Applications of Density functional theory for

modeling metal-semiconductor contacts, reaction

pathways, and calculating oxidation states

Sergei Posysaev

DFT

University of Oulu Graduate School

Reports series in Physical Sciences

Report No. 123 (2018)

APPLICATIONS

APPLICATIONS OF DENSITY FUNCTIONAL THEORY

FOR MODELING METAL-SEMICONDUCTOR CONTACTS,

REACTION PATHWAYS, AND CALCULATING OXIDATION

STATES

SERGEI POSYSAEV

University of Oulu Graduate School

University of Oulu

Faculty of Science

Nano and Molecular Systems Research Unit

Academic Dissertation to be presented with the assent of the Faculty of Science,

University of Oulu, for public discussion in the Auditorium IT 116, on December

14th, 2018 at 12 o’clock noon.

REPORTS SERIES IN PHYSICAL SCIENCES Report No. 123

OULU 2018 UNIVERSITY OF OULU

Opponent

Prof. Marcel Sluiter, Delft University of Technology, Netherlands

Reviewers

Prof. Bernardo Barbiellini, Lappeenranta University of Technology, Finland

Prof. Kalevi Kokko, University of Turku, Finland

Custos

Prof. Matti Alatalo, University of Oulu, Finland

ISBN 978-952-62-2131-1

ISBN 978-952-62-2132-8 (PDF)

ISSN 1239-4327

Juvenes print

Oulu 2018

Posysaev, Sergei, Applications of Density functional theory for modeling

metal-semiconductor contacts, reaction pathways, and calculating oxidation

states

University of Oulu Graduate School; University of Oulu, Faculty of Science,

Nano and Molecular Systems Research Unit

Acta Univ. Oul. X XXX, 2018

University of Oulu, P.O. Box 8000, FI-90014 University of Oulu, Finland

Abstract

Density functional theory (DFT) is a well-established tool for calculating the

properties of materials. The volume of DFT-related publications doubles every 5–

6 years, which has resulted in the appearance of continuously growing open

material databases, containing information on millions of compounds. Furthermore,

the results of DFT computations are frequently coupled with experimental ones to

strengthen the computational findings.

In this thesis, several applications of DFT related to physics and chemistry are

discussed. The conductivity between MoS2 and transition metal nanoparticles is

evaluated by calculating the electronic structure of two different models for the

nanoparticles. Chemical bonding of Ni to the MoS2 host is proven by the system’s

band alignment. To meet the demand for cleaner fuel, the applicability of the (103)

edge surface of molybdenum disulfide in relation to the early stages of the

hydrodesulfurization (HDS) reaction is considered. The occurrence of the (103)

edge surface of molybdenum disulfide in the XRD patterns is explained. A method

for calculating oxidation states based on partial charges using open materials

databases is suggested. We estimate the applicability of the method in the case of

mixed valence compounds and surfaces, showing that DFT calculations can be used

for the estimation of oxidation states.

Keywords: Bader charge, density functional theory, molybdenum disulfide,

oxidation state, reaction pathway

Posysaev, Sergei, Tiheysfunktionaaliteorian sovelluksia metalli-

puolijohdeliitosten, reaktiopolkujen ja hapetustilojen mallinnuksessa

Oulun yliopiston tutkijakoulu; Oulun yliopisto, Luonnontieteellinen tiedekunta

Nano- ja molekyylisysteemien tutkimusyksikkö

Acta Univ. Oul. X XXX, 2018

Oulun yliopisto, PL 8000, 90014 Oulun yliopisto

Tiivistelmä

Tiheysfunktionaaliteoria (density functional theory, DFT) on yleisesti käytetty

työkalu laskennallisessa materiaalitutkimuksessa. DFT:llä tuotettujen julkaisujen

määrä kaksinkertaistuu 5-6 vuoden välein, minkä johdosta käytettävissä on

jatkuvasti kasvava määrä avoimia materiaalitietokantoja, joihin on talletettu

miljoonien yhdisteiden ominaisuuksia. DFT-laskujen tuloksia täydennetään myös

usein kokeellisilla tuloksilla.

Tässä työssä tarkastellaan tiheysfunktionaaliteorian sovelluksia fysiikassa ja

kemiassa. MoS2:n ja metallisten nanopartikkelien välistä johtavuutta on tutkittu

mallintamalla erilaisia nanopartikkeleita. Nikkelin ja MoS2:n välinen kemiallinen

sidos selittyy systeemin energiavöiden kohdistumisella. MoS2:n (103)-pinnan

soveltuvuutta rikinpoistoreaktion varhaisissa vaiheissa on tutkittu tarkoituksena

löytää uusia menetelmiä puhtaan polttoaineen tuottamiseksi. Myös (103)-pinnan

esiintyminen röntgendiffraktiokuvissa selitetään. Työssä on myös esitetty

menetelmä hapetustilojen laskemiseksi tietokannoista löytyvien laskettujen

varausjakaumien avulla. Menetelmän soveltuvuutta on tarkasteltu erilaisille

yhdisteille ja pinnoille. Tämä tarkastelu osoittaa, että DFT-tuloksia voidaan käyttää

hapetustilojen laskemiseen.

Asiasanat: Bader-varaus, tiheysfunktionaaliteoria, molybdeenidisulfidi,

hapetustila, reaktiopolku

1

Acknowledgements

The present work was carried out in the Nano and Molecular Systems Research

Unit (NANOMO), University of Oulu. I would like to thank my supervisor Prof.

Matti Alatalo, who saw in me a possible Ph.D. student, guided all the way, taught

me the Finnish culture and became a friend. Special thanks to Prof. Marko Huttula

for being a good boss and providing me with the chance to finish my doctoral study.

This work involved many close collaborations and I would like to acknowledge

Dr. Wei Cao and Xinying Shi for not giving up on our project and not just finish it,

but hugely improve it, which led to the publication in a good journal. I thank Prof.

Talat Rahman for providing an opportunity for my research visit and for an amazing

time in Florida, Dr. Duy Le for productive collaboration. Also, big thanks go to

Lyman Baker, before knowing him I was sure that combining beer and own library

is impossible.

I want to express my gratitude to my follow-up group members: Prof. Erkki

Thuneberg, Prof. Juha Vaara, Doc. Saana-Maija Huttula for their guidance and

understanding.

I am grateful to NANOMO for the friendly and productive environment.

Particularly to Jack Lin, Nønne Prisle and especially to Georgia Michailoudi for

being a good friend and for tasty cakes.

I especially thank my beloved Olga Miroshnichenko, who brought me to Oulu

and helped me during the whole Ph.D. Thanks to Irina Ignatova and to Roman

Frolov who taught me how to stand up for oneself and that one can have both a

career and a lot of children. Anton Artamonov thank you for being Anton. A special

thanks to the Poluyanov family, for the numerous evenings filled with joy

The last thanks are for my family. Big thank to my Mother, she is the main

reason why I went to university, even though it was difficult for her to let 16 years

old son go to another city to study. All my journey wouldn’t be possible without

Father’s support. Sister, who inspired me, and whom I always want to catch up but

have not succeeded yet…Thanks to my son for all the joy he brought. He taught

me that four hours of uninterrupted sleep is one of the best things in this world.

Oulu, December 2018 Sergei Posysaev

2

Abbreviations

API application programming interface

DFT density functional theory

etc. et cetera

i.e. id est

ILC inorganic layered crystals

e.g. exempli gratia

GIGO garbage in, garbage out

GGA generalized gradient approximation

HDS hydrodesulfurization

LDA Local Density Approximation

OS oxidation state

PBC periodic boundary conditions

REST–API The REpresentational State Transfer–Application Programming

Interface

TMD transition metal dichalcogenides

XRD X-ray diffraction

3

Original publications

This thesis is based on the following publications, which are referred throughout

the text by their Roman numerals:

I Shi, X., Posysaev, S., Huttula, M., Pankratov, V., Hoszowska, J., Dousse, J.-Cl.,

Zeeshan, F., Niu, Y., Zakharov, A., Li, T., Miroshnichenko, O., Zhang, M.,

Wang, X., Huang, Z., Saukko, S., González, D. L., Dijken, S., Alatalo, M., Cao.

W., Metallic contact between MoS2 and Ni via Au nanoglue. Small, 14,

1704526 (2018). DOI: 10.1002/smll.201704526

II Posysaev, S., Alatalo, M., Surface morphology and sulfur reduction pathways

of MoS2 Mo-edges of monolayer, (100) and (103) surfaces by molecular

hydrogen: A DFT study. ACS Omega, submitted for publication

III Posysaev, S., Miroshnichenko, O., Alatalo, M., Le, D., Rahman, T., Oxidation

states of binary oxides from data analytics of the electronic structure.

Computational Materials Science, submitted for publication

All papers in the thesis are a result of teamwork. The author has been the main

writer of the Papers II and III. Theoretical calculations and data management were

done by the author for Papers I and II and partially for Paper III.

4

Papers not included in the thesis

The author contributed to the publications which are not included in the thesis

publication

I Miroshnichenko, O., Posysaev, S., Alatalo, M., A DFT study of the effect of SO4

groups on the properties of TiO2 nanoparticles. Phys. Chem. Chem. Phys.,

2016, 18, 33068-33076. DOI: 10.1039/C6CP05681D

II X. Shi, M. Huttula, V. Pankratov, J. Hoszowska, J.-Cl. Dousse, F. Zeeshan, Y.

Niu, A. Zakharov, Z. Huang, G. Wang, S. Posysaev, O. Miroshnichenko, M.

Alatalo, W. Cao. Quantification of bonded Ni atoms for Ni-MoS2 metallic

contact through X-ray photoemission electron microscopy. Microscopy and

Microanalysis, 24, 458−459 (2018). DOI: 10.1017/S1431927618014526

5

Contents

Abstract

Tiivistelmä

Acknowledgements 1

Abbreviations 2

Original publications 3

Papers not included in the thesis 4

Contents 5

1 Introduction 7

2 Quantum mechanics 9

2.1 Postulates of quantum mechanics ................................................................ 9

2.1.1 The wave function............................................................................ 10

2.1.2 Observables ...................................................................................... 10

2.1.3 Measurements .................................................................................. 11

2.1.4 Expectation Values ........................................................................... 11

2.2 Schrödinger Equation ................................................................................. 11

3 Computational methods 13

3.1 Density functional theory ........................................................................... 13

3.1.1 Born–Oppenheimer approximation ................................................ 13

3.1.2 Hohenberg–Kohn theorems ............................................................. 14

3.1.3 Kohn-Sham equations ...................................................................... 14

3.1.4 Exchange-correlation functional ..................................................... 15

3.1.5 Computational details of DFT ......................................................... 17

3.2 Application of DFT ..................................................................................... 18

3.2.1 Surface energy .................................................................................. 18

3.2.2 Calculation of reaction pathways .................................................... 20

3.2.3 Bader Charges .................................................................................. 21

3.3 Data analysis ................................................................................................ 22

3.3.1 Materials databases .......................................................................... 22

3.3.2 Data collection ................................................................................. 22

3.3.3 Data processing ................................................................................ 23

3.3.4 Data cleaning and data exploring .................................................... 23

4 Summary of articles 24

4.1 Metallic Contact between MoS2 and Ni via Au Nanoglue ....................... 24

6

4.2 Surface morphology and sulfur reduction pathways of MoS2 Mo-

edges of monolayer, (100) and (103) surfaces by molecular

hydrogen ....................................................................................................... 27

4.3 Oxidation states of binary oxides from data analytics of the

electronic structure ...................................................................................... 29

5 Conclusions 32

List of references 33

Original publications 40

7

1 Introduction

One of the greatest things a human being can do is to ask ‘Why is this so?’. Greater

yet is to search for the answer, by studying nature. A lot of time has passed and a

lot of ‘why’s’ have found answers. Thus, a scheme has been formed for justifying

new answers consisting of the following steps: observe a phenomenon, form a

hypothesis describing the phenomenon, conduct a controllable experiment to verify

the hypothesis.

To conduct an experiment is not a trivial task. Many aspects can affect the

results of experiments and one must ensure that all variables are correctly

controlled.

Nowadays with modern techniques and increasing computer power, сomputers

have enabled us to conduct experiments by computer simulation, which saves both

time and money. We can model processes from interacting quantum particles [1]

up to the formation of galaxies [2]. Even though one can in principle control

everything in modeling, it is possible to forget/skip something important and end

up with a model that will not reflect reality.

Today many computational models are available for the calculations of

material properties, on different scales of the phenomena studied. When one wants

to know the behavior of a material at the scale of electrons, Density Functional

Theory (DFT) is one of the methods that he/she should use.

Since its formulation in the 1960’s and up to nowadays of DFT has proven to

be a reliable tool [3,4]. It has been applied to chemistry, physics, and biology [5].

The importance of DFT was evidenced by the Nobel prize awarded to one of the

creators of DFT, Walter Kohn [6].

DFT is widely applied in physics for calculating properties of materials. MoS2

is one of the transition metal dichalcogenides (TMDs), which falls in the category

of inorganic layered crystals (ILC). It is a cheap material and abundant on Earth.

MoS2 exhibits semiconducting band structure and it has also been proposed as a

promising alternative to traditional silicon-based semiconductors [7,8]. However,

the lack of dangling bonds in pristine ILC material surface makes it difficult to

bond with metals, resulting in contact instability and Schottky barriers [9,10] with

high contact resistance, which restrains the performance of TMDs based devices.

One of the aims of this thesis was to prove theoretically the existence of an effective

metallic contact between MoS2 and Ni nanoparticles.

The edge planes of MoS2 are used as a catalyst for hydrodesulfurization (HDS)

reaction during which the emission of sulfur dioxide is reduced. Sulfur dioxide can

8

provoke acid rains and affects human health [11]. This is why the demand for

cleaner fuel rises each year within stricter environmental regulations [12].

Extensive studies have been performed in order to improve the processes of fuel

purification such as doping [13], novel supports [14,15], and changing the

morphology of the catalyst [16,17]. However, desired properties have not yet been

achieved. A way of possible improvement of the catalytic activity of MoS2 was the

second aim of this thesis. We studied reaction pathways of one of the surfaces which

most frequently occurs in X-ray diffraction (XRD).

Currently, the volume of DFT-related publications doubles every 5–6 years

[18]. This growth has given rise to continuously growing open materials databases

of millions of compounds. Such databases have been used for merging together

physics and machine learning techniques to show a unique possibility to predict

materials properties by intelligent algorithms trained at a large number of known

data [19] as well as for the inverse task - to find a material with desired properties

[20].

Oxidation state (OS) indicates the number of electrons an atom has lost or

gained. Properties of each material depend on OS of its atoms. DFT has not been

designed to calculate OS directly, although DFT data is routinely used for the

extraction of atomic charges [21–24]. We used materials databases to resolve the

long-standing question about the applicability of partial charges to the

determination of the OS. Empirical rules for establishing the OS in binary oxides

are discussed later in this work, based on a large number of data.

This thesis is structured as follows: Chapter 2 introduces the basics of quantum

mechanics. In Chapter 3 the foundations and applications of density functional

theory are discussed along with data analysis and materials databases. In a summary

of articles, Chapter 4, the main findings are included along with additional material

which was not included in the Papers. The last chapter consists of conclusions

based on the original publications.

9

2 Quantum mechanics

This chapter gives a brief overview of quantum mechanics. Before explaining the

theory, which describes the nature of subatomic particles and energies at the

smallest scale and to understand that quantum mechanics is not pure magic, it is

important to provide an example of quantization, as many people find it puzzling

that some physical quantities can have only discrete values.

Quantization appears only when some restriction is present in a system. An

analogous example in classical mechanics would be the potential energy of a

human in a skyscraper. The human can only be on a certain floor with a certain

potential energy, analogous to electron levels in an atom (Fig. 2.1), whereas a

falling human experiences a continuous range of potential energy. The same goes

for a free particle, it can have a continuous range of energies.

2.1 Postulates of quantum mechanics

Postulates of quantum mechanics were built taking classical mechanics as a starting

point and do not have a “real” justification. Even though for a scientist, a theory

which should be accepted by postulates usually sows a grain of doubt, quantum

mechanics has been with us about 100 years and is “the most precisely tested and

most successful theory in the history of science” [25].

Fig. 2.1 Example of discrete and continuous range of energy in quantum and classical

mechanics.

10

2.1.1 The wave function

The first postulate says that each state of the system is fully defined by a wave

function, 𝛹(𝑟, 𝑡) . The probability of finding a particle (or particles in a many

particle system) in a certain volume 𝑑3𝑟 at time 𝑡 is given by the square of the wave

function, i.e. taking the complex conjugate of the wave function multiplied by the

wave function itself and then by the volume element:

𝑃(𝑟, 𝑡) = 𝛹∗(𝑟, 𝑡)𝛹(𝑟, 𝑡)𝑑3𝑟. (2.1)

For example, if an electron is present in an infinite vacuum, then we need to

integrate the right-hand side of Eq. (2.1) over infinity to get one as our answer.

2.1.2 Observables

The second postulate says that for each observable quantity there is a linear, self-

adjoint operator.

In classical mechanics, one can define observables as quantities that can be

measured in an experiment. Given the trajectory of the system, x(t), or the phase

space trajectory, p(x), we can determine all observables of the system. For instance,

we could find the momentum, the energy, the angular momentum, etc.

Table 1 Physical observables and the corresponding quantum operators (denoted by

hat).

In quantum mechanics, the observables x(t), p, etc. are replaced with operators.

Because the state of the system is uniquely determined by a wave function instead

of the trajectory, the operators act on the wave function. The position operator acts

Classical Quantum

x(t) �̂�𝛹(�⃗⃗�, 𝑡) = 𝑥𝛹(�⃗⃗�, 𝑡)

p(t) �̂�𝛹(�⃗⃗�, 𝑡) = −𝑖ℎ𝑑

𝑑𝑥𝛹(�⃗⃗�, 𝑡)

E �̂�𝛹(�⃗⃗�, 𝑡) = [�̂�2

2𝑚+ �̂�(𝑥)] 𝛹(�⃗⃗�, 𝑡)

11

on the wave function by simple multiplication. The momentum operator acts

through differentiation etc. (Table 1).

2.1.3 Measurements

The third postulate concerns observables. It says that in any measurement, the only

possible values one can obtain are the eigenvalues 𝜆 of the operator 𝛰 , that

correspond to the observable to be measured. The operator and its eigenvalue

satisfy the eigenvalue equation:

𝛰𝛹(𝑟, 𝑡) = 𝜆𝛹(𝑟, 𝑡). (2.2)

An important point is that after the measurement, the wavefunction 𝛹(𝑟, 𝑡)

immediately collapses into the corresponding eigenstate 𝜆. Thus, the measurement

affects the state of the system.

2.1.4 Expectation Values

The fourth postulate says that if the system is characterized by a state 𝛹 , the

average of the measurements of the observable 𝛰 at time 𝑡 is given by the

expectation value of 𝛰:

⟨𝛰⟩𝑡 = ∫ 𝑑3𝑟𝐷

𝛹∗(𝑟, 𝑡)𝛰𝛹(𝑟, 𝑡). (2.3)

Expectation values are important in an experiment because an experiment is

not measuring just one copy of the system. For example, when measuring the

spectrum of a water molecule, one is measuring the spectrum of many molecules

in the sample.

2.2 Schrödinger Equation

Schrödinger equation summarizes all postulates together. The state and the time

evolution of the system can be obtained by solving the time-dependent Schrödinger

equation:

𝑖ħ𝜕

𝜕𝑡𝛹(𝑟, 𝑡) = �̂�𝛹(𝑟, 𝑡), (2.4)

12

where ħ is Planck's constant divided by 2𝜋 and �̂� is the Hamiltonian operator,

which corresponds to the total energy of the system.

The significance of the Schrödinger equation in quantum mechanics is the

same as that of Newton’s second law in classical mechanics. They are both

equations of motion. The goal here is to find 𝛹(𝑟, 𝑡) knowing an initial state

𝛹(𝑟, 0) and the solution to the time dependent Schrödinger equation describes the

dynamical behavior of the particle. In contrast to Newton’s equation, where we can

determine the exact position of the particle as a function of time, we get a wave

function, which tells us the probability of finding the particle in some region in

space.

13

3 Computational methods

This chapter gives an overview of the computational methods used in this thesis.

The Schrödinger equation can be solved analytically only for simple systems. For

resolving the many-body problem, approximations should be made. They are

discussed further along with the application of DFT. The use of materials databases

and principles of data analysis end this chapter.

3.1 Density functional theory

DFT is an ab initio method for solving the Schrödinger equation, where the

properties of the many-electron system are fully defined by the electron density that

depends only on 3 spatial coordinates.

Electrons are present as a set of n one-electron Kohn-Sham (Schrödinger like)

equations, which are evaluated in a potential (external + effective, see section 3.1.4).

Before going to the basics of DFT, first, the term “functional” needs to be defined.

A functional is related to the concept of a function, as one can guess based on the

name. A function takes a value of a variable (variables) and defines a single number

from those variables. A simple example of a function is 𝑓(𝑥) = ax + b . A

functional, denoted by 𝐹, is similar, for example 𝐹(𝑓) = ∫ 𝑓(𝑥)𝑑𝑥, but it takes a

function instead of the value of a variable and yields a single number from the

function.

3.1.1 Born–Oppenheimer approximation

The Born-Oppenheimer approximation [26] is the assumption that the nuclear

motion and the electronic motion can be separated. It leads to a molecular wave

function in terms of the electron coordinates 𝑟𝑖 and nuclear coordinates �⃗⃗�𝑗.

𝛹𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒 (𝑟𝑖 , �⃗⃗�𝑗) = 𝛹𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠(𝑟𝑖 , �⃗⃗�𝑗)𝛹𝑛𝑢𝑐𝑙𝑒𝑖(�⃗⃗�𝑗). (3.1)

Further assumptions are: 1) Nuclei are present as fixed points, creating a static

external potential V. 2) Electrons respond much faster to changes in their

surroundings than nuclei due to the greater mass of nucleus which is at least 1800

times heavier than the electrons. Hence we can assume that electrons instantly

follow the motion of the nuclei. 3) The nuclear motion sees a smeared out potential

arising from the electrons.

14

3.1.2 Hohenberg–Kohn theorems

The roots of DFT stem from two theorems proved by Hohenberg and Kohn [27].

The first theorem states: ‘The ground-state energy from Schrödinger’s equation is

a unique functional of the electron density’. This means that there is an exact

dependence between the ground-state wave function and the electron density. It

allows us to reduce complexity from 3N dimensions to just 3 dimensions (Fig. 3.1).

The first Hohenberg–Kohn theorem does not yield information on how to find

the functional. The second theorem states: ’ The electron density that minimizes the

energy of the overall functional is the true electron density corresponding to the

full solution of the Schrödinger equation’. Taking an assumption that the functional

is known, one needs to vary the electron density until the energy from the functional

is minimized.

3.1.3 Kohn-Sham equations

Solving the Schrödinger equation using just the Hohenberg–Kohn theorems is

still an intractable many-body problem of interacting electrons in a static external

potential. Kohn and Sham reduced this problem to a tractable problem of

noninteracting electrons moving in an effective potential. This led to a set of self-

consistent, single particle equations known as the Kohn-Sham equations [28]:

Fig. 3.1 Schematic illustration of transforming many-electron system to electron density

15

(−1

2𝛻2 + 𝜗𝑒𝑓𝑓(𝑟) − 𝜖𝑗) 𝜑𝑗(𝑟) = 0, (3.2)

where 𝜗𝑒𝑓𝑓 is the sum of the external, Hartree and exchange-correlation potentials,

𝜑𝑗(𝑟) is the jth Kohn-Sham orbital of energy 𝜖𝑗 . The solutions of the Kohn–Sham

equations are single-electron wave functions that depend on three spatial

coordinates. Thus, to find the effective potential, we need to define the electron

density, which is given by:

𝑛(𝑟) = ∑|𝜑𝑗(𝑟)|2

𝑁

𝑗=1

. (3.3)

The following self-consistent cycle should be used to find the ground state

energy: an initial guess for 𝜑𝑗(𝑟) which is applied to Eq. (3.3) and later on used for

finding the effective potential, which is subsequently used in Eq. (2.4). Then Kohn-

Sham equation is solved for 𝜑𝑗(𝑟) iteratively until the initial and new densities are

the same, then the ground state density has been found. Otherwise, one must select

a new trial density through minimization of the total energy and continue to repeat

the iterative procedure.

3.1.4 Exchange-correlation functional

The Hohenberg-Kohn theorems refer to a functional, which therefore needs to

be defined more accurately. As discussed briefly above, a functional is a function

of another function; in the case of DFT, it is the spatially dependent electron density.

Kohn and Sham separated the effective potential into three terms:

𝜗𝑒𝑓𝑓(𝑟) = 𝜗(𝑟) + ∫𝑛(𝑟′)

|𝑟 − 𝑟′|𝑑𝑟′ + 𝜗𝑥𝑐(𝑟), (3.4)

where the first and second terms are the external and classical electrostatic

(Hartree) potentials, which can be found exactly. The third term is a definition of

the exchange-correlation potential which contains the non-classical electrostatic

interaction energy and the difference between the kinetic energies of the interacting

and non-interacting systems.

Results of DFT calculations depend on the choice of the exchange-correlation

functional. The simplest approximation is the Local Density Approximation (LDA)

[28], where the exchange energy and correlation energy are based on uniform

16

electron gas. In this situation, the electron density is constant at all points in space;

𝑛(𝑟) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡.

LDA is not the only functional, there are several others. One of the most

popular classes of functionals uses information about the local gradient in the

electron density. This approach defines the generalized gradient approximation

(GGA), which include information from the gradient of the electron density [29–

32]. More accurate calculations became possible with including second derivatives

(meta-GGA functionals) [33–35] of the electron density. Popularity and accuracy

of DFT increased significantly after the development of hybrid functionals [36–38].

Range-separation allowed application of hybrid functionals to solids and surfaces

[39–43]. Accuracy was improved further by local hybrids [44–46], and even more

after coupling it with range-separation [47,48].

Some functionals give good results for one system and some for another, so

when reporting results of DFT calculation it is important to indicate which

functional has been used. More accurate functionals usually consume more

computer resources and nowadays it is more a trade-off between accuracy and the

speed of calculation. Perdew and co-workers [49] provide a classification of

functionals using a reference to the Biblical account of Jacob’s ladder (Fig. 3.2).

Scientists get more and more physical information, climbing by ladder and in the

top, the Schrödinger equation is solved without approximations.

Fig. 3.2 Illustration of Perdew’s classification of DFT functionals using “Jacob’s ladder”.

17

3.1.5 Computational details of DFT

There are many important parameters involved in a DFT calculation such as:

whether to perform spin-polarized or non-spin-polarized calculations, the use of

symmetry, the choice of eigensolver, structural optimization algorithms etc. These

parameters are sometimes a matter of taste, sometimes the choice depends on the

system studied. The following concepts are the most crucial ones and they need to

be discussed before diving in the applications of DFT: unit cell, periodic boundary

conditions, Brillouin zone [50] and methods for expanding the wave functions. This

is a minimum for understanding the principles of DFT calculations.

A unit cell is the most basic and least volume consuming repeating structure of

any solid. The lattice structure is created if the unit cell is repeated in all three

directions. To avoid using large systems in computations, save computational time

and prevent molecules/atoms from flying off into space, periodic boundary

conditions (PBC) are normally employed (Fig. 3.3). For example, if a molecule

leaves the red cell through the right wall, its image will enter the cell through the

left wall.

The concept of the Brillouin zone is of major importance in the band theory of

materials. The Brillouin zone is the unit cell of the reciprocal lattice. We will not

go into the details of this construction here but simply state that many parts of the

mathematical problems posed by DFT are much more convenient to solve in terms

of reciprocal space and the accuracy of a calculation depends on the number of k

points in each direction. One of the most popular solutions to split the reciprocal

space was suggested by Monkhorst and Pack [51]. Together with the number of k

Fig. 3.3 Two-dimensional schematic of periodic boundary conditions. The unit cell of

system is in the lower left corner (red).

18

points, the accuracy of a calculation depends on the kinetic energy cutoff, which is

proportional to the number of plane waves being utilized as basis functions to

represent the wavefunction.

It is worth mentioning that nowadays there are many computer programs for

atomic scale materials modelling from first principles. Examples of well-known

packages include ABINIT [52], GPAW [53], Quantum ESPRESSO [54], and VASP

[55,56].

3.2 Application of DFT

DFT is used in physics, chemistry, materials science, and biology. Due to the

increase in computer power, DFT constantly finds new applications. Results can be

obtained for a wide range of physical quantities, starting from the simple lattice

constant of a BCC metal up to the prediction of complex system behavior.

Here we concentrate on the techniques, which have been used in the original

publications.

3.2.1 Surface energy

Before discussing the surface energy, one needs to explain how to model surfaces.

The best model for a surface would be a slice of a material which is infinite in two

dimensions, which can be done using PBC. The common idea for modeling

surfaces is presented in Fig. 3.4, where four-unit cells are used along the y direction

and a vacuum layer is added on top of the five-layer slab. It is important to ensure

that the vacuum layer is thick enough to avoid interaction between the bottom and

the top layer of the slab in the z direction. Molecular surface coverage, percentages

of defects/adatoms on the surface and other structural features can be controlled by

varying the number of unit cells in the x and y direction. A number of layers in the

z direction usually depend on the material and the properties studied. A thicker slab

gives more accurate results but consumes more computational power; in other

words, there is usually a trade-off between accuracy and computational time.

19

Surface energy is the energy needed to create the surface by a cleavage along a

plane in a bulk material. This is a difficult quantity to determine experimentally

because it usually requires measuring the surface tension at the melting temperature

of the metal [57]. On the other hand, a theoretical determination is relatively easy.

The surface energy (𝐸𝑠) is calculated as

𝐸𝑠(𝑛) =1

2𝐴(𝐸𝑛 − 𝑛𝐸𝑏), (3.5)

where 𝑛 is the number of unit cells, 𝐴 is the surface area, 𝐸𝑛 is the total energy of

the relaxed surface slab and 𝐸𝑏 is the total energy of a bulk unit cell. Also, the

cleavage energy can be calculated using the same equation; in that case, 𝐸𝑛 is the

total energy of the unrelaxed slab.

One can cleavage different surfaces from the same bulk material and the surface

with the lowest surface energy is the most stable surface. It means that this surface

is the most likely one to be created under mechanical impact and the one which

most frequently occurs in nature.

Surface energy calculated by Eq. (3.5) does not properly converge with

increasing number of layers, due to numerical errors arising from differences in the

Brillouin-zone sampling [58]. There are other techniques for calculating the surface

energy which avoid these errors [58–61]. Usually, however, the surface energy

calculated by Eq. (3.4) gives an acceptable result if care is used when choosing the

k-point mesh [60,62].

Fig. 3.4 Surface slab model.

20

3.2.2 Calculation of reaction pathways

Knowledge about reaction steps and reaction barriers is crucial for the commercial

application of catalysts. Modeling of reaction pathways helps to increase the rates

of catalytical reaction and adjust conditions which leads to a cheaper product for

customers.

Before starting to model a chemical reaction, it is important to find the proper

initial and final states of the reaction with a minimum energy of the system studied.

Chemical reactions usually take place on the surface of the catalyst. Due to the

complexity of the surface structure, finding the initial and final states is not a trivial

task. If we take even a simple adsorption of the CO molecule, there are several

adsorption sites on the ideal surface of any solid (Fig. 3.5). A number of possible

adsorption sites significantly increases on real surfaces due to adatoms, steps, and

vacancies. Picking a wrong site results in a higher barrier for the reaction.

Once the initial and final states are known, the reaction/adsorption barrier can

be found. There are several methods for finding the barrier.

The nudged elastic band (NEB) method [63] uses several images between the

initial and final state during the search for the minimum reaction pathway. It

optimizes the intermediate images by adding spring forces along the band between

the images and by projecting the force perpendicular to the band. A variation of the

NEB method is the climbing image NEB (cNEB) [64], where the highest energy

image can be adjusted to the top of the saddle point (Fig. 3.6). The dimer method

[65] is another method for finding saddle points. It allows starting from any initial

21

configuration near a saddle point. The dimer method needs only one image and

searches for the saddle points in random directions. Sometimes switching from a

pre-converged NEB calculation to the dimer method helps to save computational

time and provide a possibility to use finer parameters in the calculation due to the

usage of only one image.

3.2.3 Bader Charges

There are several techniques for the estimation of the total electronic charge of an

atom [21,22,24,66]. Bader analysis [21] is one of them. It utilizes electron density

for estimating the net charge on an atom. A volume of a cell is separated in Bader

volumes when:

∇𝜌(𝑟) = 0, (3.6)

where 𝜌(𝑟) is the electron density, and the gradient is taken with respect to the

three-dimensional coordinate 𝑟. This zero flux condition effectively yields distinct

regions (Bader volumes) for each atom in a system and the integral of the density

inside them is assigned as a net charge of an atom.

22

3.3 Data analysis

Nowadays, the amount of data is rapidly growing. This growth helps to find new

patterns in all areas. In our information age, data analysis of multidimensional data

has become an essential skill for researchers in all areas. Data analysis includes

processes such as cleaning, mining, retrieving, modeling, transforming of data with

the purpose of finding useful information, patterns and to support decision-making.

3.3.1 Materials databases

Quantum mechanical calculations consume a lot of computational power of modern

computers. Not all researchers have the possibility to repeat a calculation from a

scientific article. Providing the results of your calculation to an open database

allows everyone to see full results of the calculation.

Materials databases such as AFLOW [67,68], Materials Project [69],

Engineering Virtual Organization for Cyber Design [70], and NOMAD [71]

provide easy data mining. AFLOW requires strict parameters for calculation [72],

which simplifies the comparison between the results of different calculations.

3.3.2 Data collection

Most of the databases allow a user to download data. With the data available, there

are two different ways to proceed. The first one is to find and download the

compounds needed manually through the web interface of a database. It is an easy

and straightforward way to proceed if one needs only a few compounds. The second

way is to use an application programming interface (API), which enables users to

search and download desired data for post-processing, through appropriate queries.

AFLOW provides The REpresentational State Transfer–Application Programming

Interface (REST–API) [73] For example, using REST–API a query to get all links

for titanium oxides in the database looks like this:

“http://aflowlib.duke.edu/search/API/?species('Ti', 'O'),$nspecies(2)”. It is a simple

query where you indicate the species and number of species in a compound. The

filter can be refined further by various values such as band gap, formation energy

etc.

23

3.3.3 Data processing

Downloading data from multiple sources results in files in different formats, which

contain different quantities. One needs to prepare all the collected data in one

format with the same features. In Paper III, for faster processing and analyzing of

data, a database of compounds was transformed to a database of cations, meaning

that each record in the new data set represents an atom rather than a compound (Fig.

3.7).

3.3.4 Data cleaning and data exploring

The open databases can include duplicates, errors, and missing or unnecessary

information. Before analyzing data, one needs to clean it. Cleaning is one of the

crucial components of data science. It is supported by the famous statement

‘garbage in, garbage out (GIGO)’. GIGO means that nonsense input data produces

nonsense output.

During data exploring one uses visual exploration to understand what is in a

dataset and the characteristics of the data. Usually, cleaning goes together with data

exploring. Examples of data cleaning and exploring are presented in the summary

of Paper III, where the dataset used in Paper III is used as an example dataset.

24

4 Summary of articles

4.1 Metallic Contact between MoS2 and Ni via Au Nanoglue

It was noticed that the edge plane of MoS2 is more chemically active than basal

plane [74]. A majority of computational studies only focused on the contact

between metal and monolayer TMDs [75], where resistance is high owing to the

larger physical separation between contacts. A comprehensive computational study

of metal-mTMD contacts [76] has demonstrated the high advantage of edge

contacts in terms of electron injection efficiency.

Our aim was to prove conduction between the edge surface of MoS2 and a

metallic contact. The experimental results showed two types of interfaces. In type-

I, the MoS2 and Ni are connected via Au nanoparticles, which features bulk-like

properties. In contrast, the type-II interface is nanocrystallized Au nanoparticles,

with sizes typically below 5 nm. Both types of interfaces were modeled. Each one

contains a 3×1×4 unit cell of the MoS2 host. To reach a reasonable computational

time for model-I (type-I contact), the nanocrystallized Au nanoglue was simplified

to 4 layers of Au atoms between the Ni and MoS2 sides. Two contact layers are

relaxed for structural optimization at each side, and another two fixed as Au (111).

The second model (type-II) employs the interfacial character where the Au and Ni

atoms are randomly placed to reach possible alloying. The Ni (111) surface is

connected to the interface, opposite to the MoS2.

In the present calculations, DFT structural optimization for the edge surface

slab MoS2 (Fig. 4.1a) has been carried out. Two views of the unit cell of bulk MoS2

are shown in Fig. 4.1b, c. First-principles calculations were performed, using the

projector augmented wave method as implemented in Vienna Ab initio Simulation

Package (VASP) [55,56]. The plane wave cut-off energy was set to 400 eV. The

Perdew-Burke-Ernzerhof GGA functional was used to treat the electron exchange

and correlations with van der Waals (DFT-D3) correction [32,77]. The convergence

criterion was chosen so that the maximum force on one atom is smaller than 0.02

eV/Å. The calculations were performed with periodic boundary conditions. The

vacuum region in the z-direction was set to 15 Å to avoid mirror interactions

between nearby images. During relaxation all atoms were allowed to relax except:

in model-I, two middle layers of gold, which represent the bulk of gold; and in

model-II, two bottom layers of MoS2, which represent the bulk of MoS2.

25

The Brillouin zone was sampled using a Monkhorst-Pack k-point set [51]

containing 4×1×2 k-points during the structural optimization and PDOS calculation.

30 k-points were used along with special k-point path in the Brillouin zone for band

structure calculation. To obtaining magnetic properties spin-polarized calculations

were carried out. The VESTA3 code [78] was used to plot 3D isosurfaces for the

𝛥𝜌 function. The calculations are consistent with those using other software

packages [79,80], as well as results crosschecked with experimental determinations

[81].

The partial density of states (PDOS) of Mo and S atoms subjected to the Au

adsorption is shown in Fig. 4.2a. After adsorbing gold atoms, structural relaxation

is small at the host side. The guest gold buffer tends to be adsorbed by the ideal

MoS2 edge, forming a contact with the host. The charge density difference clarifies

the charge transfer between the gold layer and the MoS2 side, as shown in Fig. 4.2b.

The non-zero PDOS (gray line in Fig. 4.2a) of the host disperses crossing the Fermi

level, denoting a total metallization via side contact after the introduction of the

gold slab. This is different from the bare MoS2 side whose PDOS (red line in Fig.

4.2a) suffers from discontinuity around the Fermi-level, as pointed out by the green

regions in the figure.

Fig. 4.1 Structure of MoS2. (a) Representation of edge (blue) and top (red) MoS2 surfaces,

(b) and (c) unit cell of bulk MoS2. The pink and yellow spheres represent the Mo and S

atoms respectively.

26

Separate spin-polarized calculations were performed for bulk Ni and with

different percentage of Ni atoms substitutions (0, 8, 16%) for Au-Ni and MoS2-Ni

interfaces. Our theoretical result for the magnetic moment of bulk nickel is 0.67 µB,

in good agreement with experimental results. Reduction of the magnetic moment

of nickel was observed on atoms in contact with Au (111) down to 0.44 µB and with

MoS2 (100) surface down to 0.2 µB. Small magnetic moments were found on Mo,

S and Au atoms as well.

Introduction of the Au-Ni alloy onto the MoS2 side offers new electron

migration channels between the metal and the semiconductor side, leading to a

metallic contact mechanism. Such TMDs possess unique properties with numerous

promising applications in the semiconductor industry.

27

4.2 Surface morphology and sulfur reduction pathways of MoS2

Mo-edges of monolayer, (100) and (103) surfaces by molecular

hydrogen

In addition to the above-mentioned applications, molybdenum disulfide is a well-

known catalyst applied in fuel purification. Various methods for improving the

catalytic activity of MoS2 have been tried [13–17]. However, chemically activity

of the (103) edge surface of MoS2, which has the highest XRD peak in several

experimental works [82–85], has not been studied.

Our aim was to shed light on the catalytic properties of the (103) edge surface

of MoS2 and discover reaction pathways. Another motivation was to find the reason

for the occurrence of the (103) surface of MoS2 in XRD patterns.

Ab initio DFT calculations were performed in spin-polarized mode, with VASP

[55,56], using the PBE functional to treat the electron exchange and correlations,

together with van der Waals (DFT-D3) correction [32,77]. The electronic density

was calculated with a 350 eV cut-off energy. The convergence criterion was chosen

so that the maximum force on one atom is smaller than 0.02 eV/Å. The calculations

were performed with periodic boundary conditions. A Monkhorst-Pack k-point set

containing 2×2×1 k-points was used to sample the Brillouin zone during the

structural optimization [51].

Transition state (TS) search was performed using two methods in the following

order: first, climbing image nudged elastic band [64] and then the dimer method

[65]. To ensure that the system is in the TS, for each TS the dynamical matrix was

calculated and verified that it contains exactly one imaginary frequency.

According to our findings, the (100) surface is less energetically favorable than

the (103) surface, which explains the higher peak of the (103) surface in the XRD

patterns, but contradicts with Refs. [86–88], where the (100) peak is higher.

Apparently, the morphology of the samples depends on the preparation route of

MoS2.

The reaction pathway for the formation of sulfur vacancy consists of the same

steps as proposed by Prodhomme et al. [89] and presented in Fig. 4.3. We described

the pathway for the (100) monolayer, (100) surface, and (103) surface. According

to our results the barriers among the three models are similar.

28

To summarize, we demonstrated that the reason for the occurrence of the (103)

surface of MoS2 in XRD patterns is its low surface energy. The difference in the

barriers between the (100) monolayer, (100) surface, and (103) surface is less than

0.07eV. It allows us to conclude that the monolayer and bulk MoS2 possess similar

catalytic properties in the HDS reaction. In addition, a new reaction pathway for

the formation of sulfur a vacancy was found.

Fig. 4.3 Reaction pathway for the formation of a sulfur vacancy. Black, green and blue

paths relate to (100) monolayer, (100) surface, and (103) surface, respectively. Paths are

shifted vertically for better readability. The energies of the S1 state are taken as

references for each reaction path. Mo atoms in turquoise, S atoms in yellow, and H

atoms are in green and in red (the two hydrogen atoms are depicted with different colors

for clarity). Geometrical structures are depicted for the (100) monolayer surface.

29

4.3 Oxidation states of binary oxides from data analytics of the

electronic structure

Several studies have tried to establish a relationship between the OS and Bader

charge, and it has proven to be a nontrivial task. According to Reeves and Kanai,

the charge difference between two consecutive OS (between Ru(II) and Ru(III))

[90] is rather marginal and it is much smaller than unity. It is natural to look for a

method which gives an integer number for OS, but is it true in nature? Can we say

that all 8 valence electrons of osmium in OsO4 go entirely to oxygen atoms? It is

well known that all ionic bonds have some features of a covalent bond. Jansen and

Wedig [91] are confident that a connection between OS and Bader charge exists

and they highlight “the need for factors other than calculated charges to be taken

into consideration in order to obtain a meaningful connection with oxidation states”.

To our knowledge, all previous work tried to connect OS with Bader charges only

in a few compounds, and the main topic in those articles was different.

Nowadays the popularity of open and big data gives rise to continuously

growing open materials databases with DFT results for millions of compounds. Our

aim was to find a dependency between OS and Bader charge, using compounds

from an open database, and to provide a possibility to determine the oxidation state

from ordinary DFT calculations. Our focus was on binary oxides, i.e. compounds

of two elements, one of which is oxygen. They were chosen for simplicity as they

contain only two species. They are among the most calculated compound types in

materials databases. We show that a correlation indeed exists between OSs and

Bader charges and discuss the applicability of determining OS by Bader charges in

surfaces and in mixed-valence compounds (MV), where an element in the

compound can be in several OSs. In addition, the applicability of results with other

exchange-correlation functionals was examined.

The data set was downloaded from the AFLOW database [67,68], a materials

properties repository which has a standard for ab initio calculations [72]. To

classify materials by their OS we need to formulate an accurate definition of the

cation’s OS. By stating that the sum of the OSs in a compound has to be neutral,

that oxygen has an OS of -2, and that the OS of the anion is a known quantity, we

calculate the OS per compound and the OS of the cation as follows:

OS per compound = −OS of anion ∗ number of anions, (4.1)

30

OS of cation =OS per compound

number of cations. (4.2)

Databases can include duplicates, errors etc., as was mentioned in chapter 3.3.4.

Also, a result for OS of cation obtained from Eq. (4.2) can be wrong in some cases.

For example, in Fig. 4.4 a unit cell is represented, where molecules of oxygen are

confined between layers of copper atoms. OS of copper atoms is -4 by Eq. (4.2),

which is wrong because copper does not have a bond with oxygen.

To ensure that all oxygen atoms present in -2 state we disregard compounds

using several rules such as: compounds were disregarded if they have the shortest

bond between cation and anion larger than 130% of the sum of covalent radii [92]

to ensure that each atom in a compound has a covalent bond. Compounds were

disregarded if there is an O-O bond length smaller than 1.65 Å. The OS of oxygens

in an O-O bond can be 0 or -1 depending on the multiplicity of the bond (double or

single). The length of the double O=O bond is 1.21 Å, the length of a single O-O

bond is 1.48 Å [93]. To take into account that DFT gives errors [94] we had to

increase the threshold bond length, thus compounds with the O-O bond smaller

than 1.65 Å were disregarded. The resulting data set includes 697 compounds and

consists of 10743 atoms.

We carried out analysis of the resulting data set and demonstrated a correlation

between OS and Bader charge for a large set of single and mixed valence systems.

The results can be summarized as below:

• Bader charge is a good parameter for predicting OS in most of the binary

oxides

31

• There is a unique linear relationship between Bader charges and OS for

each binary oxide

• The results in Figure 2 (Paper III) may be used for predicting OS

• Our approach fails to predict OS of iron in MV compounds

• Bader charge intervals are not equal between OSs and tend to shrink with

higher OS

• Oxygen in -1 or 0 OS can have Bader charge down to -0.96 e, oxygens

with lower Bader charge are in -2 OS

• With a high probability cations of the same element in binaries are in

different OS, if the maximal Bader charge difference is more than 0.1 e

• The results can be compared with those for other exchange-correlational

functionals, but an estimation of the discrepancy is needed.

We present a technique which can give at least a hint about the OS of the

cations and predict OS in MV compounds. For surfaces, clusters and more complex

systems than binary, the charges can differ.

32

5 Conclusions

Density functional theory was used to study an effective metallic contact between

MoS2 and Ni nanoparticles, the catalytic activity of the (103) edge surface of MoS2

and the dependency between Bader charges and oxidation states.

We demonstrated by investigation of the band structure of the contact region

that a metallic contact to the edge surface of molybdenum disulfide can be created

with Au-Ni alloy.

The (103) edge surface of MoS2 was proven to have similar catalytic properties

to the (100) edge surface of MoS2 and the monolayer in relation to the

hydrodesulfurization reaction. Moreover, a more energetically favorable pathway

was found.

The long-standing issue about the applicability of Bader charges to oxidation

states has been solved for binary oxides. In addition, a new method for obtaining

oxidation states in mixed-valence compounds has been suggested.

DFT took root in physics and now flourishes almost in every field of science

where materials are involved. We have shown here only a fraction of possible

applications. They continue to grow and DFT finds new ways to be applied in

various scientific problems.

Future considerations in the use of online open databases for computed

properties of materials enable the utilization of big data to find connections between

materials and their properties. For example, doping of materials has led to only a

limited set of materials with new properties, whereas creation of an artificial

intelligence framework for the automated recommendation of doping candidates

can provide a huge speed increase in finding new materials with desired properties.

33

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40

Original publications

I Reprinted with permission from Small. Shi, X., Posysaev, S., Huttula, M.,

Pankratov, V., Hoszowska, J., Dousse, J.-Cl., Zeeshan, F., Niu, Y., Zakharov,

A., Li, T., Miroshnichenko, O., Zhang, M., Wang, X., Huang, Z., Saukko, S.,

González, D. L., Dijken, S., Alatalo, M., Cao. W., Metallic contact between

MoS2 and Ni via Au nanoglue. Small, 14, 1704526 (2018). DOI:

10.1002/smll.201704526. Copyright Wiley-VCH Verlag GmbH & Co. KGaA.

Reproduced with permission.

II Posysaev, S., Alatalo, M., Surface morphology and sulfur reduction pathways

of MoS2 Mo-edges of monolayer, (100) and (103) surfaces by molecular

hydrogen: A DFT study. ACS Omega, submitted for publication

III Posysaev, S., Miroshnichenko, O., Alatalo, M., Le, D., Rahman, T., Oxidation

states of binary oxides from data analytics of the electronic structure.

Computational Materials Science, submitted for publication

Original publications are not included in the electronic version of the dissertation.

41

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