Ab-Initio Quantum Transport Simulations with Tight-Binding-Like Hamiltonians

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Doctoral Thesis

Ab-initio quantum transport simulations with tight-binding-like Hamiltonians

Author(s): Stieger, Christian

Publication Date: 2019

Permanent Link: https://doi.org/10.3929/ethz-b-000418185

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ETH Library Ab-initio quantum transport simulations with tight- binding-like Hamiltonians

Diss. ETH No. 26312

Ab-initio quantum transport simulations with tight-binding-like Hamiltonians

A dissertation submitted to ETH ZURICH

for the degree of Doctor of Sciences

presented by CHRISTIAN STIEGER Master of Physics, ETH born June 3rd, 1983 citizen of Wallisellen ZH, Switzerland

accepted on the recommendation of Prof. Dr. Mathieu Luisier, examiner Prof. Dr. Jim Greer, co-examiner

2019

Acknowledgements

First of all, I’d like to express my sincere gratitude to Prof. Dr. Math- ieu Luisier for the opportunity to work in his group at the Integrated Systems Laboratory (IIS) and for his support and guidance. I am especially thankful for his thorough revision of this thesis. I would also like to thank Prof. Dr. Jim Greer for accepting co-examination. I am thankful to all the members at the Nano-TCAD group for their company and support. Special thanks go to Dr. Petr A. Khomyakov for his valuable advice, Dr. Hamilton Carrillo Nunez and Cedric Klinkert for testing Winterface and supplying feedback, and to Hansj¨org Gisler for keeping the coﬀee machine in working order at all times.

Abstract

In this thesis a framework for quantum transport simulation from first principles is introduced, focusing on coherent electronic transport, but discussing the importance of electron-phonon interactions for transition metal dichalcogenides (TMDs) as well. The transport model is based on the non-equilibrium Green’s function (NEGF) formalism which requires a localized basis set. To obtain a suitable Hamilto- nian matrix for a given device structure, a representative unit cell is first identified and its electronic properties calculated with density- functional theory (DFT) expressed in a plane-wave basis. These results must be transformed into a localized basis set using so-called maximally localized Wannier functions (MLWFs). From the MLWF representation of the original unit cell, the device Hamiltonian can be constructed with the help of properly designed upscaling techniques. The main focus of this thesis lies in the interfacing part of the MLWF representation with a quantum transport code, OMEN in our case. To this end, a code called Winterface was developed. Elaborations of its concepts, algorithms, and general functionality are presented on the basis of a molybdenum disulfide (MoS2) monolayer structure, as well as heterostructures involving tungsten disulfide (WS2). The process of Hamiltonian upscaling was simplified through the application of a two-stage process, where the first step consists of a conversion of the MLWF Hamiltonian data into blocks corresponding to interactions along atomic bonds. This is the key innovation of this work. This does not only make the actual upscaling procedure, which represents the second step, more transparent, as opposed to an approach relying on raw MLWF data, but also allows for an investigation of the local properties of super structures. The latter can help shed light

vii viii ABSTRACT onto the inﬂuence of local features such as interface regions. On the basis of this, more complex device geometries can be generated and the corresponding Hamiltonians be approximately constructed. Ex- amples of such super structures are provided with MoS2 on top of WS2 or both 2-D materials next to each other. The developed approach can be considered completely general, restricted only by the capability of the user to accurately model the developed physics in DFT and to ”wannierize” plane-wave results, as well as the imposed computational intensity.

Finally, the importance of the inclusion of electron-phonon interactions into the transport model is demonstrated on the basis of self- heating effects in MoS2, WS2, and black phosphorous devices. Due to the versatiliy of the NEGF formalism, this can be done conveniently through additional scattering self-energy terms. The interactions for the phonon part can also be derived from first-principles using density functional perturbation theory (DFPT). The electronic part results from multiple wannierizations using the concepts already introduced for the coherent case. Dissipative calculations present themselves as natural extensions of ballistic ones, but with a significant increase in computational burden. Zusammenfassung

In dieser Arbeit wird eine Umgebung fürTranportsimulationen auf quantenmechanischer Ebene präsentiert, wobei der Fokus auf elektro- nischem Transport liegt. Allerdings wird auch auf die Wichtigkeit von Elektron-Phonon Wechselwirkung in Ubergangsmetall-Dichalcogenen¨ (TMDs) eingegangen. Das Transportmodell ist dabei basierend auf dem nichtgleichgewichts Greens Funktionen (NEGF) Formalismus, der eine lokalisierte Basis benötigt. Um eine Hamiltonianmatrix für eine spezifische Kanalregion eines Transistors zu bekommen, muss zuerst eine repräsentative Einheitszelle gewähltwerden, deren elektronische Eigenschaften anschliessend mittels Dichtefunktionaltheorie (DFT) berechnet werden können.Da dafüraber eine Basis aus ebe- nen Wellen verwendet wird, welche nicht lokalisiert ist, muss zuerst eine Transformation zu einer Basis so genannter maximallokalisierten Wannier Funktionen (MLWFs) vorgenommen werden. Von dieser aus- gehend, kann auf Grundlage der primitiven Einheitszelle und speziell dafürentwickelten Hochskalierungstechniken eine Hamiltonianmatrix generiert werden, die die gewünschte Struktur beschreibt. Der Fokus dieser Arbeit liegt daher auf der Schnittstelle zwischen der Repräsen- tation durch MLWF und einem Transportsimulator, in unserem Fall OMEN. Zu diesem Zweck wurde ein Programm namens Winterface entwickelt, dessen Konzepte, Algorithmen und generelle Funktion- alitätam Beispiel einer Einzelschicht von Molybdändisulfid (MoS2), sowie Heterostrukturen des letzteren mit Wolframdisulfid (WS2), erklärt werden. Der Prozess der Hochskalierung von Hamiltonianmatrizen konnte mittels einer Zerlegung in zwei Schritte vereinfacht werden. Im ersten Schritt wird die Konvertierung einer Hamiltonianmatrix in MLWF Repräsentation in eine andere Repräsentation basierend auf

ix x ZUSAMMENFASSUNG

Wechselwirkungen entlang Bindungen zwischen Atomen vorgenommen. Hierbei handelt es sich um die hauptsächliche Innovation dieser Arbeit, da dadurch nicht nur der zweite Schritt, die Hochskalierung von Hamiltonianmatrizen, massiv vereinfacht wird, sondern zusätzlich auch Untersuchungen von lokalen Eigenschaften gröererStrukturen ermöglicht werden, womit der Einfluss lokaler Eigenschaften auf ihre Umgebung abgeschätztwerden kann. Auf dieser Grundlage können komplexe Geometrien verwirklicht und entsprechende Hamiltonian- matrizen approximativ generiert werden. Beispiele werden in Form von Heterostrukturen von einer MoS2 Schicht auf einer aus WS2, sowie mit beiden Materialen in derselben Schicht, präsentiert. Die entwickelte Methodik kann als allgemein anwendbar betrachtet werden, eingeschränktnur durch die Näherungenin DFT, sowie der da- rauffolgenden Wannierisierung und der resultierenden rechnerischen Belastung.

Schlussendlich wird die Wichtigkeit der Berücksichtigung von Elek- tron-Phonon Wechselwirkungen in das Transportmodell anhand von Beispielen bestehend aus MoS2, WS2 und schwarzem Phosphor, disku- tiert. Aufgrund der Erweiterbarkeit des NEGF Formalismusses, können diese Effekte leicht über zusätzliche Streuungsterme einbezogen werden. Die phononischen Wechselwirkungen könnenüber Dichtefunk- tionalstörungstheorie(DFPT) erlangt werden und der elektronische Teil aus mehreren Wannierisierungen, wozu die bisher entwickelten Werkzeuge ausreichend sind. Dissipative Berechnungen stellen daher eine natürliche Erweiterung des koherenten Modells dar, allerdings auf Kosten einer signifikant erhöhten Anforderung an die Rechenleistung. Contents

Acknowledgements v

Abstract vii

Zusammenfassung ix

1 Introduction 1

2 Quantum Transport From First-Principles 5 2.1 Density-Functional Theory ...... 5 2.2 Maximally Localized Wannier Functions ...... 9 2.3 Transport in the Wavefunction Formalism ...... 12 2.4 Transport in the NEGF Formalism ...... 15 2.5 Description of the Toolchain ...... 17

3 Hamiltonian Data in Terms of Bonds 23 3.1 Atomic Lattices and Unit Cells ...... 23 3.1.1 Unit Cell, Deﬁnition and Example ...... 23 3.1.2 Basis Expansions ...... 25 3.1.3 Metric in Periodic Space ...... 28 3.1.4 Automatic Detection of Basis Expansions . . . 29 3.2 Upscaling Technique for an Ideal Case ...... 31 3.3 Generation of Bond Interactions ...... 33 3.3.1 Matching Wannier Centers to Atomic Positions 33 3.3.2 Generating Interaction Data Along Atomic Bonds 39

xi xii CONTENTS

4 Generating Hamiltonian Matrices 49 4.1 Exact Upscaling Technique and Bandstructure Calcu- lations ...... 49 4.2 Interface with OMEN ...... 58 4.2.1 Wannierization Process ...... 61 4.2.2 Error Estimation ...... 63 4.3 Approximate Upscaling Technique and Local Bandstruc- tures ...... 67 4.3.1 Bond Index Substitution ...... 69 4.3.2 Combining Multiple Wannierizations ...... 74 4.4 Results ...... 83

5 Dissipative Transport 91 5.1 Transport Model ...... 91 5.1.1 Hamiltonian and Dynamical Matrix ...... 91 5.1.2 Electron and Phonon Coupling ...... 96 5.1.3 Calculation of Observables ...... 98 5.2 Device Results ...... 100

6 Structure of the Code 111 6.1 Libmat ...... 111 6.2 Liblat ...... 114

7 Conclusion and Outlook 117

A 2-D vs. 3-D k-point Grid in Monolayer MoS2 121

B Basis Set Reduction after Wannierization 125

List of Publications 127

Curriculum Vitae 141 Chapter 1

Introduction

The inevitable end of Moore’s scaling law [1] calls for novel transistor concepts that can deliver reliable logic performance in future ultrascaled technology nodes. Although the semiconductor industry has already moved to three-dimensional FinFETs [2], innovations at the architecture and material levels will be required for next-generation devices. Besides the widely studied gate-all-around (GAA) nanowire (NW) [3, 4, 5] and ultra-thin-body [6] field-effect transistors, recent years have seen the emergence of a new class of two-dimensional (2- D) materials consisting of atomically thin layers connected by van der Waals forces. Its first and probably most famous member is graphene, which was discovered by Novoselov et. al. in 2005 [7]. Despite im- pressive carrier mobility values (> 100,000 V/ms), graphene does not lend itself to logic applications due to the lack of a band gap. How- ever, the available design space for 2-D materials is huge, according to recent theoretical investigations [8]. Currently, a strong accent is set on transition metal dichalcogenides (TMD), such as MoS2, which appear more promising as future channel materials than graphene. A transistor made of a single monolayer of MoS2 was experimentally realized in 2011 [9]. Applications involving few-layer of heterostructures of 2-D materials have already been demonstrated, e.g. light-emitting diodes [10], photodetectors [11, 12, 13], memory cells [14] or memris- tors [15]. Among the 2-D materials that have received wide attention, black phosphorous (BP) stands out [16, 17]. As compared to TMDs,

1 2 CHAPTER 1. INTRODUCTION

BPs exhibit highly anisotropic electrical and thermal properties [18], which could pave the way for other original applications.

At this point, it is not clear whether 2-D materials can compete with other technologies and if so, which component or heterostructure is the most suitable at performing a given task. Whilst such a large number of possible configurations offers exciting opportunities in terms of novel device concepts, it also requires improved solutions, both experimental and theoretical, to explore the available design space. Technology computer aided design (TCAD) represents a pow- erful and well-established approach to address this challenge. Thanks to its cost- and time-effectiveness, TCAD can help experimentalists to rapidly converge towards the most promising contenders. However, at the current nanometer scale of the transistor dimensions, classical and semi-classical simulation methods such as the drift-diffusion or Boltzmann transport equations should be replaced by a quantum mechanical treatment of the device properties. To describe the electronic structure of 2-D materials, different methods exist, from the most fundamental ones, such as density functional theory (DFT) [19], up to empirical ones, for example tight-binding (TB) [20]. When coupled with a quantum transport solver, both approaches suffer from their own limitations, such as the size of the system that can be handled for DFT, or the need to create physically meaningful parameters sets for TB. As a compromise between them, a maximally localized Wan- nier function (MLWF) representation of the DFT results can be used [21, 22]. This can be seen as a first step towards ab initio device investigations.

The idea consists of identifying a primitive unit cell that is representative of the system of interest, perform a DFT calculation of it, convert the plane-wave results into a set of MLWF, and ﬁnally scale up the obtained TB-like Hamiltonian matrix to the size of the considered device. Once the general concept is established, what remains to be developed is a technique capable of providing the necessary throughput to screen the large design space of 2-D materials. This thesis aims at providing a toolbox, called Winterface, to automatically up- scale the MLWF representation of comparatively small unit cells to desired, sometimes complex, geometries, e.g. heterostructures. Since 3 the problem to be solved depends on numerical outputs and cannot be directly formulated in terms of equations, practical examples will be provided to demonstrate the principle of the proposed approach, identify its shortcomings, and give evidence of its utility to simulate nano-devices based on 2-D materials. The atomic structures selected as testbeds will ideally illustrate the key concepts that have been implemented. Although the physics of these examples might be in- teresting, this is not the selection criterion that was applied. Here, what matters is the robustness and versatility of the created Win- terface code. So far, hundreds of material conﬁgurations have been constructed and the transport properties of all of them could be successfully investigated, thus demonstrating the quasi-universality of the methodology. However, at this point, a complete automation was not possible, because critical components are missing, in particular an initial guess projection for the MLWF [23]. Additionally, acceptable approximations depend on a compromise between computational burden and physical accuracy, which have to be prioritized by the user of the upscaling algorithms available in Winterface.

While the main focus of this work is on coherent transport, it is expected that thermally-induced phenomena are considered as well: they are expected to play a significant role in ultra-scaled structures due to the close proximity of electrons and phonons and their increased coupling [24]. Few studies mainly concerned with graphene [25, 26, 27], but also molybdenum disulfide [28, 29] and black phosphorous [30] have discussed these issues. Still, it can be generally said that the thermal behavior of 2-D materials is not completely understood yet. Another reason for the inclusion of a dissipative scattering mecha- nism is the presence of a non-physical negative differential resistance in the ballistic simulation of several TMDs [31]. Such a behaviour has never been experimentally observed at room temperature [32]. The NDR originates from the TMD bandstructure, which exhibits several narrow energy bands that cannot carry any current if the electrostatic potential undergoes large variations between the source and drain contacts [33]. It is therefore an artifact of the ballistic approximation. Accounting for electron-phonon scattering helps get rid of NDR by connecting bands that would otherwise remain independent from each other [34]. Depending on the material under investigation, 4 CHAPTER 1. INTRODUCTION scattering is therefore required for accurate modeling of the current vs. voltage characteristics.

The outline of the thesis is as follows: in Chapter 2, the toolchain mentioned above, and its constituents are introduced. Then, in Chap- ter 3 the notion of upscaling is explained on the basis of an ideal example. Subsequently, the first step of a general approach to upscaling, the generation of interactions along atomic bonds, is presented. This preliminary work greatly simplifies the actual process of interfacing plane-wave DFT results to the quantum transport code OMEN [35], and allows for additional approximate upscaling techniques dedicated to heterostructures, as discussed in Chapter 4. In Chapter 5 the aforementioned importance of self-heating effects is elaborated in detail for single-gate devices with a molydenum disulfide monolayer, tungsten disulfide, and black phosphorous as channel material. In Chapter 6, the code structure behind Winterface is briefly discussed and finally, the thesis is concluded and an outlook is provided in Chapter 7. Chapter 2

Quantum Transport From First-Principles

2.1 Density-Functional Theory

Semiconductor and other solid-state devices have long been modeled using classical theories such as the drift-diﬀusion equations. However, as the channel length of modern transistors is reaching 10nm and below, a full quantum mechanical treatment has become unavoidable. Condensed matter (solid bodies) consists of atomic nuclei (ions), usually arranged in an elastic lattice, and of electrons. To capture their properties, the Schr¨odingerequation must be solved. Restricting ourselves to the stationary problem of N charged particles, we must deal with a 3N dimensional eigenvalue problem of the form

N h 2 X i − ~ ∇2 +V (r , .., r ) Ψ (r , ..., r ) = EΨ (r , ..., r ), (2.1) 2m i 1 N e 1 N e 1 N 0 i

where Ψe(r1, ..., rN ) is the many-particle electronic wavefunction of N electrons situated at positions r1 to rN , m0 the rest mass of the electron, E the total energy, and V (r1, .., rN ) is a many-body term including all electron-electron and electron-ion interactions. Solving

5 6 CHAPTER 2. QUANTUM TRANSPORT this problem numerically is unfeasible, even for small systems, due to the exponential scaling of its complexity with the number of particles. For example, a system of just 3 particles solved on a grid of 10 points along each direction results in a matrix of size (10 · 10 · 10)(3·3) = 1027 to be diagonalized. Without approximations, any attempt at dealing with this problem is hopeless. The first step is the Born-Oppenheimer approximation [36], where the assumption is made that the motion of atomic nuclei and electrons can be separated. This simplification was already applied to Eq. (2.1), the nucleus wavefunction Ψn(R1, ..., RN ) having been removed. This allows for the overall numerical problem to be solved in two less complicated steps. Usually, the motion of atomic nuclei, along with core electrons in closed shells, is handled partly classically, while the remaining electrons receive a full quantum- mechanical treatment. Even with this approximation, the complexity of Eq. (2.1) needs to be further reduced by introducing single-particle wavefunctions φi(rj), which interact only with the mean-field of their surroundings. Here, each electron moves in a slightly different electrostatic potential, resulting from all charges in the system. According to the Hartree-Fock method [37], the many-electron wavefunction can be recovered from the single-electron wavefunctions by means of the Slater determinant

  φ1(r1) . . . φn(r1) 1 . . Ψe(r1, ..., rn) = √ det  . .  . (2.2) N!  . .  φ1(rn) . . . φn(rn) A popular alternative is a formalism called density-functional theory [19] (DFT), where the many-electron problem is formulated in terms of the spatial electron density

Z Z ∗ n(r) = N dr2... drN Ψe(r, r2, ..., rn)Ψe(r, r2, ..., rn), (2.3)

which is given as a functional of the many-electron wavefunction Ψe(r1, ..., rn). Similar to the Hartree-Fock method, a system of ﬁc- titious single-particle wavefunctions is introduced in such a way that the charge density of this simpliﬁed system given as 2.1. DENSITY-FUNCTIONAL THEORY 7

n X 2 n(r) = |φi(r)| (2.4) i is equivalent to that of the original many-body problem. It can be shown that such a system is governed by the Kohn-Sham equation:

2 − ~ ∇2 + V (r) φ (r) = ε φ (r). (2.5) 2m s i i i

In Eq. (2.5), the φi(r)’s are the independent particle Kohn-Sham orbitals and Vs(r) the single-particle eﬀective potential given by

Z 0 0 n(r ) Vs(r) = V (r) + dr 0 + Vxc(r), (2.6) |{z} |r − r | | {z } (1) | {z } (3) (2) which is the sum of the external (1), the Hartree (2), and the exchange (3) potentials. The latter is deﬁned as the functional derivative of the exchange correlation energy

δE [n(r)] V (r) = xc . (2.7) xc δn(r) By deﬁnition, the exchange correlation energy is the diﬀerence between the total energy and the quantities that can be computed exactly. Contributions from the Pauli exclusion principle and many- body Coulomb interactions must be included, but the exact form of the exchange-correlation functional is not known. It can be approximated for the case of slowly varying densities as Z LDA Exc [n↑(r), n↓(r))] = drn(r)xc[n↑(r), n↓(r)] (2.8)

with the total electron density n(r) = n↑(r) + n↓(r) as the sum of the electron spin densities and xc[n↑(r), n↓(r)] the exchange-correlation energy per particle of a homogeneous electron gas of charge densitiy n(r). This form of Exc is known as the local density approximation (LDA), which has been successfully applied to many quantum chemistry problems, but it is known to underestimate the band gap of semiconductors by as much as 50%. Improvements have been made 8 CHAPTER 2. QUANTUM TRANSPORT by taking into account the gradient of the electron density in the so-called generalized gradient approximation (GGA) whose most established form was parametrized by Perdew, Burke, and Ernzerhof [38] (PBE). In GGA, the exchange-correlation functional is written as

Z GGA Exc [n↑(r), n↓(r)] = drf(n↑(r), n↓(r), ∇n↑(r), ∇n↓(r)), (2.9)

∇n↑ being the derivative of the charge density at position r. Fur- ther improvements have been made in the form of a hybrid approach combining the GGA and the Hartree-Fock exchange correlation by Heyd, Scuseria, and Ernzerhof [39]. Using it, it was shown that the lattice constant and band gap of most semiconductors can be reproduced with a high degree of accuracy [40], but at the cost of a significant increase in computational demand.

Since the single-particle eﬀective potential in Eq. (2.6) depends on the charge density, Eqs. (2.5-2.7) form a self-consistent system of equations, which must be solved iteratively until convergence is achieved. Finally, the total energy E[n] of the system is given by

Z 1 Z Z n(r)n(r0) E[n] = drV (r)n(r) + dr dr0 + T [n] + E [n], 2 |r − r0| s xc (2.10)

where Ts is the kinetic part of the non-interacting system.

Density-Functional Theory is a very eﬃcient formalism scaling as O(N 3). Approximations enter the formalism only through the exchange-correlation energy. Simulations of systems up to 1000 atoms are nowadays pretty much standard. Furthermore, as DFT does not depend on empirical parameters, it can be used to generate meaningful predictions for novel materials from ﬁrst-principles. As such, it is better suited than empirical models such as tight-binding [41], when it comes to the investigation of heterostructures [42, 43], metal- insulator-metal junctions [44], or novel 2-D materials [8]. 2.2. MLWF 9

2.2 Maximally Localized Wannier Func- tions

The physics of a (perfect) lattice can be encapsulated entirely within one unit cell, usually, but not necessarily a primitive cell. The Hamil- tonian operator for such a system is then composed of a kinetic part and a periodic potential. This is the situation present in Eq. (2.5), and thus the Kohn-Sham orbitals form a system of orthonormal eigenfunctions known as Bloch waves. They take the form:

ikr ψn,k(r) = un,k(r) · e , (2.11)

where un,k(r) is a periodic function in real-space with band index n and wave vector k. A Bloch wave is periodic in real-space up to a phase factor eikR where R is a vector pointing to a neighboring unit cell. This representation in terms of energy eigenfunctions is very convenient for many applications because the Hamiltonian ma- ∗ trix elements obey Hmn(k) = hψn,k|H|ψm,ki = mnδmn. However, if we want to ﬁnd a Hamiltonian operator localized in a super cell, this plane-wave representation is no more convenient as Bloch waves extend over all space.

An alternative representation that is centered and localized in real- space and that can be translated along periodic images of the unit cell is better suited for many applications. Bloch waves can be transformed into such a representation through a process called wannierization [21] by which periodic wave functions are transformed into a set of orthonormal basis functions called Wannier functions, which have the desired localized properties. The process is essentially a Fourier transform where the fact that Bloch waves are defined only upto a phase factor eiφ(k) offers a large degree of freedom, which can be cast (k) into a unitary matrix Umn. Wannier functions are defined as:

V Z X w (r) = d3k U (k)ψ (r) e−ikR. (2.12) nR (2π)3 mn mk BZ m In Eq. (2.12), the set of momentum(k)-dependent wave functions ψmk(r) of band index m are replaced through a unitary transform by 10 CHAPTER 2. QUANTUM TRANSPORT

a set of Wannier functions wnR(r) with Wannier index n and assigned to the unit cell situated at vector R with respect to the origin. The integration is performed over the entire Brillouin zone and V is the volume of the atomic unit cell in real space. For the case where the (k) unitary matrix Umn is chosen such that the spread functional

X 2 2 Ω = hwn0(r)|r |wn0(r)i−|hwn0(r)|r|wn0(r)i| (2.13) n

is minimal, we speak of Maximally Localized Wannier Functions (MLWF). In this conﬁguration the Wannier functions themselves as well as the Hamiltonian matrix elements can be proved to be real. A comparison of Bloch waves and MLWF is presented in Fig. 2.1.

The translation symmetry of the lattice is expressed through R vectors, each of them corresponds to a translated (with respect to a reference unit cell), but otherwise identical Wannier function placed at position R. Due to the spacial localization, pairwise interactions among Wannier functions extend only over a ﬁnite subset of R vectors, where the origin may be set arbitrarily. This localization in space is achieved at the cost of localization in energy, i.e. Wannier functions form an orthonormal basis set, but they are not eigenfunctions of the Hamiltonian operator. Therefore, the Wannier index n is not a band index and in general the whole set of Wannier functions contributes to each band. The Hamiltonian operator can be expressed in terms of pairwise interactions among Wannier functions among a ﬁnite range of unit cells described by R vectors, i.e.

Hnm(R) = hwn0|H|wmRi . (2.14)

Each H(R) thus describes the interactions of the atoms in the unit cell shifted according to R with the atoms in the home cell at R = 0. This spatially localized representation has ”tight-binding-like” characteristics, as needed for upscaling a set of Hamiltonian matrices H(R) deﬁned on a cluster of unit cells to a set of Hamiltonian matrices H˜(R˜ ) describing interactions among a cluster of super cells. Note that what we call upscaling is the transformation that allows us to go from 2.2. MLWF 11

(a) (b) (c)

Figure 2.1: Illustration of plane-wave and Wannier representations of a quantum mechanical ground state on a one-dimensional grid of ﬁve unit cells. The parameter R gives the position of each cell relative to the center and the black dots represent atomic positions. (a) Cell- periodic part of a Bloch wave un,k(r). (b) Corresponding Bloch wave ikr ψn,k(r) = un,k(r)·e . (c) Example of a maximally localized Wannier function (MLWF). This is a real function localized in real space and centered on a speciﬁc site. Here, the solid line represents the MLWF centered at R = 0, whereas the dashed lines refer to equivalent images centered at R 6= 0, thus illustrating the periodicity of the crystal in the Wannier picture. 12 CHAPTER 2. QUANTUM TRANSPORT the DFT calculation of a small unit cell to the production of a large- scale Hamiltonian matrix corresponding to a device structure. This concept will be explained in detail in Chapters 3 and 4.

2.3 Transport in the Wavefunction For- malism

Density-functional theory, when expressed in a plane-wave basis, lends itself perfectly to electronic structure calculations of periodic structures or small molecules. The purpose of this thesis is however to eval- uate the ”current vs. voltage” characteristics of nano-devices. To do that, the atomic system of interest must be driven out-of-equilibrium by an external voltage source. Such situations can be realized by at- taching reservoirs to the simulation domain, from which electrons can be injected and collected. Furthermore, to allow for the deﬁnition of the required open boundary conditions (OBCs) [45, 46], the Hamil- tonian that describes the electronic properties of the device must be expressed in a localized basis. Here, the choice is that of MLWF, as explained in the previous Section. The Schr¨odingerequation with OBCs takes the form

h RB i C C E − HMLW F (kt) − Σ (E, kt) Ψ (E, kt) = S (E, kt). (2.15)

In the resulting Wavefunction formalism, also called Quantum Transmitting Boundary Method (QTBM), E is a diagonal matrix containing the energy of the injected electrons, HMLW F (kt) describes the Hamiltonian matrix of the isolated system in a MLWF basis, and kt is the electron momentum modeling the directions assumed periodic. RB Σ (E, kt) is the boundary self-energy matrix, S(E, kt) the corre- C sponding injection vector, and Ψ (E, kt) the wavefunction injected from contact C ∈ {L, R} at energy E and momentum kt. It contains entries for each atomic location and Wannier function. For devices with two contacts, refered to as L (left) and R (right), injections of electrons from the left (right) contact correspond to the state ΨL R L R RB (Ψ ) and the injection vector S (S ). Note that Σ (E, kt) can be computed iteratively [47], from (generalized) eigenvalue problems 2.3. WAVEFUNCTION FORMALISM 13

[48, 49, 50] or from contour integral techniques [51].

Eq. (2.15) is no longer an eigenvalue problem, but a sparse linear system of equations that must be solved for each injection energy E and transverse momentum kt. For the calculation of observables, both quantities must be discretized in such a way that integrations over them capture the energy- and momentum-dependent behavior C C,σ of Ψ (E, kt). Assuming that Ψ (E, kt, ri) is the wavefunction at positions ri for Wannier function σ, the density-of-states of contact C is deﬁned as

1 X X dE −1 gC (r ,E, k ) = |ΨC,σ(E, k , r )|2 , (2.16) i t 2π n t i dkC σ n n

C where kn refers to the momentum along the transport direction of the nth state injected from contact C at energy E and transverse momentum kt. The electron density n(ri) can then be calculated as

Z ∞ X X C C n(ri) = g (ri,E, kt) · f(E,EF ), (2.17) CBMC C kt

where the integral is performed from the conduction band mini- C mum (CBM) of each contact C and f(E,EF ) is the Fermi distribution function of the electrons in equilibrium in each contact. The energy C EF refers to the Fermi level of contact C.

The Schr¨odingerequation in Eq. (2.15) contains an external potential, whose Hartree component VH (r) must be self-consistently calculated with the Poisson equation since the charge in the device gives rise to an electrostatic potential:

ρ(r) ∇2V (r) = − , (2.18) H ε(r) where the position-dependent charge density may include several components such as the acceptor and donor concentrations as well as the electrons and hole densities. In the following, for simplicity, we 14 CHAPTER 2. QUANTUM TRANSPORT restrict ourselves to the presence of electrons only (n(r)) and donors (ND(r)). Since the charge density n(r) depends on VH (r) through the Schrödingerequation and VH (r) on n(r) through the Poisson equation, the circular dependencies must be resolved self-consistently until convergence is reached. Once the out-of-equilibrium state of the system has been determined in this way, the famous Landauer-Büttiker formula [52, 53] can be used to produce the net ballistic current flow- ing through the device from the left to the right contact or vice-versa:

Z ∞ e X dE σ L R I = − T (E, kt) · f(E,EF ) − f(E,EF ) . (2.19) ~ CBM L 2π kt,σ

σ Here kt is the transverse momentum, σ the spin, and T (E, kt) the transmission function per spin, deﬁned as

σ,L 2 |Ψ L (E, kt, xout)| −1 σ X km dE dE T (E, kt) = , (2.20) σ,L 2 R L |Ψ L (E, kt, xin)| dkn dkm n,m km where the indices run over all propagating modes in each contact σ,L and Ψ L (E, kt, xout/in) is the spin-dependent wavefunction injected km from contact L and evaluated at the ﬁrst (xin) and last (xout) point of the device along the transport direction. Coherent current through a L R device from left to right is observed if f(E,EF ) 6= f(E,EF ), i.e. if a bias is applied, and propagating modes are available in both contacts.

If the mean free path of electrons exceeds the dimensions of the device, the ballistic approach of single-particle wavefunctions is sufficient. However, for cases where scattering mechanisms such as electron- phonon interactions must be included for accurate physical modeling, a different approach relying on Green’s functions and called the non-equilibrium Green’s function (NEGF) [54, 55] formalism is called for. The great advantage of NEGF over the Wavefunction formalism is that many-body effects can be introduced in a perturbative way through scattering self-energy terms [56, 57]. However, for the moment we will focus on the ballistic NEGF case in the next Section and introduce the dissipative case later in Chapter 5. 2.4. NEGF FORMALISM 15

2.4 Transport in the NEGF Formalism

The Green’s function associated with the linear operator L(r) is de- ﬁned as

L(r)G(r, r0) = δ(r − r0), (2.21) where δ(r−r0) is the Dirac-delta function. Once the Green’s function is known, inhomogeneous diﬀerential equations of the form

L(r)f(r) = s(r) (2.22) can be solved as Z f(r) = dr0G(r, r0)s(r0). (2.23)

Eq. (2.21) for the operator in Eq. (2.15) then becomes

RB R E − HMLW F (kt) − Σ (E, kt) · G (E, kt) = 1, (2.24)

R where the Green’s function G (E, kt) is called the retarded Green’s RB R function and Σ (E) the retarded boundary self-energy. G (E, kt) is a matrix that can be calculated as

R RB −1 G (E, kt) = E − HMLW F (kt) − Σ (E, kt)) , (2.25)

upon which the wavefunction from Eq. (2.21) is recovered by C R C Ψ (E, kt) = G (E, kt)S (E, kt). Eq. (2.25) is typically solved with the help of a recursive Green’s function (RGF) algorithm that involved a forward and backward pass [58]. With the advanced Green’s † function deﬁned as the conjugate transpose, i.e. GA = GR , two additional Green’s functions can be introduced

G≶ = GRΣ≶GA, (2.26) the lesser (<) and greater (>) Green’s functions, which depend on the lesser and greater self-energies. The boundary form of the latter RB can be computed from Σ (E, kt). This is beyond the scope of this 16 CHAPTER 2. QUANTUM TRANSPORT thesis.

Green’s Functions as well as the self-energies, once expanded into an orthonormal basis {φn(r)} such as MLWF, have the following form

0 X ∗ 0 G(r, r ; E) = Gnm(E)φn(r)φm(r ). (2.27) nm

If the basis functions {φn(r)} are centered in space and well localized, i.e. φn(r) = f(r − rn), they can be approximated as orbitals containing a point charge at the center rn. These ”orbitals” which are usually distributed closely around atomic positions. Thus, the transport problem can be understood in terms of atomic positions and atomic orbitals describing their interactions. For a two-dimensional system (2-D) with transport along the x-axis, conﬁnement along the y-axis, and the z-axis assumed periodic, the following system of equations must be solved for the electron population:

( P RB R Eδli − Hil(kz) − Σ (E, kz) · G (E, kz) = δij, l il lj (2.28) ≷ P R ≷B A Gij(E, kz) = lm Gil (E, kz) · Σlm (E, kz) · Gmj(E, kz).

The Gij(E, kz)’s represent the electron Green’s Functions at energy E and momentum kz between atoms i and j situated at position ri and rj, respectively. They are of size Norb,i × Norb,j, where Norb,i is the number of orbitals (basis components) describing atom i. The Gij(E, kz)’s can be either retarded (R), advanced (A), lesser (<), or greater (>). The same conventions apply to the self-energies Σij(E, kz). The electron concentration n(ri) for each atomic position ri is given by

X Z dE n o n(r ) = −i tr G<(E, k ) , (2.29) i 2π ii z kz which can then be plugged into Poisson’s equation. After convergence, the electrical current ﬂowing between two adjacent unit cells labeled s and s + 1 of a 2-D device structure can also be extracted from the Green’s functions 2.5. TOOLCHAIN 17

Z e X X X dE n < Id,s→s+1 = tr Hij(kz) · Gji(E, kz) ~ 2π kz i∈s j∈s+1 (2.30) < o −Gij(E, kz) ·Hji(kz) .

Here ~ is Planck’s reduced constant and e the elementary charge. The calculation can be simpliﬁed by grouping atoms together into orthorhombic unit cells arranged sequentially in transport direction such that interactions exist among next neighbor cells only. An example for a 2-D device made of a MoS2 monolayer structure is depicted in Fig. 2.2. The momentum-dependent Hamiltonian matrix for the full device includes three components (blocks):

ikz ∆ −ikz ∆ H(kz) = H0 + H+e + H−e , (2.31)

all of them being deﬁned in Fig. 2.2. For quantum transport from ﬁrst-principles, the Hamiltonian matrices H0, H+ and H− must be generated using DFT. Due to the computational demand of which, only small unit cells can be simulated. They must then be upscaled to match the dimensions of a full device. Since most DFT codes use a plane-wave basis to expand the Kohn-Sham states, as mentioned ear- lier, a transformation to a suitable localized basis must be performed, here MLWF.

2.5 Description of the Toolchain

In this thesis, an infrastructure was implemented to generate tight- binding like Hamiltonian matrices for use in device simulations. Its main task is to interface the quantum transport solver OMEN [35] with the combination of a density functional theory (DFT) package such as VASP [59], Quantum Espresso [60], or Abinit [61] and the Wannier90 code [62]. It is therefore part of a toolchain that is visual- ized in Fig. 2.3 and described in the following paragraphs. 18 CHAPTER 2. QUANTUM TRANSPORT

(a) RB RB OBC OBC

y z

z (b) s-1 s s+1

Figure 2.2: Schematic of a 2-D single-gate transistor structure for quantum transport made of a MoS2 monolayer in the channel region. (a) The considered device is composed of 9 orthorhombic unit cells in sequence along the transport direction. The open boundary conditions on either side manifest themselves in terms of boundary self-energies ΣRB. The transport direction is along the x-axis, the y-axis is the direction of conﬁnement, and the z-axis is assumed periodic. (b) Zoom into the center 3 unit cells from (a), designated s − 1, s and s + 1. Interactions are such that they exist only between next-neighbor cells, i.e. Hs,s+i 6= 0 only for −1 ≤ i ≤ 1. (c) Top view of the channel region. The periodicity in z-direction is given by ∆. The small hexagonal unit cell on the left is the primitive unit cell of monolayer MoS2, whereas the larger orthorhombic unit cell on the right, is the one used as the building block for the device in (a). The Hamiltonian matrix H0 describes the self-interactions of the central slab, H+(H−) the interactions of the central slab with the one above(below) it. 2.5. TOOLCHAIN 19

(a) (b) Wannier90 Ab-inito electronic structure Transformation calculation of plane-waves into set of MLWFs generate (c) structural input

(f)

(e) generate (d) g Hamiltonian Winterface quantum transport generate device structure bandstructure

Figure 2.3: Developed toolchain to perform quantum transport simulations from first-principles. The interfacing parts (c,e) are in red. They were implemented as part of this thesis and build the core of an open source package called ”Winterface” (a) The ground state energy of the considered system is first calculated with DFT using a plane-wave representation of the wavefunctions. (b) The resulting Hamiltonian is then transformed into a tight-binding like basis with the help of MLWF and the Wannier90 code. (c) The MLWF data is analyzed and reorganized to generate structural inputs that are passed to OMEN. (d) This input is used by OMEN to create a device structure according to user specifications. (e) The Hamiltonian matrices corresponding to the channel region of this device are produced and transferred to OMEN. (f) Device simulations from first-principles are finally possible within OMEN. 20 CHAPTER 2. QUANTUM TRANSPORT

OMEN is a nanodevice simulator capable of modeling the ”current vs. voltage” characteristics of up to tens of thousands of atoms in a full-band framework and at the atomic scale. It was originally designed to rely on the semi-empirical sp3d5s∗ tight-binding model [20], but has since been modiﬁed to accept structural and Hamiltonian data supplied externally [50]. This update will allow us to use the quantum transport algorithms implemented in OMEN with Hamiltonian matrices constructed from ﬁrst-principles. Any other QT package, e.g. NanoTCAD ViDES [63] or TB Sim [64] would work as well.

VASP is a code that provides the quantum mechanical ground state of atomic systems from ﬁrst-principles, within the DFT approximation. Such calculations can be computationally very expensive and are typically limited to a small number of atoms (up to about a thou- sand). In VASP the Hamiltonian is represented in a plane-wave basis. Using its results as the basis for quantum transport simulations from ﬁrst-principles requires a switch from plane-waves to MLWF and an additional upscaling technique. Note that instead of VASP any other plane-wave DFT package providing an interface to Wannier90 can be used since the work presented here depends only on Wannier90’s outputs.

Wannier90 is a tool to efficiently transform the plane-wave representation used in DFT packages into a set of maximally localized Wannier functions based on a unitary transform. It has been designed in such a way that it only requires a couple of input files produced by DFT codes, but is otherwise completely independent of the specificities employed to derive them. The first file, called the MMN file, contains the overlaps of the periodic parts umk(r) of the Bloch states sampled on a grid in the Brillouin zone

(k,b) Mmn = humk|unk+bi. (2.32) The k points correspond to the k-grid used in the DFT calculation and the b vectors point to next neighbors. One of the great strengths of Wannier90 is that it can operate on a subset of the total number of bands used in DFT. The band indices m and n are thus elements of an 2.5. TOOLCHAIN 21 index set specified by the user. In practice we typically exclude low- lying core states and unoccupied states far above the Fermi energy. For quantum transport, we ideally include only states around the band gap. An example of this process is provided in Chapter 4, Section 4.2. The second file, called the AMN file, contains overlaps of the wave functions ψmk(r) with localized trial orbitals gn(r)

(k) Amn = hψmk|gni. (2.33)

The gn(r) in Eq. (2.33) must be specified by the user and can be placed anywhere in the unit cell. It has been observed that placing them on atomic positions is typically a successful strategy. These overlaps serve as the initial guesses for the steepest ascent method implemented in Wannier90 to find the MLWF. The quality of the results depends greatly on sensible inputs. Wannierization can be a very time-consuming process as finding a proper selection of bands in Eq. (2.32) and a suitable configuration of trial orbitals in Eq. (2.33) is a difficult task. It should be noted that an automated initial guess generator was proposed [65], but it has not yet been tested for specific application of quantum transport.

Winterface The Winterface code, the main focus of this thesis, interfaces OMEN to outputs from Wannier90 in a two-stage process. First, the outputs of Wannier90 are analyzed to produce inputs speciﬁ- cally designed for quantum transport, as implemented in OMEN (step (c) in Fig. 2.3). Second, after OMEN generated a device structure on the basis of these inputs, corresponding Hamiltonian matrices H0, H+, and H−, as introduced in Eq. (2.31), are generated (step (e) in Fig. 2.3). All steps involved will be explained in detail in Chapters 3 and 4.

Chapter 3

Hamiltonian Data in Terms of Bonds

3.1 Atomic Lattices and Unit Cells

A periodic structure such as an atomic lattice can be described in terms of a unit cell, whose repetition according to the corresponding translational symmetry allows to recover all positions within the lattice. The main purpose of this Chapter is the manipulation of unit cells, such as a transition from one translational symmetry to another, as well as the adaptation of the representation of the underlying physics. It is therefore prudent to ﬁrst introduce the notion of a unit cell and the conventions used therein.

3.1.1 Unit Cell, Deﬁnition and Example Any arrangement of atomic positions that in combination with a translational symmetry results in unique coordinates for all positions is a valid representation of an atomic lattice. A unit cell is deﬁned by the following components:

N×N • A matrix B = [b1, ..., bN ] ∈ R , where N is the dimension of

23 24 CHAPTER 3. HAMILTONIAN DATA IN TERMS OF BONDS

space and the columns bi describe the translational symmetry of the lattice. (1) (1) (N ) A matrix A = [p , p , ..., p t ] ∈ [0, 1)3×Na ⊂ N×Na con- • 1 2 Na R taining the atomic positions as columns expressed in basis B.Na is the total number of atomic positions and Nt the total number (j) of atomic types, i.e. pi is the i-th position in the unit cell and is occupied by an atom of type j.

• A list of strings id of size Nt containing information about the atomic type for each position in A.

Neither the choice of B nor of A is unique. To ensure a clear representation of all possible unit cells, the matrices B and A are normalized so that all entries of A lie within the interval [0, 1). There- fore, all atomic positions are found within the parallelepiped spanned by the columns of B. Setting the origin and sorting the atomic positions along the atomic types and coordinates completely determine the contents of a unit cell given in the basis B. Important properties of unit cells are:

• Their volume is deﬁned as vol(B) = |det(B)|. • Their orientation is deﬁned as orient(B) = sign(det(B)). • A unit cell expressed in basis B˜ is called a super cell of a unit cell expressed in B, if both matrices describe the same lattice and vol(B˜ ) > vol(B).

• In the special case where no B˜ with vol(B˜ ) < vol(B) exists, whilst describing the same lattice, the unit cell is called primitive cell and the basis B is called primitive basis.

• The ratio of the number of atomic positions for two diﬀerent bases is given by the ratio of their volumes, i.e. N˜a/Na = vol(B˜ )/vol(B). The same applies for the number of atoms belonging to each atomic type.

• The switch to Cartesian coordinates for an atomic position in (j) P (j) basis B is given by pi,cart = m pi,m · bm. In other words Acart = B · A. 3.1. ATOMIC LATTICES AND UNIT CELLS 25

The conventions introduced above allow for a unified treatment of atomic lattices since any representation of a given lattice takes the form of a unit cube [0, 1)N when viewed in its own basis. As a consequence, the categorization into Bravais lattices and corresponding symmetry groups is not needed for the applications presented in this work. The description of an infinite lattice is completed by a grid of vectors R ∈ ZN (in basis B), where R = 0 is the home unit cell and all R 6= 0 correspond to image unit cells. As an explicit example a monolayer structure of MoS2 is considered. It will serve as the main test bed throughout this work. Fol- lowing Setyawan and Curtarolo [66], the hexagonal unit cell for this structure can be defined using the basis

 a a 0 2√ √2 B = [b , b , b ] = −a 3 a 3 , (3.1) 1 2 3  2 2 0 0 0 c where a is the lattice constant and c the interlayer distance (for a monolayer c a). The matrix A containing the atomic positions is then

 2 2  0 3 3 (1) (2) (2) 1 1 A = [p1 , p2 , p3 ] = 0 3 3  , (3.2) 1 1 d 1 d 2 2 − 2c 2 + 2c where d is the vertical distance between the two sulfur atoms. The molybdenum atom resides at the origin in the xy-plane and in the middle of the unit cell in the z-direction. Finally, the choice of id = {’Mo’,’S’} completes the data set. A graphical representation is given in Fig. 3.1.

3.1.2 Basis Expansions The basis B can always be replaced by another basis B˜ such as the orthorhombic super cell in Fig. 3.1. To ensure that the new basis describes the same lattice, it is best expressed as a linear expansion of the old one

˜ X bi = cji · bj (3.3) j 26 CHAPTER 3. HAMILTONIAN DATA IN TERMS OF BONDS

Z (a) 2/3 c 1/3 b3 ~ b2 ~ c/2 d ~ 3 y b2

a¡ 3/2 0 x

-a¡3/2 a b1 a/2

Figure 3.1: Example of a unit cell for a MoS2 monolayer structure. (a) Isometric view of the primitive unit cell, where a is the lattice constant, c the interlayer distance (here displayed much shorter than in reality) and d the vertical distance between the two sulfur atoms (in cyan). The blue arrows are the three basis vectors of the unit cell and the pink fractions in the top plane of the parallelepiped correspond to the atomic positions in basis B. (b) Top view of the situation in (a). The green arrows are the basis vectors of an orthorhombic super cell. The colored atoms are either inside the blue or the green rectangle as they belong to either unit cell. The dashed lines delimit the image unit cells. (c) Same situation as in (b), but viewed from the primitive basis B. The axis c1(c2) represent the coordinates in the vectors b1(b2). From this perspective, the lattice exists on a grid of integers. 3.1. ATOMIC LATTICES AND UNIT CELLS 27

or in other words

B˜ = B · C, (3.4)

where det(C) 6= 0, cj,i ∈ Q in general, but cj,i ∈ Z if the initial unit cell is primitive. This property is immediately visible when con- sidering expansions into super cells from a primitive basis, where only expansions satisfying 2 ≤ |det(C)| ∈ Z are valid. Since a primitive unit cell is irreducible, all entries of C must be integers. For the reverse expansion back to the primitive cell 1/2 ≥ |det(C−1)| ∈ Q must hold, which can only be satisﬁed if C−1 holds at least one fraction. The transformation of coordinates into the new basis B˜ can be written as

˜ −1 −1 −1 AB˜ = B · B · AB = (B · C) · B · AB = C · AB. (3.5) The new coordinates will not necessarily lie within the unit cube described by the new basis, but might be situated in an image cell. For expansions C with |det(C)| > 1, additional atomic positions will have to be found to match the increase in volume. Forcing positions into the interval [0, 1) is done by applying the modulus defined as ( x, y = 0, mod(x, y) = (3.6) x − bx/yc · y, y 6= 0. Hence, the new coordinates are found by scanning along the grid of R vectors in basis B, applying the conversion to the new basis B˜ in Eq. (3.4) and subsequently taking the modulus of 1. Practically, this can be done iteratively by setting R to a vector of random integers at each iteration until all new positions are found. It should be noted that without the convention of coordinates in the interval [0, 1), the choice of the positions to include in a unit cell described by a basis B˜ with vol(B˜ ) > vol(B), as compared to the initial unit cell is arbitrary. An additional benefit of atomic positions in basis B is that operations such as rotations or deformations need only be applied to the basis because their effect is passed onto the atomic positions automatically when using the modified basis to transform to Cartesian coordinates. 28 CHAPTER 3. HAMILTONIAN DATA IN TERMS OF BONDS

3.1.3 Metric in Periodic Space

The last ingredient needed when working with unit cells is that the periodic space described in their terms qualiﬁes as a metric space. The dot product of two positions p and q in basis B is

> > > hp, qiB := (B · p) · (B · q) = p · B B · q, (3.7)

p implying the norm kpkB = hp, piB. A metric operating in periodic space must consider any position in the lattice. All equivalent images must therefore be taken into account to find the distance between two positions, defined as the length of the closest possible connection between them. Such a metric may then be defined as

0 d(p, q)B = min (p + R) − (q + R ) = min (p − q) + R . R,R0 ∈ZN B R∈ZN B (3.8) The range of R vectors to search can be restricted by forcing both positions into the same unit cell with the modulus operation of Eq. (3.6). In this case the shortest possible connection can be found either in the home cell or among the next-neighbor image cells. Since p − q ∈ (−1, 1)N if p, q ∈ [0, 1)N , the range of R vectors can be further restricted to the positive sector {0, 1}N by taking the absolute value over p − q. The metric is thus deﬁned as

d(p, q)B := min |p˜ − q˜| − R , p˜ = mod(p, 1). (3.9) R∈{0,1}N B

> As the product B B is positive deﬁnite, d(p, q)B ≥ 0 for all p,q. Furthermore, d(p, q)B = d(q, p)B from Eq. (3.9). What is left to demonstrate is the triangle inequality: 3.1. ATOMIC LATTICES AND UNIT CELLS 29

d(p, q) + d(q, r) = min |p˜ − q˜| − R + min |q˜ − ˜r| − R , R∈{0,1}N B R∈{0,1}N B

pq qr = |p˜ − q˜| − Rmin + |q˜ − ˜r| − Rmin , B B

pq qr ≥ |p˜ − q˜| + |q˜ − ˜r| − (Rmin + Rmin) , B

pq qr ≥ |p˜ − ˜r| − (Rmin + Rmin) , B

≥ min |p˜ − ˜r| − R , R∈{0,1}N B = d(p, r). (3.10) In conclusion:

• Any lattice can be treated in terms of positions inside the unit cube [0, 1)N and a grid of integers in terms of vectors R ∈ ZN representing the home cell at R = 0 and image cells at R 6= 0.

• Speciﬁcs of a lattice such as the volume of the unit cell enter the description of a lattice through the basis B and the metric from Eq. (3.9) that depends on it.

3.1.4 Automatic Detection of Basis Expansions Finally, an algorithm was developed, allowing for automatic detection of expansion matrices C of a primitive basis B, given a template T for the desired basis B˜ , such that B˜ = diag([λi, ..., λN ])·T with λi ∈ R>0 and vol(B˜ ) is minimal. Hence, the task is to ﬁnd the expansion C leading to the smallest possible super cell, such that the basis vectors of B˜ are equivalent up to a positive constant to the template basis vectors in T.

In order for B˜ to be a valid basis for the lattice initially expressed in B, the directions given by T must correspond to crystallographic directions, in which case T in basis B takes the form

−1 B · T = [λ1q1, ..., λN qN ], (3.11) 30 CHAPTER 3. HAMILTONIAN DATA IN TERMS OF BONDS

N with λi ∈ R and qi ∈ Q . The task of ﬁnding the smallest possible expansion is now equivalent to determining a factor fi for each column N such that fi · λiqi ∈ Z and kfi · λiqik2 minimal. Since

|λiqij| 1 ≥ ∈ Q, (3.12) maxj |λiqij| it follows that

λiqij ∀j ∃ν ∈ N such that ν · ∈ Z. (3.13) maxj |λiqij| | {z } pij

Determining the ν for each column qi, where kqik2 is minimal, corresponds to ﬁnding the least common multiple among the denominators of pij for each i, over j. Since the denominators are not known, the implementation scans through all ν ≥ 1 until either all qij are integers (up to a small numerical tolerance), or an upper limit νmax has been reached, i.e. the resulting basis would be so large as to render it useless. The νmax might be exceeded only if the initial template T was poorly chosen, in which case Eq. (3.11) is violated, or if the tolerance level was set too strict. A few points should be mentioned: • For this algorithm to be useful in real-world applications, the lattice must be aligned ’nicely’ with the Cartesian axes. For example, to ﬁnd the expansion from the primitive basis to the orthorhombic super cell in Fig. 3.1, the template T can be set to the identity. However, if the basis vectors of the primitive basis were slightly rotated such that the Cartesian axes do not lie along crystallographic directions, the template would have to be rotated in the same way to arrive at the same result. In such a case, it is more practical to determine the correct expansion by other means, rather than attempting to detect the rotation. • Small deviations from a ’nice’ alignment can be compensated by increasing the tolerance level when detecting integers. • For lattices that only slightly deviate from an exact symmetry as dictated by T, the tolerance level for detecting integers can be increased, upon which inexact expansions are automatically detected and compensated for by a deformation tensor. 3.2. UPSCALING TECHNIQUE FOR AN IDEAL CASE 31

3.2 Upscaling Technique for an Ideal Case

To illustrate the process of upscaling, consider an expansion of C = 3 on a primitive unit cell of a two-dimensional lattice. The task at hand is now to construct the set of Hamiltonian matrices H˜(R˜ ) describing the physics in terms of the super cell, from the initial set of H(R) (m) corresponding to the primitive cell. To this end, the coordinate ri ∈ {0, 1, 2}2 with 1 ≤ i ≤ 9 identifying each primitive cell inside the super cell at R˜ m is introduced. Since R = C · R˜ = 3 · R˜ , the i-th primitive cell inside the m-th super cell can be mapped onto the grid of primitive (m) ˜ (m) unit cells as Ri = 3 · Rm + ri . Thus, the relative positioning of two primitive cells is

(m,n) ˜ ˜ (m) (n) δRij = 3 · (Rm − Rn) + (ri − rj ), (3.14)

(m,n) which determines the Hamiltonian matrix H(δRij ) describing the interactions between two primitive cells. Each H˜(R˜ ) = H˜(R˜ m − R˜ n) can be constructed by scanning across all primitive cells forming each super cell. Therefore, the Hamiltonian matrices describing interactions among super cells consist of 81 blocks of Hamiltonian matrices describing interactions among the primitive cells contained in them. (m,n) Some of the data is repeated since δRij in Eq. (3.14) can have the same value for diﬀerent combinations of m, n, i, and j. A graphical illustration is presented in Fig. 3.2. The shape of the matrices in Fig. 3.2 is determined by the order in which the primitive cells are arranged in the super cell. For quantum transport, a block-tridiagonal shape is needed, but any arrangement is valid since the corresponding H˜(R˜ ) are equivalent up to a unitary transform.

It should be noted that the case presented here represents the simplest possible scenario and serves only to introduce the notion of upscaling. In general the process is more involved, for example, the concept of subcells as coordinates inside a super cell may not work when scaling to super cells belonging to a diﬀerent symmetry group, such as from the primitive hexagonal cell of MoS2 to the orthorhombic super cell in Fig. 3.1, because some of the subcells are only partially 32 CHAPTER 3. HAMILTONIAN DATA IN TERMS OF BONDS

(a) (c)

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 ~ 2 R=(1,1) 3 4 5 6 7 8 ~ c2 9 R=(-1,1) 1 ~ 2 R=(1,0) 3 4 5 6 c1 7 8 ~ ~ ~ 9 R=(-1,0) R=(0,0) R=(1,0) 1 ~ 1 4 7 1 4 7 2 R=(1,-1) (0,0) (1,0) (2,0) (b) (3,0) (4,0) (5,0) 3 4 2 5 8 2 5 8 5 (0,1) (1,1) (2,1) (3,1) (4,1) (5,1) 6 7 3 6 9 3 6 9 8 ~ (0,2) (1,2) (2,2) (3,2) (4,2) (5,2) 9 R=(-1,-1)

Figure 3.2: Upscaling technique of a set of H(R) describing interactions among unit cells in basis B, to a set of H˜(R˜ ) describing interactions among super cells in basis B˜ = 3 · B. (a) Schematic of interactions among a set of 21 primitive cells. Each tile represents a unit cell shifted according to R and the corresponding Hamiltonian matrix H(R). (b) Schematic of the home super cell and the ﬁrst neighbor super cell at R˜ = (1, 0). Each super cell holds 9 primitive unit cells from (a) arranged in a speciﬁc order, as indicated by the index 1 ≤ i ≤ 9. Inside each subcell, the mapping R = 3·R˜ +r to the grid of primitive unit cells is given. (c) Schematic of the set of H˜(R˜ ) describing interactions in the super cell. The interaction range for the super cell is next neighbor only leading to 9 H˜(R˜ ) as compared to 21 H(R) for the primitive cell. Each tile represents a whole matrix H(R) from (a) determined by the relative positioning of primitive cells given in Eq. (3.14). The blocks of H˜(R˜ ) are found by scanning over all of the primitive cells in each super cell, where a high degree of sparsity is achieved for R˜ 6= 0. 3.3. GENERATION OF BOND INTERACTIONS 33 inside the super cell. A general formalism will be introduced in the next Section and in Chapter 4.

3.3 Generation of Bond Interactions

The main purpose of this Section is the upscaling of Wannier Hamil- tonian data expressed in terms of an initial unit cell to a representation corresponding to a super cell. Two output files from Wan- nier90 are of special interest for this purpose. The main output file, called wout, contains the Wannier centers and spreads of each Wan- nier function. The second file, called hrdat, contains the Hamiltonian in a Wannier representation in the form of a list of matrix elements Hij(R) = hwi0|H|wjRi. In principle, this is all the information needed to generate a set of H˜(R˜ ) for a super cell on the basis of an initial set of H(R). However, the raw data provided by Wannier90 is not ideally suited for this task. We have simplified the process by first converting the outputs of Wannier90 into Hamiltonian interactions corresponding to bonds between atomic positions in our unit cell or an image position in a neighboring cell.

3.3.1 Matching Wannier Centers to Atomic Posi- tions

The initial DFT simulations are carried out with a description of the material in terms of atomic positions. If possible it would be convenient to keep the same positions when generating interactions along bonds. Since Wannier90 returns a set of Wannier centers, the ﬁrst step consists of establishing whether these centers can be assigned to the existing atomic positions. In many cases the Wannier centers are clustered around the atoms whose orbitals they represent, but in other cases they lie on bond centers. In fact, they can be located anywhere in space, depending on the results of the wannierization. The upscaling code should therefore be able to handle all these conﬁgurations by matching Wannier centers to appropriately chosen atomic positions using the metric introduced in Eq. (3.9). An example where atomic 34 CHAPTER 3. HAMILTONIAN DATA IN TERMS OF BONDS

(b) (c) (d) (a)

(e) (f) (h) (i)

y y

x (g) x

Figure 3.3: Wannierization of a MoS2 monolayer structure. (a) Prim- itive unit cell holding one molybdenum and two sulfur atoms. The Wannier centers, represented by the small blue marbles are tightly clustered on the atomic positions. 5 d-like orbitals are found on molybdenum and 3 p-like orbitals on each sulfur atom. (b-d) Wannier centers on sulfur. The green parallelepiped represents the primitive unit cell of MoS2 seen from above, the dashed line shows one honeycomb of the same volume. (e-g) Wannier centers on molybdenum.

positions and Wannier centers coincide is single-layer MoS2, as shown in Fig. 3.3. It should be noted that normally the only available information is the Hamiltonian expressed in a Wannier basis and the Wannier centers. Plotting the Wannier functions themselves can be extremely memory intensive because they require the Bloch states themselves rather than a number of overlaps between them. Additionally, the Wannier data may be hard to visualize. Since the focus of this thesis was on 2-D structures, a straight forward way of illustrating them is to integrate the Wannier functions along the out-of-plane direction, z R 2 in our case, i.e. dz|wn0(x, y, z)| . As a second example, graphene is considered. In Fig. 3.4, it can be seen that the atomic positions alone are insuﬃcient to describe the electronic distribution of this material. 3.3. GENERATION OF BOND INTERACTIONS 35

(b) z (c) (d) (a)

y (e) (f) x x

Figure 3.4: Wannierization of graphene. (a) Primitive unit cell holding two carbon atoms. The Wannier centers are located on both atoms, but also on the bond centers. (b-d) Wannier centers on bond centers. The green parallelepiped represents the primitive unit cell of graphene seen from above, the dashed line shows one honeycomb of the same volume. (e,f) Wannier centers on the carbon atoms.

Some Wannier centers are directly located on the bonds connecting two neighboring carbon atoms. Hence, a reasonable matching of the atomic positions and Wannier centers is not possible, especially since three Wannier centers are situated on bond centers within the primitive unit cell, where there are only two carbon atoms. Any matching scheme would be both uneven and arbitrary. This is why the concept of ﬁctitious atoms has been introduced. It corresponds to the Wannier centers that cannot be matched to real atomic positions. For certain structures a combination of both approaches is necessary, that is selected bond centers and atomic positions should be included. This is the case for a graphene-silicene heterostructure, as depicted in Fig. 3.5. For the graphene layer, the bond centers should be included for the reasons stated above, while for the silicene layer, there is no reason to do so. To deal with such situations, our upscaling tool, Winter- face, allows us to specify such an arrangement directly on the basis of the atomic type. As a cut-oﬀ radius is introduced to remove very long range bond interactions, almost any atomic arrangement can in 36 CHAPTER 3. HAMILTONIAN DATA IN TERMS OF BONDS

(a)

(b)

Figure 3.5: Wannierization of a heterostructure made of graphene on top of silicene. For graphene the bond centers must be included in the matching process. For silicene matching to the atomic centers is sufficient. The blue marbles are the Wannier centers, the blue transparent spheres are carbon atoms, while the smaller white transparent spheres represent bond centers between neighboring carbon atoms. Finally, the yellow spheres are silicon atoms. (a) Unit cell and matching centers seen from the side. (b) Same as (a), but seen from above. 3.3. GENERATION OF BOND INTERACTIONS 37 principle be covered. In cases where the Wannier center distribution does not correlate with the atomic positions at all, e.g. for a badly converged data set, all Wannier centers can be directly chosen as the new atomic grid. With the matching approaches described above, first, a set of real or fictitious atomic positions must be identified. In a second step each Wannier center is matched to its closest grid point contender. To automatize the matching procedure, an additional method where the process is reversed has been developed as well. Employing the density- based algorithm DBSCAN [67], clusters of Wannier centers are first identified and their center of mass is then used as the fictitious atom they are matched to. While this automatic technique works for any possible configuration and often provides almost identical results as compared to other approaches, it suffers from a severe drawback: it does not keep information about the chemical origin of the Wannier centers. This might be especially problematic if electron-phonon interactions are included in transport calculations: crystal vibrations at atomic positions must be linked to potentially delocalized Wan- nier centers whose mass is unknown. Additionally, the location of the Wannier centers may be somewhat deceiving as to what they actually represent. For example, a situation where the centers appear to be congregating on the bond centers, when they actually stem from the atomic positions can be encountered, as can be seen in Fig. 3.6.

The developed matching process requires basic knowledge of the chemistry of the structure under investigation and furthermore depends on the preference of the user. It should be realized that for some operations such as the calculation of the bandstructure of a given unit cell, the positions of the Wannier centers do not matter, only the periodicity of the structure. The situation is different if the charge distribution should be accounted for, as in device simulations where Poisson’s equation must be solved self-consistently with Schrödinger’s equation. If point charges are used, their exact location might influence the shape of the resulting electrostatic potential. The charge distribution in terms of Wannier centers can be followed more closely to address this issue, but at the expense of a large computational burden. 38 CHAPTER 3. HAMILTONIAN DATA IN TERMS OF BONDS

Figure 3.6: Cubic lattice with one atomic position in the middle of the unit cell. A matching according to straight forward geometrical considerations would lead to the introduction of ﬁctitious atoms on the bonds connecting two direct neighbors. A careful examination of the Wannier functions indicates however that the Wannier functions are in fact p-like with their center of mass on the real atoms. Assigning all Wannier centers to the real atomic positions appears more natural in this case even though the spread of the matching is slightly worse. Still, both matching processes are possible, depending on the user preference. 3.3. GENERATION OF BOND INTERACTIONS 39

3.3.2 Generating Interaction Data Along Atomic Bonds To illustrate the complete process of converting raw Hamiltonian data, as produced by Wannier90, into a Hamiltonian expressed in terms of bonds, the already mentioned MoS2 monolayer structure will serve as test bed. It was simulated in VASP within the generalized gradient approximation of Perdew-Burke-Ernzerhof (PBE) [38] using a lattice constant of 3.18A, a 400-eV plane-wave cutoﬀ energy, a 21x21x1 Monkhorst-Pack k-point grid, and a 45A out-of-plane vacuum separation between the structure and its closest images. Due to the periodicity of the lattice, Wannier centers can be spread out among images of the same atomic positions, as can be observed Fig. 3.7. The wannierization corresponding to the red marbles in Fig. 3.7, henceforth referred to as the red case, resulted from placing orbital projections directly onto atomic positions. To demonstrate the importance of a proper treatment of the spreading out of Wannier centers, a second wannierization is provided, corresponding to the blue marbles in Fig. 3.7 and henceforth referred to as the blue case. The spreading of the Wannier centers was enforced by placing orbital projections on images of the atomic positions. Note that such situations are fre- quently encountered when wannierizing 2-D materials and diﬃcult to avoid. Hence, they must be dealt with properly. For a given structure, there is normally only one set of data to work with. However, in terms of matching, the two situations are identical due to the metric introduced in Eq. (3.9), which takes the periodicity into account. The same does not hold for operators in Wannier representation, where the relative position between Wannier centers must be considered.

In both our examples, 5 centers are found on the molybdenum atom and 3 on each sulfur atom. Consequently, interactions between molybdenum atoms are given as 5 × 5 matrices, those of molybdenum with sulfur as 5 × 3 matrices, and 3 × 3 matrices are required to describe the coupling between two sulfur atoms. Since interactions between atoms and their images on the grid of unit cells are expressed in terms of the Wannier functions matched to them, each atom within the chosen unit cell must be considered unique, even if there are multiple instances of the same element such as the two sulfur atoms in 40 CHAPTER 3. HAMILTONIAN DATA IN TERMS OF BONDS

(a) (b)

Figure 3.7: Small section of a MoS2 monolayer structure. The primitive unit cell simulated in VASP holds one molybdenum (transparent red spheres) and two sulfur (transparent blue spheres) atoms. The small red marbles represent Wannier centers converged onto atomic positions within the parallelepiped delimited by the blue lines, while the blue marbles represent Wannier centers resulting from a diﬀerent set of orbital projections. Both wannierizations are equally valid and produce 5 Wannier centers on molybdenum and 3 on sulfur. How- ever, in the second case the centers are spread out among images in neighboring cells. (a) Isometric view. (b) View from the top. 3.3. GENERATION OF BOND INTERACTIONS 41

MoS2. To obtain a complete representation of the Wannier Hamil- tonian data in terms of bonds, the task at hand is to ﬁnd all bonds along which relevant matrix elements exist and to construct the corresponding interaction matrices. This can be done by analyzing the connections between Wannier centers matched to the individual atoms forming each bond. Based on that, the diﬀerence between the blue and the red case will clearly appear.

The Hamiltonian operator in Wannier representation is given in terms of R vectors. Its matrix elements Hmn(R) = hwn0|H|wmRi can be computed and backtransformed using a set of unitary matrices U (k) computed by Wannier90, with X H(R) = U (k)†H(k)U (k)e−ikR, (3.15) k and X H(k) = H(R)eikR. (3.16) R Note that H(k) must be diagonalized to retrieve the initial energies from DFT. Its eigenvalues are only exact on the k-point grid used therein. In between these grid points a Fourier interpolation is made. From Eq. (3.16) the basic property

H(R) = H(−R)† (3.17) of Wannier Hamiltonians follows since H(k) must be self-adjoint for all k. By the same argument, for each R vector, its spatial inverse must be included as well.

Let us assume for the moment that the set of Wannier Hamiltonian matrices for the red and for the blue case were made up of exactly the same matrix elements, but permuted relative to each other. In this case the following holds true for a Wannier functionw ˜nR(r) in the blue case with a relative shift of δR as compared to the red case with Wannier functions wnR(r):

w˜n,R(r) = wn,R(r − δRn) = wn(R+δRn)(r). (3.18) 42 CHAPTER 3. HAMILTONIAN DATA IN TERMS OF BONDS

The k-dependent Hamiltonian H˜mn(k) of the blue case is equal to that of the red case, Hmn(k), up to a phase factor

X ikR H˜mn(k) = hw˜m0|H|w˜nRi e , R X ikR = hwmδRm |H|wnR+δRn i e , R X ikR (3.19) = hwm0|H|wnR+δRn−δRm i e , R

X ikR ik(δRm−δRn) = hwm0|H|wnRi e e , R

ik(δRm−δRn) = Hmn(k) · e . From Eq. (3.19), we have

X X H˜(k) = H˜(R)eikR = V †H(R)V eikR = V †H(k)V, (3.20) R R with

ikδRm Vmn = δmne . (3.21) Since the matrix V is constant among all R vectors and unitary, the same eigenvalues are obtained with and without shifts of the Wan- nier centers. In terms of bandstructure calculations, the placement of Wannier centers does not matter at all. In fact Wannier centers can be shifted anywhere among images of the unit cell, provided that the Hamiltonian matrices are adapted appropriately and the range of R vectors is chosen such that all relevant data is preserved. Note however that in Eq. (3.19) it was implicitly assumed that for each R vector, R + δR is either included or its matrix elements vanish. Since the number of R vectors in the output of Wannier90 does not take the distribution of Wannier centers into account and is roughly the same as the number of k-points used in the DFT calculation, a relatively dense grid of k-points is required to properly treat the cases where shifts of the Wannier centers occur. 3.3. GENERATION OF BOND INTERACTIONS 43

If the goal is to ﬁnd interactions along bonds, the initial distribution of Wannier centers can no longer be ignored, since doing so would lead to a confusion between short- and long-distance neighboring interactions. A closer look at the self interaction matrix of sulfur in the red and blue cases provides some insight into this issue. For the blue case the relative positioning of the Wannier centers was once accounted for and once ignored. Overall, the following 3 × 3 matrices are found:

−5.4896 0.0000 −0.0025 Hred,S =  0.0000 −5.4920 0  −0.0025 0 −5.6947

−5.5399 + 0.0000i 0.0318 − 0.0076i 0.0836 − 0.0032i  Hblue,S =  0.0318 + 0.0076i −5.5116 + 0.0000i −0.0539 − 0.0104i 0.0836 + 0.0032i −0.0539 + 0.0104i −5.6201 + 0.0000i

−5.5399 + 0.0000i 0.2015 − 0.0510i −0.0484 + 0.0061i ˜ Hblue,S =  0.2015 + 0.0510i −5.5116 + 0.0000i −0.0539 − 0.0104i −0.0484 − 0.0061i −0.0539 + 0.0104i −5.6201 + 0.0000i

In Hblue,S (H˜blue,S) the relative positioning is taken into account (ignored). By looking at these matrices, it is not clear why Hblue,S should be preferred over H˜blue,S, but by performing a singular value decomposition and comparing the results with those of Hred,S, the importance of an appropriate treatment clearly appears.

Hred,S Hblue,S H˜blue,S s1 5.6948 5.6951 5.7342 s2 5.4920 5.4902 5.6368 s3 5.4896 5.4865 5.3007

The singular values from Hblue,S, where the relative positioning of the atoms is carefully handled, are closer to those of Hred,S. The slight diﬀerences can be explained by the fact that the blue wannierization is of lower quality than the red one, as the Hamiltonian matrices exhibit non-vanishing imaginary parts. Nonetheless, using Hblue,S, the 44 CHAPTER 3. HAMILTONIAN DATA IN TERMS OF BONDS

Hred,S data can be reproduced reasonably well. Such relative shifts may occur along the vacuum direction as well, but they can be safely ignored, because Wannier functions are periodic along this direction. A more in depth analysis of this behavior is provided in Appendix A.

Recalling p˜i = mod(pi, 1) from Eq. (3.9), a bond between two positions p˜i and p˜j inside a unit cell is deﬁned as the vector that ˜ points from one position to the other, i.e. bij = p˜j − p˜i. All other bonds in the lattice can be recovered by allowing the placement of positions in image unit cells:

˜ bij(R) = (p˜j + Rj) − (p˜i + Ri) = bij + R, (3.22) ˜ where bij is called the principal bond. This decomposition of a bond in a principal part and a relative placement among image unit cells preserves the basic property of Hamiltonians in a Wannier representation. The interactions along a bond bij(R) are given by a small Hamiltonian matrix h(i,j)(R), analogous to the Hamiltonian matrices H(R) describing interactions among unit cells. To ﬁnd the matrices h(i,j)(R), we will ﬁrst have a look at the principal bonds only. The initial situation consists of:

N • A list of (ﬁctitious) positions pi ∈ [0, 1) , i ∈ {1, ..., Na} inside the unit cell.

• An index vector Ii for each atomic position pi holding the matched Wannier indices.

(i) N • A list of Wannier centers wm ∈ R , m ∈ Ii matched to pi.

• A set of Hamiltonian matrices H(R) in a Wannier representation.

The challenge in ﬁnding the correct h(i,j) for each principal bond ˜ bij, is the treatment for the case where Wannier centers are spread out among images of the positions deﬁning the bond, such as the blue case in Fig. 3.7. To detect the placement of Wannier centers with respect to the positions inside the unit cell, a list of ’Wannier bonds’ ˜(i) {bm }, m ∈ Ii, i.e. vectors pointing from an atomic position p˜i to 3.3. GENERATION OF BOND INTERACTIONS 45

(i) the closest image w˜ m of a matched Wannier center with index m, is generated.

The shift in R vectors of a Wannier center relative to the closest image matched to a position p˜i can be derived from the corresponding ’Wannier bond’:

(i) (i) (i) ˜(i) (i) Rm = w˜ m − wm = p˜i + bm − wm . (3.23) (i) Thus, the relative position between two Wannier centers wm and (j) wn matched to the atomic positions p˜i and p˜j respectively is

(i,j) (j) (i) ˜(j) (j) ˜(i) (i) δRmn = Rn − Rm = (p˜j + bn − wn ) − (p˜i + bm − wm ). (3.24)

(i,j) Therefore, the matrix element hmn for the interaction along the ˜ (i,j) principal bond bij is found in the matrix H(δRmn ) as:

(i,j) (i,j) hmn = HIi(m),Ij (n) δRmn , (3.25) (i,j) where HIi(m),Ij (n) δRmn corresponds to the data produced by Wannier90 as introduced in Eq. (2.14). A graphical representation of this process is presented in Fig. 3.8. The extension to general bonds bij(R) is done by shifting the position pj = p˜j + R which in turn shifts the attached Wannier (j) center wn in Eq. (3.24) by the same R. This then translates into Eq. (3.25) as

(i,j) (i,j) hmn (R) = HIi(m),Ij (n) δRmn + R . (3.26)

The full data set of interactions along bonds {bij(R)} with 1 ≤ i, j ≤ Na and R such that all significant elements of H(R) are included, can be constructed by placing pi in the home cell and pj in image cells as indicated by R for all pairwise combinations of i and j. For each such pairing, Eq. (3.26) must be employed for all pairwise combinations of the matched Wannier functions as specified by Ii and Ij respectively to find all the elements of the interaction matrix h(bij(R)) associated with the bond in question. 46 CHAPTER 3. HAMILTONIAN DATA IN TERMS OF BONDS

(0,1) (1,1)

(j)

(j) wn

~ b(j) (0,0) n

~(j) ~ wn pj

(i) (1,0)

wm (i) ~ wm pi

Figure 3.8: Schematic illustrating the detection of the relative po- (i) (j) sition between two Wannier centers wm and wn matched to the atomic positions p˜i and p˜j, respectively. The green circles (squares) are atomic positions (images) and the blue circles (squares) are Wan- nier centers (images). Even though most of the Wannier centers converged onto images of the atomic positions, they all belong to the home cell at R = (0, 0). The matrix element needed for the con- (i) (i) nection between the images of these Wannier centers w˜ m and w˜ m closest to the atomic positions, is therefore found in H(R 6= 0). The (i) (j) (i) (j) shift Rm (Rn ) for the Wannier center wm (wn ) to coincide with (i) (j) the image w˜ m (w˜ n ) is given by Eq. (3.23) in terms of the ’Wannier (i) (j) (j) bonds’ b˜m (b˜n ). After shifting the Wannier center wn by the rel- (i,j) (j) (i) ative placement δRmn = Rn − Rm , it ﬁnds itself in the same unit (i) cell at R = (1, 0) as wm . The correct matrix element is therefore (i,j) located in H(δRmn ) as indicated by the large pink arrow. 3.3. GENERATION OF BOND INTERACTIONS 47

The bonds and their interaction matrices directly inherit the spatial inversion symmetry of Hamiltonians in Wannier representation from Eq. (3.17) because

(i,j) (i,j) hmn (R) = HIi(m),Ij (n) δRmn + R , (j,i) ∗ = HIj (n),Ii(m) δRnm − R , (3.27) (j,i) ∗ = hnm (−R) .

Following the scheme outlined above, we note that Na·Na principal bonds exist, one for each pair of atomic positions inside the unit cell, including vanishing bonds where both positions are the same. The latter represent the self-interactions of the atoms with themselves. Additionally, each principal bond has a list of R vectors attached to it, forming the set of general bonds {bij(R)}. The total number of bonds is then ≤ Na · Na · NR, because some of the Hamiltonian data describing long-range interactions can be neglected.

Since the main task of Hamiltonian data in terms of constructing interactions along bonds is querying for interactions with bond vectors, an eﬃcient arrangement is required. The main building block is the unit cell containing the (ﬁctitious) atomic positions generated during the matching process. Because the interactions between these positions are expressed in terms of mutually exclusive sets of Wannier functions, all atomic types are considered unique even if the chemical origin was equivalent, such as the two sulfur atoms in MoS2. As a direct consequence, this unit cell is always primitive. The interaction data in terms of bonds is arranged as follows:

• The top level consists of a sorted list of index pairs {(i, j)} with 1 ≤ i, j ≤ Na, indicating which of the positions {p˜i} form the principal bond vectors.

• Each element of {(i, j)} has a sorted list of pairs {R, h(i,j)(R)} attached to it. The R indicates which image cell the position pj = p˜j + R is located in. The interactions along the bond (i,j) bij(R) are then given by h (R). 48 CHAPTER 3. HAMILTONIAN DATA IN TERMS OF BONDS

With the interaction data completely sorted, the querying for bonds given by a starting index µ, an ending index ν and a bond vector b, is now a two stage process:

• Search for (µ, ν) in the list of {(i, j)}. If a matching entry is ˜ found, compute the corresponding principal bond bµν and sub- tract it from the bond vector b.

˜ N • If b − bµν ∈ Z up to a numeric tolerance, search among the list of R vectors. If a matching entry is found, return the corresponding interaction matrix h(µ,ν)(R).

˜ N Note that b−bµν can be assigned to an R ∈ Z up to a tolerance of 1/2. In the extreme case, this allows for a spatial mismatch of an entire unit cell. In terms of accurate physical modeling, such an extreme warping of bonds is unlikely to produce sensible results. Especially for the approximate matching algorithms discussed in Section 4.3, the tolerance must be chosen such that acceptable regions of mismatch for diﬀerent bonds do not overlap each other. Finally, since the querying algorithm above consists of sequential searches in two sorted ranges, 2 2 the running time is O(log(Na )) + O(log(NR)) = O(log(Na · NR)). Chapter 4

Generating Hamiltonian Matrices

4.1 Exact Upscaling Technique and Band- structure Calculations

The preliminary work done in Section 3.3 eliminated various pitfalls of the raw Wannier Hamiltonians produced by Wannier90. For this purpose, the original data set was transformed into a diﬀerent representation in terms of interactions along bonds. The generation of Hamiltonian matrices representing structures whose physics are encapsulated within the initial Wannier Hamiltonian is now a straightforward process. Given two sets of atomic positions {p(i)} and {p(j)}, a scan through all pairwise combinations must be done, whilst querying for the corresponding bond and then copying the interaction blocks into a large matrix container H({p(i)}, {p(j)}), as summarized in the table below.

49 50 CHAPTER 4. GENERATING HAMILTONIAN MATRICES

p(i) p(i) ... p(i) p(i) 1 2 Ni−1 Ni p(j) h(b(i,j)) h(b(i,j)) ... h(b(i,j) ) h(b(i,j) ) 1 1,1 1,2 1,Ni−1 1,Ni p(j) h(b(i,j)) h(b(i,j)) ... h(b(i,j) ) h(b(i,j) ) 2 2,1 2,2 2,Ni−1 2,Ni ...... p(j) h(b(i,j) ) h(b(i,j) ) ... h(b(i,j) ) h(b(i,j) ) Nj −1 Nj −1,1 Nj −1,2 Nj −1,Ni−1 Nj −1,Ni p(j) h(b(i,j) ) h(b(i,j) ) ... h(b(i,j) ) h(b(i,j) ) Nj Nj ,1 Nj ,2 Nj ,Ni−1 Nj ,Ni

From Eq. (3.26) we have that

H({p(i)}, {p(j)}) = H({p(j)}, {p(i)})†, (4.1)

H({p(i) + R}, {p(i)}) = H({p(i)}, {p(i) + R})†. (4.2) Note that in principle, the two sets {p(i)} and {p(j)} do not have to be of the same size to generate interaction matrices between them. Since our objective is to transform an initial set of H(R), as produced by Wannier90, into a diﬀerent set of H˜(R˜ ) representing the same lattice, but expressed in terms of a diﬀerent unit cell, the special case {p(i)} = {p(j) + R} is the one relevant for us. In the case where the new unit cell is a super cell, the Hamiltonian data must be upscaled, meaning that some of the original interactions must be repeated in the new representation. The situation in Fig. 3.2 can now be readily understood, as it represents the ideal case where all Wannier centers converged onto a single atomic position inside the initial unit cell.

For large {p(i)}, where the bond length between atomic positions drastically exceeds the interaction range of the initial Wannier Hamil- tonian, the produced device Hamiltonian matrices exhibit a high degree of sparsity whose pattern depends on the ordering of the positions in {p(i)}. The standard ordering relation for atomic positions used throughout this work is deﬁned as

pi < pj ⇔ [pi,1 < pj,1] ∨ [(pi,1 = pj,1) ∧ (pi,2 < pj,2)] ∨ ... (4.3) 4.1. EXACT UPSCALING 51

(a) (b) (c) z

R=(-1,0)

y x

Figure 4.1: Example of the spatial ordering obtained with Eq. (4.3) and the resulting Hamiltonian matrices. (a) Wannier Hamiltonian matrices H(R) on a grid of 37 R vectors (empty blocks are ommit- ted), represented using the primitive unit cell from Wannier90. (b) Atomic ordering according to Eq. (4.3). The green circles represent atomic positions with their ordering index displayed next to them. (c) Hamiltonian matrices H(R) exhibiting a block tri-diagonal sparsity pattern on a grid of 9 R vectors, represented using an orthorhombic super cell.

This ordering relation is also known as a lexicographical order, which is used in sorting, for example, names in a phone book. It leads to atomic positions organized in slices along the ﬁrst coordinate axis. For the orthorhombic unit cells typically used in quantum transport simulations, this is equivalent to cutting slices perpendicular to the x- axis, which is also the transport direction of electrons. In tandem with the limited range of Wannier functions, this ordering scheme results in a block tri-diagonal sparsity pattern in the Hamiltonian matrices, as exempliﬁed in Fig. 4.1. The generated Hamiltonian matrices and R vectors can then be used in bandstructure calculations using Eq. (3.16). To demonstrate the validity of our approach, a MoS2 monolayer structure is considered. A DFT calculation, followed by a wannierization were performed on both the primitive hexagonal cell and an orthorhombic super cell. After employing the upscaling technique on the Hamiltonian represented in the primitive cell to match the orthorhombic cell, the bandstructures extracted from either case can be compared directly. The 52 CHAPTER 4. GENERATING HAMILTONIAN MATRICES results are equivalent up to slight discrepancies coming from the transformation into Wannier functions and the Fourier interpolation of the bandstructure, as presented in Table 4.1 and Fig. 4.2.

∗ ∗ ∗ ∗ Eg(eV ) me mh m˜ e m˜ h VASP 1.648 0.4639 0.5769 - - wannier90 1.656 0.4644 0.5759 0.4619 0.5715 scaled 1.656 0.4620 0.5765 0.4596 0.5720

Table 4.1: Band gaps and eﬀective masses extracted at the K’ point (in ∗ ∗ Γ-X direction) from the bandstructures in Fig. 4.2. me and mh were ∗ ∗ calculated using parabolic ﬁts,m ˜ e andm ˜ h are derived from Hessian matrices (details given in Section 4.2.2). 4.1. EXACT UPSCALING 53

(a) (b)

VASP Orth Wannier Orth. Scaled Prim.

Figure 4.2: Comparison of the bandstuctures of the smallest orthorhombic cell of a MoS2 monolayer. The dashed blue lines have been obtained with the proposed upscaling approach, based on bond- centered Wannier Hamiltonians as described in Sections 3.3, using a primitive hexagonal cell as basis. The K’-point is where the K-point of the hexagonal symmetry is found in the orthorhombic symmetry. The red circles result from a direct simulation of an orthorhombic cell in DFT and the green lines from a wannierization of this data. The observed diﬀerences mainly come from the wannierization itself (3.22 meV on average between the red circles and the green curves), not from the upscaling method (0.42 meV between the green and the blue curves). (a) Zoom around the minimum of the conduction band and the maximum of the valence band. (b) Full bandstructure in the ﬁrst Brillouin zone. 54 CHAPTER 4. GENERATING HAMILTONIAN MATRICES

(a) (b)

x x

Figure 4.3: Van der Waals heterostructure made of a WS2 (bottom) and a MoS2 (top) monolayer. The primitive unit cell of this stack is made of 6 atoms only due to the close lattice constants. (a) Side view of the unit cell. (b) Top view of the same unit cell.

Our approach is not limited to structures made of exact repro- ductions of the initial DFT unit cell. Since the Hamiltonian data from Wannier90 is represented in terms of bonds, any domain where all or a sub-set of these bonds are present can be constructed. Since inter-atomic interactions depend on the surrounding environment, the physics may not be captured appropriately. One class of materials where removing or adding atomic layers is possible without significantly perturbing the local properties, are weakly interacting van der Waals heterostructures (vdWh) [68]. A MoS2-WS2 stack, as shown in Fig. 4.3, ideally illustrates the concept of vdWh. Both materials have practically the same lattice constant and a very similar inter-layer distance in multi layered configurations. First, we examine the coupling matrices between a single-layer of WS2 placed below a monolayer of MoS2. To estimate the interaction strength inside each layer and between them, we should measure the so-called ’total energy’ contained in each bond. As our interaction matrices are in general not quadratic, we cannot compute their eigenvalues and sum them. Instead, we can perform a singular value decomposition and add the results, as proposed in Table 4.2. We are first mainly interested in the off-diagonal elements as the diagonal ones are subject to an arbitrary shift in energy. From Ta- 4.1. EXACT UPSCALING 55

WSW,l SW,u Mo SMo,l SMo,u W 19.5859 3.4470 3.4494 0.0209 0.0684 0.0083 SW,l 3.4470 19.0588 1.7342 0.0074 0.0333 0.0065 SW,u 3.4494 1.7342 19.1133 0.0658 0.4445 0.0315 Mo 0.0209 0.0074 0.0658 21.3802 3.1517 3.1480 SMo,l 0.0684 0.0333 0.4445 3.1517 18.6284 1.4716 SMo,u 0.0083 0.0065 0.0315 3.1480 1.4716 18.5698

Table 4.2: Sum of the singular values corresponding to atomic interactions inside the unit cell in Fig. 4.3. The strongest MoS2-WS2 interactions are highlighted in blue. The indices l(u) refer to the sulfur layer situated above (below) the transition metal layer. ble 4.2 it is apparent that the intra-layer interactions are stronger in both the MoS2 and WS2 layers, than the inter-layer ones by about 2 orders of magnitude, except for the S-S connection at the layer interface. As expected, stacked MoS2 and WS2 only weakly interact with each other. However, to determine whether the properties of isolated MoS2 (WS2) can be retrieved from the Hamiltonian of the MoS2/WS2 heterostructure by removing the entries corresponding to WS2 (MoS2), the bandstructure resulting from this process must be computed. To this end two sets of Hamiltonian matrices were constructed, one for the upper layer of MoS2 and one for the lower layer of WS2. Bandstructure calculations for both sets can then be compared to their exact counterparts, i.e. DFT simulations of pure MoS2 or WS2 monolayers. Results are presented in Fig. 4.4. From these comparisons, it can be deduced that the approximation of using only one layer from the bilayer stack works well to extract the properties of an isolated 2-D material, which is also confirmed by the electron and hole effective masses extracted at the K-point, which are almost identical with less than 1% difference between the ones coming from the individual monolayers and those computed from the bilayer stack, as presented in Table 4.3. More complex device structures may thus be constructed on the basis of the bilayer stack, where in some sections both materials overlap, while in others only one compound is present. An example of such a heterojunction is given in Fig. 4.5. 56 CHAPTER 4. GENERATING HAMILTONIAN MATRICES

(a) (b) (c)

(d)

(e)

(f)

(g)

Figure 4.4: Bandstructures derived from the top layer of MoS2 and the bottom layer of WS2 from the heterostructure in Fig. 4.3 and comparison to the DFT simulations of pure MoS2 and WS2 monolayer. The thin black lines belong to the double layer simulation, the thick green (blue) lines to pure MoS2 (WS2) monolayers. (a) Full bandstructure of the double layer heterostructure and of both monolayer simulations. (b) Comparison of the MoS2 bandstructure extracted from the double layer simulation with that of the pure monolayer. (c) Same as (b), but for WS2. (d) Zoom into the conduction band from (b). (e) Zoom into the valence band from (b). (f) Zoom into the conduction band from (c). (g) Zoom into the valence band from (c).

∗ ∗ ∗ ∗ Eg(eV ) me,x me,y mh,x mh,y MoS2 extr. 1.6558 0.4600 0.4627 0.5722 0.5636 MoS2 pure 1.6557 0.4596 0.4624 0.5720 0.5632 WS2 extr. 1.7864 0.2985 0.3053 0.4130 0.3981 WS2 pure 1.7862 0.2983 0.3051 0.4129 0.3979

Table 4.3: Band gaps and eﬀective masses extracted at the K point in x and y direction from the bandstructures in Fig. 4.4. The eﬀec- tive masses were derived from Hessian matrices at the K-point (details given in Section 4.2.2). The comparison is between MoS2 (WS2) monolayers extracted from the bilayer stack in Fig. 4.3 with the pure variants of a monolayer of MoS2 (WS2). 4.1. EXACT UPSCALING 57

WS2

MoS2

Figure 4.5: MoS2/WS2 van der Waals heterostructure made of three parts: one pure MoS2 extension, one overlap region, and one pure WS2 extension. The starting point is the unit cell from Fig. 4.3, which is then upscaled to the smallest possible orthorhombic cell and then repeated 36 times. In the first 12 repetitions, the WS2 entries were removed, whereas in the last 12 the MoS2 entries were discarded. The Hamiltonian matrix (left) and the corresponding device structure (right) are plotted. The different sections clearly manifest themselves as sub-matrices of different sizes, with transition regions in between. 58 CHAPTER 4. GENERATING HAMILTONIAN MATRICES

Such layered structures are particularly appealing to create p-n diodes at the ultimate thickness limit [69] with two 2-D monolayers stacked on top of each other, as in Fig. 4.5, one p-doped, the other one n-doped. Electrical doping through additional gate-like contacts is a commonly used approach for that. With overlap only in the central part, an ultra-thin depletion region can be obtained, as demonstrated experimentally in [70]. In the speciﬁc MoS2/WS2 example, removing the Hamiltonian entries corresponding to one layer and computing the bandstructure of the remaining components produces accurate results, but the success of this procedure might strongly depend on the materials in question. Often relaxed structures in a multi-layered arrangement do not exactly match their monolayer parent, thus leading to inaccurate bandstructures for the separated individual layers. Ad- ditionally, diﬀerent materials usually do not share the same lattice constant, contrary to MoS2 and WS2. In such cases, larger unit cells are required in the DFT calculations so that the lattice mismatch can be absorbed. Nevertheless, strain is inevitable, as for example in the silicene-graphene heterostructure from Fig. 3.5. The MoS2/WS2 system is ideal and was chosen for demonstrational purposes. The separation of the Hamiltonian entries can still be applied to more complex material stacks and the algorithms presented here can toler- ate some spatial warping. The limits of our approach will be explored in Section 4.3.

4.2 Interface with OMEN

The concepts introduced in Sections 3.3 and 4.1 can now be used to manipulate outputs from Wannier90 and convert them to inputs for OMEN. Here, step (c) from Fig. 2.3 is explained in detail. An outline is given in Fig. 4.6.

The starting point is a Hamiltonian in terms of interactions along bonds, which is created on the basis of Wannier90 outputs. Since the underlying unit cell is always primitive, the algorithm introduced in Section 3.1 for automatic unit cell expansions according to a template is recalled. By convention OMEN deﬁnes the x-axis as the transport direction, the y-axis as the restricted axis (direction of conﬁnement), 4.2. INTERFACE WITH OMEN 59

y y y (b) (c) (a) z z x z x x

(d) x

Figure 4.6: Schematic of the steps involved to interface Wannier90 with OMEN, starting from a conversion of Wannier90 outputs into a Hamiltonian in terms of interactions along bonds, as introduced in Section 3.3. (a) Repeatable primitive cell (here hexagonal MoS2) from which the desired simulation domain can be constructed. (b) Cor- responding orthorhombic cell. (c) Extended orthorhombic cell such that bonds with signiﬁcant interactions do not extend beyond next- neighbor cells. (d) Schematic of the simulation domain. The unit cell in (c) is repeated both along the transport axis (x) and along the direction assumed periodic (z). Three Hamiltonian components are created, H0, which contains the interactions with neighbor cells along x and H+(H−), which includes the interactions with the neighbor cells along +z(−z) (see Eq. (4.4)). 60 CHAPTER 4. GENERATING HAMILTONIAN MATRICES

(a) y (b)

Figure 4.7: Examples of lattices with different symmetries and possible transport directions. (a) MoS2 monolayer structure with hexagonal symmetry. The smallest orthorhombic cell is depicted as a blue frame, where the transport direction can be specified along (100) in the lower left corner or along (010) in the upper right corner. (b) BiIO monolayer structure with cubic symmetry, where an infinity of orthorhombic super cells exist. In the center, the primitive cell with transport along (100) is depicted. Due to the cubic symmetry, transport along (010) is equivalent. Transport along (110), (210) or in general (mn0) with m, n ∈ N is also possible. and the z-axis as the periodic axis. Since the first step is to find the smallest possible orthorhombic super cell, a template basis adhering to this convention must be specified. Depending on the lattice symmetry, there might be multiple directions along which transport can be simulated. Examples are presented in Fig. 4.7. For the MoS2 monolayer structure in Fig. 4.7, the template bases used for each case are:

1 0 0 0 0 1 T100 = 0 0 1 , T010 = 1 0 0 . 0 1 0 0 1 0 4.2. INTERFACE WITH OMEN 61

The ﬁrst column is the transport direction, the second one the restricted direction, and the third one speciﬁes the direction assumed periodic.

The unit cell used as the basic building block for devices in OMEN must not only be of orthorhombic symmetry, but also the interactions may not exceed next neighbor cells along the periodic (z) direction. This condition is necessary such that the device Hamiltonian matrix H(kz) can be written as

ikz ∆ −ikz ∆ H(kz) = H0 + H+e + H−e . (4.4)

where H−,H+ and H0 were introduced in Fig. 4.6, ∆ is the width of the orthorhombic cell along the z-axis, and the wave vec- π π tor − ∆ ≤ kz ≤ ∆ models the periodicity of the system according to Bloch’s theorem. Expansion coefficients for the orthorhombic cell corresponding to (c) in Fig. 4.6 should be extracted directly from the Hamiltonian in terms of interactions along bonds. As the longest- range interactions can usually be neglected without significantly af- fecting the transport simulations, a compromise between the matrix bandwidth and the accuracy can be made. There are further possibilities to decrease the size of the Hamiltonian matrices for transport calculations. The first and most important one is the size of the Wan- nier basis set. Even though the construction of the basis does not belong to the Winterface functionalities, it is still relevant to briefly discuss the wannierization process itself.

4.2.1 Wannierization Process

For quantum transport, accurate bandstructure modeling is required only around the gap separating the conduction from the valence band taking about 1eV on each side of it. A suitable initial guess for Wannier90 can often be found by analyzing the character of the site- projected wave functions. In VASP this information is stored in the PROCAR ﬁle [71]. A decomposition of each band of monolayer MoS2 is presented in Fig. 4.8. 62 CHAPTER 4. GENERATING HAMILTONIAN MATRICES

(a) (b) (c)

0.49

1.64 cient [normalized]

0.95 projection coe projection

Figure 4.8: (a) Decomposition of the DFT plane-waves into 1s, 3p and 5d orbitals projected to the molybdenum atom. The region around the band gap appears to be predominantly of d-character. (b) Same as (a), but for the sulfur atoms, whose behavior around the band gap is dominated by p-like orbitals. (c) Comparison of the bandstructures resulting from two diﬀerent initial guesses on the basis of (a) and (b). The green bands were derived using an initial guess containing 5 d-orbitals on molybdenum and 3 p-orbitals on each sulfur atom, resulting in a basis set of 11 Wannier functions. The black bands were derived using only dx2-y2,dx2 and dxz orbitals on molybdenum. The accuracy of the smaller basis set (black bands) is lower than the one with more Wannier functions, especially at energies larger 0.5eV into the conduction band. 4.2. INTERFACE WITH OMEN 63

Wannier functions are constructed by mixing plane-wave eigenfunctions of the Hamiltonian operator, as deﬁned in Eq. (2.12). In general, each band receives contributions from all Wannier functions. It is not always possible to reduce the size of the basis set once the wannierization is complete. However, possibilities of neglecting some of the Wannier functions whilst accurately reproducing the bands in an energy window exist and are discussed in Appendix B.

4.2.2 Error Estimation The basic principle when generating Hamiltonian matrices of minimal size for a given device structure consists of sorting out interactions along bonds whose absence does not significantly affect the bandstructure in the critical region around the band gap. This can be done by both setting a cutoff tolerance for the interaction strength, below which the bond is discarded, or by directly setting spacial limits and discarding bonds extending beyond them. Analytically, an upper bound to the perturbation of eigenvalues is given by Weyl’s Theorem. Let νi ≥ ... ≥ νn be the eigenvalues of a Hamiltonian operator H, and µi ≥ ... ≥ µn the eigenvalues of a perturbed operator H + P , then

|νi − µi| ≤ kP k2 ∀i ∈ {1, ..., n}. (4.5) This provides us with both an upper bound for the expected perturbation and the insight that the effect of perturbations is cumula- tive. For a detailed view, bandstructures must be computed based on a perturbed and an unperturbed Hamiltonian matrix, after which the influence of the perturbation can be estimated by directly comparing the results. To test perturbative effects and the functionality of the upscaling technique introduced in Sec. 3.3 and 4.1, a second algorithm has been developed to compute the bandstructures of a super cell with Hamiltonian data belonging to a primitive cell. Such an algorithm relies on the zone folding concept. If two completely different methods, upscaling and zone folding, produce the same results, then the likelihood that both are working correctly is high. 64 CHAPTER 4. GENERATING HAMILTONIAN MATRICES

In crystal theory, the basis G spanning the reciprocal unit cell is given by

G = 2πB−T , (4.6) The ratio between the volume of the reciprocal cell coming from the primitive (|Gp|) and the super cell (|Gs|) is equal to the ratio of the super cell (|Bs|) and of the primitive cell (|Bp|) volume, i.e. |Gp|/|Gs| = |Bs|/|Bp|. Each k-point on a path deﬁned in the reciprocal cell of a supercell is therefore found multiple times in the reciprocal cell of the primitive unit cell (Brillouin Zone, BZ), as demonstrated in Fig. 4.9 The algorithm to compute a bandstructure with the zone folding method includes the following steps:

• Deﬁne a path of k-points in the reciprocal cell of the super cell. • Map the k-points in the reciprocal cell of the super cell to those in the reciprocal cell of the primitive cell using the same algorithms as to ﬁnd the atomic positions belonging to a super cell.

• Compute the bandstructure at these k-points with the Hamil- tonian directly imported from Wannier90 and corresponding to the primitive cell.

• Group the energies of the k-points that are folded into the same location in the reciprocal unit cell of the super cell.

In the example of Fig. 4.9, the number of energies that must be grouped is equal to 2, as expected from the volume ratio of |Gp|/|Gs|. For the purpose of computing the bandstructure of super cells, this approach is much faster than the upscaling method, because it solves two eigenvalue problems of dimension N for each k-point, instead of solving one eigenvalue problem of dimension 2N per k-point. The bandstructure computed with this algorithm can be considered as an exact reference for the upscaling technique. The inﬂuence of perturbations to the Hamiltonian data can thus be determined by calculating bandstructures for the same cell using ﬁrst the raw Wannier90 outputs with the folding algorithm laid out above and secondly with the 4.2. INTERFACE WITH OMEN 65

(a) (b)

K M K k 0 y M y

Figure 4.9: Schematic of a primitive hexagonal unit cell compared to an orthorhombic super cell in both real and reciprocal space. (a) Primitive hexagonal cell of MoS2 in green and orthorhombic super cell in blue. (b) Same as (a), but in reciprocal space. Symmetry points in blue (green) belong to the blue (green) cell. The Wigner-Seitz cell for each case, i.e. the first Brillouin zone, is indicated by thin dashed lines and the irreducible parts by the stronger shading. The hexagonal green unit cell is completely covered by 4 orthorhombic blue cells, where in each case only half overlaps. The blue k-point path, Y − Γ − X, is defined in the irreducible part of the blue cell, with periodic images given in the center. Each k-point of the blue path is found twice in the green unit cell, but in different sections of the Brillouin zone, due to folding. For example, the Γ-point of the blue cell appears as both the Γ-point of the green cell, but also as the M-point in the middle, i.e. the M-point is folded into the Γ-point. 66 CHAPTER 4. GENERATING HAMILTONIAN MATRICES upscaling method after sorting out certain interactions along bonds beforehand.

For quantum transport, errors in the ﬁrst and second derivatives of the bandstructure are important as well, the latter being used to calculate eﬀective mass tensors. These quantities can be computed together with the bandstructure with minimal overhead. From Eq. (3.16), the derivatives of the Hamiltonian operator can be derived analytically:

∂ X ikR H(k) = i Ri ·H(R)e , (4.7) ∂ki R

2 ∂ X ikR H(k) = − Ri · Rj ·H(R)e , (4.8) ∂ki∂kj R where the i and j indices run over the x, y, and z axes. Fur- thermore, performing the derivative on the stationary Schr¨odinger equation yields:

∂ ∂ m = n : hm| H(k) |mi = Em(k), (4.10) ∂ki ∂ki

∂ hn| ∂ H(k) |mi m 6= n : hn| |mi = ∂ki , (4.11) ∂ki Em(k) − En(k)

where the gauge freedom allows us to set hn| ∂/∂ki |ni = 0. The second derivative of the bandstructure can be computed by performing another derivative on Eq. (4.10). 4.3. APPROXIMATE UPSCALING 67

From Eqs. (4.7) through (4.12), the first and second derivatives at each k-point and for each band can be computed (even in the presence of degeneracies [72]). Since |mi and |ni are eigenstates of the Hamiltonian matrix H(k) and are typically computed at the same time as the bandstructure, since the overhead to obtain the derivative is low, involving only small matrix manipulations. With these results, the influence of Hamiltonian perturbations on the bandstructure can be precisely studied, as shown in Fig. 4.10. It can be seen that discarding the long range interactions produced by Wannier90 leads to relatively accurate results. For the bandstructure computed using the upscaling technique, the interactions were filtered such that those exceeding an expansion of C = diag([3, 2, 1]) are discarded (the full expansion where are all bonds are included is C = diag([5, 4, 1])). The user of Winterface can determine whether this accuracy is sufficient or not for his application. With the approximation presented in Fig. 4.10, the orthorhombic cell used in the quantum transport simulation is 20/6 ≈ 3.33 times smaller than if all interactions were included, which is advantageous from a computational viewpoint.

4.3 Approximate Upscaling Technique and Local Bandstructures

In this Section two advanced simulation examples will be presented. They require additional work as compared to previous cases, as their device structures must be created manually. The interfacing scheme introduced in Section 4.2 is not directly applicable to them. Never- theless, valid OMEN inputs can be produced. The purpose of these examples is to explore the limits of the proposed upscaling approach. 68 CHAPTER 4. GENERATING HAMILTONIAN MATRICES

(a) (b)

(e) (f)

Figure 4.10: Comparison of the MoS2 monolayer bandstructure around the top of the valence band along the Y − Γ − X path of the super cell Brillouin zone (see Fig. 4.9). The K’ point is where the K-point of the primitive cell is found in the super cell. The bands calculated using the folding algorithm (thick green lines) are considered the reference, whereas the bands calculated using the upscaling technique (dashed black lines) discard some of the long ranging interactions. The critical points at the top of the valence band at the Γ-point and the K’-point are highlighted. (a) Top two valence bands. (b) Error between the green and the dashed lines in (a). (c) Same as (a), but for the ﬁrst derivative of the bandstructure. (d) Error in dE/dk. (e) Same as (a), but for the second derivative of the bandstructure. (f) Error in d2E/dk2. 4.3. APPROXIMATE UPSCALING 69

Since a poorly converged wannierization is not compatible with the concepts presented here, it is assumed that the Wannier functions are maximally localized and the imaginary parts of the interactions in- signiﬁcant.

4.3.1 Bond Index Substitution The upscaling procedure explained in Section 4.1 relies on exact bonds in the sense that both the starting index i and the target index j, as well as the bond vector bij are provided. The target index is not strictly required, as a starting point and a bond vector are suﬃcient to extract an interaction matrix for the bond in question. The only criterion that should not be violated under any circumstances is that the interaction matrix along this bond has the correct dimensions. Otherwise, the Hamiltonian matrices cannot be properly generated. From this point of view, all positions with the same number of Wannier functions matched to them are potential valid targets when searching for interactions along bonds. This enables the creation of Hamilto- nian matrices that do not correspond to exact super cells of the initial primitive cell simulated in DFT. In this way, it is possible to construct complex structures for quantum transport and to investigate their local properties such as the bandstructure of a well-speciﬁed device structure.

As an illustration, we again consider a heterostructure composed of MoS2 and WS2, this time not placed on top of each other, but next to each other, as shown in Fig. 4.11 From the original unit cell in Fig. 4.11, through repetition of certain subsections, larger structures can be generated, with different configurations. For example, the intrinsic regions on both sides of the interface can be made longer. Or one material can be sandwiched between two extensions of the other. To realize such structures, the method and approximations presented in Fig. 4.12 must be followed. The challenge with atomic arrangements such as the ones in Fig. 4.12 resides in the association of the correct Wannier data with each bond. To ensure a proper Hamiltonian construction process, the matrix filling algorithm described in Section 4.1 must be modified when 70 CHAPTER 4. GENERATING HAMILTONIAN MATRICES

Figure 4.11: Top view of a MoS2-WS2 heterostructure, where both materials are placed next to each other. Due to the applied periodicity along the x-axis in DFT, the interface region appears in the middle of the structure, as well as on both sides of the unit cell delimited by the dashed orange rectangle. The small blue rectangle corresponds to a primitive unit cell of WS2, the green one to a primitive unit cell of MoS2. Each of these two unit cells is only marginally aﬀected by the presence of the other material, provided that the distance from the interface region is large enough. If wannierized, their Hamiltonian entries can be expected to match those of an isolated layer, up to a rigid energy shift. 4.3. APPROXIMATE UPSCALING 71

(a)

(b)

(c)

Figure 4.12: Examples of device structures that can be constructed after wannierizing the unit cell in Fig. 4.11. (a) WS2-MoS2 super- lattice. The black frame represents the unit cells in 4.11. The red arrow refer to interactions close to the WS2-MoS2 interface. The blue arrows represent interactions between two cells of pure WS2 or MoS2. Finally, the orange arrows connect the pure material with the interface region. (b) Structure generated by repeating the subcells in the intrinsic regions on both sides, leaving one interface region in the middle. (c) Structure generated by repeating subcells in the intrinsic region of MoS2 with an interface region to WS2 on both sides. 72 CHAPTER 4. GENERATING HAMILTONIAN MATRICES

searching for an interaction matrix with indices and vector {i, j, bij}:

• If no exact interaction matrix can be found, determine all indices jn where the number of matched Wannier functions is equal to that at index j. For each such jn, identify the interaction matrix corresponding to {i, jn, bij}.

• Repeat the previous steps for the reverse bond {j, i, bji}.

• The ﬁnal interaction matrix h(i, j, bij) is equal to the average † [h(i, jn, bij) + h(j, in, bji) ]/2, thus ensuring self-adjointedness of the resulting Hamiltonian.

A few points should be further considered:

• This scheme is not guaranteed to work as it can produce mean- ingless matrices if the coupling blocks are not chosen carefully. It is up to the user to make sure that the two small unit cells in Fig. 4.11 are distant enough from the interface region so that they describe the intrinsic properties of the targeted material.

• The number of queries per bond is increased dramatically, which slows down the whole algorithm. As the time spent in Winter- face is typically only a fraction of the overall time to simulate quantum transport, this is a minor issue.

• We must allow for some spatial tolerance when matching the supplied bond vector to those in the interaction data. For the example presented in Fig. 4.12 this is minimal. For a diﬀerent structure with distorted bonds at the interface, the situation might be more complicated.

• Because of these restrictions, it is recommended to manually check the produced Hamiltonian matrices before plugging them into a quantum transport simulator.

Due to the localized nature of Wannier functions, each part of the structure only interacts with its immediate surroundings. This fact can be exploited with the modiﬁed scheme described above to generate a set of approximate Hamiltonian matrices for all subsections 4.3. APPROXIMATE UPSCALING 73

(a) (b)

WS2 MoS2 ref Eg = 1.79 eV z WS2 ref WS2 cell 1 2 4 WS2 cell 2 MoS2 cell 3 1 3 MoS2 cell 4

x MoS2 Eg = 1.64 eV

Figure 4.13: (a) Comparison of local bandstructures coming from unit cells extracted at diﬀerent positions in Fig. 4.11. The results of pure MoS2 and WS2 are given as references, green for MoS2 and blue for WS2. The top of the valence band is set to E = 0 in all cases. The black lines refer to the bandstructures of the unit cells indicated by the shaded regions in the central unit cell. (b) Conduction (upper line) and valence (lower line) band edge as a function of the distance from the WS2-MoS2 interface.

of the full structure. Since the proposed scheme ensures the spacial inversion symmetry of Eq. (3.17), the resulting matrices can be used in bandstructure calculations. This can be very useful when charac- terizing the local properties of a large structure such as determining whether the small unit cells in Fig. 4.11 exhibit the same properties as pure MoS2 or WS2. A comparison of local bandstructures computed for diﬀerent small unit cells around the interface region in Fig. 4.11 is shown in Fig 4.13. The bandstructures of pure MoS2 and WS2 are also given as references. 74 CHAPTER 4. GENERATING HAMILTONIAN MATRICES

4.3.2 Combining Multiple Wannierizations The second extension of the upscaling technique builds on the concepts introduced above, but here the approach of generating approximate Hamiltonian matrices is taken a step further by combining interactions along bonds derived from different wannierizations. Such structures must have clearly defined interface regions where bonds belonging to different wannierizations meet. Furthermore, the region at the interface between both domains must be included in each wannierization so that it can be used to connect them. To demonstrate this approach, a large superstructure was constructed from three different parts belonging to the stacked heterostructure of MoS2 and WS2 in Fig. 4.3 on either side, and from the lateral heterostructure of MoS2 and WS2 side by side in Fig, 4.11 in the middle. The created arrangement is depicted in Fig. 4.14. The process of searching for approximate interactions remains the same as before, except that for bonds crossing an interface, the final interaction matrix will be the mix from a wannierization A (MoS2-WS2 stack) and a wannierization B (WS2-MoS2 lateral heterostructure), as illustrated in Fig. 4.15. For this scheme to produce useful Hamiltonian matrices, the interactions in terms of bonds corresponding to each wannierization must be first adjusted and made compatible. Simulations of different structures in DFT have a shift in energy relative to each other, which must be compensated before two different wannierizations can be combined into a single Hamiltonian matrix. For two Hamiltonian matrices H(k) and H˜(k) distinct only by a shift ∆ in the energy spectrum, we find

H˜(k) = U(k) · [D(k) + ∆1] · U(k)† = H(k) + ∆1 (4.13) where U(k) diagonalizes H(k). When transforming into the representation in terms of R vectors this becomes

X H˜(R) = [H(k) + ∆1] · e−ikR = R + ∆1δ(R). (4.14) k In other words, a shift in energy is found on the diagonal of H(R = 0), which in the representation in terms of interactions along 4.3. APPROXIMATE UPSCALING 75

(a) (c (b)

MoS2-WS2 stack WS2-MoS2 WS2-MoS2 stack x

Figure 4.14: Schematic of a large superstructure made of three diﬀer- ent parts wannierized individually and then interfaced together along the x-axis. Zooms of the individual regions are included above the full structure. The areas where two domains are connected to each other are shaded in red. There are two such regions. In the ﬁrst one, the WS2 layer of the MoS2-WS2 stack and of the WS2-MoS2 lateral heterostructure is chosen as common element, while MoS2 plays the same role in the second shaded area. 76 CHAPTER 4. GENERATING HAMILTONIAN MATRICES

A,B interface wannierisation A wannierisation B

Figure 4.15: Schematic describing how interaction matrices for bonds crossing an interface region between two wannierizations A and B are constructed. Each wannierization is deﬁned only in its own atomic domain. The blue atoms represent the domain for wannierization A and the green atoms the domain for wannierization B. If the index i belongs to the blue domain and the index j to the green one, an acceptable substitute for the target index must be found for both directions of the bond vector bij and bji, respectively. The ﬁnal interaction matrix is thus a mix of both wannierizations h(i, j, bij) = † [hA(i, jn, bij) + hB(j, in, bji) ]/2. 4.3. APPROXIMATE UPSCALING 77 bonds lands on the diagonals of the self-interaction matrix.

This shift can be extracted from bonds across interfaces between two sections described by diﬀerent wannierizations. If such a bond starts in wannierization A at index i and ends in wannierization B at index j, the underlying assumption is that the substitute atom used in wannierization A (B) is highly similar to the real target atom in wannierization B (A). In this way a pairwise matching of atoms, one from each atomic domain belonging to wannierization A (B), is done. For the case presented in Fig. 4.15, the atom at index i (j) in wannierization A (B) is matched with the atom at index in (jn) in wannierization B (A). Therefore, the energy shift ∆ can be extracted from the corresponding self-interactions

|hA(i, i, 0)−hB(in, in, 0)| ≈ |hA(jn, jn, 0)−hB(j, j, 0)| ≈ ∆1. (4.15)

In principle the shift can be extracted from just one pair of matched atoms, but by applying Eq. (4.15) to all bonds across an interface between Wannier domains, the mean of the energy shift is computed and the accuracy is improved. For the structure in Fig. 4.14, the standard deviation is found to be < 1meV for both connections of Wannier domains in the WS2 region on the left, as well as in the MoS2 region on the right.

Compensating for a shift in energy between different wannierizations involves modifications of the diagonal elements of the self- interaction Hamiltonian blocks. However, there is another, yet un- mentioned, degree of freedom of Wannier functions that manifests itself only in the off-diagonal elements. Wannier functions are, just like wave functions in general, defined only up to a phase factor. For maximally localized Wannier functions, as produced by Wannier90, the Hamiltonian entries do not contain any imaginary component and the aforementioned phase is rotated into the real axis, where it can turn into either 1 or -1. Since the diagonal elements of self interaction Hamiltonian blocks are of the form hwi,0|H|wi,0i, the sign cancels out, but for off-diagonal elements of the form hwi,0|H|wj,Ri, it does not. As Wannier functions form an orthonormal basis set, flipping a 78 CHAPTER 4. GENERATING HAMILTONIAN MATRICES sign is equivalent to changing a left-handed coordinate system into a right-handed one. Both are equally valid and the underlying physics may be expressed in either one, but two different systems cannot be mixed with each other. To investigate this effect for the structure in Fig. 4.14, the Hamiltonian matrices for all atoms at the interface between two domains were computed for both wannierizations involved. These Hamiltonian matrices correspond to the whole blue and green domains in Fig. 4.15. To connect two different wannierizations, these two Hamiltonian matrices must describe the same physics up to a small perturbation, with the additional freedom that the signs of the Wannier functions may vary between these two cases. To find the relative Wannier signs distribution, significant entries in each matrix are replaced by their sign, whereas entries below a certain tolerance level are set to zero. The relative distribution of Wannier signs is then given as the elementwise product of the two modified Hamilto- nian matrices. The results for both connecting areas in Fig. 4.14 are plotted in Fig. 4.16. To make the wannierizations on each side of an interface compatible, the signs of one of them must be modified to coincide with the other. Each Wannier function contributes to a Hamiltonian matrix on a whole column and a whole row. Flipping the sign of the i-th Wannier function results in multiplying both the i-th column and the i-th row by -1, thus canceling out on the diagonal. Adapting one wannierization to the other is done by simultaneously flipping the signs of full rows and columns in the sign-difference matrices (c) and (f) in Fig. 4.16 until all green dots are turned black. The solution of this process can be written as a unitary transformation matrix P of the form

±1   ..   .  ±1 whose diagonal elements are equal to ±1. The Hamiltonian matrix is transformed through H˜ = P ·H· P , which keeps the spectrum intact, since P is self-evidently orthogonal.

To ﬁnd the sign of each entry in P, a system of linear equations is 4.3. APPROXIMATE UPSCALING 79

(a) (b) (c)

(d) (e) (f)

Figure 4.16: Sign distribution for the Hamiltonian matrices corresponding to similar structures at the interface in Fig 4.14, but resulting from two different wannierizations. Green dots represent a negative sign, black dots a positive one, and white areas 0 or insignif- icant entries. (a) Sign distribution for the WS2 region on the left of the first interface in Fig. 4.14. (b) Same as (a), but for the WS2 region on the right of the interface. (c) Elementwise product of the signs in (a) and (b). (d-f) Same as (a-c), but for MoS2 in the second interface in Fig. 4.14. 80 CHAPTER 4. GENERATING HAMILTONIAN MATRICES extracted from the sign-difference matrix S shown in sub-plots (c) and (f) in Fig. 4.16. The sign-difference matrix S corresponds to two sets of Wannier functions {wi} with signs {σi} and {wj} with signs {σj}, respectively. Each of its entries represents the following equation

σi · σj = S(i, j), (4.16) which can be written as

σi − S(i, j) · σj = 0. (4.17) From the matrix S, a large overdetermined system of equations of at most rank N-1 (the entries on the diagonal are always 1) can be extracted. The system can be completed to rank N by appending appropriate entries of σi = 1 until the rank is complete, guaranteeing a non-vanishing solution. From the solution of Eq. (4.17), the signs of all interactions along bonds can be corrected for one of the two involved wannierizations. To demonstrate the validity of this approach, local bandstructures can be computed for small unit cells situated immediately around the interface, on either side, where only one wannierization is involved, as well as for a unit cell including matrix elements coming from both wannierizations. Results for this process are shown in Fig. 4.17.

The adaptations of the Wannier signs for atoms situated near tran- sitions between two wannierization domains may cause a cascade eﬀect deeper into the structure, if the concerned atoms are used as substi- tutes at several locations. In this case the same treatment must be carried out for all interface atoms. For the structure in Fig. 4.14, the middle section was constructed using the technique introduced in Fig. 4.12(b), where substitution occured, whereas the left and right sections were constructed as an exact super cell such that no substitu- tions occur. If the middle section is used as the reference to which the other parts must be adapted, the corrections among Wannier signs is minimal and no cascading is observed. However, if the middle section is adapted to the left and right ones, cascading takes place on both sides of the middle section. Since (c) and (f) in Fig. 4.16 are diﬀerent, the regions where the middle section must be adapted on either side should not overlap. Even though this is not an issue for the example 4.3. APPROXIMATE UPSCALING 81

(a) (b) (c) (d)

Figure 4.17: Comparison of local bandstructures calculated across domains described by two wannierizations in Fig. 4.14 after equalizing the shift in energy and adapting the Wannier sign distribution. (a) Bandstructure calculated just before (green), at (yellow), and after (black) the ﬁrst interface (WS2) in Fig. 4.14. (b) Regions where the bandstructure in (a) were computed. The dashed lines indicate the interface between the wannierizations. (c) Same as (a), but for the MoS2 interface in Fig. 4.14. (d) Atomic structures corresponding to (c). 82 CHAPTER 4. GENERATING HAMILTONIAN MATRICES presented here, because of the thickness of the middle section, it may cause problems in other cases.

A few closing remarks regarding the examples presented in this Section:

• Both examples assume that Wannier functions of the same character appear in the same order, which is essential to connect two domains with different wannierizations. The order of the Wan- nier functions in the output files of Wannier90 is the same as the order of the initial projections specified during the wannierization process. For interface atoms, it is therefore imperative that the initial projections are specified in the same order.

• It was implicitly assumed that the Wannier signs should not be modiﬁed within each individual wannierization. While such a treatment might be necessary in some cases, practically, this has not been observed in all applications discussed in this work. The fact that all Wannier functions originate from the same initial DFT wave functions explains this behavior.

• A complex structure such as the one in Fig. 4.14 can always be alternatively constructed from just a single wannierization, but this would dramatically increase the number of required atoms in the initial DFT simulation to include the desired structural features. Larger DFT domains could increase the workload and computational burden by orders of magnitude, due to the typically cubic scaling of the matrix diagonalization underpinning DFT.

• The connection of two wannierization sets, as shown here, is not completely universal since the adaptation of the sign distribution might be inconsistent, depending on the location of the Wannier domains. This might be the case when one wannierization partition has two or more neighbors in close proximity. In quantum transport, however, it is unlikely that an arrangement other than a sequence of Wannier domains along the transport direction will be needed. The approach could still be improved to allow for the simulation of more general layouts. 4.4. RESULTS 83

4.4 Results

To demonstrate that the Hamiltonian matrices generated using the ideas of Chapter 3 and 4 are viable for quantum transport calculations, the transmission function through several structures mentioned previ- ously was evaluated. With this quantity. the ballistic current ﬂowing through them can be computed, taking advantage of the Landauer- B¨uttiker formalism. Recalling Eq. (2.20):

σ,L 2 |Ψ L (E, kt, xout)| −1 σ X km dE dE T (E, kt) = , (4.18) σ,L 2 R L |Ψ L (E, kt, xin)| dkn dkm n,m km it follows that for devices where all unit cells are equivalent, the transmission function simply counts the number of available propagating modes at each energy E, the fraction in Eq. (4.18) being always 1. For inhomogeneous devices, the picture is more complex.

Modes that can propagate from the contacts towards a device are injected in the latter. If we consider the left contact, these correspond to the modes with dE/dk > 0 (dE/dk < 0) for electrons (holes), where E(k) is the contact bandstructure. When illustrating this quantity, only half of the Brillouin zone will be drawn due to the symmetry with respect to k = 0. Note that the wave vectors are normalized with respect to their maximum value kmax = π/∆x, where ∆x is the length of a unit cell in transport direction. In each example the transmission function in Eq. (4.18) will be given for kz = 0 and kz = π/(2∆z). The initial DFT simulations were all performed within the generalized gradient approximation of Perdew-Burke-Ernzerhof (PBE) [38], except BiIO where van der Waals interactions were included (optB88- vdW) [73, 74]. The following examples were simulated:

• A MoS2 monolayer with transport along the [100] direction in Fig. 4.18. The device length is 40.8nm and made of 32 (identical) unit cells of 48 atoms each.

• A WS2 monolayer with transport along the [001] direction in Fig. 4.19. The device length is 41.4nm, which is equal to 25 identical unit cells of 72 atoms each. 84 CHAPTER 4. GENERATING HAMILTONIAN MATRICES

• A BiIO monolayer with transport along the [101] direction in Fig. 4.20. The device length is set to 40.9nm and is composed of 24 unit cells of 108 atoms each, all identical.

• A stack of MoS2 and WS2 monolayers, as presented in Fig. 4.5, in Fig. 4.21. The full device length is 40.8nm with an overlap region of 13.6nm. The whole structure is 32 unit cells in total, of 144 atoms in the overlap region and 72 on the left and right extensions.

• A monolayer of MoS2 and WS2 with a lateral interface region (see Fig. 4.11) in Fig. 4.22. The device length is 39.8nm, which is equivalent to 18 unit cells of 96 atoms each. The ﬁrst 9 unit cells are made of WS2, the 10th holds the interface region, whereas the last 8 are made of MoS2. Note that the MoS2 and WS2 layers are rotated such that transport is aligned with the [001] direction when compared to pure monolayers of either ﬂavor.

• A stack of graphene and silicene, as shown in Fig. 3.5, in Fig. 4.23. The device starts with pure graphene on the left, continues with an overlap region in the middle, and ﬁnishes with silicene on the right. All regions have a length of 39.3nm, resulting in a total device length of 118nm or 78 unit cells. Each unit cell in the overlap region is composed of 64 Silicon atoms, 144 Carbon atoms, and 216 ﬁctitious atoms located on the centers of the bonds between Carbon atoms. Note that this material has no band gap since it is metallic. 4.4. RESULTS 85

Electron Hole = 0 z k (d) (f) (a) (b) (c) (e) ) z Δ /(2 = z k (g) (h) (i) (j) (k) (l)

Figure 4.18: Transmission function of a MoS2 monolayer with transport along the [100]-direction for kz = 0 (top row) and kz = π/(2∆z) (bottom row), for both electrons (left half) and holes (right half). (a) Conduction bands in the left contact. (b) Electron transmission function. (c) Conduction bands in the right contact. (d) Valence bands in the left contact. (e) Hole transmission function. (f) Valence bands in the right contact. (g-l) Same as (a-f) but for kz = π/(2∆z). Since all unit cells composing the structure are identical and no bias is applied, T (E, kz) counts the number of modes propagating from one contact to the other at an energy E and momentum kz. The fact that this property is satisﬁed indicates that the upscaling procedure works as intended. 86 CHAPTER 4. GENERATING HAMILTONIAN MATRICES

Electron Hole = 0 z

(d) (e) (f) k

(a) (b) (c) ) z (j) (k) (l) Δ /(2 = z k (g) (h) (i)

Figure 4.19: Same as Fig. 4.18, but for a WS2 monolayer with transport along the [001]-direction.

Electron Hole (d) (e) (f) = 0 z k

(a) (b) (c) (j) (k) (l) ) z /(2 = z (g) (h) (i) k

Figure 4.20: Same as Fig. 4.18 and Fig. 4.19, but for BiIO with transport along the [101] crystal axis. 4.4. RESULTS 87

MoS2 Electron WS2 MoS2 Hole WS2 (d) (e) (f) = 0 z k

(a) (b) (c) (k) (l)

(j) ) z /(2 = z k (g) (h) (i)

Figure 4.21: Same as Figs. 4.18 to 4.20, but for a MoS2(top)- WS2(bottom) van der Waals heterostructure with three seperate regions, one made of pure MoS2 on the left, an overlap area in the middle, and a pure WS2 monolayer on the right.

Because of the conduction and valence band offsets between MoS2 and WS2, the electron and hole transmission functions exhibit a more complex behavior than in the case of homogeneous materials. First, the transmission can only be different from 0 if at leasts one band is available on both contacts. Secondly, if m(n) bands are injected from the left (right) contacts, then, because of reflection effects, T (E, kz) ≤ min(m, n). Besides these key features, it is difficult to interpret the transmission function results in Fig. 4.21. It should however be no- ticed that the validity of the upscaling can be verified in a different way: instead of calculating the transmission function from the left to the right contact, it can be evaluated between two adjacent cells along the transport direction. Due to current continuity, the transmission from cell i to i + 1 has to be the same as between j and j + 1, where i 6= j. This property was verified for all results shown in this chapter. 88 CHAPTER 4. GENERATING HAMILTONIAN MATRICES

WS2 Electron MoS2 WS2 Hole MoS2 = 0

(d) z

(e) (f) k (a) (b) (c) )

(l) z (j) (k) Δ /(2 = z (g) (h) (i) k

Figure 4.22: Same as Figs. 4.18 to 4.21, but for a lateral MoS2-WS2 heterostructure.

In the lateral MoS2-WS2 heterostructure in Fig. 4.22, a perculiar- ity occurs at E − Ef ≈ 1.9eV and kz = 0. It can be seen in sub-plots (a) and (c) that bands are available on both the left and right contacts in this case and in spite of that, the transmission function is equal to 0. This can be attributed to the fact that the lowest energy band in (c) has an energy width smaller than the conduction band offset between MoS2 and WS2. As a consequence, a state injected from the left contact at E − EF = 1.9eV does not find any band with the same symmetry property in the right contact. It is therefore reflected back to its origin. 4.4. RESULTS 89

kz = 0 kz = /(2 z)

(a) (b) (c) (d) (e) (f) GR GR Si Si

Figure 4.23: Transmission function through a graphene (bottom, left extension) and silicene (top, right extension) van der Waals heterostructure at kz = 0 (left) and kz = π/(2∆z) (right). (a) Band- structure of the left contact (graphene). (b) Transmission function around the Fermi level. (c) Bandstructure of the right contact (silicene). (d-f) Same as (a-c), but for kz = π/(2∆z).

Chapter 5

Dissipative Transport

5.1 Transport Model

5.1.1 Hamiltonian and Dynamical Matrix For cases where the ballistic transport model introduced in Section 2.4 does not properly capture the physics at play, electron-phonon scattering should be taken into account. This is the case when, for example, the current ﬂowing through a 2-D device becomes high enough so that the energy lost by electron start to locally increase the phonon population, thus inducing so-called Joule or self-heating processes. To model them, the electron and phonon populations must be self-consistently coupled through scattering self-energies. In this chapter, 3 transistor structures, as depicted in Fig. 5.1 will be studied using this extended transport model, with the MLWF Hamiltonian produced by Winter- face. The dynamical matrix describing the phonons was also upscaled from the results of a small orthorhombic unit cell using a technique similar to the one developed for electrons. The upscaling of the dynamical matrix was however done manually. Note that MoS2 and WS2 were treated within the generalized gradient approximation (GGA) of Perdew, Burke, and Ernzerhof (PBE) [38], whereas the hybrid functional of Heyd, Scuseria, and Ernzer- hof (HSE06) [39] was employed for black phosphorus. A 25×1×25 Monkhorst-Pack k-point grid and a 500 eV plane-wave cutoﬀ energy

91 92 CHAPTER 5. DISSIPATIVE TRANSPORT

Figure 5.1: Schematic view of the single-gate field effect transistor with single-layer MoS2, WS2, or armchair-oriented black phosphorus (AC BP) as semiconducting channel. The gate length is set to Lg=15 nm, while the source and drain extensions measure Ls=Ld=12.5 nm −2 with a donor doping concentration ND=6e13 cm . Perfectly ohmic contacts are considered. The gate contact is separated from the channel by an HfO2 dielectric layer of thickness tox=3 nm and permittivity R=20. Transport occurs along the x-axis, y is the direction of confinement, and the z-axis (out-of-plane) is assumed periodic and modeled via a kz/qz momentum dependence of the physical quantities. 5.1. TRANSPORT MODEL 93

Figure 5.2: Electron (a-c) and phonon (d-f) bandstructure of single- layer MoS2 (a and d), WS2 (b and e), and black phosphorus (c and f) along their high symmetry lines. In case of electrons, the black dots are the results of DFT calculations with VASP [59], whereas the solid lines represent the bandstructures after a conversion of the plane-wave outputs of VASP into a set of maximally localized Wannier functions with the wannier90 tool [62]. The phonon modes have been computed with DFPT [75].

was enforced in all electronic structure calculations. Spin-orbit coupling was neglected. The accuracy of the plane-wave to MLWF conversion is demonstrated in Fig. 5.2(a-c) for the selected 2-D materials: 3 p-like orbitals were adopted for S, 5 d-like orbitals for Mo and W, and 4 sp3 hybrid for P.

The electron and phonon transport properties are modeled at the ab-initio level with NEGF. Their coupling is realized via scattering self-energies that ensure current and energy conservation. In this con- text, the following system of equations must be solved for the electron population 94 CHAPTER 5. DISSIPATIVE TRANSPORT

X RB RS R (Eδli − Hil(kz) − Σil (E, kz) − Σil (E, kz)) · Glj(E, kz) = δij l ≷ X R ≷B ≷S A Gij(E, kz) = Gil (E, kz) · (Σlm (E, kz) + Σlm (E, kz)) · Gmj(E, kz), lm (5.1)

RS where Eq. (2.28) was extended by Σil (E, kz), i.e. the scattering self-energy coupling electrons and phonons. To handle phonon transport through transistors with a 2-D channel, as in Fig. 5.1, the following NEGF-based system of equations must be processed (details are beyond the scope of this thesis [76]):

X 2 RB RS R (ω δli − Φil(qz) − Πil (ω, qz) − Πil (ω, qz)) · Dlj(ω, qz) = δij l ≷ X R ≷B ≷S A Dij(ω, qz) = Dil (ω, qz) · (Πlm (ω, qz) + Πlm (ω, qz)) · Dmj(ω, qz), lm (5.2)

where the D(ω, qz)’s are the phonon Green’s functions at frequency ω and momentum qz, ω is a diagonal matrix containing the B phonon frequency as single entry, the Π (ω, qz)’s are the boundary S self-energies, and the Π (ω, qz)’s the scattering ones, while Φ(qz) refers to the dynamical (Hessian) matrix of the studied system. The same Green’s Function types as for electrons also exist for phonons (retarded, advanced, lesser, and greater). All blocks involved in Eq. (5.2) are of size 3×3, which corresponds to the degrees of freedom of the crystal vibrations. The matrix Φ(qz) is constructed in an analogous way as H(kz): through density functional perturbation theory (DFPT) [75], as available in VASP, the dynamical matrix of a representative unit cell is computed from ﬁrst-principles. It is then scaled up to obtain the entries for a larger device structure, e.g. the transistor in Fig. 5.1. The phonon bandstructures of single-layer MoS2, WS2, and black phosphorus are displayed in Fig. 5.2(d-f).

With the knowledge of H(kz) and Φ(qz), electron and phonon quantum transport simulations can be performed, ﬁrst in the ballistic 5.1. TRANSPORT MODEL 95

Figure 5.3: (a) Electron transmission function, as obtained from quantum transport calculations, through single-layer MoS2 (solid red line), WS2 (dashed blue line), and AC BP (dashed-dotted green line) at kz=0, with ﬂat-band conditions, and with the conduction band minimum aligned with the energy E=0 eV. (b) Same as (a), but for phonons at qz=0.

S S limit of transport, i.e. Σ (E, kz)=Π (ω, qz)=0. Eqs. (5.1) and (5.2) can be solved with a recursive Green’s Function (RGF) algorithm [77] that produces only the desired entries of the Green’s Functions. The open boundary conditions are derived from the solution of eigenvalue problems with a shift-and-invert technique [78]. As an illustration, the ballistic electron and phonon transmission functions through 40 nm long monolayers of MoS2, WS2, and black phosphorus (in the armchair configuration) were computed and the results are plotted in Fig. 5.3 (a) and (b) for kz=0 and qz=0, respectively. A flat band potential is assumed in all cases so that a typical step-like behavior of the transmission as a function of the electron/phonon energy can be observed. Attention should be paid to the fact that the mini-gaps (regions with no bands) present in the phonon bandstructures in Fig. 5.2(d-f) are clearly visible in the transmission functions in Fig. 5.3(b). To accurately compute the current that flows through a 2-D transistor, it is not sufficient to keep one single momentum point, e.g. kz=qz=0. Here, it has been found that a total of 11 kz and qz momentum points is necessary to reliably model the periodicity of the 96 CHAPTER 5. DISSIPATIVE TRANSPORT out-of-plane direction z (no more than 1% current variations as compared to simulations with 21 momentum points). It should finally be underlined that electrons and phonons can only enter and leave the simulation domain at the source and drain contact extremities, no escape through the top or bottom oxide of the transistor in Fig. 5.1 is possible. This restriction certainly causes an overestimation of the lattice temperature [79] when the electron and phonon populations fully interact with each other, as described in the next Section.

5.1.2 Electron and Phonon Coupling To complete the picture it remains to deﬁne the scattering self-energies that couple electron and phonon transport and drive both populations out-of-equilibrium. To reduce the computational intensity, only the S diagonal blocks of the electron-phonon components Σ≷ (E, kz) are retained. It can be analytically demonstrated, as in Ref. [80], that these scattering self-energies ensure energy conservation. Each individual block is deﬁned as

X X X Z d( ω) Σ≷S(E, k ) = i ~ · nn z 2π qz l ij h ≷ ∇iHnl · Gll (E − ~ω, kz − qz) · ∇jHln× i ≷ ≷ ≷ ≷ Dln(ω, qz) − Dll (ω, qz) − Dnn(ω, qz) + Dnl(ω, qz) . (5.3)

i i In Eq. (5.3)) ∇iHnm=δHnm/δ(rm − rn) is the derivative of the Hamiltonian matrix block Hnm with respect to variations along the i=x, y, or z axis of the bond rm −rn connecting atoms n and m. Since the Hamiltonian matrix in the MLWF basis includes connections with the 30 (36) nearest-neighbors (NNs) of each atom in MoS2 and WS2 (black phosphorus), the sum over l in Eq. (5.3) has to cover the same range of interactions. Practically, summing over the 12 (MoS2 and WS2) or 13 (black phosphorus) NNs is suﬃcient. Still, although the S scattering self-energy Σ≷ (E, kz) is block diagonal only, its entries couple one atom at ri with its 12-13 NNs at rl, thus indirectly accounting for non-diagonal eﬀects. The fact that each momentum kz 5.1. TRANSPORT MODEL 97

is connected to all possible kz − qz points also contributes to the presence of non-diagonal eﬀects. Note that the ∇iHmn blocks were also constructed by upscaling the results of a small unit cell. S The phonon-electron scattering self-energy matrix Π (ω, qz) can- S not be assumed diagonal as Σ≷ (E, kz) because this would violate the energy conservation rule between electrons and phonons. It has to take the following form

X X Z dE Π≷ij(ω, q ) = −i · nn z 2π kz l n ≷ ≶ tr ∇iHln · Gnn(E + ~ω, kz + qz) · ∇jHnl · Gll (E, kz) o ≷ ≶ ∇iHnl · Gll (E + ~ω, kz + qz) · ∇jHln · Gnn(E, kz) (5.4)

for the diagonal block entries and

Z ij X dE Π≷ (ω, q ) = i · nl z 2π kz n ≷ ≶ tr ∇iHln · Gnn(E + ~ω, kz + qz) · ∇jHnl · Gll (E, kz) o ≷ ≶ ∇iHnl · Gll (E + ~ω, kz + qz) · ∇jHln · Gnn(E, kz) (5.5)

for the non-diagonal ones. In Eqs. (5.4) and (5.5), “tr {}” refers to the trace operator. Atoms l and i must be coupled with each other via a non-zero matrix element Hli. The last missing components are the retarded electron and phonon scattering self-energies, which can be approximated as

(Σ>(E, k ) − Σ<(E, k )) ΣR(E, k ) ≈ z z , (5.6) z 2 (Π>(ω, q ) − Π<(ω, q )) ΠR(ω, q ) ≈ z z . (5.7) z 2 98 CHAPTER 5. DISSIPATIVE TRANSPORT

It is obvious from Eqs. (5.3-5.5) that the scattering self-energies S bridge the electron and phonon populations because Σ (E, kz) de- S pends on D(ω, qz) and Π (ω, qz) on G(E, kz). This is also the reason why Eqs. (5.3-5.7) must be self-consistently solved till convergence is reached in the so-called Born approximation. The resulting numerical problem is particularly challenging since all energies E, frequencies ω, and momentum kz/qz are connected altogether. Such systems of equations cannot be tackled on a single computational core and must therefore be treated in parallel. To manage the distribution of the numerical tasks and the gathering of the needed data to calculate the scattering self-energies, the multi-level parallelization scheme introduced in Ref. [81] was adapted and enhanced to enable electro-thermal transport simulations of 2-D materials from ﬁrst-principles.

5.1.3 Calculation of Observables

A second self-consistent loop must be established between the solution of the Schr¨odingerand Poisson equations to properly take the elec- trostatics of the examined devices into account. After convergence the electrical current ﬂowing between two adjacent unit cells labeled s and s + 1 of the 2-D device structure

Z q X X X dE n < Id,s→s+1 = · tr Hij · Gji(E, kz)− ~ 2π kz i∈s j∈s+1 (5.8) < o Gij(E, kz) ·Hji ,

the electron component of the energy current between s and s + 1

Z 1 X X X dE n < IE−el,s→s+1 = E · tr Hij · Gji(E, kz)− ~ 2π kz i∈s j∈s+1 (5.9) < o Gij(E, kz) ·Hji ,

as well as its phonon part 5.1. TRANSPORT MODEL 99

X X X Z dω n I = ω · tr Φ · D<(ω, q )− E−ph,s→s+1 2π ~ ij ji z qz i∈s j∈s+1 (5.10) < o Dij(ω, qz) · Φji

can be computed from the electron and phonon Green’s functions. In all these equations, ~ is Planck’s reduced constant, q the elementary charge, and it is implied that atom i (j) belongs to the unit cell s (s + 1). It should be pointed out in Eqs. (5.8) to (5.10) that (energy) current conservation does not only require an integration over energy (or frequency), but also a summation over the momentum (kz or qz). Another quantity of interest is the eﬀective lattice temperature of unit cell s, Teff (s). It can be obtained by assuming that the total tot phonon population within one unit cell s, Nph (s), which is ﬁrst calculated with NEGF, locally obeys a Bose-Einstein distribution function NBose(Teff , ω) = 1/(exp(~ω/kBTeff ) − 1) with Teff as the governing temperature and kB as Boltzmann’s constant

X X Z d( ω) N tot(s) = ~ LDOS(r , ω, q )N (T (s), ω), (5.11) ph 2π i z Bose eff qz i∈s

X X Z d( ω) = i ~ tr D<(ω, q ) . (5.12) 2π ii z qz i∈s

The local density-of-states LDOS(Ri, ω, qz) is deﬁned as

> < LDOS(ri, ω, qz) = i × tr Dii (ω, qz) − Dii (ω, qz) . (5.13)

The eﬀective lattice temperature of unit cell s is then retrieved by matching Eqs. (5.11) and (5.13). The resulting non-linear system of equations can be solved, for example, with a Newton-Raphson iterative method. If necessary, an atomic resolution of Teff is also possible. 100 CHAPTER 5. DISSIPATIVE TRANSPORT

5.2 Device Results

To illustrate the influence of electro-thermal effects on the performance of n-type transistors made of a 2-D material, the single-gate structure in Fig. 5.1 was simulated with the models described in Section 5.1.1 together with single-layer MoS2, WS2, and armchair- oriented black phosphorus (AC BP) channels. The logic switch specifications include a gate contact of length Lg = 15 nm, 12.5 nm long source and drain extensions with a donor doping concentration −2 ND = 6e13 cm , perfectly ohmic contacts, a top (bottom) oxide layer of thickness tox = 3 (20) nm and relative permittivity R = 20 (3.9), and a supply voltage VDD = 0.67 V. The Id −Vgs transfer characteristics of the MoS2 device at a drain- to-source voltage Vds = VDD and OFF-current IOFF = 0.1µA/µm are reported in Fig. 5.4 for three different situations: (i) ballistic trans- S S port (Σ (E, kz) = Π (ω, qz)=0 in Eqs. (5.1-5.4)), (ii) dissipative transport in the presence of electron-phonon scattering, but with an S equilibrium phonon population (only Π (ω, qz) = 0), and (iii) in the same configuration as in (ii), but with both the electron and phonon populations driven out-of-equilibrium. The latter case incorporates electro-thermal effects, contrary to the others where the lattice temperature does not vary as a function of the electrical current and remains equal to the room temperature T0 = 300K. It might look surprising in Fig. 5.4 that the ballistic current is not the largest one. This peculiar feature has already been discussed in details in Ref. [34]: in typical transistor structures, the ballistic transmission function T (E) from left (source) to right (drain) only slightly depends on the applied Vds at a given Vgs. With strong drain-induced barrier lowering (DIBL), a shift towards lower energy of the point where T (E) turns on is expected, but no shape changes. However, with MoS2 and most other TMDs, the transmission function exhibits a strong dependence on Vds, as can be seen by comparing T (E) in Fig. 5.3 (flat-band conditions, zero electrostatic potential everywhere) and in Fig. 5.5 (linear potential drop between both device ends). Obviously, at energies close to the conduction band edge, one of the two available channels in the source (sub-bands with a positive velocity) of MoS2 stops conducting at high Vds so that T (0)=1 with a linear potential drop, while T (0)=2 with flat bands. As a conse- 5.2. DEVICE RESULTS 101

Figure 5.4: Transfer characteristics Id − Vgs at Vds = 0.67 V of the single-gate transistor in Fig. 5.1 with a single-layer MoS2 as channel. The ballistic limit of transport (dashed blue lines), the current with electron-phonon scattering, but an equilibrium phonon population (green lines with crosses), and the current with self-heating eﬀects (solid red lines) are provided. The current reduction caused by self-heating is indicated by the orange double-arrow. 102 CHAPTER 5. DISSIPATIVE TRANSPORT

Figure 5.5: Electron transmission function, as obtained from quantum transport calculations, through the same single-layer 2-D materials as in Fig. 5.3 at kz=0, but with a linear potential drop (see inset) between both structure ends instead of ﬂat band conditions. 5.2. DEVICE RESULTS 103 quence, the ballistic current is reduced. This “anomaly” is due to the speciﬁcity of the TMD bandstructures, where sub-bands with a very narrow energy width ∆E smaller than qVds can be found, as indicated in Fig. 5.2(a). These states are not conductive in the ballistic limit of transport, but become active as soon as electron-phonon scattering is switched on [34]. In this case an electron occupying a non-conductive band can be transferred to a conductive one by absorbing or emitting a phonon. This explains the rather counterintuitive reinforcement of the dissipative current with respect to the ballistic one in Fig. 5.4.

When the influence of the phonon-electron scattering self-energy S Π (ω, qz) is added to the picture, phonon emission and absorption processes induce local variations of the lattice temperature. This causes a current decrease by about 10%, as compared to the case with electron-phonon interactions only, which is marked by a double arrow in Fig. 5.4 and labeled self-heating. To better understand what happens inside the MoS2 transistor, the spectral electron and phonon current distributions in Fig. 5.6(a-b) are investigated as next step. It can be observed in sub-plot (a) that electrons lose a substan- tial portion of their energy through phonon emission after passing the maximum of the electrostatic potential energy barrier, when they are accelerated by the strong electric field on the drain side. From sub-plot (b) it can be inferred that at the location with the highest phonon generation rate (and therefore the highest lattice temperature too, as explained later) at around x = 31 nm, the phonon current is equal to 0 and the created lattice vibrations either propagate towards the drain (x = 40 nm, positive current) or the source (x = 0 nm, negative current) contact. The phonons that reach the source can interact there with electrons moving towards the drain and increase their backscattering rate [82], thus leading to the current reduction visible in Fig. 5.4. To complement this analysis, it is demonstrated in Fig. 5.6(c) that energy is conserved in the developed simulation approach. The sum of the electron and phonon energy currents, as given in Eqs. (5.9) and (5.10), is indeed constant from source to drain. This indicates that the energy lost by electrons is correctly transferred to the phonon bath and vice versa. Slight fluctuations of the total energy current still exist (less than 1%): they are due to the slow convergence of Eqs. (5.1-5.6). It has been verified that more self-consistent iter- 104 CHAPTER 5. DISSIPATIVE TRANSPORT

Figure 5.6: (a) Electron spectral current (current as a function of the position along the transport direction and of the electron energy) of the same transistor as in Fig. 5.4 at kz = 0, Vgs = 0.4 V, and Vds = 0.67 V. The case with self-heating eﬀects is plotted. Red indicates high current concentrations, green no current. The dashed blue line refers to the conduction band edge of MoS2. Energy loss can be clearly seen on the drain (right) side of the device. (b) Phonon spectral current in the same structure as before. Red indicates now a positive current (from left to right), blue a negative one, green no current. The vertical dashed line corresponds to the location with no netto phonon current, i.e. the location with the highest phonon generation rate. (c) Position-resolved electron (blue line with circles), phonon (dashed-dotted green line), and total (electron+phonon, solid red line) energy current ﬂowing through the same transistor as before. The sign change of the phonon energy current at around x = 31 nm as well as the conservation of the total energy current (almost constant solid red line) are clearly visible. 5.2. DEVICE RESULTS 105

Figure 5.7: Eﬀective lattice temperature increase ∆T = T − T0 as a function of the position along the x-axis and of the gate voltage Vgs at a ﬁxed drain bias Vds = 0.67 V for the same single-layer MoS2 transistor as in Fig. 5.6. ations between the Green’s Functions and the scattering self-energies improve the situation, at the expense of the computational time.

With Eqs. (5.11) and (5.12), the non-equilibrium phonon population can be converted into a position-resolved eﬀective lattice temperature Teff that is resolved at the unit cell level. The results are provided in Fig. 5.7 for the MoS2 transistor.

What is shown is the lattice temperature increase ∆T = Teff − T0 as a function of the location along the x-axis and as a function of the applied gate-to-source voltage Vgs. Two points should be emphasized: first, Teff peaks at the same x-coordinate as where the phonon current vanishes in Fig. 5.6(b) and where the phonon generation rate reaches its maximum. The formed hot spot is in fact situated close to the channel-drain interface, in the region with the sharpest drop of the electrostatic potential energy and therefore the highest phonon emission probability. Secondly, the ∆T values are extremely high >400 K at Vgs=0.7 V. Before approaching such temperatures, the MoS2 106 CHAPTER 5. DISSIPATIVE TRANSPORT single-layer crystal would already have oxidized (oxidation temperature: 675 K) [83] and the whole device would have broken down. As a comparison, in Ref. [84], the ∆T of a micrometer-scale monolayer MoS2 transistor was estimated through Raman thermometry mea- surements to be around 250 K for a much larger Vds = 30 V vs. 0.67 V here, but a smaller current Id ≈ 200 µA/µm vs. ∼1000 µA/µm here and a higher contact resistance Rc=5kΩ×µm vs. perfectly ohmic contacts here. This confirms that self-heating effects are very important in monolayer MoS2, but that the simulated temperature increases are overestimated, mainly because no phonon escape through the surrounding oxide layers is allowed. Such mechanisms would definitively lead to a decrease of the effective lattice temperature, but their exact contribution is difficult to quantify. Keeping in mind that ∆T is overestimated, it is still relevant to compare the electro-thermal properties of MoS2 with those of other 2- D materials, namely WS2 and AC black phosphorus, and with those of more conventional ultra-scaled Si nanowire transistors since the same set of approximations is applied everywhere, i.e. phonons escape at the source and drain only, not through oxide layers. The goal of this study is to determine what material/design is the least sensitive to self-heating and why. In Fig. 5.8(a) the electrical power dissipated inside the channel of all switches, Pdiss, is plotted as a function of the drive current Id. For the Si nanowire transistor, a <100>-oriented structure with a diameter d=3nm, a gate length Lg=15 nm, and a gate-all-around configuration was selected and the data from Ref. [82] was recalled. Here, Pdiss is defined as the difference in the electron energy current (see Eq. (5.8)) between the source and the drain. Normally, it is expected that the total dissipated power is equal to Vds × Id and that it has the same value for all structures at a given Id. However, due to the short length of the considered devices, only a fraction of the total power dissipation takes place in the channel region, the rest in the contacts. This explains why Pdiss < Vds × Id in our simulations and why the curves are different in Fig. 5.8(a). The amount of dissipated power in a given component can be put in relation with the phonon-limited mobility µph of its underlying channel material [85]: the higher µph the less phonons are emitted and the less electrical power is converted into heat. To verify whether this statement is valid or not here, the phonon-limited electron mobility 5.2. DEVICE RESULTS 107

Figure 5.8: (a) Power dissipated inside the channel of a single-layer MoS2 (red line with circles), WS2 (blue line with triangles), and AC BP (green line with squares) transistor with the same structure and dimensions as in Fig. 5.1 vs. the electrical current ﬂowing through this device. The data for a circular Si nanowire transistor with a gate-all- around conﬁguration, a diameter d=3 nm, and transport along the <100> crystal axis are also provided (dashed black line) [82]. (b) Same as (a), but for the maximum lattice temperature increase vs. drive current. (c) Combination of (a) and (b), i.e. maximum lattice temperature increase vs. dissipated power inside the channel. 108 CHAPTER 5. DISSIPATIVE TRANSPORT

Figure 5.9: Electron phonon-limited mobility of single-layer MoS2, WS2, and AC BP as a function of the carrier concentration [87] and with the same plotting conventions as in Fig. 5.3. The “dR/dL” mobility extraction method has was for that purpose [86].

of single-layer MoS2, WS2, and AC black phosphorus, as computed with the “dR/dL” method [86], is presented in Fig. 5.9 as a function of the carrier concentration. Due to the heavier atomic mass of tungsten as compared to molybdenum, the amplitude of the crystal vibrations and in turn the electron- phonon coupling strength is weaker in WS2 than in MoS2, which gives a higher mobility and a lower dissipated power at a ﬁxed current magnitude, as postulated above. This demonstration fails however when AC black phosphorus comes into play: its mobility is smaller than that of MoS2 and WS2 due to more signiﬁcant crystal vibrations, but it still dissipates less power. The reason behind this apparent inconsistency can be traced back to the distinctive bandstructure of TMDs, where sub-bands with a narrow energy width cannot carry current at high Vds, as shown in Fig. 5.5. As a result, electron-phonon interactions are needed to make those sub-bands conductive and to reach high current densities. The number of phonon emission processes occurring 5.2. DEVICE RESULTS 109

in TMDs under the application of a high Vds is therefore much more important than predicted by the mobility, which is computed under ﬂat-band conditions. While electron-phonon interactions contribute to a current increase in TMDs, they simultaneously consume electrical power and lead to the higher heat dissipation of MoS2 and WS2 seen in Fig. 5.8(a). Although the Si nanowire transistor has about the same phonon-limited electron mobility as AC black phosphorus 2 −2 (300 cm /Vs at an electron concentration nD ≈1e13 cm ), it dissipates more power. Since the crystal and device structures are quite diﬀerent, it is not possible to explain this behavior with the same line of arguments as before.

Figure 5.8(b) then reports the maximum effective lattice temperature increase extracted from the same transistors as before versus the electrical current Id. What matters in this sub-plot is no more the number of phonons that is generated (heat dissipation), but once they have been created how efficiently they can leave the active region of the logic switch. If the emitted phonons have a high group velocity, they more rapidly diffuse away from the region where they originate such that the effective lattice temperature at this location cannot rise as much as it would have if the phonon population would keep accumulating. Hence, the thermal transport properties determine to a large extend the magnitude of the self-heating effects. For example, in WS2, less power is dissipated at a given Id than in MoS2, but the lattice temperature increase ∆T is the same in both 2-D materials: less crystal vibrations are generated in WS2 due to the weaker electron-phonon coupling and lower electron effective mass, but they propagate slower due to the higher atomic mass of tungsten. The poorer thermal transport properties of single-layer WS2 are indirectly illustrated in Table 5.1, which gives the sound velocity of all studied 2-D crystals, as derived from ab-initio calculations. It can also be seen in Fig. 5.8(b) that the temperature increase is lower in AC black phosphorus and in the Si nanowire than in both TMDs.

Finally, the maximum lattice temperature increase can be represented as a function of the power dissipated inside the transistor channel. The results are shown in Fig. 5.8(c). While the curves for the 2-D materials are not identical, they all exhibit a much higher 110 CHAPTER 5. DISSIPATIVE TRANSPORT

∗ ∗ Eg (eV) me mh vs,T A vs,LA ~ωopt (meV) MoS2 1.67 0.46 0.56 4.7 7.1 58 WS2 1.8 0.3 0.38 3.9 6.2 53 BPΓ−X 1.61 0.18 0.15 4.6 5.0 57 BPΓ−Y 1.61 1.2 2.5 3.9 8.5 57

Table 5.1: Selected material parameters of single-layer MoS2, WS2, and black phosphorus along the Γ-X and Γ-Y symmetry lines: band gap (Eg), electron (me) and hole (mh) eﬀective masses at the band edges, sound velocity of the transverse (vs,T A) and longitudinal (vs,LA) acoustic phonon branches in km/s, as well as optical phonon energy (~ωopt). All data were extracted from DFT [59] and DFPT [75] calculations.

lattice temperature increase than the Si nanowire at a given Pdiss. The strong confinement of phonons in 2-D crystals leads to rather poor thermal transport properties and prevents the rapid evacuation of heat from the active region of 2-D logic switches. As mentioned ear- lier, allowing phonons to escape through the surrounding oxide layers would certainly reduce the ∆T , but the same would happen in the Si nanowire where phonons are also confined in the semiconductor [79]. A recent experimental study on single-layer MoS2 [84] clearly established that self-heating is much more significant in this 2-D material than in Si, which qualitatively agrees with the finding presented here. Chapter 6

Structure of the Code

The Winterface code is divided into two libraries libmat, a matrix library, and liblat, a library using libmat as its base and providing all algorithms described in the previous sections.

6.1 Libmat

The libmat library started out as a wrapper for calls to BLAS functions to make the code more readable. It then grew into a more versatile framework that contains all basic functions to operate on matrices used in liblat. The functionality of the library will not be discussed in detail, only the basics will be introduced to make the code in liblat comprehensible.

Two basic design choices follow from the use of algorithms dealing with crystal lattices. Matrices often play the role of containers for vectors such as the three basis vectors describing the translational symmetry of a lattice or the positions of atoms. Such matrices can be designed to work as an array of columns, imitating the general interface of C++ standard library containers, e.g. std::vector. In practice, this means that a special kind of iterator pointing to entire columns of a matrix spawned by the methods cBegin() and cEnd() must be available, as well as a variable size, where the number of

111 112 CHAPTER 6. STRUCTURE OF THE CODE rows must be speciﬁed upon creation. It is important that requesting additional space for more columns is possible at any time. Columns can then be appended using the familiar push back operator. A typical piece of code looks like this: const int M=3, N=10;

// create matrix of size MxN of random numbers fMat mat1 = rand(M,N);

// create empty matrix of size Mx0, reserve for2N columns fMat mat2(M,0); mat2.reserve(2∗N);

// copy mat1 into mat2 column by column for (auto i=mat1.ccBegin(); i!=mat1.ccEnd(); ++i) mat2.push back(∗i);

// append mat1 to mat2 in one call mat2.push back(mat1);

// the result is thus mat2=[mat1 mat1] This use of iterators over columns in loops is typical in liblat. Even though working over columns is to be preferred whenever possible due to the column-major arrangement of the physical memory of CPUs, iterators over rows are provided as well. A clever use of all these iterators in conjunction with calls to the standard library of C++ is demonstrated in the next example: const int M=8, N=12; fMat mat1 = rand(M,N), mat2(N,M);

// write the transpose of mat1 into mat2 std :: transform(mat1.ccBegin(),mat1.ccEnd(),mat2.rBegin());

// sort the columns of mat1 and the rows of mat2 std :: sort(mat1.cBegin(),mat1.cEnd(),vcmp); std :: sort(mat2.rBegin(),mat2.rEnd(),vcmp); Here, vcmp is a lambda implementation of the relation in Eq. (4.3). The second major design choice is the seamless implementation of comparators with a given tolerance level, elementwise and otherwise. This stems from the fact that matrices are used in a variety of algorithms operating on lattices where we have to consider the periodicity 6.1. LIBMAT 113 of the system and be especially careful with respect to finite input precision. Rounding errors can always happen due to the convention that atomic positions, in the coordinates of the translational symmetry, must always lie within the interval [0, 1). Practically this means that we need a tolerance level such that [0, 1) effectively becomes [−, 1 − ). Otherwise, any small error for a vector put at the origin, which is a frequent occurrence in the usual choice of atomic positions, could lead to coordinates slightly slipping into the negative. If periodicity is then enforced by applying modulus 1 on it, a correct coordinate at −δ with 0 < δ < would be forced to the other side of the unit cell by resulting in 1 − δ. In many functions, it is imperative that such translations are prevented, but when working with lattices, it is in general desirable when the positions do not appear to be jumping back and forth for no reason. In the library this is realized with the help of a tolerance level adjustable by the user and implemented as a stack of floating point numbers. At any given point in time the top of the stack is used as the current tolerance level. Consider the following example:

fMat my func(const fMat& mat1, const fMat2& mat2) noexcept { // push1e5 onto the stack, compare then pop and return set mtol(1e−5); fMat res = mat1

return res; }

int main() { // push1e-3 onto the stack set mtol(1e−3); fMat mat1 = rand(3,3), mat2 = rand(3,3);

// printa comparison using1e3 first, then1e5 inside my_func std :: cout << "\n" << (mat1

return 0; } In this way different tolerance levels can be used in different parts of the code. The upside is that the handling of the tolerance level is seamlessly taken care of in the background, allowing for a clean 114 CHAPTER 6. STRUCTURE OF THE CODE and readable code. The downside is that function calls can mess up the tolerance stack by forgetting to reset it at the end. Hence, bugs in one part of the code can be caused by other, unrelated code segments. This feature must therefore be used with great caution, especially when exceptions are thrown so that the concerned function may be closed without resetting the tolerance level. It is best practice to set and reset the tolerance explicitly around the sections where an adjustment is needed. Besides the actual matrix class, a number of additional functions are provided such as the rand function used in the examples above, as well as functions to calculate eigenvalues. A list of the available functions can be found in the file lm fn.h.

6.2 Liblat

This is the library that contains all of the functionality described in Chapter 3 and 4. It can be either directly linked to OMEN or alternatively compiled as a standalone executable ltool which oﬀers the possibility to calculate bandstructures or to deal with lattices. The concepts discussed in Section 4.3 are available in ltool as well.

The code is subdivided into two namespaces:

• aux: for a number of auxiliary functions and classes that are needed, but do not belong to the core library functionality.

• ll : for the core functionality. All classes have a preﬁx ll in order to avoid naming conﬂicts.

The most important components of these namespaces can be vi- sualized in Fig. 6.1 and are listed below:

• aux parser: class providing the basic functionality for reading member variables from a text file. By inheriting directly or indirectly from this class, member variables can be initialized by user inputs defined in a text file. A complex inheritance structure is possible, allowing user inputs to be split among multiple classes, whilst using the same text file as a central input. 6.2. LIBLAT 115

aux_parser

ll_BStest_input ll_wmatching_input

ll_wf_input ll_hbondss_input ll_BStest ll_hbondss ll_mesh ll_omen

ll_fn ll_hbonds ll_hio ll_io ll_cell ll_bonds

Figure 6.1: General organization and code structure of liblat. Boxes designate classes, bubbles collections of functions, solid arrows inheritance, while dashed arrows indicate dependencies. The central cluster of functions, ll omen connects everything together. Liblat generates the Hamiltonians along bonds, calls the bandstructure tests, and pre- pares the input ﬁles for OMEN.

• ll BStest input: input parameters for the bandstructure test. • ll wmatching input: input parameters for the matching of Wannier centers to atomic positions. • ll wf input: class combining parameters speciﬁc to the OMEN interface as well as those required for the matching process and the bandstructure test. It relies on the inheritance from the respective classes. • ll BStest: bandstructure test using a mesh or a trace in k- space. • ll mesh: class implementing a mesh in N-dimensional space. It is capable of conveniently generating meshes compatible with the result of the MATLAB function meshgrid [88], but has a more generic design. • ll cell: class constructing a unit cell from an atomic lattice, as introduced in Section 3.1. • ll bonds: class used to describe bonds in a lattice, as introduced in Section 3.3. 116 CHAPTER 6. STRUCTURE OF THE CODE

• ll hbonds: specialized version of ll bonds including interaction data along bonds generated from Wannier90 output, as introduced in Section 3.3.

• ll hbondss: class handling multiple instances of ll hbonds based on an analogous interface. Its working principle was introduced in Section 4.3.

• ll fn: collection of utility functions used all over the code. The most important ones are: – dist: functions to measure distances in periodic space. – genHam: functions to generate sets of {R, H(R)} from Wannier Bonds Hamiltonians in supercells, to be used, for example in, bandstructure calculations. – calcBS, calcFoldedBS, ...: functions to calculate bandstructures.

• ll io: functions to read data from Wannier90, VASP or OMEN outputs.

• ll hio: an abstraction layer for writing Hamiltonian matrices in diﬀerent ways. Most importantly the function hctor which is the fundamental function generating Hamiltonian matrices up to the device level using a Wannier Bonds Hamiltonian.

• ll omen: central cluster of functions, where OMEN’s inputs are generated.

As already mentioned, the outputs of Winterface can also be transferred to other quantum transport simulators, requiring only slight modifications of the code outputs to match the format of the alter- nate QT tool inputs. The code was specifically designed to allow for straightforward extensions to different formats. Chapter 7

Conclusion and Outlook

A general technique for upscaling Hamiltonians in MLWF representation of small unit cells up to the device level was established. The ﬁrst step consists of a reformulation of raw MLWF Hamiltonian data into a representation in terms of interactions along bonds. This makes the second step, the actual upscaling, much more transparent and enables more complex, approximate, upscaling techniques, as well as investigations of local properties, e.g. bandstructures. The latter can be useful when generating complex geometries. Additionally, the importance of the inclusion of scattering eﬀects in to the transport model was demonstrated on the basis of examples of MoS2, WS2 and black phosphorous monolayer devices.

Even though the focus of this thesis is on 2-D structures, the algorithms presented in Chapters 3 and 4 work equally well and without modification of the code for 1-D and 3-D structures. Additionally, once the first step of generating Hamiltonian data in terms of bonds is completed, generating Hamiltonian matrices for whole devices, as discussed in Section 4.1, is independent of the chosen basis set. An extension of Winterface to other localized bases is therefore possible, such as tight-binding coefficients or Gaussian type orbitals (GTO), as implemented, for example, in the CP2K [89] package. The concepts introduced in Section 4.3 may present different challenges, however, depending on the specificities of the selected basis set. The Hamilto-

117 118 CHAPTER 7. CONCLUSION AND OUTLOOK nian interactions do not necessarily have to be those of the electronic system. The phononic system can be described in terms of bonds as well, allowing for an upscaling procedure in the same manner. In Chapter 5, the phonon upscaling was realized manually, but it could be automated by taking advantage of the functionalities available in Winterface. We would like to draw the readers attention to the fact that for phonons, inexact upscaling, as presented in Section 4.3 might not be possible. This feature will be veriﬁed in the future.

As demonstrated in Chapter 5, self-heating eﬀects play an important role in many TMDs. Such simulations rely on the coupling between electrons and phonons, which is given by the derivatives of the MLWF Hamiltonian with respect to the atomic positions. Here, the approach chosen for the calculation of the Hamiltonian derivatives is an approximation where the structure of interest was simulated and wannierized twice, with a slight diﬀerence in the lattice constant corresponding to hydrostatic strain. Other approximations are possible, such as scaling for each axis separately or independent simulations where a single atom is displaced in each case. Which approach is preferable depends on the material and computational burden acceptable. This part is not yet included in Winterface, but the representation of the Hamiltonian in terms of interactions along bonds lends itself naturally to the computation of derivatives.

Winterface does not impose any restrictions on the geometry of the considered devices. The challenge arises at the DFT and wannierization stages, where it must be decided how the desired features are best included in a single unit cell or possibly split into multiple simulations and assembled into a Hamiltonian, as presented in Section 4.3.2. A few options to handle more complex geometries are: (i) inclusion of one or both contacts in a larger unit cell, (ii) a larger unit cell with localized defects, or (iii) inclusion of the oxide and/or the substrate to allow for a better model of heat dissipation. In each of these cases the limiting factor is the physical modeling of lattice mismatches between two materials that should be put together, the ionic relaxation at interfaces or defects and in general the computational burden involved. Nevertheless, complex structures could be wannierized and then upscaled using the concepts of Section 4.3, e.g. Ti-TiO2-MoS2 119 contact geometries, as demonstrated in Ref. [90]. To a certain extent, Winterface is able to assist in the process of generating such complex structures, as it allows for an investigation into the coupling strength along bonds. Moreover, with the concept of localized bandstructures introduced in Section 4.3, a rough estimate for the local properties of each material can be made.

Appendix A

2-D vs. 3-D k-point Grid in Monolayer MoS2

To simulate 2-D structures in DFT, the k-vector along the vacuum direction does not need to be discretized, assuming that the distance between a structure and its image is large enough. The integration in Eq. (2.12) vanishes if no k-point is specified along any direction. Although there is still mixing due to the U (k) transformation matrix, the result is a periodic rather than a localized function. Therefore, the characterization of the Wannier functions in terms of R vectors loses its meaning in the vacuum directions. Wannier centers spread across both sides of the vacuum may be shifted into the same cell inconsequentially. A comparison of a Wannier function along the vacuum direction of a MoS2 monolayer for a 2-D and a 3-D k-point grid is given in Fig. A.1. The bands along the vacuum direction turn out to be flat with no dispersion, which becomes apparent when we take the derivative ∂H/∂ki in Eq. (3.16). The latter vanish since there is no R vector along the vacuum axis. The effective masses are thus infinite and no transport can occur along this direction.

121 122 APPENDIX A. 2-D VS. 3-D K-POINT GRID

ΔL ΔL ΔL

Figure A.1: Comparison of a Wannier function located on the Molyb- denum atom in a MoS2 monolayer as calculated with 2-D (9x9x1, blue) and a 3-D (9x9x9 green) k-point grid. For simplicity, the absolute value squared of the Wannier function was considered and the components along the non-vacuum directions, x and y, were integrated, i.e.w ˜(z) = R dxdy|w(x, y, z)|2, and the result normalized such that the maximum is at 1. The plot stretches over 4 layers separated by a distance ∆L. The structure was placed at L0. The periodic nature of the 2-D case is clearly shown by identical peaks on all layers, whereas the 3-D case has only one such peak at L0. The peaks at L0 are identical, but due to the k-points along the vacuum direction, the wannierization using a 3-D grid leads to a non-periodic solution. The smaller peaks on L1 and L-1 are insigniﬁcant (note the log scale). However, the bandstructures for both cases are identical, also along the vacuum direction where the ﬂat bands are found. 123

In conclusion, it was demonstrated that 2-D and 3-D k-point grids lead to equivalent results, indicating that the two dimensional nature of the structure was not imposed by the choice of the grid, rather it is a consequence of the conﬁnement along one axis. Additionally, relative shifts among R vectors, as discussed in Chapter 3, occuring along the direction of conﬁnement, can be savely ignored.

Appendix B

Basis Set Reduction after Wannierization

The wannierization of a MoS2 monolayer, presented in Fig. 4.9, requires a basis set of 11 Wannier functions. Since the bands stretch over a much larger range in energy than needed for quantum transport, it can be envisioned that some of the Wannier functions may be neglected without disturbing the critical region around the band-gap. This possibility can be investigated by deﬁning an energy interval I = [Emin,Emax] and computing the following projector into the corresponding eigenspace for each k-point:

X † PI (k) = vi(k) · vi(k) . (B.1)

i(k)∈I

Here vi(k) is the eigenvector corresponding to the eigenvalue i(k) of the Hamiltonian matrix deﬁned in Eq. (3.16). When investigating † the projected Hamiltonian matrix H˜I (k) = PI (k) ·H(k) · PI (k) , it is often found that only a few Wannier functions are needed to reproduce the bands in the interval I, since entire rows and columns are zero in H˜. While the bands inside the interval should not be aﬀected by discarding certain Wannier functions, as indicated by the projected Hamiltonian matrix, the bands outside the interval may be disturbed to such an extent that they are folded into the interval of interest.

125 126 APPENDIX B. BASIS SET REDUCTION

(a) (b)

Figure B.1: Investigation of reduced Wannier basis sets. The goal is to accurately reproduce the bands in the interval Emin ≤ E ≤ Emax (shaded region), while using less Wannier function than resulting from the initial wannierization. (a) The energy interval around the band gap is targeted. With 2 Wannier functions less (black lines) the DFT bandstructure (green bands) is well-reproduced, but ”new” bands appear in this region of interest. (b) The top 2 conduction bands are targeted. With 3 Wannier functions less, the behavior of the DFT bands is well captured is this narrow energy interval.

The situation for two diﬀerent intervals is presented in Fig. B.1.

Using the projector-method, it is possible to ﬁnd Wannier functions that can be discarded without disturbing a given region of interest. Since the reduction of the basis set has an enormous impact on the computational burden of quantum transport simulations, discarding Wannier functions in this manner is recommended wherever possible. However, for the critical region around the band gap, this is rarely observed without causing signiﬁcant disturbances, and it is therefore most readily employed to discard bands elsewhere in the bandstructure, such as low-lying core bands or conduction bands far above the band gap (unless already excluded at the wannierization stage). List of Publications

Peer-reviewed Journal Papers

• C. Stieger, A. Szabo, T. Bunjaku, and M. Luisier, ”Ab-initio quantum transport simulation of self-heating in single-layer 2-D materials”, Journal of Applied Physics, vol. 122, p. 045708, 2017

• C. Stieger, A. Szabo, H. Carrillo Nunez, C. Klinkert, and M. Luisier, ”Winterface: a Wannier90 postprocessing tool for ab- inito quantum transport simulations”, in preparation, 2019

Conferences

• C. Stieger, A. Szabo, T. Bunjaku, and M. Luisier, ”Ab-initio modeling of self-heating in single-layer MoS2 transistors”, 75th Device Research Conference (DRC), 2017

• H. Carrillo Nunez, C. Stieger, and M. Luisier, ”Performance predictions of single-layer In-V double-gate n-and p-type ﬁeld- eﬀect transistors”, IEEE International Electron Devices Meeting (IEDM), 2016

• M. Luisier, A. Szabo, C. Stieger, C. Klinkert, S. Br¨uck, A. Jain, and L. Novotny, ”First-principles simulations of 2-D semiconductor devices: Mobility, I-V characteristics, and contact resistance”, IEEE International Electron Devices Meeting (IEDM), 2016

127 128 LIST OF PUBLICATIONS

• C. Stieger, A. Szabo, P.A. Khomyakov, A. Schenk, and M. Luisier, ”Multiscale modeling of electron transport in semiconductor nanostructures”, Materials Science and Engineering Congress (MSE), 2014 Bibliography

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Christian Stieger was born in Z¨urich, Switzerland, on June 3rd, 1983. From 2008 to 2013 he studied Physics at the Swiss Federal Institute of Technology in Z¨urich. He received his MSc. degree in 2013. In the same year he joined the Integrated Systems Laboratory, ETHZ as a research assistant in the Computational Nanoelectronics group. His research interests focus on the numerical simulations of quantum transport in nanoscale devices.

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