Electronic Properties of Mos2 Monolayers with Vacancies

UNIVERSIDADE FEDERAL DO RIO DE JANEIRO

INSTITUTO DE F´ISICA

Electronic Properties of MoS2 Monolayers with Vacancies

F´abioRangel Duarte Filho

Disserta¸cãode Mestrado apresentada ao Programa de Pós-Gradua¸cãoem F´ısicado Instituto de F´ısicada Uni- versidade Federal do Rio de Janeiro - UFRJ, como parte dos requisitos necessáriosàobten¸cãodo t´ıtulode Mestre em Ciências(F´ısica).

Orientadora: Tatiana G. Rappoport Coorientador: Rodrigo B. Capaz

Rio de Janeiro Agosto de 2020

iii

Resumo

Electronic Properties of MoS2 Monolayers with Vacancies

F´abioRangel Duarte Filho

Orientadora: Tatiana Gabriela Rappoport Coorientador: Rodrigo Barbosa Capaz

Resumo da Disserta¸cão de Mestrado apresentada ao Programa de Pós- Gradua¸cãoem F´ısicado Instituto de F´ısicada Universidade Federal do Rio de Janeiro - UFRJ, como parte dos requisitos necessários àobten¸cãodo t´ıtulode Mestre em Ciências(F´ısica).

Alavancada pela crescente busca por materiais planares, a pesquisa sobre Dicalco- genetos de Metal de Transi¸cão(TMDs) em monocamadas chamou a aten¸cãopara um composto interessante, o MoS2. Devido a sua ampla gama de propriedades mecânicase fotoeletrônicas,esse TMD se tornou um tópicorecorrente de pesquisa nos últimosanos e se mostrou útilem muitas aplica¸cões.No entanto, devido ao processo de fabrica¸cãodesse composto e sua poss´ıvel intera¸cãocom metais reativos, defeitos em forma de vacância surgem e modificam as propriedades opto-eletrônicasdesse material. Neste estudo, uma rede com milhõesde orbitais ésimulada usando o software HPC de transporte quântico KITE. A partir de parâmetrosvindos de calculos ab-initio, foram feitas simula¸cõessob um modelo de tight-binding do MoS2 com 13 a 26 orbitais em uma rede de 512x512 s´ıtiospermitindo, assim, cálculosde quantidades espectrais como DOS e condutividade com alta fidelidade. Isso nos permite entender os efeitos de defeitos das recorrentes vacânciasem sistemas com e sem acoplamento spin-órbita. iv

Palavras-chave: dicalcogenetos de metais de transi¸cão, KITE, MoS2, quantidades espectrais, materiais bidimensionais, simula¸cão,tight-binding, TMD, transporte eletrônico,

2D. v

Abstract

Electronic Properties of MoS2 Monolayers with Vacancies

F´abioRangel Duarte Filho

Advisor: Tatiana Gabriela Rappoport

Coadvisor: Rodrigo Barbosa Capaz

Abstract da Disserta¸cão de Mestrado apresentada ao Programa de Pós- Gradua¸cãoem F´ısicado Instituto de F´ısicada Universidade Federal do Rio de Janeiro - UFRJ, como parte dos requisitos necessários àobten¸cãodo t´ıtulode Mestre em Ciências(F´ısica).

Leveraged by the growing search for planar materials, research on monolayer Transition

Metal Dichalcogenides (TMDs) drew attention to a most interesting compound, MoS2. Due to its wide range of mechanical and photoelectronic properties, this TMD has become a recurrent research topic in recent years and has proved useful in many applications. However, due to the manufacturing process of this compound and its possible interaction with reactive metals, defects in the form of vacancy arise and modify the optoelectronic properties of this material. In this study, a lattice containing millions of orbitals is simulated using the KITE quantum transport HPC software. Based on parameters from ab-initio calculations, simulations were made under a MoS2 tight-binding model with 13 to 26 orbitals in a 512x512 site lattice, thus allowing calculations of spectral quantities such as DOS and conductivity with high ﬁdelity. This allows us to understand the defect eﬀects of recurring vacancies in systems with and without spin-orbit coupling. vi

Keywords: electronic tranport, KITE, monolayer material, MoS2, simulation, spectral quantities, tight-binding, TMD, transition metal dichalcogenides, 2D. vii

Agradecimentos

Agrade¸coprimeiramente aos meus antepassados cujo esfor¸cocoletivo me deu a opor- tunidade de estar aqui. Dentre eles, agrade¸coespecialmente aos meus pais, que tiveram uma influênciadireta na minha existênciae educa¸cão.Influênciaa qual perdura mesmo sem a presen¸caf´ısica. Em seguida, agrade¸coaos sábiose cientistas do passado que trouxeram o conhecimento da humanidade a tal ponto que, utilizando seus métodos, foi poss´ıvel alcan¸caros resultados desse estudo. Em especial, agrade¸coàTatiana Rappoport pela excelente orienta¸cãoe pelo incansável esfor¸copara me ajudar na minha forma¸cãointegral como cientista e f´ısico. Agrade¸cotambémRodrigo Capaz por aceitar ser o meu coorientador e ao AndréSaraiva por ter me sugerido esses como orientadores no fim da minha gradua¸cão. Têma minha gratidãoos colaboradores desse projeto – A. Molina-Sanchez da INL e L. Canonico da UFF – que contribu´ıramcom dados, tempo e instala¸cões. Com isso, agrade¸copor fim àagênciade fomento CAPES que financiou esse projeto me concedendo a bolsa de pós-gradua¸cão. viii

Contents

Contents viii

List of Figuresx

1 Introduction1

1.1 MoS2 Atomic and Electronic Structure...... 4 1.2 Disorder...... 6

2 Methods9

2.1 Tight-binding Method...... 10

2.1.1 Fundamentals...... 11

2.1.2 The MoS2 Tight-biding Model...... 13 2.2 Wannier 90...... 15

2.3 Kernel Polynomial Method...... 18

2.3.1 Chebyshev Polynomials Expansion...... 19

2.3.2 Kernel Polynomials...... 25

2.3.3 Calculation of DOS...... 27

2.3.4 Green’s Function Chebyshev Polynomial Expansion, an Alternative to KPM...... 28

2.4 KITE...... 31

2.4.1 Adding Disorder...... 32

2.4.2 Parallelization...... 33 ix

3 Calculation Procedure 36 3.1 Conﬁguration Parameters...... 36

3.1.1 The Lattice...... 37 3.1.2 Choosing the Kernel...... 37 3.1.3 Assuring Eﬃciency - System Size & Number of Divisions...... 38

3.1.4 Calculation Parameters...... 38 3.1.5 Random Vectors...... 40 3.2 Graphene Density of States...... 41

3.3 Longitudinal Conductivity...... 44 3.3.1 Kubo Formulas...... 44 3.3.2 Eﬃciency...... 46

3.3.3 Results...... 47

4 Results 49 4.1 System Parameters...... 50

4.2 MoS2 Density of States...... 52

4.3 MoS2 Conductivity...... 53 4.4 Super-cell...... 55

4.5 Discussion...... 59

5 Conclusion 65

ReferênciasBibliográficas 67

A Resolution - Broadening vs KPM 85

B Number of Moments 90

C Graphene Example Code 93 x

List of Figures

1.1 MoS2 in 1H structural phase TMD illustration (purple sphere = Mo, yellow sphere = S) of: a) and b) side view. c) top-down view with unit cell. d) Standard orientation of crystal shape showing two layers and the bonding nature - Van der Waals for the inter-layer interaction and covalent for inter-

atomic interactions (bi-color cylinder)...... 3

1.2 The red arrow indicates the bandgap jump. In a) we have the calculated

bulk band structure with the indirect bandgap between D point and C point. In b) we have the single-layer one with the direct bandgap between A point and B point...... 5

1.3 Illustration of valley coupling. Schematics of low-energy band structure for

monolayer MX2. Red (blue) curves represent bands with spin up (down).

The black dashed line shows the Fermi energy EF measured from the middle of the gap...... 6

2.1 Normalized Chebyshev Polynomials of the ﬁrst kind of degree n from 0 to 5. 20

2.2 a) Positioning the mask with a onsite disorder ”over” our system. b) Eﬀect of the onsite disorder in our lattice. The greyed-out atom represents the

Mo atom with changes in its eigen potentials, matrix diagonal terms.... 34 xi

2.3 a) Positioning the mask with a structural disorder ”over” our system. b) Eﬀect of the structural disorder in our lattice in case the atom self potentials

and hopping potentials are changed. b) Eﬀect of the disorder in our lattice in case of vacancy in such atom. The greyed-out atom with the greened- out hopping interactions represents the Mo atom with changes in its self

potentials and interactions, any of its matrix terms...... 35

3.1 Graphene lattice with vectors a = aC and b = bC . a) Up-down view (”cC ” vector), the outlined area shows the primitive unit cell. b) Birds eye view. 37

3.2 512×512 unit cell grahene system, 512 moments DOS...... 41

3.3 Illustration of the graphene structure with the same mask applied over ev-

ery unit cell. Half of the carbon atoms (brown color) are getting disordered (green color) by some amount given by the chosen distribution...... 42

3.4 Same system as in Figure (3.2) but with Gaussian on-site disorder..... 42

3.5 Removing 10% of carbon atoms of graphene lattice...... 43

3.6 Graphene with 10% vacancy...... 43

3.7 Conductivity comparison with DOS in the cases of - a) Graphene without disorder; b) 10% vacancy disordered graphene. Here the conductivity calculation was not made on the entire system, but only on the part of it near

or inside the gap...... 48

4.1 MoS2 lattice with vectors a = aMoS2 and b = bMoS2 . a) Up-down view

(”cMoS2 ” vector), the outlined area shows the primitive unit cell. b) Bird’s eye view...... 51

4.2 Comparison between the Pybinding tight-binding band structure given our parameters (green) and the DFT calculated one (blue)...... 51

4.3 a) Pristine 512×512 unit cells, 1024 moments MoS2. b) Comparison between pristine and disordered system...... 52 xii

4.4 Multiple concentrations of vacancies on Mo with concentrations of a) 0.2% to 1% disordered system. b) Highly disordered - 0.8% to 2%, the arrow

indicates the extra peak observed in high concentrations)...... 53

4.5 Multiple concentrations of vacancies on both S independently with concentrations of a) 0.2% to 1% disordered system. b) Highly disordered - 0.8%

to 2%, the arrow indicates the extra peak observed in high concentrations. 53

4.6 Multiple concentrations of vacancies on Mo and on both S independently

with concentrations of a) 0.2% to 1% disordered system. b) Highly disordered - 0.8% to 2%. Here no discernible peak shows itself in the more disordered cases, suggesting that the peaks pointed by the arrows in the

previous cases were but the interference of S vacancies on Mo atoms or vice-versa. That is, when the vacancy concentration for one speciﬁc atom is high enough, we are expected to see disorder eﬀects on unit cell level

in some occurrences. To isolate these eﬀects, we are going to consider concentrations of 1% or lower...... 54

4.7 Conductivity comparison between pristine, 1% vacancies on S, 2% vacancies on S - a) In scale with the system conduction band ity; b) Zoomed in the gap - see scale...... 54

4.8 Conductivity comparison between pristine, 1% vacancies on Mo, 2% vacancies on Mo - a) In scale with the system conduction band conductivity;

b) Zoomed in the gap...... 55

4.9 Conductivity comparison between pristine, 1% vacancies on MoS2, 2% vacancies on MoS2 - a) In scale with the system conduction band conductiv-

ity; b) Zoomed in the gap...... 55

4.10 Super-cell arrangement in comparison with primitive cell. a) 9×9 primitive

cells. b) 3×3 super-cells...... 56 xiii

4.11 Diﬀerence between hopping reach in terms of unit cell in the cases: a) primitive cells, b) super-cells. Only the hoppings for the central Mo atom

are shown not to clutter the image...... 57

4.12 Illustration of possible quasi-crystalline arrangement...... 58

4.13 Comparison between diﬀerent unit cells - primitive unit cell (original) and super-cell...... 59

4.14 Comparison between super-cell and pristine-cell systems under the same disorders; highly disordered on the right. a) and b) - disorder on S. c) and

d) - disorder on Mo. e) and f) - disorder on both. The arrows mark the point of extra peaks in the super-cell mode in comparison with pristine cell. 60

4.15 Comparison between super unit cell conductivity (with a slightly higher broadening parameter η) and pristine conductivity of a system with 1% concentration of vacancies both on S and Mo orbitals. Overall the peaks

are seen in the same positions...... 61 xiv

4.16 For a) and b) we have a band structure of MoS2 with 0% and 1.6% concentrations of S vacancy respectively calculated on a 9×9 system via DFT

with GGA approximation. The red arrow and the number (in eV) show the energy level oﬀset between unoccupied defect state and conduction band bottom for each band structure. Simulation made with the Vienna ab-initio

simulation package. Next we have atom-projected density of states for the monolayer MoS2 with: c) Mo vacancy and d) S vacancy. These were calculated on a 7×7 FETs

system via DFT with LDA approximation. The system is commposed of monolayer MoS2 encapsulated by alumina (Al2Ox) and hafnia (HfOx). Several states are introduced within the nominal bandgap and signiﬁcant

distortion of the valence band edge structure is observed. As for e) and f), we have DFT calculations for band structure (e) and partial density of states (f) for single-layer MoS2 5×5 supercell with a sulfur

vacancy . The localized states are highlighted by red lines. Green dashed line corresponds to the case without vacancy...... 62

4.17 Comparison between SOC super-cell (continuous) and no-SOC super-cell (dashed) with 0.5% vacancies on Mo (red and orange) or S(green and olive).

Simulation on a 512×512 super-cell system with 3072 moments - the system is needed to be large enough so that the SOC splitting peaks can be resolved. 63

A.1 13 orbital, 512 moments MoS2 simulation with - a) Green Function approach with decreasing η broadening parameter; b) KPM approach with Jackson’s Kernel...... 86

A.2 Comparing Green’s Kernel with KPM-Jackson’s Kernel inside the gap, i.e.

expecting DOS=0...... 87 xv

A.3 In gap comparison between Green and Jackson’s with varying Chebyshev Moments - a) 512 moments - under ﬁt; b) 1024 moments - good ﬁt; c)

1536 moments - light over ﬁt. (refer ﬁt quality to Relation (B.1)...... 87 A.4 In gap comparison between Green and Jackson’s with varying Chebyshev Moments - a) 512 sites, 1024 moments; b) 1024 sites, 2048 moments.... 88

A.5 In-gap comparison between Green and Jackson’s with varying η on a system with vacancy on S atoms...... 89

B.1 512x512 site simulation of MoS2, with 1% vacancy disorder on S atoms... 91 B.2 The same system as in Figure (B.1)...... 92

B.3 512x512 site simulation of graphene with vacancy disorder on 10% of atoms. Diﬀerent values for N - from 2−3D to 23D. Energy normalized for the hopping therm...... 92 xvi 1

Chapter 1

Introduction

Transition Metal Dichalcogenides

The discovery of graphene along with all its applications and properties led to the start of a hunt for new planar materials with similar or complementary uses [1,2]. By 1935, it was argued by Peierls and Landau that the free-standing two-dimensional sheet of atoms could not exist in nature as the long-range order would break down due to the thermal ﬂuctuations and meltdown the 2D sheet into a three-dimensional bulk [3]. Graphene, then, emerged with a diﬀerent physics that bridge quantum electrodynamics and condensed matter physics [4], which would open the path for investigation of a larger number of 2D layered materials.

Although extremely innovative, graphene still lacks an electronic bandgap, which is essential for optoelectronic applications, despite many attempts to implement it [5–7].

Those attempts of opening a gap in bent monolayer systems and multilayer graphene, however, proved themselves to be diﬃcult while trying to preserve mobility and mechanical properties. This research leveraged the chase for some other component that would display such behavior, which brought us to the layered Transition Metal Dichalcogenides (TMDs). The TMDs became one of the highly studied families of materials due to its quasi-two- dimensional crystallographic structure and its electronic and mechanical properties [8–11].

Their bandgaps include such a wide range of sizes that all visible and infrared parts of the 2 spectrum are covered [12]. Other than these, one can highlight materials like hexagonal boron nitride (h-BN), borophene, phosphorene, silicene, germanene, and MXenes, which also became heated topics in research [13–18].

TMDs are semiconductors of the form MX2, where M is a transition metal atom typically from groups 4-7 (Mo, W, and Re) and X is a chalcogen atom such as S, Se or Te.

Examples of TMDs are, therefore, molybdenum disulﬁde MoS2, molybdenum diselenide

MoSe2, tungsten disulﬁde WS2 or tungsten diselenide WSe2. In bulk form, the TMDs are layered materials with each layer of the compound consisting of three atomic layers from which the transition metal one is sandwiched between two other formed by the chalcogens as illustrated in Figure 1.1. The layers have strong intralayer bonding given by covalent chemical interactions while possessing week interlayer van der Waals-like bonding [19]. This allows the isolation or fabrication of two-dimensional TMDs via methods such as exfoliation or vapor deposition.

Much like graphene, the layered TMDs possess dramatically diﬀerent properties compared to their bulk counterparts. One of the landmark features of the semiconducting

TMDs is that they reveal a direct bandgap in monolayer form, whereas they have an indirect bandgap in bulk form with exception of few cases (GaSe and ReSX2)[20,21]. The direct bandgap, in its turn, is a highly interesting property for optical conductivity purposes once the energy needed to raise the electrons to the conduction band does not change the electronic crystal momentum, what makes the light absorption process more efficient. The transition from an indirect to direct bandgap when bulk materials are scaled down to monolayers gives rise to unique electrical and optical features that evolve from the quantum confinement and change in orbital hybridization effects [22]. The tunable bandgap in TMDs is accompanied by strong photoluminescence and large exciton binding energy, making them promising candidates for a variety of optoelectronic applications [22,23], including solar cells, photo-detectors, light-emitting diodes, and photo-transistors [24–27].

TMDs have shown a vast range of applications. From fundamental studies of val- 3

Figure 1.1: MoS2 in 1H structural phase TMD illustration (purple sphere = Mo, yellow sphere = S) of: a) and b) side view. c) top-down view with unit cell. d) Standard orientation of crystal shape showing two layers and the bonding nature - Van der Waals for the inter-layer interaction and covalent for inter-atomic interactions (bi-color cylinder).

leytronics and Raman spectroscopy [28–31] to applications in biomedicine, catalysis, electronics, photonics, power generation, power storage, and others [8,11,24–26,32–35]. Par- ticularly, these ultrathin semiconductors possess huge potential to be made into ultra- small and low power transistors that are more eﬃcient than the state-of-art silicon-based ones due to its electronic properties and mechanical ﬂexibility [11], what makes them good candidates for novel materials that cope with the ever-shrinking electronic devices.

The synthesis of TMDs and 2D materials is still an active research area with the main methods involving exfoliation and chemical vapor deposition. The main challenge is mass production, i.e. scalability while maintaining atomic thickness precision [36–39]. In later years, however, this issue has been approached with a variety of new strategies [40,41]. 4

1.1 MoS2 Atomic and Electronic Structure

The electronic structure of semiconducting layered chalcogenides has been studied for decades both theoretically and experimentally. Several experiments using angle-resolved ultraviolet photoemission spectroscopy were conducted to determine the band structure for multiple TMDs [42–44]. As for the theoretical approach, multiple authors have calculated the two-dimensional electronic structure of those materials via density functional theory (DFT) within, among others, local density approximation (LDA) both for three dimensions and two dimensions [45–49].

Of all TMDs, MoS2 deserves special attention. It is a well-established semi-conductor with low toxicity and high thermal and chemical endurance [50]. In few-layer MoS2 the size and the nature of the gap depends on the number of layers, with a transition between a direct gap in monolayer compounds of around 1.78 eV [43,51] to a smaller indirect gap of 1.29 eV [43] for two or more layers. These energies fall in the solar spectrum range around the visible light and, thus, allow the MoS2 employment for photovoltaic [52] and photocatalytic [53] applications. This diﬀerence in bandgap is shown in Figure (1.2).

Also, the electronic properties of MoS2 appear to be highly sensitive to the external pressure and strain, both of which aﬀect the insulating gap and, under particular conditions, can also induce an insulator/metal transition [54].

Single layers of compounds from the group-VIB MX2 (where M = Mo, W and X = S, Se) are of special interest for valleytronics because, among other reasons, their conduction band minima and valence band maxima are located at degenerate, but nonequivalent valleys - the K and K’ points of the hexagonal BZ - and their spin-orbit coupling is substantial in comparison with other layered materials due to the presence of heavy metal atoms [55,56].

The absence of inversion symmetry in single layer samples lifts the spin degeneracy of the energy bands in the presence of spin-orbit coupling (SOC). This forces spin-splitting 5

Figure 1.2: Image modiﬁed from Figures (2) and (3) of reference [54]. The red arrow indicates the bandgap jump. In a) we have the calculated bulk band structure with the indirect bandgap between D point and C point. In b) we have the single-layer one with the direct bandgap between A point and B point.

in opposite valleys to be opposite in order to respect time-reversal symmetry and thus makes it possible to associate a valley index for carriers in diﬀerent valleys (Figure (1.3)). When there is no inversion symmetry, the valley index can be associated with physical quantities such as the Berry curvature and orbital magnetic moment [56]. This phenomena has been studied via valley- and spin- Hall eﬀects theoretically [56–58] and observed experimentally [59–61].

MoS2 lattice parameters have also already been calculated both experimentally and theoretically via LDA or Generalized Gradient Approximation (GGA) alongside with Projected Augmented Waves (PAW) [51]. In the hexagonal conﬁguration, the lattice constant was found to be around 3.11 A,˚ the distance between Mo and S atoms was calculated as 2.37 A˚ and the distance between two S atoms as 3.11 A˚ using LDA+PAW. 6

Figure 1.3: Illustration of valley coupling. Image from Figures (1) of reference [57]. Schematics of low-energy band structure for monolayer MX2. Red (blue) curves represent bands with spin up (down). The black dashed line shows the Fermi energy EF measured from the middle of the gap.

1.2 Disorder

Disorder can be introduced in many ways such as adsorption of adatoms, vacancies, or interstitials. The introduction of disorder is bound to bring changes to the properties of the compound. Adsorption of transition metal elements can induce magnetization properties, where the group 4A elements are usually chosen due to their capability of forming stable planar honeycomb structures [62,63]. Nanomeshes of graphene and MoS 2 might be of particular interest to combine the two materials’ mechanical and optoelectronic properties [51]. Bare, single layer MoS2, which is normally a nonmagnetic, direct bandgap semiconductor, attains a net magnetic moment upon adsorption of speciﬁc transition metal atoms, as well as silicon and germanium atoms (GeS2)[51]. Similarly to graphene, native defects in TMDs, including MoS2, can also induce the formation of local magnetic moments [64–66].

The doping of TMDs is also useful for creating n-type and p-type semiconductors, 7 which can be applied in p-type semiconductor-based complementary metal-oxide semiconductors (CMOS-)FET and other p-n junctions based optoelectronic devices [23, 67].

N-type MoS2 can be achieved by hydrogen adsorption and oxygen chemisorption while niobium and nitrogen doping form p-type MoS2 [67]. In such components, if a more reactive metal is used in contact with the MoS2 layer, chemical reactions can create defects in the TMD, altering its properties [68].

Vacancy point defects in semiconductors strongly influence transport and optical properties. In contrast to the bulk material, the observed electron mobility in single-layer MoS2 is unexpectedly low. Point defects, dislocations, or extended grain boundary defects may induce changes in the electronic structure that affect the transport of free charge carriers and may, therefore, be a primary source for the unexpectedly low mobility [50, 69]. Created artificially via thermal annealing, α-particle irradiation, or electron beam irradiation, these vacancies behave like traps holding electrons, holes, or excitons. The effects of disorder increase in lower dimensions due to the tighter localization of the electron wavefunction [70].

Monolayers can be produced, following the top-down approach, from natural MoS2 crystals by micromechanical exfoliation, intercalation based exfoliation, or, on a larger scale, by liquid-exfoliation techniques. Another approach is the bottom-up with proce- dures like chemical vapor deposition (CVD) that provides a controllable growth of the material with the desired number of layers on the substrate of interest [69]. Because none of the current methods of synthesis of those layered semiconductors is perfect, disorders in the form of cationic or anionic vacancies or dislocations are expected to appear naturally in the compounds [65,66].

The natural, or possibly engineered, occurrence of these defects and their eﬀects on spectral quantities of interest motivate us to understand how TMDs behaves under those conditions. The simulations of disordered TMDs with a random distribution of defects are, however, immensely expensive computationally when using DFT methods [54,71]. In this 8 study, we follow a diﬀerent approach - the lattice and spectral quantities of the disordered layered MoS2 are reviewed by the light of the high-performance software KITE [72] using a tight-binding method, in which extensive numerical calculations can be performed on large systems with a relatively low computational cost. Thus, an accurate set of results for the system properties will be reported in the following pages.

Prospect

In the next chapters, we describe our study with the goal of reaching a good description of the MoS2 TMD properties. The methodology here developed will open the doors for a new approach for disordered simulation, as we will discuss in the conclusion. The second (2) chapter will cover the key methods for accomplishing this study: the tight-binding modeling, kernel polynomial methods along with an important polynomial family, and the software used such as wannier90 and KITE. We will go into some details such as the derivation of some of those methods and will introduce some others.

With this foundation set, we can then further go into the third (3) chapter, where these methods will be applied on graphene to explain of the procedure while tacking a simple and well-known compound as an example, guiding the reader through the entire process of simulation and reaching the desired spectral quantities with and without defects. In the fourth (4) chapter we appply the previously presented concepts and work framework for the case of the MoS2 monolayer for multiple conﬁgurations - from pristine to disordered, also covering the spin-orbit coupling case. The results and its literature com- parisons and discussion will be presented along with some of the diﬃculties faced and the solutions engineered. In this chapter, we will also introduce the super cell approach, which will be key to develop the new methodology for disordered simulations. The last chapter (5) is a conclusion of this study where the outcomes and perspectives of further research on this or other TMDs are presented. 9

Chapter 2

Methods

The following pages introduce and explain in some depth most of the methods used during this study. In this analysis, we perform real space tight-binding (TB) calculations to obtain spectral quantities of our system. The procedure here implemented starts by gathering ab-initio parameters from density functional theory (DFT) calculations of a low sized system that serves as input for achieving a description in terms of Wannier functions of our electronic wave functions. This is used then to build our lattice, with which we use kernel polynomial methods (KPM) to arrive at our spectral quantities.

This chapter contains a description of these methods and the software used. It starts by introducing the DFT and the TB methods, followed by a brief explanation of the wannier90 software localization process. This is followed by the approaches used for calculating the spectral quantities employing the KPM and Green’s function polynomial expansion. By the end of the chapter, KITE is introduced, which is the software that builds the tight-binding systems and calculates the quantities of interest using the previously mentioned methods while also introducing disorder.

The techniques presented here will be later employed in the following Chapters3 and

4 on actual system simulations. 10

Density Functional Theory Introduction

Proposed in 1964 by Hohenberg and Kohn [73], the use of the electronic density as an electronic structure calculation variable replaces the many-body electronic complex wavefunction structure calculations of 3N spatial coordinates with simply three cartesian coordinates - for the electronic density.

As it is a physical observable and it has an intuitive interpretation, the electronic density function shows itself to be a very convenient variable. In their paper [73], Hohen- berg and Kohn prove, for the ground-state electron density, the existence of a one-to-one mapping between it and the ground state many-particle wave-function as well as its minimization of the total electronic energy.

Later, in 1965, Kohn and Sham [74] introduce the framework in which the interacting electrons in a static external potential can be traceable to non-interacting electrons moving in an eﬀective potential. This alongside the use of the density as a fundamental variable forms the basis of what became to be known as Density Functional Theory (DFT) [75].

DFT together with LDA (local-density approximation) [74] became very popular for solid-state calculations because of the computational advantage brought by it and was further reﬁned during the last decades of the XX century.

Despite this method itself being too computationally costly for describing large systems, it is essential for the calculation of many preliminary quantities in this study, serving as stepping stone for the methods that follow, as described below.

2.1 Tight-binding Method

From the simulation methodologies developed as propelling tools for the discovery of new materials and applications, semi-empirical atomistic methods are the most simple and eﬀective ones to calculate the electronic properties of materials [76]. 11

2.1.1 Fundamentals

To solve the Schroedinger eigenvalue and eigenvector problem

H |ψi = E |ψi (2.1) of a given system, it is useful, in certain conditions, to use the tight-binding approach [77]. To start, in a tight-binding (TB) model, atoms nuclei of a crystal or molecule are considered way heavier in comparison to their electrons, so that the Born-Oppenheimer and the approximation of ﬁxed position for the nuclei applies. Next, as the electrons are tightly bound to atomic orbitals, it is convenient to write the electronic wave function in terms of the atomic orbitals or tight-binding orbitals such as to form a linear combination of atomic orbitals (LCAO): N X |ψi = φi |ii , (2.2) i=1 where φi are complex coeﬃcients and |ii are the atomic orbitals. Note that N does not necessarily equals the number of atoms, rather it is the total number of orbitals in our system. Our Schroedinger equation can then be written as

X ∗ X ∗ φi φj hi| H |ji = E φi φj hi|ji . (2.3) i,j i,j If we assume the orbitals are orthonormal, which is a good approximation given the nuclei of the atoms will not change their positions, as in a crystal for example, we get the equation (2.4) from equation (2.3) and we can then write the eigen-equation explicitly in matrix form as (2.5)

X hi| H |ji φj = Eφi, (2.4) j       h1| H |1i h1| H |2i · · · h1| H |Ni φ1 φ1  h2| H |1i h2| H |2i · · · h2| H |Ni   φ   φ     2   2   . . .. .   .  = E  .  , (2.5)  . . . .   .   .  hN| H |1i hN| H |2i · · · hN| H |Ni φN φN 12 where the diagonal terms are the self energies of the orbitals and the oﬀ-diagonal terms are the hopping terms.

Waves in crystals, whether vibrational waves or electron waves, are best described in the reciprocal lattice. Considering free electrons over a periodic potential given by the tight-binding assumptions, we get a periodic Hamiltonian of the form

X H = K + Vj, (2.6) j ~ where K is the kinetic energy of the free electron and Vj = V (~r − Rj) is the interac- th ~ tion between the electron and the j nucleus of position Rj, which is periodic given the translation symmetry of the crystal.

To solve this particularly symmetric Hamiltonian, Bloch’s theorem requests that the single particle states would be of the form

~ i~k·R~ ψn~k(~r + R) = e ψn~k(~r), (2.7) which is not satisﬁed by a single atomic orbital, but by a linear combination

1 X i~k·R~ ~ ψ ~ (~r) = √ e φn(~r − R). (2.8) nk N R~ √ The normalization is given by 1/ N for a N unit cell crystal, the same N in equation 2.2 in the case we have one orbital per unit cell, and being ~k the Bloch momentum.

On a second quantization formalism, one can re-write the atomic orbital state basis in terms of their respective creation and annihilation operators acting on the electron state of that particular orbital. So that our Hamiltonian of an 1D system with one orbital per unit cell would be written in the space representation as

N N X † X † † H = Eσciσciσ − t (ciσci+1,σ + ci+1,σciσ), (2.9) σ,i=1 σ,i=1 where σ is the spin index and t is the spin-independent hopping term. We see that the hoppings are restricted here to the first neighbors only by using the assumption that the 13 nucleus-atom interaction is short-ranged in comparison with the system length and will be negligible on sites further than the first neighbor ones. As a result, our matrix will have mostly zeros in the off-diagonal term, categorizing it as a sparse matrix. The same holds when we deal with further nearest neighbors as long as the distances stay small in comparison to the system size.

In the case we have multiple orbitals per site (and/or multiple atoms per site) it is convenient to write a set of operators per orbital with the proper notation so it can be identiﬁed. Commonly the notation adopted is the orbital letter indexed by its variation.

For the d orbitals, for example, one can write its operators as

† † † † † {di,3z2−r2 , di,x2−y2 , di,xy, di,xz, di,yz} along with the annihilation counterparts. Although it was discussed here the reciprocal lattice and space, and Bloch momentum deﬁnition for a periodic crystal, the system we are working with is disordered and thus does not possess the translation symmetry needed for the reciprocal lattice discussion to be of immediate relevance. Rather, it is easier for a disordered system to be solved in the real space, even though the Hamiltonian in this case quickly scales up with the system size

- for each unit cell, there must be accounted the orbitals of interest for every atom, and the desired system size tends to be the largest possible to simulate the thermodynamic limit. Such large Hamiltonians can easily reach billions of orbitals, and thus, require a great number of computational resources, what becomes a great limiting factor to real space calculations. To deal with this, it was used in this study the kernel polynomial method, which is covered in this chapter. The Bloch discussion will serve as a basis for the wannier90 method in an upcoming section.

2.1.2 The MoS2 Tight-biding Model

The analysis of the properties of TMDs and other crystals can be performed with the use of tight-binding Hamiltonians, which are simpler compared to DFT calculations. They 14 are also easily scalable in the case of layered materials, once that to analyze multilayer compounds, the single-layer TB description can be used as a fundamental block to be repeated along with the addition of simple interlayer hopping terms.

The TB method for describing our system can prove itself more convenient than ﬁrst- principle methods such and Density Functional Theory when it comes to systems with a very large number of atoms. Although DFT is already able to deal with systems with thousands of orbitals, with the SIESTA method for example [78], its scalability can be computationally challenging and demanding. Thus for calculations on systems with millions of orbitals, which is the purpose of this current study, the TB method is essential.

Such a tight-biding model could be used, for example, to study direct-indirect bandgap transition from multilayer compounds to single-layer systems if the orbitals are well-chosen

[54]. To define a tight-binding model for our single-layer MoS2, first, there is the need to introduce the Hilbert space on which we are going to work. The bandgap properties defin- ing orbitals relevant for the calculation are the 5 d orbitals (d3z2−r2 , dx2−y2 , dxy, dxz, dyz) from Mo and the 3 p orbitals (px, py, pz) from S as pointed in Ref [54]. In analyzing the orbital character of each energy level at the main high-symmetry points of the Bril- louin Zone, as calculated by DFT, it is noticeable that an accurate description of such electronic states on the conduction and valence bands involves at least the Mo orbitals d3z2−r2 , dx2−y2 , dxy, and the S orbitals px, py. This suggests that a 5-band tight-binding model would be enough to describe the system [58], and a further simplification was done where the S and 3p orbitals are omitted, resulting in a 3-band TB model [79].

These simplistic TB models can prove ﬂawed in a more comprehensive description especially when trying to describe the system under point defect disorder, as it will be discussed later by the end of Chapter4. In this situation the other d orbitals from Mo and p orbitals from S prove themselves essential.

In the current study, however, the TB model used has a total of 13 orbitals per unit cell, namely the d3z2−r2 , dx2−y2 , dxy, dxz, dyz from Mo and the s, px, py, pz for each of the 15 two S atoms. The S atoms s orbitals are included for completeness. Taking this into account, we can deﬁne our working Hilbert space by means of the

13-fold vector:

φi = (si,t, pi,x,t, pi,y,t, pi,z,t, di,3z2−r2 , di,x2−y2 , di,xy, di,xz, di,yz, si,b, pi,x,b, pi,y,b, pi,z,b), (2.10) where di,α creates an electron in the respective α orbital of the Mo atom in the i-unit cell, the si,α,t, pi,α,t, si,α,b, pi,α,b, creates an electron in the α orbital in the top (t) or bottom (b) S atom of the i unit cell respectively. Because here we are dealing with a 2D lattice, the equation 2.9 changes for a 2D description adding a index j for identifying the second degree of freedom in the lattice ({i} → {i, j}). We are also not dealing with one single orbital, rather the set described by the vector 2.10. For each unit cell composed of 1 Mo atom and 2 S atoms, the hopping terms include interaction up to the third neighbors, making so that we expect further hopping therms. We can rewrite the equation 2.9 in our case to: X X H = E α† α − t (α† β + β† α ), with: α,σ ijσ ijσ ijσ i+hi,j+hj ,σ i+hi,j+hj ,σ ijσ α,σ,i,j α,β,σ,i,j,hi,hj

α, β ∈ {st, px,t, py,t, pz,t, d3z2−r2 , dx2−y2 , dxy, dxz, dyz, sb, px,b, py,b, pz,b}, 1 1 σ ∈ , − , 2 2 i, j ∈ {1, ··· ,N}, ( {1, 2, 3} , if α = β, hi + hj ∈ {0, 1, 2, 3} , if α 6= β. (2.11) Completing the tight-binding space representation of our system Hamiltonian.

2.2 Wannier 90

Wannier functions are a complete set of orthogonal functions that serve as a representation of our orbital linear combination that is orthogonal between diﬀerent sites of a crystal. Those functions are of interest because, if well deﬁned, they can be maximally localized, which is useful for acquiring the lattice properties of our system. 16

Given the Bloch bands functions ψnk(~r) deﬁned by the equation 2.8, one can build ~ ~ Wannier functions wnR~ = wn(~r − R) labelled by the Bravais lattice vectors R as [80] Z " N # V X ~ ~ ~ |w i = d~k U (k) |ψ i e−ik·R, (2.12) nR~ (2π)3 mn m~k BZ m=1 where U (~k) and the Brillouin Zone integral are unitary transformations. The U (~k) transformation gives the freedom for multiple possible Wannier functions. Those which are maximally localized in the orbitals are reached by the Marzari-Vanderbilt (MV) approach, that consists of ﬁnding the Wannier functions, i.e., choosing the U (~k), such that the sum of the quadratic spread of the Wannier function around its center is minimized:

N X 2 Ω = h(~r − r¯n) in n N X 2 2 = hr − 2~r · r¯n + |r¯n| in (2.13) n N X 2 2 = hr in − |r¯n| n ˆ ¯ ˆ where hOin = On = hwn0| O |wn0i. ˜ This Ω can further be split into a gauge-invariant part ΩI and the rest Ω, which is the part corresponding to the minimization of an isolated set of bands [80]. Discretizing the ~r and r2 expectation values in reciprocal space, and by the deﬁnition (~k,~b) ~ of an overlap matrix Mmn = h~um~k|~un,~k+~bi, where b are vectors connecting the mesh point ~ −i~k·~r k to its nearest neighbors and ~un~k(~r) = e ψn~k(~r) is the periodic part of the Bloch function, one can derive

N " N # 1 X X X (~k,~b) 2 ΩI = ωb 1 − |Mmn | , Nkp ~k,~b m=1 n=1 (2.14) " N N # 2 ˜ 1 X X (~k,~b) ~ X (~k,~b) 2 Ω = ωb −Im lnMnn − b · r¯n + |Mmn | . Nkp ~k,~b n=1 m6=n

With Nkp being the number of k-points in the Monkhorst-Pack grid, ωb a weight associated ~ th to the b shell andr ¯n the center of the n Wannier function. These are enough to evolve 17

(k) Umn towards maximum localization of the (2.12) equation for an isolated set of bands. In the case of entanglement in energy bands such as the conduction bands, we work with ΩI as a functional of the N dimension Bloch subspace at each k-point. The latter is deﬁned

(k) via a unitary transformation among the Nwin states that fall into a chosen energy window where there is degeneracy of states:

opt X dis(~k) |~u i = U |~u ~ i , (2.15) n~k mn mk (~k) m∈Nwin

dis(~k) (~k) where U is a unitary rectangular Nwin × N matrix. With this subspace defined, one can find the ΩI projection on this subspace as a measure of the degree of mismatch between different Bloch momenta subspaces. The MV localization procedure can be repeated here, thus completely minimizing Ω.

Since we are dealing with unitary transformations U (~k) and U dis(~k), the interpolation with the target system is straightforward:

W ~ (~k) † dis(~k) † dis(~k) (~k) H (k) = (U ) (U ) H~kU U (2.16)

And we can work with its Fourier transformations to compute the interpolated band energies and the real space grid.

This whole procedure is done by the wannier90 software. The maximum localized Wannier functions themselves are never explicitly constructed unless when required for visualization or post-processing purposes. The minimized U (~k) and U dis(~k) along with the M (~k,~b) are enough to deﬁne the centers and spreads of those Wannier functions.

Given the unit cell atoms cartesian coordinates and a set of k-points in terms of the lattice vectors, the wannier90 software is able to determine the Wannier functions and iteratively calculate their maximum localized form. Given the shape of the electronic functions, it can calculate the interactions between diﬀerent orbitals. These results give us the value for the hoppings up to any distance the TB cell we will build. 18

In this study, the wannier90 software is supplied with ab-initio DFT results for describing a small monolayer MoS2 system, as will be explained in Chapter4 in more detail.

The resulting tight-binding conﬁguration is used as an input to the KITE software to build the system on a larger scale with the possibility of adding disorder, as explained later in this chapter. With the system built, next comes the simulation of its spectral properties.

2.3 Kernel Polynomial Method

It was seen in the previous sections that, to obtain the desired properties of our system, we would need to solve a real space Hamiltonian with a large number of terms for our system is disorder and does not possess the translation symmetry needed to work in the Bloch momentum space. In a system with N sites each one having d relevant orbitals, the eﬀective Hamiltonian to be solved will be of the size D × D, where D = N · d, and with D → ∞ on the attempt to reach the thermodynamic limit.

Canonically, the eigenvalues and eigenvectors of a Hamiltonian are found through direct diagonalization. And despite providing all the information of the system, this method has a major drawback: the computational cost scales with the cube of the system size D while the memory size scales quadratically. For this reason, this naive approach is more adequate for small systems [81], which is not our case.

By the end of the past century many eﬀorts had been made to circumvent this problem at the cost of dropping the requirements for a complete and exact knowledge of the spectrum. A natural approach was already considered from the early days of quantum mechanics - to characterize the spectral quantities in terms of its moments, which can be very eﬃciently calculated by iteration. To remedy the numerical instabilities risen by the iteration utilizing ordinary power moments of the energy, polynomials of the energy were suggested instead. The polynomials worked to solve the weight the power moments of the energy would bring to the boundaries of the spectrum, but alongside, they brought another complication - the Gibb’s oscillations. 19

The Gibb’s oscillations occur when a polynomial series describing a nonsmooth function is truncated at a ﬁnite value. Due to the overshooting nature of the polynomial expansions, oscillations emerge near the not continuously diﬀerentiable points. This became known as the Gibbs phenomenon. To deal with this, by the year 1994 the kernel polynomial method (KPM) was coined [81, 82]. The kernel would serve as a wrapping function, a regularization factor to assure the oscillations were being taken care of in the right places. The kernel is responsible for smoothing the function’s discontinuous points to improve the convergence of the polynomials approximation [83].

For the KPM, the most time-consuming step requires only multiplications of the considered matrix by a small set of vectors, allowing the calculation of spectral properties and correlation functions that scale linearly with the system size D in the case of sparse matrices or quadratically otherwise. Our matrix, despite big in size, can be considered sparse granting good scalability when using the KPM because our interaction range is small in comparison with the system size. This low resource scaling is the reason why

KPM is now used to approach a variety of problems involving correlation, response or Green’s functions of large matrices, e.g. density of states, static correlations and dynamic correlations [84–86]. It also became a component for multiple methods such as Monte Carlo simulations and Cluster Perturbation Theory (CPT) [81].

In parallel to these attempts, other methods were being developed during the same period such as the Lanczos recursion method [87].

2.3.1 Chebyshev Polynomials Expansion

Introducing a function scalar product between integrable functions in a certain interval

Z b hf|gi = dx w(x)f(x)g(x) (2.17) a 20

Figure 2.1: Normalized Chebyshev Polynomials of the ﬁrst kind of degree n from 0 to 5.

where w(x) is a weight function, one can ﬁnd a complete set of polynomials pn(x) such that they fulﬁll orthogonality relations

hpn|pmi = δn,m/hn (2.18)

with hn = 1/ hpn|pni granting the normalization. The completeness of this set of polynomials allow us to express, through this scalar product, a function f(x) in terms of the pn(x): ∞ X f(x) = cnpn(x) with cn = hpn|fi hn. (2.19) n=0 In principle, for a function expansion, any type of orthogonal polynomial can be used, but, for the kernel polynomial method, some have been better studied than others [81,82]. From all the rising families of polynomials in the 90’s studies, the Chebyshev polynomials proved to be of outstanding fast convergence. Not only that but this family of polynomials had a close relation to the Fourier transform [88], which is an important characteristic for derivation of optimal kernels for ﬁnite order expansions which have not been derived for other sets of polynomials.

Both sets of Chebyshev polynomials are deﬁned in the interval [a, b] = [−1, 1] through 21 the expressions

Tn(x) = cos[n arccos(x)] (2.20) √ for the ﬁrst kind with the w(x) = (π 1 − x2)−1 scalar product weight function, and

sin[(n + 1) arccos(x)] U (x) = (2.21) n sin[arccos(x)] √ for the second kind with the w(x) = π 1 − x2 scalar product weight function.

From these deﬁnitions, one can derive the recursion relations

T0(x) = 1,T1(x) = x (2.22) Tm(x) = 2xTm−1(x) − Tm−2(x), (m ≥ 2) for the ﬁrst kind, and

U0(x) = 1,U−1(x) = 0 (2.23) Um(x) = 2xUm−1(x) − Um−2(x), (m ≥ 1) for the second kind.

Although we have those two kinds available, the first kind of the Chebyshev polynomials is the one commonly used for polynomial expansion because they are bounded in the defined interval, while the second kind diverges at its boundaries. The closure relation for the Chebyshev polynomials of the first kind is given by

∞ ! 2 1 X δ(x − y) = √ + Tm(x)Tm(y) . (2.24) 2 2 π 1 − x m=1

A function f(x) can then be expressed in terms of the series of this kind via

∞ ! 2 µ0 X f(x) = √ + µmTm(x) , (2.25) 2 2 π 1 − x m=1 with the coefficients Z 1 µm = dx f(x)Tm(x) (2.26) −1 Naturally, because the Chebyshev polynomials are defined only in the [−1, 1] interval, we need to rescale our quantities to fit in the same interval, ie rescal our Hamiltonian. 22

Because our Hamiltonian depends on the eigenvalues E of our matrix, to ﬁt the [−1, 1] interval a simple linear transformation of the Hamiltonian and all energy scales is needed:

H˜ = (H − b)/a, (2.27) E˜ = (E − b)/a.

Given the extremal eigenvalues of the Hamiltonian, Emin and Emax, the scaling factors above read E − E a = max min , 2 − (2.28) E + E b = max min , 2 the parameter being a small cutoﬀ to avoid stability problems near the boundaries of the interval [−1, 1]. It can be adapted as a function of the resolution of the expansion, ie, for a ﬁnite N order expansion, a good can be given by = 1/N.

Calculation of Moments

With our inner product deﬁnition, our moments will be of two types. First we have simple expectation values of the Chebyshev expansion

˜ µn = hβ| Tn(H) |αi , (2.29) where |αi and |βi are states of the system. Straightforwardly expanding the H˜ Chebyshev terms, we can deﬁne iteratively the outcoming states as a function of the original probing state |αi: ˜ |αni = Tn(H) |αi , (2.30) which, from the equation (2.22), one can determine that ˜ |α0i = T0(H) |αi = |αi , ˜ ˜ |α1i = T1(H) |αi = H |α0i , . (2.31) . ˜ |αni = 2H |αn−1i − |αn−2i . 23

With this, our equation (2.29) simply yields

µn = hβ|αni . (2.32)

In the case we have |βi = |αi, the iterations can be further simpliﬁed using a relation derivable from the recursion relations (2.22):

2Tm(x)Tn(x) = Tm+n + Tm−n (2.33) to write

µ2n = 2 hαn|αni − µ0, (2.34) µ2n+1 = 2 hαn+1|αni − µ1.

Overall, for the case of a sparse H˜ , the matrix-vector multiplication operation cost is of the order O(D), increasing linearly with the number of moments to be calculated. For the calculation of N moments, there will be needed O(ND) computational resources

(processing power, time). In the case of equations (2.34) the numerical eﬀort is reduced by a factor of 2.

Stochastic Evaluation of Traces

The second case is when our moments are of the form of

˜ µn = Tr[ATn(H)], (2.35) where we have a trace over the polynomials and a linear operator A that multiply our Hamiltonian for us to get the quantities of interest. Following the previous method, the numerical eﬀort needed to perform these calculations, would be of the order of O(D2), but extremely good approximations have been made in the past to evaluate the traces of large matrices [81,89,90]. The method consists of evaluating the trace trough random vectors. 24

For an arbitrary basis {|ii}, random vectors are deﬁned as

 x  ξ1 ξx   2  |xi =  .  , x = 1, 2, ··· ,R (2.36)  .  x ξD with each of its components being a complex number satisfying the relations:

x hhξi ii = 0,

x y hhξi ξj ii = 0, (2.37)

x? y hhξi ξj ii = δi,jδx,y, being the hh· · ·ii the statistical average operation. With that, one can evaluate the trace of a Hermitian operator B deﬁned in the same basis {|ii}, ie with matrix elements Bij = hi| B |ji. Its trace is then given by

R 1 X Θ = lim hx| B |xi , (2.38) R→∞ R x=1 which can be statistically averaged to get

** R ++ R D 1 X 1 X X x? x hhΘii = lim hx| B |xi = lim hhξi ξj ii Bij R→∞ R R→∞ R x=1 x=1 i,j=1 (2.39) D X = Bii = Tr(B). i=1 The approximation comes when we do not use R → ∞ random vectors to determine our momentum, rather we use a small number of random vectors R D which causes ﬂuctuations that can be found by calculating (δΘ)2 = hhΘ2ii − hhΘii2. We get for the ﬂuctuation [81,91]

D ! 1 X (δΘ)2 = Tr(B2) + (hh|ξx|4ii − 2) B2 , (2.40) R i jj j=1 from which one can determine the relative error

δΘ 1 ≈ √ . (2.41) Θ DR 25

That means that not only the error decreases with the number of random vectors used R, but the dimension D of our system too. We ﬁnd that for our case then, because our system is very large, most of the times very few random vectors will be needed, granting fast calculations of the momenta.

2.3.2 Kernel Polynomials

We have seen in the previous sections that in the case of ﬁnite number of Chebyshev moments, the Gibbs oscillations become a concern for the polynomial expansion of the function of interest near the points with discontinuity. To solve that problem, we adopt the kernel approach, that is, we module our approximation with another function that is responsible for smoothing near points with discontinuity. In terms of the expansion, this translates into alter our expansion coeﬃcients, our µn moments, by multiplying then to the gn kernel term. From equation (2.25), we have that

N ! 2 g0µ0 X fKPM (x) = √ + gnµnTn(x) . (2.42) 2 2 π 1 − x n=1

Now we need to find the optimal kernel, i.e., coefficients gn, for our application. One simple and classical early kernel is the one suggested by Fejérin 1904, where he shows that for continuous functions, the kernel coefficients

n gF = 1 − (2.43) n N makes the approximation converge uniformly in the interval [−1 + , 1 − ]. In the limit

F N → ∞ we then get that the fKPM (x) → f(x). The uniform convergence is obtained under very general conditions, suﬃcing that [81]:

1. The kernel coeﬃcients gn are positive.

2. The convolution of the function with the kernel does not aﬀect normalization, or,

the kernel is normalized. 26

3. As N → ∞, the kernel coeﬃcient g1 approach 1.

Those conditions are also very useful for practical purposes. The ﬁrst assures the conser- vation of the signal of our quantities - positive quantities remain positive throughout the interval. The second one conserves the expanded function integral, its normalization

Z 1 Z 1 dx fKPM (x) = dx f(x). (2.44) −1 −1

Jackson Kernel

Tightening the third condition for uniform convergence, we try to ﬁnd a kernel so that the g1 approaches unity at a optimal rate, that is, a kernel with optimal resolution [81,92]. If we write, for arbitrary aν ∈ R, N X gn = aνaν+n, (2.45) ν=1 then, by the methods of Lagrangian multiplier, we get for the constraint

δQ δC = λ , (2.46) δanu δanu with Q = g0 = g1 and the constraint C = g0 − 1 = 0. Setting the boundary conditions a0 = aN = 0, we arrive at an eigenvalue problem of many harmonic oscillators with ﬁxed boundaries: πk(ν + 1) a =a ¯ sin . (2.47) ν N + 1

The solution for the gn in equation 2.45 then yields

a¯2 πkn πkn πk g = (N − n + 1) cos + sin cot , (2.48) n 2 N + 1 N + 1 N + 1 which normalization is granted bya ¯2 = 2/(N + 1), and the optimal value for Q is then given by πk Q = g − g = 1 − cos (2.49) 0 1 N + 1 27 obtained making k=1. From this we deﬁne a new optimal kernel derived by Jackson in his PhD thesis in 1911 [92] with the kernel coeﬃcients given by πn πn π (N − n + 1) cos + sin + cot N + 1 N + 1 N + 1 gJ = . (2.50) n N + 1 This kernel variance being s π σJ = 1 − cos , (2.51) N + 1 that, for large N can be approximated to π σJ ≈ . (2.52) N + 1 2.3.3 Calculation of DOS

We have seen in the previous sections that one of the basic applications of the Chebyshev polynomials expansion is the calculation of the spectral properties of large matrices [81, 93]. In this case, we are interested in the spectral density of our Hamiltonian, which corresponds to its density of states. Considering the calculation of the DOS ρ(E) of an D × D Hamiltonian H deﬁned as D 1 X ρ(E) = δ(E − E ) (2.53) D k k=1 where the En are the eigenergies. To perform the expansion, we redeﬁne the intervals as in equations (2.27) and (2.28), so that we can write the DOS in its polynomial expansion representation using equation (2.25)

D 1 X ρ˜(E˜) = δ(E˜ − E˜ ) D k k=1 . (2.54) ∞ ! 2 µ0 X ˜ = √ + µmTm(E) 2 2 π 1 − x m=1 ˜ Being Tm(E) the Chebyshev polynomials of the ﬁrst kind (2.22) and the µm its moments. The KPM considers, instead, smooth truncations of this expressions of the form N ! ˜ 2 µ0g0 X ˜ ρ˜(E) = √ + µmgmTm(E) , (2.55) 2 2 π 1 − x m=1 28

where N is the ﬁnite number of moments and the gm are the Gibbs damping factors [93], or kernel coeﬃcients. The moments are then given by equation (2.26) as Z 1 ˜ ˜ ˜ µm = dE ρ˜(E)Tm(E) −1 D 1 X Z 1 = dET˜ (E˜)δ(E˜ − E˜ ) D m k k=1 −1 D 1 X = T (E˜ ) (2.56) D m k k=1 D 1 X = hk| T (H˜ ) |ki D m k=1 1 = Tr[T (H˜ )]. D m This is the trace (2.35) that, as seen in the previous sections, can be calculated via the stochastic vector approach (2.39).

2.3.4 Green’s Function Chebyshev Polynomial Expansion, an Alternative to KPM

An approach to the calculation of the spectral functions is through the computation of a single-particle Green’s function in frequency domain, which can be achieved using the single-particle Hamiltonian Chebyshev polynomials expansion. The diﬀerence from KPM arises from the choice of the function of the system to be analyzed. Contrary to what was seen in equations 2.56 of the previous section, this method will not use delta functions to analyze the spectral properties of the system, rather an approach based on the Density Matrix Renormalization Group (DMRG) [94] will be adopted in which we make use of a

Chebyshev expansion of the function

Z ±∞ ± i(±z−x)t 1 fz (x) = −i dt e = , (2.57) 0 ±z − x

With x, Rez = ω ∈ R and Im z = η > 0. With this function, one has control over the η broadening while maintaining the Green’s function integrity. The spectral function can then latter be constructed in the thermodynamic limit where η → 0+. 29

The Gibbs oscillations from the Chebyshev expansion reconstruction of our spectral function are not dealt with the KPM ﬁlter kernels in this method, rather the problem is dealt with by keeping track of the resolvent broadening. This method application results on obtaining the spectral functions with a well controlled broadening and, at the same time, it gives access to the full Green’s function of the problem [84,85].

For the calculation of the G Green function in frequency domain we evaluate an expression of the form

G± (z) = hψ | Aˆ[E − Hˆ ± z]−1Bˆ |ψ i , (2.58) A,ˆ Bˆ 0 0 0

ˆ ˆ with z = ω + iη, η > 0 and |ψ0i is an eigenstate of H with H |ψ0i = E0 |ψ0i. The ’+’ and ’-’ parts are the retarded and advanced Green functions in frequency representation.

To deal with this Green function expansion, let us ﬁrst expand the equation (2.57) in terms of Chebyshev polynomials. In a similar fashion as in equation (2.25), we expand our DMRG function as ∞ ± X ± fz = αn (z)Tn(x). (2.59) n=0

The coeﬃcients, as in given by equation (2.26) with the appropriate weight function discussed previously, are given by

2 Z 1 T (x) 1 α± = dx √ n n 2 π(1 + δn,0) −1 1 − x ±z − x −2i Z ±∞ Z 1 T (x) = dt e±izt dx √ n e−ixt 2 π(1 + δn,0) 0 −1 1 − x n+1 Z ±∞ 2(−i) ±izt (2.60) = dte Jn(t) 1 + δn,0 0 2 1 = √ √ !n r (1 + δn,0) z2 z2 − 1 1 (±z)n+1 1 + 1 − z2 z2

where Jn(t) is the Bessel function of the first kind [85] . Note that the finite broadening η is kept. Like it was done in the equation (2.27,2.28) we now rescale our Green function 30 to be contained in the Chebyshev definition domain of [−1, 1]:

G± (ω) = f (H¯ − E ) A,ˆ Bˆ ±(ω+iη) 0 (2.61) ¯ = af±a(ω+iη)−b a(H − E0) − b . ¯ Now for the expectation values of Tn(a(H − E0) − b), the µn momenta of the Chebyshev polynomial expansion of G, we have

ˆ ¯ ˆ µn = hψ0| ATn(a(H − E0) − b)B |ψ0i , (2.62) which ﬁnally yields ∞ X G± (ω) = a α±(±a(ω + iη) − b)µ . (2.63) A,ˆ Bˆ n n n=0 For a ﬁnite moment expansion

∞ N ± X ± X ± G ∝ αn µn ≈ αn µn, (2.64) n=0 n=0 the number of moments N needed to obtain a reliable expansion can be estimated by considering the properties of the Bessel functions from equation (2.60) as [85]

−1 N & (aη) . (2.65)

We have seen that a is inversely proportional to the extreme eigenvalues diﬀerence and, thus, inversely proportional to the system size. From this we can rewrite the Equation

2.65 as α∆E N & η (2.66) α∆E η & N being α some proportionality scalar. This relation show more clearly that η behaves as a resolution for our calculation using Green’s expansion and the right side of the last equation shows us that this resolution must be greater than the one from Chebyshev expansion momenta. This means that our broadening η has to be chosen small enough so that we are able to resolve narrow structures in the energy spectrum - that is η has to 31 be small in comparison to the width of the spectral structure of interest, in our case the possible peaks in the gap. This means that η upper bound is ﬁxed by the width in energy of our structures of interest, but what about the lower bound?

As η decreases, we are expected to have a better resolutions of our system, but this remains true only if it is accompanied by the growth of N as shown in the Equation (2.66). One can imply from this that η must be large enough so that the Chebyshev expansion

Gibb’s oscillations do not end up being resolved. We will see in Equation (B.1), however, that that N cannot grow indeﬁnitely, what implies in a lower bound for the value of η.

The lower limit for our broadening is given by the Gibbs oscillations scale. That is, the size of η must be high enough to encompass the oscillation period. This makes it so that the role of this imaginary part is to reduce the oscillation amplitude by averaging the oscillations down to a smooth function in an interval. As the number of Chebyshev polynomials vary, these oscillations period also vary as it will be seen in the Appendix (A) and a optimal value for both η and N can be derived. It may be the case, however, that a great amount of Chebyshev polynomials are to be used (and consequently a large system) to have a good simulation of our quantity, ultimately bumping up the computational cost of our calculations.

A discussion of how the the method presented here will be best used for our calculations is presented in Appendix (A).

2.4 KITE

KITE is a general-purpose open-source tight-binding software for accurate real-space simulations of electronic structure and quantum transport properties of large-scale systems with tens of billions of atomic orbitals. Written in C++ and with a Python-based interface, this software is optimized for shared memory multi-node CPU architectures with good scalability. It uses the methods described in this this chapter to perform large scale calculations of multiple spectral quantities. The calculations of DOS, LDOS and 32 conductivity in this study were all made using this software [72].

There are other software that deal with the modeling of electronic structure and quantum transport properties of empirical tight-binding models [95–98]. Those numerical implementations of real-space quantum transport, however, have so far been limited to mesoscopic structures with up to ten millions of orbitals [99]. KITE brings improve- ments to the spectral expansion methods due to its original approach to disorder and parallelization techniques.

The KITE code computes target functions with spectral algorithms based on numeri- cally exact Chebyshev polynomial expansion of Green’s function, [85] as worked out in the previous sections. Its memory management improves data aﬃnity on local interactions.

This plus the multi-threading pre-deﬁned partitions in real space through domain decomposition reduces computation time when dealing with quantum transport calculations or complex structures with large unit cells.

The software is designed and optimized to maximize system size, improve self-averaging and to achieve fine energy resolution. The large systems we are dealing with can generate sparse tight-binding Hamiltonians, that, depending on the number of nearest neighbors of interest, can become too large to be stored. KITE, however, benefits from the periodic nature of the underlying lattice and avoid the storage of the superfluous periodic part of the Hamiltonian by using a set of pattern rules to encode it.

2.4.1 Adding Disorder

One of KITE’s most innovative method is how the software deals with disordered systems. As we have seen, once the system is disordered, one can no longer rely on the translational symmetry that would make our Hamiltonian so simple, rather, to describe the Hamiltonian, one must write it in terms of all possible orbitals. KITE approach, however, bypasses the need of such a large matrix by storing the disorder site information apart in the form of a mask. 33

Once the disorder is defined, it is known in which orbital it is and, from the previous definitions of hoppings, one can also store the neighbors location which will be in range of its hopping potential. Given that, one can create a Hamiltonian as if it was purely crystalline and after, on top of it, add the hopping effects on the precise sites. This procedure is called ”adding a mask”, where to the pristine Hamiltonian it is added another matrix that encompass all the distortions brought by the disorder, as illustrated in the Figures (2.2- a)) and (2.3- a)). With this, not only is the effective matrix smaller and thus easier to be stored, but the calculations are mostly independent which means they are free to be parallelized in chunks.

Two types of disorder can be added through the KITE python interface - the onsite disorder and the structural disorder. The effect of the onsite disorder is rather straightforward - on the specific orbital chosen as disordered, the onsite energy is shifted. In terms of the mask above described, simply adding to the diagonal term will suffice. The structural disorder however, includes not only the effect of the onsite disorder, but also to the binding terms. One example of such structural disorder, and the one that will be mostly used in this study, is a vacancy. In the case of the vacancy, not only the diagonal terms are to be changed, but also are the hopping-connecting off-diagonal terms.

In Figures (2.2) and (2.3), we can see illustrations of the eﬀect of masks in the case of onsite disorder and structural disorder respectively.

In the next chapter we will cover the steps for the calculation of these disorders as well as a basic example to help illustrate these concepts - a graphene layer will be simulated in its pristine and disordered conﬁguration.

2.4.2 Parallelization

The inter-thread dependency for the Chebyshev calculations is dealt by adding ever- updating ghost cells to each subdomain - an extra layer at the domain border that holds copies of the elements from the neighbor domains - so that the full iteration can be 34

Figure 2.2: a) Positioning the mask with a onsite disorder ”over” our system. b) Eﬀect of the onsite disorder in our lattice. The greyed-out atom represents the Mo atom with changes in its eigen potentials, matrix diagonal terms. performed independently. The ghost synchronization update is not parallelizable, which makes it a bottleneck for the run eﬃciency. Despite it being a non-parallel component of the algorithm, the bottleneck scales with a border/volume ratio, becoming negligible for large systems.

A smaller subdomain scale deﬁned in the compilation, TILE, tiles the subdomain to control the size of memory chunks so that the matrix-vector multiplication can be vec- torized, in other words, it minimizes the transfers and cache misses in the memory cache.

The TILE compilation parameter is highly dependent on the hardware architecture, which means it can be optimized for each machine. 35

Figure 2.3: a) Positioning the mask with a structural disorder ”over” our system. b) Eﬀect of the structural disorder in our lattice in case the atom self potentials and hopping potentials are changed. b) Eﬀect of the disorder in our lattice in case of vacancy in such atom. The greyed-out atom with the greened-out hopping interactions represents the Mo atom with changes in its self potentials and interactions, any of its matrix terms. 36

Chapter 3

Calculation Procedure

This chapter presents the details of calculation using the methods discussed in the previous section.

In the next sections, this procedure will be presented alongside understandable examples using the well-known graphene structure in order to make these steps clear for the reader. The same concepts will be used in the next chapter with our system of interest

- the monolayer MoS2 TMD. The code for the simulation of graphene with a vacancy is available in AppendixC.

3.1 Conﬁguration Parameters

The first step is to supply the configuration parameters for the KITE software. Firstly, we set the lattice as described in the following subsection - the unit cell, which holds the information about the on-site energies and inter and intra-cell hopping elements, together with the number of unit cells in each direction. For now it is only possible to have periodic boundary conditions using KITE, that is the sites of one extremity of the system are connected to those in the opposite extreme. But because we are attempting to simulate an extended system and not a finite one, the periodic conditions are actually more convenient. This, along with the disorder configuration, sets the system Hamiltonian. 37

3.1.1 The Lattice

For the case of the graphene, we build the hexagonal carbon lattice using the vectors π π π ˚ aC =(0, 2 sin( 3 ))×aCC and bC =(1 + cos( 3 ), -sin( 3 ))×aCC , aCC = 1.42A. The lattice is shown with hoppings up to the ﬁrst neighbors in Figure (3.1).

Figure 3.1: Graphene lattice with vectors a = aC and b = bC . a) Up-down view (”cC ” vector), the outlined area shows the primitive unit cell. b) Birds eye view.

3.1.2 Choosing the Kernel

The calculation of DOS using KITE unrolls in a similar fashion of the 2.3.3 subsection of the last chapter except for the kernel used. Instead of using a kernel from start, the KITE software makes sure to have the Chebyshev function polynomial expansion parameters free to be set accordingly to the need. That is - ﬁrst the more generic moments coeﬃcients

µm given in equation (2.26) are calculated and, if needed, those are reduced to the proper coeﬃcients of a given kernel.

Given that, KITE offers as a post-processing tool the choice between Jackson’s kernel given in equation (2.50) or the Green’s moments from equation (2.60). Now because of the discussion at the end of the 2.3.4 section, the best approach for a final result is to be decided. The differences between both options are discussed in the Appendix(A). 38

3.1.3 Assuring Eﬃciency - System Size & Number of Divisions

For maximum efficiency, the number of repetitions of our unit cell shall be such that it can be equally divided between all available TILEs. As mentioned in the previous chapter, the TILE is a subdomain defined for memory optimization. The TILE is set to be a 64×64 unit cell domain in the real space, so for that reason, the number of repetitions of unit cells must be at least 64 in each direction. For each processor thread, at least one TILE is to be processed. Because we are interested in an isotropic simulation, in both linearly independent directions there should be the same number of TILEs. The availability of threads now defines how big can our system get. Given that, the most efficient form to arrange our system is to work with a number of cores that is a square of an integer number and equally divide the number of calculated TILEs between them.

The calculations in this work took place in the UFF cluster managed by L. Canonico,

R. Muniz, and D. Soares. The cluster disposed of machines with 24 cores each, which means, by the thought process above, we can work simultaneously with 16 (42) threads, on a subspace of 4 TILEs in each direction. Because the minimum quantity of unit cells per TILE is 64×64, 4×4 TILEs give us at least a 256×256 unit cell system.

For a basic example, a graphene lattice of 512×512 unit cells will grant us a high enough resolution for our calculation. Because graphene is a very simple structure composed of only two diﬀerent orbitals per unit cell, the increase the system size does not cost us much in computational terms, but a bigger lattice will not be necessary for this example.

3.1.4 Calculation Parameters

Next, the spectrum range is provided to the software. Because the Chebyshev polynomials are defined between -1 and 1 (see equation (2.20)), it is paramount to provide the code with the electronic energy range of our system so that it can re-scale our quantities as illustrated in the equations 2.27. If the provided range does not encompass the whole 39 spectrum, the polynomial expansion of our spectral quantities will overflow and fail to give a meaningful expansion. It is OK, however, to overshoot the range size, once it would still encompass the whole energy structure. In fact, we might actually want to do this in case on-site energy disorders are added to our orbitals because in this case some of the bands shift in energy. To take in account this shifting, we purposely overshoot the spectrum range limits, giving the polynomials expansion a breathing range. It is unadvised, however, to provide the code with an arbitrarily large range, for the number of calculated points in each fit would also need to increase proportionally to maintain the same resolution to the expansion. To save the computational cost of unnecessarily calculating points which will never hold a value for such spectral quantity, the user should provide a proper energy range for their system.

The band structure provides us with the energy range of our spectrum by checking the highest energy and lowest energy band. If the user does not have access to the band structure or does not know the energy limits and leaves this quantity unassigned, the KITE software automatically determines a spectrum range that encompasses the system with the cost of a brief calculation.

Another calculation parameter is the number of energy points calculated, which determines the number of equally spaced energy values in the given spectrum range that are to be calculated. In the case of symmetrical range, it is important to input an odd number for the number of points if the 0eV point calculation is of interest. A new calculation for a different set of points smaller than the one provided in the configuration file can be easily performed by the post-processing tools, for, as we have seen, the expansion moments are the core for the calculations as well as the heavier load. This energy dot resolution however does not necessarily correspond to the real resolution for our calculation, despite being the lower limit for that one as explained in the previous chapter (2.3.4, 2.3.2).

Regarding the graphene, we can use the KITE’s automatic spectrum range identiﬁer.

It will take a bit longer to perform the calculation but will save us from having to put 40 the right numbers by hand. When dealing with more complex systems it may be faster to provide the input if it is previously known, especially when dealing with a disordered system.

Another important parameter is the number of Chebyshev expansion moments used. This is discussed deeply in Appendix (B), where it is set 512 moments for the graphene simulations.

3.1.5 Random Vectors

For the calculation of traces, we have seen in the previous chapter how the stochastic method is the best approach. From equation (2.40) we can see that the trace ﬂuctuation is given as a function of the random vector distribution, which implies that it may be pertinent to specify the choice for the distribution of the random variables. In the case of

KITE, the random vector distribution is given by an exponential phase-type distribution.

Given the equation (2.41) no more than 10 random vectors are needed for attaining enough precision given the system size of 512, which, by this equation, gives us a relative error of 0.014. The time of execution however scales linearly with the number of random vectors used - for each new distributed variables, all the calculations must be redone.

Due to the instability of the cluster facilities power provider, the extended calculation time of increasing random vectors may compromise the whole calculation - if some error occurs during any of those calculations all progress is lost. An approach to this issue is to perform multiple calculations using one single random vector. This can be done with no complications since, from analyzing the equation (2.39), we conclude that calculations using a number N of random vectors are the same as performing an average of the result from N calculations using 1 random vector. With this, any inconvenience will disturb one only random vector run from a batch of N and not all N calculations simultaneously. 41

3.2 Graphene Density of States

Following the above procedure and the calculation of section 2.3.3, we can perform the calculation for the density of states (DOS) in graphene.

Initially, the results for the pristine graphene will be presented. For the DOS, we get the classic ﬁgure for the single-atom honeycomb lattice as in Figure (3.2).

Figure 3.2: 512×512 unit cell grahene system, 512 moments DOS.

Next, let us see the eﬀects of the on-site structural disorder mentioned in the previous chapter (2.4.1). We apply to one of the carbon atoms inside its unit cell a shift in its on-ste energy by a Gaussian distributed random value with average 1.5 eV and standard deviation of 1 eV. In Figure (3.3) we see this mask being applied to all unit cells of the system. From Figure (3.4) we can see the expected eﬀect: the system energies are shifted up, but not uniformly.

At last, let us look into the case of structural disorders. As explained in the previous chapter, this disorder does not only modify the on-site potentials but also with the hop- 42

Figure 3.3: Illustration of the graphene structure with the same mask applied over every unit cell. Half of the carbon atoms (brown color) are getting disordered (green color) by some amount given by the chosen distribution.

Figure 3.4: Same system as in Figure (3.2) but with Gaussian on-site disorder. pings between orbitals. For this example, we took away 10% of all carbon in our system and its respective interactions. A 10% disordered system is heavily disordered and it is 43 not expected that this concentration would represent a system with natural defects. Here, a high concentration of native disorder was chosen to emphasize its eﬀect on the system, helping the illustration of the example. The result in our lattice can be seen in Figure (3.5) and the result for our DOS can be seen in Figure (3.6).

Figure 3.5: Removing 10% of carbon atoms of graphene lattice.

Figure 3.6: Graphene with 10% vacancy. 44

In the next section, we will observe the eﬀects of disorder in the conductivity - in particular, we are going to probe if the electronic states associated to this new peak are conductive or not.

3.3 Longitudinal Conductivity

The conductivity tensor is an important quantity to be calculated for it connects the theoretical models to experimental transport measurements. Instead of a scalar quantity, or a one-point correlation function, as calculated in the previous sections for the DOS, the conductivity tensor is a two-point correlation function. A tool for the calculation of such quantity is the Kubo’s Formula and its variations.

3.3.1 Kubo Formulas

The Kubo Formula

Kubo’s formula outputs the linear response of an observable in the face of a time- independent perturbation. The Kubo’s formula for the conductivity tensor is, then, a current-current correlation function [100]

1 Z ∞ Z ∞ σα,β(µ, T ) = dt dλ hf(µ, T, H)jαjβ(t + i~λ)i , (3.1) Ω 0 0 being j the electronic current operator, f(µ, T, H) ≡ 1/(1 + e−(µ−H)/kB T ) the Fermi-Dirac distribution for a chemical potential µ and temperature T in a Ω volume sample.

Kubo-Bastin Formula

The above formula is obtained under very general conditions with the only requirement that the electrical ﬁeld is weak enough so that the linear-response theory is valid. We can further simplify this relation by ignoring the electron-electron interactions. After performing this approximation and subsequent steps, the equation (3.1) can be written 45 as the Kubo-Bastin formula: [101]

2 Z E+ + − ie ~ dG () dG () σα,β(µ, T ) = d f()Tr vαδ( − H)vβ − vα vβδ( − H) , (3.2) Ω E− d d with H and v being the non-interaction Hamiltonian and velocity operators. The velocity operator v can be expressed in real space with the help of the position operators |Rii , i = 1, ·D and the Hamiltonian matrix elements via the Heisenberg relation

D 1 1 X v = [R,H] = (R − R )H |R i hR | . (3.3) i i i j ij i j ~ ~ i,j=1 This form ﬁnds itself useful in the case of sparse tight-binding Hamiltonians like ours. To perform a Chebyshev expansion, we follow the steps in a similar fashion of the

DOS expansion. In ﬁrst place, the system is re-scaled in the [−α,α] interval so that the ˜ ˜ re-scaled conductivityσ ˜α,β(˜µ, T ) is expressed in terms of the re-scaled Green function, G, and δ function, δ˜. Those last two can be represented in their polynomial form as [86].

N ˜ ˜ 2 X Tn ˜ δ(˜ − H) = √ gn Tn(H), 2 δ + 1 π 1 − ˜ n=0 n,0 N ±in arccos(˜) ˜± ˜ 2i X e ˜ G (˜, H) = ∓√ gn Tn(H), so that (3.4) 2 δ + 1 ˜ n=0 n,0 ˜± ˜ N ±in arccos(˜) ! dG (˜, H) X gn d e ˜ = ∓2i √ Tn(H). d˜ δ + 1 d˜ 2 n=0 n,0 ˜ Inserting these in the re-scaled version of the equation (3.2), one obtains, for the conductivity, the following

4e2 4 Z 1 f(˜) X σ˜ (µ, T ) = ~ d˜ Γ (˜)µαβ , (3.5) α,β πΩ ∆E2 (1 − ˜2)2 nm nm −1 n,m where g g h i µαβ = n m Tr v T (H˜ )v T (H˜ ) , nm (1 + δ )(1 + δ ) α n β m n0√ m0 √ (3.6) 2 in arccos(˜) 2 −in arccos(˜) Γnm(˜) = (˜ − im 1 − ˜ )e Tn(˜) + (˜ + in 1 − ˜ )e Tm(˜). Now, regarding the numerical integration of the equation (3.5) integral, there are two possible approaches. Either one performs the integration before expanding the function 46 in terms of Chebyshev polynomials or one expands before the integration. Those two routes outcome results with different convergence rates and accuracy [93]. Although it is more convergence-efficient to have the integration done in before the expansion, it will be preferred to do the opposite. The Kubo-Bastin equation (3.5) can be further simplified for T = 0 by performing the energy integration analitically, which gives the Kubo-Streda formula. The Kubo-Streda can be expressed as a contribution of two different terms which, in the case of Chebyshev expansions, have different convergence rates causing for inaccuracy.

Kubo-Greenwood Formula

For many applications, and in all cases in this study, the conductivity is calculated in the special case of longitudinal conductivity. This grants a way to simplify the equations (3.5) in making α = β we obtain the Kubo-Greenwood formula

Z df(, µ T ) σ (µ, T ) = d σ (), (3.7) αα d 0 where σ0 is the conductivity at T = 0:

2 σ0() = 2πe ~Tr[vαδ( − H)vαδ( − H)]. (3.8)

Re-scaling and expanding the δ-functions in this equation, one obtains the Kubo-Greenwood formula in terms of the Chebyshev polynomials [81]

2πe2 4 X σ (˜ ) = ~ µ T (˜ )T (˜ ), (3.9) αα F Ω π2(1 − ˜2 ) mn m F n F F m,n where ˜F is re-scaled the Fermi energy.

3.3.2 Eﬃciency

As it can be seen in the equations (3.5) and (3.9), due to the many index sums of multiplication of the Chebyshev polynomials, the computational cost scales quadratically with the 47 total number of moments N. In the case of the Kubo-Greenwood formula, the eﬃciency can be taken to a linear scale by re-writing the longitudinal conductivity as

i i i σαα(F ) = hΨα(F )|Φα(F )i , (3.10) where

i i |Ψα(F )i ≡ vαδ(f − H) |Rii , |Φα(F )i ≡ δ(f − H)vα |Rii . (3.11)

These can be expanded independently via the KPM as

∞ ! i 2 g0 X |Ψα(F )i = q vα + gnTn(F )vαTn(H) |Rii , 2 2 π 1 − F n=1 (3.12) ∞ ! i 2 g0 X |Φα(F )i = q vα + gnTn(F )Tn(H)vα |Rii , 2 2 π 1 − F n=1 which scale linearly with the system size. This, along with the stochastic evaluation of traces expression for the position vector from the section (2.3.1), that is taking |Rni → |ri, results in ﬁnally expressing the longitudinal conductivity as

R X r r σαα(F ) = hΨα(F )|Φα(F )i . (3.13) r In this case, the entire polynomial expansion is made for every energy point. Therefore if the number of calculated points is large enough, it might be the case that it actually becomes less eﬃcient to perform this independent expansion [102].

3.3.3 Results

Here basically two methods of conductivity calculation will be used to attain information via KITE - the regular conductivity calculation that uses the principle of Equation (3.5) and thus scale quadratically with N and the ”single-shot” longitudinal conductivity that uses the linear scalability of Equation (3.13). The ﬁrst can be used as a probing tool for our system, that is, to obtain qualitative information for a similar small system. The second gives us freedom of doing focused calculations in some speciﬁc energy points in 48 our system - in this case the vicinity of the bandgap. The subsequent calculations will show the results for the ”single-shot” approach.

Let us recover our graphene example and show the eﬀect of the vacancy structural disorder on its conductivity. We can see the result in Figure (3.7). For the case of graphene, the vacancy peak originated at the gap showed itself to be conductive for this system size.

Figure 3.7: Conductivity comparison with DOS in the cases of - a) Graphene without disorder; b) 10% vacancy disordered graphene. Here the conductivity calculation was not made on the entire system, but only on the part of it near or inside the gap.

These results for the disordered graphene DOS and longitudinal conductivity are compatible to the studies of references [103–105], where they are discussed and explained in more detail. 49

Chapter 4

Results

With the discussion of the methods and the simulation procedure set, let us now approach the objective of this study. - the monolayer MoS2 compound. In this chapter, the results of the simulations of our system will be presented. The lattice and intra-lattice interactions were acquired by the combined use of the Quantum Expresso and wannier90 software. This lattice was then simulated via repetition of its unit cell using the software KITE and the results for the spectral quantities of DOS and conductivity are shown in the next sections.

Subsequently, we considered the use of a super-cell description of our lattice for the simulation because, as we will see, this will imply in a more precise and less approximated description of our hoppings. The calculations and results are again performed for this case, results that can be used for comparison with outcomes of previous studies of this system present in the literature.

The comparison is followed by subsequent simulations of our system accounting for the spin-orbit eﬀect, by doubling the number of orbitals used before up to 26 per unit cell. We calculated the DOS which revealed the spin-related nature of the vacancy peaks. 50

4.1 System Parameters

First, we shall review the parameters needed for our calculations set in the last chapter for the graphene, but now in the case of MoS2.

Lattice

To build the lattice, we follow the process described in Chapter2. We have here a similar process to the one described at Reference [106] as we share the same initial lattice data.

Using the Quantum Expresso DFT numerical package [107], we an ab-initio calculation under an LDA approximation of a monolayer MoS2 super-cell composed of 3×3 unit cells with the orbitals described by the vector in equation (2.10). In sequence, this result was used as input for the wannier90 software, where a 6×6 super-cell was simulated providing us with all the hopping terms within this range. These results were available thanks to a collaboration with A. Molina-Sanchez from the International Iberian Nanotechnology Laboratory.

Some of the lattice parameters for the hexagonal MoS2 obtained from this process are the lattice constant a = 3.102A,˚ the Mo-S distance dMo−S = 2.378A˚ and the S-S distance dS−S = 3, 129A,˚ which are compatible with the previous measurements, as mentioned in the introduction [51]. Along with this, the hopping parameters between atomic orbitals were derived. For implementation in KITE, only hoppings terms with energy higher than 8.1% of the maximum hopping value were considered due to a numerical issue that limited the number of neighbors for intercell hoppings. With these parameters, one can deﬁne the

unit cell, as shown in Figure (4.1), that will be repeated in the lattice vectors aMoS2 =(3.102 ˚ ˚ ˚ ˚ A, 0 A) and bMoS2 =(−1.551 A, 2.687 A) directions using the KITE software.

With this unit cell deﬁned, we can calculate the band structure via Pybinding [108] and compare with the DFT band structure as shown in Figure (4.2). We can see that there is a good agreement between them. 51

Figure 4.1: MoS2 lattice with vectors a = aMoS2 and b = bMoS2 . a) Up-down view

(”cMoS2 ” vector), the outlined area shows the primitive unit cell. b) Bird’s eye view.

Figure 4.2: Comparison between the Pybinding tight-binding band structure given our parameters (green) and the DFT calculated one (blue).

Calculation Parameters

This unit cell is then repeated in both lattice vector directions to achieve a 512×512 lattice system. With this size, it is safe to choose 1024 Chebyshev polynomial moments to perform our calculation with precision as discussed in AppendixB.

For the spectral range, a symmetrical range between -15.5 to 15.5 eV was chosen. In this interval, we calculated the density of states in a total of 5001 points, giving us a dot 52 energy resolution of 6 meV.

The other parameters for our simulation were already set in the previous chapter.

4.2 MoS2 Density of States

With the graphene example in mind, we can have a better intuition for the case of DOS in MoS2. First, we see the pristine DOS of our 512×512 unit cells system with 1024 Chebyshev moments side-by-side with the same system but with some sulfur vacancies, as can be seen in Figure (4.3), which has the same type of disorder as in Figure (2.3) and (3.6). Here we can get the idea of where to focus our analysis.

Figure 4.3: a) Pristine 512×512 unit cells, 1024 moments MoS2. b) Comparison between pristine and disordered system.

Let us now pay more careful attention to what happens inside the gap between valence and conduction bands, where the peak caused by our vacancy appears.

In the following ﬁgures we see what are the eﬀects of increasing vacancy concentration in three cases - with vacancies only on molybdenum (Mo - Figure(4.4)) sites, vacancies only on sulfur (S - Figure(4.5)) sites and vacancy on both (Figure(4.6)) sites. All three masks act in the same system of Figure(4.3- a) as base. These results will be discussed in a later section. 53

Figure 4.4: Multiple concentrations of vacancies on Mo with concentrations of a) 0.2% to 1% disordered system. b) Highly disordered - 0.8% to 2%, the arrow indicates the extra peak observed in high concentrations).

Figure 4.5: Multiple concentrations of vacancies on both S independently with concentrations of a) 0.2% to 1% disordered system. b) Highly disordered - 0.8% to 2%, the arrow indicates the extra peak observed in high concentrations.

4.3 MoS2 Conductivity

For the MoS2 conductivity, we have the results for our three cases of disorder in Figures (4.7),(4.8) and (4.9).

As explained in the Appendix (A), we sacriﬁce some of the smoothness of the function to be able to attain a better resolution for our points. The oscillations depend on the broadening parameter η set for our Green function expansion and the ones in the pristine 54

Figure 4.6: Multiple concentrations of vacancies on Mo and on both S independently with concentrations of a) 0.2% to 1% disordered system. b) Highly disordered - 0.8% to 2%. Here no discernible peak shows itself in the more disordered cases, suggesting that the peaks pointed by the arrows in the previous cases were but the interference of S vacancies on Mo atoms or vice-versa. That is, when the vacancy concentration for one specific atom is high enough, we are expected to see disorder effects on unit cell level in some occurrences. To isolate these effects, we are going to consider concentrations of 1% or lower.

Figure 4.7: Conductivity comparison between pristine, 1% vacancies on S, 2% vacancies on S - a) In scale with the system conduction band ity; b) Zoomed in the gap - see scale.

case can be loosely interpreted as base noise. The interesting aspect of the plots is that we clearly see a diﬀerence between disordered systems and the pristine one, conﬁrming that our vacancy peaks show a conductive behavior. 55

Figure 4.8: Conductivity comparison between pristine, 1% vacancies on Mo, 2% vacancies on Mo - a) In scale with the system conduction band conductivity; b) Zoomed in the gap.

Figure 4.9: Conductivity comparison between pristine, 1% vacancies on MoS2, 2% vacancies on MoS2 - a) In scale with the system conduction band conductivity; b) Zoomed in the gap.

4.4 Super-cell

Because of how KITE is optimized, some of the hoppings between sites in diferent unit cells further than the second neighbors are removed by the cutoﬀ and thus the system simulation does not account for every interaction obtained by the wannier90 software. As we are working with hoppings up to the second nearest unit cell in some cases, to better represent the bandstructure of our system, we needed a re-deﬁnition of the unit 56

Figure 4.10: Super-cell arrangement in comparison with primitive cell. a) 9×9 primitive cells. b) 3×3 super-cells.

cell. That is - instead of working with the unit cell defined in the previous sections, a three-by-three super-cell built out of 9 repetitions of the primitive cell was defined. A unit cell composed of one Mo atom and two S atoms can be alternatively be called as primitive cell, as a distinction from super-cell unit cells. A 3×3 grid of primitive cells assures us that hoppings to the previously second or third nearest neighbor will now only fall into the first neighbor unit cell case, as seen in Figure (4.11). The redefinition of the lattice in this way demands a redefinition of the hoppings between sites - for each orbital inside the unit cell, it must be considered all orbitals to which we expect to have hoppings and not only to the neighbor unit cells. That is, we have consider the hoppings to those orbitals inside of the very same super-cell which does not belong to the same atom or primitive cell.

Now, because we are approaching the calculations with the super-cell as described above, the 256×256 super-cell space corresponds to a 768×768 primitive unit cell Hamil- tonian. Using the 13-orbital tight-binding model of the previous chapter given in equation (2.10), we are working with a 9984×9984 Hamiltonian. As this is already a large number and enough for the intentions of this study, we decided to keep the TILEs per thread at its minimum. 57

Figure 4.11: Diﬀerence between hopping reach in terms of unit cell in the cases: a) primitive cells, b) super-cells. Only the hoppings for the central Mo atom are shown not to clutter the image.

This transformation makes applying disorder no longer straightforward as described in section 2.4.1, rather we have to account for the eﬀect of considering a large unit cell as the subject of disorder on the eﬀective concentration and distribution. This is so because our disorder-adding mechanism is unit-cell oriented, so that we have to take into account the new interactions inside the super-cell.

Because we are working with a 3×3 super-cell, that is a square of three primitive cells in each direction, simply adding a vacancy in place of one of the atoms of our super-cell, will result in an overall disorder with concentration nine times smaller. At ﬁrst, this could be addressed simply by increasing the concentration of the disorder in the whole system by nine times. This however may cause some issues regarding the nature of the vacancy distribution.

Suppose the disorder is structured in such a way that there is a molybdenum vacancy centered in the super-cell. When imposing this same structure of disorder for all the super-cells of the system, a repeated structure will arise - a bundle of sites with pristine atoms with a disordered atom in the center. This along with the aggravating factor of the higher concentration due to the proposed ﬁx may cause a crystalline disorder structure in which a disorder is always surrounded by eight pristine unit cells as illustrated in Figure 58

(4.12). This thought experiment suggests that the vacancy structure should be such that all the atoms of interest (in this case molybdenum) of the super-cell are subject to being removed and form a vacancy, not merely one. In practice, this translates in having multiple disorder masks acting at the same time over the unit cells. Each atom of interest is associated with a vacancy mask with the original unaltered concentration. With this, not only we get rid of the concentration complications, but also deal with the issue of the arising crystalline structure by homogenizing the distribution.

Figure 4.12: Illustration of possible quasi-crystalline arrangement.

DOS on Super-cell

As we see in Figure (4.13), there are some noticeable changes in our density of states. We can see that the in-gap peaks are better resolved and have higher amplitudes and also the outer bands had their shape corrected, but no positions have changed nor new peaks have arisen. Let us analyze if the previously observed behaviors of vacancy levels remain. In Figure (4.14), we can see that for higher concentrations some diﬀerences appear in the form of peaks, which could be hidden before, but now are better resolved due to more prominent and better-resolved peaks. About conductivity, no extra meaningful information can be obtained in analyzing the super-cell system conductivity for the nature of conduction of the vacancy peaks shall not change with the new unit-cell structure and the conductivity value was obtained primarily 59

Figure 4.13: Comparison between diﬀerent unit cells - primitive unit cell (original) and super-cell. for qualitative purposes. To prove this point, we calculated the conductivity using the super-cell arrangement in Figure (4.15).

4.5 Discussion

Despite our results have included the simulations for Mo vacancies, the sulfur vacancies are more likely to appear during the fabrication than other types of point defects [109,110].

The results achieved here have shown agreement with previous results in literature.

Illustrated in Figure (4.16), the theoretical DFT results for the monolayer MoS2 on a substrate that show the DOS for vacancy peaks lying in proximity with our peaks in terms of energy. The molybdenum vacancy is less studied given it is also less common, but there are also DFT results in literature [110, 111] compatible with our simulations. The fact that the real space TB simulations of a large system presented in this study is compatible with precise DFT calculations shows that real-space simulations where the tight-binding parameters originate from ﬁrst principle calculations can capture many of the features of 60

Figure 4.14: Comparison between super-cell and pristine-cell systems under the same disorders; highly disordered on the right. a) and b) - disorder on S. c) and d) - disorder on Mo. e) and f) - disorder on both. The arrows mark the point of extra peaks in the super-cell mode in comparison with pristine cell. the original model and be used in realistic large-scale simulations. Despite not having a precise quantitative agreement with DFT results, it shows qualitative agreement with the expected results. 61

Figure 4.15: Comparison between super unit cell conductivity (with a slightly higher broadening parameter η) and pristine conductivity of a system with 1% concentration of vacancies both on S and Mo orbitals. Overall the peaks are seen in the same positions.

Other than that, many references study the eﬀect of vacancies on other TMDs and, given the similarity of their structures, the compounds are expected to behave similarly when under vacancy-type disorder. And in fact they do as shown in references [65, 66,

113–115] that study TMDs such as MoSe2, MoTe2, WS2 and WSe2. A diverging result can be seen in reference [116], where a 6-band tight-binding model is used to simulate the monolayer MoS2 with spin-orbit coupling (SOC). Despite displaying the vacancy peaks inside the gap, the positions were quite oﬀ, almost the opposite of the ones achieved by this and other studies. Nonetheless, that study was pioneering in the simulations for this TMD using a TB model.

Experimentally, it has been observed that the mobility of monolayer MoS2 is usually one order of magnitude below the theoretically expected one [67, 109, 112, 117]. The defect states created by low concentrations of sulfur vacancies do not make the TMD act as an electron donor, in other words, they do not contribute to an n-type behavior 62

Figure 4.16: Figures extracted from references [67][a) and b)], [110][c) and d)] and [112][e) and f)]. For a) and b) we have a band structure of MoS2 with 0% and 1.6% concentrations of S vacancy respectively calculated on a 9×9 system via DFT with GGA approximation. The red arrow and the number (in eV) show the energy level oﬀset between unoccupied defect state and conduction band bottom for each band structure. Simulation made with the Vienna ab-initio simulation package. Next we have atom-projected density of states for the monolayer MoS2 with: c) Mo vacancy and d) S vacancy. These were calculated on a 7×7 FETs system via DFT with LDA approximation. The system is commposed of monolayer MoS2 encapsulated by alumina (Al2Ox) and hafnia (HfOx). Several states are introduced within the nominal bandgap and signiﬁcant distortion of the valence band edge structure is observed. As for e) and f), we have DFT calculations for band structure (e) and partial density of states (f) for single-layer MoS2 5×5 supercell with a sulfur vacancy . The localized states are highlighted by red lines. Green dashed line corresponds to the case without vacancy.

MoS2 [118, 119]. Due to the proximity to the conduction band, these states act more as an electron acceptor, a p-type behavior. Only for sulfur vacancy densities superior to

6.3%, the peak is broad enough so that it manifests states with lower energy. The greater 63 energy diﬀerence from the conduction band makes it possible to observe n-type behavior on MoS2 in this case [67]. In fact this state broadening can be seen here in Figure (4.14 - (b)).

Reference [112] discusses the eﬀect of the temperature in the conductivity of TMDs. It concludes that, as the temperature decreases, hoppings over a longer distance but with close energy spacing are more favorable, so it would be expected to observe the electron- donating behavior even for small concentrations if the temperature is small enough. This also opens the discussion to the spatial distribution of the disorder inﬂuencing the conductivity, as discussed in reference [109].

Spin-orbit Coupling

Calculations were also run for the SOC case, with twice as much orbitals (26 orbitals) compared to the previous calculations (13 orbitals), to gather spin-related information of our in-gap states and better compare with some previous results of reference [116].

Figure 4.17: Comparison between SOC super-cell (continuous) and no-SOC super-cell (dashed) with 0.5% vacancies on Mo (red and orange) or S(green and olive). Simulation on a 512×512 super-cell system with 3072 moments - the system is needed to be large enough so that the SOC splitting peaks can be resolved.

In Figure (4.17) we can see a shift in energy in the SOC case and the splitting of the peaks: the 2 leftmost Mo vacancy peaks and the leftmost S vacancy peak. Given 64 the overall compatibility of our previous results, it is plausible to conclude that the low orbital approximations lack in providing a good description, and the more orbitals the

ﬁner is the information we get from the simulations. This result was ﬁrst observed experimentally in TMDs in Reference [120] along with some theoretical calculations. Our result shows accordance with both the ab-initio calculations and the experimental result. Moreover, they suggest that the chalcogen vacancies concentration could be used to tune the spin-valley polarization in TMDs due to the hybridization between defect-localized in-gap states and delocalized, pristine-like states.

This could potentially induce single-photon emissions. 65

Chapter 5

Conclusion

This study presented the process and the tools to simulate and analyze spectral quantities of crystal systems with two dimensions of a considerably large size and number of orbitals under any arbitrary disorder. This serves as a stepping stone not only for the prospects for this study but also for a multiplicity of other problems in Condensed Matter Physics.

This document started with an introduction (Chapter1) to the TMDs and its disordered cases with focus on the MoS2. Subsequently, Chapter2 presented the methods that details, among others, the Chebyshev expansions, which are used in our KPM and

Green’s expansion calculations. Next, (Chapter3) presented the procedure that is used in our analysis using graphene as an example. Finally, Chapter4 presented the results of the calculations on vacancy disordered MoS2 with and without spin-orbit coupling. Indeed, the results here presented, bring a new look to real-space calculations of disordered MoS2 TMD spectral quantities with an unprecedented level of precision. But the main outcome of this study was the development of the tools that structure the tight-binding ab-initio-derived system with disorder on a super-cell conﬁguration while on KITE environment. This unravels the possibility for a very similar process to take place when accounting for direct ab-initio parameterization of disordered systems.

Given that the first principle calculations are computationally costly, the system constructed by these methods cannot be very large. They are usually descriptions of super- cells composed of around N × N unit cells, with sizes of the order of the super-cell 66 developed here in Chapter4. The calculations via first principles of such super-cells containing vacancies describe the resulting disorder effects with more fidelity than would the simulation with the simple removal of an atom, for it would also account for other effects such as the structural relaxation . But now, with the toolset here developed, one can bring the entire ab-initio precision on the effects of the disorder to build a super-cell mask. This new mask is built in such a way that it would reproduce the very details about the hoppings and energy brought by the ab-initio calculations on our system. Now, by having it under the KITE environment, it becomes possible to build disordered systems hundred of thousands of times larger while keeping the ab-initio fidelity. This toolset is composed of all the methods described in this system study - such as the wannier90 software that connects the ab-initio DFT calculations with the KPM KITE simulations, and the effect of a mask on the system, which was explained with the help of the simple graphene system. It also includes the building of a TMD super-cell with as much as 234 interconnected orbitals and the analysis made around its vacancy position when facing the reproducibility of such disorder in the system.

Perspectives

This collection of methods opens the way for simulating large systems with diﬀerent kinds of ab-initio structured disorders. And, because KITE is developed to calculate quantities such as the density of states, longitudinal conductivity, linear and non-linear optical conductivity, hall eﬀects, and others, the scope of disordered systems studies becomes overall greatly increased.

All these analyses can also be translated, but not only, to the other TMDs such as MoSe2 or WS2, which are systems under intense research in the past few years [65, 66, 113–115], expanding the scope of this study. 67

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Appendix A

Resolution - Broadening vs KPM

Two approaches to the expansion of our spectral functions in terms of Chebhyshev polynomials were introduced in the Chapter2. Let us discuss them here and compare their behavior on expanding a function and determine the best approach to perform our calculations.

First it was suggested the kernel polynomial method to behave as conduit for the expansion. From the kernels suggested, the Jackson’s kernel was introduced as one with a good convergence rate and good precision. Later it was introduced the Green’s function expansion method, which would approximate not the original function itself, but less abruptly approximate its Green function keeping a broadening parameter to hold the approximation in check while maintaining all the information stored in the green’s function.

The approach will prove itself valuable when compared in terms of convergence for, as we have seen, the moments of expansion are the bottleneck of our calculations. That method which achieve the most reasonable result with the least amount of moments will be the one chosen for our simulations.

For this study, it was simulated a simple MoS2 system of 512x512 sites which were occupied with the conventional unit cell. According to the equation (B.1), the number of Chebyshev polynomials needed for a good resolution using the KPM Jackson’s kernel should be around 1024. 86

Let us compare side by side the simulations for the DOS of the KPM and the Green’s Function expansions in Figure (A.1).

Figure A.1: 13 orbital, 512 moments MoS2 simulation with - a) Green Function approach with decreasing η broadening parameter; b) KPM approach with Jackson’s Kernel.

We can see that as the broadening parameter decreases and approach its lower bound as given by Equation (2.66 the resolution increases and the spectral function gets closer to the Jackson’s Kernel KPM approach. It is clear, however, that the Green’s Function expansion struggles to ﬁt the gap by itself. As the broadening parameter η from equation (2.63) decreases, we see a better ﬁt, as expected for this indicates we are reaching the thermodynamic limit.

Let us compare then inside the region of interest in Figure (A.2) - the gap between the valency and conduction bands - with values of η around its lower bound.

We see that for values of η close or lower than the Chebyshev momenta resolution (∆E/N = 31/1024 ≈ 0.03), Gibb’s oscillations are still being resolved. Once we give it a little boost, we can see that it stop resolving the oscillations, but, despite having it being somewhat close to the expected in-gap value (zero), it is clearly not precise, especially near the gap borders where it occurs the formation of a shoulder. Nonetheless, Gibb’s oscillations should be more manageable with the increasing number of polynomials. Let us then analyze the eﬀect f varying the number of polynomials for this system in the next 87

Figure A.2: Comparing Green’s Kernel with KPM-Jackson’s Kernel inside the gap, i.e. expecting DOS=0.

ﬁgure, Figure (A.3).

Figure A.3: In gap comparison between Green and Jackson’s with varying Chebyshev Moments - a) 512 moments - under fit; b) 1024 moments - good fit; c) 1536 moments - light over fit. (refer fit quality to Relation (B.1).

We can see that even though the number of moments increases in this ﬁxed system, the oscillations do not behave better. In fact, once we go further than the 2N suggested limit, the oscillations actually increase in amplitude in frequency. We can try by increasing the size of our system so that we still obey the 2N limit in the next ﬁgure. Yet it is relevant to study if the amount of polynomials needed to achieve 88 the convergence is worth the precision boost brought by this approach if any.

Figure A.4: In gap comparison between Green and Jackson’s with varying Chebyshev Moments - a) 512 sites, 1024 moments; b) 1024 sites, 2048 moments.

It is clear that even by doubling the size of our system so that more moments could be calculated, no considerable improvement was observed in our function. From this, we can conclude two things.

First that with 512 sites, our simulation is already getting close to the thermodynamic limit, and thus increasing the system will not only be more costly computational but also will not give us any better results.

Second is that the Green Function approach is not reliable for precision measurements.

It is still useful to hold extra information about our system that can be later be calculated with the proper KPM and in cases where the Green’s function is needed, as for the Kubo’s formulas in conductivity calculation for example. The case where we are probing our system and interested in qualitative information. But for overall DOS calculations, Jackson’s kernel KPM approach will be preferred over the Green’s function for it is better precision and convergence.

We can further analyze in Figure (A.5) what are the implications of these oscillations when facing disorder that aﬀects our gap.

Because now we are looking for peaks in our gap, the oscillations are not welcome 89

Figure A.5: In-gap comparison between Green and Jackson’s with varying η on a system with vacancy on S atoms. and we have to make it so that η cannot resolve these oscillations anymore. This results in having poor precision in our region of interest, but it is enough for a qualitative measurement. 90

Appendix B

Number of Moments

For a continuous function, the discussions about the function expansion of the Chapter 2 holds completely, but we must have in mind that, in our case, we are trying to approximate a function that is not continuous, rather it is a function defined in many points. In the case of trying to approximate a continuous function to a set of points, if we make the number of moments N → ∞, we no longer guarantee the best convergence. In fact, once the expansion resolution surpasses the grid resolution, the expansion starts to over-fit the data and thus we lose the fidelity of the expansion. In order to know the maximum number of moments allowed not to overflow the expansion, we can simply compare the resolutions. Given a system of size D, the number of sites can behave as an effective grid resolution for its reciprocal quantity, which is the case for spectral functions. Comparing it to the expansion resolution that is proportional to N −1, we can derive the maximum number of moments. For the case of the Jackson kernel, using the equation (2.52) for the kernel resolution, one can write 1 π ≈ , D N + 1 N + 1 ≈ πD, (B.1)

N ≈ πD. Given this, it is safe to use N = 2D to keep a good resolution and to be sure not to over-ﬁt the data. In the case of 512 sites, the number of moments is then set to N = 1024. 91

We witness the clear effects of this study when we compare the DOS of our system in different configurations as seen below.

Figure B.1: 512x512 site simulation of MoS2, with 1% vacancy disorder on S atoms.

In Figure (B.1) we can see the closer we get to N = 2D better the resolution of our simulation. Not only the DOS inside the gap assumes quantities closer to zero, but also the peaks both from the disorder and the bands have a higher resolution.

For illustration, we can dispense some calculation time to analyze what would happen in the case we extrapolate the number of moments further from what we have calculated in Equation (B.1). The result is seen in Figure (B.2) shows that for N = 4D, which is larger than N = πD, there already are some points where the predicted over-ﬁtting shows itself. While no extra information about the system is brought to the analysis, we are subjected to misinformation.

For graphene, we can see from Figure (B.3) that the same happens. Few moments cannot ﬁt the data with precision and larger number of moments produce oscillations even in the valance and conduction bands. As the graphene examples do not require high precision and the number of moments being the bottleneck of our simulation, it was 92

Figure B.2: The same system as in Figure (B.1).

Figure B.3: 512x512 site simulation of graphene with vacancy disorder on 10% of atoms. Diﬀerent values for N - from 2−3D to 23D. Energy normalized for the hopping therm. preferred to use 512 moments for the graphene calculations. 93

Appendix C

Graphene Example Code

This appendix presents the python-based KITE interface for the graphene with vacancy example. We can see here concepts discussed in Chapter3.

""" ##################################################################### #KITE | Release1.0# # # # Kite home: quantum-kite.com# # # # Developed by: SimaoM. Joao, JoaoV. Lopes, TatianaG. Rappoport,# # Misa Andelkovic, Lucian Covaci, Aires Ferreira, 2018-2019# # # ##################################################################### """ """ Honeycomb lattice with vacancy disorder

Lattice: Honeycomb1[nm] interatomic distance andt=1[eV] hopping; Disorder: StructuralDisorder, vacancy with concentration0.1 inside A and0.1 insideB sublattices; Configuration: size of the system 512x512, without domain decomposition(nx=ny=1), periodic boundary conditions, double precision, automatic scaling; Calculation: dos;

""" import kite import numpy as np import pybinding as pb def honeycomb_lattice(onsite=(0, 0)): """Makea honeycomb lattice with nearest neighbor hopping 94

Parameters ------onsite: tuple or list Onsite energy at different sublattices. """ ""

# define lattice vectors theta= np.pi/3 a1= np.array([1+ np.cos(theta), np.sin(theta)]) a2= np.array([0, 2* np.sin(theta)])

# createa lattice with2 primitive vectors lat= pb.Lattice( a1=a1, a2=a2 )

# Add sublattices lat.add_sublattices( # name, position, and onsite potential (’A’,[0, 0], onsite[0]), (’B’,[1, 0], onsite[1]) )

# Add hoppings lat.add_hoppings( # inside the main cell, between which atoms, and the value ([+0,+0],’A’,’B’,- 1), # between neighboring cells, between which atoms, and the value ([-1,+0],’A’,’B’,- 1), ([-1,+1],’A’,’B’,- 1) )

return lat lattice= honeycomb_lattice((-0.0, 0.0))

# Add vacancy disorder as an object ofa class StructuralDisorder. In this manner we can distribute vacancy disorder # ona specific sublattice witha specific concentration. # unless you would like the same pattern of disorder at both sublatices, # each realisation should be specified asa separate object struc_disorder_A= kite.StructuralDisorder(lattice, concentration=0.1) struc_disorder_A.add_vacancy(’A’) struc_disorder_B= kite.StructuralDisorder(lattice, concentration=0.1) struc_disorder_B.add_vacancy(’B’) disorder_structural=[struc_disorder_A, struc_disorder_B] # loada honeycomb lattice and structural disorder nx= ny=1 lx= 512 ly= 512 95

# make config object which caries info about #- the number of decomposition parts[nx, ny], #- lengths of structure[lx, ly] #- boundary conditions, setting True as periodic boundary conditions, and False elsewise, #- info if the exported hopping and onsite data should be complex, #- info of the precision of the exported hopping and onsite data,0- float,1- double, and2- long double. #- scaling, if None it’s automatic, if present select spectrum_bound=[ e_min, e_max] configuration= kite.Configuration(divisions=[nx, ny], length=[lx, ly], boundaries=[True, True], is_complex=False, precision=1) # require the calculation of dos num_moments= 512 calculation= kite.Calculation(configuration) calculation.dos(num_points=1000, num_moments=num_moments, num_random=1, num_disorder=1) # configure the*.h5 file kite.config_system(lattice, configuration, calculation, filename=’ vacancies.h5’, disorder_structural=disorder_structural)