicon nanowires with various disorder. Bibliography

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